Mechanical Properties of Materials
Chapter 7 and 8
ari.cankaya.edu.tr/~ebiber/ie114/w
eek-8.ppt, 13 dec 2010
Stress and Strain: If a load applied to the material is static or changing
slowly with time and it is applied uniformly on the surface of interest,
then we can test the behavior of the material under applied load by a
test called “stress-strain” test. The ways of applying load are
summarized in the figure below:
Tension test is the most common. A typical specimen is:
Typically L=4 x d (diameter)
L=2 in (50 mm)
Test continues usually till the specimen is
permanantly deformed or fractured.
The load and deformation relationship depends on the geometrical
factors of the specimen; therefore, normalization of them to
geometric dimensions are helpful in comparing the materials.
Engineering stress:
where F= perpendicular force applied to the surface
uniformly.
A0= the original cross sectional area before loading
unit of σ is N/m2 or lbf/in2 (psi).
Engineering strain:
where li =instantaneous length
l0=initial length
strain is unitless and it is sometimes expressed in percentage by multiplying
the value with 100.
Compression test is not very common. It is usually used if the material
application consists of compressive force system or if the material is
brittle under tensile force.
Compressive force is negative by convention yielding negative stress
and strain.
Shear and torsional tests:
Shear stress:
where F=force parallel to the surface
shear stress
shear strain is γ, calculated by tangent of
strain angle θ shown in the figure.
Torsion is a variation of pure shear,
wherein a structural member is
twisted like in machine axles and
drive shafts.
 is the angle of twist.
 is a function of torque T while  is
a function of .
This test is usually performed on
cylindrical shafts or tubes.
Stress state is a function of the
orientations of the planes. For
example;
pp’ plane is oriented at an angle of ө. The stress on this plane is not a pure tensile
stress anymore.
Elastic Deformation: is observed
when stress and strain are
proportional.
Hooke’s Law:
E=modulus of elasticity or
Young’s modulus (GPa or psi)
for metals E=45-407 GPa.
for polymers E=0.007-4 GPa
E is a measure of material’s stiffness or materials resistance to elastic deformation.
The greater the modulus, the stiffer the material or the smaller the elastic strain that
results from the application of a given stress.
Elastic deformation is nonpermanent.
There are materials (gray cast iron, concrete, and many polymers) for which
this initial elastic portion of the stress-strain curve is not linear.
In this case either tangent or secant modulus is normally used (shown in the
figure above).
Elastic strain is due to small changes
in interatomic spacing and
streching the interatomic bonds.
Therefore the magnitude of E is a
measure of the resistance to
separation of adjacent
atoms/ions/molecules. This
modulus is proportional to the
slope of the F versus r curve:
The imposition of compressive, shear
or torsional stresses evokes
elastic behavior as well. For low
shear stress levels:
G= shear modulus
We assume that the elastic deformation is time independent. However, there is a
time-dependent elastic strain component. Elastic deformation will continue after the
stress application and during the complete recovery. This time-dependent elastic
behavior is known as anelasticity.
For metals anelastic component is normally small and is usually neglected.
For polymers, its magnitude may be significant (viscoelastic behavior).
If the material is isotropic and applied stress is uniaxial (only in z diraction):
x = y
= Poisson’s ratio (theoretical value =0.25;
for many metals b/w 0.25-0.35.)
For isotropic materials:
Many materials are elastically anisotropic. This means the elastic behavior changes
with crystallographic direction. Therefore to characterize the elastic properties of the
material, several E values should be reported for specific directions. In fact even for
isotropic materials, at least two constants should be given.
Mechanical Behavior of Materials
Metals: Elastic deformation of metallic materials is usually upto strains of about
0.005. Beyond this point, the stress and strain are no longer proportional and
deformation of the material becomes permanent and nonrecoverable. This is called
plastic deformation.
Plastic deformation corresponds to the breaking of bonds with original atom
neighbors and reforming bonds with new neighbors. This permanent deformation for
metals is accomplished by means of a process called slip, which involves the motion
of dislocations.
Tensile properties of Metals:
1)
Yielding and yield strength
Most structures are designed to ensure that only elastic deformation will result
when a stress is applied. Therefore it is useful to know the stress level at
which plastic deformation begins or where the yielding occurs.
If the transition from elastic to plastic behavior is gradual, the point of
yielding may be determined as the initial departure from linearity
(P=proportional limit). In cases where it is difficult to determine this
point (P point) precisely, a conventional approach is used. A straight
line is constructed parallel to the elastic deformation line at a strain
offset usually 0.002. The stress correspoding to this point is yield
strength (y).
For the materials having nonlinear elastic region, yield strength is
defined as the stress required to produce some amount of strain
(=0.005).
For the materials showing a behavior like in Figure 7.10b, the yield
strength is the average of the upper and lower limits.
The magnitude of yield strength is a measure of material’s resistance to
plastic deformation. Yield strength may range from 35 MPa to 1400
MPa.
