Abstract
We consider a three-dimensional elastic body whose material response function depends not only on the gradient of the deformation, but also on its second gradient. Using the elastic energy-momentum tensor as derived by Eshelby [2] we generalize a well-known uniqueness result of Knops and Stuart [8] for a Dirichlet boundary value problem associated with this response function.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
B. Dacorogna, Direct Methods in the Calculus of Variations. Springer, Berlin (1989).
J.D. Eshelby, The elastic energy-momentum tensor. J. Elasticity 5 (1975) 321–335.
L.C. Evans, Partial Differential Equations, Berkeley Mathematics Lecture Notes 3B. Berkeley (1994).
A.E. Green, On some general formulae in finite elastostatics. Arch. Rational Mech. Anal. 50 (1973) 73–80.
M.E. Gurtin, An Introduction to Continuum Mechanics. Academic Press, San Diego (1981).
R. Hill, Energy-momentum tensors in elastostatics: some reflexions on the general theory. J. Mech. Phys. Solids 34(3) (1986) 305–317.
F. John, Uniqueness of non-linear elastic equilibrium for prescribed displacements and sufficiently small strains. Comm. Pure Appl. Math. 25 (1972) 617–634.
R.J. Knops and C.A. Stuart, Quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity. Arch. Rational Mech. Anal. 86 (1984) 233–249.
J.K. Knowles and E. Sternberg, On a class of conservation laws in linearized and finite elastostatics. Arch. Rational Mech. Anal. 44 (1972) 187–211.
A. Mareno, Global continuation in higher-gradient nonlinear elasticity. Thesis, Cornell University (2002).
G.A. Maugin and C. Trimarco, Pseudomomentum and material forces in nonlinear elasticity: variational formulations and application to brittle fracture. Acta Mechanica 94 (1992) 1–28.
N.G. Meyers, Quasi-convexity and lower semi-continuity of multiple variational integrals of any order. Trans. Amer. Math. Soc. 119 (1965) 125–149.
R.D. Mindlin, Second gradient of strain and surface-tension in linear elasticity. Internat. J. Solids Struct. 1 (1965) 417–438.
C.B. Morrey, Jr., Multiple Integrals in the Calculus of Variations. Springer, Berlin (1966).
N. Triantafyllidis and S. Bardenhagen, Derivation of higher order gradient continuum theories in 2, 3-D non-linear elasticity from periodic lattice models. J. Mech. Phys. Solids 42 (1994) 111–139.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Mareno, A. Uniqueness of Equilibrium Solutions in Second-Order Gradient Nonlinear Elasticity. Journal of Elasticity 74, 99–107 (2004). https://doi.org/10.1023/B:ELAS.0000033865.03618.e0
Issue Date:
DOI: https://doi.org/10.1023/B:ELAS.0000033865.03618.e0