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Analytically solving for the effect of sand grains on the overall shrinkage of the cement paste-sand composite is possible, as far as is known, only in the dilute limit where the volume fraction of sand is small enough to ignore interactions between sand grains. In this limit, which is reasonable for a sand volume fraction of less than about 6-8%, a single inclusion problem can be solved in the following way.
Consider three concentric spheres of radii a, s, and b, with a < s < b, with the origin taken at the common center:
where Ki and Gi are the bulk and shear moduli of the i'th phase and εio is the isotropic shrinkage strain of the i'th phase. Each phase is assumed to be elastically isotropic. The unrestrained shrinkage (eigenstrain) tensor for the i'th phase is then:
In this spherically symmetric problem only the displacement u(r) is non-zero, where u is the radial component of the elastic displacement vector. Only the radial equilibrium equation then needs to be solved, ∂σrr/∂r = 0. The general solution for u is ui(r) = αi r + βi/r2 , where ui(r) is the radial displacement in the i'th phase, and αi and βi are unknown coefficients which will be determined by the boundary conditions. We note that the unrestrained shrinkage strains of each phase do not come into the equilibrium equation, since they are constants that disappear after the derivative ∂ / ∂r is taken. However, they do come into the stress-strain relations, σi = Cij ( εj − εjo), used to match boundary conditions.
There are five boundary conditions for this problem: the radial stress and u(r) are
continuous at r=a and r=s, and the radial stress equals zero at r=b. Once the solution for the
displacements are obtained, the overall shrinkage strain is just
ε* = u(r=b)/b.
The final result is fairly complicated. The composite shrinkage strain, ε* , normalized by ε1o, the unrestrained shrinkage strain of the bulk cement paste matrix (phase 1), is given by:
To do the identical problem in 2-D, simply replace the factor 4/3 in Equations (7)-(9) by 1, and replace V2 and V3 by A2 and A3, where A2 and A3 are the interfacial zone and sand areas, respectively. This problem has recently been generalized to the case of a gradient of elastic and shrinkage properties surrounding a spherical aggregate [43,44].