Variational Analysis of Fracture Mechanics
Summary
Variational analysis of fracture mechanics employs the calculus of variations to model the initiation and propagation of cracks through energy minimisation principles. In this framework, a solid body is described by displacement fields together with a set of discontinuities representing crack surfaces. The total energy comprises bulk elastic contributions and surface energies associated with newly created fracture surfaces. A key development is the formulation of free-discontinuity functionals, which allow for both smooth deformation and sharp cracks within a unified variational setting. Γ-convergence then provides a rigorous tool to pass from discrete or nonlinear models to effective continuum descriptions, capturing small-strain limits and homogenisation in heterogeneous materials. This approach has deepened understanding of Griffith’s criterion for brittle fracture, enabled multiscale analysis of composite and random media, and paved the way for computational methods that track evolving crack networks in complex geometries.
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Technical terms
Variational analysis: A mathematical framework that seeks the configuration of a system by minimising an energy functional over admissible fields and discontinuities.
Γ-convergence: A notion of convergence for functionals ensuring that minimisers of approximating problems converge to minimisers of the limit problem.
Free-discontinuity functional: An energy functional defined on pairs of deformation fields and crack sets, allowing the discontinuity set to vary freely in the minimisation process.
Homogenisation: The process of deriving effective macroscopic material properties by averaging microscopic heterogeneities, often via asymptotic analysis.
Griffith energy: The sum of bulk elastic energy and surface energy proportional to crack surface area, forming the basis of a variational criterion for brittle fracture.
References
- Γ-Convergence and Stochastic Homogenization of Second-Order Singular Perturbation Models for Phase Transitions. Journal of Nonlinear Science (2024).
- A Unified Model for Stress-Driven Rearrangement Instabilities. Archive for Rational Mechanics and Analysis (2020).
- A global method for deterministic and stochastic homogenisation in BV. Annals of PDE (2022).
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