Micromechanical analysis of hyperelastic composites with localized damage using a new low-memory Broyden-step-based algorithm

A multiscale (micro-to-macro) analysis is proposed for the prediction of the finite-strain behavior of composites with hyperelastic constituents and embedded localized damage. The composites are assumed to possess periodic microstructure and be subjected to a remote field. At the microscale, finite-strain micromechanical analysis based on the homogenization technique for the (intact) composite is employed for the prediction of the effective deformation. At the macroscale, a procedure, based on the representative cell method and the associated higher-order theory, is developed for the determination of the elastic field in the damaged composite. The periodic composite is discretized into identical cells and then reduced to the problem of a single cell by application of the discrete Fourier transform. The resulting governing equations, interfacial and boundary conditions in the Fourier transform domain are solved by employing the higher-order theory in conjunction with an iterative procedure to treat the effects of damage and material nonlinearity. The initial conditions for the iterative solution are obtained using the weakly nonlinear material limit and a natural fixed-point iteration. A locally convergent low-memory Quasi-Newton solver is then employed. A new algorithm for the implementation of the solver is proposed, which allows storing in the memory directly the vector function history sequence, which may be advantageous for convergence control based on specific components of the objective vector function. The strong-form Fourier transform-based approach employed here, in conjunction with the new solver, enables to extend the application of the method to nonlinear materials and may have computational efficiency comparable or possibly advantageous to that of standard approaches.