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CLM Theorem

Recently, a new exact theorem has been proven for elastic composite problems in 2-D. There can be general anisotropy, and any number of phases, with perfect bonding between phases [22, 32,33]. For this manual, we will consider only isotropic phases. Assume that the i 'th phase has bulk and shear moduli K_i and G_i, and the overall effective moduli are K and G. Now make a transformation in each phase of the following form: (1 / K_i)' = (1 / K_i) − C, (1 / G_i' = (1 /G_i) + C, where C is an constant such that the new elastic moduli in each phase are still positive. Then the theorem says that the new effective moduli, K' and G', are given in terms of the old moduli by (1 / K)' = (1 / K) − C, and (1 / G)' = (1 / G) + C. This theorem can then be used to test elastic programs like ELAS2D.F.

The same microstructure can be used as in the Hashin-Rosen example, Sec. 5.9. The phase moduli were originally K₁ = 1, G₁ = 0.5, K₂ = 3, and G₂ = 1. The values of the effective moduli using these phase moduli are: K = 1.6627, G = 0.711, obtained using the program ELAS2D.F. Using a value of C = 0.2, the transformed phase moduli are K₁ = 1.25, G₁ = 5/11, K₂ = 7.5, and G₂ = 5/6, and so the transformed effective moduli are expected to be: K' = 2.492 and G' = 0.6225. The numerical values of the moduli, again obtained using ELAS2D.F, are: K' = 2.499 and G' = 0.6092, an error of only 0.3% in K, and 2.1% in G. This error is a direct measure of how well pixels can resolve the true interface, since the CLM theorem assumes a perfect interface.