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Tensile Testing
Plane stress and Plane strain
bull Plane stress and Plane strainbull Plane Stress State of stress in which one or
two of the pairs of faces on an elementalbull cube are free of any stresses eg Torsion of thin
wall tube Expansion of a thin walledbull spherical shell under internal pressurebull Plane strain State of tress where one of the
pair of faces on an elemental undergoesbull zero strain eg Torsion of thin walled tube Piece
of material being compressed in a die
Mechanical properties
The mechanical properties of a material are obtained by subjecting a specimen to prescribed loads and then measuring the resulting deformation
Usually the test is carried out on a special machine that is specifically designed for this purpose The measurements of the load and of the deformation are carried out
Tensile Test Machine (Instron)
Electronic Extensometer
ExtensometerGrip
Specimen
Grip
Tensile Testing(stretching or pulling)
Determine the behavior of the engineering material under load
Tensile Test -Tensile test determines the strength of the material when subjected to a simple stretching operation -Standard dimension test samples are pulled slowly and at uniform rate in a testing machin-The strain ( the elongation of the sample) is defined as
Engineering Strain = = (change in length)(original length) = L0
-The stress ( the applied force divided by the original cross-sectional area) is defined as
Engineering Stress = = (applied force)(original area) =PA0
Stress-Strain Curve
bull Elastic Limit Greatest stress a material is capable of
developing without a permanent setbull Note elastic limit for metals do not differ widely from the
values of the proportionalitybull Elastic limit may be taken as that stress at which there is
a permanent set of 02It is therefore higher than limit of proportionality (suggested by some authors)
bull Hooke srsquo law Stress is directly proportional to strain in the elastic range
bull Young srsquo Modulus It is ratio between stress and reversible strain (stiffness)
bull It is in fact a measure of the inter atomic bonding forces
Yield strength Proof stress usually defined as the stress which produces a measurable amount of permanent strain ie 02 or 01
Tensile strength - the maximum stress applied to the specimen
Failure stress - the stress applied to the specimen at failure (usually less than the maximum tensile strength because necking reduces the cross-sectional area)
Ductility
Elongation elongation is a measure of ductility which is given by
elongation =100 (Lo - Lf) Lo
whereLo = Initial lengthLf = Final Length
Ductility Reduction in Area reduction in area is a measure of ductility which is given by
reduction in area =100 (Ao - Af) Ao
whereAo = Initial areaAf = Final area
Poissonrsquos ratio
When pulled in tension (X) a sample getslonger and thinner ie a contraction in thewidth (Y) and breadth (Z)Poissonrsquos ratio when strained in the (X) direction how much strain occurs in the lateral directions (Y amp Z)For most metals this value is 0333
bull This is the Poisson effect Poissonrsquos ratio which is thebull negative ratio of the contraction over the extension isbull also an elastic constantbull Poissons Ratio laterial strainbull longitudinal strainbull εbull νbull εbull = minus = minus perpbull 1048614bull x unloadedbull ybull zbull loadedbull Pbull Pbull bullMetals ν = 03 ndash 035bull bullCeramics ν = 02 ndash 025bull bullPolymers ν = 025 ndash 0 5
Modulus of elasticity - the initial slope of the curve related directly to the strength of the atomic bonds
bull Resilience The ability of a material to absorb energy with in elastic limit
bull Measure by the modulus of resilience
-Which is strain energy per unit volume
-Stress the material from zero stress to the yield stress
bull Energy Force multiplied by the distance over which it acts
Modulus of resilience - the area under the linear part of the curve measuring the stored elastic energy
Toughness
The ability of a material to absorb energy in the plastic range
Toughness - The total area under the curve which measures the energy absorbed by the specimen in the process of breaking
Stress-Strain Curve
True Stress and True Strainbull True stress is the loadP divided by
the instantaneous area of the specimen Ai
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives AoLo AiLi
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking begins
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
True Strain
)1ln(
ln
e
LLL
L
L
L
dLd
oi
o
i
i
i
Assumes constant volumeValid for all strains up to point where necking begins
Engineering Vs True Stress-Strain Curves
Stress
Strain
True Stress - Strain Curve
Engineering Stress - Strain Curve
Ultimate Tensile Strength
Fracture
Fracture
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull IN Stress-Strain Curvesbull Plastic deformation is uniform and
permanent between the elastic limit and the UTS
bull bull Plastic deformation becomes non-uniform once the UTS is exceeded
bull In tension this non-uniform deformation manifests itself
bull as ldquoneckingrdquo
Criterion for necking
Increase in true stress (due to reduction in cross-sectional area) as the specimen elongates is more than to load carrying capacity due to strain hardening
Criteria for Instability
Rate of Geometrical Softening and Rate of Work Hardening
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10
Criteria for Necking
02 Offset Yield StrengthOffset Yield Strength
Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point The elastic portion of the curve is not perfectly linear and microscopic amounts of deformation can occur As a matter of practical convenience the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 02 from the origin The intersection of this line and the measured stress-strain line is used as an approximation of the materials yield strength called the 02 offset yield
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
Plane stress and Plane strain
bull Plane stress and Plane strainbull Plane Stress State of stress in which one or
two of the pairs of faces on an elementalbull cube are free of any stresses eg Torsion of thin
wall tube Expansion of a thin walledbull spherical shell under internal pressurebull Plane strain State of tress where one of the
pair of faces on an elemental undergoesbull zero strain eg Torsion of thin walled tube Piece
of material being compressed in a die
Mechanical properties
The mechanical properties of a material are obtained by subjecting a specimen to prescribed loads and then measuring the resulting deformation
Usually the test is carried out on a special machine that is specifically designed for this purpose The measurements of the load and of the deformation are carried out
Tensile Test Machine (Instron)
Electronic Extensometer
ExtensometerGrip
Specimen
Grip
Tensile Testing(stretching or pulling)
Determine the behavior of the engineering material under load
Tensile Test -Tensile test determines the strength of the material when subjected to a simple stretching operation -Standard dimension test samples are pulled slowly and at uniform rate in a testing machin-The strain ( the elongation of the sample) is defined as
Engineering Strain = = (change in length)(original length) = L0
-The stress ( the applied force divided by the original cross-sectional area) is defined as
Engineering Stress = = (applied force)(original area) =PA0
Stress-Strain Curve
bull Elastic Limit Greatest stress a material is capable of
developing without a permanent setbull Note elastic limit for metals do not differ widely from the
values of the proportionalitybull Elastic limit may be taken as that stress at which there is
a permanent set of 02It is therefore higher than limit of proportionality (suggested by some authors)
bull Hooke srsquo law Stress is directly proportional to strain in the elastic range
bull Young srsquo Modulus It is ratio between stress and reversible strain (stiffness)
bull It is in fact a measure of the inter atomic bonding forces
Yield strength Proof stress usually defined as the stress which produces a measurable amount of permanent strain ie 02 or 01
Tensile strength - the maximum stress applied to the specimen
Failure stress - the stress applied to the specimen at failure (usually less than the maximum tensile strength because necking reduces the cross-sectional area)
Ductility
Elongation elongation is a measure of ductility which is given by
elongation =100 (Lo - Lf) Lo
whereLo = Initial lengthLf = Final Length
Ductility Reduction in Area reduction in area is a measure of ductility which is given by
reduction in area =100 (Ao - Af) Ao
whereAo = Initial areaAf = Final area
Poissonrsquos ratio
When pulled in tension (X) a sample getslonger and thinner ie a contraction in thewidth (Y) and breadth (Z)Poissonrsquos ratio when strained in the (X) direction how much strain occurs in the lateral directions (Y amp Z)For most metals this value is 0333
bull This is the Poisson effect Poissonrsquos ratio which is thebull negative ratio of the contraction over the extension isbull also an elastic constantbull Poissons Ratio laterial strainbull longitudinal strainbull εbull νbull εbull = minus = minus perpbull 1048614bull x unloadedbull ybull zbull loadedbull Pbull Pbull bullMetals ν = 03 ndash 035bull bullCeramics ν = 02 ndash 025bull bullPolymers ν = 025 ndash 0 5
Modulus of elasticity - the initial slope of the curve related directly to the strength of the atomic bonds
bull Resilience The ability of a material to absorb energy with in elastic limit
bull Measure by the modulus of resilience
-Which is strain energy per unit volume
-Stress the material from zero stress to the yield stress
bull Energy Force multiplied by the distance over which it acts
Modulus of resilience - the area under the linear part of the curve measuring the stored elastic energy
Toughness
The ability of a material to absorb energy in the plastic range
Toughness - The total area under the curve which measures the energy absorbed by the specimen in the process of breaking
Stress-Strain Curve
True Stress and True Strainbull True stress is the loadP divided by
the instantaneous area of the specimen Ai
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives AoLo AiLi
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking begins
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
True Strain
)1ln(
ln
e
LLL
L
L
L
dLd
oi
o
i
i
i
Assumes constant volumeValid for all strains up to point where necking begins
Engineering Vs True Stress-Strain Curves
Stress
Strain
True Stress - Strain Curve
Engineering Stress - Strain Curve
Ultimate Tensile Strength
Fracture
Fracture
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull IN Stress-Strain Curvesbull Plastic deformation is uniform and
permanent between the elastic limit and the UTS
bull bull Plastic deformation becomes non-uniform once the UTS is exceeded
bull In tension this non-uniform deformation manifests itself
bull as ldquoneckingrdquo
Criterion for necking
Increase in true stress (due to reduction in cross-sectional area) as the specimen elongates is more than to load carrying capacity due to strain hardening
Criteria for Instability
Rate of Geometrical Softening and Rate of Work Hardening
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10
Criteria for Necking
02 Offset Yield StrengthOffset Yield Strength
Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point The elastic portion of the curve is not perfectly linear and microscopic amounts of deformation can occur As a matter of practical convenience the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 02 from the origin The intersection of this line and the measured stress-strain line is used as an approximation of the materials yield strength called the 02 offset yield
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
Mechanical properties
The mechanical properties of a material are obtained by subjecting a specimen to prescribed loads and then measuring the resulting deformation
Usually the test is carried out on a special machine that is specifically designed for this purpose The measurements of the load and of the deformation are carried out
Tensile Test Machine (Instron)
Electronic Extensometer
ExtensometerGrip
Specimen
Grip
Tensile Testing(stretching or pulling)
Determine the behavior of the engineering material under load
Tensile Test -Tensile test determines the strength of the material when subjected to a simple stretching operation -Standard dimension test samples are pulled slowly and at uniform rate in a testing machin-The strain ( the elongation of the sample) is defined as
Engineering Strain = = (change in length)(original length) = L0
-The stress ( the applied force divided by the original cross-sectional area) is defined as
Engineering Stress = = (applied force)(original area) =PA0
Stress-Strain Curve
bull Elastic Limit Greatest stress a material is capable of
developing without a permanent setbull Note elastic limit for metals do not differ widely from the
values of the proportionalitybull Elastic limit may be taken as that stress at which there is
a permanent set of 02It is therefore higher than limit of proportionality (suggested by some authors)
bull Hooke srsquo law Stress is directly proportional to strain in the elastic range
bull Young srsquo Modulus It is ratio between stress and reversible strain (stiffness)
bull It is in fact a measure of the inter atomic bonding forces
Yield strength Proof stress usually defined as the stress which produces a measurable amount of permanent strain ie 02 or 01
