Chapter 2 Mechanical Properties 1-8-2012

8/2/2019 Chapter 2 Mechanical Properties 1-8-2012

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MATE 202: Introduction to Materials Engineering

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MechanicalProperties

Chapter 2

Optical microscope images of Rockwell B indents made in four different metallic alloys:

A: Normalized AISI 1018 plain carbon steel;B: Oil Quenched and Tempered AISI 4140 low alloy carbon steel;

C: Annealed Titanium Alloy Ti6Al4V;

D: Precipitation Hardened Aluminum Alloy 6061-T6.

Hardness measurement is a standardized technique used widely in industry to test a materials response

to localized plastic deformation. Hardness is not a true material property, but for some steels (non-

austenitic) hardness correlates very well with tensile strength. The response of non-metallic materials toindentation can also tell volumes about their mechanical properties; for example many brittle ceramic

materials tend to fracture during indentation.

J

ohnA.

Nychka


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Chapter 2: Mechanical Properties

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Mechanical properties are dictated by the structure of the material and its processing history. In engineer-

ing design standardized properties and their testing methods are required to compare materials against

each other in order to select the appropriate candidate materials. Knowing the limitations of the testingmethod and the implications of the values associated with various property data is critical for practicing

engineers, regardless of their discipline.

Issues to address:Stress and strain what are they and why are the used instead of load and displacement?

Tension testing

Elastic and plastic behaviour

Toughness and ductilityHardness measurements

How to interpret hardness? Is there correlation with other material responses?

Variability of materials properties


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2.1 Properties of materials (& those of interest in Mat E 202)

2.1.1 Properties of materials

Physical/Mechanical (here is where we will spend most of our time in Mat E 202)

Chemical

Thermal

Electrical

Magnetic

Optical

Deteriorative (and here!)

True properties are independent of the volume of material being tested.

2.1.2 Mechanical Properties

We will discuss properties such as:

Stiffness (Youngs Modulus of Elasticity, E)

Strength (Yield strength, y orys) Poissons ratio () Fracture toughness (KIC K-one-C)

Another related material response we will discuss is hardness (which is not defined as a material

property). Hardness is related to strength, and wear resistance (what you may already know as

scratch resistance).

Mass density () is an important physical property which we will discuss later in Chapter 4Here are some examples of the importance of physical properties to the deformation of

aircraft wings from the forces of the air pressure over the wing:

Figure 2.1 Some illustration of mechanical properties that we will study the defor-

mation shown may be exaggerated for the purposes of illustration. (source: Ashby).


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Example 2.1

a) A steel cable in a suspension bridge (10 cm diameter, 20 m long) is under a tensile load of

3140 kN. Calculate the engineering stress on the cable.

b) If the strain on the cable is 0.2% calculate the final length of the cable.

a) From the statement:

P=3140000N

Ao = d2/4 = 7853.98mm2

= P/Ao, so, = 3140000N/7853.98mm2 = 400 Mpa

TIP: if you calculate area in mm2, and use load in N, stress will be in MPa (1 N/mm2 = 1 MPa)

b) From the statement:

= 0.2% = 0.002lo = 20 m

= l/lo = (lf lo)/loWe want to solve for lf, so

lf = * lo + lo = 0.002*(20 m) + 20 m) = 0.04 + 20 = 20.04 m2.4 Linear Elasticity

Linear elasticity applies to metals and ceramics, but not usually polymers.

2.4.1 Hooke's Law (stress is directly proportional to strain in the elastic strain regime)

where E is Youngs modulus

also = G, where G is the shear modulus.

= E

Fig. 2.3

The constant relation between stress and

strain implied by Hookes law. It is also

observed that the strain is independent of

time at low temperatures (< 0.3-0.4 Tm).

(Source: W.D. Callister)


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2.4.2 Poissons ratio (perpendicular strain and parallel strain are proportional in the linear

elastic strain regime)

Define Poisson's ratio a material property. (Use , nu, for Poissons ratio),

NOTE: is positive* for metals and ceramics, and generally ranges from 0.3-0.4.

