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Theories of Failure

• Failure: Every material has certain strength, expressed in terms of stress or strain, beyond

which it fractures or fails to carry the load.

• Failure Criterion: A criterion (standard/principle/measure/gauge/norm) is used to

hypothesize (imagine/assume/theory/visualize) the failure.

• Failure Theory: A Theory behind a failure criterion.

Why we need failure theories?

• To design structural components/elements and calculate margin of safety.

• To guide in materials development.

• To determine weak and strong directions.

Failure Mode

• Yielding: a process of global permanent plastic deformation. Change in the geometry of the

object.

• Low stiffness: excessive elastic deflection.

• Fracture: a process in which cracks grow to the extent that the component breaks apart.

• Buckling: the loss of stable equilibrium. Compressive loading can lead to bucking in

columns.

• Creep: a high-temperature effect. Load carrying capacity drops.

Theories OF Failure

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Four important failure theories, namely (1) maximum shear stress theory, (2) maximum

normal stress theory, (3) maximum strain energy theory, and (4) maximum distortion

energy theory.

Out of these four theories of failure, the maximum normal stress theory or Rankins’s

theory is only applicable for brittle materials, and the remaining three theories are

applicable for ductile materials.

Following are the important common features for all the theories.

1. In predicting failure, the limiting strength values obtained from the uniaxial testing are

used.

2. The failure theories have been formulated in terms of three principal normal stresses

(S1, S2, S3) at a point.

3. For any given complex state of stress (sx, sy, sz, txy, tyz, tzx), we can always find its

equivalent principal normal stresses (S1, S2, S3). Thus the failure theories in terms of

principal normal stresses can predict the failure due to any given state of stress.

4. The three principal normal stress components S1, S2, & S3, each which can be

comprised of positive (tensile), negative (compressive) or zero value.

5. When the external loading is uniaxial, that is S1= a positive or negative real value,

S2=S3=0, then all failure theories predict the same as that has been determined from

regular tension/compression test.

6. The material properties are usually determined by simple tension or compression tests

7. Some structural members are subjected to biaxial or triaxial stresses.

8. To determine whether a component/element will fail or not, some failure theories are

proposed which are related to the properties of materials obtained from uniaxial

tension or compression tests.

9. Ductile materials usually fail by yielding and hence the limiting strength is the yield

strength of material as determined from simple tension test which is assumed the same

in compression also. For brittle materials limiting strength of material is ultimate tensile

strength intension or compression.

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Conservative ( traditional/old fashioned/ conventonal)

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Problem solving strategy for Failure thoery:

Syp- yield strength , Sut and Suc=ultimate tensile strength and ultimate compressive

strengths

Failure Theories

1. Failure under load can occur due to excessive elastic deflections or due to excessive

stresses.

2. Failure prediction theories due to excessive stresses fall into two classes: Failure when

the loading is static or the number of load cycles is one or quite small, and failure due

to cyclic loading when the number of cycles is large often in thousands of cycles.

Failure under static load

Parts under static loading may fail due to:

a) Ductile behavior: Failure is due to bulk yielding causing permanent deformations that are

objectionable. These failures may cause noise, loss of accuracy, excessive vibrations, and

eventual fracture. In machines, bulk yielding is the criteria for failure. Tiny areas of yielding

are OK in ductile behavior in static loading.

b) Brittle behavior: Failure is due to fracture. This occurs when the materials (or conditions)

do not allow much yielding such as ceramics, grey cast iron, or heavily cold-worked parts or

concrete.

End of Lecture……..

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The yield point, alternatively called the elastic limit, marks the end of elastic behaviour and the

beginning of plastic behaviour. When stresses less than the yield point are removed, the

material returns to its original shape. For many materials that do not have a well-defined yield

point, a quantity called yield strength is substituted. Yield strength is the stress at which a

material has undergone some arbitrarily chosen amount of permanent deformation, often 0.2

percent.

Any increase in the stress beyond the yield point causes greater permanent deformation and

eventually fracture.

A point at which Maximum load or stress required to initiate the plastic deformation of

material such point is called as Upper yield point. And a point at which minimum load

or stress required to maintain the plastic behavior of material such a point is called as Lower

yield point.

