# Homogenization of many-body structures subject to large deformations

ESAIM: Control, Optimisation and Calculus of Variations (2012)

- Volume: 18, Issue: 1, page 91-123
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topStelzig, Philipp Emanuel. "Homogenization of many-body structures subject to large deformations." ESAIM: Control, Optimisation and Calculus of Variations 18.1 (2012): 91-123. <http://eudml.org/doc/272763>.

@article{Stelzig2012,

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.},

author = {Stelzig, Philipp Emanuel},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {homogenization; large deformations; contact mechanics; noninterpenetration; many-body structure; cord-belt; polyconvexity; brittle fracture; Γ-convergence; -convergence; Ciarlet-Nečas condition},

language = {eng},

number = {1},

pages = {91-123},

publisher = {EDP-Sciences},

title = {Homogenization of many-body structures subject to large deformations},

url = {http://eudml.org/doc/272763},

volume = {18},

year = {2012},

}

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

PB - EDP-Sciences

VL - 18

IS - 1

SP - 91

EP - 123

AB - 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.

LA - eng

KW - homogenization; large deformations; contact mechanics; noninterpenetration; many-body structure; cord-belt; polyconvexity; brittle fracture; Γ-convergence; -convergence; Ciarlet-Nečas condition

UR - http://eudml.org/doc/272763

ER -

## References

top- [1] R.A. Adams and J.J.F. Fournier, Sobolev spaces, Pure and Applied Mathematics (Amsterdam) 140. Elsevier/Academic Press, Amsterdam, 2nd edition (2003). Zbl1098.46001MR2424078
- [2] L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs, Oxford University Press, Oxford (2000). Zbl0957.49001MR1857292
- [3] M. Barchiesi and G. Dal Maso, Homogenization of fiber reinforced brittle materials : the extremal cases. SIAM J. Math. Anal.41 (2009) 1874–1889. Zbl1203.35025MR2564198
- [4] A. Braides, Γ-convergence for beginners, Oxford Lecture Series in Mathematics and its Applications 22. Oxford University Press, Oxford (2002). Zbl1198.49001MR1968440
- [5] 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. Zbl1098.35014MR2223360
- [6] A. Braides, A. Defranceschi and E. Vitali, Homogenization of free discontinuity problems. Arch. Rational Mech. Anal.135 (1996) 297–356. Zbl0924.35015MR1423000
- [7] P.G. Ciarlet, Mathematical elasticity. Three-dimensional elasticity I, Studies in Mathematics and its Applications 20. North-Holland Publishing Co., Amsterdam (1988). Zbl0648.73014MR936420
- [8] P.G. Ciarlet and J. Nečas, Injectivity and self-contact in nonlinear elasticity. Arch. Rational Mech. Anal.97 (1987) 171–188. Zbl0628.73043MR862546
- [9] G. Cortesani and R. Toader, A density result in SBV with respect to non-isotropic energies. Nonlinear Anal.38 (1999) 585–604. Zbl0939.49024MR1709990
- [10] G. Dal Maso, An introduction to Γ-convergence, Progress in Nonlinear Differential Equations and their Applications 8. Birkhäuser Boston Inc., Boston, MA (1993). Zbl0816.49001MR1201152
- [11] G. Dal Maso and G. Lazzaroni, Quasistatic crack growth in finite elasticity with non-interpenetration. Ann. Inst. Henri Poincaré Anal. Non Linéaire27 (2010) 257–290. Zbl1188.35205MR2580510
- [12] G. Dal Maso and C.I. Zeppieri, Homogenization of fiber reinforced brittle materials : the intermediate case. Adv. Calc. Var.3 (2010) 345–370. Zbl1200.49011MR2734192
- [13] L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions. Studies in Advanced Mathematics, CRC Press, Boca Raton, FL (1992). Zbl0804.28001MR1158660
- [14] G.A. Francfort and C.J. Larsen, Existence and convergence for quasi-static evolution in brittle fracture. Commun. Pure Appl. Math.56 (2003) 1465–1500. Zbl1068.74056MR1988896
- [15] A. Giacomini and M. Ponsiglione, Non-interpenetration of matter for SBV deformations of hyperelastic brittle materials. Proc. R. Soc. Edinb. Sect. A138 (2008) 1019–1041. Zbl1151.74034MR2477450
- [16] G.A. Iosifýan, Homogenization of problems in the theory of elasticity with Signorini boundary conditions. Mat. Zametki75 (2004) 818–833. Zbl1216.74015MR2085809
- [17] 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). Zbl0685.73002MR961258
- [18] D. Lukkassen, G. Nguetseng and P. Wall, Two-scale convergence. Int. J. Pure Appl. Math.2 (2002) 35–86. Zbl1061.35015MR1912819
- [19] A. Mikelić, M. Shillor and R. Tapiéro, Homogenization of an elastic material with inclusions in frictionless contact. Math. Comput. Model.28 (1998) 287–307. Zbl1126.74480
- [20] O. Pantz, The modeling of deformable bodies with frictionless (self-)contacts. Arch. Ration. Mech. Anal.188 (2008) 183–212. Zbl1138.74038MR2385740
- [21] L. Scardia, Damage as Γ-limit of microfractures in anti-plane linearized elasticity. Math. Models Methods Appl. Sci.18 (2008) 1703–1740. Zbl1157.74003MR2463778
- [22] L. Scardia, Damage as the Γ-limit of microfractures in linearized elasticity under the non-interpenetration constraint. Adv. Calc. Var.3 (2010) 423–458. Zbl1200.49013MR2734196
- [23] A. Signorini, Sopra alcune questioni di statica dei sistemi continui. Ann. Scuola Norm. Sup. Pisa Cl. Sci.2 (1933) 231–251. Zbl59.0738.01MR1556704JFM59.0738.01
- [24] 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. Zbl1303.74034MR2887929
- [25] J.Y. Wong, Theory of ground vehicles. John Wiley & Sons Inc., New York (2001).

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.