Elastic Theory Has Zero Radius of Convergence

TL;DR

This paper demonstrates that nonlinear elastic theory's power series expansion has zero radius of convergence due to crack instability, providing an asymptotic form for the series coefficients related to thermal crack nucleation.

Contribution

It establishes the zero radius of convergence for elastic series and derives an explicit asymptotic formula for the series coefficients considering thermal crack nucleation.

Findings

Series coefficients grow as C n^{1/2} asymptotically

Elastic series is only asymptotic, not convergent

Explicit formula for the growth constant C

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