8.2 Yield criteria and failure theories

Yield Criteria and Failure Theories

Yield criteria and failure theories provide the mathematical framework for predicting when a material will begin to deform permanently or fracture under load. Without them, you'd have no systematic way to check whether a complex, real-world stress state is safe or dangerous.

These tools connect what you already know about stress analysis (Mohr's circle, principal stresses) to actual design decisions. The core question is always the same: given a multi-axis stress state, how do you compare it to a simple uniaxial yield strength from a tensile test?

Yield Criteria for Material Failure

Concept and Role of Yield Criteria

A yield criterion is a mathematical rule that takes a general stress state and tells you whether the material has started to yield (deform plastically). The problem it solves is straightforward: tensile test data gives you yield strength under uniaxial loading, but real components experience combined stresses in multiple directions. Yield criteria bridge that gap.

• The yield strength ($\sigma_y$) is the uniaxial stress at which plastic deformation begins. Every yield criterion ultimately compares some function of the applied stresses back to this single value.
• Yield criteria work with principal stresses, which strip away shear and let you evaluate the most critical normal stress combinations.
• The right criterion to use depends on whether the material is ductile or brittle, and on the type of loading (uniaxial, biaxial, or triaxial).

Common Yield Criteria

Three yield criteria appear most often:

• Tresca (maximum shear stress criterion)
• von Mises (maximum distortion energy criterion)
• Drucker-Prager (a pressure-dependent extension, used for soils and some polymers)

Each has different assumptions about what physical mechanism drives yielding. Tresca focuses on shear stress alone; von Mises accounts for the total distortion energy from all three principal stress differences; Drucker-Prager adds sensitivity to hydrostatic pressure.

Tresca vs. von Mises Yield Criteria

Tresca Yield Criterion

The Tresca criterion says yielding begins when the maximum shear stress in the material equals the shear stress at yielding in a simple tensile test. Since the maximum shear stress in uniaxial tension is $\frac{\sigma_y}{2}$, the criterion becomes:

$\max(|\sigma_1 - \sigma_2|,\; |\sigma_2 - \sigma_3|,\; |\sigma_3 - \sigma_1|) = \sigma_y$

where $\sigma_1$, $\sigma_2$, and $\sigma_3$ are the principal stresses.

• Geometrically, the Tresca yield surface is a hexagonal prism in principal stress space.
• It's the more conservative of the two criteria: if Tresca says you're safe, von Mises will agree.
• Tresca is best suited for ductile materials where shear-driven slip is the dominant yielding mechanism.

von Mises Yield Criterion

The von Mises criterion says yielding begins when the distortion energy (the portion of strain energy that changes shape, not volume) reaches the value it would have at yielding in uniaxial tension:

$(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2 = 2\sigma_y^2$

An equivalent and often more convenient form defines the von Mises equivalent stress:

$\sigma_v = \sqrt{\frac{1}{2}\left[(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2\right]}$

Yielding occurs when $\sigma_v \geq \sigma_y$.

• Geometrically, the von Mises yield surface is a cylinder in principal stress space (it circumscribes the Tresca hexagon).
• It fits experimental data for most ductile metals better than Tresca because it accounts for the combined effect of all three principal stress differences, not just the largest one.
• It applies to isotropic, ductile materials with roughly equal tensile and compressive yield strengths.

Applying Tresca and von Mises Criteria

Here's the step-by-step process for checking whether yielding occurs:

1. Determine the stress state. Identify all normal and shear stress components ($\sigma_x$, $\sigma_y$, $\tau_{xy}$, etc.) at the point of interest.

2. Find the principal stresses. Use Mohr's circle or the eigenvalue method to compute $\sigma_1$, $\sigma_2$, and $\sigma_3$. Order them so that $\sigma_1 \geq \sigma_2 \geq \sigma_3$.

3. Substitute into the criterion.

   • Tresca: Compute $|\sigma_1 - \sigma_3|$ and compare to $\sigma_y$.
   • von Mises: Compute $\sigma_v$ and compare to $\sigma_y$.

4. Interpret the result. If the criterion value meets or exceeds $\sigma_y$, the material is predicted to yield at that point.

Quick example: A steel shaft ($\sigma_y = 250 \text{ MPa}$) has principal stresses $\sigma_1 = 200 \text{ MPa}$, $\sigma_2 = 50 \text{ MPa}$, $\sigma_3 = -30 \text{ MPa}$.