In the diagram below, is the ratio between the (negative) strain in the x-direction

(perpendicular or lateral strain) and the strain in the z-direction (parallel, axial, or longitudinal

strain). The elastic strains have been exaggerated for the purposes of illustration. Note that is

purely an elastic property and does not apply to plastic deformations (plastic strain).

* Some polymeric materials actually have a negative Poissons ratio. When pulled in tension, these so-called auxeticmaterials actually get larger in diameter! An example is polyester fibers produced in a certain way. For more infor-mation see the research article: N.Ravirala, K.L.Alderson, P.J.Davies, V.R.Simkins, and A.Alderson, Negative Pois-

sons Ratio Polyester Fibers, Textile Research Journal July 2006 vol 76 no. 7 p 540-546.doi: 10.1177/0040517506065255

parallel

larperpendicu

=

Figure 2.4. The change in dimensions associatedwith elastic deformation. The size of the strains is

exaggerated for the purposes of illustration.

(Source: W.D. Callister)


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Example 2.2

A 10mm diameter steel rod is put in tension with a load of 26 kN. The Poissons ratio of the

steel is 0.27, what is the diameter of the rod under load? E = 210GPa

= P/Ao, so, = 26000N/78.54mm2 = 331 MPa

Using Hookes Law, the strain is found

= E, so = 0.001576 (the parallel or longitudinal strain along the axis of therod)

= - perpendicular/parallel, we know that = 0.27, and parallel = 0.001576

solving forperpendicular = -0.000426

perpendicular = (df do)/do , solving for df (we know it should be smaller than dobecause the strain is negative) we find:

df = 9.996 mm - a very small change, but still a smaller diameter!

2.4.3 Elastic properties of materials

Normally as engineers we want stiff materials to minimize shape change under load. For materials like metals and ceramics the magnitude of the elasticity is directly related to

the strength and density of primary atomic bonds. For polymers it is related to the strength

and density of the secondary bonds.

For metals and ceramics that have crystalline structure, the structure is repeated almost per-fectly so the elastic properties dont change more than 1%. For polymers, with their more

random structure, the properties can change significantly (more than a factor of 2).

When we load the materials the atoms move farther apart and develop increased interatomicforces to return them to their normal spacing thus E is a measure of the size of the atomic

forces and the number of bonds being stretched. In polymers, since E depends primarily on the secondary bonds, E is typically much smallerthan in metals since the bonds are so much weaker.

Sometimes we want lower stiffness as exists in elastomers (a special type of polymer), toabsorb energy or provide flexibility as in automobile tires, hoses, O rings or conveyor belts.

Since structure affects the number of bonds/area, it turns out that E can vary with directionin the structure in some metals (anisotropy). e.g. in Fe the E varies by two times with direc-

tion in the single crystal. (Crystallinity is a regular, repeated packing: we will treat this

soon).

Normally, however, we regard E for metals as being an almost invariant property (manysmall crystals in many orientations averages out any anisotropy). When we strengthen met-

al, for example by strain hardening, the E remains nearly the same even when yield strength

and tensile strength can increase by a factor of 3. In polymers E depends on the secondary bonds and the number and spacing of these are var-

iable since the packing is not strictly crystalline and hence varies with processing and mo-

lecular weight and hence there is a significant range of E.

E is directly proportional to the bond stiffness, So (we will talk a lot more about this in Chapter

4), and also depends on the number of bonds/area.


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Bond stiffness and strength can also be related to the melting temperature of a material. Solids

melt when thermal energy, kT, becomes comparable to the bonding energy (Eo) .

The bond stiffness is related to the type of atomic or intermolecular bond, and later you

will see just how many more properties the bonding can explain! The force and bon-energy

curves in Figure 2.5 are representations of the interatomic forces and energy between a pair of

atoms as a function of their separation distance.

Different types of bonds have different associated force and energy curves, and the dif-

ferences in shapes accounts for differences in properties.

