Linear elasticity obtained from finite elasticity by Γ-convergence under weak coerciveness conditions
Annales de l'I.H.P. Analyse non linéaire, Volume 29 (2012) no. 5, p. 715-735

The energy functional of linear elasticity is obtained as Γ-limit of suitable rescalings of the energies of finite elasticity. The quadratic control from below of the energy density W(v) for large values of the deformation gradient ∇v is replaced here by the weaker condition W(v)|v| p , for some p>1. Energies of this type are commonly used in the study of a large class of compressible rubber-like materials.

@article{AIHPC_2012__29_5_715_0,
     author = {Agostiniani, Virginia and Dal Maso, Gianni and DeSimone, Antonio},
     title = {Linear elasticity obtained from finite elasticity by $\Gamma$-convergence under weak coerciveness conditions},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {29},
     number = {5},
     year = {2012},
     pages = {715-735},
     doi = {10.1016/j.anihpc.2012.04.001},
     zbl = {1250.74008},
     mrnumber = {2971028},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2012__29_5_715_0}
}
Agostiniani, Virginia; Dal Maso, Gianni; DeSimone, Antonio. Linear elasticity obtained from finite elasticity by Γ-convergence under weak coerciveness conditions. Annales de l'I.H.P. Analyse non linéaire, Volume 29 (2012) no. 5, pp. 715-735. doi : 10.1016/j.anihpc.2012.04.001. http://www.numdam.org/item/AIHPC_2012__29_5_715_0/

[1] V. Agostiniani, A. Desimone, Γ-convergence of energies for nematic elastomers in the small strain limit, Contin. Mech. Thermodyn. 23 no. 3 (2011), 257-274 | MR 2795609 | Zbl 1272.76028

[2] V. Agostiniani, A. Desimone, Ogden-type energies for nematic elastomers, Internat. J. Non-Linear Mech. 47 (2012), 402-412

[3] S. Conti, G. Dolzmann, Γ-convergence for incompressible elastic plates, Calc. Var. Partial Differential Equations 34 no. 4 (2009), 531-551 | MR 2476423 | Zbl 1161.74038

[4] S. Conti, G. Dolzmann, S. Müller, Kornʼs second inequality and geometric rigidity with mixed growth conditions, arXiv:1203.1138 | MR 3194689

[5] G. Dal Maso, M. Negri, D. Percivale, Linearized elasticity as Γ-limit of finite elasticity, Set-Valued Anal. 10 (2002), 165-183 | MR 1926379 | Zbl 1009.74008

[6] G. Dal Maso, An Introduction to Γ-Convergence, Birkhäuser, Boston (1993) | MR 1201152

[7] A. Desimone, L. Teresi, Elastic energies for nematic elastomers, Eur. Phys. J. E 29 (2009), 191-204

[8] G. Friesecke, R.D. James, S. Müller, A theorem on geometric rigidity and the derivation on nonlinear plate theory from three-dimensional elasticity, Comm. Pure Appl. Math. 55 (2002), 1461-1506 | MR 1916989 | Zbl 1021.74024

[9] G.A. Holzapfel, Nonlinear Solid Mechanics: A Continuum Approach for Engineering, Wiley, Chichester (2000) | MR 1827472 | Zbl 0980.74001

[10] S. Müller, M. Palombaro, Derivation of a rod theory for biphase materials with dislocations at the interface, arXiv:1201.4290v1 | MR 3116013 | Zbl 1274.74064

[11] L. Scardia, C.I. Zeppieri, Gradient theory for plasticity as the Γ-limit of a nonlinear dislocation energy, http://cvgmt.sns.it/paper/1575/

[12] B. Schmidt, Linear Γ-limits of multiwell energies in nonlinear elasticity theory, Contin. Mech. Thermodyn. 20 no. 6 (2008), 375-396 | MR 2461715 | Zbl 1160.74321