# Derivation of a Linearised Elasticity Model from Singularly Perturbed Multiwell Energy Functionals

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## Abstract

Linear elasticity can be rigorously derived from finite elasticity under the assumption of small loadings in terms of Gamma-convergence. This was first done in the case of one-well energies with super-quadratic growth and later generalised to different settings, in particular to the case of multi-well energies where the distance between the wells is very small (comparable to the size of the load). In this paper we study the case when the distance between the wells is independent of the size of the load. In this context linear elasticity can be derived by adding to the multi-well energy a singular higher order term which penalises jumps from one well to another. The size of the singular term has to satisfy certain scaling assumptions whose optimality is shown in most of the cases. Finally, the derivation of linear elasticty from a two-well discrete model is provided, showing that the role of the singular perturbation term is played in this setting by interactions beyond nearest neighbours.

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## References

- 1.Agostiniani, V., Blass, T., Koumatos, K.: From nonlinear to linearized elasticity via \(\Gamma \)-convergence: the case of multiwell energies satisfying weak coercivity conditions. Math. Models Methods Appl. Sci.
**25**, 1–38 (2015)MathSciNetCrossRefzbMATHGoogle Scholar - 2.Agostiniani, V., Dal Maso, G., DeSimone, A.: Linearized elasticity obtained from finite elasticity by \(\Gamma \)-convergence under weak coerciveness conditions. Ann. Inst. H. Poincaré Anal. Non Linéaire
**29**, 715–735 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 3.Alicandro, R., Cicalese, M.: A general integral representation result for the continuum limits of discrete energies with superlinear growth. SIAM J. Math. Anal.
**36**, 1–37 (2004)MathSciNetCrossRefzbMATHGoogle Scholar - 4.Alicandro, R., Lazzaroni, G., Palombaro, M.: On the effect of interactions beyond nearest neighbours on non-convex lattice systems. Calc. Var. Partial Differ. Equ.
**56**, 42 (2017)MathSciNetCrossRefzbMATHGoogle Scholar - 5.Braides, A., Solci, M., Vitali, E.: A derivation of linear elastic energies from pair-interaction atomistic systems. Netw. Heterog. Media
**2**, 551–567 (2007)MathSciNetCrossRefzbMATHGoogle Scholar - 6.Chaudhuri, N., Müller, S.: Rigidity estimate for two incompatible wells. Calc. Var. Partial Differ. Equ.
**19**, 379–390 (2004)MathSciNetCrossRefzbMATHGoogle Scholar - 7.Conti, S., Schweizer, B.: Rigidity and gamma convergence for solid-solid phase transitions with \(SO(2)\) invariance. Commun. Pure Appl. Math.
**59**, 830–868 (2006)MathSciNetCrossRefzbMATHGoogle Scholar - 8.Dal Maso G.:
*An Introduction to*\(\Gamma \)*-Convergence. Progress in Nonlinear Differential Equations and Their Applications*, vol. 8. Birkhäuser, Boston (1993)Google Scholar - 9.Dal Maso, G., Negri, M., Percivale, D.: Linearized elasticity as \(\Gamma \)-limit of finite elasticity. Set-Valued Analysis
**10**, 165–183 (2002)MathSciNetCrossRefzbMATHGoogle Scholar - 10.De Lellis, C., Székelyhidi Jr., L.: Simple proof of two-well rigidity. C. R. Math. Acad. Sci. Paris
**343**, 367–370 (2006)MathSciNetCrossRefzbMATHGoogle Scholar - 11.Dolzmann, G., Müller, S.: Microstructures with finite surface energy: the two-well problem. Arch. Ration. Mech. Anal.
**132**, 101–141 (1995)MathSciNetCrossRefzbMATHGoogle Scholar - 12.Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton (1992)zbMATHGoogle Scholar
- 13.Friesecke, G., James, R.D., Müller, S.: A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. Commun. Pure Appl. Math.
**55**, 1461–1506 (2002)MathSciNetCrossRefzbMATHGoogle Scholar - 14.Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford Mathematical Monographs. Oxford Science Publications, New York (1993)zbMATHGoogle Scholar
- 15.Kato T.:
*Perturbation Theory for Linear Operators*. Springer, Berlin (1980) (second ed.; first ed., 1966)Google Scholar - 16.Modica, L., Mortola, S.: Un esempio di \(\Gamma \)-convergenza. Boll. Un. Mat. Ital. B
**5**(14), 285–299 (1977)MathSciNetzbMATHGoogle Scholar - 17.Schmidt, B.: Linear \(\Gamma \)-limits of multiwell energies in nonlinear elasticity theory. Contin. Mech. Thermodyn.
**20**, 375–396 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 18.Schmidt, B.: On the derivation of linear elasticity from atomistic models. Netw. Heterog. Media
**4**, 789–812 (2009)MathSciNetCrossRefzbMATHGoogle Scholar - 19.Ziemer W.P.:
*Weakly Differentiable Functions. Graduate Texts in Mathematics*, vol. 120. Springer, New York (1989)Google Scholar