Effect Of Temperature & Strain Rate On
Flow Properties
1
• The stress-strain curve and the flow and fracture
properties of a material are strongly dependent on:
- strain rate
- temperature at which the test was conducted.
• In general strength decreases and ductility increases as:
- strain rate is decreased, or
- the test temperature is increased.
2
Figure 2-1. Yield strength changes as a function of (a) temperature
and (b) strain
3
Figure 2-2. Effect of strain rate and temperature on stressstrain curves.
4
Figure 2-3. Changes in engineering stress-strain curve of mild steel
with temperature.
5
• This general behavior may not take place in certain
temperature ranges if structural changes such as
precipitation, strain aging, or recrystallization
occur.
• The above thermally activated processes can assist
deformation and reduce or increase strength at
elevated temperatures.
• When materials are deformed at high temperatures
and/or long exposure, structural changes can occur
resulting in time-dependent deformation or creep.
6
Figure 2-4. Effect of temperature on the yield strength of bodycentered cubic Ta, W, Mo, Fe, and face-centered cubic Ni
7
Note
• For the bcc metals (see Fig. 2.4), the yield stress increases
rapidly with decreasing temperature.
• For Ni and most fcc metals, the yield stress is only slightly
temperature dependent.
• Fig. 2.4 can also be used to understand why most bcc metals
exhibit brittle fracture at low temperatures.
• A comparison of the flow stress of two materials at elevated
temperature requires a correction for the effect of temperature
on Elastic Modulus.
8
• The temperature dependence of flow stress at constant strain and
strain rate can be given by:
Q

  C2 exp RT 

  ,

2-1
where Q is the activation energy for plastic flow, C2 is a constant,
T is the testing temperature and R is the universal gas constant
• A plot of ln versus 1/T will give a straight line with a slope Q/R
• The activation energy Q can be determined by performing two
tensile tests at two temperatures, T1 and T2 and at a constant
strain rate.
  1  T1T2
Q  R ln  
  2  T2  T1
2-2
9
• Equation 2.1 can also be written as:


  f Z   f   exp( H RT ) 


2-3
where H is an activation energy (calorie per mole). It is related
to the activation energy of Eq. 2.1 by Q = m H, where m is the
strain rate sensitivity.
• Z is the Zener-Hollomon parameter or temperature-modified
strain rate.

Z   exp H


RT
2-4
10
• The above equation can be written in a different form for hotworking conditions:
  A(sinh  ) n ' exp  Q RT 



2-5
where A, , and n’ are experimentally determined constants
• At low stresses ( < 1.0), Eq. 2.5 reduces to:
  A1 n ' exp  Q RT 



2-6
The power law equation (Eq. 2.6) can be used to describe
creep, and superplasticity to some extent.
11
• At high stresses ( > 1.2), Eq. 2.5 reduces to:
  A2 exp(n' ) exp  Q RT 


2-7

The constants  and n’ can be determined from tests at high
and low stresses.
12
Strain Rate Effects
• Lowest range of strain rates  Creep and Stress Relaxation
• Intermediate range 10-4 <  < 10-2  Hot working/Tensile test
• Highest range  shock wave or explosive test

• Stress-strain curves can be sensitive to strain rate 
– flow stress increases with strain rate
– work hardening rate may also increase with strain rate
13
• Two parameters used to describe the above effects are:
- Strain rate sensitivity (m), and this is given as:
and
 ln 
m

 ln 
(2.8)
 ,T
 ln w
s

 ln   ,T
where
d
w
d
 ,T
(2.9)
14
• Equations 2.8 and 2.9 can be expressed as
m
  K
(2.10)
d
' s
 K
d
(2.11)
• It is possible to determine m from tensile tests by changing the
strain rate suddenly and by measuring the instantaneous
change in stress. This technique is illustrated in Fig. 2.5.
15
Figure 2-5. Strain-rate changes during tensile test. Four strain
rates are shown: 10-1, 10-2, 10-3, and 10-4s-1.
16
• Applying Equation 2.10 and 2.11 to two strain rates and
eliminating K, we have:
ln 2 /  1 
m
 
ln  2 / 1 


(2.12)
• One can easily obtain m from the strain rate changes in Figure 2-5
• The parameter m is important in accessing the superplasticity of
materials
17
Constitutive Equations
• Describe the relations between stress and strain in terms of the
variables of strain rate and temperature
• Early concept: f(,,,T) = 0
– Analogous to equilibrium in thermodynamics system which
states that:
f(P, V, T) = 0
• There are several forms of constitutive relations, including the
simple power law relation (Hollomon equation) and it’s
variants.
18
Other Examples of Constitutive Relations
•
 = f(Z) = f(eH/RT) 
(2.3)
where Z is called the Zener-Hollomon parameter, H is an
activation energy (calorie per mole), of which Q = m H
•
 = A(sinh)n` e-Q/RT
(2.5)
where A, , and n` are experimentally determined constants
19
Constitutive Relations (cont)
• At low stresses ( < 1.0) :

n ' Q / RT
  A1 e
(2.6)
where A, , and n` are experimentally determined constants.
• At high stresses ( > 1.2), and the equation reduces to:

  A2 exp(  )e
Q / RT
(2.7)
The constants  and n` are related by  = n`
20
Figure 2-6. Stress-strain curves for AISI 1040 steel subjected to
different heat treatments; curves obtained from tensile test.
21