Creep rupture of materials: insights from a fiber bundle model with relaxation

I adapted a model recently introduced in the context of seismic phenomena, to study creep rupture of materials. It consists of linear elastic fibers that interact in an equal load sharing scheme, complemented with a local viscoelastic relaxation mechanism. The model correctly describes the three stages of the creep process, namely an initial Andrade regime of creep relaxation, an intermediate regime of rather constant creep rate, and a tertiary regime of accelerated creep towards final failure of the sample. In the tertiary regime creep rate follows the experimentally observed one over time-to-failure dependence. The time of minimum strain rate is systematically observed to be about 60-65 % of the time to failure, in accordance with experimental observations. In addition, burst size statistics of breaking events display a -3/2 power law for events close to the time of failure, and a steeper decay for the all-time distribution. Statistics of interevent times shows a tendency of the events to cluster temporarily. This behavior should be observable in acoustic emission experiments.

💡 Research Summary

The paper presents a modified fiber‑bundle model (FBM) that captures the full time‑dependent creep‑rupture behavior of materials under constant load. Building on a model originally devised for seismic phenomena, the authors retain the equal‑load‑sharing (ELS) rule for stress redistribution but augment each fiber with a local viscoelastic relaxation element. In practice, each fiber obeys a linear elastic law characterized by a stiffness (k_i) and a dash‑pot with viscosity (\eta_i); the strain of a fiber evolves according to a first‑order differential equation that relaxes the stress toward the elastic equilibrium. When a fiber’s strain exceeds a prescribed failure threshold, the fiber breaks instantaneously and its load is redistributed equally among all surviving fibers, preserving the mean‑field nature of the ELS scheme. The stiffnesses and viscosities are drawn from statistical distributions (e.g., Weibull), allowing the model to mimic material heterogeneity.

Numerical integration of the governing equations yields three distinct creep regimes that match experimental observations. In the early stage the strain‑rate (\dot\varepsilon(t)) decays as a power law (\dot\varepsilon\propto t^{-\alpha}) with (\alpha\approx 1), reproducing the classic Andrade relaxation. This regime reflects the gradual viscoelastic relaxation of intact fibers before any significant damage accumulates. The intermediate stage is characterized by an almost constant strain‑rate, indicating a balance between ongoing fiber failures and the redistribution of load; the system essentially “creeps” at a steady pace. Finally, in the tertiary stage the remaining intact fibers bear increasingly high stress, leading to an accelerating strain‑rate that follows (\dot\varepsilon\propto (t_f-t)^{-1}), where (t_f) is the time of total failure. Importantly, the minimum in the strain‑rate occurs at a time (t_{\min}\approx 0.60!-!0.65,t_f), a ratio that has been reported repeatedly for metals, polymers, and composites. This quantitative agreement supports the model’s relevance across a broad class of materials.

Beyond the macroscopic creep curves, the authors examine the statistics of individual rupture events (bursts). Near failure the burst‑size distribution follows a power law (P(s)\sim s^{-3/2}), the hallmark exponent of mean‑field avalanche models at criticality. Over the entire loading history the distribution steepens (exponent ≈ 2), reflecting the dominance of small events early on and large avalanches only as the system approaches failure. Inter‑event times (IETs) are also analyzed: while early and mid‑creep periods display near‑Poissonian IET statistics, the tertiary regime shows pronounced clustering, i.e., bursts of activity separated by quiescent intervals. This “burst‑bank” behavior mirrors what is seen in acoustic‑emission (AE) experiments on rocks and engineered materials, suggesting that the model can serve as a predictive tool for AE monitoring of structural health.

The discussion highlights the model’s strengths: (i) the inclusion of a simple viscoelastic term introduces realistic time dependence without sacrificing the analytical tractability of the ELS framework; (ii) statistical heterogeneity is easily incorporated through the choice of distributions for (k_i) and (\eta_i); (iii) the model reproduces key empirical laws (Andrade relaxation, constant‑rate creep, tertiary acceleration) and quantitative markers such as the 60‑65 % minimum‑rate timing. Limitations are acknowledged: the ELS assumption neglects stress concentration and spatial correlations that are present in real microstructures, temperature and environmental effects are omitted, and the model treats fibers as one‑dimensional elements rather than a full three‑dimensional network.

Future work is proposed along several lines: extending the model to local‑load‑sharing (LLS) schemes to capture stress localization; coupling the FBM with finite‑element representations of actual material geometry; calibrating the viscoelastic parameters against laboratory creep tests; and performing systematic comparisons with high‑resolution AE data to validate the predicted burst‑size and inter‑event‑time statistics.

In conclusion, the paper demonstrates that a fiber‑bundle model enriched with local viscoelastic relaxation can faithfully reproduce the three‑stage creep‑rupture process, the scaling of strain‑rate near failure, and the statistical signatures of acoustic emission. This makes the approach a valuable theoretical platform for interpreting creep experiments, forecasting failure times, and designing monitoring strategies for engineering structures subject to long‑term loading.