Our starting point is a parameterized family of functionals (a 'theory') for which we are interested in approximating the global minima of the energy when one of these parameters goes to zero. The goal is to develop a set of increasingly accurate asymptotic variational models allowing one to deal with the cases when this parameter is 'small' but finite. Since Gamma-convergence may be non-uniform within the 'theory', we pose a problem of finding a uniform approximation. To achieve this goal we propose a method based on rectifying the singular points in the parameter space by using a blow-up argument and then asymptotically matching the approximations around such points with the regular approximation away from them. We illustrate the main ideas with physically meaningful examples covering a broad set of subjects from homogenization and dimension reduction to fracture and phase transitions. In particular, we give considerable attention to the problem of transition from discrete to continuum when the internal and external scales are not well separated, and one has to deal with the so-called 'size' or 'scale' effects.

Braides, A., Truskinovsky, L. (2008). Asymptotic expansions by Gamma-convergence. CONTINUUM MECHANICS AND THERMODYNAMICS, 20(1), 21-62 [10.1007/s00161-008-0072-2].

Asymptotic expansions by Gamma-convergence

BRAIDES, ANDREA;
2008-01-01

Abstract

Our starting point is a parameterized family of functionals (a 'theory') for which we are interested in approximating the global minima of the energy when one of these parameters goes to zero. The goal is to develop a set of increasingly accurate asymptotic variational models allowing one to deal with the cases when this parameter is 'small' but finite. Since Gamma-convergence may be non-uniform within the 'theory', we pose a problem of finding a uniform approximation. To achieve this goal we propose a method based on rectifying the singular points in the parameter space by using a blow-up argument and then asymptotically matching the approximations around such points with the regular approximation away from them. We illustrate the main ideas with physically meaningful examples covering a broad set of subjects from homogenization and dimension reduction to fracture and phase transitions. In particular, we give considerable attention to the problem of transition from discrete to continuum when the internal and external scales are not well separated, and one has to deal with the so-called 'size' or 'scale' effects.
2008
Pubblicato
Rilevanza internazionale
Articolo
Sì, ma tipo non specificato
Settore MAT/05 - ANALISI MATEMATICA
English
Asymptotic expansions; Fracture; Gamma-convergence; Homogenization; Multi-scale modelling; Phase transitions; Uniformity
Braides, A., Truskinovsky, L. (2008). Asymptotic expansions by Gamma-convergence. CONTINUUM MECHANICS AND THERMODYNAMICS, 20(1), 21-62 [10.1007/s00161-008-0072-2].
Braides, A; Truskinovsky, L
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/27004
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