1. 8. Stress-Strain Relations
ME 612 Metal Forming and Theory of Plasticity
8. Stress-Strain Relations
Assoc.Prof.Dr. Ahmet Zafer Şenalp
Mechanical Engineering Department
Gebze Technical University

2. Mechanical Engineering Department, GTU
8. Stress-Strain Relations
Experiments have shown that in uniaxial loading strain corresponding to certain stress is composed of two parts:
Recoverable elastic strain
Irrecoverable plastic strain
Experiments have shown that elastic strain can be related to stress by linear elastic equations. The equations valid for isotropic solid materials are:
(8.1)
(8.2)
(8.3)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

3. Mechanical Engineering Department, GTU
8. Stress-Strain Relations
: Poisson’s ratio E : Young elasticity modulus G : Shear elasticity modulus The above ex,ey and ez equations can be rearranged to express in terms of hydrostatic and deviatoric stresses:
(8.4)
(8.5)
(8.6)
(8.7)
(8.8)
(8.9)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

4. Mechanical Engineering Department, GTU
8. Stress-Strain Relations
Here sm: hydrostatic stress: is deviatoric stress: In terms of indicial notation:
(8.10)
(8.11)
(8.12)
(8.13)
=1 if i=j
=0 if i j
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

5. 8. Stress-Strain Relations
Figure 8.1. Elastic and plastic strains
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

6. Mechanical Engineering Department, GTU
8. Stress-Strain Relations
Theory of plasticity involves with irrecoverable plastic strain.
In multiaxial loading general strain term can be decomposed into elastic and plastic parts:
: Total strain
: Elastic strain component
: Plastic strain component
In differential form;
(8.14)
(8.15)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

7. Mechanical Engineering Department, GTU
8.1. Prandl-Reuss Equations
8. Stress-Strain Relations
Reuss assumed that the plastic strain increment is at any instant proportional to the instantaneous stress deviation and shear stresses, thus: In terms of indicial notation: : is an instantaneous non-negative constant of proportionality : deviatoric stress The above equation can be expressed in terms of principal stress directions: These equations give only ratio but does not give information about quantity.
(8.16)
(8.17)
(8.18)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

8. Mechanical Engineering Department, GTU
8.1. Prandl-Reuss Equations
8. Stress-Strain Relations
These equations are called Prandl-Reuss equations and can be written this form:
(8.19)
(8.20)
(8.21)
(8.22)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

9. Mechanical Engineering Department, GTU
8.2. Levy-Mises Equations
8. Stress-Strain Relations
Levy-Mises equations can be defined as a special case of Prandl-Reuss equations. These are In terms of total strains
(8.23)
(8.24)
(8.25)
(8.26)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

10. Mechanical Engineering Department, GTU
8.2. Levy-Mises Equations
8. Stress-Strain Relations
As seen Levy-Mises equations discard elastic behavior. Hence when elastic deformation is important Prandl-Reuss equations should be used.
(8.27)
(8.28)
(8.29)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU