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Degeneration
Integral Functional
Taylor Expansion
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On the Commutability of Homogenization and Linearization in Finite Elasticity
On the Commutability of Homogenization and Linearization in Finite Elasticity,10.1007/s0020501104387,Archive for Rational Mechanics and Analysis,St
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On the Commutability of Homogenization and Linearization in Finite Elasticity
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Citations: 1
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Stefan Müller
,
Stefan Neukamm
We consider a family of nonconvex integral functionals $$\frac{1}{h^2}\int_\Omega W(x/\varepsilon,{\rm Id}+h\nabla g(x))\,\,{\rm d}x,\quad g\in W^{1,p}({\Omega};{\mathbb R}^n)$$ where W is a Carathéodory function periodic in its first variable, and nondegenerate in its second. We prove under suitable conditions that the Γlimits corresponding to linearization (h → 0) and homogenization ($${\varepsilon\rightarrow 0}$$) commute, provided W is minimal at the identity and admits a quadratic
Taylor expansion
at the identity. Moreover, we show that the homogenized integrand, which is determined by a multicell homogenization formula, has a quadratic
Taylor expansion
with a quadratic term that is given by the homogenization of the second variation of W.
Journal:
Archive for Rational Mechanics and Analysis  ARCH RATION MECH ANAL
, vol. 201, no. 2, pp. 465500, 2011
DOI:
10.1007/s0020501104387
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References
(15)
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R. Abeyaratne
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N. Triantafyllidis
Journal:
Journal of Applied Mechanicstransactions of The Asme  J APPL MECH
, vol. 51, no. 3, 1984
Homogenization and twoscale convergence
(
Citations: 496
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G. Allaire
Published in 1992.
The limit behavior of a family of variational multiscale problems
(
Citations: 5
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Margarida Ba ´ i
,
Irene Fonseca
Journal:
Indiana University Mathematics Journal  INDIANA UNIV MATH J
, vol. 56, no. 1, pp. 150, 2007
Homogenization of Multiple Integrals
(
Citations: 154
)
A. Braides
,
A. Defranceschi
Published in 1998.
A derivation of linear elastic energies from pairinteraction atomistic systems
(
Citations: 4
)
Andrea Braides
,
Margherita Solci
,
Enrico Vitali
Published in 2007.
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