Effective particle mapping

It has been long known that a spherical particle, surrounded by a spherical shell of different elastic moduli, can be exactly mapped into a new, uniform property spherical particle, which is as large as the old particle plus shell combined [33,22,34]. This can also be done for a circular particle surrounded by a circular shell. Of course, in concrete some of the shells overlap. In this effective particle mapping, overlaps are ignored, so that each particle is assumed to have a complete, isolated shell around it.

Let the interior particle be phase 3, with diameter b, and the shell material be phase 2, with outer diameter a. The phase label 1 is reserved for the matrix. Figure 1 shows a schematic of such a mapping. Ref.  [34] contains the formulas for such a mapping in 3-D, for both the effective properties K and G, and for G in 2-D. The 2-D mapping for Kis also included below. Note that in Ref. [34], in 3-D, label i is the same as 3 here, and label m is the same as 2 here. In 2-D, label f is the same as 3 here, and label m is the same as 2 here. Also, in both 2-D and 3-D, $\nu_i$_i is the Poisson's ratio for phase i.

The 3-D results are presented first. The effective G is the solution to the following quadratic equation (the positive square root is the physical choice),

A(G/G₂ ) ² + 2 B (G/G ₂ ) + C = 0 (11)

with the coefficients given by:

$$8 z (4 - 5\nu_2)\eta_{\alpha} p^{10/3} - 2[63 z \eta_{\beta} + 2 \eta_{\alpha} \eta_{\gamma}] p^{7/3} + 252 z \eta_{\beta} p^{5/3} -$$ A =

$$50 z (7 - 12 \nu_2 + 8 \nu_2^2) \eta_{\beta} p + 4 (7 - 10 \nu_2) \eta_{\beta} \eta_{\gamma},$$

$$-2 z (1 - 5\nu_2)\eta_{\alpha} p^{10/3} + 2[63 z \eta_{\beta} + 2 \eta_{\alpha} \eta_{\gamma}] p^{7/3}$$ B =

$$- 252 z \eta_{\beta} p^{5/3} + 75 z (3 - \nu_2) \eta_{\beta} \nu_2 p + \frac{3}{2} (15 \nu_2 - 7) \eta_{\beta} \eta_{\gamma},$$

$$4 z (5\nu_2 - 7)\eta_{\alpha} p^{10/3} - 2[63 z \eta_{\beta} + 2 \eta_{\alpha} \eta_{\gamma}] p^{7/3}$$ C =

$$+ 252 z \eta_{\beta} p^{5/3} + 25 z (\nu_2^2 - 7) \eta_{\beta} p - (7 + 5 \nu_2) \eta_{\beta} \eta_{\gamma},$$

$$\eta_{\alpha} z (7 - 10 \nu_2)(7 + 5 \nu_3) + 105 (\nu_3 - \nu_2),$$ =

$$\eta_{\beta} z (7 + 5 \nu_3) + 35 (1 - \nu_3),$$ =

$$\eta_{\gamma} z (8 - 10 \nu_2) + 15 (1 - \nu_2),$$ =

z = G₃/G₂ - 1,

p = (b/a)³. (12)

The effective K is given by:

$$K = K_2 + \frac{c (K_3 - K_2)}{1 + (1 - c) \frac{(K_3 - K_2)}{(K_2 + \frac{4}{3} G_2)}}$$ (13)

For 2-D, the plane strain shear modulus, µ ₂₃ , is used from Ref. [34], but with the notation of this paper. The effective G is given by the solution of the following quadratic equation (in this case the negative square root is the physical choice),

A (G/G₂)² + 2 B (G/G₂) + C = 0 (14)

with the coefficients given by:

$$3 p (1 - p)^2 (r - 1)(r + \eta_3)+[r \eta_2 + \eta_2 \eta_3 - (r \eta_2 - \eta_3) p^3][p \eta_2 (r - 1) - (r \eta_2 + 1)],$$ A =

$$-3 p (1 - p)^2 (r - 1)(r + \eta_3)+\frac{1}{2}[r \eta_2 + (r - 1) p + 1][(\eta_2 - 1)(r +\eta_3)-2(r \eta_2 - \eta_3)p^3]$$ B =

$$+ \frac{p}{2}(\eta_2 + 1)(r - 1)[r + \eta_3+(r \eta_2 - \eta_3)p^3],$$

$$3 p (1 - p)^2 (r - 1)(r + \eta_3)+[r \eta_2 + (r - 1) p + 1][r + \eta_3 + (r \eta_2 - \eta_3)p^3],$$ C =

$$\eta_i 3 - 4 \nu_i, \quad\hbox{for}\quad i=2,3,$$ =

r = G₃/G₂,

p = (b/a)². (15)

The effective K, derived independently from Ref. [34], is given by:

$$K = \frac{p (K_2 + G_2)K_3 + (1 - p)(K_3 + G_2) k_2}{p (K_2 + G_2) + (1 - p)(K_3 + G_2)}.$$ (16)