Elastic theory has zero radius of convergence

In nonlinear elastic theory, the inverse bulk modulus K, for example, depends on the compression P: 1/K(P) = c(0) + c(1) P + c(2)P(2) +...+ c(n)P(n) +.... Elastic materials that allow cracks are unstable at finite temperature with respect to fracture under a stretching load. As a result, the above power series has zero radius of convergence: it is an asymptotic series. For a two-dimensional isotropic elastic medium allowing cracks we compute the asymptotic form c(n+1)/c(n) --> Cn(1/2) as n --> infinity. We present an explicit formula for C as a function of temperature and material properties.