We study stochastic homogenization by Γ-convergence of nonconvex integrals of the calculus of variations in the space of functions of bounded deformation.
Y.Abddaimi, G.Michaille and C.Licht, Stochastic homogenization for an integral functional of a quasiconvex function with linear growth, Asymptot. Anal.15(2) (1997), 183–202.
2.
M.A.Akcoglu and U.Krengel, Ergodic theorems for superadditive processes, J. Reine Angew. Math.323 (1981), 53–67.
3.
L.Ambrosio, A.Coscia and G.Dal Maso, Fine properties of functions with bounded deformation, Arch. Rational Mech. Anal.139(3) (1997), 201–238. doi:10.1007/s002050050051.
4.
N.Ansini and F.Bille Ebobisse, Homogenization of periodic multi-dimensional structures: The linearly elastic/perfectly plastic case, Adv. Math. Sci. Appl.11(1) (2001), 203–225.
5.
O.Anza Hafsa, N.Clozeau and J.-P.Mandallena, Homogenization of nonconvex unbounded singular integrals, Ann. Math. Blaise Pascal24(2) (2017), 135–193. doi:10.5802/ambp.367.
6.
O.Anza Hafsa and J.-P.Mandallena, Γ-Convergence of nonconvex integrals in Cheeger-Sobolev spaces and homogenization, Adv. Calc. Var.10(4) (2017), 381–405. doi:10.1515/acv-2015-0053.
7.
O.Anza Hafsa and J.P.Mandallena, Γ-Limits of functionals determined by their infima, J. Convex Anal.23(1) (2016), 103–137.
8.
A.Arroyo-Rabasa, G.De Philippis and F.Rindler, Lower semicontinuity and relaxation of linear-growth integral functionals under pde constraints, 2017, arXiv:1701.02230.
9.
J.-F.Babadjian, Traces of functions of bounded deformation, Indiana Univ. Math. J.64(4) (2015), 1271–1290. doi:10.1512/iumj.2015.64.5601.
10.
G.Bouchitté, Convergence et relaxation de fonctionnelles du calcul des variations à croissance linéaire. Application à l’homogénéisation en plasticité, Ann. Fac. Sci. Toulouse Math. (5)8(1) (1986/87), 7–36. doi:10.5802/afst.628.
11.
G.Bouchitté and M.Bellieud, Regularization of a set function – application to integral representation, Ricerche Mat.49(suppl) (2000), 79–93, Contributions in honor of the memory of Ennio De Giorgi (Italian).
12.
G.Bouchitté, I.Fonseca and L.Mascarenhas, A global method for relaxation, Arch. Rational Mech. Anal.145(1) (1998), 51–98. doi:10.1007/s002050050124.
13.
F.Cagnetti, G.Dal Maso, L.Scardia and C.Ida Zeppieri, Stochastic homogenisation of free-discontinuity problems, Arch. Ration. Mech. Anal.233(2) (2019), 935–974. doi:10.1007/s00205-019-01372-x.
14.
M.Caroccia, M.Focardi and N.Van Goethem, On the integral representation of variational functionals on , SIAM J. Math. Anal.52(4) (2020), 4022–4067. doi:10.1137/19M1277564.
15.
A.Cristina Barroso, I.Fonseca and R.Toader, A relaxation theorem in the space of functions of bounded deformation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)29(1) (2000), 19–49.
16.
R.De Arcangelis and G.Gargiulo, Homogenization of integral functionals with linear growth defined on vector-valued functions, NoDEA Nonlinear Differential Equations Appl.2(3) (1995), 371–416. doi:10.1007/BF01261182.
17.
G.De Philippis and F.Rindler, On the structure of -free measures and applications, Ann. of Math. (2)184(3) (2016), 1017–1039. doi:10.4007/annals.2016.184.3.10.
18.
G.De Philippis and F.Rindler, Characterization of generalized Young measures generated by symmetric gradients, Arch. Ration. Mech. Anal.224(3) (2017), 1087–1125. doi:10.1007/s00205-017-1096-1.
19.
G.De Philippis and F.Rindler, Fine properties of functions of bounded deformation – an approach via linear pdes, 2019, arXiv:1911.01356.
20.
I.Fonseca and S.Müller, Quasi-convex integrands and lower semicontinuity in , SIAM J. Math. Anal.23(5) (1992), 1081–1098. doi:10.1137/0523060.
21.
