4. Multiaxial fatigue criteria

Uniaxial S-N curves are only valid when the stress state is proportional and the principal directions remain fixed. Real CAE components—crankshaft fillets, suspension knuckles, turbine disks, welded brackets—experience combined bending, torsion, and axial loading with phase shifts and non-proportional histories. Under these conditions, the amplitude and direction of the principal stresses rotate through the cycle, and a single equivalent stress no longer predicts crack initiation reliably. Multiaxial fatigue criteria exist to collapse a complex stress tensor history into a scalar damage parameter that can be compared against uniaxial fatigue data.

This chapter covers the four families most frequently implemented in commercial CAE post-processors: critical-plane criteria, Findley's stress-based criterion, McDiarmid's shear-stress criterion, and invariant-based (stress-invariant) methods. The emphasis is on how each criterion is formulated, what material data it requires, and how it is actually evaluated on a finite element stress history.

Why the Critical Plane Concept Matters

Observations of fatigue cracks in ductile metals show that initiation occurs on specific crystallographic planes, typically those experiencing the highest resolved shear stress amplitude combined with a tensile normal stress that opens the crack. The critical plane approach formalizes this: instead of computing a single scalar from the tensor, it searches over all candidate plane orientations and evaluates a damage parameter on each plane. The plane that yields the maximum damage parameter is the critical plane, and its value drives the fatigue life prediction.

For a plane defined by its unit normal n, the traction vector is t = σ · n. The normal stress on the plane is σ_n = n · t, and the resolved shear stress vector is τ = t − σ_n n. The shear stress amplitude τ_a is obtained from the history of τ using a minimum circumscribed circle or a convex-hull-based algorithm. The normal stress amplitude σ_n,a and mean σ_n,m are computed from the scalar history of σ_n.

Findley's Criterion

Findley proposed a linear combination of shear stress amplitude and maximum normal stress on the critical plane:

τ_a + k · σ_n,max = f

where k is a material constant determined from two uniaxial fatigue limits (typically fully reversed bending and torsion), and f is the allowable shear stress amplitude. The normal stress term uses the maximum value over the cycle, not the amplitude, which captures the mean-stress sensitivity of the material. Findley's criterion is well suited to ductile metals and is relatively insensitive to phase angle because it does not require the shear and normal peaks to coincide.

To calibrate k and f, use the fully reversed bending fatigue limit σ−1 and the fully reversed torsion fatigue limit τ−1. For bending, the critical plane is the plane of maximum shear at 45° to the surface, giving τ_a = σ−1/2 and σ_n,max = σ−1/2. For torsion, τ_a = τ−1 and σ_n,max = 0. Solving the two equations yields:

k = (2τ<sub>−1</sub> − σ<sub>−1</sub>) / σ<sub>−1</sub>
f = τ<sub>−1</sub>

For a typical steel with σ−1 = 400 MPa and τ−1 = 240 MPa, k = 0.20 and f = 240 MPa. A CAE implementation must scan plane orientations—commonly 10° increments in both spherical angles, giving roughly 1,800 planes—and evaluate the parameter at each. The maximum over all planes is the Findley equivalent stress.

McDiarmid's Criterion

McDiarmid distinguishes between two failure modes: Case A (crack initiation on the surface, shear-dominated) and Case B (crack initiation at a stress concentration, tensile-dominated). For Case A, the criterion is:

τ_a / τ<sub>−1</sub> + σ_n,max / (2σ<sub>u</sub>) = 1

where σu is the ultimate tensile strength. The second term accounts for the mean normal stress effect. McDiarmid requires only the torsion fatigue limit and the ultimate strength, making it attractive when bending fatigue data is unavailable. However, it is more sensitive to the choice of critical plane than Findley because the normal stress term is weighted by 1/(2σu), which is typically small.

In CAE, McDiarmid is often applied to welded joints and cast components where the surface condition differs from the polished specimen used to obtain τ−1. A knock-down factor on τ−1 is common practice.

Invariant-Based Methods

Invariant-based criteria avoid the plane search by using stress invariants or combinations of the second deviatoric invariant J2 and the first invariant I1. The most widely implemented is the Sines criterion:

√(J<sub>2,a</sub>) + α · I<sub>1,m</sub> = β

where J2,a is the amplitude of the second deviatoric invariant, I1,m is the mean hydrostatic stress, and α, β are material constants. Sines is computationally cheap—no plane scanning—but it cannot distinguish between proportional and non-proportional loading, and it tends to be non-conservative for out-of-phase torsion-bending. The Crossland criterion is similar but uses the maximum hydrostatic stress instead of the mean.

Invariant methods are best used for proportional loading or as a screening tool before running a critical-plane analysis. They are also useful when the stress history is long and the plane search would be prohibitively expensive.

Implementing the Criteria in a CAE Post-Processor

The practical workflow for a critical-plane analysis in a CAE environment follows a consistent pattern regardless of which criterion is selected. First, extract the stress tensor history at each node or integration point from the FE results. For linear static subcase combinations, the history is a superposition of scaled load cases; for transient or modal superposition, it is a time series. Second, if the loading is non-proportional, apply a cycle counting algorithm—rainflow on the equivalent shear stress history is common—to identify cycles and their amplitudes. Third, for each candidate plane, compute the shear stress amplitude using the minimum circumscribed circle method, and the normal stress amplitude and maximum. Fourth, evaluate the criterion and retain the maximum value over all planes.

A common pitfall is the treatment of mean stress. Findley and McDiarmid both include a normal stress term that partially accounts for mean stress, but neither is a full mean-stress correction. For high mean stresses, a Goodman or Gerber correction on the final equivalent stress is often applied in addition. Another pitfall is the plane search resolution: 10° increments are standard, but for strongly non-proportional loading, 5° may be necessary to capture the true maximum. The computational cost scales linearly with the number of planes, so a 5° search is roughly four times slower.

Comparison and Selection Guidance

In a CAE workflow, the choice is often driven by available material data and the nature of the loading. For a crankshaft subjected to combined bending and torsion with a phase shift, Findley or McDiarmid on a critical plane is appropriate. For a bracket under proportional combined loading, Sines is sufficient and avoids the plane search cost. The key is to verify the criterion against a known failure location or a test correlation before applying it to a full model.

Once the equivalent stress is computed, it is compared to the uniaxial S-N curve using the same damage accumulation rules as in uniaxial fatigue—Miner's rule for variable amplitude, or a strain-life curve if local plasticity is significant. The multiaxial criterion only replaces the equivalent stress calculation; the rest of the fatigue workflow remains unchanged.