In the present paper, theoretical foundations of redundancy in spatially continuous, elastostatic, and linear representations of structures are derived. Adopting an operator-theoretical perspective, the redundancy operator is introduced, inspired by the concept of redundancy matrices, previously described for spatially discrete representations of structures. Studying symmetry, trace, rank, and spectral properties of this operator as well as revealing the relation to the concept of statical indeterminacy, a continuum-mechanical theory of redundancy is proposed. Here, the notion “continuum-mechanical” refers to the representation being spatially continuous. Apart from the theory itself, the novel outcome is a clear concept providing information on the distribution of statical indeterminacy in space and with respect to different load carrying mechanisms. The theoretical findings are confirmed and illustrated within exemplary rod, plane beam, and plane frame structures. The additional insight into the load carrying behavior may be valuable in numerous applications, including robust design of structures, quantification of imperfection sensitivity, assessment of adaptability, as well as actuator placement and optimized control in adaptive structures.

在本文中，导出了结构的空间连续、弹性和线性表示中冗余的理论基础。采用算子理论视角，受冗余矩阵概念的启发，引入了冗余算子，该概念之前描述了结构的空间离散表示。研究该算子的对称性、迹、秩和谱性质，并揭示与静态不确定性概念的关系，提出了冗余的连续介质力学理论。这里，概念“连续机械”指的是空间连续的表示。除了理论本身，新的成果是一个清晰的概念，提供了关于空间静态不确定性分布和不同承载机制的信息。理论发现在示例性杆、平面梁和平面框架结构中得到证实和说明。对承载行为的额外了解在许多应用中可能很有价值，包括结构的稳健设计、缺陷敏感性的量化、适应性评估，以及自适应结构中的致动器放置和优化控制。