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, 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.