Time evolution of damage due to environmentally assisted aging in a fiber bundle model

Damage growth in composite materials is a complex process which is of interest in many fields of science and engineering. We consider this problem in a fiber bundle model where fibers undergo an aging process due to the accumulation of damage driven by the locally acting stress in a chemically active environment. By subjecting the bundle to a constant external load, fibers fail either when the load on them exceeds their individual intrinsic strength or when the accumulated internal damage exceeds a random threshold. We analyze the time evolution of the breaking process under low external loads where aging of fibers dominates. In the mean field limit, we show analytically that the aging system continuously accelerates in a way which can be characterized by an inverse power law of the event rate with a singularity that defines a failure time. The exponent is not universal; it depends on the details of the aging process. For localized load sharing, a more complex damage process emerges which is dominated by distinct spatial regions of the system with different degrees of stress concentration. Analytical calculations revealed that the final acceleration to global failure is preceded by a stationary accumulation of damage. When the disorder is strong, the accelerating phase has the same functional behavior as in the mean field limit. The analytical results are verified by computer simulations.

Figure 1

Damage event rate $n$ as a function of time to failure $t_f-t$ for different exponents $\gamma$ of the damage law. The results were obtained by ELS simulations of single samples with ${10}^7$ fibers. The straight lines show the analytical results of Eq. (11) for comparison.

Figure 2

Lifetime $t_f$ of samples as a function of load for different values of $\gamma$. The continuous lines represent power laws of exponent $\gamma$. Excellent agreement can be observed with the analytic predictions [Eq. (12)]; slight deviations occur solely in the vicinity of the critical load. The offset between the theoretical lines and the synthetic data is cosmetic (so the lines can be seen).

Figure 3

Waiting time distributions obtained by three ELS simulations of a single sample with ${10}^7$ fibers corresponding to different values of $\gamma$ as indicated in the key. The straight lines show the analytical results of Eq. (13) for comparison. The offset between the theoretical lines and the synthetic data is cosmetic (so the lines can be seen).

Figure 4

Subset of a FBM with $N=401\times 401$, $W=0.1$, $a=1$, $\gamma =1$, and $\sigma =0.01$ after (a) 100, (b) 2000, (c) 20 000, and (d) 80 000 broken fibers due to the aging process. Dark blue fields mark intact fibers, light blue fields mark fibers initially broken due to the applied stress, yellow fields mark fibers which broke due to the aging process (silent events), and red fields mark fibers which broke due to the local stress redistribution.

Figure 5

Average damage $\langle D\rangle$ due to the aging process (dashed blue curves) and due to local stress redistribution (dotted red curve) as a function of time $t$ for a FBM with $N=401\times 401$, $a=1$, $W=0.1$, and (a), (b) $\gamma =1$; (c), (d) $\gamma =2$; and (e), (f) $\gamma =3$, as well as $\sigma =0.002$ (left-hand side) and $\sigma =0.01$ (right-hand side). The full black curves are fits according to Eq. (18) [for one (1$n$) or two (2$n$) broken neighbors] and Eq. (22) [for independent breaking (0$n$)] with fit parameter $d_n$ given by (a) 7 (1$n$), 435 (0$n$); (b) 6 (2$n$), 146 (1$n$), 2150 (0$n$); (c) 13 (1$n$); (d) 12 (2$n$), 279 (1$n$); (e) 38 (1$n$); and (f) 12 (2$n$). The green curves are the corresponding triggered damage according to Eqs. (28) and (29), and the turquoise curves are the triggered damage according to Eq. (30).