Upper yield point is the point after which the plastic deformation starts. This is due to the fact

that the dislocations in the crystalline structure start moving. But after a while, the dislocations

become too much in number and they restrict each other’s movement. This is called strain

hardening and lower yield point is the point after which strain hardening begins.

Dislocations are defects present in crystal areas where atoms are out of position (irregular

alignment).

Why the lower yield point stress value of mild steel is consider as a strength of material

instead of upper yield point stress?

Failure of mechanical component means it fail To perform it's operations efficiently for

example consider shaft which transmits rotational motion ,now when shaft is unable to

transmit motion efficiently then it will fail. Basically their are three types of failure in case of

mechanical component i.e

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1) failure due to elastic deformation

2) failure due to plastic deformation

3) failure due to fracture

When component deforms elastically it's dimensions changes and it fails. And this failure is

known as failure due to elastic deformation

When component undergoes plastic deformation it's dimension changes permanently and

failure takes place this is known as failure due to plastic deformation.

For ductile metals elastic failure is criteria of failure because ductile metals undergo elastic

deformation before failure. And elastic deformation starts at lower yield point.

As Mild steel is ductile material we consider lower yield point

Upper yield point is not constant it varies with shape of specimen and rate of loading

Lower yielding point is constant for all shapes and rate of loading because of its consistency

lower yielding point is taken as yield stress of mild steel

Upper yield point corresponds to the load that is required to initiate yielding. Lower yield point

corresponds to the min load that is required to maintain yield.

Normally we use the lower yield point to determine the yield strength of the material being

tested, cause the upper yield is momentary.

Upper yield point is the max load at which deformation starts, starting of deformation means

dislocations are started moving in the material.

So this type of phenomenon is called permanent deformation by slip ( slip mechanism).

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As the slip is taking place in the material, it offers less resistance to the material and hence

curve falls slightly ( stress is the measurement of resistance offered by the material during the

application of load).

And it reaches to some stress ( lower yiled point stress) which is the minimum stress required

to maintain the deformation in the mateial.. And at the lower yield point for the low carbon

steels ( mild steels) the stress strain cure is in some wave nature , this is because to break bonds

with impurites while dislocations are moving out of the material , hence resistance increases

and decreases periodically after that strain hardening takes place which increases resistance

slowly by increasing of dislocations in the material...

What is strain softening and strain hardening?

Work hardening, also known as strain hardening is the strengthening of a metal by plastic

deformation. This strengthening occurs because of dislocation movements and dislocation

generation within the crystal structure of the material.

Reason for Work hardening: As the deformation of the material occur in the plastic region,

the dislocation of the material increases. The dislocation interaction is repulsive in nature. As

the dislocation density increases the further deformation of the material become difficult, this

is called Work Hardening or Strain Hardening.

some materials exhibit an elevation in yield stress along with plastic strain, sometimes strain

rate or some internal variables which is known as hardening and if it shows a decrease in yield

stress with plastic strain, it is called softening.

Strain hardening is the process of increasing the hardness and strength of a metal by plastic

deformation and is a cold working process. Strain hardening is due to the increased resistance

to dislocation movement through a crystal lattice.

No crystal lattice is perfect, it has some crystallographic defects called dislocations

( Dislocation).

The dislocation movement is along the slip plane (plane of greatest atomic density and direction

is along the closest packed direction within the slip plane).Slip will occur when the shear stress

along the crystallographic plane reaches a critical value, which leads to movement of

dislocations.

What is Plasticity?

The theory of linear elasticity is useful for modelling materials which undergo small

deformations and which return to their original configuration upon removal of load. Almost all

real materials will undergo some permanent deformation, which remains after removal of

load. With metals, significant permanent deformations will usually occur when the stress

reaches some critical value, called the yield stress, a material property. Elastic deformations

are termed reversible; the energy expended in deformation is stored as elastic strain energy

and is completely recovered upon load removal. Permanent deformations involve the

dissipation of energy; such processes are termed irreversible, in the sense that the original state

can be achieved only by the expenditure of more energy.

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The classical theory of plasticity grew out of the study of metals in the late nineteenth century.

It is concerned with materials which initially deform elastically, but which deform plastically

upon reaching a yield stress.

In metals and other crystalline materials the occurrence of plastic deformations at the micro-

scale level is due to the motion of dislocations and the migration of grain boundaries on the

micro-level..

