Thermal Strains and Element of the
Theory of Plasticity
Thermal Strains
• Thermal strain is a special class of Elastic strain that
results from
– expansion with increasing temperature, or
– contraction with decreasing temperature
• Increased temperature causes the atoms to vibrate by
large amount. In isotropic materials, the effect is the
same in all directions.
• Over a limited range of temperatures, the thermal
strains at a given temperature T, can be assumed to be
proportional to the change, T.
   T  T0    T 
(A8-1)
where T0 is the reference temperature ( = 0 at T0). The
coefficient of thermal expansion, , is seen to be in units
of 1/oC, thus making strain dimensionless.
• Since uniform thermal strains occur in all directions in
isotropic material, Hooke’s law for 3-D can be
generalized to include thermal effects.
1
 x   x    y   z    T 
E
1
 y   y    x   z    T 
E
1
 z   z    x   y    T 
E
(A8-2)
• The theory of plasticity is concerned with a number of
different types of problems. It deals with the behavior
of metals at strains where Hooke’s law is no longer
valid.
• From the viewpoint of design, plasticity is concerned
with predicting the safe limits for use of a material
under combined stresses. i.e., the maximum load which
can be applied to a body without causing:
– Excessive Yielding
– Flow
– Fracture
• Plasticity is also concerned with understanding the
mechanism of plastic deformation of metals.
• Plastic deformation is not a reversible process, and
depends on the loading path by which the final state is
achieved.
• In plastic deformation, there is no easily measured
constant relating stress to strain as with Young’s modulus
for elastic deformation.
• The phenomena of strain hardening, plastic
anisotropy, elastic hysteresis, and Bauschinger effect
can not be treated easily without introducing
considerable mathematical complexity.
Figure 8-1(a). Typical true stress-strain curves for a ductile metal.
Hooke’s law is followed up to the yield stress 0, and beyond 0,
the metal deforms plastically.
Figure 8-1b. Same curve as 8-1a, except that it shows what happens
during unloading and reloading - Hysteresis. The curve will not be
exactly linear and parallel to the elastic portion of the curve.
Figure 8-1c. Same curve as 8-1a, but showing Bauschinger effect.
It is found that the yield stress in tension is greater than the yield
stress in compression.
Figure 8-2. Idealized flow curves. (a) Rigid ideal plastic material
Figure 8-2b. Ideal plastic material with elastic region
Figure 8-2c. Piecewise linear ( strain-hardening) material.
• A true stress-strain curve is frequently called a flow
curve, because it gives the stress required to cause the
metal to flow plastically to any given strain.
• The mathematical equation used to describe the stressstrain relationship is a power expression of the form:
  k n
(8-1)
where K is the stress at  = 1.0 and n, the strainhardening coefficient, is the slope of a log-log of
Eq. 8-1
(8-2)
log   log K  n log 
That is,
FAILURE CRITERIA: FLOW/YIELD and FRACTURE
• The Flow, Yield or Failure criterion must be in terms
of stress in such a way that it is valid for all states of
stress.
• A given material may fail by either yielding or fracture
depending on its properties and the state of stress.
• A number of different failure criteria are available,
some of which predict failure by yielding, and others
failure by fracture.
• The terms flow criterion, yield criterion and failure
criterion have different meanings.
– Yield criterion applies mainly to materials that are in the
annealed condition (usually ductile materials).
– Failure criterion applies to both ductile and brittle
materials. However, it is mainly used for brittle
materials (fracture criterion), in which the limit of
elastic deformation coincides with failure.
– Flow criterion applies to materials that have been
previously processed via work hardening (usually
ductile materials).
• In applying a yielding criterion, the resistance of a
material is given by its yield strength.
• In applying a fracture criterion, the ultimate tensile
strength is usually used.
• Failure criterion for isotropic materials can be
expressed in the following mathematical form:
f 1, 2 , 3    c
(8-3)
where failure (yielding or fracture) is predicted to occur
when a specific mathematical function f of the principal
normal stresses is equal to the failure strength of the
material, c, from a uniaxial tension test.
• The failure strength is either the yield strength o, or the
ultimate strength u, depending on whether yielding or
fracture is of interest.
• Let us define an effective stress, , which is a single
numerical value
that characterizes the state of applied
_
stress. If    c ( failure occurs)
where c is a known material
property
_
Failure is not expected if    c (no failure)
The safety factor against failure is given as: X   c
_

