基于极小化型变分公式的复合水凝胶结构瞬态稳定性分析
Transient stability analysis of composite hydrogel structures based on a minimization-type variational formulation
AI总结：
本文基于双场极小化变分框架和Q$_1$RT$_0$有限元离散，通过增量势局部稳定性准则及刚度矩阵正定性判定，研究了约束条件下水凝胶瞬态溶胀引发的起皱失稳，并分析了临界屈曲载荷和模态对几何与材料参数的依赖。
摘要：
我们采用一个规范变分框架，对几何约束下扩散驱动的瞬态溶胀过程中水凝胶结构失稳进行预测性表征。有限弹性与Fick扩散耦合的变分公式具有双场极小化结构，其中变形映射和流体体积通量作为时间离散势能的极小化子获得，该势能包含内部和外部能量贡献。空间离散后，利用协调的Q$_1$RT$_0$有限元设计实现该极小化原理，对流体体积通量采用最低阶Raviart-Thomas型插值。为分析满足极小化原理的凝胶某一平衡态的结构稳定性，我们在增量势上应用局部稳定性准则，该准则基于如下思想：稳定平衡态在无穷小邻域内所有可能状态中具有最低势能。利用该准则可知，当耦合的全局有限元刚度矩阵失去正定性时，分叉型结构失稳被激活。随后将该概念应用于确定一对代表性薄膜-基底水凝胶系统中起皱失稳的发生和性质。特别地，我们分析了临界屈曲载荷和模态形状对系统几何和材料参数的依赖关系。
英文摘要：
We employ a canonical variational framework for the predictive characterization of structural instabilities that develop during the diffusion-driven transient swelling of hydrogels under geometrical constraints. The variational formulation of finite elasticity coupled with Fickian diffusion has a two-field minimization structure, wherein the deformation map and the fluid-volume flux are obtained as minimizers of a time-discrete potential involving internal and external energetic contributions. Following spatial discretization, the minimization principle is implemented using a conforming Q$_1$RT$_0$ finite-element design, making use of the lowest-order Raviart-Thomas-type interpolations for the fluid-volume flux. To analyze the structural stability of a certain equilibrium state of the gel satisfying the minimization principle, we apply the local stability criterion on the incremental potential, which is based on the idea that a stable equilibrium state has the lowest potential energy among all possible states within an infinitesimal neighborhood. Using this criterion, it is understood that bifurcation-type structural instabilities are activated when the coupled global finite-element stiffness matrix loses its positive definiteness. This concept is then applied to determine the onset and nature of wrinkling instabilities occurring in a pair of representative film-substrate hydrogel systems. In particular, we analyze the dependencies of the critical buckling load and mode shape on the system geometry and material parameters.
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