Coupled electrochemical and mechanical modelling is essential to investigate stress generation and to evaluate its effect on battery performance. However, prior models are mostly focused on a single, isolated particle. While these models are very useful to consider stress and fracture inside a particle, they cannot be used for problems such as fracture in-between particles since particle interaction is omitted. Moreover, the electrode level model coupling of mechanics and electrochemistry is lacking. There have been several attempts to couple the mechanical model of isolated particles with the porous electrode model to analyze the distribution of intercalation stresses inside different particles across the electrode. In those attempts, however, the particles are considered isolated with no interaction between them. It should be noted that particle interaction can result in significant stress level comparable to the stress generated from the concentration gradient.

Mechanical stress can change the electrochemical potential of a solid, and therefore (i) affects the diffusion in the solid, and (ii) affects the electrochemical reaction between the solid and the electrolyte. Although the first effect has been generally addressed, the second effect of mechanical stress on the electrochemical reaction rate has not been included in most models. Including the second effect is necessary when modeling particle interaction since stress gradient in the electrode leads to spatial-dependent interaction stress between particles. It is necessary to account for different intercalation/de-intercalation currents for particles under different stress states that are dependent on their interaction with neighbor particles.

**Fig. 1** illustrates our concept to model particle interaction and to link the particle level and the continuum electrode level. Each spatial point in the continuum level corresponds to a particle level representative volume element (RVE) consisting of many particles and porous volume occupied by the electrolyte. The shade of color in the particle represents lithium concentration.

At the particle level, solid diffusion is modeled using a generalized electrochemical potential, which captures the effects of phase transition and mechanical stress. The stress in a particle is a superposition of the stress from lithium concentration gradient inside the particle (*σ _{ij }^{c}*) and the stress from particle interaction (

*σ*). The stress

_{ij }^{i}*σ*can be calculated by the lithium concentration distribution inside the particle using an analogy to thermal stress. Regarding the interaction stress, each particle can be considered to be an inclusion embedded in a matrix composed of all other particles. According the Eshelby’s inclusion theory, the stress in a spherical inclusion resulting from its interaction with the matrix is uniform. Thus, the interaction stress,

_{ij }^{c }*σ*, is uniform inside the particle.

_{ij }^{i}At the electrode level, we consider coupled electrochemical reaction and stress field both in the electrode thickness direction and in an in-plane direction, which can be called pseudo-3D model in comparison to the classical pseudo-2D model. The electrode level stress (*Σ _{ij}*) is calculated on the continuum with effective mechanical properties and the eigenstrains of the RVE. Conceptually, the electrode level stress at a spatial point in the electrode can be viewed as the mechanical loading on an RVE at that point. A relation between the electrode level stress (

*Σ*) and the particle interaction stress (

_{ij}*σ*) has been developed using volume average approach. The effect of stress on the intercalation current is captured by a stress-dependent over-potential.

_{ij }^{i}An important finding we have obtained with this model is a new degradation mechanism, where small electrochemically inactive regions can potentially cause major degradation. We find that the tensile stress generated in the vicinity of an inactive region is large enough to easily break the binder between particles, resulting in fracture or growth of the inactive region. The result points to an approach to reduce degradation by improving the homogeneity of the electrode.