Samer Adeeb

Stress: Stress Based Failure Criteria

In traditional engineering materials (e.g. steel and traditional concrete), the strength of the material is isotropic, i.e., independent of the direction of the applied load. When designing a component using such materials, a measure of the maximum stress applied throughout the lifetime of the component is guaranteed by the designer to be less than or equal to a corresponding measure of the maximum strength of the material. While the stress matrix has six independent components, an isotropic stress based failure criterion is based on one number, usually a real valued function of these six components. A natural choice of the measure of the maximum stress is one of the stress invariants studied previously. The real valued function of the six stress components will either give me a number indicating that the stress state is safe or that the stress state is critical. If we imagine the six dimensional vector space of the components \sigma_{11}, \sigma_{22}, \sigma_{33}, \sigma_{12}, \sigma_{13} and \sigma_{23}, then, the critical state of stress can be viewed as a surface. This surface is termed the “Yield Surface“. A point inside the surface represents a non-critical state of stress, while a point on the surface represents a critical state of stress. A point outside the surface indicates the hypothetical case that the stress is higher than the failure stress.
In the following, different isotropic failure criteria (yield surfaces) are presented.

Maximum Shear Stress (Tresca)

The Tresca or maximum shear stress criterion states that a metal will yield when the maximum shear stress at any point reaches a maximum (critical) positive value termed \tau_{\max}\in\mathbb{R}^+. As shown in the maximum shear stress section, given a state of stress, the value of the maximum shear stress is half the maximum difference between the eigenvalues (principal stresses) of the stress matrix. Given \sigma \in \mathbb{M}^3 with
eigenvalues \sigma_1, \sigma_2 and \sigma_3, then according to Tresca yield criterion, the material will fail or yield when:

    \[ \max\left\{{{|\sigma_1-\sigma_2|} \over 2},{{|\sigma_1-\sigma_3|} \over 2},{{|\sigma_2-\sigma_3|} \over 2}\right\}=\tau_{\max} \]

Where \tau_{\max} is a positive constant that is dependent on the type of material.

The material constant \tau_{\max} can be determined experimentally by performing a uniaxial state of stress test. In this case, \sigma_{11} is the only nonzero component in the stress matrix \sigma and the material will yield or fail when \sigma_{11} reaches the yield stress \sigma_y. Then:

    \[ \sigma= \left(\begin{matrix} \sigma_y & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}\right) \]

In this case, the eigenvalues of \sigma are \sigma_y, 0,0 and the material constant can be determined as follows:

    \[ \tau_{\max}=\max\left\{{{|\sigma_y-0|} \over 2},{{|\sigma_y-0|} \over 2},{{|0-0|} \over 2}\right\}={|\sigma_y|\over 2} \]

The Graphical Representation of the Tresca Yield Surface in a Two Dimensional (Plane) State of Stress

Let \sigma\in\mathbb{M}^3 and assume that one of the eigenvalues of \sigma, namely \sigma_3 is equal to zero. Therefore, the material will yield when:

    \[ \max\left\{{{|\sigma_1-\sigma_2|} \over 2},{{|\sigma_1-0|} \over 2},{{|\sigma_2-0|} \over 2}\right\}=\tau_{\max} \]

Rearranging yields:

    \[ \max\left\{{{|\sigma_1-\sigma_2|} },{{|\sigma_1|} },{{|\sigma_2|} }\right\}=2\tau_{\max} \]

Given a numerical value for \tau_{\max}, the above relationship can be viewed graphically as a “yield surface” on a plane whose horizontal and vertical axes represent \sigma_1 and \sigma_2 respectively (Figure 1a). A point inside the surface has coordinates \sigma_1 and \sigma_2 satisfying:

    \[ \max\left\{{{|\sigma_1-\sigma_2|} },{{|\sigma_1|} },{{|\sigma_2|} }\right\}<2\tau_{\max} \]

On the other hand, a point on the solid line in (Figure 1a) represents a critical point where the equality between the maximum shear stress and \tau_{\max} hold.

The Graphical Representation of the Tresca Yield Surface in a Three Dimensional State of Stress

Similarly, the relationship:

    \[ \max\left\{{{|\sigma_1-\sigma_2|} },{{|\sigma_1-\sigma_3|} },{{|\sigma_2-\sigma_3|} }\right\}=2\tau_{\max} \]

can be used to draw a graphical yield surface in a three dimensional space (Figure 1b) whose 3 axes represent the eigenvalues \sigma_1, \sigma_2 and \sigma_3. A point inside the surface represent a non-critical state of stress and its coordinates satisfy the following relationship:

    \[ \max\left\{{{|\sigma_1-\sigma_2|} },{{|\sigma_1-\sigma_3|} },{{|\sigma_2-\sigma_3|} }\right\}<2\tau_{\max} \]

while a point on the surface represents a critical point where the equality between the maximum shear stress and \tau_{\max} hold.