Tensile strength - the maximum stress applied to the specimen
Failure stress - the stress applied to the specimen at failure (usually less than the maximum tensile strength because necking reduces the cross-sectional area)
Ductility
Elongation elongation is a measure of ductility which is given by
elongation =100 (Lo - Lf) Lo
whereLo = Initial lengthLf = Final Length
Ductility Reduction in Area reduction in area is a measure of ductility which is given by
reduction in area =100 (Ao - Af) Ao
whereAo = Initial areaAf = Final area
Poissonrsquos ratio
When pulled in tension (X) a sample getslonger and thinner ie a contraction in thewidth (Y) and breadth (Z)Poissonrsquos ratio when strained in the (X) direction how much strain occurs in the lateral directions (Y amp Z)For most metals this value is 0333
bull This is the Poisson effect Poissonrsquos ratio which is thebull negative ratio of the contraction over the extension isbull also an elastic constantbull Poissons Ratio laterial strainbull longitudinal strainbull εbull νbull εbull = minus = minus perpbull 1048614bull x unloadedbull ybull zbull loadedbull Pbull Pbull bullMetals ν = 03 ndash 035bull bullCeramics ν = 02 ndash 025bull bullPolymers ν = 025 ndash 0 5
Modulus of elasticity - the initial slope of the curve related directly to the strength of the atomic bonds
bull Resilience The ability of a material to absorb energy with in elastic limit
bull Measure by the modulus of resilience
-Which is strain energy per unit volume
-Stress the material from zero stress to the yield stress
bull Energy Force multiplied by the distance over which it acts
Modulus of resilience - the area under the linear part of the curve measuring the stored elastic energy
Toughness
The ability of a material to absorb energy in the plastic range
Toughness - The total area under the curve which measures the energy absorbed by the specimen in the process of breaking
Stress-Strain Curve
True Stress and True Strainbull True stress is the loadP divided by
the instantaneous area of the specimen Ai
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives AoLo AiLi
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking begins
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
True Strain
)1ln(
ln
e
LLL
L
L
L
dLd
oi
o
i
i
i
Assumes constant volumeValid for all strains up to point where necking begins
Engineering Vs True Stress-Strain Curves
Stress
Strain
True Stress - Strain Curve
Engineering Stress - Strain Curve
Ultimate Tensile Strength
Fracture
Fracture
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull IN Stress-Strain Curvesbull Plastic deformation is uniform and
permanent between the elastic limit and the UTS
bull bull Plastic deformation becomes non-uniform once the UTS is exceeded
bull In tension this non-uniform deformation manifests itself
bull as ldquoneckingrdquo
Criterion for necking
Increase in true stress (due to reduction in cross-sectional area) as the specimen elongates is more than to load carrying capacity due to strain hardening
Criteria for Instability
Rate of Geometrical Softening and Rate of Work Hardening
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10
Criteria for Necking
02 Offset Yield StrengthOffset Yield Strength
Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point The elastic portion of the curve is not perfectly linear and microscopic amounts of deformation can occur As a matter of practical convenience the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 02 from the origin The intersection of this line and the measured stress-strain line is used as an approximation of the materials yield strength called the 02 offset yield
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
Tensile Test Machine (Instron)
Electronic Extensometer
ExtensometerGrip
Specimen
Grip
Tensile Testing(stretching or pulling)
Determine the behavior of the engineering material under load
Tensile Test -Tensile test determines the strength of the material when subjected to a simple stretching operation -Standard dimension test samples are pulled slowly and at uniform rate in a testing machin-The strain ( the elongation of the sample) is defined as
Engineering Strain = = (change in length)(original length) = L0
-The stress ( the applied force divided by the original cross-sectional area) is defined as
Engineering Stress = = (applied force)(original area) =PA0
Stress-Strain Curve
bull Elastic Limit Greatest stress a material is capable of
developing without a permanent setbull Note elastic limit for metals do not differ widely from the
values of the proportionalitybull Elastic limit may be taken as that stress at which there is
a permanent set of 02It is therefore higher than limit of proportionality (suggested by some authors)
bull Hooke srsquo law Stress is directly proportional to strain in the elastic range
bull Young srsquo Modulus It is ratio between stress and reversible strain (stiffness)
bull It is in fact a measure of the inter atomic bonding forces
Yield strength Proof stress usually defined as the stress which produces a measurable amount of permanent strain ie 02 or 01
Tensile strength - the maximum stress applied to the specimen
Failure stress - the stress applied to the specimen at failure (usually less than the maximum tensile strength because necking reduces the cross-sectional area)
Ductility
Elongation elongation is a measure of ductility which is given by
elongation =100 (Lo - Lf) Lo
whereLo = Initial lengthLf = Final Length
Ductility Reduction in Area reduction in area is a measure of ductility which is given by
reduction in area =100 (Ao - Af) Ao
whereAo = Initial areaAf = Final area
Poissonrsquos ratio
When pulled in tension (X) a sample getslonger and thinner ie a contraction in thewidth (Y) and breadth (Z)Poissonrsquos ratio when strained in the (X) direction how much strain occurs in the lateral directions (Y amp Z)For most metals this value is 0333
bull This is the Poisson effect Poissonrsquos ratio which is thebull negative ratio of the contraction over the extension isbull also an elastic constantbull Poissons Ratio laterial strainbull longitudinal strainbull εbull νbull εbull = minus = minus perpbull 1048614bull x unloadedbull ybull zbull loadedbull Pbull Pbull bullMetals ν = 03 ndash 035bull bullCeramics ν = 02 ndash 025bull bullPolymers ν = 025 ndash 0 5
Modulus of elasticity - the initial slope of the curve related directly to the strength of the atomic bonds
bull Resilience The ability of a material to absorb energy with in elastic limit
bull Measure by the modulus of resilience
-Which is strain energy per unit volume
-Stress the material from zero stress to the yield stress
bull Energy Force multiplied by the distance over which it acts
Modulus of resilience - the area under the linear part of the curve measuring the stored elastic energy
Toughness
The ability of a material to absorb energy in the plastic range
Toughness - The total area under the curve which measures the energy absorbed by the specimen in the process of breaking
Stress-Strain Curve
True Stress and True Strainbull True stress is the loadP divided by
the instantaneous area of the specimen Ai
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives AoLo AiLi
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking begins
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
True Strain
)1ln(
ln
e
LLL
L
L
L
dLd
oi
o
i
i
i
Assumes constant volumeValid for all strains up to point where necking begins
Engineering Vs True Stress-Strain Curves
Stress
Strain
True Stress - Strain Curve
Engineering Stress - Strain Curve
Ultimate Tensile Strength
Fracture
Fracture
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull IN Stress-Strain Curvesbull Plastic deformation is uniform and
permanent between the elastic limit and the UTS
bull bull Plastic deformation becomes non-uniform once the UTS is exceeded
bull In tension this non-uniform deformation manifests itself
bull as ldquoneckingrdquo
Criterion for necking
Increase in true stress (due to reduction in cross-sectional area) as the specimen elongates is more than to load carrying capacity due to strain hardening
Criteria for Instability
Rate of Geometrical Softening and Rate of Work Hardening
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10
Criteria for Necking
02 Offset Yield StrengthOffset Yield Strength
Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point The elastic portion of the curve is not perfectly linear and microscopic amounts of deformation can occur As a matter of practical convenience the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 02 from the origin The intersection of this line and the measured stress-strain line is used as an approximation of the materials yield strength called the 02 offset yield
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
Electronic Extensometer
ExtensometerGrip
Specimen
Grip
Tensile Testing(stretching or pulling)
Determine the behavior of the engineering material under load
Tensile Test -Tensile test determines the strength of the material when subjected to a simple stretching operation -Standard dimension test samples are pulled slowly and at uniform rate in a testing machin-The strain ( the elongation of the sample) is defined as
Engineering Strain = = (change in length)(original length) = L0
-The stress ( the applied force divided by the original cross-sectional area) is defined as
Engineering Stress = = (applied force)(original area) =PA0
Stress-Strain Curve
bull Elastic Limit Greatest stress a material is capable of
developing without a permanent setbull Note elastic limit for metals do not differ widely from the
values of the proportionalitybull Elastic limit may be taken as that stress at which there is
a permanent set of 02It is therefore higher than limit of proportionality (suggested by some authors)
bull Hooke srsquo law Stress is directly proportional to strain in the elastic range
bull Young srsquo Modulus It is ratio between stress and reversible strain (stiffness)
bull It is in fact a measure of the inter atomic bonding forces
Yield strength Proof stress usually defined as the stress which produces a measurable amount of permanent strain ie 02 or 01
Tensile strength - the maximum stress applied to the specimen
Failure stress - the stress applied to the specimen at failure (usually less than the maximum tensile strength because necking reduces the cross-sectional area)
Ductility
Elongation elongation is a measure of ductility which is given by
elongation =100 (Lo - Lf) Lo
whereLo = Initial lengthLf = Final Length
Ductility Reduction in Area reduction in area is a measure of ductility which is given by
reduction in area =100 (Ao - Af) Ao
whereAo = Initial areaAf = Final area
Poissonrsquos ratio
When pulled in tension (X) a sample getslonger and thinner ie a contraction in thewidth (Y) and breadth (Z)Poissonrsquos ratio when strained in the (X) direction how much strain occurs in the lateral directions (Y amp Z)For most metals this value is 0333
bull This is the Poisson effect Poissonrsquos ratio which is thebull negative ratio of the contraction over the extension isbull also an elastic constantbull Poissons Ratio laterial strainbull longitudinal strainbull εbull νbull εbull = minus = minus perpbull 1048614bull x unloadedbull ybull zbull loadedbull Pbull Pbull bullMetals ν = 03 ndash 035bull bullCeramics ν = 02 ndash 025bull bullPolymers ν = 025 ndash 0 5
Modulus of elasticity - the initial slope of the curve related directly to the strength of the atomic bonds
bull Resilience The ability of a material to absorb energy with in elastic limit
bull Measure by the modulus of resilience
-Which is strain energy per unit volume
-Stress the material from zero stress to the yield stress
bull Energy Force multiplied by the distance over which it acts
Modulus of resilience - the area under the linear part of the curve measuring the stored elastic energy
Toughness
The ability of a material to absorb energy in the plastic range
Toughness - The total area under the curve which measures the energy absorbed by the specimen in the process of breaking
Stress-Strain Curve
True Stress and True Strainbull True stress is the loadP divided by
the instantaneous area of the specimen Ai
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives AoLo AiLi
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking begins
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
True Strain
)1ln(
ln
e
LLL
L
L
L
dLd
oi
o
i
i
i
Assumes constant volumeValid for all strains up to point where necking begins
Engineering Vs True Stress-Strain Curves
Stress
Strain
True Stress - Strain Curve
Engineering Stress - Strain Curve
Ultimate Tensile Strength
Fracture
Fracture
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull IN Stress-Strain Curvesbull Plastic deformation is uniform and
permanent between the elastic limit and the UTS
bull bull Plastic deformation becomes non-uniform once the UTS is exceeded
bull In tension this non-uniform deformation manifests itself
bull as ldquoneckingrdquo
Criterion for necking
Increase in true stress (due to reduction in cross-sectional area) as the specimen elongates is more than to load carrying capacity due to strain hardening
Criteria for Instability
Rate of Geometrical Softening and Rate of Work Hardening
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10
Criteria for Necking
02 Offset Yield StrengthOffset Yield Strength
Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point The elastic portion of the curve is not perfectly linear and microscopic amounts of deformation can occur As a matter of practical convenience the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 02 from the origin The intersection of this line and the measured stress-strain line is used as an approximation of the materials yield strength called the 02 offset yield
