The asymptotic structure of transient elastodynamic fields at the tip of a stationary crack

Abstract

The asymptotic structure of the transient elastodynamic near-tip fields around a stationary crack is investigated for all three fracture modes. The transient fields are obtained as the sum of their quasi-static counterparts and corresponding transient correction terms, in terms of variable-separable expansions. By allowing the coefficients of terms in the quasi-static expansion to deviate from their quasi-static restrictions, the correction terms are shown to be the particular solutions of a set of first order (for mixed mode I and II) or second order (for mode III) ordinary differential equations with constant coefficients and non-homogeneous terms involving only sine and cosine functions of the independent variable. It is found that the transient effects of dynamic loading on the near-tip fields are to alter the universal angular variations of the quasi-static field quantities for the fifth and higher order terms in their variable-separable expansions; thus the first four terms in the expansions have the same angular variations under both quasi-static and dynamic loading conditions. This seems to suggest that transient effects on the crack-tip fields are in general less severe for a stationary crack than for a propagating crack where only the first two terms in the expansions hold the same angular variations under both steady-state and transient crack growth conditions. Furthermore, the transient higher order terms for a stationary crack do not depend on the time-rate of the stress intensity factors; in fact, they only relate to the even order time-derivatives of the instantaneous values of the coefficients of the terms in the quasi-static expansions. This is also in contrast with the case of transient crack propagation where the time rates of the dynamic stress intensity factors play important roles in the higher order transient terms. Explicit expressions for the transient near-tip stress and displacement fields are provided.