Variational Theory of Balance Systems

Continua with non-local constitutive laws: exploitation of entropy inequality

In this paper, we consider a system of balance laws sufficiently general to contain the equations describing the thermomechanics of a one-dimensional continuum; this system involves some constitutive functions depending on the elements of the so called state space assumed to contain the spatial gradients of some of the unknown fields. The compatibility of the constitutive equations with an entropy-like principle is considered via an extended Liu procedure by using as constraints both the balance equations and some of their gradient extensions. This procedure is then applied to the equations of…

Multisymplectic Theory of Balance Systems and Entropy Principle

In this paper we are presenting the theory of balance equations of the Continuum Thermodynamics (balance systems) in a geometrical form using Poincare-Cartan formalism of the Multisymplectic Field Theory. A constitutive relation $\mathcal{C}$ of a balance system $B_{C}$ is realized as a mapping between a (partial) 1-jet bundle of the configurational bundle $π:Y\to X$ and the dual bundle similar to the Legendre mapping of the Lagrangian Field Theory. Invariant (variational) form of the balance system $B_{C}$ is presented in three different forms and the space of admissible variations is defined…

Balance Systems and the Variational Bicomplex

In this work we show that the systems of balance equations (balance systems) of continuum thermodynamics occupy a natural place in the variational bicomplex formalism. We apply the vertical homotopy decomposition to get a local splitting (in a convenient domain) of a general balance system as the sum of a Lagrangian part and a complemental "pure non-Lagrangian" balance system. In the case when derivatives of the dynamical fields do not enter the constitutive relations of the balance system, the "pure non-Lagrangian" systems coincide with the systems introduced by S. Godunov [Soviet Math. Dokl.…

Hilbert's energy-momentum tensor extended

A variational derivative of a Lagrangian with regard to the metric tensor is used in classical field models to define Hilbert's energy-momentum tensor for a matter field. In solid-state physics, constitutive relationships between fundamental field variables are a topic that is covered by a broad variety of models. In this context, a constitutive tensor of higher order replaces the of the second-order metric tensor. For the classical field models of gravity and electrodynamics, a similar premetric description with a linear constitutive relation has recently presented. In this paper, we analyze …

梯度流之外的变分结构：宏观涨落理论视角

Variational structures beyond gradient flows: a macroscopic fluctuation-theory perspective

本研究从宏观涨落理论出发，通过引入通量拓展密度描述，构建抽象框架将变分结构从满足细致平衡的耗散系统推广至更广泛的随机粒子系统，给出分量分解与广义正交关系，并验证了理论的适用性。

非线性电弹性静力学的三种变分公式及其平衡方程与扰动

On equilibrium equations and their perturbations using three different variational formulations of nonlinear electroelastostatics

本文用三种变分公式推导非线性电弹性静力学平衡与分岔方程，揭示麦克斯韦应力自然涌现，并指出基于电场或电位移的公式更利于稳定性分析。