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Homogenization of many-body structures subject to large deformations

Published online by Cambridge University Press:  23 December 2010

Philipp Emanuel Stelzig
Philipp Emanuel Stelzig*
Affiliation:
Lehrstuhl für numerische Mathematik und Steuerungstheorie, Zentrum Mathematik, Technische Universität München, Boltzmannstrasse 3, 85748 Garching bei München, Germany. stelzig@ma.tum.de Dipartimento di Matematica, Università degli Studi di Trento, via Sommarive 14, 38050 Povo (TN), Italy
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Abstract

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

Homogenizationlarge deformationscontact mechanicsnoninterpenetrationmany-body structurecord-beltpolyconvexitybrittle fractureΓ-convergence
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 1 , January 2012 , pp. 91 - 123
Copyright
© EDP Sciences, SMAI, 2010

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References

R.A. Adams and J.J.F. Fournier, Sobolev spaces, Pure and Applied Mathematics (Amsterdam) 140. Elsevier/Academic Press, Amsterdam, 2nd edition (2003).
L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs, Oxford University Press, Oxford (2000).
Barchiesi, M. and Dal Maso, G., Homogenization of fiber reinforced brittle materials : the extremal cases. SIAM J. Math. Anal. 41 (2009) 1874–1889. Google Scholar
A. Braides, Γ -convergence for beginners, Oxford Lecture Series in Mathematics and its Applications 22. Oxford University Press, Oxford (2002).
A. Braides, and V. Chiadò Piat, Another brick in the wall, in Variational problems in materials science, Progr. Nonlinear Differential Equation Appl. 68, Birkhäuser, Basel (2006) 13–24.
Braides, A., Defranceschi, A. and Vitali, E., Homogenization of free discontinuity problems. Arch. Rational Mech. Anal. 135 (1996) 297–356. Google Scholar
P.G. Ciarlet, Mathematical elasticity. Three-dimensional elasticity I, Studies in Mathematics and its Applications 20. North-Holland Publishing Co., Amsterdam (1988).
Ciarlet, P.G. and Nečas, J., Injectivity and self-contact in nonlinear elasticity. Arch. Rational Mech. Anal. 97 (1987) 171–188. Google Scholar
Cortesani, G. and Toader, R., A density result in SBV with respect to non-isotropic energies. Nonlinear Anal. 38 (1999) 585–604. Google Scholar
G. Dal Maso, An introduction to Γ -convergence, Progress in Nonlinear Differential Equations and their Applications 8. Birkhäuser Boston Inc., Boston, MA (1993).
Dal Maso, G. and Lazzaroni, G., Quasistatic crack growth in finite elasticity with non-interpenetration. Ann. Inst. Henri Poincaré Anal. Non Linéaire 27 (2010) 257–290. Google Scholar
Dal Maso, G. and Zeppieri, C.I., Homogenization of fiber reinforced brittle materials : the intermediate case. Adv. Calc. Var. 3 (2010) 345–370. Google Scholar
L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions. Studies in Advanced Mathematics, CRC Press, Boca Raton, FL (1992).
Francfort, G.A. and Larsen, C.J., Existence and convergence for quasi-static evolution in brittle fracture. Commun. Pure Appl. Math. 56 (2003) 1465–1500. Google Scholar
Giacomini, A. and Ponsiglione, M., Non-interpenetration of matter for SBV deformations of hyperelastic brittle materials. Proc. R. Soc. Edinb. Sect. A 138 (2008) 1019–1041. Google Scholar
Iosifýan, G.A., Homogenization of problems in the theory of elasticity with Signorini boundary conditions. Mat. Zametki 75 (2004) 818–833. Google Scholar
N. Kikuchi and J.T. Oden, 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).
Lukkassen, D., Nguetseng, G. and Wall, P., Two-scale convergence. Int. J. Pure Appl. Math. 2 (2002) 35–86. Google Scholar
Mikelić, A., Shillor, M. and Tapiéro, R., Homogenization of an elastic material with inclusions in frictionless contact. Math. Comput. Model. 28 (1998) 287–307. Google Scholar
Pantz, O., The modeling of deformable bodies with frictionless (self-)contacts. Arch. Ration. Mech. Anal. 188 (2008) 183–212. Google Scholar
Scardia, L., Damage as Γ-limit of microfractures in anti-plane linearized elasticity. Math. Models Methods Appl. Sci. 18 (2008) 1703–1740. Google Scholar
Scardia, L., Damage as the Γ-limit of microfractures in linearized elasticity under the non-interpenetration constraint. Adv. Calc. Var. 3 (2010) 423–458. Google Scholar
Signorini, A., Sopra alcune questioni di statica dei sistemi continui. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 2 (1933) 231–251. Google Scholar
P.E. Stelzig, 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.
J.Y. Wong, Theory of ground vehicles. John Wiley & Sons Inc., New York (2001).