Mechanical Properties of Materials Chapter 7 and 8 Stress

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Mechanical Properties of Materials

• Chapter 7 and 8

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• 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

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• Tension test is the most common. A typical
  specimen is

Typically L4 x d (diameter) L2
in (50 mm)
Test continues usually till the specimen
is permanantly deformed or fractured.
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• 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 s is N/m2 or lbf/in2 (psi).
Engineering strain
where li instantaneous length l0initial
length strain is unitless and it is sometimes
expressed in percentage by multiplying the value
with 100.
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• 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 Fforce parallel to the surface
shear stress
shear strain is ?, calculated by tangent
of strain angle ? shown in the figure.
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• 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.

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• Stress state is a function of the orientations of
  the planes. For example

pp plane is oriented at an angle of ?. The
stress on this plane is not a pure tensile
stress anymore.
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• Elastic Deformation is observed when stress and
  strain are proportional.

Hookes Law
Emodulus of elasticity or Youngs modulus (GPa
or psi) for metals E45-407 GPa. for polymers
E0.007-4 GPa
E is a measure of materials 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.
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• 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).
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• 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

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• 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).
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If the material is isotropic and applied stress
is uniaxial (only in z diraction)

• Poissons ratio (theoretical value 0.25
• for many metals b/w 0.25-0.35.)

?x ?y
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• 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.
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• Tensile properties of Metals
• 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.

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• If the transition from elastic to plastic
  behavior is gradual, the point of yielding may be
  determined as the initial departure from
  linearity (Pproportional 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
  materials resistance to plastic deformation.
  Yield strength may range from 35 MPa to 1400 MPa.

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• 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.
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• 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.
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• 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.

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• 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
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• 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 materials 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.

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• 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.
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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.
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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).
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• 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.
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• Hardness is a measure of a materials resistance
  to localized plastic deformation.
• Measured hardnesses are relative (not absolute).
  Hardness tests are
• simple and inexpensive
• test is nondestructive
• 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.
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• 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.
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• Knoop and Vickers test A very small diamond
  indenter having pyramidal geometry is forced into
  the surface of the specimen.Applied loads1-1000
  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.

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• Hardness Conversion

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• Correlations between hardness and tensile
  strength
• Hardness and tensile strength are indicators of a
  metals resistance to plastic deformation.

For most steels
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• Variability of material properties There are
  numbers of factors causing uncertainities in the
  measured data
• measurement method
• variations in the specimen fabrication procedures
• operator
• calibration of the apparatus.
• These variabilites affect the masurements
  accuracy and consistency.

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• 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.
• Ngt1

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.
N1.2-4.0