Plastic deformations are normally rate independent, that is, the stresses induced are

independent of the rate of deformation (or rate of loading).

Plasticity theory began with Tresca, when he undertook an experimental program into the

extrusion of metals and published his famous yield criterion discussed later on. Further

advances with yield criteria and plastic flow rules were made in the years which followed by

Saint-Venant, Levy, Von Mises, Hencky and Prandtl.

Imp Points:

• Permanent deformation that cannot be recovered after load removal

• Hookes law (linear relation between stress and strain) not valid

• Beyond Hooke’s law to failure is Plastic behaviour

• Tensile test to study plastic behaviour

• Elastic properties may be of interest, but these are measured ultrasonically much more

accurately that by tension testing.

• Plasticity theory deals with yielding of materials under complex stress states

• Plastic deformation is a non-reversible process where Hooke’s law is no longer valid.

• One aspect of plasticity in the viewpoint of structural design is that it is concerned with

predicting the maximum load, which can be applied to a body without causing excessive

yielding.

Plasticity vs elasticity

Plasticity is a property of a material or a system that allows it to deform irreversibly. Elasticity

is a property of a system or a material that allows it to deform reversibly.

Elasticity is a concept directly connected with the deformation of materials. When an exterior

stress is applied to a solid body, the body tends to pull itself apart. This causes the distance

between atoms in the lattice to increase. Each atom tries to pull its neighbor as close as possible.

This causes a force trying to resist the deformation. This force is known as strain.

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If a graph of stress versus strain is plotted, the plot will be a linear one for some lower values

of strain. This linear area is the zone which the object is deformed elastically. Elastic

deformation is always reversible. It is calculated using Hooke’s law.

The Hooke’s law states that for the elastic range of the material applied stress is equal to the

product of the Young’s modulus and the strain of the material. The elastic deformation of a

solid is a reversible process, when the applied stress is removed the solid returns to its original

state.

Plasticity is a concept which is connected with the plastic deformation. When the plot of stress

versus strain is linear, the system is said to be in the elastic state. However, when the stress is

high the plot passes a small jump on the axes. This limit is when it becomes a plastic

deformation. This limit is known as the yield strength of the material.

Plastic deformation occurs mostly due to the sliding of two layers of the solid. This sliding

process is not reversible. The plastic deformation is sometimes known as the irreversible

deformation, but actually some modes of plastic deformation are reversible.

After the yield strength jump, the stress versus strain plot becomes a smooth curve with a peak.

The peak of this curve is known as the ultimate strength. After the ultimate strength, the

material begins to “neck” making unevenness of the density over length. This makes very low

density areas in the material making it easily breakable. Plastic deformation is used in metal

hardening to pack the atoms thoroughly.

What is the difference between Plasticity and Elasticity?

• Plasticity is the property that causes irreversible deformations on an object or a system.

Such deformations can be caused by forces and impact.

• Elasticity is a property of objects or systems that allows them to deform reversibly. Elastic

deformations can be caused by forces and impacts.

• An object must pass the elastic deformation stage in order to enter the plastic deformation

stage.

Assumptions of linear elasticity:

1) continuity of material,

2) homogenity(just one material) and isotropy (properties are the same in all directions),

3) linear elasticity (valid Hook´s law),

4) the small deformation theory,

5) static loading,

6) no initial state of stress

A solid is a continuum, it has got its volume without any holes, gaps or any interruptions. Stress

and strain is a continuous function. Homogeneous material has got physical characteristics

identical in all places (concret, steel, timber). Combination of two or more materials ( concret

+ steel) is not homogeneous material. Isotropy means that material has got characteristics

undependent on the direction – (concret, steel – yes, timber – not). Elasticity is an ability of

material to get back after removing the couses of changes (for example load) into the original

state. If there is a direct relation between stress and strain than we talk about Hooke´s law =

this is called physical linearity.

Small deformations theory:

Changes of a shape of a (solid) structure are small with aspect to its size (dimensions). Then

we can use a lot of mathematical simplifications, which usually lead to linear dependency

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Static loading:

It means gradually growing of load (not dynamic effects)

In the initial state there are all stresses equal zero. (Inner tension e.g. from the production).

