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Equal shear modulus

A result that can be very useful in testing elastic programs is the equal shear modulus result, true for any microstructure [21,22]. If there are only two phases present, with equal shear moduli G but different bulk moduli K₁ and K₂, then the effective shear modulus of the entire system is just G, and the effective bulk modulus K of the system is, in 2-D,

We choose the same cubic image as was used for the small contrast in conductivity case to test the equal shear moduli result. A 22 x 22 x 22 pixel cube (phase 2) is centered in a 30 x 30 x 30 unit cell, so that the volume fractions are c₁ = 0.60563 and c₂ = 0.39437. The two shear moduli are taken equal to unity, G₁ = G₂ = 1, and the two bulk moduli are K₁ = 1, and K₂ = 20. The exact answer, according to eq. (53), is K = 2.263250, while the numerical answer is K = 2.269684, a difference of only 0.3%.

In 2-D, a test of the equal shear moduli result can be combined with a test of the effect of digital resolution by considering the checkerboard problem. The shear moduli are both equal to 2, and there is a ratio of 10 between the bulk moduli (K₁ = 1, K₂ = 10) (see Fig. 6 for a picture of the microstructure). The effective bulk modulus is computed as a function of system size L x L, where each "check" is L/2 x L/2. The exact value of K is 2.8 using eq. (60). Table 9 displays the data from this test. For L > 16, the error in the effective bulk modulus is about 1% or less.