On Ellipticity Conditions Within Gradient Poroelasticity Under Finite Deformations = ОБ УСЛОВИЯХ ЭЛЛИПТИЧНОСТИ В ГРАДИЕНТНОЙ ТЕОРИИ ПОРОУПРУГОСТИ ПРИ КОНЕЧНЫХ ДЕФОРМАЦИЯХ

In this paper, we discuss strong ellipticity conditions within nonlinear elasticity. We consider a Cauchy-type simple material model and strain gradient elasticity. The constitutive equations of a simple material are formulated using the strain energy density, which is given as a function of the deformation gradient. In strain gradient elasticity, the strain energy density depends on the first and second gradients of strain. For strain gradient elasticity of third order, it depends on the first, second and third deformation gradients. We formulate strong ellipticity conditions and analyze their relation to infinitesimal stability. These conditions are defined using the strain energy density and its convexity with respect to a particular class of deformations. In gradient elasticity, the strong ellipticity conditions, as defined in the theory of partial differential equations, constrain the form of the constitutive equations and the deformations themselves, depending on the deformation gradient of the highest order. Infinitesimal stability is defined as the positive definiteness of the second variation of potential energy with respect to admissible displacements. We consider the relationship between ellipticity and infinitesimal stability for the first boundary-value problem, which is a boundary-value problem with Dirichlet boundary conditions. We demonstrate an essential difference between the considered models. For example, in a simple material, strong ellipticity implies stability of affine deformations within the first boundary-value problem. However, within strain-gradient elasticity, this statement is generally incorrect. A series of inequalities serves as sufficient conditions. As a particular case of gradient elasticity, we discuss gradient poroelasticity. In this theory, the strong ellipticity conditions are not met; however, there is still an ellipticity property in the sense of Douglis – Nirenberg.