Failure Criteria
Intuition
A machine part fails when it stops performing its intended function. This simple definition conceals most of the difficulty. Failure might mean yielding (permanent, plastic deformation that leaves the part out of spec), fracture (separation into pieces), buckling, fatigue at stresses below the static yield point, creep at elevated temperature, or excessive deflection under load. A failure criterion is a predictive rule — a scalar function of the stress (or strain) state whose crossing a threshold signals that a particular failure mode is imminent. It is a model of the material's limit, calibrated against experiment, and it is never the same thing as the physical mechanism of failure itself.[fn::The conflation of criterion with mechanism is the single most common conceptual error in undergraduate strength-of-materials. Tresca does not say that shear causes yield; it says that the maximum shear stress correlates with the onset of yielding well enough to be useful, which is a weaker and more defensible claim. The distinction matters because correlation-based criteria can be empirically excellent within their calibration domain and disastrous outside it.]
The central intellectual challenge is that real stress states are tensors — six independent components in 3D (or three principal values, once diagonalized) — while a go/no-go engineering decision is scalar. One must collapse a multi-axial state onto a single number and pick a threshold. Different collapse rules encode different physical hypotheses about what the material "cares about," and the choice of rule is, in the end, an empirical question settled by which one tracks the yield locus of a given material class.[fn::One could regard the entire field as an exercise in dimensionality reduction under adversarial conditions: the adversary being the material, which is under no obligation to make its failure locus convex, smooth, isotropic, or even a function of stress alone.]
This note concerns itself with the two criteria that dominate ductile-metal practice — von Mises (distortion energy) and Tresca (maximum shear stress) — and the brittle regime handled by Mohr-Coulomb. It also insists on the distinction between a failure criterion (a prospective, normative model: "under these conditions, this material will yield") and failure analysis (a retrospective, diagnostic activity: "this specific part, found in this specific field, failed because of X"). The two activities share vocabulary and tools but answer different questions; conflating them is an error that propagates into warranty claims and, occasionally, litigation.[fn::The prospective/retrospective distinction is borrowed, loosely, from the statistics literature on prediction versus inference. It maps cleanly here: a yield criterion is a classifier, failure analysis is forensics. Asking a classifier to do forensics — "von Mises says this shaft should have held" — is a category error, though engineers commit it constantly.]
First principles
The stress tensor and its decomposition
The state of stress at a point is the symmetric second-rank tensor $\sigma_{ij}$, which (by the spectral theorem) always admits a diagonal form in the basis of its three principal stresses $\sigma_1 \geq \sigma_2 \geq \sigma_3$. Any yield criterion worth the name is expressed in principal coordinates, because the material itself — assuming isotropy — cannot see the laboratory frame.
The decomposition that turns out to matter most for ductile yielding is the split into hydrostatic (volumetric) and deviatoric (distortional) parts:
$$ \sigma_{ij} = \underbrace{\sigma_m \delta_{ij}}_{\text{hydrostatic}} + \underbrace{S_{ij}}_{\text{deviatoric}}, \qquad \sigma_m = \tfrac{1}{3}(\sigma_1 + \sigma_2 + \sigma_3), \quad S_{ij} = \sigma_{ij} - \sigma_m \delta_{ij}. $$
The hydrostatic part changes volume without changing shape; the deviatoric part changes shape without changing volume.[fn::This is exact for the infinitesimal strain tensor; for finite strain the clean volumetric/distortional split becomes convention-dependent and subtly poisoned by the choice of strain measure. For the elastic-plastic regime of ordinary mechanical design the small-strain decomposition is adequate, and one should not be seduced by its apparent elegance into believing it is physically fundamental.] Bridgman's high-pressure experiments established that, for metals under sufficiently high hydrostatic tension or compression, yielding is essentially not triggered by the pressure itself — the material "cares about" the deviator, not the mean. This is the empirical bedrock of the distortion-energy hypothesis.[fn::Bridgman's 1952 Physics of High Pressure remains the touchstone. The result is striking: steels can sustain hydrostatic pressures of hundreds of GPa without yielding, while the same material yields at a deviatoric shear of perhaps 300 MPa. The ratio is enormous. One would expect, then, that yield criteria which ignore the hydrostatic component are not approximations but asymptotically correct — and for ductile metals this is essentially true. If anything, understated: the insensitivity is the reason von Mises works, not merely a convenience.]
