A Unified Gradient Theory for Frame-Indifferent Rates of Tensorial Internal Variables

Abstract

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.