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
ExtensometerGrip
Specimen
Grip
Tensile Testing(stretching or pulling)
Determine the behavior of the engineering material under load
Tensile Test -Tensile test determines the strength of the material when subjected to a simple stretching operation -Standard dimension test samples are pulled slowly and at uniform rate in a testing machin-The strain ( the elongation of the sample) is defined as
Engineering Strain = = (change in length)(original length) = L0
-The stress ( the applied force divided by the original cross-sectional area) is defined as
Engineering Stress = = (applied force)(original area) =PA0
Stress-Strain Curve
bull Elastic Limit Greatest stress a material is capable of
developing without a permanent setbull Note elastic limit for metals do not differ widely from the
values of the proportionalitybull Elastic limit may be taken as that stress at which there is
a permanent set of 02It is therefore higher than limit of proportionality (suggested by some authors)
bull Hooke srsquo law Stress is directly proportional to strain in the elastic range
bull Young srsquo Modulus It is ratio between stress and reversible strain (stiffness)
bull It is in fact a measure of the inter atomic bonding forces
Yield strength Proof stress usually defined as the stress which produces a measurable amount of permanent strain ie 02 or 01
Tensile strength - the maximum stress applied to the specimen
Failure stress - the stress applied to the specimen at failure (usually less than the maximum tensile strength because necking reduces the cross-sectional area)
Ductility
Elongation elongation is a measure of ductility which is given by
elongation =100 (Lo - Lf) Lo
whereLo = Initial lengthLf = Final Length
Ductility Reduction in Area reduction in area is a measure of ductility which is given by
reduction in area =100 (Ao - Af) Ao
whereAo = Initial areaAf = Final area
Poissonrsquos ratio
When pulled in tension (X) a sample getslonger and thinner ie a contraction in thewidth (Y) and breadth (Z)Poissonrsquos ratio when strained in the (X) direction how much strain occurs in the lateral directions (Y amp Z)For most metals this value is 0333
bull This is the Poisson effect Poissonrsquos ratio which is thebull negative ratio of the contraction over the extension isbull also an elastic constantbull Poissons Ratio laterial strainbull longitudinal strainbull εbull νbull εbull = minus = minus perpbull 1048614bull x unloadedbull ybull zbull loadedbull Pbull Pbull bullMetals ν = 03 ndash 035bull bullCeramics ν = 02 ndash 025bull bullPolymers ν = 025 ndash 0 5
Modulus of elasticity - the initial slope of the curve related directly to the strength of the atomic bonds
bull Resilience The ability of a material to absorb energy with in elastic limit
bull Measure by the modulus of resilience
-Which is strain energy per unit volume
-Stress the material from zero stress to the yield stress
bull Energy Force multiplied by the distance over which it acts
Modulus of resilience - the area under the linear part of the curve measuring the stored elastic energy
Toughness
The ability of a material to absorb energy in the plastic range
Toughness - The total area under the curve which measures the energy absorbed by the specimen in the process of breaking
Stress-Strain Curve
True Stress and True Strainbull True stress is the loadP divided by
the instantaneous area of the specimen Ai
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives AoLo AiLi
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking begins
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
True Strain
)1ln(
ln
e
LLL
L
L
L
dLd
oi
o
i
i
i
Assumes constant volumeValid for all strains up to point where necking begins
Engineering Vs True Stress-Strain Curves
Stress
Strain
True Stress - Strain Curve
Engineering Stress - Strain Curve
Ultimate Tensile Strength
Fracture
Fracture
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull IN Stress-Strain Curvesbull Plastic deformation is uniform and
permanent between the elastic limit and the UTS
bull bull Plastic deformation becomes non-uniform once the UTS is exceeded
bull In tension this non-uniform deformation manifests itself
bull as ldquoneckingrdquo
Criterion for necking
Increase in true stress (due to reduction in cross-sectional area) as the specimen elongates is more than to load carrying capacity due to strain hardening
Criteria for Instability
Rate of Geometrical Softening and Rate of Work Hardening
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10
Criteria for Necking
02 Offset Yield StrengthOffset Yield Strength
Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point The elastic portion of the curve is not perfectly linear and microscopic amounts of deformation can occur As a matter of practical convenience the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 02 from the origin The intersection of this line and the measured stress-strain line is used as an approximation of the materials yield strength called the 02 offset yield
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
Tensile Testing(stretching or pulling)
Determine the behavior of the engineering material under load
Tensile Test -Tensile test determines the strength of the material when subjected to a simple stretching operation -Standard dimension test samples are pulled slowly and at uniform rate in a testing machin-The strain ( the elongation of the sample) is defined as
Engineering Strain = = (change in length)(original length) = L0
-The stress ( the applied force divided by the original cross-sectional area) is defined as
Engineering Stress = = (applied force)(original area) =PA0
Stress-Strain Curve
bull Elastic Limit Greatest stress a material is capable of
developing without a permanent setbull Note elastic limit for metals do not differ widely from the
values of the proportionalitybull Elastic limit may be taken as that stress at which there is
a permanent set of 02It is therefore higher than limit of proportionality (suggested by some authors)
bull Hooke srsquo law Stress is directly proportional to strain in the elastic range
bull Young srsquo Modulus It is ratio between stress and reversible strain (stiffness)
bull It is in fact a measure of the inter atomic bonding forces
Yield strength Proof stress usually defined as the stress which produces a measurable amount of permanent strain ie 02 or 01
Tensile strength - the maximum stress applied to the specimen
Failure stress - the stress applied to the specimen at failure (usually less than the maximum tensile strength because necking reduces the cross-sectional area)
Ductility
Elongation elongation is a measure of ductility which is given by
elongation =100 (Lo - Lf) Lo
whereLo = Initial lengthLf = Final Length
Ductility Reduction in Area reduction in area is a measure of ductility which is given by
reduction in area =100 (Ao - Af) Ao
whereAo = Initial areaAf = Final area
Poissonrsquos ratio
When pulled in tension (X) a sample getslonger and thinner ie a contraction in thewidth (Y) and breadth (Z)Poissonrsquos ratio when strained in the (X) direction how much strain occurs in the lateral directions (Y amp Z)For most metals this value is 0333
bull This is the Poisson effect Poissonrsquos ratio which is thebull negative ratio of the contraction over the extension isbull also an elastic constantbull Poissons Ratio laterial strainbull longitudinal strainbull εbull νbull εbull = minus = minus perpbull 1048614bull x unloadedbull ybull zbull loadedbull Pbull Pbull bullMetals ν = 03 ndash 035bull bullCeramics ν = 02 ndash 025bull bullPolymers ν = 025 ndash 0 5
Modulus of elasticity - the initial slope of the curve related directly to the strength of the atomic bonds
bull Resilience The ability of a material to absorb energy with in elastic limit
bull Measure by the modulus of resilience
-Which is strain energy per unit volume
-Stress the material from zero stress to the yield stress
bull Energy Force multiplied by the distance over which it acts
Modulus of resilience - the area under the linear part of the curve measuring the stored elastic energy
Toughness
The ability of a material to absorb energy in the plastic range
Toughness - The total area under the curve which measures the energy absorbed by the specimen in the process of breaking
Stress-Strain Curve
True Stress and True Strainbull True stress is the loadP divided by
the instantaneous area of the specimen Ai
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives AoLo AiLi
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking begins
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
True Strain
)1ln(
ln
e
LLL
L
L
L
dLd
oi
o
i
i
i
Assumes constant volumeValid for all strains up to point where necking begins
Engineering Vs True Stress-Strain Curves
Stress
Strain
True Stress - Strain Curve
Engineering Stress - Strain Curve
Ultimate Tensile Strength
Fracture
Fracture
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull IN Stress-Strain Curvesbull Plastic deformation is uniform and
permanent between the elastic limit and the UTS
bull bull Plastic deformation becomes non-uniform once the UTS is exceeded
bull In tension this non-uniform deformation manifests itself
bull as ldquoneckingrdquo
Criterion for necking
Increase in true stress (due to reduction in cross-sectional area) as the specimen elongates is more than to load carrying capacity due to strain hardening
Criteria for Instability
Rate of Geometrical Softening and Rate of Work Hardening
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10
Criteria for Necking
02 Offset Yield StrengthOffset Yield Strength
Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point The elastic portion of the curve is not perfectly linear and microscopic amounts of deformation can occur As a matter of practical convenience the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 02 from the origin The intersection of this line and the measured stress-strain line is used as an approximation of the materials yield strength called the 02 offset yield
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
Tensile Test -Tensile test determines the strength of the material when subjected to a simple stretching operation -Standard dimension test samples are pulled slowly and at uniform rate in a testing machin-The strain ( the elongation of the sample) is defined as
Engineering Strain = = (change in length)(original length) = L0
-The stress ( the applied force divided by the original cross-sectional area) is defined as
Engineering Stress = = (applied force)(original area) =PA0
Stress-Strain Curve
bull Elastic Limit Greatest stress a material is capable of
developing without a permanent setbull Note elastic limit for metals do not differ widely from the
values of the proportionalitybull Elastic limit may be taken as that stress at which there is
a permanent set of 02It is therefore higher than limit of proportionality (suggested by some authors)
bull Hooke srsquo law Stress is directly proportional to strain in the elastic range
bull Young srsquo Modulus It is ratio between stress and reversible strain (stiffness)
bull It is in fact a measure of the inter atomic bonding forces
Yield strength Proof stress usually defined as the stress which produces a measurable amount of permanent strain ie 02 or 01
Tensile strength - the maximum stress applied to the specimen
Failure stress - the stress applied to the specimen at failure (usually less than the maximum tensile strength because necking reduces the cross-sectional area)
Ductility
Elongation elongation is a measure of ductility which is given by
elongation =100 (Lo - Lf) Lo
whereLo = Initial lengthLf = Final Length
Ductility Reduction in Area reduction in area is a measure of ductility which is given by
reduction in area =100 (Ao - Af) Ao
whereAo = Initial areaAf = Final area
Poissonrsquos ratio
When pulled in tension (X) a sample getslonger and thinner ie a contraction in thewidth (Y) and breadth (Z)Poissonrsquos ratio when strained in the (X) direction how much strain occurs in the lateral directions (Y amp Z)For most metals this value is 0333
bull This is the Poisson effect Poissonrsquos ratio which is thebull negative ratio of the contraction over the extension isbull also an elastic constantbull Poissons Ratio laterial strainbull longitudinal strainbull εbull νbull εbull = minus = minus perpbull 1048614bull x unloadedbull ybull zbull loadedbull Pbull Pbull bullMetals ν = 03 ndash 035bull bullCeramics ν = 02 ndash 025bull bullPolymers ν = 025 ndash 0 5
Modulus of elasticity - the initial slope of the curve related directly to the strength of the atomic bonds
bull Resilience The ability of a material to absorb energy with in elastic limit
bull Measure by the modulus of resilience
-Which is strain energy per unit volume
-Stress the material from zero stress to the yield stress
bull Energy Force multiplied by the distance over which it acts
Modulus of resilience - the area under the linear part of the curve measuring the stored elastic energy
Toughness
The ability of a material to absorb energy in the plastic range
Toughness - The total area under the curve which measures the energy absorbed by the specimen in the process of breaking
Stress-Strain Curve
True Stress and True Strainbull True stress is the loadP divided by
the instantaneous area of the specimen Ai
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives AoLo AiLi