von Mises: the distortion-energy hypothesis
The hypothesis, due to Huber (1904) and rediscovered independently by von Mises (1913) and Hencky (1924), states that yielding begins when the elastic distortional strain energy per unit volume reaches a material-specific critical value. From the elastic strain energy density
$$ U = \frac{1}{2E}\big[(\sigma_1^2 + \sigma_2^2 + \sigma_3^2) - 2\nu(\sigma_1\sigma_2 + \sigma_2\sigma_3 + \sigma_3\sigma_1)\big], $$
subtract the volumetric part $U_v = \frac{1-2\nu}{6E}(\sigma_1+\sigma_2+\sigma_3)^2$ to obtain the distortional energy $U_d$. Setting $U_d$ equal to its uniaxial-yield value ($\sigma_3 = 0$, $\sigma_1 = \sigma_2 = 0$, $\sigma_1 = S_y$) and simplifying yields the familiar equivalent stress
$$ \sigma' = \sqrt{\frac{1}{2}\big[(\sigma_1-\sigma_2)^2 + (\sigma_2-\sigma_3)^2 + (\sigma_3-\sigma_1)^2\big]} \;\leq\; S_y. $$
In invariant form, $\sigma' = \sqrt{3 J_2}$, where $J_2 = \tfrac{1}{2} S_{ij} S_{ij}$ is the second invariant of the deviator. The yield surface $\sigma' = S_y$ in principal-stress space is a circular cylinder of radius $\sqrt{2/3}\, S_y$, with its axis along the hydrostatic line $\sigma_1 = \sigma_2 = \sigma_3$. The cylinder's symmetry about the hydrostatic axis encodes the statement "the material ignores pressure."
Tresca: maximum shear stress
Tresca (1864), predating von Mises by half a century and working without the formalism of invariants, proposed a far simpler rule: yield occurs when the maximum shear stress on any plane reaches half the uniaxial yield strength. From Mohr's circle, $\tau_{\max} = \tfrac{1}{2}(\sigma_{\max} - \sigma_{min})$ where $\sigma_{\max}$ and $\sigma_{\min}$ are the algebraic largest and smallest principal stresses. The criterion is
$$ \tau_{\max} = \tfrac{1}{2}\big|\sigma_{\max} - \sigma_{\min}\big| \;\leq\; \frac{S_y}{2}, \qquad\text{i.e.}\quad |\sigma_{\max} - \sigma_{\min}| \leq S_y. $$
Geometrically, the Tresca locus in $\pi$-space (the deviatoric plane perpendicular to the hydrostatic axis) is a hexagon inscribed in the von Mises circle, the six vertices touching the circle and the six sides cutting inside it. Tresca is therefore the more conservative criterion — it predicts yielding at lower load for any state other than pure uniaxial or equibiaxial tension, where the two coincide. The maximum discrepancy is in pure shear ($\sigma_1 = -\sigma_3 = \tau$, $\sigma_2 = 0$): von Mises gives $\sigma' = \sqrt{3}\,\tau$ so yield at $\tau_y = S_y/\sqrt{3} \approx 0.577\,S_y$, while Tresca gives $\tau_y = S_y/2 = 0.500\,S_y$. The ratio is $2/\sqrt{3} \approx 1.155$, a 15.5% gap — non-trivial in margin-sensitive design.[fn::This 15% is the figure one quotes to justify choosing von Mises over Tresca in margin-critical ductile design. It is also the figure that tempts one to overstate the case: in many design codes the safety factor absorbs the difference, and Tresca's conservatism is purchased "for free" within the factor of safety. The real argument for von Mises is not the 15% but the shape: von Mises tracks the experimental yield locus of annealed copper, aluminum, and mild steel to within a percent or two across the entire deviatoric plane, whereas Tresca's hexagonal corners are a poor fit to the smoothly-curved experimental data. See Taylor & Quinney 1931, still cited because still correct.]
