Approximation results by difference schemes of fracture energies: the vectorial case.

Focardi, Matteo; Gelli, M. S.

doi:10.1007/s00030-003-1002-4

We provide a variational approximation, in the sense of De Giorgi's $\Gamma$-convergence, by finite-difference schemes of functionals of the type \[ \int_\Omega \psi(\nabla u)\dx+\displaystyle\int_{J_u}g\left(u^+-u^-,\nu_u\right) dH^2 \] defined for $u\in SBV(\Om;\rr^N)$, where $\Om$ is an open set in $R^3$. These energies are related to variational models in fracture mechanics for non-linear elastic materials. The approximating functionals are of the form $$ \int_{\ticaleps\cap\Om}\psi_\e\left(\nabla u(x)\right)\dx $$ where $\psi_\e$ is an interaction potential taking into account a separation of scales, $\ticaleps$ is a suitable regular triangulation of $R^3$ and $u$ is affine on each element of the assigned triangulation.

Approximation results by difference schemes of fracture energies: the vectorial case. / M. Focardi; M.S. Gelli. - STAMPA. - 10(2003), pp. 469-495.

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Titolo:	Approximation results by difference schemes of fracture energies: the vectorial case.
Autori di Ateneo:	FOCARDI, MATTEO M. S. Gelli mostra contributor esterni nascondi contributor esterni
Autori:	FOCARDI, MATTEO; M. S. Gelli
Data di pubblicazione:	2003
Rivista:	NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS
Volume:	10
Pagina iniziale:	469
Pagina finale:	495
Abstract:	We provide a variational approximation, in the sense of De Giorgi's $\Gamma$-convergence, by finite-difference schemes of functionals of the type \[ \int_\Omega \psi(\nabla u)\dx+\displaystyle\int_{J_u}g\left(u^+-u^-,\nu_u\right) dH^2 \] defined for $u\in SBV(\Om;\rr^N)$, where $\Om$ is an open set in $R^3$. These energies are related to variational models in fracture mechanics for non-linear elastic materials. The approximating functionals are of the form $$ \int_{\ticaleps\cap\Om}\psi_\e\left(\nabla u(x)\right)\dx $$ where $\psi_\e$ is an interaction potential taking into account a separation of scales, $\ticaleps$ is a suitable regular triangulation of $R^3$ and $u$ is affine on each element of the assigned triangulation.
Handle:	http://hdl.handle.net/2158/344569
Appare nelle tipologie:	1a - Articolo su rivista

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