Determination of the amplitudes in analytical asymptotic solutions of a tensile elastic-plastic crack by variational method

Abstract

Multi-parameter asymptotic solutions of an elastic-plastic mode I crack were proposed by several authors. The unknown parameters, such as the amplitudes in asymptotic ex pansion, were usually estimated by using the “point matching” technique. In this paper, this technique is evaluated by a parallel variational method recently developed. The analytical solutions are included in a statically admissible full stress field, and the amplitudes are adjusted by minimizing the complementary energy of the structure. It is shown that the variational method can provide accurate values of the amplitudes. The results show that in certain cases, the four-term analytical solution can reproduce adequately the stress field in the vicinity of the crack tip. However, in some other cases, even the four-term analytical solution cannot describe the near-tip field if the amplitudes are calculated in the frame of the full stress field. More higher order terms are needed. In these cases, some approximate solutions, such as the J- Q theory, may be more advantageous in engineering applications.