Mohr-Coulomb: the brittle regime
For materials in which the hydrostatic component does matter — concrete, rock, cast iron, soils, most ceramics — the yield (or, more properly, fracture) locus is not a cylinder but a cone opening toward the compressive side. The Mohr-Coulomb criterion states that shear failure occurs on a plane when the shear stress on that plane reaches a linear function of the normal stress acting on it:
$$ |\tau| = c + \sigma_n \tan\phi, $$
where $c$ is cohesion and $\phi$ is the angle of internal friction. In principal-stress space this becomes (assuming $\sigma_1 \geq \sigma_2 \geq \sigma_3$, compression negative):
$$ \sigma_1 = \sigma_3 \, N_\phi + 2c\sqrt{N_\phi}, \qquad N_\phi = \frac{1+\sin\phi}{1-\sin\phi}. $$
The crucial qualitative difference from the ductile criteria is the dependence on $\sigma_3$ (the confining pressure): compressive confinement strengthens the material. This is why concrete's compressive strength is an order of magnitude greater than its tensile strength — not because the chemistry changes, but because the failure criterion is pressure-sensitive.[fn::The temptation to model everything with von Mises and move on is strong, and most CAD post-processors default to it. For a steel shaft in torsion, this is fine. For a cast-iron pump housing under internal pressure, defaulting to von Mises is a quiet error: the housing will fail in tension long before von Mises predicts, because cast iron's locus is shifted strongly toward compression. The honest fix is Mohr-Coulomb or the modified Mohr (Coulomb-Mohr with the tensile cutoff $\sigma_1 = S_{ut}$), but few practicing engineers apply it; instead they use ad hoc knockdown factors on the tensile side and pretend the criterion is unchanged.]
Why von Mises beats Tresca for ductile metals — and when it doesn't
For annealed isotropic ductile metals whose yield is governed by dislocation motion along slip systems, von Mises is the empirically correct criterion. The reason is subtle: a polycrystal has grains in many orientations, and a single slip system activates when its resolved shear stress exceeds a critical value (Schmid's law, a Tresca-like local criterion). Averaged over many grains with randomly distributed orientations, the macroscopic yield locus smooths to the $J_2$ cylinder. Tresca's hexagonal corners correspond to a single-crystal orientation family; von Mises is the polycrystalline average. This is the deep reason for the 15% gap and the smooth fit, and it predicts exactly when von Mises will /fail/[fn::Pun intended.]:
- Strong texture (preferred grain orientation, as in drawn wire, rolled sheet in the rolling direction, forgings) breaks the random-orientation assumption and produces yield loci that are neither circular nor hexagonal but some anisotropic intermediate shape — often better fit by Hill's 1948 anisotropic yield criterion, which generalizes von Mises with additional measured coefficients.
- Cyclic loading with mean stress, or non-proportional loading where principal directions rotate, can drive kinematic hardening that the simple $J_2$ theory ignores. Bauschinger effects make the "yield surface" a moving target.
- / high/ triaxial tension (e.g. at a sharp notch root or a crack tip) produces hydrostatic tensile stress large enough that void growth and cleavage, than shear-driven yielding, control failure. The von Mises cylinder extends indefinitely along the hydrostatic axis, but the real material does not. This is exactly where the distortion-energy hypothesis overstates safety, and where one needs either a porous plasticity model (Gurson–Tvergaard–Needleman) or a fracture-mechanics treatment.
- Materials whose deformation is not dislocation-mediated (polymers below $T_g$, most glasses, all ceramics, cast irons with their graphite flakes acting as internal notches) lie outside the theory's domain entirely.
Worked example
Consider a solid circular shaft of diameter $d = 40\,\text{mm}$, made of AISI 1045 cold-drawn steel with $S_y = 490\,\text{MPa}$ and $S_{ut} = 565\,\text{MPa}$, carrying a bending moment $M = 750\,\text{N·m}$ and torque $T = 1{,}200\,\text{N·m}$ applied steadily (ignoring fatigue for this example).
Step 1 — stresses at the critical surface point
At the outer fibre, the bending stress is
$$ \sigma_x = \frac{32 M}{\pi d^3} = \frac{32(750)}{\pi(0.040)^3} = 298.4\,\text{MPa} \quad (\text{tension, at the top fibre}). $$
The torsional shear is
$$ \tau_{xy} = \frac{16 T}{\pi d^3} = \frac{16(1200)}{\pi(0.040)^3} = 95.5\,\text{MPa}. $$
The element at the critical point is in plane stress ($\sigma_z = \tau_{xz} = \tau_{yz} = 0$), with $\sigma_x = 298.4$, $\sigma_y = 0$, $\tau_{xy} = 95.5$ (MPa throughout).
