**Journal of Theoretical**

and Applied Mechanics

and Applied Mechanics

**42**, 2, pp. 295-314, Warsaw 2004

### A fatigue failure criterion for multiaxial loading with phase shift and mean value

A new criterion based on the critical plane approach has been developed for multiaxial non-proportional fatigue failure. The criterion correctly takes into account the influence of phase shift and mean values under combined bending and torsion loading. From a certain point of view, the criterion with such a defined non-proportionality measure can be understood as a combination of the two approaches: critical plane and integral approach. The criterion has the following form

$ \tau_{\alpha^*(eqnp)} = (\tau_{\alpha^*(a)} + c_1 \sigma_{\alpha^*(a)} + c_2\sigma_{\alpha^*(m)}) \Bigl(1 + \frac{t_{-1}}{b_{-1}}H^n\Bigr) \leq c_3 $

where the multiplicand of the equivalent shear stress $ \tau_{\alpha^*(eqnp)}$ contains the amplitude of the shear stress $ \tau_{\alpha^*(a)}$, the amplitude $ \sigma_{\alpha^*(a)}$ and mean value $ \sigma_{\alpha^*(m)}$ of the normal stress acting in the critical plane. The multiplier contains the loading non-proportionality measure $ H$. Taking into account the fact of different sensitivity of various materials to loading non-proportionality, the equation also includes the material data: $ t_{-1}$ – fatigue limit in torsion, $ b_{-1}$ – fatigue limit in bending.

The predictive capability of the criterion was demonstrated by analyzing 67 experimental results from the literature. The predicted results are generally in good agreement with the experimental ones.

$ \tau_{\alpha^*(eqnp)} = (\tau_{\alpha^*(a)} + c_1 \sigma_{\alpha^*(a)} + c_2\sigma_{\alpha^*(m)}) \Bigl(1 + \frac{t_{-1}}{b_{-1}}H^n\Bigr) \leq c_3 $

where the multiplicand of the equivalent shear stress $ \tau_{\alpha^*(eqnp)}$ contains the amplitude of the shear stress $ \tau_{\alpha^*(a)}$, the amplitude $ \sigma_{\alpha^*(a)}$ and mean value $ \sigma_{\alpha^*(m)}$ of the normal stress acting in the critical plane. The multiplier contains the loading non-proportionality measure $ H$. Taking into account the fact of different sensitivity of various materials to loading non-proportionality, the equation also includes the material data: $ t_{-1}$ – fatigue limit in torsion, $ b_{-1}$ – fatigue limit in bending.

The predictive capability of the criterion was demonstrated by analyzing 67 experimental results from the literature. The predicted results are generally in good agreement with the experimental ones.

*Keywords*: high cycle fatigue; multiaxial fatigue; non-proportional loading; out-of-phase loading; mean value