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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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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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