That is the applied stress can be increased by a factor of
X before failure occurs.
Maximum Normal Stress Criterion (Rankine)
• Yielding (Plastic flow) takes place when the greatest
principal stress in a complex state of stress reaches the
flow stress in a uniaxial tension.
• Since 1 > 2 > 3, Flow occurs when
Flow stress in uniaxial tension
0 (tension) = 1
(8-4)
Maximum normal stress in a
complex stress state.
Compressive strength is usually greater than tensile
strength.
 0 tension    1   0 compressio n 
Where  0 is the flow stress of the material.
• The great weakness of this criterion is that it predicts plastic
flow of a material under a hydrostatic state of stress;
however, this is impossible, as shown by the example below.
• It is well known that tiny shrimp can live at very
great depths. The hydrostatic pressure due to water is
equivalent to 1 atm (10-5 N/m-2) for every 10 m; at 1000 m
below the surface the shrimp would be subjected to a
hydrostatic stress of 10-7 N/m-2.
Hence,
 p   1   2   3  107 N / m2
• Experiment to determine the yield stress of the shrimp
(defined as the stress at which the amplitude of the tail
wiggling would have becomes less than a critical value)
when crushed between two fingers showed that it occurred
at a stress of about 10-5 N/m-2 (14.5 psi).
Hence,
 0  105 N / m2
Rankine’s criterion predicts that shrimp failure would occur at
p   0  105 N / m2
This corresponds to a depth of only 10m. Fortunately for all
lovers of crustaceans, this is not the case, and hydrostatic
stresses do not contribute to plastic flow.
Maximum-Shear-Stress or Tresca Criterion
• This yield criterion assumes that yielding occurs when
the maximum shear stress in a complex state of stress
equals the maximum shear stress at the onset of flow in
uniaxial-tension.
• From Eq,(2.21), the maximum shear stress is given by:
 max 
1   3
(8.5)
2
Where  1 is the algebraically largest and  3 is the
algebraically smallest principal stress.
For uniaxial tension,  1   0, 2   3  0 , and the maximum
shearing yield stress  0 is given by:
0 
0
2
Substituting in Eq. (8.3), we have
 max 
1   3
2
 0 
0
2
Therefore, the maximum-shear-stress criterion is given by:
1   3   0
(8.6)
• This criterion corresponds to taking the differences
between 1 and 3 and making it equal to the flow stress
in uniaxial tension.
• This criterion does not predict failure under hydrostatic
stress, because we would have 1 = 3 = p and no
resulting shear stress.
von Mises’ or Distortion-Energy Criterion
• This criterion is usually applied to ductile material
• von Mises’ proposed that yielding would occur when the
second invariant of the stress deviator J2 exceeds some
critical value.
J2  k 2
1
2
2
2
(8.7)
where J 2   2   3    1   2    3   1 
6


for yielding in uniaxial tension  1   0 ;  2   3  0

 02   02  6k 2

0 
3k
(8.8)
Substituting Eq. 8-8 into 8-7, we obtain the usual form of
von Mises’ yield criterion. When the expression


1
2
2
2 1/ 2
( 1   2 )  ( 2   3 )  ( 3   1 )
 o
2
(8.9)
then the material will flow. The above expression above is
known as effective stress. It is now accepted that it
expresses the critical value of the distortion (or shear)
component of the deformation energy of a body.
Additional Failure Criteria
• Octahedral Shear Stress Yield Criteria: This is another
yield criteria often used for ductile metals. It states that
yielding occurs when the shear stress on the octahedral
planes reaches a critical value.
• Mohr-Coulomb Failure Criterion: This is used for
brittle metals, and is a modified Tresca criterion.
• Griffith Failure Criterion: Another criterion used for
brittle metals. It simply states that failure will occur
when the tensile stress tangential to an ellipsoidal cavity
and at the cavity surface reaches a critical level 0.
• McClintock-Walsh Criterion: Another criterion used for
brittle metals. It is an extension of the Griffith’s
criterion, and considers a frictional component acting on
the flaw faces that had to be overcome in order for them
to grow.
Example
A region on the surface of a 6061-T4 aluminum alloy
component has strain gage attached, which indicate the
following stresses:
11 = 70 MPa
22 = 120 MPa
12 = 60 MPa
Determine the yielding for both Tresca’s and von Mises’
criteria, given that 0 = 150 MPa (the yield stress).
Solution
Since we were given the value of 12, we must therefore
first establish the principal stresses. Invoke Eq. 4-37.
 1 , 2 
Hence,
 11   22
2

   11   22  2

2

  12 



2



1 = 160 MPa;
2 = 30 MPa;
1 = 0
According to Tresca, max = (160 - 0)/2 = 80 MPa
For yielding in uniaxial tension:
0/2 = 75 MPa
Since the 80 MPa > 75 MPa, Tresca criterion would be
unsafe.
The von Mises criterion can be invoked from Eq. 8-9.


1
2
2
2 1/ 2
o 
( 1   2 )  ( 2   3 )  ( 3   1 )
2
The L.H.S. of the above Eq. gives a value of 175 MPa.
This criterion predicts that the material will not fail (flow),
unlike the Tresca criterion, which predicts that the material
will flow.
Therefore, the Tresca criterion is more conservative than
the von Mises’ criterion in predicting failure.