In this chapter, 2D Computational Grains (CGs) with elastic inclusions or voids and 3D CGs with spherical/ellipsoidal inclusions/voids or without inclusions/voids are developed for micromechanical modeling of composite and porous materials. A compatible displacement field is assumed along the outer boundary of each CG. Independent displacement fields in the CG are assumed as characteristic-length-scaled T-Trefftz trial functions. Muskhelishvili’s complex functions are used for 2D CGs, and Papkovich-Neuber solutions are used for 3D CGs to construct the T-Trefftz trial displacement fields. The Papkovich-Neuber potentials are linear combinations of spherical/ellipsoidal harmonics. To develop CG stiffness matrices, multi-field boundary variational principles are used to enforce all the conditions in a variational sense. Through numerical examples, we demonstrate that the CGs developed in this chapter can estimate the overall material properties of heterogeneous materials, and compute the microscopic stress distributions quite accurately, and the time needed for computing each SERVE is far less than that for the finite element method.

In this chapter, Computational Grains are developed for the direct micromechanical modeling of heterogeneous materials reinforced with coated particulate inclusions. Each CG is treated as a three-phase particle/coating/matrix grain, wherein the exact internal displacement field is assumed in terms of the P-N solutions that are further represented by the spherical harmonics. The Computational Grain program generates accurate homogenized moduli as well as exact local interphase stress distributions, with good agreement to the very fine-mesh FE technique and the CSA (Composite Sphere Assemblage) model. The effects of the material properties as well as the thickness of the coating system on the effective properties and localized stress concentrations are also examined for the CGs, where the former parameters play more important roles than the latter ones in altering the response of composite materials. Finally, a simpler implementation of periodic boundary conditions on the SERVEs is developed through the surface-to-surface constraints of the displacement field on the opposite faces. The developed CGs provide accurate and efficient computational tools in the direct modeling of the micromechanical behavior of the particulate composites reinforced with coatings/interphases, which cannot be easily accomplished by the off-the-shelf FE packages and classical models.

In this chapter, viscoelasticity effects in composites are studied. Three-dimensional CGs with linear viscoelastic matrices, containing linearly elastic spherical inclusions with or without interphases/coatings are treated. For each CG, the independent displacement fields are developed by the characteristic-length-scaled Papkovich-Neuber solutions and spherical harmonics. A compatible boundary displacement field is also assumed with Wachspress coordinates as nodal shape functions on each of the polygonal faces. Multi-field boundary variational principles are used to develop the CG stiffness matrices. After the establishment of CGs in Laplace transform domain, the homogenized and localized responses are transformed back to the time domain using the Zakian technique. With different kinds of models to describe the property of the viscoelastic polymers, the generated homogenized moduli and localized stress distributions are validated against the experimental data, simulations by commercial FE software, and predictions by composite spherical assemblage models. Parametric studies are also carried out to investigate the influence of material and geometric parameters on the behavior of viscoelastic composites. Finally, the viscoelastic CGs are also used to study the effect of the negative Young’s modulus of particles on the stability and loss tangent of viscoelastic composites.

In this chapter, a new kind of Computational Grain (CG) with embedded cylindrical elastic fibers is developed for the micromechanical modeling of fiber-reinforced composites. The trial displacement fields within the CGs are assumed using Papkovich-Neuber solutions. Cylindrical harmonics scaled by characteristic lengths are employed as the P-N potentials. A compatible displacement field is assumed at elemental surfaces and fiber–matrix interfaces, and the stiffness matrices of CGs are derived by a newly developed multi-field boundary variational principle.

Through numerical simulations, we demonstrate that the developed CGs have high computational efficiency, and they can accurately capture the localized stress distributions under various loadings. Computational Grains are also effective for estimating the effective material properties of fiber-reinforced composites, as validated by comparing with experimental results in the literature. Moreover, with the use of parallel computation, the time required for CGs is significantly decreased. Thus, we consider that the kind of CGs developed in this study is an accurate and efficient tool for the micromechanical modeling of fiber composites. Such a tool of micromechanical modeling can also be combined with meso- and macro-scale finite elements for the multi-scale analysis of laminates and composite parts, which will be given in Chapter 12.