Triaxial Fatigue: An Engineering Overview

This article considers triaxial fatigue. For help, with this and other fatigue-related problems, take a look at PDE’s Life & Reliability services at Life & Reliability Engineering | Fatigue, Fracture & FEA Services or, alternatively, contact us directly as below.

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Fatigue failure under multiaxial stress states is a critical consideration in the design of mechanical components subjected to real-world service loads. Unlike uniaxial fatigue, which involves cyclic loading along a single axis, triaxial fatigue arises when stresses vary simultaneously in multiple directions — often with complex phase relationships. This is common in shafts, pressure vessels, rotating equipment, and structural junctions.

Sectioned view of a marine diesel engine in a ferry engine room, highlighting components like the crankshaft where triaxial fatigue commonly develops due to complex cyclic stresses.

Modern standards such as ASTM E2207 and ISO 12112 guide multiaxial fatigue testing, providing consistent procedures for strain-controlled and load-controlled fatigue under combined axial-torsional conditions. From a design perspective, accurately evaluating fatigue life under triaxial stress requires decomposing the stress state and applying robust multiaxial fatigue criteria.

Stress States and Principal Stress Decomposition

Any general stress state at a material point is represented by the 3D stress tensor with normal and shear components. Principal stress analysis reduces this to three orthogonal directions with only normal stresses:

$\sigma_{1,2} = \frac{\sigma_x + \sigma_y}{2} \pm \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}$

This applies for plane stress. For full triaxial states, principal stresses are the roots of the characteristic equation of the stress tensor. Once known, we define:

$\sigma_H = \frac{\sigma_1 + \sigma_2 + \sigma_3}{3}$

$\sigma_{eq} = \sqrt{\frac{(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2}{2}}$

The triaxiality ratio is:

$T = \frac{\sigma_H}{\sigma_{eq}}$

This ratio influences both plasticity and fatigue damage mechanisms. High triaxiality typically promotes brittle failure and reduces fatigue resistance.

Multiaxial Fatigue Categories

Two common multiaxial load types are:

Proportional loading: all stress components vary in phase
Non-proportional loading: components vary out of phase, often inducing extra hardening

Fatigue may be shear-dominated or normal stress-dominated, depending on the stress path. Shear-driven fatigue often leads to crack initiation on maximum shear planes (~45° to loading), whereas tension-driven fatigue initiates on perpendicular planes.

Multiaxial Fatigue Criteria

Several criteria account for the combined effect of shear and normal stresses. These include invariant-based and critical-plane models. Below are three commonly applied methods.

Crossland Criterion

An invariant-based approach that considers the amplitude of the second invariant of the deviatoric stress and the maximum hydrostatic stress:

$\sqrt{J_{2,a}} + \alpha , \sigma_{H,\max} \leq C$

Where:

$J_{2,a}$ is the amplitude of the second deviatoric stress invariant
$\sigma_{H,\max}$ is the peak hydrostatic stress in the cycle
$\alpha, C$ are material constants derived from uniaxial and torsional fatigue limits

Dang–Van Criterion

A critical-plane criterion evaluated in the time domain:

$\tau(t) + \alpha , \sigma_H(t) \leq \beta \quad \text{for all } t$

Where:

$\tau(t)$ is the resolved shear stress at time $t$
$\sigma_H(t)$ is the hydrostatic stress
$\alpha, \beta$ are material parameters

This method is conservative and particularly suitable for high-cycle fatigue with complex stress histories.

Findley Criterion

A critical-plane method considering shear stress range and normal stress:

$\tau_a + k , \sigma_{n,\max} \leq \tau_{\text{lim}}$

Where:

$\tau_a$ is shear stress amplitude on the critical plane
$\sigma_{n,\max}$ is the maximum normal stress on that plane
$k$ is a material constant
$\tau_{\text{lim}}$ is the fatigue limit in shear

Findley’s model is widely used in automotive and aerospace sectors for its physical relevance and computational efficiency.

Worked Example

Consider a hollow shaft subjected to fully-reversed axial and torsional stresses:

Axial stress amplitude: $\sigma_a = 100\text{MPa}$
Torsional stress amplitude: $\tau_a = 80\text{MPa}$
Material limits: $\sigma_e = 200\text{MPa}, \tau_e = 150\text{MPa}$

Using Crossland’s criterion:

First, calculate constants.

Under pure tension:

$\frac{\sigma_e}{\sqrt{3}} + \alpha \cdot \frac{\sigma_e}{3} = \tau_e$

Substitute:

$\frac{200}{\sqrt{3}} + \alpha \cdot \frac{200}{3} = 150$

Solve for $\alpha$ :

$\alpha = \frac{150 - \frac{200}{\sqrt{3}}}{\frac{200}{3}} \approx 0.52$

Now apply the criterion:

Compute hydrostatic stress:

$\sigma_H = \frac{\sigma_1 + \sigma_2 + \sigma_3}{3} = \frac{144.3 - 44.3 + 0}{3} = 33.3,\text{MPa}$

Compute $\sqrt{J_{2,a}}$ using principal stresses:

$\sqrt{J_{2,a}} = \sqrt{\frac{1}{6}[(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2]} \approx 98.7,\text{MPa}$

Apply Crossland:

$98.7 + 0.52 \cdot 33.3 = 116.0 < 150$

Safe.

Apply Findley criterion with $k = 0.3$ :

$\tau_a + k \cdot \sigma_{n,\max} = 80 + 0.3 \cdot 44.3 = 93.3,\text{MPa}$

Since $93.3 < 150$ , fatigue life is not exceeded.

Conclusion

Triaxial fatigue must be addressed in modern mechanical design where multiaxial loading is present. By decomposing stress states and applying models like Crossland, Dang–Van, or Findley — each with their own strengths — engineers can predict fatigue life conservatively and reliably. Standards such as ASTM E2207 ensure consistent material data for use with these criteria. With proper methodology, components can be designed to endure complex loading scenarios without premature fatigue failure.

For a more-in-depth overview of this and related topics, including online calculators for multiaxial fatigue, see eFatigue – Multiaxial. As a reminder, for project support, take a look at our Life & Reliability services at Life & Reliability Engineering | Fatigue, Fracture & FEA Services.

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Stress States and Principal Stress Decomposition

Multiaxial Fatigue Categories