We give a first contribution to the homogenization of many-body structures that are exposed to large deformations and obey the noninterpenetration constraint. The many-body structures considered here resemble cord-belts like they are used to reinforce pneumatic tires. We establish and analyze an idealized model for such many-body structures in which the subbodies are assumed to be hyperelastic with a polyconvex energy density and shall exhibit an initial brittle bond with their neighbors. Noninterpenetration of matter is taken into account by the Ciarlet-Nečas condition and we demand deformations to preserve the local orientation. By studying Γ-convergence of the corresponding total energies as the subbodies become smaller and smaller, we find that the homogenization limits allow for deformations of class special functions of bounded variation while the aforementioned kinematic constraints are conserved. Depending on the many-body structures' geometries, the homogenization limits feature new mechanical effects ranging from anisotropy to additional kinematic constraints. Furthermore, we introduce the concept of predeformations in order to provide approximations for special functions of bounded variation while preserving the natural kinematic constraints of geometrically nonlinear solid mechanics.

Keywords: homogenization, large deformations, contact mechanics, noninterpenetration, many-body structure, cord-belt, polyconvexity, brittle fracture, Γ-convergence

@article{COCV_2012__18_1_91_0, author = {Stelzig, Philipp Emanuel}, title = {Homogenization of many-body structures subject to large deformations}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {91--123}, publisher = {EDP-Sciences}, volume = {18}, number = {1}, year = {2012}, doi = {10.1051/cocv/2010052}, mrnumber = {2887929}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv/2010052/} }

TY - JOUR AU - Stelzig, Philipp Emanuel TI - Homogenization of many-body structures subject to large deformations JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2012 DA - 2012/// SP - 91 EP - 123 VL - 18 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2010052/ UR - https://www.ams.org/mathscinet-getitem?mr=2887929 UR - https://doi.org/10.1051/cocv/2010052 DO - 10.1051/cocv/2010052 LA - en ID - COCV_2012__18_1_91_0 ER -

%0 Journal Article %A Stelzig, Philipp Emanuel %T Homogenization of many-body structures subject to large deformations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2012 %P 91-123 %V 18 %N 1 %I EDP-Sciences %U https://doi.org/10.1051/cocv/2010052 %R 10.1051/cocv/2010052 %G en %F COCV_2012__18_1_91_0

Stelzig, Philipp Emanuel. Homogenization of many-body structures subject to large deformations. ESAIM: Control, Optimisation and Calculus of Variations, Volume 18 (2012) no. 1, pp. 91-123. doi : 10.1051/cocv/2010052. http://www.numdam.org/articles/10.1051/cocv/2010052/

[1] Sobolev spaces, Pure and Applied Mathematics (Amsterdam) 140. Elsevier/Academic Press, Amsterdam, 2nd edition (2003). | MR | Zbl

and ,[2] Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs, Oxford University Press, Oxford (2000). | MR | Zbl

, and ,[3] Homogenization of fiber reinforced brittle materials : the extremal cases. SIAM J. Math. Anal. 41 (2009) 1874-1889. | MR | Zbl

and ,[4] Γ-convergence for beginners, Oxford Lecture Series in Mathematics and its Applications 22. Oxford University Press, Oxford (2002). | MR | Zbl

,[5] Another brick in the wall, in Variational problems in materials science, Progr. Nonlinear Differential Equation Appl. 68, Birkhäuser, Basel (2006) 13-24. | MR | Zbl

, and ,[6] Homogenization of free discontinuity problems. Arch. Rational Mech. Anal. 135 (1996) 297-356. | MR | Zbl

, and ,[7] Mathematical elasticity. Three-dimensional elasticity I, Studies in Mathematics and its Applications 20. North-Holland Publishing Co., Amsterdam (1988). | MR | Zbl

,[8] Injectivity and self-contact in nonlinear elasticity. Arch. Rational Mech. Anal. 97 (1987) 171-188. | MR | Zbl

and ,[9] A density result in SBV with respect to non-isotropic energies. Nonlinear Anal. 38 (1999) 585-604. | MR | Zbl

and ,[10] An introduction to Γ-convergence, Progress in Nonlinear Differential Equations and their Applications 8. Birkhäuser Boston Inc., Boston, MA (1993). | MR | Zbl

,[11] Quasistatic crack growth in finite elasticity with non-interpenetration. Ann. Inst. Henri Poincaré Anal. Non Linéaire 27 (2010) 257-290. | Numdam | MR | Zbl

and ,[12] Homogenization of fiber reinforced brittle materials : the intermediate case. Adv. Calc. Var. 3 (2010) 345-370. | MR | Zbl

and ,[13] Measure theory and fine properties of functions. Studies in Advanced Mathematics, CRC Press, Boca Raton, FL (1992). | MR | Zbl

and ,[14] Existence and convergence for quasi-static evolution in brittle fracture. Commun. Pure Appl. Math. 56 (2003) 1465-1500. | MR | Zbl

and ,[15] Non-interpenetration of matter for SBV deformations of hyperelastic brittle materials. Proc. R. Soc. Edinb. Sect. A 138 (2008) 1019-1041. | MR | Zbl

and ,[16] Homogenization of problems in the theory of elasticity with Signorini boundary conditions. Mat. Zametki 75 (2004) 818-833. | MR | Zbl

,[17] Contact problems in elasticity : a study of variational inequalities and finite element methods, SIAM Studies in Applied Mathematics 8. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1988). | MR | Zbl

and ,[18] Two-scale convergence. Int. J. Pure Appl. Math. 2 (2002) 35-86. | MR | Zbl

, and ,[19] Homogenization of an elastic material with inclusions in frictionless contact. Math. Comput. Model. 28 (1998) 287-307. | Zbl

, and ,[20] The modeling of deformable bodies with frictionless (self-)contacts. Arch. Ration. Mech. Anal. 188 (2008) 183-212. | MR | Zbl

,[21] Damage as Γ-limit of microfractures in anti-plane linearized elasticity. Math. Models Methods Appl. Sci. 18 (2008) 1703-1740. | MR | Zbl

,[22] Damage as the Γ-limit of microfractures in linearized elasticity under the non-interpenetration constraint. Adv. Calc. Var. 3 (2010) 423-458. | MR | Zbl

,[23] Sopra alcune questioni di statica dei sistemi continui. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 2 (1933) 231-251. | JFM | Numdam | MR

,[24] Homogenization of many-body structures subject to large deformations and noninterpenetration. Ph.D. Thesis, Technische Universität München (2009). Available electronically at http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:91-diss-20091214-797081-1-9. | Numdam | MR

,[25] Theory of ground vehicles. John Wiley & Sons Inc., New York (2001).

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