Gradient elasticity and nonstandard boundary conditions

• Published in: International journal of solids and structures Vol. 40; no. 26; pp. 7399 - 7423
• Main Author: Polizzotto, Castrenze
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier Ltd 01.12.2003; Elsevier Science
• Subjects: Boundary conditions; Exact sciences and technology; Fundamental areas of phenomenology (including applications); Gradient elasticity; Nonlocal continua; Physics; Solid mechanics; Static elasticity; Static elasticity (thermoelasticity...); Structural and continuum mechanics; Gradient elasticity; Boundary conditions; Nonlocal continua
• ISSN: 0020-7683; 1879-2146
• DOI: 10.1016/j.ijsolstr.2003.06.001

Gradient elasticity for a second gradient model is addressed within a suitable thermodynamic framework apt to account for nonlocality. The pertinent thermodynamic restrictions upon the gradient constitutive equations are derived, which are shown to include, besides the field (differential) stress–strain laws, a set of nonstandard boundary conditions. Consistently with the latter thermodynamic requirements, a surface layer with membrane stresses is envisioned in the strained body, which together with the above nonstandard boundary conditions make the body constitutively insulated (i.e. no long distance energy flows out of the boundary surface due to nonlocality). The total strain energy is shown to include a bulk and surface strain energy. A minimum total potential energy principle is provided for the related structural boundary-value problem. The Toupin–Mindlin polar-type strain gradient material model is also addressed and compared with the above one, their substantial differences are pointed out, particularly for what regards the constitutive equations and the boundary conditions accompanying the solving displacement equilibrium equations. A gradient one-dimensional bar sample in tension is considered for a few applications of the proposed theory.