# Tresca Yields Criterion

For Tresca yield criterion, yielding is initiated when the maximum shear stress, i.e. the largest of the three developed maximum shear stresses, exceeds the threshold value of material.

FIGURE 6.1 Perfect plastic and plastic with linear work hardening.

Let define maximum shear stress as it is in Eq. (2.44), where «г, and <тз are the maximum and minimum principal stresses respectively. Also, let *к* be the threshold value that defines whether the developed shear stress is enough to make the body initiate yielding. Thus, at yielding point:

Consider the case where only tensile stress instead of shear stress developed in the body. Express Eq. (6.1) in the following form:

The yield stress of material, *a _{y}* is determined through uniaxial load test. Therefore,

*c*should equal to the yield strength of material,

*a-±*and <тз are zero at yielding point. By applying the condition to equation above yields the follows:

Express *к* in term of *a _{y}* produces the expression below:

By Tresca yield criterion, the stress required for a body to yield in shear is half the amount of that required for a body to yield in tension.

# Von Mises Yields Criterion

Von Mises yield criterion is devised based on distortion energy theory. According to this criterion, yielding is initiated when the second deviatoric invariant exceeds the threshold value of material.

The equation below is obtained by substituting the expressions of *I _{t}* and /2 (Eqs. (2.35) and (2.36)) into

*J*

*2*(Eq. (2.47)):

After expanding the equation above it becomes:

Simplify the equation above yields the follows:

Express the equation above in terms of principal stresses, where *a _{x}* =

_{y}=

*a-*= <тч and r

_{vv}= r

_{vz}=

*t*0 and we get:

_{xz}=

Factorization of the equation above with | gives us:

Rearrange the equation above yields the follows:

Factorise each grouped quadratic equation leads to this:

Let *к* be the threshold value that defines whether the developed shear stress is enough to make the body initiate yielding. Also, let *a* and *a _{2}* be the coupled shear stresses that achieve threshold value:

At yielding point, by applying the relationship in Eqs. (6.3) to (6.2) becomes: Simplify the equation above and we get:

_{2} =

By simplifying the equation above we obtain:

Equalise the Eqs. (6.4) and (6.5) yields the follows:

Square root both sides in the equation above results in the following expression:

By Von Mises yield criterion, the stress required for a body to yield in shear is times of the amount required for a body to yield in tension.