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking begins
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
True Strain
)1ln(
ln
e
LLL
L
L
L
dLd
oi
o
i
i
i
Assumes constant volumeValid for all strains up to point where necking begins
Engineering Vs True Stress-Strain Curves
Stress
Strain
True Stress - Strain Curve
Engineering Stress - Strain Curve
Ultimate Tensile Strength
Fracture
Fracture
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull IN Stress-Strain Curvesbull Plastic deformation is uniform and
permanent between the elastic limit and the UTS
bull bull Plastic deformation becomes non-uniform once the UTS is exceeded
bull In tension this non-uniform deformation manifests itself
bull as ldquoneckingrdquo
Criterion for necking
Increase in true stress (due to reduction in cross-sectional area) as the specimen elongates is more than to load carrying capacity due to strain hardening
Criteria for Instability
Rate of Geometrical Softening and Rate of Work Hardening
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10
Criteria for Necking
02 Offset Yield StrengthOffset Yield Strength
Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point The elastic portion of the curve is not perfectly linear and microscopic amounts of deformation can occur As a matter of practical convenience the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 02 from the origin The intersection of this line and the measured stress-strain line is used as an approximation of the materials yield strength called the 02 offset yield
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
Stress-Strain Curve
bull Elastic Limit Greatest stress a material is capable of
developing without a permanent setbull Note elastic limit for metals do not differ widely from the
values of the proportionalitybull Elastic limit may be taken as that stress at which there is
a permanent set of 02It is therefore higher than limit of proportionality (suggested by some authors)
bull Hooke srsquo law Stress is directly proportional to strain in the elastic range
bull Young srsquo Modulus It is ratio between stress and reversible strain (stiffness)
bull It is in fact a measure of the inter atomic bonding forces
Yield strength Proof stress usually defined as the stress which produces a measurable amount of permanent strain ie 02 or 01
Tensile strength - the maximum stress applied to the specimen
Failure stress - the stress applied to the specimen at failure (usually less than the maximum tensile strength because necking reduces the cross-sectional area)
Ductility
Elongation elongation is a measure of ductility which is given by
elongation =100 (Lo - Lf) Lo
whereLo = Initial lengthLf = Final Length
Ductility Reduction in Area reduction in area is a measure of ductility which is given by
reduction in area =100 (Ao - Af) Ao
whereAo = Initial areaAf = Final area
Poissonrsquos ratio
When pulled in tension (X) a sample getslonger and thinner ie a contraction in thewidth (Y) and breadth (Z)Poissonrsquos ratio when strained in the (X) direction how much strain occurs in the lateral directions (Y amp Z)For most metals this value is 0333
bull This is the Poisson effect Poissonrsquos ratio which is thebull negative ratio of the contraction over the extension isbull also an elastic constantbull Poissons Ratio laterial strainbull longitudinal strainbull εbull νbull εbull = minus = minus perpbull 1048614bull x unloadedbull ybull zbull loadedbull Pbull Pbull bullMetals ν = 03 ndash 035bull bullCeramics ν = 02 ndash 025bull bullPolymers ν = 025 ndash 0 5
Modulus of elasticity - the initial slope of the curve related directly to the strength of the atomic bonds
bull Resilience The ability of a material to absorb energy with in elastic limit
bull Measure by the modulus of resilience
-Which is strain energy per unit volume
-Stress the material from zero stress to the yield stress
bull Energy Force multiplied by the distance over which it acts
Modulus of resilience - the area under the linear part of the curve measuring the stored elastic energy
Toughness
The ability of a material to absorb energy in the plastic range
Toughness - The total area under the curve which measures the energy absorbed by the specimen in the process of breaking
Stress-Strain Curve
True Stress and True Strainbull True stress is the loadP divided by
the instantaneous area of the specimen Ai
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives AoLo AiLi
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking begins
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
True Strain
)1ln(
ln
e
LLL
L
L
L
dLd
oi
o
i
i
i
Assumes constant volumeValid for all strains up to point where necking begins
Engineering Vs True Stress-Strain Curves
Stress
Strain
True Stress - Strain Curve
Engineering Stress - Strain Curve
Ultimate Tensile Strength
Fracture
Fracture
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull IN Stress-Strain Curvesbull Plastic deformation is uniform and
permanent between the elastic limit and the UTS
bull bull Plastic deformation becomes non-uniform once the UTS is exceeded
bull In tension this non-uniform deformation manifests itself
bull as ldquoneckingrdquo
Criterion for necking
Increase in true stress (due to reduction in cross-sectional area) as the specimen elongates is more than to load carrying capacity due to strain hardening
Criteria for Instability
Rate of Geometrical Softening and Rate of Work Hardening
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10
Criteria for Necking
02 Offset Yield StrengthOffset Yield Strength
Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point The elastic portion of the curve is not perfectly linear and microscopic amounts of deformation can occur As a matter of practical convenience the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 02 from the origin The intersection of this line and the measured stress-strain line is used as an approximation of the materials yield strength called the 02 offset yield
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
bull Elastic Limit Greatest stress a material is capable of
developing without a permanent setbull Note elastic limit for metals do not differ widely from the
values of the proportionalitybull Elastic limit may be taken as that stress at which there is
a permanent set of 02It is therefore higher than limit of proportionality (suggested by some authors)
bull Hooke srsquo law Stress is directly proportional to strain in the elastic range
bull Young srsquo Modulus It is ratio between stress and reversible strain (stiffness)
bull It is in fact a measure of the inter atomic bonding forces
Yield strength Proof stress usually defined as the stress which produces a measurable amount of permanent strain ie 02 or 01
Tensile strength - the maximum stress applied to the specimen
Failure stress - the stress applied to the specimen at failure (usually less than the maximum tensile strength because necking reduces the cross-sectional area)
Ductility
Elongation elongation is a measure of ductility which is given by
elongation =100 (Lo - Lf) Lo
whereLo = Initial lengthLf = Final Length
Ductility Reduction in Area reduction in area is a measure of ductility which is given by
reduction in area =100 (Ao - Af) Ao
whereAo = Initial areaAf = Final area
Poissonrsquos ratio
When pulled in tension (X) a sample getslonger and thinner ie a contraction in thewidth (Y) and breadth (Z)Poissonrsquos ratio when strained in the (X) direction how much strain occurs in the lateral directions (Y amp Z)For most metals this value is 0333
bull This is the Poisson effect Poissonrsquos ratio which is thebull negative ratio of the contraction over the extension isbull also an elastic constantbull Poissons Ratio laterial strainbull longitudinal strainbull εbull νbull εbull = minus = minus perpbull 1048614bull x unloadedbull ybull zbull loadedbull Pbull Pbull bullMetals ν = 03 ndash 035bull bullCeramics ν = 02 ndash 025bull bullPolymers ν = 025 ndash 0 5
Modulus of elasticity - the initial slope of the curve related directly to the strength of the atomic bonds
bull Resilience The ability of a material to absorb energy with in elastic limit
bull Measure by the modulus of resilience
-Which is strain energy per unit volume
-Stress the material from zero stress to the yield stress
bull Energy Force multiplied by the distance over which it acts
Modulus of resilience - the area under the linear part of the curve measuring the stored elastic energy
Toughness
The ability of a material to absorb energy in the plastic range
Toughness - The total area under the curve which measures the energy absorbed by the specimen in the process of breaking
Stress-Strain Curve
True Stress and True Strainbull True stress is the loadP divided by
the instantaneous area of the specimen Ai
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives AoLo AiLi
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking begins
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
True Strain
)1ln(
ln
e
LLL
L
L
L
dLd
oi
o
i
i
i
Assumes constant volumeValid for all strains up to point where necking begins
Engineering Vs True Stress-Strain Curves
Stress
Strain
True Stress - Strain Curve
Engineering Stress - Strain Curve
Ultimate Tensile Strength
Fracture
Fracture
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull IN Stress-Strain Curvesbull Plastic deformation is uniform and
permanent between the elastic limit and the UTS
bull bull Plastic deformation becomes non-uniform once the UTS is exceeded
bull In tension this non-uniform deformation manifests itself
bull as ldquoneckingrdquo
Criterion for necking
Increase in true stress (due to reduction in cross-sectional area) as the specimen elongates is more than to load carrying capacity due to strain hardening
Criteria for Instability
Rate of Geometrical Softening and Rate of Work Hardening
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10
Criteria for Necking
02 Offset Yield StrengthOffset Yield Strength
Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point The elastic portion of the curve is not perfectly linear and microscopic amounts of deformation can occur As a matter of practical convenience the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 02 from the origin The intersection of this line and the measured stress-strain line is used as an approximation of the materials yield strength called the 02 offset yield
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
Yield strength Proof stress usually defined as the stress which produces a measurable amount of permanent strain ie 02 or 01
Tensile strength - the maximum stress applied to the specimen
Failure stress - the stress applied to the specimen at failure (usually less than the maximum tensile strength because necking reduces the cross-sectional area)
Ductility
Elongation elongation is a measure of ductility which is given by
elongation =100 (Lo - Lf) Lo
whereLo = Initial lengthLf = Final Length
Ductility Reduction in Area reduction in area is a measure of ductility which is given by
reduction in area =100 (Ao - Af) Ao
whereAo = Initial areaAf = Final area
Poissonrsquos ratio
When pulled in tension (X) a sample getslonger and thinner ie a contraction in thewidth (Y) and breadth (Z)Poissonrsquos ratio when strained in the (X) direction how much strain occurs in the lateral directions (Y amp Z)For most metals this value is 0333
bull This is the Poisson effect Poissonrsquos ratio which is thebull negative ratio of the contraction over the extension isbull also an elastic constantbull Poissons Ratio laterial strainbull longitudinal strainbull εbull νbull εbull = minus = minus perpbull 1048614bull x unloadedbull ybull zbull loadedbull Pbull Pbull bullMetals ν = 03 ndash 035bull bullCeramics ν = 02 ndash 025bull bullPolymers ν = 025 ndash 0 5
Modulus of elasticity - the initial slope of the curve related directly to the strength of the atomic bonds
bull Resilience The ability of a material to absorb energy with in elastic limit
bull Measure by the modulus of resilience
-Which is strain energy per unit volume
-Stress the material from zero stress to the yield stress
bull Energy Force multiplied by the distance over which it acts
Modulus of resilience - the area under the linear part of the curve measuring the stored elastic energy
Toughness
The ability of a material to absorb energy in the plastic range
Toughness - The total area under the curve which measures the energy absorbed by the specimen in the process of breaking
Stress-Strain Curve
True Stress and True Strainbull True stress is the loadP divided by
the instantaneous area of the specimen Ai
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives AoLo AiLi
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking begins
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
True Strain
)1ln(
ln
e
LLL
L
L
L
dLd
oi
o
i
i
i
Assumes constant volumeValid for all strains up to point where necking begins
Engineering Vs True Stress-Strain Curves
Stress
Strain
True Stress - Strain Curve
Engineering Stress - Strain Curve
Ultimate Tensile Strength
Fracture
Fracture
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull IN Stress-Strain Curvesbull Plastic deformation is uniform and
permanent between the elastic limit and the UTS
bull bull Plastic deformation becomes non-uniform once the UTS is exceeded
bull In tension this non-uniform deformation manifests itself
bull as ldquoneckingrdquo
Criterion for necking
Increase in true stress (due to reduction in cross-sectional area) as the specimen elongates is more than to load carrying capacity due to strain hardening
