Stress, Strain, and the von Mises Yield Criterion
Table of Contents
1. What Stress and Strain Really Mean
Stress and strain are the language of structural FEM. Every contour plot you see in Abaqus, Ansys, or NASTRAN is showing you one or more components of the stress and strain field. Understanding what these quantities actually mean physically — not just mathematically — is essential for interpreting results correctly and catching modeling errors.
The key insight: stress and strain are defined at a point, not averaged over a volume. FEM computes stresses at Gauss integration points inside each element and extrapolates to nodes. The "nodal averaged stress" you see in contour plots is a post-processed quantity — differences between neighboring element Gauss point values tell you about mesh quality.
In my FEM results I see von Mises stress everywhere. What is it really? I know it's like the "effective stress" but I've never understood where it comes from.
Von Mises stress is a single number that combines all six stress components — σ_xx, σ_yy, σ_zz, τ_xy, τ_yz, τ_xz — into one "effective" value you can compare directly to your material's yield strength from a uniaxial tensile test. The physical idea: yielding in metals is driven by shear stress on slip planes, not hydrostatic pressure. Von Mises captures the "distortional energy" that drives plastic deformation. When von Mises stress = tensile yield strength from your coupon test, the material is at the point of yielding regardless of the stress state — pure tension, pure shear, or any combination. It's brilliant because it lets you use one simple test result to predict yielding in arbitrarily complex 3D stress states.
2. Normal Stress: σ = F/A
Normal stress acts perpendicular to a surface. Tensile stress is positive; compressive stress is negative (by the standard mechanics convention):
This definition assumes $F$ is uniformly distributed over area $A$. In reality, stress concentrations cause local peaks that can exceed the average by factors of 2–10. The key concept: FEM resolves this spatial variation by computing stress at many points, while hand calculations only give the average.
Stress Concentration Factor Kt
The ratio of peak local stress to nominal stress: $\sigma_{\text{max}} = K_t \cdot \sigma_{\text{nominal}}$. For a circular hole in a wide plate: $K_t = 3$. This means the stress at the edge of a small circular hole is 3× the applied stress far from the hole — FEM naturally captures this without needing a separate $K_t$ lookup, as long as the mesh is fine enough near the hole.
| Geometry | Stress Concentration Kt | Location of Peak Stress |
|---|---|---|
| Circular hole in infinite plate (tension) | 3.0 | Hole edge, perpendicular to load |
| Sharp notch (ρ → 0) | → ∞ (theoretical) | Notch root (use fracture mechanics) |
| Fillet radius r/d = 0.1 (shaft shoulder) | ~2.5 | Fillet root |
| Fillet radius r/d = 0.3 | ~1.5 | Fillet root |
| Elliptical hole (a/b = 2, tension) | 5.0 | Hole end aligned with load direction |
3. Strain: ε = ΔL/L
Normal strain is the fractional change in length — dimensionless:
For engineering applications (small strains), this is the familiar engineering strain. For large deformations (rubber, soft tissue, crash), the logarithmic (true) strain is preferred:
In Abaqus, the strain output variable LE (logarithmic strain) is used for large deformation analyses. For small strains (<1–2%), engineering strain and logarithmic strain are essentially identical. For large plastic strains typical in forming (30–100%), the difference is significant.
4. The Stress-Strain Curve
The stress-strain curve from a tensile test encodes all material behavior used in FEM:
- Elastic region: Linear slope = Young's modulus $E$
- Yield point $\sigma_y$: End of elastic behavior; where plasticity begins
- Strain hardening: Plastic region slope = tangent modulus $E_T$ (much smaller than $E$)
- Ultimate tensile strength $\sigma_{UTS}$: Peak load; necking begins after this
- Fracture strain $\varepsilon_f$: Elongation to fracture (ductility)
| Material | E (GPa) | σy (MPa) | σUTS (MPa) | εf (%) |
|---|---|---|---|---|
| Mild steel (S235) | 210 | 235 | 360–510 | 26 |
| High-strength steel (S690) | 210 | 690 | 770–940 | 14 |
| Aluminum 6061-T6 | 69 | 276 | 310 | 12 |
| Titanium Ti-6Al-4V | 114 | 880 | 950 | 14 |
| CFRP (0° uniaxial) | 135 | N/A | 1500 (T) | 1.1 |
5. Shear Stress and Shear Strain
Shear stress acts parallel to a surface. For a simple shear scenario:
The shear modulus $G$ relates shear stress to shear strain:
For steel: $G = 210/(2 \times 1.3) = 80.8$ GPa. Shear stiffness is lower than axial stiffness by the factor $2(1+\nu)$.