Table 2.2 Relationship between bond strength and melting temperature for different types

of bonds. Adapted from Ashby).

Table 2.1 Some

Youngs moduli

and their relation

to the underlying

atomic force

stiffness.

(Source: Ashby)


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We can related the shape and depth of the potential energy curves to the material properties.

The shape of the energy curve is often called a potential well in reference to the shape of a

hole dug for the purposes of finding water (i.e., a well). The deeper the potential well the

stronger the bond (i.e. more energy is needed to separate the atoms). The Youngs modulus is

related to the slope of the force curve at the equilibrium separation position between atoms, and

this slope increases when the well is narrower (large second derivative of energy with respect to

distance).

2.4.4 Response of Materials (mainly metals) to stress

Elastic deformation is recovered quickly and completely when the load is removed, inmetals the elastic deformation is small (only 2 parts in 1000 or less), but is always present

under load and is important. Elastic deformation comes from the bonds lengthening and

shortening in the structure as load is applied and removed.

It is most accurate to calculate the elastic strain from Hookes law.

Figure 2.5. a) The

dependence of repulsive,

attractive and total forces on

interatomic separation for the

interaction of two atoms.b) The dependence with

respect to separation of the

energies associated with theseinteratomic forces. Recall

that force is the derivative of

energy with respect to

distance and hence energy is

the integral of force overdistance. The minimum

energy has to occur at a

derivative of zero. (Source:

W.D. Callister)


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SCHEMATICS OF BONDS UNDER STRESS ELASTIC DEFORMATION

Plastic deformation is permanent i.e. the metal gets longer under a tensile load and thenstays longer when the load is removed. Typically elastic deformations in metals and ceram-

ics are small and plastic deformations are larger, but of course it is possible to have just a

little plastic deformation. Metals deform elastically below the yield strength and elastically

plus plastically above the yield strength. For large plastic deformation we sometimes ig-

nore the elastic because it is small, but it is always there. Here plastic is an adjective de-

scribing a type of strain and not the material loosely named plastic which tends to exhibit

lots of plasticity under the correct conditions. In metals (but not polymers) this plasticity isshear and we will see later is caused by defects known as dislocations gliding within the

grains.

SCHEMATICS OF BONDS UNDER STRESS PLASTIC+ELASTIC DEFORMATION

Metals tend to work harden or strain harden (get stronger with plastic deformation) so if weload above the yield strength but below the tensile strength, the metal will deform and get

stronger, but will not break. This phenomenon is again related to defect structure and is avery useful since it can avoid or prolong failure in certain loading conditions.

F

bondsstretch

return toinitial

Figure 2.6

The response of at-

oms to stresses in

the elastic regionshown schematical-

ly.

Figure 2.7The response of atoms to

stresses when the yield

strength is exceeded. The

plastic deformation isshown, there is also the re-

versible elastic stain re-

sponse, which is frequently

smaller than the plastic

strain.

planesstillshear

F

elastic + plastic

bondsstretch& planesshear

plastic


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2.5 Tension Testing

TENSION TESTING (TO MEASURE MECHANICAL PROPERTIES)

The specimen is shaped (dog bone) so that plastic deformation occurs only in thegauge section (deformation is only elastic at the grips) since this larger area has thus has

a lower stress.

Figure 2.9 ASTM (American Society for Testing and Materials) diagram showing the required shape anddimension for tensile specimens in flat and round forms. (Source: ASTM Designation E8/E8M-09).

Tensile machine grips the specimen and applies a constant rate of elongation. (See inlab) From the machines load cell (gives V proportional to load) we get a plot of load

versus time (but time is proportional to L, which is proportional to strain)

Figure 2.8 Schematic representation of

the tensile test machine. As the threadedshafts rotate, the crosshead moves down

elongating the specimen at a fixed rate.

