1. 7. Yield Criteria of Metals
ME 612 Metal Forming and Theory of Plasticity
7. Yield Criteria of Metals
Assoc.Prof.Dr. Ahmet Zafer Şenalp
Mechanical Engineering Department
Gebze Technical University

2. Mechanical Engineering Department, GTU
7. Yield Criteria of Metals
Recall that we say that the material yields when it exhibits an irreversible straining which is sustained once a certain level of the stress distribution is reached. A yield criterion indicates for which combination of stress components transition from elastic (recoverable) to plastic (permanent) deformations occurs. In one-dimension (Fig. 7.1(a)) yielding occurs when the uniaxial stress reaches the value of the yield stress Y in tension, i.e. at σ = Y . When does ‘yielding’ occur in multi-axial stress states? (Fig. 1(b))? The answer is given with phenomenological theories called ‘yield criteria’. Instead of presenting the requirements and constraints for a general form of a yield criterion, we will here only examine the two most important yield criteria for isotropic materials.
Figure 7.1. (a) To define yielding in one-dimensional stress states, we compare the uniaxial
stress σ with the yield stress in tension Y . (b) When does yielding occurs in multidimensions?
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

3. Mechanical Engineering Department, GTU
7. Yield Criteria of Metals
The most general form of yield criteria is: Or in terms of principal stresses; For most ductile metals that are isotropic, the following assumptions are invoked since they have been observed in many instances: 1. There is no Bauschinger effect, thus the yield strengths in tension and compression are equivalent. 2. Constancy of volume prevails so the plastic equivalent of Poisson’s ratio is The magnitude of mean normal stress; (hydrostatic stress) does not influence yielding.
(C= a constant)
(7.1)
(7.2)
(7.3)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

4. Mechanical Engineering Department, GTU
7. Yield Criteria of Metals
In Figure 7.2 two different stress states is given where the only difference is the hydrostatic stress. These two cases are equivalent in terms of yield criteria.
Figure 7.2. Mohr’s circles showing two stress states which differ by only a hydrostatic stress.
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

5. Mechanical Engineering Department, GTU
7. Yield Criteria of Metals
Yield crteria can be expressed as:
(7.4)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

6. Mechanical Engineering Department, GTU
7.1. The Tresca Criterion or
the Maximum Shear Stress Criterion
7. Yield Criteria of Metals
This criterion postulates that yielding will occur when the largest shear stress reaches a critical value. (C= Constant) and this equality is Tresca criterion. (For principal stresses rule is valid) To find the value of C uniaxial tension test case can be handled. In this test conditions apply and yield occurs when value is equal to yield strength value; Hence or
(7.5)
(7.6)
(7.7)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

7. Mechanical Engineering Department, GTU
7.1. The Tresca Criterion or
the Maximum Shear Stress Criterion
7. Yield Criteria of Metals
Considering pure shear case: conditions apply and yield occurs when maximum shear stress value is equal to shear strength value. In this case; then and Tresca criterion is
(7.8)
(7.9)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

8. 7.1. The Tresca Criterion or the Maximum Shear Stress Criterion
7. Yield Criteria of Metals
Figure 7.3. Tresca yield locus.
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

9. 7.2. Examples of the application of the Tresca criterion
7. Yield Criteria of Metals
Example 1:
Equi-biaxial tension;
σ1=σ2= and σ3=0 (Figure 7.4)
The principal stresses are:
Figure 7.4. Equi-biaxial tension σ1=σ2= ve σ3=0
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

10. Mechanical Engineering Department, GTU
7.2. Examples of the application
of the Tresca criterion
7. Yield Criteria of Metals
In this case stress state according to Figure 7.3 is at the intersection point of regions 1 and 2. Both regions can be used. we conclude that yielding in equi-biaxial tension occurs when
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

11. Mechanical Engineering Department, GTU
7.2. Examples of the application
of the Tresca criterion
7. Yield Criteria of Metals
Example 2:
Hydrostatic pressure case;
σ1=σ2=σ3=-p
are principal stresses.
is obtained and this is never equal to Y. Then in pure hydrostatic stress case yield never happens.
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

12. Mechanical Engineering Department, GTU
7.2. Examples of the application
of the Tresca criterion
7. Yield Criteria of Metals
Example 3:
Stress state is given as:
The material of this parts does not yield.
If additional 50 MPa fluid pressure is applied to the part will there be yielding?
At the beginning;
Fluid pressure adds extra -50 MPa to the above stresses.
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

13. Mechanical Engineering Department, GTU
7.2. Examples of the application
of the Tresca criterion
7. Yield Criteria of Metals
New situation is then; This case yields the same diameters with the initial one in Mohr’s circle. This example shows that hyrostatic pressure does not effect yield.
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

14. Mechanical Engineering Department, GTU
7.2. Examples of the application
of the Tresca criterion
7. Yield Criteria of Metals
Example 4:
A metal has a yield stregth of 280 MPa. This metal is loaded such that 300 MPa, 200 MPa and 50 MPa principal stresses appear. Will this metal yield according to Tresca criterion?
According to given stress values;
max = 300 MPa σmin = 50 MPa
If replaced in Tresca criterion;
< Y=280 MPa means that metal will not yield.
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

