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