Step 2 — principal stresses
$$ \sigma_{1,2} = \frac{\sigma_x + \sigma_y}{2} \pm \sqrt{\Big(\frac{\sigma_x-\sigma_y}{2}\Big)^2 + \tau_{xy}^2} = \frac{298.4}{2} \pm \sqrt{(149.2)^2 + (95.5)^2} = 149.2 \pm \sqrt{22{,}261 + 9{,}120} = 149.2 \pm 177.1. $$
So $\sigma_1 = 326.3\,\text{MPa}$, $\sigma_2 = -27.9\,\text{MPa}$, and the third principal stress (out of the surface) is $\sigma_3 = 0$.[fn::Note the convention $\sigma_1 \geq \sigma_2 \geq \sigma_3$ requires care: here we have $\sigma_1 = 326.3$, $\sigma_2 = 0$, $\sigma_3 = -27.9$. The "in-plane" principal values are not automatically the algebraically ordered ones when plane stress leaves a zero in the middle. A common student error is to apply Tresca using the two in-plane principals and forget that $\sigma_2 = 0$ is sandwiched between them — which it is not here, so no harm, but in other geometries it bites.]
Step 3 — von Mises
$$ \sigma' = \sqrt{\sigma_1^2 - \sigma_1\sigma_3 + \sigma_3^2} = \sqrt{(326.3)^2 - (326.3)(-27.9) + (-27.9)^2}\,\text{MPa} $$ $$ = \sqrt{106{,}472 + 9{,}104 + 778} = \sqrt{116{,}354} = 341.1\,\text{MPa}. $$
(Since $\sigma_2 = 0$ sits between $\sigma_1$ and $\sigma_3$, the reduced plane-stress form $\sigma' = \sqrt{\sigma_A^2 - \sigma_A\sigma_B + \sigma_B^2}$ with $\sigma_A = \sigma_1$, $\sigma_B = \sigma_3$ is the one to use.)
Design check: $n = S_y / \sigma' = 490 / 341.1 = 1.44$ — yielding-safe under a modest safety factor.
Step 4 — Tresca
With $\sigma_{\max} = 326.3$, $\sigma_{\min} = -27.9$:
$$ |\sigma_{\max} - \sigma_{\min}| = 326.3 - (-27.9) = 354.2\,\text{MPa} \;\leq\; S_y = 490\,\text{MPa}. $$
So $n_{\text{Tresca}} = 490 / 354.2 = 1.38$.
Step 5 — comparison
| Criterion | $\sigma_{\text{eq}}$ (MPa) | $n$ |
|---|---|---|
| von Mises | 341.1 | 1.44 |
| Tresca | 354.2 | 1.38 |
Tresca is, as expected, more conservative — by about 3.8% in equivalent stress, here, not the full 15% (which is the worst-case gap in pure shear; this loading is bending-dominated). Both criteria clear the part, so for a quick sizing pass the choice is immaterial; for a margin-critical aerospace or automotive part, the 0.06 in safety factor is real money, and the question becomes whether the steel is well-enough annealed to deserve the von Mises assumption.[fn::If the shaft is cold-drawn, as the AISI 1045 spec above admits, the answer is "mostly yes but the Bauschinger effect under reversed torsion will surprise you." Cold drawing introduces a texture and a kinematic-hardening component that the textbook $J_2$ theory does not see. For a one-shot monotonic design pass this is irrelevant; for a shaft under reversing service torque (think: a vehicle driveline), it is the dominant consideration and motivates moving to a cyclic plasticity model — which is a topic for Fatigue & Fracture, not here.]
Pitfalls
1. Applying von Mises to cast iron. Cast iron's tensile strength is governed by graphite-flake-initiated fracture, not dislocation glide; its failure locus is shifted strongly toward the compressive quadrant and is far better described by modified Mohr–Coulomb with $S_{ut} \ll S_{uc}$. Using von Mises will systematically over-predict the tensile-side strength, often by a factor of 2–4. This is one of the more expensive silent errors in machine design, because the von Mises contour plot in the FEA post-processor will show a reassuringly low number while the part is on its way to cracking.