Maximum Shear Stress in Practice
The maximum shear stress in a 3D stress state: $\tau_{\max} = (\sigma_1 - \sigma_3)/2$ (Tresca criterion). This is the driving force for crystal slip and plastic deformation. The Tresca criterion ($\sigma_{\text{eff}} = \sigma_1 - \sigma_3 \leq \sigma_y$) is slightly conservative compared to von Mises and is used in some pressure vessel codes (ASME Boiler Code).
6. The Full Stress Tensor in 3D
At any point in a 3D body, the complete stress state requires six independent components (the stress tensor is symmetric: $\tau_{ij} = \tau_{ji}$):
In FEM output, these are typically reported as S11, S22, S33, S12, S13, S23 (Abaqus notation) or SX, SY, SZ, SXY, SXZ, SYZ. Understanding which component corresponds to which physical direction requires knowing your model's coordinate system orientation — a common source of errors in post-processing.
7. Von Mises Yield Criterion
The von Mises equivalent (or effective) stress:
Yielding occurs when $\sigma_{\text{vM}} \geq \sigma_y$. In terms of principal stresses $\sigma_1, \sigma_2, \sigma_3$:
Physical Interpretation: Distortional Energy
Von Mises stress is proportional to the square root of the distortional (deviatoric) strain energy density — the strain energy that causes shape change, not volume change. Since yielding in ductile metals is driven by dislocation slip (shear-based shape change), and hydrostatic pressure doesn't cause yielding (a material under 1 GPa of equal hydrostatic compression still hasn't yielded), von Mises criterion captures the physics correctly.
My FEM shows a von Mises stress of 280 MPa at a notch root, and the material yield strength is 350 MPa. Does that mean I'm safe? The actual stress might be higher than the mesh is showing, right?
Good question — and you're right to be suspicious. If the mesh at the notch root is coarse, the FEM significantly underestimates the peak stress. A key test: check if the stress field is converged. Refine the mesh at the notch by 2× and see if the peak stress changes. If it jumps from 280 to 350 MPa, your original mesh was too coarse. For sharp notches (very small fillet radius), theoretical stress approaches infinity — at some point, fracture mechanics using stress intensity factors K_I is more appropriate than a von Mises yield check. The transition typically occurs when the plastic zone at the notch root is small compared to other dimensions — below about 1 mm root radius, switch to fracture mechanics.
8. Principal Stresses and Mohr's Circle
Principal stresses are the normal stresses on planes where shear stress is zero — they represent the maximum and minimum normal stresses at a point. For 2D (plane stress):
Mohr's Circle is a graphical construction representing the transformation of stress components with rotation of axes. The circle's center is at $(\sigma_{xx}+\sigma_{yy})/2$ and radius $R = \sqrt{((\sigma_{xx}-\sigma_{yy})/2)^2 + \tau_{xy}^2}$. Principal stresses are at the leftmost and rightmost points of the circle; maximum shear stress is at the top.
In FEM post-processing, principal stresses (S1, S2, S3 in Abaqus) are useful for understanding crack propagation (cracks grow perpendicular to the maximum principal tensile stress) and for interpreting composite material failures where orientation matters.
When should I look at principal stresses vs. von Mises stress in my FEM results?
Use von Mises for ductile metal yielding and fatigue — it's the right metric for those failure modes. Use maximum principal stress (S1) for brittle materials like ceramics, cast iron, and concrete — these fracture by tensile cleavage and the maximum tensile stress governs failure. For composite materials, you need to check stress in each fiber and matrix direction against their respective allowables — neither von Mises nor principal stress alone is sufficient. For fatigue crack growth direction, look at the orientation of maximum principal stress: cracks grow perpendicular to the maximum tensile principal stress direction. This tells you where to look on weld toes, holes, and notches for crack initiation.
9. Cross-Topics
| Topic | Connection | Link |
|---|---|---|
| Hooke's Law | σ=Eε is Hooke's Law in continuum form — the basis of linear elastic FEM | Springs & Hooke's Law |
| Fluid Pressure | Hydrostatic pressure is a special case of isotropic stress state | Fluid Pressure and Buoyancy |
| Thermal Expansion | Thermal strain = αΔT; constrained thermal expansion causes thermal stress | Thermal Expansion |
| Work & Energy | Strain energy density = ½σ:ε — drives topology optimization | Work and Energy |
| Materials Mechanics | Plasticity models, fatigue criteria, fracture mechanics | Materials Mechanics |
| Mohr's Circle Tool | Interactive Mohr's circle for any 2D stress state | Mohr's Circle Calculator |