Unified mean-field modeling of viscous short-fiber suspensions and solid short-fiber reinforced composites

Abstract

Mean-field homogenization is an established and computationally efficient method estimating the effective linear elastic behavior of composites. In view of short-fiber reinforced materials, it is important to homogenize consistently during process simulation. This paper aims to comprehensively reflect and expand the basics of mean-field homogenization of anisotropic linear viscous properties and to show the parallelism to the anisotropic linear elastic properties. In particular, the Hill–Mandel condition, which is generally independent of a specific material behavior, is revisited in the context of boundary conditions for viscous suspensions. This study is limited to isothermal conditions, linear viscous and incompressible fiber suspensions and to linear elastic solid composites, both of which consisting of isotropic phases with phase-wise constant properties. In the context of homogenization of viscous properties, the fibers are considered as rigid bodies. Based on a chosen fiber orientation state, different mean-field models are compared with each other, all of which are formulated with respect to orientation averaging. Within a consistent mean-field modeling for both fluid suspensions and solid composites, all considered methods can be recommended to be applied for fiber volume fractions up to 10%. With respect to larger, industrial-relevant, fiber volume fractions up to 20%, the (two-step) Mori–Tanaka model and the lower Hashin–Shtrikman bound are well suited.

Influence of flow–fiber coupling during mold-filling on the stress field in short-fiber reinforced composites

“…In order to realize the flow-fiber coupling, the effective viscosity tensor V of the fiber suspension is formulated in terms of fiber orientation tensors N and N. Throughout this study, the effective Cauchy stress tensor σ refers to a generalized incompressible Newtonian fluid. Furthermore, the Mori-Tanaka model [26,[38][39][40] is used reading…”

“…The polarization tensor P 0 contains information about the fiber geometry and the matrix material and can be computed analytically for spheroidal fibers. In this work the expressions given in Ponte Castañeda and Willis [41] are adapted for viscosity and simplified for incompressible matrix behavior following Karl and Böhlke [40]. The inversion P −1 0 is defined on the symmetric deviatoric subset (symDev) for the considered case of incompressible suspensions with P −1 0 P 0 = P 2 representing the identity on symDev for second-order tensors.…”

“…(3) the matrix viscosity V M is considered both Newtonian and non-Newtonian in the context of a generalized Newtonian fluid. Within the present study, the shear-thinning Cross-model [45] for the polymer matrix is implemented as follows to be compared against the Newtonian behavior [40,46]…”

“…For clarity, the basics regarding mean-field modeling are omitted here and the reader is referred to further literature, e.g., Nemat-Nasser and Hori [47] or Kanaun and Levin [48]. Based on the Mori-Tanaka model [26], the effective stiffness tensor reads C = C M + R with the abbreviation R defined as follows [27,40,49]…”

“…Further information can be found in the literature, e.g., in Nemat-Nasser and Hori [47] or in Torquato [52]. Regarding the uniform meanfield homogenization of viscous fiber suspensions and solid short-fiber reinforced composites, reference is made to the recent work of Karl and Böhlke [40].…”