  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.
