Yielding Criteria for Ductile Metals

At present, there are two generally accepted theories for predicting the onset of yielding in ductile metals:

1. Maximum Shear Stress Theory or Tresca Criterion
2. von Mises Criterion or Distortion Energy Criterion

1. Maximum-Shear-Stress Theory or Tresca Criterion

The maximum-shear-stress theory, sometimes called the Tresca yield criterion, states that yielding will occur when the maximum shear stress reaches a critical value k. In terms of principal stresses \sigma_1, \sigma_2, \sigma_3, we can write:

If \sigma_1\geq \sigma_2\geq \sigma_3, we previously showed that, the maximum shear stress is given by

For uniaxial tension \sigma_1 = \sigma_{yp}, \sigma_2 = \sigma_3 = 0, where \sigma_{yp} is the yield strength in simple tension. Therefore, the shearing yield stress for simple tension \tau_{yp} is equal to one-half of the tensile yield stress.

Substituting these values into the equation for the maximum shear stress results in

or

This is sometimes written as

where \sigma_1' and \sigma_3' are the deviators of the principal stresses and k is the yield stress for pure shear, i.e., the stress at which yielding occurs in torsion, where \sigma_1 = -\sigma_3.

The maximum-shear-stress theory is in good agreement with experimental results, being slightly on the safe side, and is widely used by designers for ductile metals.

The Tresca yielding condition is:

However, in certain plasticity problems this simple form cannot be used as the yielding condition since it is not known which of the three principal stresses is the largest. A proper yield function must not depend on the arbitrary labeling "1, 2, 3".

Tresca’s actual statement is:

Thus, instead of assuming a particular ordering, we must write a form that treats all three differences symmetrically.

Symmetric Algebraic Form of Tresca Criterion

Tresca requires that one (not necessarily all) of the following be equal to 2k:

A compact way to enforce the condition “at least one of these equals 2k” is to write

This expression is:

• symmetric in the principal stresses,
• equal to zero if any pair difference satisfies |\sigma_i - \sigma_j| = 2k.

Thus, (9) is a proper analytic representation of Tresca’s condition.

Reuss’ Invariant Form

To write the yield criterion in terms of stress invariants, Reuss transformed the symmetric form (9) into an invariant expression involving the second and third invariants of the deviatoric stress tensor:

• J_2 = \dfrac12 \sigma'_{ij}\sigma'_{ij} (second invariant of the deviatoric stress),
• J_3 = \det(\sigma'_{ij}) (third invariant of the deviatoric stress).

Reuss showed that the product expression (9) is equivalent to the polynomial:

This is a fully invariant form of the Tresca criterion, valid without assuming any ordering of the principal stresses. Obviously, such a complex relation will result in very cumbersome mathematics. It is for this reason that the yielding criterion that is discussed next is preferred in most theoretical work.

Plane Stress Problems

In plane stress problems where σ₃ = 0, the Tresca criterion, which states that yielding begins when the maximum shear stress reaches σ_yp/2, simplifies to:

max {|σ₁ – σ₂|, |σ₁|, |σ₂|}  =  σ_yp

Since we do not know if σ₁ or σ₂ is the largest principal stress, this expression takes different forms in different quadrants of the σ₁–σ₂ plane.

Quadrants I and III (Same Signs)
When σ₁ and σ₂ have the same sign, the difference |σ₁ – σ₂| is always smaller than either |σ₁| or |σ₂| (or both). Therefore, in the first and third quadrants, the expression max{|σ₁ – σ₂|, |σ₁|, |σ₂|} is equal to either |σ₁| or |σ₂|.

• First Quadrant: Both stresses are positive. Below the line σ₂ = σ₁ (where σ₁ > σ₂), the criterion gives σ₁ = σ_yp. Above this bisection line, we must have σ₂ = σ_yp.
• Third Quadrant: Both stresses are negative. By the same analogy, below the bisection line (where the magnitude of σ₁ is larger), we must have σ₁ = –σ_yp. Above the bisection, we must have σ₂ = –σ_yp.

Quadrants II and IV (Opposite Signs)
When the stresses have opposite signs, the term |σ₁ – σ₂| represents the sum of the absolute values, which is larger than either individual magnitude. Therefore, the difference term controls the yielding.

• Second Quadrant: σ₁ is negative and σ₂ is positive. Therefore:
  max{|σ₁ – σ₂|, |σ₁|, |σ₂|} = |σ₁ – σ₂| = σ₂ – σ₁ = σ_yp
  This describes a line connecting (–σ_yp, 0) to (0, σ_yp).

