Strong Ellipticity and Infinitesimal Stability within Nth-Order Gradient Elasticity

We formulate a series of strong ellipticity inequalities for equilibrium equations the gradient elasticity up to Nth order. Within this model continuum, there exists deformation energy introduced as an objective function gradients As result, constitute system 2N-order nonlinear partial differential (PDEs). Using these boundary-value problem with Dirichlet boundary conditions, we prove positive definiteness second variation functional total energy. In other words, establish sufficient conditions infinitesimal instability. Here, restrict ourselves particular class deformations which includes affine deformations.