The load required to do this is measured

in the load cell, typically output as a volt-age that increases linearly with load. The

strain can be estimated by the time and a

knowledge of the fixed rate of crossheadspeed. For accurate measurement of

strain the extensometer affixed to the

specimen gives a voltage proportional toits extension. Adapted from Structure

and Properties Vol. III by Hayden Moffatt

and Wulff.


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We convert load vs. L to vs. as follows:

Ao= (thickness) x (width in gauge section prior to deformation).

To convert L or time to strain we need only find strain at one point since strain is linear withtime orL. One strain that we can measure is the strain to fracture, which we find by fitting

together the broken specimen and measuring the separation of the 50 mm gauge marks we put

there before the test, this strain is reasonably accurate except for very small strains.

For very accurate strains we rely on the output from the extensometer that is a voltageproportional to the extension of the extensometer (specimen).

Figure 2.10. A typical engineering stress strain curve for a metal. The plastic stain shown hereis much smaller (100X) than the typical 10-25% expected in most metals.

From the engineering stress-strain curve we can determine:

Youngs Modulus of Elasticity,E: the slope of elastic portion of- curve. E is a measure

of material "stiffness";need to use extensometer to get accurate values.

Resilience Modulus, Ur: the specific elastic energy [J/m3] absorbed by a material up to the

elastic/proportional limit; the area under the stress strain curve in the elastic regime. Resili-

ent materials can withstand a great deal of elastic energy up until they permanently deform,

and can return the elastic energy in a reversible fashion.

Yield Strength, y (aka: yield point, proportional limit, elastic limit): the stress at whichplastic deformation begins to occur (i.e., permanent deformation).

The stress at which the deformation transforms from being initially elastic

(straight line) to plastic (curved line). Atomic defects called dislocations start

moving at this stress, which causes shearing of the crystals.

= LOADAo


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Ceramics dont usually have a yield stress in tension (but a tensile strength), as

no plastic deformation usually occurs because dislocation dont move (are ses-

sile) with the bond types exhibited by ceramics (ionic or covalent).

Polymers have a yield strength, but a different atomistic mechanism is responsi-

ble (polymers dont really have any dislocations).

0.2% offset yield strength (engineering yield strength): is "a measure of strength" one

of the common methods of specifying yield for FCC metals (since the transition from elastic

to plastic deformation is not very well pronounced in stress-strain curves).

How to determine Engineering Yield Strength:

1. Measure 0.002 strain (0.2%) checkthe units of strain on the axis!!!! (e.g., %,

mm/mm)

2. Draw line starting at = 0.002 with a slope of E (parallel to Hookes law in theelastic region)

3. Extend line to intersect the s-e curve, then read corresponding stress

4. The value of the stress at the intersection is the engineering yield strength (0.2%; proof stress; YS).

5. Observe that after unloading the permanent deformation is 0.002.

Strain to fracture, f(aka: plastic strain to failure): measure the L of the fractured speci-

mens gauge length and divide by Lo. This is a measure ofplastic strain to failure (also

called tensile ductility).

(Ultimate) tensile strength (UTS; TS): the highest engineering stress (in tension) on the

curve.

For metals: necking is insipient at the UTS

For ceramics: tensile strength is where cracks start to propagate

For Polymers: tensile strength is where polymer chains align and the chains are about to

break

% reduction in area (%RA or %AR): a measure ofductility, or ability of a material to

undergo plastic deformation. Regarded as being more sensitive than strain to fracture. It is

the measure of ductility commonly used for steels.

Measure thickness and width (or diameter) right at fracture in the neck to determine

area at fracture, Af :

Note that there is a continually increasing load in the plastic region up to the tensile strengthbecause metal strain hardens, i.e. the yield strength property is changing during the test.

When we determine the yield strength, we determine it as it was before the test (not as it is

in the gauge length after the test).

After the tensile strength is reached and necking begins, the load decreases because the re-duced area at the neck more than compensates for the increases in strength from work hard-

ening.

