Multiscale Simulations of Composites with Non-Local Damage-Enhanced Mean-Field Homogenization

The mean-field homogenization (MFH) approach is an attractive framework for multiscale
methods, as it provides predictions of the macroscopic behavior of particle or fiber reinforced
composites at a reasonable computational cost. Efficient MFH methods have been available
for a long time for linear elastic problems, using for example the Mori-Tanaka scheme [2],
but they can also be extended in the non-linear regime after linearization of the constitutive
behavior at the current strain state, as for the incremental approach, e.g. [1].
In this work, the application of ductile-damage theories to a multiscale analysis of continuous
fiber reinforced composites is considered. Toward this end, the incremental MFH approach is
extended to account for the damage behavior happening in the matrix material at the microscale and to derive the effective properties of particle or fiber reinforced composites.
However, capturing the degradation, damage or failure of material happening at the microscopic scale could lead to loss of uniqueness in the solution as the governing partial differential equations may lose ellipticity at a given level of loading corresponding to the strain-softening onset. Thus, in order to avoid the strain/damage localization caused by matrix material softening, the gradient-enhanced formulation [3] is adopted to describe the material behavior of the matrix during the homogenization process, as we have recently proposed [4].
As illustration, the behavior of a fiber re-enforced elasto-plastic matrix is considered. The
properties of the matrix correspond to an elasto-plastic material experiencing damage, with
a non-local form of Lemaitre Chaboche model. The fibers are assumed linear elastic, see [4]
for details. A loading-unloading cycle is applied in the direction transverse to the fibers. A
maximal deformation of 10 % is reached before the unloading proceeds to zero-transverse
deformation. The effective behavior predicted by the MFH models is compared to the solutions
obtained by finite element computations performed on a unit periodic cell and on RVE
where the micro-structure is fully meshed. The results for three fiber volume ratios are presented in Fig. 1. For the three fiber volume ratios, the homogenized property is dominated
by the properties of the matrix, with an obvious elasto-plastic behavior exhibiting softening.
For vI = 15% and 30%, rather good predictions are given by the MFH model, with, as expected, higher macroscopic stress and damage predicted by the MFH due to the incremental
formulation. However for vI = 50%, the MFH model overestimates the macroscopic stress
considerably. This error comes from the assumption of Mori - Tanaka based MFH.
As it is shown to be an efficient multi-scale approach, the developed gradient enhanced MFH
formulation presented can now be used to model the behavior of composite laminates experiencing damage.