Learning topic
Strength Theories (Failure Criteria)
Compare failure criteria for multiaxial stress: Tresca maximum shear stress, von Mises distortion energy, Mohr-type criteria, and equivalent stress.
Under a multiaxial stress state, a material is subjected simultaneously to several stress components. Failure criteria reduce such a state to an equivalent measure that can be compared with material strength obtained from simpler tests.
Why an equivalent stress is needed
A single principal stress does not always represent the severity of a complex stress state. For ductile metals, differences between principal stresses and the deviatoric part of the stress state are especially important. This motivates the widespread use of the Tresca and von Mises criteria.
For ductile isotropic materials under a multiaxial stress state, the Tresca and von Mises criteria are widely used.
Tresca:
$$\sigma_{\mathrm{eq,T}}=\max\left(|\sigma_1-\sigma_2|,|\sigma_2-\sigma_3|,|\sigma_3-\sigma_1|\right).$$
von Mises:
$$\sigma_{\mathrm{eq,VM}}=\sqrt{\frac{(\sigma_1-\sigma_2)^2+(\sigma_2-\sigma_3)^2+(\sigma_3-\sigma_1)^2}{2}}.$$
- $\sigma_1$, $\sigma_2$, $\sigma_3$ — principal stresses;
- $\sigma_{\mathrm{eq,T}}$ — Tresca equivalent stress;
- $\sigma_{\mathrm{eq,VM}}$ — von Mises equivalent stress.
In allowable-stress design, the corresponding equivalent stress is compared with an allowable value consistent with the material properties and the adopted design method.
Tresca criterion
The maximum-shear-stress criterion associates yielding with the largest difference between principal stresses. It is straightforward for hand calculations and is generally somewhat more conservative than von Mises for many loading states.
von Mises criterion
The distortion-energy criterion associates yielding with the energy of shape change. It is widely used for ductile isotropic metals in engineering calculations and numerical analysis.
Plane stress
When $\sigma_z=0$, the von Mises equivalent stress can be written directly in terms of the plane-stress components:
$$\sigma_{\mathrm{eq,VM}}=\sqrt{\sigma_x^2-\sigma_x\sigma_y+\sigma_y^2+3\tau_{xy}^2}.$$
This form allows an assessment without first calculating the principal stresses.
Materials with different tensile and compressive strengths
For brittle materials, or materials whose tensile and compressive strengths differ substantially, other criteria may be more appropriate, including Mohr-type approaches. The selected relation and allowable material characteristics must follow the adopted code or calculation method.
Historical criteria
Educational courses also discuss the maximum-normal-stress and maximum-normal-strain criteria. They are useful for understanding the development of failure theories but are not universal criteria for modern design of ductile metals.
Example
For principal stresses $\sigma_1=100\ \text{MPa}$, $\sigma_2=40\ \text{MPa}$, and $\sigma_3=0$, Tresca gives $\sigma_{\mathrm{eq,T}}=100\ \text{MPa}$. Von Mises gives:
$$\sigma_{\mathrm{eq,VM}}=\sqrt{\frac{60^2+40^2+100^2}{2}}\approx87.2\ \text{MPa}.$$
The criterion must be selected according to the material, expected failure mechanism, and the rules of the specific design method.
About this topic
Strength theories (failure criteria) allow engineers to evaluate structural safety under complex multiaxial stress states by comparing them to simple uniaxial tensile testing. This section details maximum shear stress theory (Tresca criterion), distortion energy theory (von Mises criterion), and Mohr-Coulomb failure criterion for materials with asymmetric tensile and compressive strengths.