2. Conflating a yield criterion with a fracture criterion. Von Mises and Tresca are yield criteria: they predict the onset of plastic deformation, which for a ductile part under monotonic load is not failure in the catastrophic sense. Fracture — especially brittle fracture initiated at a defect — is governed by fracture mechanics ($K_{Ic}$, $J$-integral, $G$), which depends on crack size in a way that yield criteria simply cannot. A part can satisfy von Mises with a safety factor of 3 and still fail in brittle fracture from a 1 mm crack if $K_{Ic}$ is low. The two checks answer different questions and both must be performed.
3. Ignoring the hydrostatic component in brittle and porous materials. The whole point of von Mises is hydrostatic-independence, and that point is only valid for dense ductile metals. For concrete, rock, soils, sintered or additively-manufactured porous metals, and any material where void growth matters, the hydrostatic stress drives the failure mechanism, and a pressure-insensitive criterion is not a conservative approximation but a category mistake. Use Mohr–Coulomb, Drucker–Prager, or Gurson as appropriate.[fn::A particular trap: AM (laser-powder-bed) Inconel, which on a datasheet looks like a perfectly ductile nickel superalloy, often has residual porosity of 0.1–0.5% that makes its high-triaxial-tension response distinctly non-von-Mises. The datasheet tensile test (uniaxial, low triaxiality) will not reveal this. The lesson is that the material is not the same object as the datasheet, and criteria are calibrated against the latter.]
4. Using plane-stress principal formulas for a 3D problem. The reduced form $\sigma' = \sqrt{\sigma_A^2 - \sigma_A\sigma_B + \sigma_B^2}$ is only valid when one principal stress is zero (plane stress). In a thick shaft under combined tension, internal pressure, and torsion, all three principals are nonzero, so the two-term formula underestimates $\sigma'$. The safe habit is to always compute three principals and use the full $J_2$ expression.
5. Confusing the failure criterion with failure analysis. A criterion is a prospective model: "for this material, under this loading, the margin against yield is 1.4." Failure analysis is a retrospective investigation of a specific broken part, involving fractography, metallurgy, service-load reconstruction, manufacturing-record review, and often detective work about whether the part was loaded as designed or abused. A criterion tells you nothing about why a specific part failed, and a failure analysis cannot by itself give you a calibrated criterion. The two activities inform each other but are not the same, and treating them as interchangeable produces bad design and bad forensics. See Failure Analysis Methodology.
6. Forgetting that principal stresses require algebraic ordering. The convention $\sigma_1 \geq \sigma_2 \geq \sigma_3$ is not cosmetic: Tresca's $|\sigma_{\max}-\sigma_{\min}|$ and Mohr–Coulomb's branch logic both depend on it. Mixing up the ordering — for example, labelling an in-plane compressive stress as $\sigma_3$ when a zero out-of-plane stress is algebraically larger — silently corrupts the answer. Code that sorts principals is doing real work; code that doesn't, isn't.
References
- Shigley's Mechanical Engineering Design, 11th ed. — Budynas & Nisbett, McGraw-Hill, 2020. The standard undergraduate-and-practitioner reference; chapter 5 covers the failure criteria discussed here with worked examples and a healthy dose of the conservative simplifications one expects from a code-oriented text.
- Norman E. Dowling, /Mechanical Behavior of Materials/, 2nd ed. — Prentice Hall, 1999. Stronger on the materials-science side than Shigley; the chapter on yield criteria is paired with a thorough discussion of anisotropic yield (Hill), and the connection between single-crystal Schmid and polycrystalline $J_2$ is made explicit, which most meche texts dodge.
- P. W. Bridgman, /The Physics of High Pressure/ — Bell, 1952. The empirical foundation for the hydrostatic-insensitivity assumption underlying von Mises; dated but not superseded for the basic point.
- G. I. Taylor & H. Quinney, "The Plastic Distortion of Metals" — /Phil. Trans. R. Soc. A/, 230 (1931)]. The experimental locus that decisively favored von Mises over Tresca for copper and aluminum; the reason a 1931 paper is still cited is that the experiment has not needed re-running.
- H. Hoffman & G. Sachs, /Theory of Plasticity for Engineers/ — McGraw-Hill, 1953. A mid-century text that treats Tresca and von Mises with equal seriousness and is useful for the historical sense of how the criteria were argued before they were settled.