All these assumptions enable to use principal of superposition which is based on linearity of

all mathematic relationship.

Saint - Venant principle of local effect

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Saint - Venant principle of local is not valid in these cases:

Assumptions of Plasticity Theory

In formulating a basic plasticity theory the following assumptions are usually made:

(1) the response is independent of rate effects

(2) the material is incompressible in the plastic range

(3) there is no Bauschinger effect

(4) the yield stress is independent of hydrostatic pressure

(5) the material is isotropic

The Bauschinger effect refers to a property of materials where the material's stress/strain

characteristics change as a result of the microscopic stress distribution of the material. For

example, an increase in tensile yield strength occurs at the expense of compressive yield

strength.

Bauschinger effect represents loss of isotropic behavior in strength-strain behavior produced

due to deformation produced in metallic materials. When steel is loaded in tension, it starts

deforming first elastically but later plastically. Plastic deformation occurs due to dislocation

movement. However, dislocation entangles during movement which requires more stress for

the further movement. This is known as work hardening.

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When the direction of stress is reversed, say from tensile to compressive, dislocation movement

can start at lower strength resulting in a decrease of strength in compression. This phenomena

is known as Bauschinger effect.

Bouschinger Effect is also known as strain softening.First observe the figure given below,

Region OA -This region is Elastic Region in tension.Within this reagion, if we unload the material it will follow the same path in the reverse direction i. e. From A to O.

Region OZ- This region is Elastic Region in compression. Within this region, if we unload the material it will follow the same path in the reverse direction i.e.From Z to O.

Region AB- Due to increase in load,tensile stresses overcome the bond strength. Dislocation starts moving towards grain boundary. Material starts yielding due to

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movement of these dislocations.Accumulation of dislocations near grain boundary creates a back pressure, because same type of dislocations repel each other.

Region BC- Immediate unloading will take curve from B to C. Elastic recovery takes place in this region. Length OC represents the permanent deformation of material.

Region CD- Compression of material takes place from C to D.

Region DZ- As in case of tension, back pressure opposes the movement of dislocations i. e. this back pressure resists the tensile load. Same back pressure will now assist the compressive load. Due to combined effect of compression and this back pressure, a curvature is observed from D to Z.

Region ZE- Due to further increase in compressive load, material starts yielding in compression. Again a back pressure is created. Now this back pressure will resist the compressive load but will assist the tensile load.

Region EF - Represents removal of compressive load.

Region FG- Again we apply a tensile load.

Region GA- Due to combined effect of tensile load and back pressure created during compression, a curvature can be observed here also.

These curvatures represents the Strain softening and this effect is know as Bouschinger Effect.

The stress-strain behaviour of steel in compression is identical to that in tension.

However, if the steel is stressed into the inelastic range in uniform tension, unloaded, then

subjected to uniform compression in the opposite direction, it is found that and the stress-strain

curve becomes nonlinear at a stress much lower than the initial yield strength [Fig.].This is

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referred to as the ‘Bauschinger effect’.In this case, the hysteresis loop is also more pronounced.

In inelastic deformation processes involving continual reversal of stress (such as metal

working, high intensity reversed seismic loading, etc), the Bauschinger effect is very important

and cannot be ignored. In other cases, where there is in general no more than one stress reversal,

the Bauschinger effect can safely be neglected.

Structural members are likely to subjected to reversal of stresses. While the mild steel in

compression behaves same as like in tension upto the yield point. However actual behavior is

different and indicates an apparently reduced yield stress in compression. This occurs only

when change in direction of strain changes. The divergence from ideal path is called

Bauschinger effect.

End of Lecture…..

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Most metals can be regarded as isotropic. After large plastic deformation however, for example

in rolling, the material will have become anisotropic: there will be distinct material directions

and asymmetries.

Theories of ductile failure (yielding)

Yielding is a shear stress phenomenon. That means materials yield because the shear stresses

on some planes causes the lattice crystals to slide like a deck of cards. In pure tension or

compression, maximum shear stresses occur on 45-degree planes – these stresses are

responsible for yielding and not the larger normal stresses.

The best predictor of yielding is the maximum distortion energy theory (DET). This theory

states that yielding occurs when the Von Mises stress reaches the yield strength. The more

conservative predictor is the maximum shear stress theory (MST), which predicts yielding to

occur when the shear stresses reach Sy/2.

