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.
Titolo: | Approximation results by difference schemes of fracture energies: the vectorial case. |
Autori di Ateneo: | |
Autori: | FOCARDI, MATTEO; M. S. Gelli |
Data di pubblicazione: | 2003 |
Rivista: | |
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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