Nonuniform Transformation Field Analysis: a Reduced Model for Multiscale Nonlinear Problems in Solid Mechanics

This chapter is devoted to the Nonuniform Transformation Field Analysis which is a reduction technique introduced in the realm of Multiscale Problems in Nonlinear Solid Mechanics to achieve scale transition for materials exhibing a nonlinear behaviour. It is indeed well recognized that the nonlinearity introduces a strong coupling between the problems at the different scales which, in full rigor, remain coupled. To avoid the computational cost of the scale coupling, reduced models have been developed. To improve on the predictions of Transformation Field Analysis where the plastic strain field is assumed to be uniform in each domain, the authors (Michel and Suquet) have proposed another reduced model, called the Nonuniform Transformation Field Analysis, where the plastic strain fields follow shape functions which are not piecewise uniform. The model is presented for individual phases exhibiting an elastoviscoplastic behavior. A brief account on the reduction technique is given first. Then the time-integration of the model at the level of a macroscopic material point is performed by means of a numerical scheme. This reduced model is applied to structural problems. The implementation of the model in a Finite Element code is discussed. It is shown that the model predicts accurately the effective behavior of nonlinear composite materials with just a few internal variables. Another worth-noting feature of the method is that the local stress and strain fields can be determined simply by postprocessing the output of the structural (macroscopic) computation performed with the model. The flexibility and accuracy of the method are illustrated by assessing the lifetime of a plate subjected to cyclic four-point bending. Using the distribution in the structure of the energy dissipated locally in the matrix by viscoplasticity as fatigue indicator, the life-time prediction for the structure is seen to be in good agreement with large scale computations taking into account all heterogeneities.