I.Fonseca and S.Müller, Relaxation of quasiconvex functionals in for integrands , Arch. Rational Mech. Anal.123(1) (1993), 1–49. doi:10.1007/BF00386367.
22.
G.A.Francfort and J.-J.Marigo, Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids46(8) (1998), 1319–1342. doi:10.1016/S0022-5096(98)00034-9.
23.
G.Dal Maso, An Introduction to Γ-Convergence, Progress in Nonlinear Differential Equations and Their Applications, Vol. 8, Birkhäuser Boston Inc., Boston, MA, 1993.
24.
G.Dal Maso and L.Modica, Nonlinear stochastic homogenization, Ann. Mat. Pura Appl. (4)144 (1986), 347–389. doi:10.1007/BF01760826.
25.
G.Dal Maso and L.Modica, Nonlinear stochastic homogenization and ergodic theory, J. Reine Angew. Math.368 (1986), 28–42.
26.
W.Han and B.Daya Reddy, Plasticity: Mathematical Theory and Numerical Analysis, 2nd edn, Interdisciplinary Applied Mathematics, Vol. 9, Springer, New York, 2013.
27.
M.Jesenko and B.Schmidt, Homogenization and the limit of vanishing hardening in Hencky plasticity with non-convex potentials, Calc. Var. Partial Differential Equations57(1) (2018), 2. doi:10.1007/s00526-017-1261-2.
28.
K.Kosiba and F.Rindler, On the relaxation of integral functionals depending on the symmetrized gradient, Proc. Roy. Soc. Edinburgh Sect. A151(2) (2021), 473–508. doi:10.1017/prm.2020.22.
29.
U.Krengel, Ergodic Theorems, De Gruyter Studies in Mathematics, Vol. 6, Walter de Gruyter & Co., Berlin, 1985, With a supplement by Antoine Brunel.
30.
C.Licht and G.Michaille, Global-local subadditive ergodic theorems and application to homogenization in elasticity, Ann. Math. Blaise Pascal9(1) (2002), 21–62. doi:10.5802/ambp.149.
31.
H.Matthies, G.Strang and E.Christiansen, The saddle point of a differential program, in: Energy Methods in Finite Element Analysis, Wiley, Chichester, 1979, pp. 309–318.
32.
K.Messaoudi and G.Michaille, Stochastic homogenization of nonconvex integral functionals. Duality in the convex case, Sém. Anal. Convexe21:Exp. No. 14 (1991), 32.
33.
K.Messaoudi and G.Michaille, Stochastic homogenization of nonconvex integral functionals, RAIRO Modél. Math. Anal. Numér.28(3) (1994), 329–356. doi:10.1051/m2an/1994280303291.
34.
J.Nečas and I.Hlaváček, Mathematical Theory of Elastic and Elasto-Plastic Bodies: An Introduction, Studies in Applied Mechanics., Vol. 3, Elsevier Scientific Publishing Co., Amsterdam–New York, 1980.
35.
F.Rindler, Lower semicontinuity for integral functionals in the space of functions of bounded deformation via rigidity and Young measures, Arch. Ration. Mech. Anal.202(1) (2011), 63–113. doi:10.1007/s00205-011-0408-0.
36.
P.-M.Suquet, Sur un nouveau cadre fonctionnel pour les équations de la plasticité, C. R. Acad. Sci. Paris Sér. A–B286(23) (1978), A1129–A1132.
37.
P.-M.Suquet, Un espace fonctionnel pour les équations de la plasticité, Ann. Fac. Sci. Toulouse Math. (5)1(1) (1979), 77–87. doi:10.5802/afst.531.
38.
R.Temam, Mathematical problems in plasticity theory, in: Variational Inequalities and Complementarity Problems, Proc. Internat. School, Erice, 1978, Wiley, Chichester, 1980, pp. 357–373.
39.
R.Temam, Problèmes mathématiques en plasticité, Méthodes Mathématiques de l’Informatique [Mathematical Methods of Information Science], Vol. 12, Gauthier-Villars, Montrouge, 1983.
40.
R.Temam and G.Strang, Existence de solutions relaxées pour les équations de la plasticité: étude d’un espace fonctionnel, C. R. Acad. Sci. Paris Sér. A–B287(7) (1978), A515–A518.
41.
R.Temam and G.Strang, Functions of bounded deformation, Arch. Rational Mech. Anal.75(1) (1980/81), 7–21. doi:10.1007/BF00284617.
