Multiscale Problems in Nonlinear Elasticity

Introduction

Quantitative Homogenization for small loads

In [NS] we study the problem of quantitative homogenization for nonlinearly elastic, periodic composites in a small load regime. The first result in this direction is the observation that under certain assumptions on the energy density and the (periodic) micro-structure, the multi-cell homogenization formula (2) simplifies to a single cell formula in an open neighborhood of the set of rotations. We deduce on that homogenization and linearization commute (on the level of integrands) in an open neighborhood of the set of rotations. Moreover, we establish a quantitative two-scale expansion for small loads, which is the first result in quantitative homogenization that applies to geometrically nonlinear elasticity. Our argument relies on the construction of a \textit{matching convex lower bound} for the stored energy function perturbed by a Null-Lagrangian. This construction has been established in a discrete setting in [FT,CDKM].

Commutability of Homogenization and Linearization

Simultaneous homogenization and dimension reduction (rods and plates)

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Publications

• [MN] Stefan Müller and Stefan Neukamm, On the commutability of homogenization and linearization in finite elasticity, Arch. Ration. Mech. Anal., 201 (2011), pp. 465–500.
• [GN] Antoine Gloria and Stefan Neukamm, Commutability of homogenization and linearization at identity in finite elasticity and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), pp. 941–964.
• [HNV] Peter Hornung, Stefan Neukamm and Igor Velcic, Derivation of a homogenized nonlinear plate theory from 3d elasticity, Calc. Var. Partial Differ. Equ., 51 (2014), pp. 677–699.
• [N] Stefan Neukamm, Rigorous derivation of a homogenized bending-torsion theory for inextensible rods from three-dimensional elasticity, Arch. Ration. Mech. Anal., 206 (2012), pp. 645–706.
• [NV] Stefan Neukamm and Igor Velcic, Derivation of a homogenized von-Karman plate theory from 3d nonlinear elasticity, Math. Models Methods Appl. Sci. 23 (2013), pp. 2701–2748.
• [NS] Stefan Neukamm and Mathias Schäffner, Quantitative homogenization in nonlinear elasticity for small loads, arXiv

References

• [B] Andrea Braides, Homogenization of some almost periodic functionals, Rend. Accad. Naz. Sci. XL, 103 (1985), pp. 313–322.
• [CDKM] Sergio Conti, Georg Dolzmann, Bernd Kirchheim and Stefan Müller, Sufficient conditions for the validity of the Cauchy-Born rule close to , J. Eur. Math. Soc. (JEMS), 8 (2006), pp. 515–539.
• [FT] Gero Friesecke and Florian Theil, Validity and failure of the Cauchy-Born hypothesis in a two-dimensional mass-spring lattice, J. Nonlinear Sci., 12 (2002), pp. 445–478.
• [GMT] Giuseppe Geymonat, Stefan Müller and Nicolas Triantafyllidis, Homogenization of non-linearly elastic materials, microscopic bifurcation and macroscopic loss of rank-one convexity, Arch. Rational Mech. Anal., 122 (1993), pp. 231–290.
• [M] Stefan Müller, Homogenization of nonconvex integral functionals and cellular elastic materials, Arch. Ration. Mech. Anal., 99 (1987), pp. 189–212.