This work concerns the continuum basis and numerical formulation for deformable materials with viscous dissipative mechanisms. We derive a viscohyperelastic modeling framework based on fundamental thermomechanical principles. Since most large deformation problems exhibit the isochoric property, our modeling work is constructed based on the Gibbs free energy in order to develop a continuum theory using the pressure-primitive variables, which is known to be well-behaved in the incompressible limit. With a general theory presented, we focus on a family of free energies that leads to the so-called finite deformation linear model. Our derivation elucidates the origin of the evolution equations of that model, which was originally proposed heuristically. In our derivation, the thermodynamic inconsistency is clarified and rectified. We then discuss the relaxation property of the non-equilibrium stress in the thermodynamic equilibrium limit and its implication on the form of free energy. A modified version of the identical polymer chain model is then proposed, with a special case being the model proposed by G. Holzapfel and J. Simo. Based on the consistent modeling framework, a provably energy stable numerical scheme is constructed for incompressible viscohyperelasticity using inf-sup stable elements. In particular, we adopt a suite of smooth generalization of the Taylor-Hood element based on Non-Uniform Rational B-Splines (NURBS) for spatial discretization. The temporal discretization is performed via the generalized-alpha scheme. We present a suite of numerical results to corroborate the proposed numerical properties, including the nonlinear stability, robustness under large deformation, and the stress accuracy resolved by the higher-order elements.
翻译:这项工作涉及以粘力分解机制的直径变形材料的连续基础和数字配方; 我们根据基本的热力机械原理, 产生一个粘合性弹性模型框架; 由于大多数大的变形问题显示出异化特性, 我们的建模工作以Gibbs免费能源为基础, 以便利用压力- 压力- 压力- 压力- 压力- 温度- 温度的变数( 众所周知, 压力- 压力- 压力- 压力的极限 ) 来发展一个连续的理论; 提出一个总体理论, 我们集中关注一个自由能源的大家庭, 导致所谓的固定变形线模型。 我们的推论说明了该模型的变异性( 该模型最初被提议) 的变异性公式的起源。 在我们的衍生过程中, 热力- 动力- 不一致性变异变异性( 以稳定的数值- 不变的内值- 数值- ) 以一个稳定的模型为基础, 以稳定的数值- 以稳定的内值- 以稳定的内值- 以稳定的内值- 数字- 结构- 以稳定的模型为基础, 以稳定的数值- 以稳定的数值- 以稳定的内- 以稳定的数值- 基- 以稳定的内- 以稳定的内制成为制成为基- 以稳定的内- 以稳定的数值- 以稳定的内- 以稳定的内- 制成为制成为基 以稳定的内- 。