Multi-scale simulation of plant tissue deformation using a model for individual cell mechanics

We present a micro-macro method for the simulation of large elastic deformations of plant tissue. At the microscopic level, we use a mass-spring model to describe the geometrical structure and basic properties of individual plant cells. The macroscopic domain is discretized using standard finite elements, in which the macroscopic material properties (the stress-strain relation) are not given in analytical form, but are computed using the microscopic model in small subdomains, called representative volume elements (RVEs), centered around the macroscopic quadrature points. The boundary conditions for these RVEs are derived from the macroscopic deformation gradient. The computation of the macroscopic stress tensor is based on the definition of virial stress, as defined in molecular dynamics. The anisotropic Eulerian elasticity tensor is estimated using a forward finite difference approximation for the Truesdell rate of the Cauchy stress tensor. We investigate the influence of the size of the RVE and the boundary conditions. This multi-scale method converges to the solution of the full microscopic simulation, for both globally and adaptively refined finite element meshes, and achieves a significant speedup compared to the full microscopic simulation.