In a truly multiscale analysis of multilayered composites, the underlying phenomena are represented and their effect on the overall behavior is determined considering the interaction between the different phases and between the laminas. The analysis gets more involved when multiple phenomena are considered since in this case not only the direct effects play a role but also the coupled effects contribute to the distribution of the local fields and the overall response. In a fibrous composite laminate reinforced with piezoelectric filaments, for example, passing an electric field in the fibers generates stresses and strains which propagate through the composite medium due to constraints that exist both at the micromechanical, ply level, and the macromechanical, laminate level. Pyroelectricity is another coupling phenomenon in which a temperature change is caused by an electric field, and hence leads to changes in the stress and strain fields throughout the composite medium.
The above phenomena have been considered by the authors in a unified, transformation field analysis (TFA) approach in which stresses and strains which cannot be removed by mechanical unloading are treated as transformation fields. Due to mutual constraints of the phases and the bonded plies, local transformations generate stresses at the micro and macro levels, which are computed by means of influence functions which depend on material geometry and properties. Treatment of damage follows the same scheme but the transformation fields are instead determined such that the local stresses in the affected phase are removed.
In the present paper, implementation of the TFA approach in a general purpose finite element code is described. This expands the multiscale analysis outlined above to composite structures where complex geometries can be modeled and the effect of local phenomena can be considered. This naturally comes at a much larger cost of the computations compared to finite element analysis with homogenized models but the benefit of obtaining a more realistic response is clear. Moreover, the availability of high performance computing and parallel processing overcomes the computation time barrier. In the present paper however, simple examples of laminated structures are given as proof of concept in which the results are compared to those of standalone routines. Since the TFA approach centers on treating the composite medium as elastic with induced local transformations, implementation in the finite element framework does not require generation of an overall instantaneous stiffness matrix, which saves tremendously on the computation time. Instead, overall transformation strains, or stresses, are computed through a multiscale model, which is implemented as a user routine, and treated in the general finite element solution as nonmechanical strains in the same way thermal strains are treated.
Special-purpose elements to impose Periodic Boundary Conditions for multiscale computational homogenization of composite materials with the explicit Finite Element Method