2.1 Introduction: Why Multi-Axial Yield Criteria Matter

In Chapter 1 we established that real machine elements — gear roots, crankshaft fillets, pressure-vessel walls, bolt threads, and welded joints — carry three-dimensional stress states. A uniaxial yield strength $S_y$ measured from a tension coupon answers only one question: *when does a bar in simple tension yield?* It does not tell you when a shaft under combined bending and torsion yields, when a thick-walled cylinder yields under triaxial pressure, or when a soil foundation fails under confining pressure.

A yield criterion is a scalar function $f(\boldsymbol{\sigma})$ that partitions stress space into an elastic domain ($f<0$) and a plastic domain ($f\ge 0$). The yield surface $f=0$ is a closed convex surface in principal-stress space (or invariant space). For ductile isotropic metals, calibration is typically done from a single uniaxial tension test: $S_y$ is the yield stress at which $f$ first reaches zero.

This chapter develops the principal criteria used in machine design and failure analysis:

1. Tresca (maximum shear stress) — conservative hexagon in plane stress.
2. von Mises (distortion energy / $J_2$) — ellipse, standard for ductile metals.
3. Mohr–Coulomb and Drucker–Prager — pressure-sensitive criteria for soils, concrete, and brittle materials.
4. Limit analysis — bounding the collapse load of structures once plasticity spreads.
5. Safety factor design — translating yield surfaces into design checks.

2.2 Hydrostatic and Deviatoric Stress Decomposition

Any Cauchy stress tensor decomposes uniquely into a hydrostatic (volumetric) part and a deviatoric (shape-changing) part:

$$\boldsymbol{\sigma} = \sigma_m \mathbf{I} + \mathbf{s}, \qquad \sigma_m = \frac{1}{3}\mathrm{tr}\,\boldsymbol{\sigma} = \frac{1}{3}(\sigma_1+\sigma_2+\sigma_3).$$

The deviatoric tensor is $\mathbf{s} = \boldsymbol{\sigma} - \sigma_m\mathbf{I}$, with $\mathrm{tr}\,\mathbf{s}=0$.

2.2.1 Physical meaning

• Hydrostatic stress $\sigma_m$ changes volume (dilatation) but not shape. Under pure hydrostatic compression, ductile metals can sustain enormous pressures without yielding (Bridgman experiments).
• Deviatoric stress $\mathbf{s}$ distorts the lattice and provides the resolved shear stresses that drive dislocation slip — the microscopic mechanism of plastic yielding in metals.

2.2.2 Deviatoric invariants

The second deviatoric invariant is

$$J_2 = \frac{1}{2}s_{ij}s_{ij} = \frac{1}{6}\big[(\sigma_1-\sigma_2)^2 + (\sigma_2-\sigma_3)^2 + (\sigma_3-\sigma_1)^2\big].$$

The third deviatoric invariant $J_3 = \det(\mathbf{s})$ controls the Lode angle $\theta$, which distinguishes axisymmetric tension from pure shear at the same $J_2$ level. Tresca and von Mises differ only in how they treat the Lode-angle dependence.

2.2.3 Implication for yield criteria

Valid isotropic metal yield criteria for ductile materials depend only on $J_2$ (and hence are independent of hydrostatic stress). Pressure-sensitive materials (concrete, rock, cast iron in tension vs compression) require criteria that include $\sigma_m$ or $I_1$.

2.3 Tresca Criterion (Maximum Shear Stress)

The Tresca (or Guest) criterion states that yielding occurs when the maximum shear stress in the material reaches the value attained at yield in uniaxial tension:

$$\tau_{\max} = \frac{S_y}{2}.$$

In terms of ordered principal stresses $\sigma_1 \ge \sigma_2 \ge \sigma_3$:

$$\boxed{\max\big(|\sigma_1-\sigma_2|,\; |\sigma_2-\sigma_3|,\; |\sigma_3-\sigma_1|\big) = S_y}$$

Equivalently, yielding occurs when any of the three pairwise differences equals $S_y$.

