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Symmetric Div-Quasiconvexity and the Relaxation of Static Problems


We consider problems of static equilibrium in which the primary unknown is the stress field and the solutions maximize a complementary energy subject to equilibrium constraints. A necessary and sufficient condition for the sequential lower-semicontinuity of such functionals is symmetric \(\mathrm{div}\)-quasiconvexity; a special case of Fonseca and Müller’s \(\mathcal {A}\)-quasiconvexity with \(\mathcal {A}= \mathrm{div}\) acting on \(\mathbb {R}^{n\times n}_\mathrm {sym}\). We specifically consider the example of the static problem of plastic limit analysis and seek to characterize its relaxation in the non-standard case of a non-convex elastic domain. We show that the symmetric \(\mathrm{div}\)-quasiconvex envelope of the elastic domain can be characterized explicitly for isotropic materials whose elastic domain depends on pressure p and Mises effective shear stress q. The envelope then follows from a rank-2 hull construction in the (pq)-plane. Remarkably, owing to the equilibrium constraint, the relaxed elastic domain can still be strongly non-convex, which shows that convexity of the elastic domain is not a requirement for existence in plasticity.

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  1. 1.

    Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems, Mathematical Monographs. Oxford University Press, Oxford 2000

  2. 2.

    Conti, S., Müller, S., Ortiz, S.: Data-driven problems in elasticity. Arch. Rational Mech. Anal. 229(1), 79–123, 2018

  3. 3.

    Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton 1992

  4. 4.

    Friesecke, G., James, R., Müller, S.: A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity. Commun. Pure Appl. Math55, 1461–1506, 2002

  5. 5.

    Fonseca, I., Müller, S.: \({\cal{A}}\)-quasiconvexity, lower semicontinuity, and Young measures. SIAM J. Math. Anal. 30(6), 1355–1390, 1999

  6. 6.

    Faraco, D., Székelyhidi, L.: Tartar’s conjecture and localization of the quasiconvex hull in \({\mathbb{R}}^{2\times 2}\). Acta Math. 200(2), 279–305, 2008

  7. 7.

    Garroni, A., Nesi, V.: Rigidity and lack of rigidity for solenoidal matrix fields. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460(2046), 1789–1806, 2004

  8. 8.

    Gurson, A.L.: Continuum theory of ductile rupture by void nucleation and growth: Part i–yield criteria and flow rules for porous ductile materials. J. Eng. Mater. Technol. 99, 2–15, 1977

  9. 9.

    Kristensen, J.: On the non-locality of quasiconvexity. Ann. Inst. H. Poincaré Anal. Non Linéaire16(1), 1–13, 1999

  10. 10.

    Lubliner, J.: Plasticity Theory. Macmillan, New York, London 1990

  11. 11.

    Meade, C., Jeanloz, R.: Effect of a coordination change on the strength of amorphous \(\text{ SiO }_2\). Science241(4869), 1072–1074, 1988

  12. 12.

    Müller, S., Palombaro, M.: On a differential inclusion related to the Born-Infeld equations. SIAM J. Math. Anal. 46(4), 2385–2403, 2014

  13. 13.

    Maloney, C.E., Robbins, M.O.: Evolution of displacements and strains in sheared amorphous solids. J. Phys. Conden. Matter20(24), 244128, 2008

  14. 14.

    Murat, F.: Compacité par compensation: condition necessaire et suffisante de continuite faible sous une hypothèse de rang constant. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 8, 69–102, 1981

  15. 15.

    Palombaro, M., Ponsiglione, M.: The three divergence free matrix fields problem. Asymptot. Anal. 40(1), 37–49, 2004

  16. 16.

    Palombaro, M., Smyshlyaev, V.P.: Relaxation of three solenoidal wells and characterization of extremal three-phase \(H\)-measures. Arch. Ration. Mech. Anal. 194(3), 775–722, 2009

  17. 17.

    Schill, W., Heyden, S., Conti, S., Ortiz, M.: The anomalous yield behavior of fused silica glass. J. Mech. Phys. Solids113, 105–125, 2018

  18. 18.

    Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton 1970

  19. 19.

    Schofield, A.N., Wroth, C.P.: Critical State Soil Mechanics. McGraw-Hill, New York City 1968

  20. 20.

    Stein, E.M., Weiss, G.: Introduction to Fourier analysis on Euclidean spaces. Princeton University Press, Princeton, N.J., Princeton Mathematical Series, No. 32. 1971

  21. 21.

    Tartar, L.: Compensated compactness and applications to partial differential equations. Nonlinear analysis and mechanics: Heriot–Watt Symp., Vol. 4, Edinburgh 1979, Res. Notes Math. 39, 136–212, 1979

  22. 22.

    Tartar, L.: The compensated compactness method applied to systems of conservation laws. In Systems of nonlinear partial differential equations (Oxford, 1982), volume 111 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., pages 263–285. Reidel, Dordrecht, 1983

  23. 23.

    Tartar, L.: Estimations fines des coefficients homogénéisés. In Ennio De Giorgi colloquium (Paris, 1983), volume 125 of Res. Notes in Math., pp. 168–187. Pitman, Boston, MA, 1985

  24. 24.

    Šverák, V.: Rank-one convexity does not imply quasiconvexity. Proc. Roy. Soc. Edinburgh Sect. A120(1–2), 185–189, 1992

  25. 25.

    Šverák, V.: On Tartar’s conjecture. Ann. Inst. H. Poincaré Anal. Non Linéaire10(4), 405–412, 1993

  26. 26.

    Zhang, K.: A construction of quasiconvex functions with linear growth at infinity. Ann. Scuola. Norm. Sup. Pisa Cl. Sci. (4)19(3), 313–326, 1992

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This work was partially supported by the Deutsche Forschungsgemeinschaft through the Sonderforschungsbereich 1060 “The mathematics of emergent effects”, project A5, and through the Hausdorff Center for Mathematics, GZ 2047/1, project-ID 390685813.

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Correspondence to M. Ortiz.

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Conti, S., Müller, S. & Ortiz, M. Symmetric Div-Quasiconvexity and the Relaxation of Static Problems. Arch Rational Mech Anal 235, 841–880 (2020).

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