AI 中文总结
研究张量内变量的与参考系无关速率，通过引入生成器统一共旋和上随体输运，从动量和微力平衡推导自由能不平衡，经分解得出本构限制，构建耦合梯度理论，扩展了含张量内变量的经典理论。
AI 中文摘要
我们为具有张量值内变量的弱非局部连续体开发了一个热力学一致的框架。引入生成器\(\boldsymbol{\Gamma}_{\alpha}=\boldsymbol{W}+\alpha\boldsymbol{D}\)统一共旋和上随体输运。从线动量和微力平衡出发，推导了不可压缩等温过程的局部自由能不平衡。在各向同性和继承对称性假设下，该不平衡可分解。最后构建了结合粘弹性和约束取向序的耦合梯度理论，扩展了经典理论。
英文摘要
We develop a thermodynamically consistent framework for weakly nonlocal continua with tensor-valued internal variables. Let $\boldsymbol{L}=\operatorname{grad}\boldsymbol{v}$, with stretching tensor $\boldsymbol{D}=\operatorname{sym}\boldsymbol{L}$ and spin tensor $\boldsymbol{W}=\operatorname{skw}\boldsymbol{L}$. We introduce the generators $\boldsymbolΓ_α=\boldsymbol{W}+α\boldsymbol{D}$, $α\in\{0,1\}$, which unify corotational and upper-convected transport. The resulting kinematic structure induces canonical frame-indifferent evolutions of both the internal variable and its spatial gradient, thereby providing a closure for gradient-dependent theories. Starting from the balances of linear momentum and microforces, together with an internal power expenditure depending on the internal variable and its gradient, we derive a local free-energy imbalance for incompressible isothermal processes. Under isotropy and inherited symmetry assumptions, this imbalance admits a canonical decomposition into contributions associated with $\boldsymbol{D}$, $\operatorname{grad}\boldsymbol{L}$, the generator-induced rate $\mathfrak{D}_α\boldsymbol{J}$, and its gradient $\mathfrak{D}^{\nabla}_α(\operatorname{grad}\boldsymbol{J})$. This decomposition yields explicit constitutive restrictions ensuring thermodynamic consistency and identifies the induced higher-order stress contributions arising from gradient dependence. Finally, we construct a coupled gradient theory combining viscoelasticity and constrained orientational order, in which distinct internal variables evolve under different transport mechanisms. The framework extends classical theories with tensorial internal variables, including Oldroyd-B and Landau-de Gennes-type models.
