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. 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 in a finite-element context, 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.

我们采用规范的变分框架来预测结构不稳定性的预测特征，该结构不稳定性是在几何约束下水凝胶的扩散驱动瞬态溶胀过程中形成的。有限弹性​​与Fickian扩散相结合的变分公式具有两场最小化结构，其中，变形图和流体通量是作为涉及内部和外部能量贡献的时间离散电位的最小化函数而获得的。为了分析满足最小化原理的凝胶某些平衡态的结构稳定性，我们对增量电势应用了局部稳定性标准，该标准基于以下观点：在所有可能的状态下，稳定的平衡态具有最低的势能一个无穷无尽的邻居。在有限元环境中使用此准则，可以理解，当耦合的全局有限元刚度矩阵失去其正定性时，会激活分叉型结构不稳定性。然后将该概念应用于确定在一对代表性的膜-基质水凝胶体系中发生的起皱不稳定性的发生和性质。特别是，我们分析了临界屈曲载荷和模态形状对系统几何形状和材料参数的依赖性。然后将该概念应用于确定在一对代表性的膜-基质水凝胶体系中发生的起皱不稳定性的发生和性质。特别是，我们分析了临界屈曲载荷和模态形状对系统几何形状和材料参数的依赖性。然后将该概念应用于确定在一对代表性的膜-基质水凝胶体系中发生的起皱不稳定性的发生和性质。特别是，我们分析了临界屈曲载荷和模态形状对系统几何形状和材料参数的依赖性。