Universality behind Basquin’s Law of Fatigue

Basquin’s law of fatigue states that the lifetime of the system has a power-law dependence on the external load amplitude, $t_f\sim {\sigma }_0^{-\alpha }$, where the exponent $\alpha$ has a strong material dependence. We show that in spite of the broad scatter of the exponent $\alpha$, the fatigue fracture of heterogeneous materials exhibits universal features. We propose a generic scaling form for the macroscopic deformation and show that at the fatigue limit the system undergoes a continuous phase transition. On the microlevel, the fatigue fracture proceeds in bursts characterized by universal power-law distributions. We demonstrate that the system dependent details are contained in Basquin’s exponent for time to failure, and once this is taken into account, remaining features of failure are universal.

Figure 1

Scaling plot of deformation-time curves measured experimentally on asphalt specimens [12]. The scaling function $S$, obtained by rescaling the two axis, has a power-law dependence on the time-to-failure.

Figure 2

Characteristic time scales $t_r$ and $t_f$ of the system. The complete FBM corresponding to Eq. (1) which includes immediate breaking and healing is solved numerically. For ${\sigma }_0>{\sigma }_l$ we see Basquin’s law and both models provide a very good fit of the lifetime data of asphalt. The fatigue limit ${\sigma }_l$ is indicated by the vertical dashed line.

Figure 3

The relaxation time $t_r$ and lifetime $t_f$ as a function of $|{\sigma }_l-{\sigma }_0|$ for different disorder distributions and $\gamma$ exponents. The straight lines have slopes $-1/3$ and $-2/3$.

Figure 4

Rescaled plot of waiting time distributions $P(T)$ obtained by FBM simulations. Inset: Corresponding results of two-dimensional DEM simulations.