• Tresca: $|\sigma_1 - \sigma_3| = |200 - (-30)| = 230 \text{ MPa}$. Since $230 < 250$, no yielding predicted.
• von Mises: $\sigma_v = \sqrt{\frac{1}{2}[(200-50)^2 + (50-(-30))^2 + ((-30)-200)^2]}$ $= \sqrt{\frac{1}{2}[22500 + 6400 + 52900]} = \sqrt{40900} \approx 202 \text{ MPa}$. Since $202 < 250$, no yielding predicted.

Notice that von Mises gives a lower equivalent stress than Tresca here, which is typical. Tresca is always equal to or more conservative than von Mises.

Principal Stresses and Failure Theories

Principal Stresses

Principal stresses are the normal stresses on planes where shear stress is zero. They represent the extreme values of normal stress at a point.

• The three principal stresses are ordered $\sigma_1 \geq \sigma_2 \geq \sigma_3$, where $\sigma_1$ is the maximum and $\sigma_3$ is the minimum.
• You find them using Mohr's circle (graphical) or by solving the characteristic equation of the stress tensor (analytical).
• Every yield and failure criterion is written in terms of principal stresses because they capture the most critical stress combinations regardless of how you originally set up your coordinate system.

Relationship to Failure Theories

Principal stresses feed directly into failure theories in different ways:

• Tresca uses the difference $\sigma_1 - \sigma_3$ (the largest principal stress difference, which equals twice the maximum shear stress).
• von Mises uses all three differences symmetrically.
• Maximum normal stress (Rankine) compares $\sigma_1$ directly to the ultimate tensile strength, or $|\sigma_3|$ to the ultimate compressive strength.
• Mohr-Coulomb uses both $\sigma_1$ and $\sigma_3$ along with separate tensile and compressive strengths.

One important physical insight: the hydrostatic stress, defined as $\sigma_h = \frac{\sigma_1 + \sigma_2 + \sigma_3}{3}$, changes volume but does not cause shape distortion. For most metals, hydrostatic stress alone does not cause yielding. That's exactly why the von Mises criterion works well for metals: it isolates the distortion (shape-changing) part of the stress state.

Selecting Failure Theories for Materials

Factors Influencing Failure Theory Selection

Choosing the right failure theory comes down to matching the theory's assumptions to your actual situation:

• Material behavior: Is it ductile (significant plastic deformation before fracture) or brittle (little to no plastic deformation)?
• Strength symmetry: Does the material have similar strengths in tension and compression, or are they very different?
• Loading type: Is the load static, or does it cycle repeatedly (fatigue)?
• Isotropy: Does the material behave the same in all directions, or is it anisotropic (like composites or wood)?

Failure Theories for Different Materials and Loading Conditions

Ductile materials (most metals: steel, aluminum, copper):

• Use the von Mises criterion as the default. It matches experimental results well for isotropic metals with similar tensile and compressive yield strengths.
• Tresca is a simpler, slightly more conservative alternative. Some design codes prefer it for that reason.

Brittle materials (ceramics, cast iron, concrete, some polymers):

• Use the Maximum Normal Stress (Rankine) criterion. Failure is predicted when $\sigma_1$ reaches the ultimate tensile strength ($S_{ut}$) or when $|\sigma_3|$ reaches the ultimate compressive strength ($S_{uc}$).
• The Mohr-Coulomb criterion is better when tensile and compressive strengths differ significantly (which is common for brittle materials). It accounts for the interaction between normal stress and shear stress on the failure plane.

Cyclic or fatigue loading:

• Static yield criteria don't apply directly. Instead, use fatigue-specific theories like the Goodman criterion or Soderberg criterion, which account for both the mean stress and the alternating stress amplitude over many cycles.

Anisotropic materials (fiber composites, rolled metals, wood):

• Standard isotropic criteria won't work. Use the Hill criterion (for metals with directional properties from rolling) or the Tsai-Wu criterion (widely used for fiber-reinforced composites).

Importance of Validation

No failure theory is perfect. Each one is a model built on simplifying assumptions. Whenever possible, validate your chosen theory against experimental data for the specific material and loading you're designing for. If test data shows the theory is non-conservative (predicts safety when failure actually occurs), you need a different theory or an added safety factor.