Abstract—

The fundamentals of the theory of objective tensor mechanical characteristics are presented: the general concept of objectivity, types of objectivity, simple tensor analogs of mechanical characteristics and simple commutative diagrams connecting them. For objective tensors of the second rank, a general classification of simple diagrams is given, examples of diagrams are given.

The mappings connecting objective mechanical tensor processes of various ranks and types of objectivity are considered. The concept of mappings independent of the frame of reference is introduced, which form the basis of the theory of constitutive relations of media, and a theorem on the necessary and sufficient conditions for such independence is established. Examples are given.

For two arbitrary diagrams of objective tensor processes (generally speaking, of different ranks), a complete set (package) of related mappings (conductors) of all possible analogs of one diagram into all possible analogs of another diagram is considered—a package of conductors, or a mapping of diagrams. For coinciding diagrams (of the same rank with the same intertwining operators), a generalizing concept of objective derivatives is introduced that make up a subpackage of conductors connecting tensors of the same (all possible) types of objectivity.

General equations for the objective derivatives of scalars, vectors, and second-rank tensors are presented, presented in Lagrangian and Eulerian forms. It is shown that the well-known derivatives of Zaremba–Jaumann, Oldroyd, Cotter–Rivlin, Truesdell, Hill, Sedov, Dienes, Gordon–Schowalter are covered by these equations.

The objective integration operator is constructed as the inverse operator to the objective derivative.

Examples of the use of objective derivatives and objective integrals in the construction of the constitutive relations of media at finite deformations are given.

中文翻译：

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经典连续体力学中的目标张量及其映射

摘要-

提出了客观张量机械特性理论的基础：客观性的一般概念，客观性的类型，简单的机械特性张量类似物以及连接它们的简单交换图。对于第二等级的客观张量，给出了简单图的一般分类，并给出了图的示例。

考虑了连接各种等级和客观类型的客观机械张量过程的映射。介绍了独立于参照系的映射概念，该概念构成了媒体本构关系理论的基础，并建立了关于这种独立性的充要条件的定理。举了例子。

对于目标张量过程的两个任意图（通常而言，不同等级），考虑一个图的所有可能类似物到另一图的所有可能的类似物的相关映射（导体）的完整集合（包装）-导体的包装，或图表的映射。对于重合图（在相同等级下具有相同的交织算子），引入了目标导数的一般化概念，该概念构成了连接相同（所有可能）客观类型的张量的导体的子包。

给出了标量，向量和第二张量的目标导数的一般方程，以拉格朗日和欧拉形式表示。结果表明，Zaremba–Jaumann，Oldroyd，Cotter–Rivlin，Truesdell，Hill，Sedov，Dienes，Gordon–Schowalter的著名导数已被这些方程式覆盖。

目标积分算子被构造为目标导数的逆算子。

给出了在有限变形下构造本构关系时使用目标导数和目标积分的例子。