Abstract
Peridynamics is a nonlocal model in Continuum Mechanics, and in particular Elasticity, introduced by Silling (2000). The nonlocality is reflected in the fact that points at a finite distance exert a force upon each other. If, however, those points are more distant than a characteristic length called horizon, it is customary to assume that they do not interact. We work in the variational approach of time-independent deformations, according to which, their energy is expressed as a double integral that does not involve gradients. We prove that the \(\Gamma \)-limit of this model, as the horizon tends to zero, is the classical model of hyperelasticity. We pay special attention to how the passage from the density of the non-local model to its local counterpart takes place.
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Aksoylu, B., Mengesha, T.: Results on nonlocal boundary value problems. Numer. Funct. Anal. Optim. 31, 1301–1317 (2010)
Andreu-Vaillo, F., Mazón, J.M., Rossi, J.D., Toledo-Melero, J.J.: Nonlocal diffusion problems, vol. 165 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (2010)
Antman, S.S.: Nonlinear problems of elasticity, vol. 107 of Applied Mathematical Sciences. Springer, New York (1995)
Attouch, H., Buttazzo, G., Michaille, G.: Variational analysis in Sobolev and BV spaces, vol. 6 of MPS/SIAM Series on Optimization. SIAM, Philadelphia (2006)
Ball, J.M.: Some recent developments in nonlinear elasticity and its applications to materials science. In: Aston, P. (ed.) Nonlinear mathematics and its applications (Guildford, 1995), pp. 93–119. Cambridge Univ. Press, Cambridge (1996)
Ball, J.M.: Some open problems in elasticity. In: Newton, P., Holmes, P., Weinstein, A. (eds.) Geometry, mechanics, and dynamics, pp. 3–59. Springer, New York (2002)
Bellido, J.C., Mora-Corral, C.: Existence for nonlocal variational problems in peridynamics. SIAM J. Math. Anal. 46, 890–916 (2014)
Bourgain, J., Brezis, H., Mironescu, P.: Another look at Sobolev spaces. In: Menaldi J.L., Rofman E., Sulem A. (eds.) Optimal Control and Partial Differential Equations, pp. 439–455. IOS Press (2001)
Braides, A.: A handbook of \({\Gamma }\)-convergence. In: Chipot M., Quittner P. (eds.) Handbook of Differential Equations: Stationary Partial Differential Equations, vol. 3, pp. 101–213. North-Holland (2006)
Brezis, H.: Analyse fonctionnelle. Masson, Paris (1983)
Ciarlet, P.G.: Mathematical elasticity. Vol. I, vol. 20 of Studies in Mathematics and its Applications. North-Holland Publishing Co., Amsterdam (1988)
Dacorogna, B.: Quasiconvexity and relaxation of nonconvex problems in the calculus of variations. J. Funct. Anal. 46, 102–118 (1982)
Dacorogna, B.: Direct methods in the calculus of variations. Applied Mathematical Sciences, vol. 78, 2nd edn. Springer, New York (2008)
Du, Q., Gunzburger, M., Lehoucq, R.B., Zhou, K.: Analysis of the volume-constrained peridynamic Navier equation of linear elasticity. J. Elasticity 113, 193–217 (2013)
Du, Q., Gunzburger, M., Lehoucq, R.B., Zhou, K.: A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws. Math. Models Methods Appl. Sci. 23, 493–540 (2013)
Elbau, P.: Sequential lower semi-continuity of non-local functionals. http://arxiv.org/abs/1104.2686
Emmrich, E., Weckner, O.: On the well-posedness of the linear peridynamic model and its convergence towards the Navier equation of linear elasticity. Commun. Math. Sci. 5, 851–864 (2007)
Evans, L.C., Gariepy, R.F.: Measure theory and fine properties of functions. CRC Press, Boca Raton (1992)
Fonseca, I., Leoni, G.: Modern methods in the calculus of variations: \(L^p\) spaces. Springer Monographs in Mathematics. Springer, New York (2007)
Fonseca, I., Müller, S., Pedregal, P.: Analysis of concentration and oscillation effects generated by gradients. SIAM J. Math. Anal. 29, 736–756 (1998)
Gobbino, M., Mora, M.G.: Finite-difference approximation of free-discontinuity problems. Proc. R. Soc. Edinburgh Sect. A 131, 567–595 (2001)
Gunzburger, M., Lehoucq, R.B.: A nonlocal vector calculus with application to nonlocal boundary value problems. Multiscale Model. Simul. 8, 1581–1598 (2010)
Hinds, B., Radu, P.: Dirichlet’s principle and wellposedness of solutions for a nonlocal \(p\)-Laplacian system. Appl. Math. Comput. 219, 1411–1419 (2012)
Lehoucq, R.B., Silling, S.A.: Force flux and the peridynamic stress tensor. J. Mech. Phys. Solids 56, 1566–1577 (2008)
Lipton, R.: Dynamic brittle fracture as a small horizon limit of peridynamics. J. Elasticity 117, 21–50 (2014)
Mengesha, T., Du, Q.: On the variational limit of a class of nonlocal convex functionals (Preprint)
Pedregal, P.: Variational methods in nonlinear elasticity. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2000)
Ponce, A.C.: An estimate in the spirit of Poincaré’s inequality. J. Eur. Math. Soc. 6, 1–15 (2004)
Ponce, A.C.: A new approach to Sobolev spaces and connections to \(\Gamma \)-convergence. Calc. Var. Partial Differ. Equ. 19, 229–255 (2004)
Silling, S.A.: Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48, 175–209 (2000)
Silling, S.A., Epton, M., Weckner, O., Xu, J., Askari, E.: Peridynamic states and constitutive modeling. J. Elasticity 88, 151–184 (2007)
Silling, S.A., Lehoucq, R.B.: Convergence of peridynamics to classical elasticity theory. J. Elasticity 93, 13–37 (2008)
Silling, S.A., Lehoucq, R.B.: Peridynamic theory of solid mechanics. In: Aref H., van der Giessen E. (eds.) Advances in Applied Mechanics, vol. 44 of Advances in Applied Mechanics, pp. 73–168. Elsevier (2010)
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Communicated by L. Ambrosio.
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Bellido, J.C., Mora-Corral, C. & Pedregal, P. Hyperelasticity as a \(\Gamma \)-limit of peridynamics when the horizon goes to zero. Calc. Var. 54, 1643–1670 (2015). https://doi.org/10.1007/s00526-015-0839-9
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DOI: https://doi.org/10.1007/s00526-015-0839-9