Stress average rule derived through the principle of virtual power
Stress average rule derived through the principle of virtual power
复制标题
通过虚拟功率原理推导出的应力平均规则
DOI：
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发表时间：
2022
期刊：
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通讯作者：
C. Hellmich
中科院分区：
文献类型：
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作者：
Nabor Jiménez Segura;B. Pichler;C. Hellmich
应力和应变平均规则是材料连续介质微观力学的重要概念支柱。上述规则表示，在微观有限代表体积单元（RVE）中的（微观）应力和（微观）应变场的空间平均值等于与相应的宏观无限小体积单元（宏观材料点）相关的（宏观）应力和（宏观）应变值。根据Hashin的著名贡献，应力和应变平均规则分别从平衡和协调条件以及与均匀（宏观）应变和（宏观）应力相关的（微观）位移和（微观）牵引边界条件导出。然而，严格地说，在边界上只能描述位移或牵引力，剩下的平均规则只是一个定义。我们在这里提出了一种方法，不需要这样的定义，通过诉诸虚拟力量（PVP）的原则作为保证机械平衡的手段：在RVE的边界处，我们规定虚拟（微观）速度，其与任意但均匀的虚拟（宏观）速度和（宏观）应变率相关联，而后者也以多线性方式相关联，与RVE内部的微观虚拟应变率场进行比较。在这些条件下，考虑到内力和外力的虚功率密度的宏观和微观表达式的等价性，得到了众所周知的应力平均规则，在微观均匀力场的情况下，得到了体积力平均规则。同样的策略应用于在原子质量点之间承载单个力的RVE，容易产生宏观的“内部维里应力张量”。
Stress and strain average rules are the key conceptual pillars of the wide field of continuum micromechanics of materials. The aforementioned rules express that the spatial average of (micro‐)stress and (micro‐)strain fields throughout a microscopically finite representative volume element (RVE) are equal to the (macro‐)stress and (macro‐)strain values associated with the corresponding macroscopically infinitesimal volume element (macroscopic material point). According to the famous contribution of Hashin, stress and strain average rules are derived from equilibrium and compatibility conditions, together with (micro‐)displacement and (micro‐)traction boundary conditions associated with homogeneous (macro‐)strains and (macro‐)stresses, respectively. However, as, strictly speaking, only displacements or tractions can be described at the boundary, the remaining average rule turns out as a mere definition. We here suggest a way to do without such a definition, by resorting to the principle of virtual power (PVP) as a means to guarantee mechanical equilibrium: at the boundary of the RVE, we prescribe virtual (micro‐)velocities, which are linked to arbitrary, but homogeneous virtual (macro‐)velocities and (macro‐)strain rates, while the latter are also linked, in a multilinear fashion, with the microscopic virtual strain rate fields inside the RVE. Considering, under these conditions, equivalence of the macroscopic and the microscopic expressions for the virtual power densities of the internal and the external forces yields the well‐known stress average rule and, in case of microscopically uniform force fields, a volume force average rule. The same strategy applied to an RVE hosting single forces between atomistic mass points, readily yields the macroscopic “internal virial stress tensor.”