Note that in static loading and ductile behavior, stress concentrations are harmless as they only

create small localized yielding which do not lead to any objectionable dimensional changes.

The material “yielding” per se is not harmful to materials as long as it is not repeated too many

times.

Theories of brittle failure There are two types of theories for brittle failure. The classical theories assume that the material

structure is uniform. If the material structure is non-uniform, such as in many thick-section

castings, and that the probability of large flaws exist, then the theory of fracture mechanics

predicts the failure much more accurately.

An important point to remember is that brittle materials often show much higher ultimate

strength in compression than in tension. One reason is that, unlike yielding, fracture of brittle

materials when loaded in tension is a normal stress phenomenon. The material fails because

eventually normal tensile stresses fracture or separate the part in the direction normal to the

plane of maximum normal stress.

In compression the story is quite different. When a brittle material is loaded in compression,

the normal stress cannot separate the part along the direction normal to the plane of maximum

normal stress. In the absence of separating normal stresses, shear stresses would have to do the

job and separate or fracture the material along the direction where the shear stresses are

maximum.

In pure compression, this direction is at 45 degrees to the plane of loading. Brittle materials,

however, are very strong in shear. The bottom line is that it takes a lot more compressive normal

stress to create a fracture.

We only discuss these theories for a 2D state of stress – 3D is similar but is more formula-

based.

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Failure Theories for Isotropic Materials: Strength and stiffness are independent of the

direction. Failure in metallic materials is characterized by Yield Strength.

1. Maximum principal stress theory.

2. Maximum principal strain theory.

3. Quadratic or Distortional Energy Theory.

What is the definition of Failure?

Obviously fracture but in some components yielding can also be considered as failure, if

yielding distorts the material in such a way that it no longer functions properly

Which stress causes the material to fail?

Usually ductile materials are limited by their shear strengths.

While brittle materials (ductility < 5%) are limited by their tensile strengths.

Theories of Failure or Yield Criteria

It is known from the results of material testing that when bars of ductile materials are

subjected to uniform tension, the stress-strain curves show a linear range within which

the materials behave in an elastic manner and a definite yield zone where the materials

undergo permanent deformation.

In the case of the so-called brittle materials, there is no yield zone. However, a brittle

material, under suitable conditions, can be brought to a plastic state before fracture

occurs.

In general, the results of material testing reveal that the behavior of various materials

under similar test conditions, e.g. under simple tension, compression or torsion, varies

considerably.

In the process of designing a machine element or a structural member, the designer has

to take precautions to see that the member under consideration does not fail under

service conditions. The word ‘failure’ used in this context may mean either fracture or

permanent deformation beyond the operational range due to the yielding of the member.

We know that the state of stress at any point can be characterized by the six rectangular

stress components—three normal stresses and three shear stresses. Similarly, the state

of strain at a point can be characterised by the six rectangular strain components.

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When failure occurs, the question that arises is: what causes the failure? Is it a particular

state of stress, or a particular state of strain or some other quantity associated with stress

and strain? Further, the cause of failure of a ductile material need not be the same as

that for a brittle material.

Any one of the above or some other factors might have caused the yielding.

Further, as pointed out earlier, the factor that causes a ductile material to yield might be

quite different from the factor that causes fracture in a brittle material under the same

loading conditions.

Consequently, there will be many criteria or theories of failure. It is necessary to

remember that failure may mean fracture or yielding. Whatever may be the theory

adopted, the information regarding it will have to be obtained from a simple test, like

that of a uniaxial tension or a pure torsion test. This is so because the state of stress or

strain which causes the failure of the material concerned can easily be calculated.

The critical value obtained from this test will have to be applied for the stress or strain

at a point in a general machine or a structural member so as not to initiate failure at that

point.

There are six main theories of failure. Another theory, called Mohr’s theory, is slightly

different in its approach

Significance of the Theories of Failure The mode of failure of a member and the factor that is responsible for failure depend

on a large number of factors such as the nature and properties of the material, type of

loading, shape and temperature of the member, etc.

We have observed, for example, that the mode of failure of a ductile material differs

from that of a brittle material.

While yielding or permanent deformation is the characteristic feature of ductile

materials, fracture without permanent deformation is the characteristic feature of brittle

materials.

