Continuous and discrete fracture models with energy gradient property
Prof. Masato Kimura, Kanazawa University
Sala conferenze IMATI-CNR - Tuesday, September 15, 2015 h.15:00
Abstract. We consider quasi-static fracture models of brittle material in
continuous (PDE) setting and in discrete (P1-FEM) one. In each case,
Francfort-Marigo type energy which is based on the classical Griffith
theory is introduced and a fracture/crack propagation model is derived
as a gradient flow of the energy of a damage variable with an
irreversible constraint. In the continuous setting, the sharp crack
profile is approximated by using the idea of Ambrosio-Tortorelli
regularization.
On the other hand, in the discrete setting, the elastic material is
descretized by means of P1 finite element method and the obtained
discrete system is regarded as a spring-block system. We introduce a
damage variable on each virtual spring and derive an ODE system of time
evolution of the damage variable as a gradient flow of a total energy.
We establish some fundamental mathematical property of the discrete
model, such as local and global existence and uniqueness of the solution,
some regularity estimates, energy gradient property, uniform estimates
of the several energies and crack length. On dimensional fracture-
vibration model is also discussed briefly.
Moreover, some interesting results on the consistency of the spring-
block system to the linear elasticity equation are presented. We reveal
a hidden symmetry of the elasticity tensor and a relation between the
positivity of the spring constant and the Poisson ratio of the elastic
material.
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