Elastic Theory Has Zero Radius of Convergence

Abstract

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 $n\to \infty$. We present an explicit formula for $C$ as a function of temperature and material properties.

5 pages, no Postscript figures

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