Figure 1: Showing the system analyzed in the paper: (1) matrix, s < r < b (2) shell, a < r < s,
and
(3) aggregate,
0 < r < a, where r is the radial coordinate. The i'th phase has bulk modulus Ki,
shear
modulus Gi,
and expansive strain εio.
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Figure 1 shows a cut through the center of the spherical aggregate, and defines the terminology used. We take s = a+h, where a is the radius of the aggregate, h is the thickness of the layer around the aggregate, and b is the overall radius of the composite system being considered. The layer around the aggregate is not necessarily to be thought of as the interfacial transition zone, but only as a parameter of the problem to be discussed. Figure 1 is not to scale, since in this dilute limit the quantity c = a3/b3, the volume fraction of the aggregate phase, should be small (less than 0.01-0.03).
Figure 1: Showing the system analyzed in the paper: (1) matrix, s < r < b (2) shell, a < r < s,
and
(3) aggregate,
0 < r < a, where r is the radial coordinate. The i'th phase has bulk modulus Ki,
shear
modulus Gi,
and expansive strain εio.
Now suppose that at least one of the phases has a non-zero value of expansive strain, so that displacements and stresses will be set up in the system. In spherical polar coordinates, the radial component of displacement, denoted u, will be the only non-zero displacement and will be a function of r only. The origin is taken at the center of the aggregate. In terms of u, the three diagonal components of the strain tensor (all shear strains are zero) are: εrr = ∂ u / ∂ r and εθθ = εφφ = u/r, where θ and φ are the spherical polar coordinate angular variables. In the i'th phase, the two independent (σθθ = σφφ) diagonal components of the stress tensor are (all shear stresses are zero):
It has been shown before [5,6,7] that in phase i the displacement u will have the form u = αi r + βi /r 2 , where αi and βi are constants, different for each phase. Of course in phase 3, the aggregate, β3 = 0, because the displacement must not diverge. So there are five coefficients that must satisfy five equations, in order for there to be a solution to this problem. The five equations come from the fact that the displacement and the radial stress must be continuous at the interfaces (r = a and r = a+h), and the radial stress must be zero at the free boundary (r = b). These five equations are:
Equation (3) comes from continuity of u at r = a, eq. (4) is from continuity of displacement at r = a+h, eq. (5) represents continuity of radial stress at r = a, eq. (6) is from continuity of radial stress at r = a+h, and eq. (7) comes from the vanishing of radial stress at r = b. Taking various choices of the parameters, these equations can easily, though a bit tediously, be solved for the values of α and β in each phase. The stress, strain, and displacement anywhere can then be easily found. For mathematical convenience, the Poisson's ratio of each phase is taken to be 0.2, so that in each phase, Ki = 4Gi / 3, and K1=K2=1, and K3=4 (arbitrary units). The qualitative aspect of the solutions are not affected by this (physically reasonable) choice. It is easier to solve eqs. (3)-(7) for a given choice of parameters, than to develop a general solution into which different choices of parameters are substituted. The general solution of eqs. (3)-(7) is rather complicated [5].