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Mathematical Analysis of Dilute Limit for Thermal Expansion

The case of thermal expansion, in the linear regime, is handled using a modified stress-strain equation:

where C_ij is the elastic modulus tensor and the volumetric expansion terms are e₁ = e₂ = e₃ = e, and the shear terms are e₄ = e₅ =e₆ = 0. In the case of thermal expansion, e would be proportional to temperature, while in the case of moisture-induced expansion or shrinkage, e would depend on relative humidity [16]. The usual Voigt notation is used, where 1 = xx, 2 = yy, 3 = zz, 4 = xz, 5 = yz, and 6 = xy.

The geometry is the same as for the previous cases, except that now we introduce R_N + 1 as being the radius of the outermost layer of the matrix plus inclusion system. For free expansion, we need to take the normal stress to be equal to zero at this boundary. There is no applied strain at infinity, so some other condition is needed to be able to determine the problem. The case where the inclusion stops at r = R_N, and R_N + 1 >> R_N , is the dilute case, since the volume fraction of aggregate will be c = R₁³ / R_N+1³ << 1. Again, the only displacement is the radial displacement, and the same form is assumed for the displacement in each shell. All notation is the same as the bulk modulus case. The continuity of the displacement is also identical to the case of the bulk modulus, but now, with the extra thermal expansion terms present in the stress strain relationship, the radial stress in the nth layer becomes:

where e_n is the linear expansion of the nth layer.

The equation connecting the coefficients in the (n + 1)th layer to those in the nth layer is then somewhat different from the bulk modulus case as well:

where Z_n is a vector of the form:

Now, this equation can be iterated to connect the n = 1 coefficients to the n =N + 1 coefficients. Care must be taken when iterating, as the resulting expression is more complicated than in the bulk modulus case, because of the presence of the Z_n terms. The result is

which then can be written as:

Using the fact that B₁ = 0 and A_N + 1 and B_{N + 1} are related via the zero stress condition at R_N + 1, this equation becomes two linear equations for two variables, A_N + 1 and A₁. These two variables are then:

and

The overall thermal expansion is then just how much the size of the system has changed, so that the effective thermal expansion, e_tot, is the displacement at r =R_N + 1 divided by the original radius of the complete system, R_N + 1: