On ellipticity of static equations of strain gradient elasticity and infinitesimal stability

Within the framework of strain gradient elasticity under finite deformations we formulate strong ellipticity conditions equilibrium equations. model a energy density is function first and second deformation gradients. Ellipticity involves certain constraints on tangent elastic moduli. It also closely related to infinitesimal stability which defined as positive definiteness variation potential functional. Here consider boundary-value problem, that with Dirichlettype boundary conditions. For one-dimensional determine necessary sufficient instability. The latter constitute two inequalities for