In this work, we provide several insights which are related to the physical origins and the mathematical formulation of the principle of material frame-indifference in solid mechanics. The salient feature of the present approach relies crucially on the concrete understanding that when one models the ambient space as a rigid Euclidean space—one with a constant metric—misses important geometrical information; this information is related to the spacetime structure, the possible spacetime transformations and the spacetime–matter interaction. Upon abandoning this point of view and considering the ambient space as a flat manifold, we can see that material frame-indifference is a spacetime property which follows naturally from a physical postulate, namely that of Leibniz equivalence. We also analyze the connection which exists between material frame-indifference and relativity principles; this analysis vindicates Noll’s initial intuition to call the principle “principle of isotropy of space”. Finally, upon considering the spatial metric as a dynamical object which incorporates the deformation field, we present a formal introduction to the concept of general covariance in a constitutive theory.

中文翻译：

---

固体力学中的时空结构，时空变换和材料框架差异

在这项工作中，我们提供了一些与物理起源和实体力学中材料框架-差异原理的数学表述有关的见解。本方法的显着特征在很大程度上取决于具体的理解，即当将环境空间建模为刚性欧几里得空间（具有恒定度量）时，它会丢失重要的几何信息。这些信息与时空结构，可能的时空转换以及时空与物质的相互作用有关。抛弃这一观点并将环境空间视为平面流形时，我们可以看到，物质框架的差异是时空性质，它自然地遵循一种物理假设，即莱布尼兹当量。我们还分析了物质框架差异和相对论原理之间的联系。该分析证明了诺尔的最初直觉，称其原理为“空间各向同性原理”。最后，在将空间度量视为包含变形场的动力学对象之后，我们对本构理论中的一般协方差概念进行了正式介绍。