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A result by Hashin and Rosen [30], which is true for both 2-D and 3-D, exactly relates, for the case of two phases, the effective bulk modulus K and the effective thermal strain e when the two phases do not have the same elastic moduli. The isotropic thermal strain for phase i is ei, and the effective isotropic thermal strain for the composite system is denoted e. The result for the value of e is
To illustrate the accuracy with which the finite element programs obey the Hashin-Rosen result, eq. (74), a 200 x 200 pixel system was set up. A total of 160 overlapping circles, 15 pixels in diameter, were thrown down randomly, such that the matrix phase had an area fraction of c1 = 0.4779, and the inclusion phase had an area fraction c2 = 0.5221. The matrix phase had E1 = 1, ν1 = 0.2, and a thermal strain of 0.1 in the x and y directions (e = 0.1). The inclusion phase had E2 = 0.3, ν2 = 0.1, and a thermal strain of 0.4 in the x and y directions (e = 0.4). Using the Hashin-Rosen equation, an overall expansion of 0.2109 is predicted. The program THERMAL2D.F gave a value of 0.2103 in the x- direction, 0.2112 in the y-direction, and an average expansion of 0.2108. This average value is within 0.05% of the exact value. The small difference between the x and y directions (< 0.5%) is due to the fact that this fairly small numerical system is not exactly isotropic.