On the role of the strong ellipticity condition in nonlinear elasticity
A class of nonlinear elastic materials is defined by an inequality (I), suggested by Gårding's inequality for elliptic systems. Some integral inequalities (J) involving the strain energy function W are derived from (I), assuming strong ellipticity of the material. Under additional restrictions on W, bounds on weak solutions of the boundary value problem of place (BVP) are obtained from (J). For special non-elliptic materials, inequalities are derived which would imply Lyapunov instability of solutions of the (BVP), according to results of \textit{R. J. Knops} and \textit{B. Straughan} [Trends Appl. pure Math. Mech., Collect. pap. present. Conf. Univ. Lecce 1975, 187-206 (1976; Zbl 0352.73044)]. The question whether these results apply to any reasonably realistic class of constitutive equations is not discussed.
Gårding's inequality for elliptic systems
scientific article; zbMATH DE number 3843229
scientific article