References
Agmon, S., On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems. Comm. Pure Appl. Math. 15 (1962), 119–147.
Agmon, S., A. Douglis, & L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II. Comm. Pure Appl. Math. 17 (1964), 35–92.
Ball, J. M., Constitutive inequalities and existence theorems in nonlinear elastostatics. Nonlinear Analysis and Mechanics Vol. I (R. J. Knops, ed.). London: Pitman, 1977.
Ball, J. M., & R. D. James, Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal. 100 (1987), 13–52.
Ball, J. M., & J. E. Marsden, Quasiconvexity at the boundary, positivity of the second variation, and elastic stability. Arch. Rational Mech. Anal. 86 (1984), 251–277.
Biot, M. A., Mechanics of Incremental Deformations. New York: Wiley, 1965.
Chen, Y.-C., Stability of pure homogeneous deformations of an elastic plate with fixed edges. Q. J. Mech. Appl. Math. 41 (1988), 249–264.
De Figueiredo, D. G., The coerciveness problem for forms over vector valued functions. Comm. Pure Appl. Math. 16 (1963), 63–94.
Duffin, R. J., Crystal's theorem on differential equation systems. J. Math. Anal. Appl. 8 (1963), 325–331.
Gurtin, M. E., Two-phase deformations of elastic solids. Arch. Rational Mech. Anal. 84 (1983), 1–29.
Hadamard, J., Leçons sur la Propagation des Ondes et les Equations de l'Hydrodynamique. Paris: Hermann, 1903.
Hayes, M., & R. S. Rivlin, Propagation of a plane wave in an isotropic elastic material subjected to pure homogeneous deformation. Arch. Rational Mech. Anal. 8 (1961), 15–22.
Hutchinson, J. W.. & V. Tvergaard, Surface instabilities on statically strained plastic solids. Int. J. Mech. Sci. 22 (1980), 339–354.
Meyers, N. G., Quasi-convexity and lower semi-continuity of multiple variational integrals of any order. Trans. Amer. Math. Soc. 119 (1965), 125–149.
Morrey, C. B., Quasi-convexity and the lower semicontinuity of multiple integrals. Pacific J. Math. 2 (1952), 25–53.
Nowinski, J. L., Surface instability of a half-space under high two-dimensional compression. J. Franklin Inst. 288 (1969), 367–376.
Reddy, B. D., Surface instabilities on an equibiaxially stretched elastic half-space. Math. Proc. Camb. Phil. Soc. 91 (1982), 491–501.
Simpson, H. C., & S. J. Spector, On failure of the complementing condition and nonuniqueness in linear elastostatics. J. Elasticity 15 (1985), 229–331.
Simpson, H. C., & S. J. Spector, On the positivity of the second variation in finite elasticity. Arch. Rational Mech. Anal. 98 (1987), 1–30.
Thompson, J. L., Some existence theorems for the traction boundary value problem of linearized elastostatics. Arch. Rational Mech. Anal. 32 (1969), 369–399.
Truesdell, C., & W. Noll, The Non-linear Field Theories of Mechanics. Handbuch der Physik III/3 (S. Flügge, ed.). Berlin Heidelberg New York: Springer-Verlag, 1965.
Van Hove, L., Sur l'extension de la condition de Legendre du calcul des variations aux intégrales multiples à plusieurs fonctions inconnues. Proc. Koninklijke Nederlandsche Akademie Van Wetenschappen 50 # 1 (1947), 18–23.
Author information
Authors and Affiliations
Additional information
Dedicated to Bernard Coleman on the occasion of his 60th birthday
Rights and permissions
About this article
Cite this article
Simpson, H.C., Spector, S.J. Necessary conditions at the boundary for minimizers in finite elasticity. Arch. Rational Mech. Anal. 107, 105–125 (1989). https://doi.org/10.1007/BF00286496
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00286496