A relaxation process for bifunctionals of displacement-Young measure state variables: A model of multi-material with micro-structured strong interface

Bessoud, Anne Laure; Krasucki, Françoise; Michaille, Gérard

Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 2, pp. 447-469.

The gradient displacement field of a micro-structured strong interface of a three-dimensional multi-material is regarded as a gradient-Young measure so that the stored strain energy of the material is defined as a bifunctional of displacement-Young measure state variables. We propose a new model by computing a suitable variational limit of this bifunctional when the thickness and the stiffness of the strong material are of order ε and $\frac{1}{ϵ}$ respectively. The stored strain energy functional associated with the model in pure displacements living in a Sobolev space is obtained as the marginal map of the limit bifunctional. We also obtain a new asymptotic formulation in terms of Young measure state variable when considering the other marginal map.

Classification : 49J45, 74N15, 35B40
Mots clés : Micro structures, Young measures, Variational convergences, Γ-convergence

[1] E. Acerbi, G. Buttazzo, D. Percivale, Thin inclusions in linear elasticity: A variational approach, J. Reine Angew. Math. 386 (1988), 99-115 | EuDML | MR | Zbl

[2] O. Anza Hafsa, J.P. Mandallena, Interchange of infimum and integral, Calc. Var. Partial Differential Equations 18 (2003), 433-449 | MR | Zbl

[3] O. Anza Hafsa, J.P. Mandallena, The nonlinear membrane energy: Variational derivation under the constraint “$\mathrm{𝑑𝑒𝑡}\nabla u\ne 0^{″}$, J. Math. Pures Appl. (9) 86 no. 2 (2006), 100-115 | MR | Zbl

[4] O. Anza Hafsa, J.P. Mandallena, The nonlinear membrane energy: Variational derivation under the constraint “ $\mathrm{𝑑𝑒𝑡}\nabla u>0^{″}$, Bull. Sci. Math. 132 no. 4 (2008), 272-291 | MR | Zbl

[5] H. Attouch, Variational Convergence for Functions and Operators, Appl. Math. Ser., Pitman Advanced Publishing Program (1985) | MR | Zbl

[6] H. Attouch, G. Buttazzo, G. Michaille, Variational Analysis in Sobolev and BV Space: Application to PDEs and Optimization, MPS/SIAM Ser. Optim. vol. 6 (December 2005) | MR

[7] E.J. Balder, Lectures on Young measures theory and its applications in economics, Workshop di Teoria della Misura e Analisi Reale Grado, 1997 Rend. Istit. Mat. Univ. Trieste 31 no. Suppl. 1 (2000), 1-69 | Zbl

[8] J.M. Ball, A version of the fundamental theorem for Young measures, M. Rascle, D. Serre, M. Slemrod (ed.), PDE's and Continuum Models of Phase Transitions, Lecture Notes in Phys. vol. 344, Springer-Verlag, New York (1989), 207-215

[9] J.M. Ball, R.D. James, Fine phase mixtures as minimizers of energy, Arch. Ration. Mech. Anal. 100 (1987), 13-52 | MR | Zbl

[10] A.L. Bessoud, F. Krasucki, G. Michaille, Multimaterial with strong interface: Variational modelings, Asymptot. Anal. 61 no. 1 (2009), 1-19 | MR | Zbl

[11] K. Bhattacharya, R.D. James, A theory of thin films of martensitic materials with applications to microactuators, J. Mech. Phys. Solids 47 (1999), 531-576 | MR | Zbl

[12] M. Bocea, I. Fonseca, A Young measure approach to a nonlinear membrane model involving the bending moment, Proc. Roy. Soc. Edinburgh Sect. A 134 no. 5 (2004), 845-883 | MR | Zbl

[13] C. Castaing, P. Raynaud De Fitte, M. Valadier, Young Measure on Topological Spaces with Applications in Control Theory and Probability Theory, Mathematics and Its Applications vol. 571, Kluwer Academic Publisher (2004) | MR | Zbl

[14] C. Castaing, M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Math. vol. 580 (1977) | MR | Zbl

[15] B. Dacorogna, Direct Methods in the Calculus of Variations, Appl. Math. Sci. vol. 78, Springer-Verlag, Berlin (1989) | MR | Zbl

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

[17] L. Fredi, R. Paroni, The energy density of martensitic thin films via dimension reduction, Interfaces Free Bound. 6 (2004), 439-459 | MR | Zbl

[18] I. Fonseca, S. Müller, P. Pedregal, Analysis of concentration and oscillation effects generated by gradients, SIAM J. Math. Anal. 29 no. 3 (1998), 736-756 | MR | Zbl

[19] D. Kinderlehrer, P. Pedregal, Characterization of Young measures generated by gradients, Arch. Ration. Mech. Anal. 119 (1991), 329-365 | MR | Zbl

[20] H. Le Dret, A. Raoult, The nonlinear membrane model as Variational limit in nonlinear three-dimensional elasticity, J. Math. Pures Appl. (9) 74 no. 6 (1995), 549-578 | MR | Zbl

[21] M.L. Leghmizi, C. Licht, G. Michaille, The nonlinear membrane model: A Young measure and varifold formulation, ESAIM Control Optim. Calc. Var. 11 (July 2005), 449-472 | EuDML | Numdam | MR

[22] E. Mascolo, L. Migliaccio, Relaxation methods in control theory, Appl. Math. Optim. 20 no. 1 (1989), 97-103 | MR | Zbl

[23] P. Pedregal, Parametrized Measures and Variational Principle, Birkhäuser, Boston (1997) | MR | Zbl

[24] M.A. Sychev, A new approach to Young measure theory, relaxation and convergence in energy, Ann. Inst. H. Poincaré 16 no. 6 (1999), 773-812 | EuDML | Numdam | MR | Zbl

[25] L. Tartar, H-measures, a new approach for studying homogenization, oscillations and concentration effects in partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A 115 (1990), 193-230 | Zbl

[26] M. Valadier, Young measures, A. Cellina (ed.), Methods of Nonconvex Analysis, Lecture Notes in Math. vol. 1446, Springer-Verlag, Berlin (1990), 152-188 | MR | Zbl

[27] M. Valadier, A course on Young measures, Workshop di Teoria della Misura e Analisi Reale Grado, September 19–October 2, 1993 Rend. Istit. Mat. Univ. Trieste 26 no. Suppl. (1994), 349-394 | Zbl