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.

The preprocessing of Computational Grains (CGs) is introduced in Chapter 3, and several types of CGs have been developed for the micromechanical modeling of different kinds of composites with particulates, fibers, and so on in Chapters 5–11. A multi-scale analysis framework of composite structures by using the CGs and the standard FEM is developed in this chapter, based on the homogenization of composite materials at the microlevel, and slender or shell structures at the meso- and macro-levels. The specific process of the multi-scale algorithm is illustrated with an example of a stiffened composite panel. The results show the multi-scale analysis method is an accurate and efficient tool for large composite structures, not only simulating the overall structural responses in a bottom-up fashion, but also obtaining the detailed stresses at multiple scales in the dehomogenization process.

In this chapter, a new type of Computational Grains is proposed to study the micro-electro-mechanical behavior of composite piezoelectric materials. This method is based on a new hybrid variational principle, and independently assumed displacements and electric potentials in each CG. Each CG can efficiently model a single physical grain of the composite material, thus saving a significant time of generating complex FEM mesh. Computational Grains can also model porous and composite piezoelectric materials even if the distribution of voids/inclusions is not symmetrical (which is assumption used with all unit cell models). Because the trial solutions are complete but do not satisfy the governing differential equations a priori, the formulation is very simple, and can accurately account for the local field concentrations efficiently and accurately. This is illustrated using different examples where the fields along the void/inclusion periphery are calculated, and the effective material properties of porous and composite materials are predicted, and compared with other analytical and computational models. The proposed CG in this chapter is expected to become a very powerful tool of direct numerical simulations of the micro/meso mechanics of composite piezoelectric materials, and can possibly lead to efficient multi-scale modeling of piezoelectric devices.

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, a Computational Grain is developed for direct numerical modelling of composites with nanoscale inclusions considering both interface stretching and bending effects, using a large number of CGs in a representative volume element. The CGs developed in this chapter are by far the first and the only numerical tool for direct numerical modelling of nanocomposites with a large number of inclusions with Steigman-Ogden matrix/inclusion interfaces. By using a new boundary-type multi-field variational principle together with Papkovich-Neuber potentials, the stiffness matrices of CGs can be directly evaluated and assembled. Together with the parallel algorithms, it is found that very efficient simulations of nanocomposites can be realized, for example, a SERVE containing 10,000 nano inclusions only takes fifty minutes on a sixteen-core computer. The influence of spatial distributions of the nano inclusions on the overall properties of nanocomposites is also investigated in this chapter. We also study the influence of interface bending resistance parameters on the effective modulus of nanocomposites. Numerical results show that interface bending resistance parameters affect the shear modulus of nanocomposites but their effect on the bulk modulus is negligible.

In this chapter, Trefftz trial functions which satisfy identically all the governing equations of linear elasticity in 2D and 3D problems are summarized. These Trefftz functions are later used in conjunction with boundary variational principles (since all the field equations are satisfied identically inside the Voronoi cell elements), to construct planar and 3D Computational Grains to directly model statistically equivalent representative volume elements (SERVEs) of heterogeneous materials at the microscale. In as much as the Trefftz functions are used as trial solutions, this modeling captures the correct and accurate stress solutions in the matrix, inclusions, and at their interfaces. The presented Trefftz solutions include: (1) Muskhelishvili’s complex functions for 2D problems,(2) Papkovich-Neubar solutions for 3 D problems,and (3) Harmonic functions in spherical coordinates, cylindrical coordinates, and ellipsoidal coordinates.

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.

This chapter discusses some general algorithms which are useful in the computational homogenization using the Computational Grains (CGs) method. First, an algorithm for generating a statistically equivalent representative volume element (SERVE) is presented. Then, an algorithm to divide the SERVE in to Voronoi cells (polygons in 2D and polyhedrons in 3D), and using a CG in each Voronoi cell is discussed. The role of parallel computation is also discussed.

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.

By rearranging the weakly singular boundary, integral equations developed by Han and Atluri, an SGBEM-CG, which is abbreviated as CG, is developed in this chapter. The CG, representing a single grain of a material, can include arbitrarily shaped voids, inclusions (of a different material), and microcracks. The CG has a stiffness matrix and a load vector, which have similar physical meanings to the traditional displacement FEM. The stiffness matrix is symmetric, positive-definite, and has the same number of rigid-body modes. Different CGs, each with different isotropic material properties, can be directly coupled by the assembly procedure, and are used to directly and efficiently model the microstructure of heterogeneous composite materials. Some examples are also presented, with microcracks interacting with inclusions and holes. This provides some insight of a possible future study of the micro-cracking and damage of heterogeneous material. By introducing stochastic variations of the shapes of CG, and stochastic variations of the properties of the constituent materials, the realistic statistical bounds on the overall properties of composite materials will be determined in future studies.