Criteria for Instability
Rate of Geometrical Softening and Rate of Work Hardening
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10
Criteria for Necking
02 Offset Yield StrengthOffset Yield Strength
Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point The elastic portion of the curve is not perfectly linear and microscopic amounts of deformation can occur As a matter of practical convenience the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 02 from the origin The intersection of this line and the measured stress-strain line is used as an approximation of the materials yield strength called the 02 offset yield
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
Tensile strength - the maximum stress applied to the specimen
Failure stress - the stress applied to the specimen at failure (usually less than the maximum tensile strength because necking reduces the cross-sectional area)
Ductility
Elongation elongation is a measure of ductility which is given by
elongation =100 (Lo - Lf) Lo
whereLo = Initial lengthLf = Final Length
Ductility Reduction in Area reduction in area is a measure of ductility which is given by
reduction in area =100 (Ao - Af) Ao
whereAo = Initial areaAf = Final area
Poissonrsquos ratio
When pulled in tension (X) a sample getslonger and thinner ie a contraction in thewidth (Y) and breadth (Z)Poissonrsquos ratio when strained in the (X) direction how much strain occurs in the lateral directions (Y amp Z)For most metals this value is 0333
bull This is the Poisson effect Poissonrsquos ratio which is thebull negative ratio of the contraction over the extension isbull also an elastic constantbull Poissons Ratio laterial strainbull longitudinal strainbull εbull νbull εbull = minus = minus perpbull 1048614bull x unloadedbull ybull zbull loadedbull Pbull Pbull bullMetals ν = 03 ndash 035bull bullCeramics ν = 02 ndash 025bull bullPolymers ν = 025 ndash 0 5
Modulus of elasticity - the initial slope of the curve related directly to the strength of the atomic bonds
bull Resilience The ability of a material to absorb energy with in elastic limit
bull Measure by the modulus of resilience
-Which is strain energy per unit volume
-Stress the material from zero stress to the yield stress
bull Energy Force multiplied by the distance over which it acts
Modulus of resilience - the area under the linear part of the curve measuring the stored elastic energy
Toughness
The ability of a material to absorb energy in the plastic range
Toughness - The total area under the curve which measures the energy absorbed by the specimen in the process of breaking
Stress-Strain Curve
True Stress and True Strainbull True stress is the loadP divided by
the instantaneous area of the specimen Ai
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives AoLo AiLi
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking begins
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
True Strain
)1ln(
ln
e
LLL
L
L
L
dLd
oi
o
i
i
i
Assumes constant volumeValid for all strains up to point where necking begins
Engineering Vs True Stress-Strain Curves
Stress
Strain
True Stress - Strain Curve
Engineering Stress - Strain Curve
Ultimate Tensile Strength
Fracture
Fracture
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull IN Stress-Strain Curvesbull Plastic deformation is uniform and
permanent between the elastic limit and the UTS
bull bull Plastic deformation becomes non-uniform once the UTS is exceeded
bull In tension this non-uniform deformation manifests itself
bull as ldquoneckingrdquo
Criterion for necking
Increase in true stress (due to reduction in cross-sectional area) as the specimen elongates is more than to load carrying capacity due to strain hardening
Criteria for Instability
Rate of Geometrical Softening and Rate of Work Hardening
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10
Criteria for Necking
02 Offset Yield StrengthOffset Yield Strength
Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point The elastic portion of the curve is not perfectly linear and microscopic amounts of deformation can occur As a matter of practical convenience the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 02 from the origin The intersection of this line and the measured stress-strain line is used as an approximation of the materials yield strength called the 02 offset yield
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
Failure stress - the stress applied to the specimen at failure (usually less than the maximum tensile strength because necking reduces the cross-sectional area)
Ductility
Elongation elongation is a measure of ductility which is given by
elongation =100 (Lo - Lf) Lo
whereLo = Initial lengthLf = Final Length
Ductility Reduction in Area reduction in area is a measure of ductility which is given by
reduction in area =100 (Ao - Af) Ao
whereAo = Initial areaAf = Final area
Poissonrsquos ratio
When pulled in tension (X) a sample getslonger and thinner ie a contraction in thewidth (Y) and breadth (Z)Poissonrsquos ratio when strained in the (X) direction how much strain occurs in the lateral directions (Y amp Z)For most metals this value is 0333
bull This is the Poisson effect Poissonrsquos ratio which is thebull negative ratio of the contraction over the extension isbull also an elastic constantbull Poissons Ratio laterial strainbull longitudinal strainbull εbull νbull εbull = minus = minus perpbull 1048614bull x unloadedbull ybull zbull loadedbull Pbull Pbull bullMetals ν = 03 ndash 035bull bullCeramics ν = 02 ndash 025bull bullPolymers ν = 025 ndash 0 5
Modulus of elasticity - the initial slope of the curve related directly to the strength of the atomic bonds
bull Resilience The ability of a material to absorb energy with in elastic limit
bull Measure by the modulus of resilience
-Which is strain energy per unit volume
-Stress the material from zero stress to the yield stress
bull Energy Force multiplied by the distance over which it acts
Modulus of resilience - the area under the linear part of the curve measuring the stored elastic energy
Toughness
The ability of a material to absorb energy in the plastic range
Toughness - The total area under the curve which measures the energy absorbed by the specimen in the process of breaking
Stress-Strain Curve
True Stress and True Strainbull True stress is the loadP divided by
the instantaneous area of the specimen Ai
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives AoLo AiLi
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking begins
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
True Strain
)1ln(
ln
e
LLL
L
L
L
dLd
oi
o
i
i
i
Assumes constant volumeValid for all strains up to point where necking begins
Engineering Vs True Stress-Strain Curves
Stress
Strain
True Stress - Strain Curve
Engineering Stress - Strain Curve
Ultimate Tensile Strength
Fracture
Fracture
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull IN Stress-Strain Curvesbull Plastic deformation is uniform and
permanent between the elastic limit and the UTS
bull bull Plastic deformation becomes non-uniform once the UTS is exceeded
bull In tension this non-uniform deformation manifests itself
bull as ldquoneckingrdquo
Criterion for necking
Increase in true stress (due to reduction in cross-sectional area) as the specimen elongates is more than to load carrying capacity due to strain hardening
Criteria for Instability
Rate of Geometrical Softening and Rate of Work Hardening
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10
Criteria for Necking
02 Offset Yield StrengthOffset Yield Strength
Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point The elastic portion of the curve is not perfectly linear and microscopic amounts of deformation can occur As a matter of practical convenience the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 02 from the origin The intersection of this line and the measured stress-strain line is used as an approximation of the materials yield strength called the 02 offset yield
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
Ductility
Elongation elongation is a measure of ductility which is given by
elongation =100 (Lo - Lf) Lo
whereLo = Initial lengthLf = Final Length
Ductility Reduction in Area reduction in area is a measure of ductility which is given by
reduction in area =100 (Ao - Af) Ao
whereAo = Initial areaAf = Final area
Poissonrsquos ratio
When pulled in tension (X) a sample getslonger and thinner ie a contraction in thewidth (Y) and breadth (Z)Poissonrsquos ratio when strained in the (X) direction how much strain occurs in the lateral directions (Y amp Z)For most metals this value is 0333
bull This is the Poisson effect Poissonrsquos ratio which is thebull negative ratio of the contraction over the extension isbull also an elastic constantbull Poissons Ratio laterial strainbull longitudinal strainbull εbull νbull εbull = minus = minus perpbull 1048614bull x unloadedbull ybull zbull loadedbull Pbull Pbull bullMetals ν = 03 ndash 035bull bullCeramics ν = 02 ndash 025bull bullPolymers ν = 025 ndash 0 5
Modulus of elasticity - the initial slope of the curve related directly to the strength of the atomic bonds
bull Resilience The ability of a material to absorb energy with in elastic limit
bull Measure by the modulus of resilience
-Which is strain energy per unit volume
-Stress the material from zero stress to the yield stress
bull Energy Force multiplied by the distance over which it acts
Modulus of resilience - the area under the linear part of the curve measuring the stored elastic energy
Toughness
The ability of a material to absorb energy in the plastic range
Toughness - The total area under the curve which measures the energy absorbed by the specimen in the process of breaking
Stress-Strain Curve
True Stress and True Strainbull True stress is the loadP divided by
the instantaneous area of the specimen Ai
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives AoLo AiLi
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking begins
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
True Strain
)1ln(
ln
e
LLL
L
L
L
dLd
oi
o
i
i
i
Assumes constant volumeValid for all strains up to point where necking begins
Engineering Vs True Stress-Strain Curves
Stress
Strain
True Stress - Strain Curve
Engineering Stress - Strain Curve
Ultimate Tensile Strength
Fracture
Fracture
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull IN Stress-Strain Curvesbull Plastic deformation is uniform and
permanent between the elastic limit and the UTS
bull bull Plastic deformation becomes non-uniform once the UTS is exceeded
bull In tension this non-uniform deformation manifests itself
bull as ldquoneckingrdquo
Criterion for necking
Increase in true stress (due to reduction in cross-sectional area) as the specimen elongates is more than to load carrying capacity due to strain hardening
Criteria for Instability
Rate of Geometrical Softening and Rate of Work Hardening
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10
Criteria for Necking
02 Offset Yield StrengthOffset Yield Strength
Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point The elastic portion of the curve is not perfectly linear and microscopic amounts of deformation can occur As a matter of practical convenience the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 02 from the origin The intersection of this line and the measured stress-strain line is used as an approximation of the materials yield strength called the 02 offset yield
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
Ductility Reduction in Area reduction in area is a measure of ductility which is given by
reduction in area =100 (Ao - Af) Ao
whereAo = Initial areaAf = Final area
Poissonrsquos ratio
When pulled in tension (X) a sample getslonger and thinner ie a contraction in thewidth (Y) and breadth (Z)Poissonrsquos ratio when strained in the (X) direction how much strain occurs in the lateral directions (Y amp Z)For most metals this value is 0333
bull This is the Poisson effect Poissonrsquos ratio which is thebull negative ratio of the contraction over the extension isbull also an elastic constantbull Poissons Ratio laterial strainbull longitudinal strainbull εbull νbull εbull = minus = minus perpbull 1048614bull x unloadedbull ybull zbull loadedbull Pbull Pbull bullMetals ν = 03 ndash 035bull bullCeramics ν = 02 ndash 025bull bullPolymers ν = 025 ndash 0 5
Modulus of elasticity - the initial slope of the curve related directly to the strength of the atomic bonds
bull Resilience The ability of a material to absorb energy with in elastic limit
bull Measure by the modulus of resilience
-Which is strain energy per unit volume
-Stress the material from zero stress to the yield stress
bull Energy Force multiplied by the distance over which it acts
Modulus of resilience - the area under the linear part of the curve measuring the stored elastic energy
Toughness
The ability of a material to absorb energy in the plastic range
Toughness - The total area under the curve which measures the energy absorbed by the specimen in the process of breaking
Stress-Strain Curve
True Stress and True Strainbull True stress is the loadP divided by
the instantaneous area of the specimen Ai
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives AoLo AiLi
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking begins
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
True Strain
)1ln(
ln
e
LLL
L
L
L
dLd
oi
o
i
i
i
Assumes constant volumeValid for all strains up to point where necking begins
Engineering Vs True Stress-Strain Curves
Stress
Strain
True Stress - Strain Curve
Engineering Stress - Strain Curve
