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首页> 外文期刊>International Journal of Solids and Structures >A second strain gradient elasticity theory with second velocity gradient inertia - Part I: Constitutive equations and quasi-static behavior
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A second strain gradient elasticity theory with second velocity gradient inertia - Part I: Constitutive equations and quasi-static behavior

机译：第二速度梯度惯性的第二应变梯度弹性理论-第一部分：本构方程和准静态行为

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摘要

A multi-cell homogenization procedure with four geometrically different groups of cell elements (respectively for the bulk, the boundary surface, the edge lines and the corner points of a body) is envisioned, which is able not only to extract the effective constitutive properties of a material, but also to assess the "surface effects" produced by the boundary surface on the near bulk material. Applied to an unbounded material in combination with the thermodynamics energy balance principles, this procedure leads to an equivalent continuum constitutively characterized by (ordinary, double and triple) generalized stresses and momenta. Also, applying this procedure to a (finite) body suitably modelled as a simple material cell system, in association with the principle of the virtual power (PVP) for quasi-static actions, an equivalent structural system is derived, featured by a (macro-scale) PVP having the typical format as for a second strain gradient material model. Due to the surface effects, the latter model does work as a combination of two subsystems, i.e. the bulk material behaving as a Cauchy continuum, and the boundary surface operating as a membrane-like boundary layer, each subsystem being in (local and global) equilibrium by its own. Further, the applied (ordinary) boundary traction splits into two (response-dependent) parts, i.e. the "Cauchy traction" transmitted to the bulk material and the "Gurtin-Murdoch traction" acting, together with all other boundary tractions, upon the boundary layer. The role of the boundary layer as a two-dimensional manifold enclosing a Cauchy continuum is elucidated, also with the aid of a discrete model. A strain gradient elasticity theory is proposed which includes a minimum total potential energy principle featuring the relevant boundary-value problem for quasi-static loads and its (unique) solution. A simple application is presented. Two appendices are included, one reports the proof of the global equilibrium of the boundary layer, the other is concerned with double and triple stresses. The paper is complemented by a companion Part II one on dynamics. Previous findings by the author [Polizzotto, C.; 2012. A gradient elasticity theory for second-grade materials and higher order inertia. Int. J. Solids Struct. 49, 2121-2137] are improved and extended.
机译：设想了具有四个几何上不同的单元元素组（分别针对主体的体积，边界表面，边缘线和拐角点）的多单元均质化过程，该过程不仅能够提取结构单元的有效本构特性，一种材料，还可以评估边界表面在近大块材料上产生的“表面效应”。结合热力学能量平衡原理，将其应用于无边界材料，此过程将导致一个等效连续体，其特征在于（普通，双倍和三倍）广义应力和动量。同样，将此程序应用于适当建模为简单材料单元系统的（有限）物体，并结合用于准静态作用的虚拟功率（PVP）原理，得出了等效的结构系统，其特征为（宏）尺寸）具有典型的格式（如第二应变梯度材料模型）的PVP。由于表面效应，后一种模型的确可以作为两个子系统的组合工作，即散装材料表现为柯西连续体，而边界表面则充当膜状边界层，每个子系统都位于（局部和全局）自身平衡。此外，所施加的（普通）边界牵引力分为两个（取决于响应）部分，即传递给散装物料的“ Cauchy牵引力”和“ Gurtin-Murdoch牵引力”与所有其他边界牵引力一起作用于边界上。层。借助离散模型，也阐明了边界层作为封闭柯西连续体的二维流形的作用。提出了一种应变梯度弹性理论，该理论包括一个最小总势能原理，其特征是准静态载荷的相关边界值问题及其（唯一）解。提出了一个简单的应用程序。包括两个附录，一个附录报告了边界层的整体平衡的证明，另一个附录涉及双重和三次应力。这篇论文还辅以第二部分有关动力学的内容。作者先前的发现[Polizzotto，C .; 2012年。二级材料和高阶惯性的梯度弹性理论。诠释J.固体结构。 49，2121-2137]进行了改进和扩展。

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