Stochasticity of fatigue failure times in sheared glasses

Abstract

Fatigue failure occurs when a solid is subjected to repeated, cyclic loading. Glasses subjected to cyclic to shear deformation have recently been investigated using computer simulations and theoretical models, to characterize and rationalize the dependence of the number of cycles to failure, depending on the properties of the glasses, and the deformation amplitude. The average number of cycles to failure has been observed to diverge as the strain amplitude approaches the so-called fatigue limit from above. In this work, rather than the average times themselves, we investigate by computer simulations the distribution of fatigue failure times, in model glasses subjected to cyclic shear deformation and in an elasto-plastic model. In particular, we observe in atomistic simulations that the standard deviation of the logarithm of failure times are proportional to their mean values, with the proportionality constant decreasing as the system size increases, indicating a sharper distribution of failure times. Using a finite-element-based elasto-plastic model, we observe similar behavior and perform a system-size analysis showing that the ratio of the standard deviation to the mean tends toward zero in the thermodynamic limit. Such distributions, rather than arising solely from the distribution of disorder in the samples that have been subjected to cyclic deformation, appear to arise from the intrinsic stochasticity of the failure process, which we analyze through a stochastic damage accumulation model.

Figures (9)

• Figure 1: Distribution of failure times: (a) Probability distribution $P(t_f)$ and (b) cumulative distribution $C(t_f)$ of failure times $t_f$ for $N=4000$, $e_{IS}=-7.00$ and $\gamma_{max}=0.086$ using $400$ samples. Lines are fit to different functional forms: Weibull, generalized gamma, Frechet, Gumbel, lognormal, and inverse Gaussian [See table \ref{['Tab:different_fitting_forms']} for the details of each distribution's form]. (b) shares the colour code with (a).
• Figure 2: Scaling of the distribution of the failure times: The cumulative distribution of $\ln{t_f}$ for (a) KA-BMLJ model with $N=4000$, $e_{IS}=-7.00$$(T_p=0.435)$ and (b) CP model with $N=46656$, $T_p=0.36$ for several strain amplitudes. The solid lines represent fits to the Gaussian distribution. Collapse of data for (c) KA-BMLJ model and (d) CP model for different $\gamma_{max}$ when $\ln{t_f}$ (the $x$-axis) is rescaled with its mean value $\langle\ln{t_f}\rangle$. The solid lines are Gaussian fits to the data points across different $\gamma_{max}$.
• Figure 3: Distribution of failure times for different system sizes (KA-BMLJ): (a) Cumulative distribution of logarithm of failure times $C(\ln{t_f})$ for a system of size $N=64000$ with $e_{IS}=-7.00$ for different strain amplitudes. Lines are fit to the normal distribution. (b) Collapse of data for different $\gamma_{max}$ when $\ln{t_f}$ is rescaled with its mean value. The solid line through data points represents Eq. (\ref{['eq:scaling_dist_tf']}). (c) and (d) show the same quantities as in (a) and (b), respectively, but for a larger system size of $N=128000$. (e) Scaled cumulative distribution functions (obtained from the fitting) for four different system sizes: $N=4000$, $16000$, $64000$, and $128000$. (Data points are suppressed for the clarity). The distributions become progressively sharper as the system size increases. (f) Scaled standard deviation $\sigma_{\ln{t_f}}/\mu_{\ln{t_f}}$vs system size.
• Figure 4: Scaling of failure time distributions for different degrees of annealing: (a) Cumulative distribution as a function of scaled variable $\ln t_f/\langle \ln t_f \rangle$ for $N=4000$, $\gamma_{max}=0.1$, and several degrees of annealing. (b) $C(\ln t_f)$vs.$\ln t_f/\langle \ln t_f \rangle$ for $N=64000$ for three degrees of annealing ($e_{IS}=-6.96, -7.00, -7.05$). Solid lines are fit to the data for the respective degree of annealing. For $N=64000$, data are collected across different $\gamma_{max}$.
• Figure 5: Results for different initial conditions: (a) The cumulative distribution of failure times for runs starting with different initial configurations and with the same initial configurations with different initial velocities. (b) Mean failure time against $\gamma_{max}-\gamma_{max}^Y$. Solid line indicates power law behaviour with exponent $-2$. The error bars represent the sample-to-sample standard deviation of the failure times. (c) Accumulated damage till failure $D^{acc}_f$ grows with failure time $t_f$ with $3/4$ exponent for both cases. For clarity, data for different velocities are shifted towards the right. ($N=64000$, $e_{IS}=-7.00$.)