Extremely large deformation with polygonal and polyhedral elements

Soft organic materials are of great engineering interest, but challenging to model with standard finite elements. The challenges arise primarily because their nonlinear elastic response is characterized by non-convex stored-energy functions, they are incompressible or nearly incompressible, and oftentimes possess complex microstructures. Recently, polygonal finite elements have been found to possess several advantages over standard finite elements when studying the mechanics of such materials under finite deformations. This chapter summarizes a new approach, using polygonal and polyhedral elements in nonlinear elasticity problems involving extremely large and heterogeneous deformations. We present both displacementbased and two-field mixed variational principles for finite elasticity, together with the corresponding lower- and higher-order polygonal and polyhedral finite element approximations. We also utilize a gradient correction scheme that adds minimal perturbations to the gradient field at the element level in order to restore polynomial consistency and recover (expected) optimal convergence rates when the weak form integrals are evaluated using quadrature rules. With the gradient correction scheme, optimal convergence of the numerical solutions for displacement-based and mixed formulations with both lower- and higher-order displacement interpolants is confirmed by numerical studies of several boundary-value problems in finite elasticity. For demonstration purposes, we deploy the proposed polygonal discretization to study the nonlinear elastic response of rubber filled with random distributions of rigid particles considering interphasial effects. These physically motivated examples illustrate the potential of polygonal finite elements to simulate the nonlinear elastic response of soft organic materials with complex microstructures under finite deformations.