2) Tensile strength:
M is the stress at the maximum
point of the stress-strain curve.
F is the fracture point.
necking
Tensile strength may vary between 50 MPa to as high as 3000 MPa.
3) Ductility: It is a measure of
the degree of plastic
deformation that has been
sustained at fracture. A
material that experiences
very little or no plastic
deformation upon fracture is
termed brittle.
Quantitatively:
lf and Af are length and area at the fracture.
Ductility of materials is important for at least two reasons: (i) it indicates the
degree to which a structure will deform plastically, (ii) it specifies the degree of
allowable deformation during fabrication. Fracture strain of brittle materials is about 5%.
The mechanical properties of the materials are sensitive to
any prior deformation,
presence of impurities,
heat treatment.
The modulus of elasticity is one mechanical parameter that is insensitive to
these treatments. Similar to modulus of elasticity, the magnitudes of both
yield and tensile strengths decline with increasing temperature.
4) Resilience: is the capacity of a material to absorb energy when it is deformed
elastically and then upon unloading to have this energy recovered.
Modulus of resilience (Ur) = strain energy per unit volume required to stress a
material from an unloaded state up to the point of yielding.
assuming a linear elastic region:
5) Toughness: is a mechanical term. It is a measure of the ability of a
material to absorb energy up to fracture.
For dynamic loading conditions and when a notch is present, notch
toughness is assessed. Fracture toughness is material’s resistance
to fracture when a crack is present.
For low strain rate situation, the area under - curve up to the point of
fracture corresponds to toughness.
For a material to be tough, it must display both strength and ductility
and often ductile materials are tougher than brittle ones.
True stress and strain:
This decrease is not
because of reducing
strength, it is because
of changing geometric
properties.
Sometimes it is more meaningful to use a true stress-true strain curve.
True stress:
True strain:
If there is no change in volume:
these equations are valid up to the onset
of necking, beyond necking actual stress
or strain has to be calculated using actual
load and area/length.
With the formation of necking, axial stress is no longer axial instead we observe
a complex stress state within the neck region. As a result the correct stress (axial)
within the neck is slightly lower than the stress computed from the applied load
and neck X-sectional area.
for some metals and alloys, the region of true stress and ture strain curve from the
onset of plastic deformation to the beginning of necking:
K and n are constants (Table 7.3).
Elastic strain recovery:
parallel to elastic deformation line
For compression loadings, there will be no maximum since no necking occurs.
The mode of fracture is different for this case.
Hardness: is a measure of a material’s resistance to localized plastic
deformation.
Measured hardnesses are relative (not absolute). Hardness tests are;
1)
simple and inexpensive
2)
test is nondestructive
3)
other mechanical properties can be estimated from hardness
data.
Rockwell Hardness tests: (ASTM standard E 18)
Several different scales can be used from possible combinations of various
indenters and different loads.
Indenters: spherical and hardened steel balls (1/16, 1/8, ¼, ½ in. diameter) and
a conical diamond (Brale) intender.
Hardness number is determined by the difference in depth of penetration
resulting from the application of an initial minor load followed by a larger load.
On the basis of minor and major loads there are two tests: Rockwell and
superficial Rockwell tests.
For Rockwell: minor load is 10 kg and major loads are 60, 100, and 150 kg.
For superficial Rockwell: minor load is 3 kg and 15, 30, and 45 kg are
major loads.
80 HRB = Rockwell hardness
of 80 on b scale
60 HR30W= superficial
hardness of 60 on 30W scale.
For each scale, hardness may
range up to 130, however,
as hardness number rise above
100 or drop below 20 on any scale,
the accuracy of test decreases.
Knoop and Vickers test: A very small diamond indenter having
pyramidal geometry is forced into the surface of the
specimen.Applied loads=1-1000 g. the impression is analyzed by
microscope and measured. The measurement is then converted to
hardness number.
Brinell hardness tests: The diameter of the hardened steel or tungsten
carbide indenter is 10 mm. Applied Loads= 500-3000 kg.The
diameter of resulting indentation on the surface is measured using a
special low power microscope. the measurement is converted to
hardness number.
Hardness Conversion:
Correlations between hardness and tensile strength:
Hardness and tensile strength are indicators of a metal’s resistance to
plastic deformation.
For most steels:
Variability of material properties: There are numbers of factors causing
uncertainities in the measured data:
1)
measurement method
2)
variations in the specimen fabrication procedures
3)
operator
4)
calibration of the apparatus.
These variabilites affect the masurements accuracy and consistency.
Design and Safety Factors: In addition to variabilities of the material
properties, the applications on the material also have many uncertanities.
As a result, design allowances must be made to protect against
unanticipated failure.
Design stress: is calculated by multiplying calculated stress (using the
maximum load) by a design factor N’.
N’>1
Select a material with a yield strength at least as high as d.
Safe stress or working stress can be used as an alternative to design stress.
N=1.2-4.0