An asymptotic finite-deformation analysis of the elastostatic field near the tip of a crack

Abstract

This paper contains an asymptotic treatment, consistent with the fully nonlinear equilibrium theory of compressible elastic solids, of the stresses and deformations near the tip of a traction-free crack in a slab of all-around infinite extent under conditions of plane strain. The loading applied at infinity is taken to be one of uniform uniaxial tension at right angles to the faces of the crack. For the particular class of elastic materials considered the tensile stress in large homogeneous uni-axial extension is asymptotic to a continuously adjustable power of the corresponding principal stretch. The asymptotic analysis of the foregoing crack problem is reduced to a nonlinear eigenvalue problem, the solution of which is established in closed form, in terms of elementary functions and a transcendental integral of such functions. This solution involves two arbitrary constants, one of which governs the amplitude of the ensuing elastostatic field near the tip of the crack. A precise estimate of the amplitude parameter, valid at sufficiently small load intensities, is deduced with the aid of a known conservation law. The remaining arbitrary constant, which is left indeterminate by the present lowest-order asymptotic analysis, does not affect the dominant behavior of the field quantities of primary physical interest. II-lustrative numerical results, appropriate to both hardening and softening materials, are presented.