Ultimate Tensile Strength
Fracture
Fracture
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull IN Stress-Strain Curvesbull Plastic deformation is uniform and
permanent between the elastic limit and the UTS
bull bull Plastic deformation becomes non-uniform once the UTS is exceeded
bull In tension this non-uniform deformation manifests itself
bull as ldquoneckingrdquo
Criterion for necking
Increase in true stress (due to reduction in cross-sectional area) as the specimen elongates is more than to load carrying capacity due to strain hardening
Criteria for Instability
Rate of Geometrical Softening and Rate of Work Hardening
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10
Criteria for Necking
02 Offset Yield StrengthOffset Yield Strength
Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point The elastic portion of the curve is not perfectly linear and microscopic amounts of deformation can occur As a matter of practical convenience the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 02 from the origin The intersection of this line and the measured stress-strain line is used as an approximation of the materials yield strength called the 02 offset yield
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
Poissonrsquos ratio
When pulled in tension (X) a sample getslonger and thinner ie a contraction in thewidth (Y) and breadth (Z)Poissonrsquos ratio when strained in the (X) direction how much strain occurs in the lateral directions (Y amp Z)For most metals this value is 0333
bull This is the Poisson effect Poissonrsquos ratio which is thebull negative ratio of the contraction over the extension isbull also an elastic constantbull Poissons Ratio laterial strainbull longitudinal strainbull εbull νbull εbull = minus = minus perpbull 1048614bull x unloadedbull ybull zbull loadedbull Pbull Pbull bullMetals ν = 03 ndash 035bull bullCeramics ν = 02 ndash 025bull bullPolymers ν = 025 ndash 0 5
Modulus of elasticity - the initial slope of the curve related directly to the strength of the atomic bonds
bull Resilience The ability of a material to absorb energy with in elastic limit
bull Measure by the modulus of resilience
-Which is strain energy per unit volume
-Stress the material from zero stress to the yield stress
bull Energy Force multiplied by the distance over which it acts
Modulus of resilience - the area under the linear part of the curve measuring the stored elastic energy
Toughness
The ability of a material to absorb energy in the plastic range
Toughness - The total area under the curve which measures the energy absorbed by the specimen in the process of breaking
Stress-Strain Curve
True Stress and True Strainbull True stress is the loadP divided by
the instantaneous area of the specimen Ai
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives AoLo AiLi
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking begins
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
True Strain
)1ln(
ln
e
LLL
L
L
L
dLd
oi
o
i
i
i
Assumes constant volumeValid for all strains up to point where necking begins
Engineering Vs True Stress-Strain Curves
Stress
Strain
True Stress - Strain Curve
Engineering Stress - Strain Curve
Ultimate Tensile Strength
Fracture
Fracture
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull IN Stress-Strain Curvesbull Plastic deformation is uniform and
permanent between the elastic limit and the UTS
bull bull Plastic deformation becomes non-uniform once the UTS is exceeded
bull In tension this non-uniform deformation manifests itself
bull as ldquoneckingrdquo
Criterion for necking
Increase in true stress (due to reduction in cross-sectional area) as the specimen elongates is more than to load carrying capacity due to strain hardening
Criteria for Instability
Rate of Geometrical Softening and Rate of Work Hardening
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10
Criteria for Necking
02 Offset Yield StrengthOffset Yield Strength
Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point The elastic portion of the curve is not perfectly linear and microscopic amounts of deformation can occur As a matter of practical convenience the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 02 from the origin The intersection of this line and the measured stress-strain line is used as an approximation of the materials yield strength called the 02 offset yield
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
bull This is the Poisson effect Poissonrsquos ratio which is thebull negative ratio of the contraction over the extension isbull also an elastic constantbull Poissons Ratio laterial strainbull longitudinal strainbull εbull νbull εbull = minus = minus perpbull 1048614bull x unloadedbull ybull zbull loadedbull Pbull Pbull bullMetals ν = 03 ndash 035bull bullCeramics ν = 02 ndash 025bull bullPolymers ν = 025 ndash 0 5
Modulus of elasticity - the initial slope of the curve related directly to the strength of the atomic bonds
bull Resilience The ability of a material to absorb energy with in elastic limit
bull Measure by the modulus of resilience
-Which is strain energy per unit volume
-Stress the material from zero stress to the yield stress
bull Energy Force multiplied by the distance over which it acts
Modulus of resilience - the area under the linear part of the curve measuring the stored elastic energy
Toughness
The ability of a material to absorb energy in the plastic range
Toughness - The total area under the curve which measures the energy absorbed by the specimen in the process of breaking
Stress-Strain Curve
True Stress and True Strainbull True stress is the loadP divided by
the instantaneous area of the specimen Ai
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives AoLo AiLi
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking begins
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
True Strain
)1ln(
ln
e
LLL
L
L
L
dLd
oi
o
i
i
i
Assumes constant volumeValid for all strains up to point where necking begins
Engineering Vs True Stress-Strain Curves
Stress
Strain
True Stress - Strain Curve
Engineering Stress - Strain Curve
Ultimate Tensile Strength
Fracture
Fracture
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull IN Stress-Strain Curvesbull Plastic deformation is uniform and
permanent between the elastic limit and the UTS
bull bull Plastic deformation becomes non-uniform once the UTS is exceeded
bull In tension this non-uniform deformation manifests itself
bull as ldquoneckingrdquo
Criterion for necking
Increase in true stress (due to reduction in cross-sectional area) as the specimen elongates is more than to load carrying capacity due to strain hardening
Criteria for Instability
Rate of Geometrical Softening and Rate of Work Hardening
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10
Criteria for Necking
02 Offset Yield StrengthOffset Yield Strength
Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point The elastic portion of the curve is not perfectly linear and microscopic amounts of deformation can occur As a matter of practical convenience the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 02 from the origin The intersection of this line and the measured stress-strain line is used as an approximation of the materials yield strength called the 02 offset yield
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
Modulus of elasticity - the initial slope of the curve related directly to the strength of the atomic bonds
bull Resilience The ability of a material to absorb energy with in elastic limit
bull Measure by the modulus of resilience
-Which is strain energy per unit volume
-Stress the material from zero stress to the yield stress
bull Energy Force multiplied by the distance over which it acts
Modulus of resilience - the area under the linear part of the curve measuring the stored elastic energy
Toughness
The ability of a material to absorb energy in the plastic range
Toughness - The total area under the curve which measures the energy absorbed by the specimen in the process of breaking
Stress-Strain Curve
True Stress and True Strainbull True stress is the loadP divided by
the instantaneous area of the specimen Ai
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives AoLo AiLi
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking begins
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
True Strain
)1ln(
ln
e
LLL
L
L
L
dLd
oi
o
i
i
i
Assumes constant volumeValid for all strains up to point where necking begins
Engineering Vs True Stress-Strain Curves
Stress
Strain
True Stress - Strain Curve
Engineering Stress - Strain Curve
Ultimate Tensile Strength
Fracture
Fracture
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull IN Stress-Strain Curvesbull Plastic deformation is uniform and
permanent between the elastic limit and the UTS
bull bull Plastic deformation becomes non-uniform once the UTS is exceeded
bull In tension this non-uniform deformation manifests itself
bull as ldquoneckingrdquo
Criterion for necking
Increase in true stress (due to reduction in cross-sectional area) as the specimen elongates is more than to load carrying capacity due to strain hardening
Criteria for Instability
Rate of Geometrical Softening and Rate of Work Hardening
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10
Criteria for Necking
02 Offset Yield StrengthOffset Yield Strength
Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point The elastic portion of the curve is not perfectly linear and microscopic amounts of deformation can occur As a matter of practical convenience the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 02 from the origin The intersection of this line and the measured stress-strain line is used as an approximation of the materials yield strength called the 02 offset yield
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
bull Resilience The ability of a material to absorb energy with in elastic limit
bull Measure by the modulus of resilience
-Which is strain energy per unit volume
-Stress the material from zero stress to the yield stress
bull Energy Force multiplied by the distance over which it acts
Modulus of resilience - the area under the linear part of the curve measuring the stored elastic energy
Toughness
The ability of a material to absorb energy in the plastic range
Toughness - The total area under the curve which measures the energy absorbed by the specimen in the process of breaking
Stress-Strain Curve
True Stress and True Strainbull True stress is the loadP divided by
the instantaneous area of the specimen Ai
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives AoLo AiLi
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking begins
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
True Strain
)1ln(
ln
e
LLL
L
L
L
dLd
oi
o
i
i
i
Assumes constant volumeValid for all strains up to point where necking begins
Engineering Vs True Stress-Strain Curves
Stress
Strain
True Stress - Strain Curve
Engineering Stress - Strain Curve
Ultimate Tensile Strength
Fracture
Fracture
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull IN Stress-Strain Curvesbull Plastic deformation is uniform and
permanent between the elastic limit and the UTS
bull bull Plastic deformation becomes non-uniform once the UTS is exceeded
bull In tension this non-uniform deformation manifests itself
bull as ldquoneckingrdquo
Criterion for necking
Increase in true stress (due to reduction in cross-sectional area) as the specimen elongates is more than to load carrying capacity due to strain hardening
Criteria for Instability
Rate of Geometrical Softening and Rate of Work Hardening
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10
Criteria for Necking
02 Offset Yield StrengthOffset Yield Strength
Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point The elastic portion of the curve is not perfectly linear and microscopic amounts of deformation can occur As a matter of practical convenience the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 02 from the origin The intersection of this line and the measured stress-strain line is used as an approximation of the materials yield strength called the 02 offset yield
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
Modulus of resilience - the area under the linear part of the curve measuring the stored elastic energy
Toughness
The ability of a material to absorb energy in the plastic range
Toughness - The total area under the curve which measures the energy absorbed by the specimen in the process of breaking
Stress-Strain Curve
True Stress and True Strainbull True stress is the loadP divided by
the instantaneous area of the specimen Ai
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives AoLo AiLi
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking begins
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
True Strain
)1ln(
ln
e
LLL
L
L
L
dLd
oi
o
i
i
i
Assumes constant volumeValid for all strains up to point where necking begins
Engineering Vs True Stress-Strain Curves
Stress
Strain
True Stress - Strain Curve
Engineering Stress - Strain Curve
Ultimate Tensile Strength
Fracture
Fracture
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull IN Stress-Strain Curvesbull Plastic deformation is uniform and
permanent between the elastic limit and the UTS
bull bull Plastic deformation becomes non-uniform once the UTS is exceeded
bull In tension this non-uniform deformation manifests itself
bull as ldquoneckingrdquo
Criterion for necking
Increase in true stress (due to reduction in cross-sectional area) as the specimen elongates is more than to load carrying capacity due to strain hardening
Criteria for Instability