Further, if the loading conditions are suitably altered, a brittle material may be made to

yield before failure.

Even ductile materials fail in a different manner when subjected to repeated loadings

(such as fatigue) than when subjected to static loadings

Any rational procedure of design of a member requires the determination of the mode

of failure (either yielding or fracture), and the factor (such as stress, strain and energy)

associated with it.

If tests could be performed on the actual member, subjecting it to all the possible

conditions of loading that the member would be subjected to during operation, then one

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could determine the maximum loading condition that does not cause failure. But this

may not be possible except in very simple cases.

Consequently, in complex loading conditions, one has to identify the factor associated

with the failure of a member and take precautions to see that this factor does not exceed

the maximum allowable value. This information is obtained by performing a suitable

test (uniform tension or torsion) on the material in the laboratory.

In discussing the various theories of failure, we have expressed the critical value

associated with each theory in terms of the yield point stress σy obtained from a uniaxial

tensile stress.

This was done since it is easy to perform a uniaxial tensile stress and obtain the yield

point stress value. It is equally easy to perform a pure torsion test on a round specimen

and obtain the value of the maximum shear stress τy at the point of yielding.

Consequently, one can also express the critical value associated with each theory of

failure in terms of the yield point shear stress τy.

In a sense, using σy or τy is equivalent because during a uniaxial tension, the maximum

shear stress τ at a point is equal to 1/2 σ; and in the case of pure shear, the normal

stresses on a 45° element are σ and –σ, where σ is numerically equivalent to τ.

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Use Of Factor Of Safety In Design

In designing a member to carry a given load without failure, usually a factor of safety N is

used. The purpose is to design the member in such a way that it can carry N times the actual

working load without failure.

It has been observed that one can associate different factors for failure according to the

particular theory of failure adopted. Consequently, one can use a factor appropriately reduced

during the design process.

Let X be a factor associated with failure and let F be the load. If X is directly proportional

to F, then designing the member to safely carry a load equal to NF is equivalent to designing

the member for a critical factor equal to X/N.

However, if X is not directly proportional to F, but is, say, proportional to F2, then designing

the member to safely carry a load to equal to NF is equivalent to limiting the critical factor to

√X /N .

Hence, in using the factor of safety, care must be taken to see that the critical factor

associated with failure is not reduced by N, but rather the load-carrying capacity is increased

by N.

As remarked earlier, when a factor of safety N is prescribed, we may consider two ways of

introducing it in design:

(i) Design the member so that it safely carries a load NF.

(ii) If the factor associated with failure is X, then see that this factor at any point in the member

does not exceed X/N.

But the second method of using N is not correct, since by the definition of the factor of

safety, the member is to be designed for N times the load. So long as X is directly proportional

to F, whether one uses NF or X/N for design analysis, the result will be identical. If X is not

directly proportional to F, method (ii) may give wrong results.

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The Flow Curve

• True stress-strain curve for typical ductile materials, i.e., aluminium, show that the stress -

strain relationship follows up the Hooke’s law up to the yield point, σo.

• Beyond σo, the metal deforms plastically with strain-hardening. This cannot be related by any

simple constant of proportionality.

• If the load is released from straining up to point A, the total strain will immediately decrease

from ε1 to ε2. by an amount of σ/E.

• The strain ε1-ε2 is the recoverable elastic strain. Also there will be a small amount of the

plastic strain ε2-ε3 known as inelastic behaviour which will disappear by time. (neglected in

plasticity theories.)

Usually the stress-strain curve on unloading from a plastic strain will not be exactly linear and

parallel to the elastic portion of the curve.

• On reloading the curve will generally bend over as the stress pass through the original value

from which it was unloaded.

• With this little effect of unloading and loading from a plastic strain, the stress-strain curve

becomes a continuation of the hysteresis behavior. (But generally neglected in plasticity

theories.)

• If specimen is deformed plastically beyond the yield stress in tension (+), and then in

compression (-), it is found that the yield stress on reloading in compression is less than the

original yield stress. The dependence of the yield stress on loading path and direction is called

the Bauschinger effect. (However it is neglected in plasticity theories and it is assumed that

the yield stress in tension and compression are the same).

