The elasticity-conductivity connection for materials with cracks of arbitrary shapes and orientation distributions

The cross-property connection between the effective elastic and conductive properties of a solid with cracks has been first given by Bristow (1960) for randomly oriented cracks, assuming that cracks have circular (penny) shapes. However, cracks in materials typically have "irregular" shapes and this factor, generally, has strong effect on the said properties (the strong stiffening effect produced by contacts between crack faces is an example). In addition, shapes of 3D cracks (for example, the statistics of the mentioned contacts) may be largely unknown. These factors make it difficult to estimate the effect of cracks on the elastic properties. We show that flat cracks of arbitrary shapes and orientation distributions produce similar effects on the elastic and conductive properties, so that changes of stiffnesses can be monitored by changes in electric conductivities. Extension to transversely isotropic materials is given. Possible extensions to nonflat crack shapes are discussed.