Elastic Theory Has Zero Radius of Convergence
Alex Buchel, James P. Sethna · Physical Review Letters · 1996
In nonlinear elastic theory, the inverse bulk modulus $K$, for example, depends on the compression $P$: $1/K(P){\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}c}_{0}{+c}_{1}{P+c}_{2}{P}^{2}+\ensuremath{\cdots}{+c}_{n}{P}^{n}+\ensuremath{\cdots}$ . 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: it is an asymptotic series. For a two-dimensional isotropic elastic medium allowing cracks we compute the asymptotic form ${c}_{n+1}/{c}_{n}\ensuremath{\rightarrow}{\mathrm{Cn}}^{1/2}$ as $n\ensuremath{\rightarrow}\ensuremath{\infty}$. We present an explicit formula for $C$ as a function of temperature and material properties.