Next-generation lithium-ion batteries with silicon anodes have positive characteristics due to higher energy densities compared to state-of-the-art graphite anodes. However, the large volume expansion of silicon anodes can cause high mechanical stresses, especially if the battery active particle cannot expand freely. In this article, a thermodynamically consistent continuum model for coupling chemical and mechanical effects of electrode particles is extended by a change in the boundary condition for the displacement via a variational inequality. This switch represents a limited enlargement of the particle swelling or shrinking due to lithium intercalation or deintercalation in the host material, respectively. For inequality constraints as boundary condition a smaller time step size is need as well as a locally finer mesh. The combination of a primal-dual active set algorithm, interpreted as semismooth Newton method, and a spatial and temporal adaptive algorithm allows the efficient numerical investigation based on a finite element method. Using the example of silicon, the chemical and mechanical behavior of one- and two-dimensional representative geometries for a charge-discharge cycle is investigated. Furthermore, the efficiency of the adaptive algorithm is demonstrated. It turns out that the size of the gap has an significant influence on the maximal stress value and the slope of the increase. Especially in two dimension, the obstacle can cause an additional region with a lithium-poor phase.

翻译：下一代采用硅负极的锂离子电池相比当前最先进的石墨负极具有更高的能量密度，因而展现出积极特性。然而，硅负极的巨大体积膨胀会引发高机械应力，尤其是当电池活性颗粒无法自由膨胀时。本文通过变分不等式对位移边界条件进行改进，扩展了热力学一致的电极颗粒化学-力学耦合连续介质模型。该边界条件的转换分别表征了因锂离子在宿主材料中嵌入/脱出导致的颗粒有限膨胀或收缩。对于以不等式约束为边界条件的情况，需要更小的时间步长以及局部更精细的网格。将原始-对偶有效集算法（可视为半光滑牛顿法）与空间-时间自适应算法相结合，实现了基于有限元法的高效数值研究。以硅为例，研究了一维和二维代表性几何结构在充放电循环中的化学与力学行为。此外，还验证了自适应算法的计算效率。结果表明，间隙尺寸对最大应力值及其增长斜率具有显著影响。尤其在二维情况下，障碍物可能诱发额外的锂贫相区域。