15. Mechanical Engineering Department, GTU
7.3. The von-Mises Yield Criterion
7. Yield Criteria of Metals
Another function that can be used for yield is To find the value of C uniaxial tension test case can be handled. In this test when value is equal to the yield stregth value yielding occurs; If these values are placed in the above equation; is obtained. In pure shear case; is obtained. Finally von Mises yield criteria is obtained as;
(7.10)
(7.11)
(7.12)
(7.13)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

16. Mechanical Engineering Department, GTU
7.3. The von-Mises Yield Criterion
7. Yield Criteria of Metals
For a general form the stress components can be written in terms of x, y, z coordinate system: PS: The two equations given above are equivalent. If von Mises yield criteria is rearranged; is obtained. The amounts of principal stresses in this equation is just at yield limit. If the above equation is generalized Y should be replaced with which indicates equivalent stress. The following equation is called as equivalent stress equation;
(7.14)
(7.15)
(7.16)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

17. 7.4. Examples Of The Application Of The von-Mises Yield Criterion
7. Yield Criteria of Metals
Example 1:
Equi-biaxial tension;
σ1=σ2= and σ3=0 (Figure 7.4)
Substitution in the expression for equivalent stress
So for this case yielding occurs when σ = Y. For this example von Mises criterion yields the same result with Tresca criterion.
Figure 7.4. Equi-biaxial tension σ1=σ2= ve σ3=0
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

18. Mechanical Engineering Department, GTU
7.4. Examples Of The Application
Of The von-Mises Yield Criterion
7. Yield Criteria of Metals
Example 2:
Hydrostatic pressure case;
σ1=σ2=σ3=-p
are the principal stresses.
Substitution in the expression for equivalent stress;
So as it was the case with the Tresca criterion, yielding does not occur in the von-Mises criterion for the case of hydrostatic pressure.
< Y
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

19. Mechanical Engineering Department, GTU
7.5. The von-Mises Criterion For the
General Plane Stress State (σ2=0)
7. Yield Criteria of Metals
Let us apply the von-Mises criterion in the general case of plane stress conditions: σ2=0 Substitution into the expression for equivalent stress gives the following: Thus according to the von-Mises yield criterion, yielding occurs when: If the above equation is shown on σ1-σ3 plane, ellipse is obtained (Figure 7.5).
(7.17)
(7.18)
(7.19)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

20. 7.5. The von-Mises Criterion For the General Plane Stress State (σ2=0)
7. Yield Criteria of Metals
Figure .7.5 von-Mises yield locus
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

21. Mechanical Engineering Department, GTU
7.5. The von-Mises Criterion For the
General Plane Stress State (σ2=0)
7. Yield Criteria of Metals
Figure 7.5. The Tresca and von-Mises yield loci for the same value of Y showing certain loading paths (i.e. for varying α = σ3/σ1).
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

22. 7.6. 3D Yield Surfaces 7. Yield Criteria of Metals
In Figure 7.6 drawings of Tresca and von Mises criterions in principal stress space are shown. Tresca is in the form of octagon prizm and von Mises is in the form of cylinder. Both are centered in a line that is the same as the direction cosines. If yielding occurs the vector sum of any combination of should touch to the yield surface. In Figure 7.7 a plane defined by is passed through the surface shown in Figure 7.6.
Figure D Yield surfaces for Tresca and von Mises
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

23. Mechanical Engineering Department, GTU
7.6. 3D Yield Surfaces
7. Yield Criteria of Metals
Figure Representation of yield criteria on plane
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

24. Mechanical Engineering Department, GTU
7.7. Isotropic Work Hardening
of Materails
7. Yield Criteria of Metals
The definitions about work hardening are given before. In the above section von-Mises yield criterion is given as: Additionally equivalent stress equation is extracted as; The principal stresses calculated in any kind of loading is placed into this equation to calculate equivalent stress. This equivalent stress is such a value that it can be directly compared with yield strength (Y) of the material. If =Y yield starts, >Y material yields, <Y material does not yield.
(7.20)
(7.21)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

25. Mechanical Engineering Department, GTU
7.7. Isotropic Work Hardening
of Materails
7. Yield Criteria of Metals
After obtaining equivalent stress equation a similar definition for strain is also needed. This is eqivalent strain definition. To reach equivalent strain definition we shpuld remember that equivalent stress is a function of plastic work; It should be noted that these expressions are valid for isotropic materials with no Bauschinger effect.. Yield criterion involves that hyrostatic stress components does not do plastic work and volume is constant. Plastic work increment per unit volume: Here is equivalent plastic strain icrement. From constancy of volume:
(7.22)
(7.23)
(7.24)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

26. Mechanical Engineering Department, GTU
7.7. Isotropic Work Hardening
of Materails
7. Yield Criteria of Metals
From Eq(7.23) With the method that details are given in Reference [1] the plastic strain increment is If elastic strains are comparitively too much small with respect to plastic strains the above equation can be expressed in terms of total strains: Using Eq (7.21) and (7.26) the equivalent stress-equivalent strain graph is plotted. The area under the curve yields work term. After the materail tests by using the data obtained equivalent stress-equivalent strain graph should be plotted.
(7.25)
(7.26)
(7.27)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

27. 7.7. Isotrop Work Hardening of Materails
7. Yield Criteria of Metals
Figure 7.6. Equivalent stress-equivalent strain graph
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU