Fracture of disordered solids in compression as a critical phenomenon. III. Analysis of the localization transition

The properties of the Hamiltonian developed in Paper II are studied showingthat at a particular strain level a ``localization'' phase transition occurscharacterized by the emergence of conjugate bands of coherently orientedcracks. The functional integration that yields the partition function is thenperformed analytically using an approximation that employs only a subset ofstates in the functional neighborhood surrounding the most probable states.Such integration establishes the free energy of the system, and upon taking thederivatives of the free energy, the localization transition is shown to becontinuous and to be distinct from peak stress. When the bulk modulus of thegrain material is large, localization always occurs in the softening regimefollowing peak stress, while for sufficiently small bulk moduli and atsufficiently low confining pressure, the localization occurs in the hardeningregime prior to peak stress. In the approach to localization, the stress-strain relation for the wholerock remains analytic, as is observed both in experimental data and in simplermodels. The correlation function of the crack fields is also obtained. It has acorrelation length characterizing the aspect ratio of the crack clusters thatdiverges as (\xi \sim (\ep_{c}-\ep)^{-2}) at localization.Comment: 11 pages, 3 figure