An initially exponential distribution of cracks that grow in size and nucleate additional cracks is analyzed, leading to an expression for the statistical distribution of cracks as a function of time in closed form. The results are used to derive a reduced modulus for the cracked material. An approach to three-dimensional calculations of fragmentation is also discussed.
The current generation of high speed computers makes it possible to simulate explosion and impact phenomena in considerable detail by numerical integration of the equations of continuum mechanics. Frequently, however, our knowledge of continuum properties if so limited that it is not possible to make successful simulations. This is particularly the case in rock mechanics, where micro-fracturing may dominate. Plasticity can be made to account for rock deformation only with great difficulty, and does not provide information on crack statistics, on rubble size. This information is particularly needed to analyze in-situ retorting of oil shale obtain moderately priced fuels. We believe that numerical simulation of bed preparation may make it possible to obtain more efficient plosives and more uniform beds and, consequently, more efficient retorting, but existing material models do not appear to be capable of detailed simulation. We have been particularly interested in determining the influence of the bedding planes, which we characterize by representing the material as transversely isotropic. In the analysis that follows we have attempted to show how statistical ideas can be used to develop material properties in a form that can be used in numerical modeling. Though the work is motivated by our interest in oil shale retorting, it may have applications in other processes that involve fragmentation. Of particular interest is the possibility of calculating oriented cracks that can transform an initially isotropic material into an anisotropic one.
