In this thesis we consider the chemo-mechanical response of a multi-material lithium-ion battery anode nano-particle at equilibrium. The stresses are induced by lithium being intercalated into the nano-particle, causing the anode materials to expand. The presence of several anode materials expanding to different extents induces stress even at equilibrium. The model we primarily study is a linear elasticity model derived under the assumption that the stress-free expansion of each anode material is small. We couple the mechanical model to a diffusion-based model governing the transport of lithium in each material through stress-assisted diffusion. At equilibrium, the chemical potential of the lithium, which depends both on the lithium concentration and the hydrostatic stress, is uniform throughout the nano-particle, governing the distribution of lithium between the materials. Silicon is a promising anode material with a high capacity for lithium, but it expands to around 380% its original size when fully lithiated, leading to several issues during battery usage. We apply our equilibrium model to a nano-particle consisting of a silicon core and a surrounding graphite shell. The aim of using both materials is to mitigate the expansion of silicon without sacrificing too much capacity. We analyse the lithium concentration, displacement and stresses within each material, and use the results to optimise the volume of the silicon core based on several measures of success. Void spaces and porous silicon are introduced into the nano-particle design, using the method of multiple scales, to improve on the simple core–shell design, to moderate success. We extend the model to a geometrically nonlinear elastic model, and analyse this after finding that solutions cease to exist above a certain state of charge. Finally, we incorporate yielding and fracture of both materials into the linear model, based on perfect plasticity.