• A true stress – strain curve provides the stress required to cause the material to flow plastically

at any strain is often called a ‘flow curve’.

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Note: higher σo means greater elastic region, but less ductility (less plastic region).

True stress and true strain

• The engineering stress – strain curve is based entirely on the original dimensions of the

specimen means This cannot represent true deformation characteristic of the material.

• The true stress – strain curve is based on the instantaneous specimen dimensions.

True strain or natural strain (first proposed by Ludwik) is the change in length referred to the

instantaneous gauge length.

The true stress is the load divided by the instantaneous area.

What is Strain Hardening?

Consider the following key experiment, the tensile test, in which a small, usually

cylindrical, specimen is gripped and stretched, usually at some given rate of stretching. The

force required to hold the specimen at a given stretch is recorded.

If the material is a metal, the deformation remains elastic up to a certain force level, the

yield point of the material. Beyond this point, permanent plastic deformations are induced.

On unloading only the elastic deformation is recovered and the specimen will have

undergone a permanent elongation (and consequent lateral contraction).

In the elastic range, the force-displacement behaviour for most engineering materials

(metals, rocks, plastics, but not soils) is linear. After passing the elastic limit (point A),

further increases in load are usually required to maintain an increase in displacement; this

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phenomenon is known as work-hardening or strain-hardening. In some cases the force-

displacement curve decreases, as in some soils; the material is said to be softening. If the

specimen is unloaded from a plastic state (B) it will return along the path BC shown, parallel

to the original elastic line. This is elastic recovery.

What remains is the permanent plastic deformation. If the material is now loaded again, the

force-displacement curve will re-trace the unloading path CB until it again reaches the

plastic state. Further increases in stress will cause the curve to follow BD.

Two important observations concerning the above tension test are the following:

(1) After the onset of plastic deformation, the material will be seen to undergo negligible

volume change, that is, it is incompressible.( assumption of plasticity)

(2) the force-displacement curve is more or less the same regardless of the rate at which the

specimen is stretched (at least at moderate temperatures).

Nominal and True Stress and Strain

There are two different ways of describing the force F which acts in a tension test. First,

normalizing with respect to the original cross sectional area of the tension test specimen Ao ,

one has the nominal stress or engineering stress,

Alternatively, one can normalize with respect to the current cross-sectional area A, leading to

the true stress,

in which F and A are both changing with time. For very small elongations, within the elastic

range say, the cross-sectional area of the material undergoes negligible change and both

definitions of stress are more or less equivalent. Similarly, one can describe the deformation in

two alternative ways. Denoting the original specimen length by lo and the current length by l,

one has the engineering strain

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Alternatively, the true strain accounts for the fact that the “original length” is continually

changing; a small change in length dl leads to a strain increment dε = dl / l and the total strain

is defined as the accumulation of these increments:

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Resilience and Toughness

Ability of absorb energy in the elastic range and release it when stress is removed is called

Resilience. High carbon steel has high resilience

Ability to absorb energy in plastic range is called Toughness. Spider silk has high toughness.

Too little carbon content leaves (pure) iron quite soft, ductile, and weak. Carbon contents

higher than those of steel make a brittle alloy commonly called pig iron.

• Flow rule is what path material follows during plastic deformation to achieve new

position according to hardening rule

Theories of Failure

In the case of multidimensional stress at a point we have a more complicated situation

present. Since it is impractical to test every material and every combination of stresses

, a failure theory is needed for making predictions on the basis of a

material’s performance on the tensile test., of how strong it will be under any other

conditions of static loading.

The “theory” behind the various failure theories is that whatever is responsible for failure

in the standard tensile test will also be responsible for failure under all other conditions

of static loading.

Brittle and ductile materials – different modes of failures – mode of failure – depends on

loading

Ductile materials – exhibit yielding – plastic deformation before failure

Brittle materials – no yielding – sudden failure

Multi-axial stress state – six stress components – one representative value

Define effective / equivalent stress – combination of components of multi-axial stress state

Equivalent stress reaching a limiting value – property of material – yielding occurs – Yield

criteria

Yield criteria define conditions under which yielding occurs

Single yield criteria – doesn’t cater for all materials

Material yielding depends on rate of loading – static & dynamic

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Parameters in uniaxial tension

End of lecture…………….