A gradient elasticity theory for second-grade materials and higher order inertia

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• 作者：Castrenze Polizzotto ^{castrenze.polizzotto@unipa.it}
• 关键词：Gradient elasticity ; Higher order inertia ; Continuum thermodynamics ; Dynamics ; Wave dispersion
• 刊名：International Journal of Solids and Structures
• 出版年：2012
• 期刊代码：93_00207683
• 类别：et
• 出版时间：August, 2012
• 卷：49
• 期：15-16
• 页码：2121-2137
• 文件大小：573 K

摘要

Second-grade elastic materials featured by a free energy depending on the strain and the strain gradient, and a kinetic energy depending on the velocity and the velocity gradient, are addressed. An m>inertial energy balance principlem> and a m>virtual work principle for inertial actionsm> are envisioned to enrich the set of traditional theoretical tools of thermodynamics and continuum mechanics. The state variables include the m>body momentumm> and the m>surface momentumm>, related to the velocity in a nonstandard way, as well as the concomitant m>mass-accelerationsm> and m>inertial forcesm>, which do intervene into the motion equations and into the force boundary conditions. The boundary traction is the sum of two parts, i.e. the m>Cauchy tractionm> and the m>Gurtin-Murdoch tractionm>, whereas the traction boundary condition exhibits the typical format of the equilibrium equation of a material surface (as known from the principles of surface mechanics) whereby the Gurtin-Murdoch traction (incorporating the inertial surface force) plays the role of applied surfacial force density. The body鈥檚 boundary surface constitutes a m>thin boundary layerm> which is in global equilibrium under all the external forces applied on it, a feature that makes it possible to exploit the traction Cauchy theorem within second-grade materials. This means that a second-grade material is formed up by two sub-systems, that is, the bulk material operating as a classical Cauchy continuum, and the thin boundary layer operating as a Gurtin-Murdoch material surface. The classical linear and angular momentum theorems are suitably extended for higher order inertia, from which the local motion equations and the moment equilibrium equations (stress symmetry) can be derived. For an isotropic material featured by four constants, i.e. the Lam茅 constants and two length scale parameters (Aifantis model), the dynamic evolution problem is characterized by a Hamilton-type variational principle and a solution uniqueness theorem. Closed-form solutions of the wave dispersion analysis problem for beam models are presented and compared with known results from the literature. The paper indicates a correct thermodynamically consistent way to take into account higher order inertia effects within continuum mechanics.