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Elastic displacements caused by a propagating crack in an infinite medium: An exact solution

DEPARTMENT OF EARTH AND PLANETARY SCIENCES MASSACHUSETTS INSTITUTE OF TECHNOLOGY, CAMBRIDGE, MASSACHUSETTS 02139

Abstract

Elastic displacements in a two-dimensional finite space caused by an infinitesimal elastic dislocation with a step function as the time function are obtained analytically with the Cagniard method. The limit of the solution, as the time goes to infinity, reduces to the known static solution. By introducing the delay time (depending on the velocity of propagation of the crack) and making use of the superposition principle in our linear system, the exact elastic displacements caused by a propagating crack may be obtained by integrating our solution over the crack length. The integration is carried out analytically, and computer-generated plots of displacements versus time for an observation point near the middle of the crack final length are obtained and serve as a check on the final solution. The cases of subshear, supershear, subsonic and supersonic propagation of the crack are considered. Applications of the solution to earthquake problems are discussed and the introduction of a more realistic rise-time function is also considered to facilitate comparison between the observed and calculated displacements.

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