Discrete particle models (DPMs) have been widely used as an alternative to classical continuum mechanics to describe the mechanical behaviour of solids, fluids and granular matter. Heretofore, the types of interaction forces considered in DPMs have been very limited; consisting mainly of linear pairwise interactions or empirically motivated generalizations thereof. Thus DPMs have been able to describe a very limited range of macroscopic constitutive behaviour and the properties have generally been strongly dependent on the discretization.

We present a new paradigm for incorporating multi-body interactions in DPMs by relating the deformation of discrete particles to that of the continuum using a mesoscale analog of the Cauchy–Born rule. We consider a Delaunay triangulation of an irregular lattice of particles comprising the body in its reference configuration. We calculate linear transformations to map the particle positions of each triangle in the current configuration to its reference configuration. Invoking the analogy of the Cauchy–Born rule, we identify the linear transformations with the continuum deformation gradient. Using this we express the continuum constitutive free energy of the body, which is a function of the deformation gradient, in terms of the current positions of the particles. The neighbour interaction forces on each of the particles are taken to be the gradients of this energy. The equations of motion for each particle are then solved numerically to obtain the particle trajectories and the overall deformation of the body. This approach is validated using comparisons with classical results of stress distribution in isotropic and anisotropic material plates with circular holes. Next, the model is applied to describe the behaviour of materials undergoing structural phase transformations. Several behaviours which arise from the underlying discreteness are described such as thermal hysteresis behaviour, detwinning and pseudoelastic response.