2.3.1 Plane stress ($\sigma_3 = 0$)

The yield locus in the $\sigma_1$–$\sigma_2$ plane is a regular hexagon with vertices at $(\pm S_y, 0)$, $(0, \pm S_y)$, and $(\pm S_y/2, \pm S_y/2)$ along the symmetry lines. The hexagon is inscribed in the von Mises ellipse.

2.3.2 Properties

• Piecewise linear — easy to implement in limit analysis and hand calculations.
• Always conservative relative to von Mises for the same $S_y$ calibration (Tresca lies inside or on the von Mises surface).
• Maximum deviation (~15.5%) occurs at pure shear ($\sigma_1 = -\sigma_3$, $\sigma_2 = 0$): Tresca predicts yield at $\tau = S_y/2$ while von Mises at $\tau = S_y/\sqrt{3}$.

2.4 von Mises Criterion (Distortion Energy / $J_2$ Flow)

The von Mises (Huber–Hencky–von Mises) criterion equates the distortion strain energy at yield to that in uniaxial tension:

$$\boxed{\bar{\sigma}_{\mathrm{vm}} = \sqrt{\frac{3}{2}s_{ij}s_{ij}} = \sqrt{3J_2} = S_y}$$

In principal form:

$$\bar{\sigma}_{\mathrm{vm}} = \sqrt{\frac{1}{2}\big[(\sigma_1-\sigma_2)^2 + (\sigma_2-\sigma_3)^2 + (\sigma_3-\sigma_1)^2\big]} = S_y.$$

2.4.1 Plane stress form

With $\sigma_3 = 0$:

$$\sigma_1^2 - \sigma_1\sigma_2 + \sigma_2^2 = S_y^2.$$

This is an ellipse in the $\sigma_1$–$\sigma_2$ plane, circumscribing the Tresca hexagon.

2.4.2 Design practice

For ductile steels, aluminum alloys, and copper, von Mises is the default criterion in FEA post-processors, ASME pressure vessel codes (with adjustments), and aerospace structures. It correlates well with experimental data for polycrystalline metals under proportional loading.

2.5 Comparison of Tresca and von Mises

For combined bending and torsion (shafts), both reduce to checking $\sqrt{\sigma^2 + 3\tau^2} \le S_y$ (von Mises) vs $\sqrt{\sigma^2 + 4\tau^2} \le S_y$ (Tresca). The difference is typically 10–15% in safety factor at the same stress state.

Forensic note: If a failed part shows shear lips at 45° to the principal tension direction, the stress state was likely near pure shear — the largest Tresca–von Mises discrepancy occurs here.

2.6 Mohr–Coulomb Criterion (Pressure-Sensitive Materials)

Brittle materials (concrete, rock, cast iron in compression-dominated states) and granular media (soils) exhibit different strengths in tension and compression. The Mohr–Coulomb criterion is a piecewise linear envelope in the Mohr plane:

$$\boxed{\tau = c + \sigma_n \tan\phi}$$

where $c$ is cohesion and $\phi$ is the angle of internal friction.

2.6.1 Principal-stress form

In 3D, Mohr–Coulomb can be written as:

$$f = \sigma_1 - \sigma_3 N_\phi + 2c\sqrt{N_\phi} = 0, \qquad N_\phi = \frac{1+\sin\phi}{1-\sin\phi}.$$

For frictionless materials ($\phi=0$): $f = \sigma_1 - \sigma_3 - 2c = 0$, which reduces to Tresca with $S_y = 2c$.

2.6.2 Cast iron and concrete

Cast iron has low tensile strength but high compressive strength. A simplified approach uses Rankine (maximum principal stress) for tension: yield when $\sigma_1 = S_{ut}$, and Mohr–Coulomb or maximum principal compression for the compressive limit. Never apply von Mises alone to gray cast iron without verifying the tensile principal is below $S_{ut}$.

2.7 Drucker–Prager Criterion (Smooth Mohr–Coulomb Approximation)

Mohr–Coulomb has corner singularities at the hexagonal vertices in deviatoric plane, causing numerical difficulties in FEA plasticity. The Drucker–Prager criterion replaces the hexagon with a smooth cone:

$$\boxed{f = \sqrt{J_2} + \alpha I_1 - k = 0}$$

where $I_1 = \sigma_1+\sigma_2+\sigma_3 = 3\sigma_m$.