Remarks:
  • The yield surface when viewed along the vector u=\{1,1,1\} has the shape of a hexagon
  • The yield surface does not differentiate between tension and compression
  • The intersection of the three dimensional yield surface with the plane representing the axis \sigma_1 and \sigma_2 gives exactly the two dimensional yield surface presented in the above section
  • The yield surface has a cylindrical like shape with its longitudinal axis being the family of vectors of the form u=\{\alpha,\alpha,\alpha\} with \alpha\in\mathbb{R}. In other words, the line in the plane when the components are all equal, i.e., \sigma_1=\sigma_2=\sigma_3 is always inside the yield surface indicating that this yield criterion assumes that a material under a state of stress with equal eigenvalues will never fail! This can also be state as that with this failure criterion a material under any hydrostatic state of stress, tension or compression, will never fail. This can be concluded directly from the equation of the yield surface by substituting equal values for the eigenvalues of \sigma

Figure 1. Graphical representation of the Tresca yield criterion (a) state of plane stress, and (b) three dimensional state of stress

Figure 1. Graphical representation of the Tresca yield criterion (a) state of plane stress, and (b) three dimensional state of stress

Von Mises Yield Criterion (J2 Flow Theory)

The von Mises yield criterion states that a material will yield when the von Mises stress at any point reaches a maximum (critical) positive value termed \sigma_{\max}\in\mathbb{R}^+. Given \sigma\in\mathbb{M}^3, then according to the von Mises yield criterion, the material will yield when:

    \[ \sigma_{vM}=\sqrt{{(\sigma_{11}-\sigma_{22})^2+(\sigma_{11}-\sigma_{33})^2+(\sigma_{22}-\sigma_{33})^2+6\sigma_{12}^2+6\sigma_{13}^2+6\sigma_{23}^2}\over 2} =\sigma_{\max} \]

where \sigma_{\max} is a positive constant that is dependent on the type of material.

The material constant \sigma_{\max} can be determined experimentally by performing a uniaxial state of stress test. In this case, \sigma_{11} is the only nonzero component in the stress matrix \sigma and the material will yield or fail when \sigma_{11} reaches the yield stress \sigma_y. Then:

    \[ \sigma= \left(\begin{matrix} \sigma_y & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}\right) \]

In this case, the material constant can be determined as follows:

    \[ \sigma_{\max}=\sqrt{{(\sigma_y-0)^2+(\sigma_y-0)^2+(0-0)^2+6(0)^2+6(0)^2+6(0)^2}\over 2}=|\sigma_y| \]

The Graphical Representation of the von Mises Yield Surface in a Two Dimensional (Plane) State of Stress

Let \sigma\in\mathbb{M}^3 and assume that one of the eigenvalues of \sigma, namely \sigma_3 is equal to zero. Therefore, the material will yield when:

    \[ \sigma_{vM}=\sqrt{{(\sigma_1-\sigma_2)^2+(\sigma_1)^2+(\sigma_2)^2}\over 2}=\sqrt{\sigma_1^2-\sigma_1\sigma_2+\sigma_2^2} =\sigma_{\max} \]

Given a numerical value for \sigma_{\max}, the above relationship can be viewed graphically (Figure 2a) as a “yield surface” on a plane whose horizontal and vertical axes represent \sigma_1 and \sigma_2 respectively. A point inside the surface has coordinates \sigma_1 and \sigma_2 satisfying the following relationship:

    \[ \sigma_{vM}=\sqrt{\sigma_1^2-\sigma_1\sigma_2+\sigma_2^2} <\sigma_{\max} \]

On the other hand, a point on the solid line in Figure 2a represents a critical point where the equality between the von Mises stress \sigma_{vM} and \sigma_{\max} hold.