% . .R AA A

AX

o f

o

=

100%


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Because the strain after the neck forms is concentrated in one location, the strain to fracture will

vary with gauge length (the strain to fracture gets smaller as the gauge length increases). We

should always specify the gauge length when we give the strain to fracture; whereas, this is not

necessary for % reduction in area, although that measure may differ between round and flat

specimens.

Notice in the above graphs that the strain to fracture is much larger in the ductile specimen, as

would be the %RA (%AR). Ductile materials tend to have a larger strain to fracture than brittle

materials. Material classification based on ductility:

f < 5 % Brittle materialf > 5 % Ductile materialf > 200 % Materials with superplasticity

Figure 2.11. Typical engi-

neering stress-strain behav-

iour to fracture, at point F.

The tensile strength TS isindicated at point M. The

circular insets represent the

geometry of the gauge sec-tion of the specimen at points

along the curve. The point M

is the onset of necking andfailure occurs within the

neck. After M the ongoing

deformation is entirely within

the neck region, which is

decreasing in cross sectionalarea. (adapted from W.D.

Callister).

Figure 2.12 The schematic differences in stress-strain curves between a brittle and a ductile metal. At

the right we see how the length, L and area, A of the gauge section change during the test for a ductile

material.

Engineering tensile strain, [mm/mm]

Engineeringtensile

str

ess,[

MPa]

brittle

ductileLo Lf

AoAf


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Example 2.3

For the stress strain diagram in

Figure 2.13:

1. What is E?

2. What is the elastic limit?

3. What is the material?

4. How much elastic and plastic

strain is present if the total strain

is 0.005?

= E => E = / = 500 MPa / 0.0024 = 208 GPa

Elastic limit is where slope changes from linear to non-linear (elastic to plastic), so at e = 0.0019. This

exact point is hard to determine accurately and is rarely used in engineering, except where there is a yield

point drop as in some steels.

Steel has an elastic modulus of ~208 GPa, so it is steel!

Draw a line up to curve at 0.005 strain - from the intersection point drop a line (with slope equal to E)

down to strain axis. The total amount of strain is: total = plastic + elastic , where the plastic and elasticstrains (p, e respectively) are shown above. tot = 0.005 so p is seen to be 0.003 and e is seen to be0.002 both in mm/mm. This is an example where only a small amount of plastic strain has occurred. For

a metal to show this behaviour it would have to be very brittle since f is less than 1%.

While you will not be assessed on Section 2.5.1 you should be aware of the following:2.5.1 True stress and true strain

During the tension test, the specimen elongates plastically, and as it does, because of the

Poisson conservation of volume during plasticity, the lateral dimensions get smaller. In

reality then, the instantaneous stress within the gauge length is increasing because the

load is supported by less area. The stress taking into account the instantaneous area is

thetrue stress . In practice we can use the true stress (T), which is always higher thanthe engineering stress for tensile loading, once the elastic limit has been reached (see

Figure 2.14).

T = P/A instantaneous = P/Ai

True strain, T, is:

T = ln (linstantaneous/loriginal) = ln(li/lo)

We can convert engineering stress (s) and strain (e) to true stress and strain through the

following relations:

T = s(1 + e)T = ln (1 + e)

Figure 2.13. Working figure for the example 2.3


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2.5.2 Toughness (e.g., KIC - critical stress intensity in Mode I crack opening ) [MPam]Toughness can also be determined from a tension test, but more reliable tests exist fordetermining toughness.

Ductile materials are said to be tougher than brittle materials because more energy per

unit volume (approximated by the area under the entire stress-strain curve) is required to

cause fracture. Recall that resilience was the elastic energy absorption of a material

toughness includes the resilience and the plastic energy absorption to failure. You will

find that for metals the plastic energy is orders of magnitude larger than the elastic. It is

important that you realize the distinction between resilience and toughness; resilient

materials are not always tough, and tough materials are not always resilient.

Brittle materials are not tough (seems obvious enough), but they can be resilient.

Toughness takes into account elastic and plastic strain energy to fracture (entire area

under the stress-strain curve).

