In an alkali-aggregate reaction, the gel may form inside the aggregate, in aggregate pores or cracks. The expansive forces then act only on the aggregate, not the cement paste matrix. We can model this case by setting ε1o = ε2o = 0, ε3o = ε, and letting h = 0 to remove the layer. Equations (3)-(7) then reduce to three equations for α1, β1, and α3, with the result
The radial stress is found to compressive everywhere. The tangential stress σθθ is compressive in the aggregate but tensile in the matrix, with its maximum tensile value at r = a. Therefore any cracking will be radially outward from the aggregate, and so no open gaps around the aggregates should in general be observed. In actual instances where expansive gel forms inside the aggregate, there will be parts of the aggregate under tensile stress, since the whole aggregate will not uniformly expand, so that there could be cracking within the aggregate as well. Treatment of random expansive centers within the aggregate cannot be done analytically, and must be handled by a numerical method [10,11].