Abstract
Strong local minimizers with surfaces of gradient discontinuity appear in variational problems when the energy density function is not rank-one convex. In this paper we show that the stability of such surfaces is related to the stability outside the surface via a single jump relation that can be regarded as an interchange stability condition. Although this relation appears in the setting of equilibrium elasticity theory, it is remarkably similar to the well-known normality condition that plays a central role in classical plasticity theory.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
The choice of the orientation of the unit normal is unimportant as long as it is smooth. By convention, the unit normal points to the region labeled “\(+\)”.
The choice of unit ball as a support of the test function \(\varvec{\phi }\) is arbitrary. \(\varvec{\phi }\) can be supported in any bounded domain of \(\mathbb { R}^{d}\), see Ball (1976).
Projection onto the second component defines the same surface because of the symmetry of equations under the phase interchange \(\varvec{ F}_{+}\rightarrow \varvec{ F}_{-}\), \(\varvec{ F}_{-}\rightarrow \varvec{ F}_{+}\), \(\varvec{ n}\rightarrow -\varvec{ n}\).
In fact, \(\varvec{ P}_\mathrm{tot}(t)=t\varvec{ P}_{+}+(1-t)\varvec{ P}_{-}\), as was shown in Ball et al. (2000).
References
Ball, J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63(4):337–403 (1976/77)
Ball, J.M.: Some open problems in elasticity. In: Geometry, Mechanics, and Dynamics, pp. 3–59. Springer, New York (2002)
Ball, J.M., Kirchheim, B., Kristensen, J.: Regularity of quasiconvex envelopes. Calc. Var. Part. Differ. Equ. 11(4), 333–359 (2000)
Ball, J.M., Marsden, J.E.: Quasiconvexity at the boundary, positivity of the second variation and elastic stability. Arch. Ration. Mech. Anal. 86(3), 251–277 (1984)
Dacorogna, B.: Quasiconvexity and relaxation of nonconvex problems in the calculus of variations. J. Funct. Anal. 46(1), 102–118 (1982)
Dacorogna, B.: Direct Methods in the Calculus of Variations. Springer, New York (1989)
Erdmann, G.: Über die unstetige Lösungen in der Variationsrechnung. J. Reine Angew. Math. 82, 21–30 (1877)
Eshelby, J.D.: Energy relations and energy momentum tensor in continuum mechanics. In: Kanninen, M., Adler, W., Rosenfeld, A., Jaffee, R. (eds.) Inelastic Behavior of Solids, pp. 77–114. McGraw-Hill, New York (1970)
Fosdick, R., Volkmann, E.: Normality and convexity of the yield surface in nonlinear plasticity. Quart. Appl. Math. 51, 117–127 (1993)
Grabovsky, Y., Kucher, V.A., Truskinovsky, L.: Weak variations of lipschitz graphs and stability of phase boundaries. Contin. Mech. Thermodyn. (2010)
Grabovsky, Y., Truskinovsky, L.: Roughening instability of broken extremals. Arch. Ration. Mech. Anal. 200(1), 183–202 (2011)
Grabovsky, Y., Truskinovsky, L.: Marginal material stability. J. Nonlinear Sci. 23(5), 891–969 (2013)
Graves, L.M.: The weierstrass condition for multiple integral variation problems. Duke Math. J. 5(3), 656–660 (1939)
Grinfeld, M.A.: Stability of heterogeneous equilibrium in systems containing solid elastic phases. Dokl. Akad. Nauk SSSR 265(4), 836–840 (1982)
Grinfeld, M.A.: Stability of interphase boundaries in solid elastic media. Prikl. Mat. Mekh. 51(4), 628–637 (1987)
Gurtin, M.E.: Two-phase deformations of elastic solids. Arch. Ration. Mech. Anal. 84(1), 1–29 (1983)
Hadamard, J.: Mémoire sur le problème d’analyse relatif à l’équilibre des plaques élastiques encastrées, volume 33 of Mem. Acad. Sci. Paris. Imprimerie nationale (1908)
Hill, R.: Energy momentum tensors in elastostatics:some reflections on the general theory. J. Mech. Phys. Solids 34, 305–317 (1986)
Kinderlehrer, D., Pedregal, P.: Characterizations of Young measures generated by gradients. Arch. Ration. Mech. Anal. 115(4), 329–365 (1991)
Lubliner, J.: Plasticity Theory. Macmillan Publishing Company, New York (1990)
McShane, E.J.: On the necessary condition of weierstrass in the multiple integral problem of the calculus of variations. Ann. Math. 32(3), 578–590 (1931)
Morrey, J., Charles, B.: Multiple Integrals in the Calculus of Variations. Springer, New York Inc, New York (1966). Die Grundlehren der mathematischen Wissenschaften, Band 130
Pedregal, P.: Laminates and microstructure. Eur. J. Appl. Math. 4(6), 121–149 (1993)
Salman, O., Truskinovsky, L.: On the critical nature of plastic flow: one and two dimensional models. Int. J. Eng. Sci. 59, 219–254 (2012)
Šilhavý, M.: Maxwell’s relation for isotropic bodies. In: Mechanics of Material Forces, Volume 11 of Adv. Mech. Math., pp. 281–288. Springer, New York (2005)
Simpson, H.C., Spector, S.J.: On the positivity of the second variation in finite elasticity. Arch. Ration. Mech. Anal. 98(1), 1–30 (1987)
Simpson, H.C., Spector, S.J.: Some necessary conditions at an internal boundary for minimizers in finite elasticity. J. Elast. 26(3), 203–222 (1991)
Tartar, L.: Étude des oscillations dans les équations aux dérivées partielles non linéaires. In: Trends and Applications of Pure Mathematics to Mechanics (Palaiseau, 1983), pp. 384–412. Springer, Berlin (1984)
Acknowledgments
The authors are grateful to the anonymous referee for valuable comments and corrections. We also thank Bob Kohn for his suggestions. This material is based on work supported by the National Science Foundation under Grant 1008092 and the French ANR Grant EVOCRIT (2008–2012).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Robert V. Kohn.
Rights and permissions
About this article
Cite this article
Grabovsky, Y., Truskinovsky, L. Normality Condition in Elasticity. J Nonlinear Sci 24, 1125–1146 (2014). https://doi.org/10.1007/s00332-014-9213-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00332-014-9213-x