Next: Basic equations
There are many ways in which a concrete can be damaged by chemical action, including various kinds of sulfate-attack, alkali-aggregate reaction, and others [1,2,3]. Often, the chemical attack causes one or more phases in the concrete material, either in the paste or in the aggregates, to grow physically. This growth induces tensile strains that cause cracking that can eventually lead to severe damage to the material. The kinds of cracking that are produced depend on which constituent has been induced to grow. Often the crack pattern that is seen under the microscope is used to try to diagnose which deleterious mechanism was responsible for the damaged concrete. There seems to have been some confusion in recent years concerning what kinds of cracking imply what kinds of physical mechanisms.
The simple, analytically soluble case of a single isolated spherical aggregate, surrounded by a shell of arbitrary thickness, all embedded in a uniform matrix, where each of the three phases can have arbitrary elastic moduli and expansive strains, can be illustrative of what sort of stresses and expansions one would expect to see given various choices of elastic and expansive parameters. In a real concrete, the close positioning of many aggregate particles will of course play a role, but the isolated aggregate case will dominate the overall qualitative features of the stress and displacements that will be seen in the real material. In this paper, we show the general equations for this case, and illustrate the kinds of stress and displacement patterns that arise from different choices of the elastic moduli and expansive strain parameters. Some of these calculations, without the shell phase, have been done before .