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)0ex0ex=0ex0exc₀+c₁P+c₂P²+⋯+cₙPⁿ+⋯ . 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 cₙ₊₁/cₙ→Cn1/2 as n→∞. We present an explicit formula for C as a function of temperature and material properties.