The purpose of this paper is to demonstrate how the general problem of interconversion between parallel and orthogonal superposition protocols can be treated using the kernels in a Fréchet series expansion about the base viscometric flow. Such series differ from Fréchet series expanded about the rest history which are encountered in the theory of Green–Rivlin materials and the simple fluid theory of Coleman and Noll, in that nonlinear response of the material is captured at first order. Unlike first and second-order functional derivatives evaluated at the rest history, the derivatives we discuss require more than one kernel in their integral representation. However, all the kernels are inter-related. The strategy in the paper involves identifying the kernels which specify the components in the first and second order Fréchet derivatives for the nonlinear constitutive models to be used in the interconversion. Interconversion between parallel and orthogonal protocols can then be effected by establishing the relationships between the kernels. Step-strain perturbations are treated by allowing differentiation in the sense of distributions. The theory is illustrated throughout by evaluating the kernels and superposition moduli associated with the incompressible corotational Maxwell and Oldroyd models.

本文的目的是演示如何使用基本粘度流的Fréchet级数展开式中的内核来解决并行和正交叠加协议之间的相互转换的一般问题。这样的级数不同于Green-Rivlin材料理论以及简单的Coleman和Noll流体理论中遇到的剩余历史的Fréchet级数，因为材料的非线性响应是一阶捕获的。与在剩余历史记录中评估的一阶和二阶函数导数不同，我们讨论的导数在其整体表示中需要多个内核。但是，所有内核都是相互关联的。本文中的策略涉及识别内核，这些内核指定了要在互转换中使用的非线性本构模型的一阶和二阶Fréchet导数。然后，可以通过建立内核之间的关系来实现并行协议和正交协议之间的相互转换。通过允许在分布意义上的差异来处理阶跃摄动扰动。通过评估与不可压缩的偶极Maxwell和Oldroyd模型相关的核和叠加模量，对这一理论进行了全面的说明。通过允许在分布意义上的差异来处理阶跃摄动扰动。通过评估与不可压缩的偶极Maxwell和Oldroyd模型相关的核和叠加模量，对这一理论进行了全面的说明。通过允许在分布意义上的差异来处理阶跃摄动扰动。通过评估与不可压缩的偶极Maxwell和Oldroyd模型相关的核和叠加模量，对这一理论进行了全面的说明。