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Quantitative homology metrics are proposed for characterizing the thermal–elastic response of polycrystalline materials. Simulations for a calcite-based polycrystal, marble, are used as an illustrative example. The homology metrics are based on topological measurements, such as the number of components and the number of handles of the thermal–elastic response fields for a complex microstructure. These homology metrics are applied to characterize not only the elastic energy density and maximum principal stress response fields in a polycrystal but also the correlated grain-boundary misorientation distributions that influenced the formation of these response fields. It is demonstrated that the topological analysis can quantitatively distinguish between different types of grain-boundary misorientations, as well as between differences in the resulting response fields.

Using measurements of the geometric and crystallographic characteristics of approximately 104 triple junctions in a MgO polycrystal, the relative grain boundary energy has been determined as a function of five macroscopic parameters. The relative energy of a particular grain boundary is inversely correlated with its frequency of occurrence. At all misorientations, grain boundaries with {1 0 0} interface planes have relatively low energies. For low misorientation angle grain boundaries, the results are consistent with the predictions of dislocation models. At high misorientation angles, the population and energy are correlated to the sum of the energies of the free surfaces that comprise the boundary.

A semi-automated method has been used to measure all five macroscopically observable parameters of 4.1×10 6 boundary plane segments making up 5.4 mm 2 of boundary area in a hot-pressed magnesia polycrystal. The observations allow a complete description of the distribution of crystal orientations, grain boundary misorientations, and the crystallographic orientations of grain boundary planes. Among the low misorientation angle grain boundaries, there is a preference for tilt boundaries, especially those with boundary plane normals in the direction. At all fixed misorientations, there is a preference for boundaries with a boundary plane normal in the direction. These boundaries are generally asymmetric and occur at least twice as frequently as the average boundary for each fixed misorientation.