The Graphical Representation of the von Mises Yield Surface in a Three Dimensional State of Stress

Similarly, the relationship:

    \[ \sigma_{vM}=\sqrt{{(\sigma_{1}-\sigma_{2})^2+(\sigma_{1}-\sigma_{3})^2+(\sigma_{2}-\sigma_{3})^2}\over 2} =\sigma_{\max} \]

can be used to draw a graphical yield surface in a three dimensional space (Figure 2b) whose 3 axes represent the eigenvalues \sigma_1, \sigma_2 and \sigma_3. A point inside the surface represents a non-critical state of stress whose coordinates satisfy the following relationship:

    \[ \sigma_{vM}=\sqrt{{(\sigma_{1}-\sigma_{2})^2+(\sigma_{1}-\sigma_{3})^2+(\sigma_{2}-\sigma_{3})^2}\over 2} <\sigma_{\max} \]

while a point on the surface represents a critical point where the equality between \sigma_{vM} and \sigma_{\max} hold.

Remarks:
  • The von Mises yield criterion is sometimes called the maximum distortion energy criterion. This will make more sense after studying the energy of deformation section
  • The von Mises yield criterion is sometimes called the J2 flow theory or the J2 plasticity theory. This is because the von Mises stress is a function of the second invariant of the deviatoric stress tensor
  • There is a very subtle difference between the Tresca and von Mises yield criteria
  • The yield surface when viewed along the vector u=\{1,1,1\} has the shape of a circle whose radius is equal to \sqrt{2\over 3}\sigma_y. Can you reach this conclusion from the equations of the von Mises yield criterion?
  • The yield surface does not differential between tension and compression
  • The intersection of the three dimensional yield surface with the plane representing the axis \sigma_1 and \sigma_2 gives exactly the two dimensional yield surface presented in the previous section
  • The yield surface has a cylindrical like shape with its longitudinal axis being the family of vectors of the form u=\{\alpha,\alpha,\alpha\} with \alpha\in\mathbb{R}. In other words, the line in the plane when the components are all equal, i.e., \sigma_1=\sigma_2=\sigma_3 is always inside the yield surface indicating that this yield criterion assumes that a material under a state of stress with equal eigenvalues will never fail! This can also be state as that with this failure criterion a material under any hydrostatic state of stress, tension or compression, will never fail. This can be concluded directly from the equation of the yield surface by substituting equal values for the eigenvalues of \sigma

Figure 2. Graphical representation of the von Mises yield criterion (a) state of plane stress, and (b) three dimensional state of stress

Figure 2. Graphical representation of the von Mises yield criterion (a) state of plane stress, and (b) three dimensional state of stress

Simple Criterion for Brittle Materials

Another criterion that is more suitable for brittle materials that are also isotropic is to find the tensile strength in tension \sigma_t and the compressive strength in compression \sigma_c from unaxial tests. The material is safe when all the principal stresses \sigma_1, \sigma_2, and \sigma_3 satisfy the following relationship:

    \[ \forall i: \sigma_c<\sigma_i<\sigma_t \]

Other Isotropic Yield Criteria

The Tresca and von Mises yield criteria do not differentiate between tension and compression. These criteria are very successful in describing failure of metals, however for other materials, where failure depends on whether the material is “on average” under tension or compression require more sophisticated failure criteria. It was shown that the Tresca and the von Mises yield criteria give yield surfaces that have a cylindrical shape in the three dimensional vector space of the eigenvalues \sigma_1, \sigma_2 and \sigma_3. The longitudinal axis of this cylinder is the family of vectors \u=\{\alpha,\alpha,\alpha\} with \alpha\in\mathbb{R}. In other words, the von Mises and the Tresca yield criteria are independent of the hydrostatic stress. The next stage of yield criteria introduces expressions of the hydrostatic stress. Two prominent yield criteria are available:

Mohr-Coulomb Yield Criterion

This criterion predicts that a material will fail when the maximum shear stress reaches a critical value that is dependent on the average stress:

    \[ \left|{{\sigma_i-\sigma_j}\over 2}\right|=-{{\sigma_i+\sigma_j}\over 2} \sin\phi+c \cos\phi \]

where, \phi and c are material constants while \sigma_i and \sigma_j are two eigenvalues of \sigma with i\neq j. Note that setting \phi=0 and c=\tau_{\max} yields the Tresca yield criterion.

Drucker-Prager Yield Criterion

This criterion predicts that the material will fail when \sigma_{vM} reaches a critical value that is dependent on the hydrostatic (average) stress:

    \[ \sigma_{vM}=A+B I_1(\sigma) \]

where A and B are material constants and I_1(\sigma) is the first invariant of the stress matrix. Note that setting A=\sigma_{\max} and B=0 yields the von Mises yield criterion.

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