This paper presents a new family of objective (Lagrangian and Eulerian) continuous strain-consistent convective (corotational and non-corotational) tensor rates. Objective strain-consistent convective tensor rates are defined as having the property that there exist objective (Lagrangian and Eulerian) strain tensors from the Hill family such that the considered rates of these strain tensors are equal to the rotated/standard (Lagrangian/Eulerian) stretching (strain rate) tensors. The family of such Eulerian strain-consistent convective tensor rates was introduced by Bruhns et al. (Proc. R. Soc. Lond. A 460:909–928, 2004 ). On the one hand, the family of continuous strain-consistent convective tensor rates presented in this paper extends the Bruhns et al. family by including Lagrangian tensor rates and, on the other hand, it is narrower than the Bruhns et al. family due to the continuity requirement imposed on tensor rates, which is necessary for applications. The theorem that any strain tensors only from the Doyle–Ericksen family (which is a subfamily of the Hill family) provide sufficient continuity conditions for strain-consistent convective tensor rates was formulated and proved. The expressions obtained by proving this theorem for convector tensors show that each tensor from the Doyle–Ericksen family is associated with a single strain-consistent convective tensor rate (in the Bruhns et al. family, any strain tensor from the Hill family generates infinitely many convector tensors associated with infinitely many strain-consistent convective tensor rates). In addition, a new family of Hooke-like isotropic hypoelastic material models based on objective continuous strain-consistent convective rates of the rotated/standard (Lagrangian and Eulerian) Kirchhoff stress tensors was constructed. The theorem that any material model from this family is self-consistent provided that the Lamé parameter λ = 0 $\lambda =0$ and/or the deformation of the body is isochoric was formulated and proved. By self-consistent hypoelastic material models are meant those models for which constitutive hypoelastic relations are counterparts of constitutive relations for Cauchy/Green elasticity. Some material models from the new family were tested by solving the simple shear problem. Both new and well-known solutions of this problem for material models were obtained using an approach that takes into account the self-consistency property of material models from the new family.

中文翻译：

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连续应变一致对流张量率族及其在类钩各向同性低弹性中的应用

本文提出了一系列新的客观（拉格朗日和欧拉）连续应变一致对流（共旋和非共旋）张量率。客观应变一致对流张量速率被定义为具有以下特性：存在来自 Hill 族的客观（拉格朗日和欧拉）应变张量，使得这些应变张量的考虑速率等于旋转/标准（拉格朗日/欧拉）拉伸（应变率）张量。这种欧拉应变一致对流张量率的家族是由 Bruhns 等人引入的。(Proc. R. Soc. Lond. A 460:909-928, 2004)。一方面，本文提出的连续应变一致对流张量率系列扩展了 Bruhns 等人的研究。家庭通过包括拉格朗日张量率，另一方面，它比 Bruhns 等人的要窄。由于对张量率的连续性要求，这是应用程序所必需的。制定并证明了仅来自 Doyle-Ericksen 家族（希尔家族的一个亚家族）的任何应变张量为应变一致对流张量速率提供足够连续性条件的定理。通过证明对流张量的这个定理获得的表达式表明，来自 Doyle-Ericksen 族的每个张量都与一个单一的应变一致对流张量率相关（在 Bruhns 等人的家族中，来自 Hill 族的任何应变张量产生无限多的与无限多的应变一致对流张量率相关联的对流张量）。此外，基于旋转/标准（拉格朗日和欧拉）基尔霍夫应力张量的客观连续应变一致对流率，构建了一个新的类虎克各向同性低弹性材料模型。如果 Lamé 参数 λ = 0 $\lambda =0$ 和/或体的变形是等容的，则该系列中的任何材料模型都是自洽的定理被制定并证明。自洽低弹性材料模型是指那些本构低弹性关系是柯西/格林弹性本构关系对应物的模型。新系列中的一些材料模型通过解决简单的剪切问题进行了测试。