Multiscale models predicting crack nucleation and propagation under thermal and rate effects
Résumé
Fracture and decohesion phenomena are widely studied across various disciplines due to their fundamental theoretical significance and broad range of applications. One of the most established and widely employed frameworks in this field is the energetic approach, encapsulated by Griffith’s energy criterion. While analytical and experimental studies have confirmed the effectiveness of this criterion in describing the propagation of preexisting cracks, it remains inadequate for capturing several critical aspects relevant to structural design. Notably, it fails to predict crack nucleation and presents significant challenges in rigorously incorporating temperature effects within an analytical framework. This study investigates the influence of temperature on crack nucleation and propagation in material failure and decohesion. Building upon Griffith’s energy criterion, we propose a simplified model focused on mode I fracture, extending the classical criterion to account for crack nucleation and the role of thermal fluctuations. By leveraging tools from equilibrium statistical mechanics, we integrate entropic contributions into the overall energy balance. Additionally, we adopt a multiscale approach, simultaneously formulating both discrete and continuum (limit) models. This methodology provides deeper insight into the intricate mechanisms governing fracture and decohesion, elucidating how microscopic-scale phenomena influence meso- and macroscopic behavior. Despite the simplicity of the proposed models, they allow for analytical tractability and a more profound understanding of the underlying physics. Our energetic approach effectively captures the competition among external loading, elastic deformation, fracture energy, and entropic effects. Specifically, our model predicts crack nucleation and quantifies the influence of thermal fluctuations on this process. Furthermore, the framework accommodates different fracture and decohesion scenarios, including cases where fracture propagates from one end or where the damaged region remains confined within the system, such as in the presence of multiple bubbles (e.g., DNA denaturation bubbles). In the latter case, our model predicts the coalescence of these bubbles prior to complete failure. Interestingly, our approach uncovers a classical critical behavior, wherein the critical load decreases with increasing temperature following the relation (1 − T/Tc)^(1/2). Consequently, at the critical temperature Tc, the system undergoes a phase transition, leading to complete rupture even in the absence of an applied mechanical load. In addition to temperature effects, our prototypical model also captures rate-dependent behavior, i.e., the system’s response under a time-dependent loading rate. Unlike temperature effects, which we analyze using equilibrium statistical mechanics, rate effects necessitate a departure from equilibrium conditions. Preliminary results indicate that the system’s dynamics are significantly influenced by the applied loading rate. Specifically, as the loading rate increases, the fracture propagation velocity also increases, whereas a decrease in the loading rate leads to a softening of the system’s mechanical response. The findings regarding both temperature and rate effects are particularly promising, as they align with experimentally observed behaviors in materials that have been challenging to describe rigorously through analytical approaches. This study introduces a relatively simple yet powerful model that captures these complex phenomena without compromising mathematical tractability and analytical rigor.
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