Figure 2.14 A comparison

of typical engineering

stress-strain and true stress

-strain behaviours. Neck-

ing begins at point M on

the engineering curve ,which corresponds to M

on the true curve. The

corrected true stress strain

curve (dotted) takes into

account the triaxial stress

state that develops in the

neck as it grows. (Source:

W.D. Callister).

Engineering

tensilestress,[

MPa]

Smaller toughness(Unreinforced polymers)

Engineering tensile strain, [mm/mm]

Smaller toughness(ceramics)

Larger toughness

Figure 2.15 Schematic tensile stress strain curves showing how toughness depends

on both tensile strength and elongation to fracture, which both affect the area under

the stress-strain curve.


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CAUTION:

Material properties are specific to the conditions under which they were determined.

The intelligent use of available physical constants for material selection and design re-

quires that the designer understand:

the test conditions under which the constant(s) were determined the nature of each testing method, and its limitations the units associated with each quantity (SI, CGS, English, etc.). In engineering

it is critical that you pay close attention to the units!!!

2.6 Hardness (recall the images on the chapter title page)

Definition of Hardness: Resistance to localized plastic deformation from penetration by an in-

denter.

H = Force/Area = Pressure [usually in GPa]

Most industrial hardness scales are dimensionless (e.g., Rockwell, Brinell, Vickers,

Shore).

Hardness can be measured by:1) area of penetration (more accurate) such as in Vickers, Knoop, or Brinell hardness

tests. Figure 2.16 below shows how a Brinell test is made.

2) depth of penetration such as in a Rockwell test.

Hardness correlates to strength (but not linearly for Rockwell see Figure 2.17) the hardness

test is easier and cheaper to perform than the tensile test.

Hardness can predict (for some materials):

The tensile strength (Brinell Hardness, HB = 0.294 x Tensile Strength (for non-austeniticsteels))

Wear resistance due to friction Fracture toughness (brittle materials only; KIC is proportional to 1/H)

Hardness is NOT a material property if hardness tests are conducted at different

loads different hardness values will likely be measured.

Figure 2.16 Schematic of us-ing a hardened sphere to make

Brinell impressions on the flat

surface of a steel sample. Thediameter of the impression and

the load employed are used to

calculate the Brinell hardness

of the steel.


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"Rockwell Hardness": most commonly used test method in North America except in the steel

industry, where Brinell is usually preferred.

Hardness is based on depth of indentation:

Minor load to establish indenter reference position

Major load pushes indenter into sample

After loading, the machine then gives a hardness number, which decreases with increasing

depth of penetration.

Important:

Take average of the least 3 readings and discard any that are widely different. Initial indent should be placed at least 4 indent diameters away from free surfaces and

edges.

Subsequent indents, especially in metals, must be at least 3 indent diameters apart fromeach other to insure that the plastic zone surrounding previous indents does not artifi-

cially inflate the next indents (i.e., work hardening increases the yield strength of the

metal surrounding the indents due to plastic deformation in metals)

Other types of damage in ceramics can cause artificial lowering of hardness valuesin indents placed too close to previous indents.

One must use the appropriate scale: Rockwell B, Indenter: 1/16" hardened steel ball (or tungsten carbide), 100 kg major

load, used for brass (non-ferrous) and mild steel, HRB

Rockwell C, Indenter: Brale (Diamond cone) and 150 kg major load, used for hard-ened steel and ceramics, HRC not accurate below HRC 20.

Sample should be flat and free from dirt or scale Important to use calibration block to check out tester, ensures load and indenter are un-

damaged and yield accurate readings.

Figure 2.17

Relationships between hardness

and tensile strength for steels,

brass and cast iron. (From ASM

Metals Handbook Volumes 1and 2 9th Edition). Note that the

Brinell scale is linear, but the

Rockwell scales are non-linear.

We see that the range of HRB

and HRC do not really overlap

so both tests are needed depend-

ing on the actual hardness. Bri-

nell can be used over a wider

range. (Source: W.D. Callister)


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Chapter 2 Mechanical Properties 1-8-2012