Strong Ellipticity and Infinitesimal Stability within N th-Order Gradient Elasticity

Victor A. Eremeyev ()
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Victor A. Eremeyev: Department of Civil and Environmental Engineering and Architecture (DICAAR), University of Cagliari, Via Marengo, 2, 09123 Cagliari, Italy

Mathematics, 2023, vol. 11, issue 4, 1-10

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

Keywords: strong ellipticity; strain gradient elasticity; infinitesimal stability; gradient elasticity of the Nth order; Dirichlet boundary conditions (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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