New incremental secant linearization method for mean-field homogenization approach of elasto-viscoplastic microscopic heterogeneous materials

Abstract Unmatched time scales of elastic and viscoplastic responses are not reasonably considered in linearizing the constitutive laws of constituent phases in elasto-viscoplastic microscopic heterogeneous materials, which makes the interaction among the constituent phases very difficult to be described by homogenization approaches. To address this issue, a new incremental secant linearization method is developed by solving linearized equations of stress, strain and time increments obtained from the Hooke’s law and the Taylor’s expansion of stress increment function. Subsequently, the new linearization method is implemented into the Mori-Tanaka’s (M−T) and self-consistent (SC) homogenization approaches. Finally, the stress–strain responses of elasto-viscoplastic microscopic heterogeneous materials (including the composites and polycrystalline materials) under different loading conditions are predicted by the incremental secant linearization-based M−T and SC approaches, and the predicted results are compared with the results obtained by other approaches, such as finite element, fast Fourier transform and generalized affine linearization methods. The comparison shows that the new secant linearization takes an important role in the accurate and effective simulations of the stress–strain responses of elasto-viscoplastic microscopic heterogeneous materials, and the predictions are independent of loading step size if the step size is not too large. Meanwhile, the homogenization approaches of elasto-viscoplastic and elasto-plastic microscopic heterogeneous materials are expected to be unified since the new secant linearization method provides the same mathematical structure for the linearized elasto-viscoplastic constitutive model as that for the elasto-plastic one.