Martedì 15 Settembre 2015, alle ore 15 Prof. Masato Kimura (Kanazawa University)


Martedì 15 Settembre 2015, alle ore 15 precise, presso la sala
conferenze dell'IMATI-CNR di Pavia, il

Prof. Masato Kimura (Kanazawa University)

terrà un seminario dal titolo:

"Continuous and discrete fracture models with energy gradient property"

nell'ambito del Seminario di Matematica Applicata (IMATI-CNR e
Dipartimento di Matematica, Pavia).

Al termine della conferenza sarà organizzato un piccolo rinfresco.


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

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

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