• Fourth Quadrant: σ₁ is positive and σ₂ is negative. Therefore:
  max{|σ₁ – σ₂|, |σ₁|, |σ₂|} = |σ₁ – σ₂| = σ₁ – σ₂ = σ_yp
  This describes a line connecting (σ_yp, 0) to (0, –σ_yp).

2. Von Mises, or Distortion-energy, Theory

In the previous section, we explained that the yield criterion for a material that is hydrostatic pressure insensitive, the yield surface takes the form

where J_2 and J_3 are the second and the third invariants of the stress deviator, C is a material constant. The simplest form of the above equation is J_2=C, which is often written as

The development of this yield criterion is associated with the names of Von Mises, Hencky, Maxwell, and Huber, and now is often known as the von Mises criterion. Von Mises proposed this criterion in the invariant form given by the above equation primarily because it was mathematically simpler than the invariant form of the maximum-shear-stress theory given by Eq. (10). Subsequent experiments showed that Eq. (11) provides better over-all agreement with combined stress-yielding data than the maximum-shear-stress theory.

From the uniaxial tension test, \sigma_1=\sigma_{\rm yp}, and J_2=\frac{1}{3}\sigma_{ \rm yp}^2. Therefore, from this test, the constant k is determined as

From a pure shear test, J_2=\tau_{\rm yp}^2, and thus

Therefore, using the von Mises criterion, we imply that the tensile and shear yield strengths of a ductile material are related through \tau_{ \rm yp}=\sigma_{ \rm yp}/\sqrt{3}\approx 0.577 \sigma_{ \rm yp}.

The equation in the box can thus be written as

• The above equation can also be written as

  \sigma_{ \rm yp} = \frac{1}{\sqrt{2}} [(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2]^{1/2} \tag{14}

• An equivalent way to express this criterion is by introducing the von Mises (equivalent) stress, \sigma_{\mathrm{v}} = \sqrt{3J_2},\tag{15} so that yielding occurs when \sigma_{\mathrm{v}} = \sigma_{ \rm yp}. Thus, the conditions J₂ = k² and σ_v  =  σ_yp are simply two different forms of the same yield criterion.

Physical Meaning

A number of attempts have been made to provide physical meaning to the von Mises yield criterion. One commonly accepted concept is that this yield criterion expresses the strain energy of distortion. On the basis of the distortion-energy concept, yielding will occur when the strain energy of distortion per unit volume exceeds the strain energy of distortion per unit volume for a specimen strained to the yield stress in uniaxial tension or compression. The derivation of Eq. (6) on the basis of distortion energy is given below. Another common physical interpretation of Eq. (6) is that it represents the critical value of the octahedral shear stress (discussed later).

The total elastic strain energy per unit volume (U_0=\frac{1}{2}\sigma_{ij}\epsilon_{ij}) can be divided into two components, the strain energy of distortion, U_d, and the strain energy of volume change, U_v.

We begin by decomposing the strain and stress tensors into their volumetric (mean) and deviatoric components:

where \delta_{ij} is the Kronecker tensor (\delta_{ij}=0 if i\neq j and \delta_{ij}=1 if i=j).

The mean (volumetric) strain and stress are defined as

and

with summation taken over repeated indices.

Because the traces of deviatoric tensors are zero, we have

since the trace of a deviatoric tensor is zero.

Recall that

where G=\frac{E}{2(1+\nu)} is the shear modulus and

where K=\frac{E}{3(1-2\nu)} is the bulk modulus.

Therefore,

and

Let's define

So U_0=U_v+U_d, and distortion energy is associated purely with shape change, while U_v​ is associated with volume change.

The maximum distortion energy states that the yielding begins when U_d reaches a critical value U_d^y equal to the distortion energy at yielding in a uniaxial test:

Therefore:

Plane Stress Problems (von Mises)

Similar to the Tresca criterion, the von Mises criterion simplifies significantly for cases of plane stress, where one principal stress is zero (let \sigma_3 = 0).

We start with the general von Mises criterion expressed in terms of principal stresses (Eq. 14):

Squaring both sides provides a slightly easier form to work with:

Substituting the plane stress condition \sigma_3 = 0 into this equation yields:

Dividing the entire equation by 2 results in the governing equation for von Mises yielding under plane stress:

This is the equation of an ellipse in the \sigma_1–\sigma_2 plane, with its major axis oriented at a 45° angle to the \sigma_1 and \sigma_2 axes.