ON ELLIPTICITY OF EQUILIBRIUM EQUATIONS WITHIN NONLINEAR GRADIENT ELASTICITY OF N-th ORDER

Within the framework of strain gradient elasticity of n-th order we introduce ellipticity and strong ellipticity conditions. In this theory a density of potential energy depends on deformation gradients up to n-th order. As a result, static equations constitute a system of three nonlinear partial differential equations of order 2n with respect to vector of displacements. This model is used for description of long-range interactions, which are essential at small scales. Indeed, the strain gradient elasticity describes size-effects observed at the micro- and nano-scales. In nonlinear theory of elasticity the conditions of ordinary ellipticity and strong ellipticity could be treated as constitutive inequalities among others. In particular, ellipticity is related to an infinitesimal material instability. From the mathematical point of view, ellipticity is a natural property of equations of statics, which guarantees certain properties of corresponding boundary-value problems, such as, for example, smoothness of solutions, solvability, properties of spectrum. In comparison to nonlinear theory of elasticity the conditions of strong ellipticity in gradient media are less examined. Here ellipticity conditions imply constraints on the dependence of constitutive equations on the deformation gradient of n-th order. Precisely, ellipticity implies constraints on the tangent stiffness moduli of highest order and does not imply any on the dependence of deformation gradients of smaller order. Strain gradient elasticity of n-th order could be treated as a certain gradient regularization of the model of order n-1 for any number n. From this point of view one can avoid the violation of ellipticity considering deformation gradients of higher order.

gradient elasticity, ellipticity, strong ellipticity