In this work, following Agoras et al. , we develop a homogenization procedure for particulate composites with elasto-viscoplastic constituents. By employing a standard time-discretization scheme for the local stress rate, we show that the associated time-incremental homogenization problem may be expressed in terms of a variational principle for a composite material with both nonlinear and point-wise heterogeneous phases. The latter formulation of the problem may serve as a basis for the application
of suitable “linear comparison composite” (LCC) techniques, which allow estimating the macroscopic response of the composite with nonlinear and non-uniform constituents from that of an LCC with the ame microstructure and with linear and uniform phases. In this work, we explore the possibility of using the LCC method of Lahellec et al.  to deal with the nonuniformity of the phase properties along with either the variational LCC procedure of Ponte Casta˜neda  or the second-order LCC procedure of Furer and Ponte Casta˜neda  to handle the part of the problem associated with the constitutive nonlinearities.
The local properties of the LCC involved in these procedures are given in terms of appropriately constructed optimality conditions, while its effective properties are computed by making use of suitable estimates for linear composites that are available from the literature. In the present work, we also consider specific applications to two-phase composite materials made out of an isotropic elasto-viscoplastic matrix reinforced (or weakened) by a family of isotropic elastic particles. The particles are taken to have identical spheroidal shapes and to be distributed randomly and with spheroidal symmetry in the matrix, leading to overall transversely isotropic behavior for the composite. The variational and second-order estimates delivered by the proposed homogenization procedure for the macroscopic response of these composites under cyclic axisymmetric, in-plane and anti-plane loading conditions are compared with corresponding numerical FEM results that were carried out for the purpose of this work, as well as with FFT results that are available from the literature. Both types of estimates are found to be in good agreement with the numerical results and to capture correctly the associated elasto-plastic transient regimes and the Bauschinger effect. In addition, the second-order estimates are found to be more accurate tan the variational estimates.
 Agoras, M., Avazmohammadi, R., Ponte Casta˜neda, P. (2016). Incremental variational procedure
for elasto-viscoplastic composites and application to polymer- and metal-matrix composites reinforced
by spheroidal elastic particles. International Journal of Solids and Structures 97–98, 668–
 Furer, J., Ponte Casta˜neda, P. (2018). A symmetric fully optimized second-order method for nonlinear
homogenization. Journal of Applied Mathematics and Mechanics 98, 222–254.
 Lahellec, N., Ponte Casta˜neda, P., Suquet, P. (2011). Variational estimates for the effective response
and field statistics in thermoelastic composites with intra-phase property fluctuations. Proc. R. Soc.
Lond. A 467, 2224–2246.
 Ponte Casta˜neda, P. (1992). New variational principles in plasticity and their application to composite
materials. J. Mech. Phys. Solids 40, 1757–1788.