  We study a class of models for brittle fracture: elastic theory models which
allow for cracks but not for plastic flow. We show that these models exhibit,
at all finite temperatures, a transition to fracture under applied load similar
to that at a first order liquid-gas transition. We study this transition at low
temperature for small tension. We discuss the appropriate thermodynamic limit
of these theories: a large class of boundary conditions is identified for which
the energy release for a crack becomes independent of the macroscopic shape of
the material. Using the complex variable method in a two-dimensional elastic
theory, we prove that the energy release in an isotropically stretched material
due to the creation of an arbitrary curvy cut is the same to cubic order as the
energy release for the straight cut with the same end points. We find the
normal modes and the energy spectrum for crack shape fluctuations and for crack
surface phonons, under a uniform isotropic tension. For small uniform isotropic
tension in two dimensions we calculate the essential singularity associated
with fracturing the material in a saddle point approximation including
quadratic fluctuations. This singularity determines the lifetime of the
material (half-life for fracture), and also determines the asymptotic
divergence of the high-order corrections to the zero temperature elastic
coefficients. We calculate the asymptotic ratio of the high-order elastic
coefficients of the inverse bulk modulus and argue that the result is unchanged
by nonlinearities --- the ratio of the high-order nonlinear terms are
determined solely by the linear theory.
