On the Ellipticity of Static Equations of Strain Gradient Elasticity and Infinitesimal Stability

Conditions for the strong ellipticity of equilibrium equations are formulated within strain gradient elasticity under finite deformations. In this model, the strain energy density is a function of the first and second gradients of the position vector (deformation gradient). Ellipticity imposes certain constraints on the tangent elastic moduli. It is also closely related to infinitesimal stability, which is defined as the positive definiteness of the second variation of the potential-energy functional. The work considers the first boundary-value problem (with Dirichlet boundary conditions). For a 1D deformation, necessary and sufficient conditions for infinitesimal stability are determined, which are two inequalities for elastic moduli.