2.7.1 Calibration to Mohr–Coulomb

Two common matchings:

• Inscribed cone (conservative): $\alpha = \frac{2\sin\phi}{\sqrt{3}(3-\sin\phi)}$, $k = \frac{6c\cos\phi}{\sqrt{3}(3-\sin\phi)}$.
• Circumscribed cone (less conservative): $\alpha = \frac{2\sin\phi}{\sqrt{3}(3+\sin\phi)}$.

Drucker–Prager is standard in geotechnical FEA (ABAQUS, PLAXIS) and concrete dam analysis.

2.8 Safety Factor Design with Yield Criteria

The factor of safety against yielding is defined as the ratio of the limiting stress (from the criterion) to the applied equivalent stress:

$$n = \frac{S_y}{\bar{\sigma}_{\mathrm{vm}}} \quad \text{(von Mises)}$$

or for Tresca:

$$n = \frac{S_y}{\sigma_{\mathrm{Tresca}}}, \qquad \sigma_{\mathrm{Tresca}} = \max(|\sigma_1-\sigma_2|, |\sigma_2-\sigma_3|, |\sigma_3-\sigma_1|).$$

2.8.1 Design allowable

The design allowable is $\sigma_{\mathrm{allow}} = S_y/n_{\mathrm{design}}$. Typical design factors for ductile machine parts: $n = 1.5$–$2.5$ depending on consequence of failure, load uncertainty, and fatigue interaction.

2.8.2 Combined loading — shaft example

A solid shaft of diameter $d$ under bending moment $M$ and torque $T$ experiences at the outer fiber:

$$\sigma = \frac{32M}{\pi d^3}, \qquad \tau = \frac{16T}{\pi d^3}.$$

von Mises: $\bar{\sigma}_{\mathrm{vm}} = \sqrt{\sigma^2 + 3\tau^2} \le \frac{S_y}{n}$.

This is the workhorse formula for shaft sizing in Shigley and machine design texts.

2.9 Introduction to Limit Analysis

Limit analysis (plastic collapse analysis) asks: *what is the maximum load a structure can carry once a plastic mechanism has formed?* Unlike elastic stress checks, limit analysis accounts for stress redistribution after local yielding.

2.9.1 Lower bound theorem (static theorem)

If a stress field is in equilibrium, satisfies traction boundary conditions, and nowhere violates the yield criterion, then the associated load is a lower bound on the collapse load.

2.9.2 Upper bound theorem (kinematic theorem)

If a kinematically admissible velocity field is chosen and the rate of external work equals the rate of internal plastic dissipation, the associated load is an upper bound on the collapse load.

2.9.3 Applications

• Thick-walled cylinders under internal pressure: autofrettage design.
• Bolted joints: bearing yield lines around holes.
• Beams: plastic hinge formation and collapse mechanisms.
• Soil mechanics: bearing capacity of foundations (Terzaghi, Prandtl mechanisms).

For machine design, limit analysis provides the ultimate capacity while elastic yield criteria provide the serviceability limit (no permanent deformation).

2.10 FEA Implementation and Common Pitfalls

Modern FEA solvers offer von Mises, Tresca, and Mohr–Coulomb/Drucker–Prager as post-processing and material models. Key pitfalls:

1. Using $S_y$ from tension for a compressive-dominant state in brittle materials — use Mohr–Coulomb or separate tensile/compressive limits.
2. Ignoring stress concentrations — yield criteria apply to local hotspot stresses, not nominal section averages.
3. Confusing von Mises stress with a principal — $\bar{\sigma}_{\mathrm{vm}}$ is a scalar invariant, not a physical stress component.
4. Plane stress vs plane strain — thick sections constrain lateral strain, raising effective yield (plane strain is ~15% higher than plane stress for von Mises).
5. Non-proportional loading — yield surfaces are calibrated for proportional loading; cyclic non-proportional paths may require kinematic hardening models (Ch.4).

2.11 Summary and Selection Guide

Design takeaway: Always reduce the 3D stress state to principals, compute the appropriate equivalent stress, and compare to $S_y/n$. For pressure-sensitive materials, never use von Mises alone — include the mean stress effect via Mohr–Coulomb or Drucker–Prager.