A mixed FFT-Galerkin approach for incompressible or slightly compressible hyperelastic solids under finite deformation

Abstract

In this study, we propose a mixed Fast Fourier Transform (FFT)-based homogenization approach to analyze finite deformations of heterogeneous solids with different incompressible or slightly compressible hyperelastic phases. The proposed mixed formulations are constructed from the minimization of the specialized total elastic strain energy using trigonometric polynomials and an adapted Green’s operator. Unlike the original form of the standard strain-based FFT-Galerkin approach, our proposed formulation using both the deformation gradient and hydrostatic pressure as unknowns successfully solves the convergence issue corresponding to the incompressibility limit of elastic materials, without loss of accuracy and efficiency. A general explicit matrix form of the system of linear equations to be solved during the Newton–Raphson iteration procedure for the mixed FFT-Galerkin approach is also provided, making the algorithm easy to implement. Examples of simple shear and uniaxial traction of heterogeneous solids consisting of slightly compressible phases, incompressible phases, or a hybrid of incompressible/totally compressible phases validated the developed FFT-Galerkin approach, demonstrating its potential for possible applications in accurately calculating the mechanical responses of materials such as composites, polymer foams, hydrogels, and other meta-materials consisting of incompressible phases.