Structural Properties of Stress Relaxation and Convergence from Viscoelasticity to Polyconvex Elastodynamics
Structural Properties of Stress Relaxation and Convergence from Viscoelasticity to Polyconvex Elastodynamics
复制标题
应力松弛的结构性质以及从粘弹性到多凸弹性动力学的收敛
DOI：
10.1007/s00205-005-0404-3
复制
发表时间：
2006
影响因子：
2.5
通讯作者：
A. Tzavaras
中科院分区：
文献类型：
--
作者：
Corrado Lattanzio;A. Tzavaras
我们考虑了一个近似于弹性动力学方程的应力松弛模型。给出了该模型具有全局自由能和具有正熵产生的充要条件。所得到的类既允许凸的平衡势，也允许非凸的平衡势。对于凸平衡势，我们证明了一个强耗散估计和两个相对能量估计：松弛过程的相对熵和调制的相对能量。这两种方法都给出了光滑解的收敛结果。对于多凸平衡势，多凸弹性动力学系统的扩充和零-拉格朗日结构提出了一个适当的相对能量概念。在极限解保持光滑的条件下，证明了多凸弹性动力学的粘性逼近的收敛性质。对于多凸情形，我们还得到了调制的相对能量，它表明了松弛近似的稳定性。
We consider a model of stress relaxation approximating the equations of elastodynamics. Necessary and sufficient conditions are derived for the model to be equipped with a global free energy and to have positive entropy production. The resulting class allows for both convex and non-convex equilibrium potentials. For convex equilibrium potentials, we prove a strong dissipation estimate and two relative energy estimates for: the relative entropy of the relaxation process and the modulated relative energy. Both give convergence results to smooth solutions. For polyconvex equilibrium potentials, an augmenting of the system of polyconvex elastodynamics and the null-Lagrangian structure suggest an appropriate notion of relative energy. We prove convergence of viscosity approximations to polyconvex elastodynamics in the regime where the limit solution remains smooth. A modulated relative energy is also obtained for the polyconvex case which shows stability of relaxation approximations.