有理力学中变形应力场和本构方程的表述
Formulation of Deformation Stress Fields and Constitutive Equations in Rational Mechanics
摘要：
在连续介质力学中，应力概念起着至关重要的作用。对于复杂材料，不同的应力概念在使用中存在歧义或理解不一。在几何上，材料单元由任意形状的封闭区域表示。内部区域由距离相关力（内部体力）作用，而表面由表面力作用。此外，作为一个整体的单元是在一个物理背景（外部区域），这是由连续的单元嵌入（外部体力）。在物理上，总能量可以叠加地分解为三个部分：内部区域能量、表面能和背景能量。但是，作为力，它们不能直接相加。在给出物理场的一般形式后，引入形变张量来表示形变引起的力的变化。由于力的变化由变形张量表示，变形应力的概念是很好的公式化。此外，作为一种自然的结果，加法分解给出了静态连续体的定义，它决定了本构方程中的材料参数。通过外微分，得到了一般形式的本构方程。在适当的情况下，本文将相关结果简化为经典结果，以便于理解。
英文摘要：
In continuum mechanics, stress concept plays an essential role. For complicated materials, different stress concepts are used with ambiguity or different understanding. Geometrically, a material element is expressed by a closed region with arbitral shape. The internal region is acted by distance dependent force (internal body force), while the surface is acted by surface force. Further more, the element as a whole is in a physical background (exterior region) which is determined by the continuum where the element is embedded (external body force). Physically, the total energy can be additively decomposed as three parts: internal region energy, surface energy, and the background energy. However, as forces, they cannot be added directly. After formulating the general forms of physical fields, the deformation tensor is introduced to formulate the force variations caused by deformation. As the force variation is expressed by the deformation tensor, the deformation stress concept is well formulated. Furthermore, as a natural result, the additive decomposition gives out the definition of static continuum, which determines the material parameters in constitutive equations. Through using the exterior differentials, the constitutive equations are formulated in general form. Throughout the paper, when it is suitable, the related results are simplified to classical results for easier understanding.
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