Homogenization of nonlinearly elastic materials, microscopic bifurcation and macroscopic loss of rank-one convexity

Geymonat, Giuseppe; Müller, Stefan; Triantafyllidis, Nicolas

doi:10.1007/bf00380256

Cited by 317 publications

(335 citation statements)

References 23 publications

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“…In this sense, we can say that Theorem 3.2.9 is a F-convergence-type result. On the link between F-convergence and Bloch waves, related results have been obtained in [17].…”

Section: G~(2)(sj) --- Gk(2)(sj) In [L2(f~)nl K~ Strongly As S ~ Omentioning

confidence: 87%

“…Recall that for the isotropic case (i.e., A = eI) we were able to obtain an explicit formula for ~or(2) (see (17)). In order to describe Yf(2) forj + 0, let us denote by 0 < )~ < • • -__< )~.~ the eigenvalues, and by (e~)l _<,, < u the corresponding eigenvectors of the Bloch-homogenized matrix A t. The spectral family :~:(2) (for j 4= 0) is defined by…”

Section: Denotes That Of S K Then For Any (St)o =< T _< K-1 ~ [L2(~mentioning

confidence: 99%

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Bloch-wave homogenization for a spectral problem in fluid-solid structures

Allaire

Conca

1996

Arch. Rational Mech. Anal.

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This paper is concerned with the study of the vibrations of a coupled fluid-solid periodic structure. As the period goes to zero, an asymptotic analysis of the spectrum (i.e., the set of eigenfrequencies) is performed with the help of a new method, the so-called Bloch-wave homogenization method (which is a blend of two-scale convergence and Bloch-wave decomposition). The limit spectrum is made of three parts: the macroscopic or homogenized spectrum, the microscopic or Bloch spectrum, and the boundary-layer spectrum. The two first parts are completely characterized: The homogenized and the Bloch spectra are purely essential, and have a band structure. The boundary-layer spectrum is shown to be empty in the special case of periodic boundary condition.

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“…In this sense, we can say that Theorem 3.2.9 is a F-convergence-type result. On the link between F-convergence and Bloch waves, related results have been obtained in [17].…”

Section: G~(2)(sj) --- Gk(2)(sj) In [L2(f~)nl K~ Strongly As S ~ Omentioning

confidence: 87%

Section: Denotes That Of S K Then For Any (St)o =< T _< K-1 ~ [L2(~mentioning

confidence: 99%

Bloch-wave homogenization for a spectral problem in fluid-solid structures

Allaire

Conca

1996

Arch. Rational Mech. Anal.

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“…In particular, W T and W AB are strongly elliptic as the sum of a strongly elliptic function on M 3 + (W AB andW T , respectively) and of an elliptic function (the Helmholtz energy density, which is polyconvex and hence elliptic). The strong ellipticity of W V is a much more delicate question, since strong ellipticity can be lost by homogenization in nonlinear elasticity (see in particular the very inspiring contribution of Geymonat, Müller, and Triantafyllidis in [13]). Provided some technical assumptions (which are supported by numerical simulations) and using the specific structure ofW c , the first author has shown in [14] that W V is strongly elliptic on M 3…”

Section: Remarkmentioning

confidence: 99%

Foundation, analysis, and numerical investigation of a variational network-based model for rubber

Gloria

Tallec

Vidrascu

2012

Continuum Mech. Thermodyn.

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“…for more details on multiscale modeling of failure, damage, and crack propagation and Refs. [430] and [447][448][449][450][451][452][453][454][455][456] for background on modeling instability phenomena such as buckling in the context of multiscale modeling.…”

Section: Beyond Purely Elastic Problemsmentioning

confidence: 99%

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

Saeb

Steinmann

Javili

2016

Applied Mechanics Reviews

190

138

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The objective of this contribution is to present a unifying review on strain-driven computational homogenization at finite strains, thereby elaborating on computational aspects of the finite element method. The underlying assumption of computational homogenization is separation of length scales, and hence, computing the material response at the macroscopic scale from averaging the microscopic behavior. In doing so, the energetic equivalence between the two scales, the Hill-Mandel condition, is guaranteed via imposing proper boundary conditions such as linear displacement, periodic displacement and antiperiodic traction, and constant traction boundary conditions. Focus is given on the finite element implementation of these boundary conditions and their influence on the overall response of the material. Computational frameworks for all canonical boundary conditions are briefly formulated in order to demonstrate similarities and differences among the various boundary conditions. Furthermore, we detail on the computational aspects of the classical Reuss' and Voigt's bounds and their extensions to finite strains. A concise and clear formulation for computing the macroscopic tangent necessary for FE 2 calculations is presented. The performances of the proposed schemes are illustrated via a series of two-and three-dimensional numerical examples. The numerical examples provide enough details to serve as benchmarks.

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Homogenization of nonlinearly elastic materials, microscopic bifurcation and macroscopic loss of rank-one convexity

Cited by 317 publications

References 23 publications

Bloch-wave homogenization for a spectral problem in fluid-solid structures

Bloch-wave homogenization for a spectral problem in fluid-solid structures

Foundation, analysis, and numerical investigation of a variational network-based model for rubber

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

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