Analytic approximations for the elastic moduli of two-phase materials

Based on the models of series and parallel connections of the two phases in a composite, analytic approximations are derived for the elastic constants (Young's modulus, shear modulus, and Poisson's ratio) of elastically isotropic two-phase composites containing second phases of various volume fractions, shapes, and regular distributions. Comparison with a plentitude of finite element simulations and numerous previous experimental investigations shows a large consistency between the results and the analytic expressions derived, confirming the adequacy of the present approach. Compared with previous classical models, the present model has several advantages, including its simplicity, accuracy of prediction, and universal applicability.

Figure 1

Illustrations of the various connection sequences [(a) and (b)] for joining the dismounted cubes and the elastic moduli calculated accordingly (c) (with $E_2=2\mathrm{GPa}$ for the inclusion and $E_1=120\mathrm{GPa}$ for the matrix), in which the designations Pxyz/Sxyz represent connection series starting with a parallel or series connection and blue and red numbers mark the number of elements added by parallel and series connection, respectively.

Figure 2

Schematic representation of the unit cell of the two-phase composites where a cube of a second phase is embedded in a cube of matrix material (a) and a cuboid of a second phase in a cuboid of matrix material (b). The integration method for calculating the elastic moduli of the composite is illustrated in (a) as well.

Figure 3

Strain distributions obtained by FEM for composites with cubic unit cells of matrix material containing a cubic inclusion for different volume fractions of the second phase $(f_2=0.05$ or $0.5)$ under pure elastic loading. For an easier viewing of the strains in the unit cell, part of it has been removed on the face facing right. Two different cases are illustrated: in the left column the matrix is more compliant than the inclusion (compliant matrix) and in the right column the matrix is stiffer than the inclusion (stiff matrix).

Figure 4

Comparison between the proposed analytical model and the results of FEM simulations for different particular arrangements of unit cell of two-phase composites: (a) 2D interpenetrating case (square cross section in square), (b) 3D embedding case (cube in cube); a FEM simulation of a spherical inclusion in a matrix cube is also included, (c) rectangular cross section in rectangle and (d) circular cross section(s) in square matrix. The results in (a)—(c) confirm the adequacy of the derived analytical expressions [Eqs. (9), (10), and (12)]; (d) proves the rationality of the consideration that a basic unit cell can represent the elastic properties of a composite with regularly (periodically) arranged inclusions.

Figure 5

Comparisons between several model predictions and the analytical expressions derived here with reported experimental results for elastic properties of two-phase composites: (a)–(g) Young's modulus, (h) shear modulus, and (i) Poisson′s ratio. It shows that the suggested model leads to more concise and accurate predictions than previous bounds.