Energy minimization for a special class of incompressible elastic materials
- Univ. of Arizona, Tucson, AZ (United States)
It is well-known that the minimization of the potential energy in finite elasticity gives rise to several necessary conditions on any minimizing deformation field y({center_dot}):{Omega} {improper_subset}R{sup 3}{yields}R{sup 3}. Chief among these are the Euler-Lagrange (equilibrium) equations and the Weierstrass condition. Among other things, this latter condition distinguishes minimizing fields from other extrema and governs the coexistent phases allowable in the field y({center_dot}). Let D be an open subset of the set T{sup +} of second order tensors on R{sup 3} with positive determinant. For the special class of compressible elastic materials given by W({center_dot}): D {yields}R, W(F)= {omega}(J), J={vert_bar}F{vert_bar}, {omega}{center_dot}: T {yields} R, T= (J{vert_bar}J= {vert_bar}F{vert_bar}, F {element_of} D) {improper_subset} R{sup +}, Dunn & Fosdick found necessary and sufficient conditions on the generating function {omega}({center_dot}) for the Weierstrass condition W(F`)+W{sub F}(F`) {center_dot} a {direct_product} b {le} W(F` + a {direct_product} b) to hold at a given deformation gradient F` The connection between ordinary convexity for W({center_dot}) and the Weierstrass condition was also analyzed. In the present talk, we discuss the modifications of the methods and results of that are required if the materials of (*) are in fact incompressible, i.e., if the stored energy W({center_dot}) is now defined on D {improper_subset} U, U= (F {element_of} T{sup +} {vert_bar}det F= 1). This modification of the underlying constitutive domain q forces significant changes in the results of which we give in detail and which we then use to arrive at conditions that ensure that certain equilibrium states in these materials are in fact energy minimizers.
- OSTI ID:
- 175440
- Report Number(s):
- CONF-950686--
- Country of Publication:
- United States
- Language:
- English
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