Elastic Theory Has Zero Radius of Convergence

Nonlinear elastic theory studies the elastic constants of a material (such as Young's modulus or bulk modulus) as a power series in the applied load. The inverse bulk modulus K, for example depends on the compression P: ${1/ K(P)} = c_0 + c_1 P + c_2 P^2 \cdots + c_n P^n + \cdots$. 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 and thus can at best be an asymptotic series. Considering thermal nucleation of cracks in a two-dimensional isotropic, linear--elastic material at finite temperature we compute the asymptotic form ${ c_{n+1}/ c_n}\to C n^{1/2}$ as $n \rightarrow \infty$. We present an explicit formula for $C$ as a function of temperature and material properties.