Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userApproximation of a Brittle Fracture Energy with a Constraint of Non-Interpenetration
99
0
0.0
(
0
)
Authors
Antonin Chambolle
Ask ChatGPT about the research
No Arabic abstract
Linear fracture mechanics (or at least the initiation part of that theory) can be framed in a variational context as a minimization problem over a SBD type space. The corresponding functional can in turn be approximated in the sense of $Gamma$-convergence by a sequence of functionals involving a phase field as well as the displacement field. We show that a similar approximation persists if additionally imposing a non-interpenetration constraint in the minimization, namely that only nonnegative normal jumps should be permissible. 2010 Mathematics subject classification: 26A45
rate research
Read More
This paper concerns the analysis and implementation of a novel iterative staggered scheme for quasi-static brittle fracture propagation models, where the fracture evolution is tracked by a phase field variable. The model we consider is a two-field variational inequality system, with the phase field function and the elastic displacements of the solid material as independent variables. Using a penalization strategy, this variational inequality system is transformed into a variational equality system, which is the formulation we take as the starting point for our algorithmic developments. The proposed scheme involves a partitioning of this model into two subproblems; phase field and mechanics, with added stabilization terms to both subproblems for improved efficiency and robustness. We analyze the convergence of the proposed scheme using a fixed point argument, and find that under a natural condition, the elastic mechanical energy remains bounded, and, if the diffusive zone around crack surfaces is sufficiently thick, monotonic convergence is achieved. Finally, the proposed scheme is validated numerically with several bench-mark problems.
This article proposes an open-source implementation of a phase-field model for brittle fracture using a recently developed finite element toolbox, Gridap in Julia. The present work exploits the advantages of both the phase-field model and Gridap toolbox for simulating fracture in brittle materials. On one hand, the use of the phase-field model, which is a continuum approach and uses a diffuse representation of sharp cracks, enables the proposed implementation to overcome such well-known drawbacks of the discrete approach for predicting complex crack paths as the need for re-meshing, enrichment of finite element shape functions and an explicit tracking of the crack surfaces. On the other hand, the use of Gridap makes the proposed implementation very compact and user-friendly that requires low memory usage, and provides a high degree of flexibility to the users in defining weak forms of partial differential equations. A test on a notched beam under symmetric three-point bending and a set of tests on a notched beam with three holes under asymmetric three-point bending is considered to demonstrate how the proposed Gridap based phase-field Julia code can be used to simulate fracture in brittle materials.
We analyze large sets of energy-release data created by stress-induced brittle fracture in a pure sapphire crystal at close to zero temperature where stochastic fluctuations are minimal. The waiting-time distribution follows that observed for fracture in rock and for earthquakes. Despite strong time correlations of the events and the presence of large-event precursors, simple prediction algorithms only succeed in a very weak probabilistic sense. We also discuss prospects for further cryogenic experiments reaching close to single-bond sensitivity and able to investigate the existence of a transition-stress regime.
We report moment distribution results from a laboratory earthquake fault experiment consisting of sheared elastic plates separated by a narrow gap filled with a two dimensional granular medium. Local measurement of strain displacements of the plates at over 800 spatial points located adjacent to the gap allows direct determination of the moments and their spatial and temporal distributions. We show that events consist of localized, larger brittle motions and spatially-extended, smaller non-brittle events. The non-brittle events have a probability distribution of event moment consistent with an $M^{-3/2}$ power law scaling. Brittle events have a broad, peaked moment distribution and a mean repetition time. As the applied normal force increases, there are more brittle events, and the brittle moment distribution broadens. Our results are consistent with mean field descriptions of statistical models of earthquakes and avalanches.
We prove that special functions of bounded deformation with small jump set are close in energy to functions which are smooth in a slightly smaller domain. This permits to generalize the decay estimate by De Giorgi, Carriero, and Leaci to the linearized context in dimension n and to establish the closedness of the jump set for local minimizers of the Griffith energy.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
