Abstract

The present paper aims to develop geometrical approach for finite incompatible deformations arising in growing solids. The phenomena of incompatibility is modeled by specific affine connection on material manifold, referred to as material connection. It provides complete description of local incompatible deformations for simple materials. Meanwhile, the differential-geometric representation of such connection is not unique. It means that one can choose different ways for analytical definition of connection for single given physical problem. This shows that, in general, affine connection formalism provides greater potential than is required to the theory of simple materials (first gradient theory). For better understanding of this inconsistence it is advisable to study different ways for material connection formalization in details. It is the subject of present paper. Affine connection endows manifolds with geometric properties, in particular, with parallel transport on them. For simple materials the parallel transport is elegant mathematical formalization of the concept of a materially uniform (in particular, a stress-free) non-Euclidean reference shape. In fact, one can obtain a connection of physical space by determining the parallel transport as a transformation of the tangent vector, which corresponds to the structure of the physical space containing shapes of the body. One can alternatively construct affine connection of material manifold by defining parallel transport as the transformation of the tangent vector, in which its inverse image with respect to locally uniform embeddings does not change. Utilizing of the conception of material connections and the corresponding methods of non-Euclidean geometry may significantly simplify formulation of the initial-boundary value problems of the theory of incompatible deformations. Connection on the physical manifold is compatible with metric and Levi-Civita relations holds for it. Connection on the material manifold is considered in three alternative variants. The first leads to Weitzenböck space (the space of absolute parallelism or teleparallelism, i.e., space with zero curvature and nonmetricity, but with non-zero torsion) and gives a clear interpretation of the material connection in terms of the local linear transformations which transform an elementary volume of simple material into uniform state. The second one allows to choose the Riemannian space structure (with zero torsion and nonmetricity, but nonzero curvature) in material manifold and it is the most convenient way for deriving of field equations. The third variant is based on Weyl manifold with specified volume form and non-vanishing nonmetricity.

中文翻译：

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用于增长固体的材质仿射连接

摘要

本文旨在为增长的固体中产生的有限不相容变形开发几何方法。不相容现象是通过在材料歧管上的特定仿射连接（称为材料连接）来建模的。它提供了简单材料局部不兼容变形的完整描述。同时，这种连接的微分几何表示不是唯一的。这意味着对于一个给定的物理问题，人们可以选择不同的方式来分析定义连接。这表明，仿射连接形式主义通常提供比简单材料理论（第一梯度理论）所需的更大潜力。为了更好地理解这种不一致，建议详细研究材料连接形式化的不同方法。这是本文的主题。仿射连接使歧管具有几何特性，尤其是在其上进行并行传输。对于简单的材料，平行传输是对材料均匀（尤其是无应力）的非欧几里得基准形状的概念的精确数学形式化。实际上，可以通过将平行传输确定为切线向量的变换来获得物理空间的连接，该切线向量对应于包含人体形状的物理空间的结构。通过将平行传输定义为切向量的变换，可以选择构造材料歧管的仿射连接，在该切向量中，其相对于局部均匀嵌入的逆像不变。利用材料连接的概念和非欧几里得几何学的相应方法可以显着简化不相容变形理论的初边值问题的表述。物理歧管上的连接与公制兼容，因此Levi-Civita关系适用。物料歧管上的连接可考虑为三种替代方案。第一个引出到Weitzenböck空间（绝对平行或遥平行的空间，即曲率和非测度为零，但扭转为非零的空间），并根据局部线性变换对材料连接进行了清晰的解释，该局部线性变换将一个基本材料的基本体积变为均匀状态。第二种方法允许选择材料流形中的黎曼空间结构（具有零扭转和非度量，但曲率非零），这是导出场方程的最便捷方法。第三个变体基于具有指定体积形式和不消失的非对称性的Weyl流形。