Rate of Geometrical Softening and Rate of Work Hardening
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10
Criteria for Necking
02 Offset Yield StrengthOffset Yield Strength
Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point The elastic portion of the curve is not perfectly linear and microscopic amounts of deformation can occur As a matter of practical convenience the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 02 from the origin The intersection of this line and the measured stress-strain line is used as an approximation of the materials yield strength called the 02 offset yield
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
Toughness
The ability of a material to absorb energy in the plastic range
Toughness - The total area under the curve which measures the energy absorbed by the specimen in the process of breaking
Stress-Strain Curve
True Stress and True Strainbull True stress is the loadP divided by
the instantaneous area of the specimen Ai
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives AoLo AiLi
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking begins
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
True Strain
)1ln(
ln
e
LLL
L
L
L
dLd
oi
o
i
i
i
Assumes constant volumeValid for all strains up to point where necking begins
Engineering Vs True Stress-Strain Curves
Stress
Strain
True Stress - Strain Curve
Engineering Stress - Strain Curve
Ultimate Tensile Strength
Fracture
Fracture
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull IN Stress-Strain Curvesbull Plastic deformation is uniform and
permanent between the elastic limit and the UTS
bull bull Plastic deformation becomes non-uniform once the UTS is exceeded
bull In tension this non-uniform deformation manifests itself
bull as ldquoneckingrdquo
Criterion for necking
Increase in true stress (due to reduction in cross-sectional area) as the specimen elongates is more than to load carrying capacity due to strain hardening
Criteria for Instability
Rate of Geometrical Softening and Rate of Work Hardening
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10
Criteria for Necking
02 Offset Yield StrengthOffset Yield Strength
Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point The elastic portion of the curve is not perfectly linear and microscopic amounts of deformation can occur As a matter of practical convenience the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 02 from the origin The intersection of this line and the measured stress-strain line is used as an approximation of the materials yield strength called the 02 offset yield
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
Toughness - The total area under the curve which measures the energy absorbed by the specimen in the process of breaking
Stress-Strain Curve
True Stress and True Strainbull True stress is the loadP divided by
the instantaneous area of the specimen Ai
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives AoLo AiLi
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking begins
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
True Strain
)1ln(
ln
e
LLL
L
L
L
dLd
oi
o
i
i
i
Assumes constant volumeValid for all strains up to point where necking begins
Engineering Vs True Stress-Strain Curves
Stress
Strain
True Stress - Strain Curve
Engineering Stress - Strain Curve
Ultimate Tensile Strength
Fracture
Fracture
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull IN Stress-Strain Curvesbull Plastic deformation is uniform and
permanent between the elastic limit and the UTS
bull bull Plastic deformation becomes non-uniform once the UTS is exceeded
bull In tension this non-uniform deformation manifests itself
bull as ldquoneckingrdquo
Criterion for necking
Increase in true stress (due to reduction in cross-sectional area) as the specimen elongates is more than to load carrying capacity due to strain hardening
Criteria for Instability
Rate of Geometrical Softening and Rate of Work Hardening
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10
Criteria for Necking
02 Offset Yield StrengthOffset Yield Strength
Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point The elastic portion of the curve is not perfectly linear and microscopic amounts of deformation can occur As a matter of practical convenience the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 02 from the origin The intersection of this line and the measured stress-strain line is used as an approximation of the materials yield strength called the 02 offset yield
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
Stress-Strain Curve
True Stress and True Strainbull True stress is the loadP divided by
the instantaneous area of the specimen Ai
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives AoLo AiLi
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking begins
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
True Strain
)1ln(
ln
e
LLL
L
L
L
dLd
oi
o
i
i
i
Assumes constant volumeValid for all strains up to point where necking begins
Engineering Vs True Stress-Strain Curves
Stress
Strain
True Stress - Strain Curve
Engineering Stress - Strain Curve
Ultimate Tensile Strength
Fracture
Fracture
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull IN Stress-Strain Curvesbull Plastic deformation is uniform and
permanent between the elastic limit and the UTS
bull bull Plastic deformation becomes non-uniform once the UTS is exceeded
bull In tension this non-uniform deformation manifests itself
bull as ldquoneckingrdquo
Criterion for necking
Increase in true stress (due to reduction in cross-sectional area) as the specimen elongates is more than to load carrying capacity due to strain hardening
Criteria for Instability
Rate of Geometrical Softening and Rate of Work Hardening
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10
Criteria for Necking
02 Offset Yield StrengthOffset Yield Strength
Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point The elastic portion of the curve is not perfectly linear and microscopic amounts of deformation can occur As a matter of practical convenience the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 02 from the origin The intersection of this line and the measured stress-strain line is used as an approximation of the materials yield strength called the 02 offset yield
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
True Stress and True Strainbull True stress is the loadP divided by
the instantaneous area of the specimen Ai
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives AoLo AiLi
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking begins
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
True Strain
)1ln(
ln
e
LLL
L
L
L
dLd
oi
o
i
i
i
Assumes constant volumeValid for all strains up to point where necking begins
Engineering Vs True Stress-Strain Curves
Stress
Strain
True Stress - Strain Curve
Engineering Stress - Strain Curve
Ultimate Tensile Strength
Fracture
Fracture
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull IN Stress-Strain Curvesbull Plastic deformation is uniform and
permanent between the elastic limit and the UTS
bull bull Plastic deformation becomes non-uniform once the UTS is exceeded
bull In tension this non-uniform deformation manifests itself
bull as ldquoneckingrdquo
Criterion for necking
Increase in true stress (due to reduction in cross-sectional area) as the specimen elongates is more than to load carrying capacity due to strain hardening
Criteria for Instability
Rate of Geometrical Softening and Rate of Work Hardening
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10
Criteria for Necking
02 Offset Yield StrengthOffset Yield Strength
Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point The elastic portion of the curve is not perfectly linear and microscopic amounts of deformation can occur As a matter of practical convenience the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 02 from the origin The intersection of this line and the measured stress-strain line is used as an approximation of the materials yield strength called the 02 offset yield
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking begins
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
True Strain
)1ln(
ln
e
LLL
L
L
L
dLd
oi
o
i
i
i
Assumes constant volumeValid for all strains up to point where necking begins
Engineering Vs True Stress-Strain Curves
Stress
Strain
True Stress - Strain Curve
Engineering Stress - Strain Curve
Ultimate Tensile Strength
Fracture
Fracture
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull IN Stress-Strain Curvesbull Plastic deformation is uniform and
permanent between the elastic limit and the UTS
bull bull Plastic deformation becomes non-uniform once the UTS is exceeded
bull In tension this non-uniform deformation manifests itself
bull as ldquoneckingrdquo
Criterion for necking
Increase in true stress (due to reduction in cross-sectional area) as the specimen elongates is more than to load carrying capacity due to strain hardening
Criteria for Instability
Rate of Geometrical Softening and Rate of Work Hardening
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10
Criteria for Necking
02 Offset Yield StrengthOffset Yield Strength
Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point The elastic portion of the curve is not perfectly linear and microscopic amounts of deformation can occur As a matter of practical convenience the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 02 from the origin The intersection of this line and the measured stress-strain line is used as an approximation of the materials yield strength called the 02 offset yield
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
True Strain
)1ln(
ln
e
LLL
L
L
L
dLd
oi
o
i
i
i
Assumes constant volumeValid for all strains up to point where necking begins
Engineering Vs True Stress-Strain Curves
Stress
Strain
True Stress - Strain Curve
Engineering Stress - Strain Curve
Ultimate Tensile Strength
Fracture
Fracture
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull IN Stress-Strain Curvesbull Plastic deformation is uniform and
permanent between the elastic limit and the UTS
bull bull Plastic deformation becomes non-uniform once the UTS is exceeded
bull In tension this non-uniform deformation manifests itself
bull as ldquoneckingrdquo
Criterion for necking
Increase in true stress (due to reduction in cross-sectional area) as the specimen elongates is more than to load carrying capacity due to strain hardening
Criteria for Instability
Rate of Geometrical Softening and Rate of Work Hardening
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10
Criteria for Necking
02 Offset Yield StrengthOffset Yield Strength
Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point The elastic portion of the curve is not perfectly linear and microscopic amounts of deformation can occur As a matter of practical convenience the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 02 from the origin The intersection of this line and the measured stress-strain line is used as an approximation of the materials yield strength called the 02 offset yield
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
True Strain
)1ln(
ln
e
LLL
L
L
L
dLd
oi
o
i
i
i
Assumes constant volumeValid for all strains up to point where necking begins
Engineering Vs True Stress-Strain Curves
Stress
Strain
True Stress - Strain Curve
Engineering Stress - Strain Curve
Ultimate Tensile Strength
Fracture
Fracture
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull IN Stress-Strain Curvesbull Plastic deformation is uniform and
permanent between the elastic limit and the UTS
bull bull Plastic deformation becomes non-uniform once the UTS is exceeded
bull In tension this non-uniform deformation manifests itself
bull as ldquoneckingrdquo
Criterion for necking
Increase in true stress (due to reduction in cross-sectional area) as the specimen elongates is more than to load carrying capacity due to strain hardening
Criteria for Instability
Rate of Geometrical Softening and Rate of Work Hardening
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10
Criteria for Necking
02 Offset Yield StrengthOffset Yield Strength
Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point The elastic portion of the curve is not perfectly linear and microscopic amounts of deformation can occur As a matter of practical convenience the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 02 from the origin The intersection of this line and the measured stress-strain line is used as an approximation of the materials yield strength called the 02 offset yield
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
Engineering Vs True Stress-Strain Curves
Stress
Strain
True Stress - Strain Curve
Engineering Stress - Strain Curve
Ultimate Tensile Strength
Fracture
Fracture
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull IN Stress-Strain Curvesbull Plastic deformation is uniform and
permanent between the elastic limit and the UTS
bull bull Plastic deformation becomes non-uniform once the UTS is exceeded
bull In tension this non-uniform deformation manifests itself
bull as ldquoneckingrdquo
Criterion for necking
Increase in true stress (due to reduction in cross-sectional area) as the specimen elongates is more than to load carrying capacity due to strain hardening
Criteria for Instability
Rate of Geometrical Softening and Rate of Work Hardening
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10
Criteria for Necking
02 Offset Yield StrengthOffset Yield Strength
Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point The elastic portion of the curve is not perfectly linear and microscopic amounts of deformation can occur As a matter of practical convenience the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 02 from the origin The intersection of this line and the measured stress-strain line is used as an approximation of the materials yield strength called the 02 offset yield
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull IN Stress-Strain Curvesbull Plastic deformation is uniform and
permanent between the elastic limit and the UTS
bull bull Plastic deformation becomes non-uniform once the UTS is exceeded
bull In tension this non-uniform deformation manifests itself
bull as ldquoneckingrdquo
Criterion for necking
Increase in true stress (due to reduction in cross-sectional area) as the specimen elongates is more than to load carrying capacity due to strain hardening
Criteria for Instability
Rate of Geometrical Softening and Rate of Work Hardening
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10
Criteria for Necking
02 Offset Yield StrengthOffset Yield Strength
Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point The elastic portion of the curve is not perfectly linear and microscopic amounts of deformation can occur As a matter of practical convenience the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 02 from the origin The intersection of this line and the measured stress-strain line is used as an approximation of the materials yield strength called the 02 offset yield
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull IN Stress-Strain Curvesbull Plastic deformation is uniform and
permanent between the elastic limit and the UTS
bull bull Plastic deformation becomes non-uniform once the UTS is exceeded
bull In tension this non-uniform deformation manifests itself
bull as ldquoneckingrdquo
Criterion for necking
Increase in true stress (due to reduction in cross-sectional area) as the specimen elongates is more than to load carrying capacity due to strain hardening
Criteria for Instability
Rate of Geometrical Softening and Rate of Work Hardening
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10
Criteria for Necking
02 Offset Yield StrengthOffset Yield Strength
Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point The elastic portion of the curve is not perfectly linear and microscopic amounts of deformation can occur As a matter of practical convenience the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 02 from the origin The intersection of this line and the measured stress-strain line is used as an approximation of the materials yield strength called the 02 offset yield
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
bull c
bull IN Stress-Strain Curvesbull Plastic deformation is uniform and
permanent between the elastic limit and the UTS
bull bull Plastic deformation becomes non-uniform once the UTS is exceeded
bull In tension this non-uniform deformation manifests itself
bull as ldquoneckingrdquo
Criterion for necking
Increase in true stress (due to reduction in cross-sectional area) as the specimen elongates is more than to load carrying capacity due to strain hardening
Criteria for Instability
Rate of Geometrical Softening and Rate of Work Hardening
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10
Criteria for Necking
02 Offset Yield StrengthOffset Yield Strength
Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point The elastic portion of the curve is not perfectly linear and microscopic amounts of deformation can occur As a matter of practical convenience the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 02 from the origin The intersection of this line and the measured stress-strain line is used as an approximation of the materials yield strength called the 02 offset yield
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
bull IN Stress-Strain Curvesbull Plastic deformation is uniform and
permanent between the elastic limit and the UTS
bull bull Plastic deformation becomes non-uniform once the UTS is exceeded
bull In tension this non-uniform deformation manifests itself
bull as ldquoneckingrdquo
Criterion for necking
Increase in true stress (due to reduction in cross-sectional area) as the specimen elongates is more than to load carrying capacity due to strain hardening
Criteria for Instability
Rate of Geometrical Softening and Rate of Work Hardening
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10
Criteria for Necking
02 Offset Yield StrengthOffset Yield Strength
Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point The elastic portion of the curve is not perfectly linear and microscopic amounts of deformation can occur As a matter of practical convenience the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 02 from the origin The intersection of this line and the measured stress-strain line is used as an approximation of the materials yield strength called the 02 offset yield
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
Criterion for necking
Increase in true stress (due to reduction in cross-sectional area) as the specimen elongates is more than to load carrying capacity due to strain hardening
Criteria for Instability
Rate of Geometrical Softening and Rate of Work Hardening
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10
Criteria for Necking
02 Offset Yield StrengthOffset Yield Strength
Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point The elastic portion of the curve is not perfectly linear and microscopic amounts of deformation can occur As a matter of practical convenience the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 02 from the origin The intersection of this line and the measured stress-strain line is used as an approximation of the materials yield strength called the 02 offset yield
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
Criteria for Instability
Rate of Geometrical Softening and Rate of Work Hardening
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10
Criteria for Necking
02 Offset Yield StrengthOffset Yield Strength
Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point The elastic portion of the curve is not perfectly linear and microscopic amounts of deformation can occur As a matter of practical convenience the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 02 from the origin The intersection of this line and the measured stress-strain line is used as an approximation of the materials yield strength called the 02 offset yield
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
Rate of Geometrical Softening and Rate of Work Hardening
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10
Criteria for Necking
02 Offset Yield StrengthOffset Yield Strength
Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point The elastic portion of the curve is not perfectly linear and microscopic amounts of deformation can occur As a matter of practical convenience the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 02 from the origin The intersection of this line and the measured stress-strain line is used as an approximation of the materials yield strength called the 02 offset yield
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
Holloman Petch Relation ship
An empirical relationship was proposed by Holloman in 1945 to describe the shape of the stress-strain curve
σ = Kεn
bull σ true stress ε is true strain K is strength coefficient and n is the strain-hardening exponent
bull Thus one can obtain n from a log-log plot of σ
versus ε K is the true stress at ε = 10
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10
Criteria for Necking
02 Offset Yield StrengthOffset Yield Strength
Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point The elastic portion of the curve is not perfectly linear and microscopic amounts of deformation can occur As a matter of practical convenience the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 02 from the origin The intersection of this line and the measured stress-strain line is used as an approximation of the materials yield strength called the 02 offset yield
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
bull n = 0 for perfectly plastic solids
bull n = 1 for perfectly elastic solids
bull n = 01 ndash 05 for most metals
bull c
bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10
Criteria for Necking
02 Offset Yield StrengthOffset Yield Strength
Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point The elastic portion of the curve is not perfectly linear and microscopic amounts of deformation can occur As a matter of practical convenience the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 02 from the origin The intersection of this line and the measured stress-strain line is used as an approximation of the materials yield strength called the 02 offset yield
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
bull c
bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10
Criteria for Necking
02 Offset Yield StrengthOffset Yield Strength
Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point The elastic portion of the curve is not perfectly linear and microscopic amounts of deformation can occur As a matter of practical convenience the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 02 from the origin The intersection of this line and the measured stress-strain line is used as an approximation of the materials yield strength called the 02 offset yield
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10
Criteria for Necking
02 Offset Yield StrengthOffset Yield Strength
Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point The elastic portion of the curve is not perfectly linear and microscopic amounts of deformation can occur As a matter of practical convenience the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 02 from the origin The intersection of this line and the measured stress-strain line is used as an approximation of the materials yield strength called the 02 offset yield
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
Criteria for Necking
02 Offset Yield StrengthOffset Yield Strength
Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point The elastic portion of the curve is not perfectly linear and microscopic amounts of deformation can occur As a matter of practical convenience the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 02 from the origin The intersection of this line and the measured stress-strain line is used as an approximation of the materials yield strength called the 02 offset yield
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
02 Offset Yield StrengthOffset Yield Strength
Defining the yield stress as the point separating elastic from plastic deformation is easier than determining that point The elastic portion of the curve is not perfectly linear and microscopic amounts of deformation can occur As a matter of practical convenience the yield strength is determined by constructing a line parallel to the initial portion of the stress-strain curve but offset by 02 from the origin The intersection of this line and the measured stress-strain line is used as an approximation of the materials yield strength called the 02 offset yield
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
bull Stress-Strain Curvesbull bull Plastic deformation is uniform andbull permanent between the elastic limit andbull the UTSbull bull Plastic deformation becomes non-uniformbull once the UTS is exceeded In tension thisbull non-uniform deformation manifests itselfbull as ldquoneckingrdquobull Uniform plastic strain Non-uniform plastic strainbull L3bull 2bull Page 47bull Uniform Plastic Flowbull bull The stress-strain curve (ie flow curve) inbull the region of uniform plastic deformationbull does not increase proportionally withbull strain The material is said to workbull harden (or strain harden)bull bull An empirical mathematical relationshipbull was advanced by Holloman in 1945 tobull describe the shape of the engineeringbull stress-strain curvebull σ = Kεnbull where is the σ true stress ε is true strainbull K is strength coefficient and n is thebull strain-hardening exponent Thus onebull can obtain n from a log-log plot of σbull versus ε K is the true stress at ε = 10bull Page 48
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
bull True strain is determined from the rate of change in gauge length with respect to the instantaneous gauge length Li
bull Up to strain where necking begins specimen deforms with a constant volume in gauge section
bull Constant Volume gives
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
True Stress P
Ai Ai AoLo
Li
PLiAoLo
S P
Ao
Li Lo L
LiLo
Lo LLo
(1 e)
S (1 e)
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
Special Case True Fracture Stress
f PfA f
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
bull Stress The true stress is defined as the ratio of the applied load to the instantaneous cross-sectional area
bull True stress can be related to the engineering stress if we assume that there is no volume change in the specimen Under this assumption
bull which leads to
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
True Strain
d dLiLi
lnLiLo
Li Lo L
ln(1 e)
Special Case True Fracture Strain
f
of A
Aln
Assumes constant volumeValid for all strains up to point where necking beginsHence valid for S lt Su
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
True Strain
bull True Strain Change in gage length with respect to the instantaneous gage length over which the change occurs
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
bull εtrue = ln(1 + εe) bull σtrue = (1 + εe)(σe)
02 Offset Yield Strength
02 Offset Yield Strength
