The [formula omitted] concept for elastic-plastic material behavior

Abstract

THE CONSIDERATION of conservation laws within the theory of elasticity lead to the observation by Eshelby[l] that the generalized (material) force on a defect in an elastic solid could be expressed as a surface integral of the energy momentum tensor. Cherepanov[2] and independently Rice[3] introduced path-independent integrals for the evaluation of two-dimensional crack problems. Rice’s work especially gave a significant impulse towards the application of the J integral in cases beyond the limits of linear elastic fracture mechanics (LEFM) based on the K concept. The starting point of Rice’s considerations was the assumption of elastic (linear or non-linear) material behavior. This implies that the stresses can be derived from a potential, the strain energy density, which is a unique function of stresses or strains. On that basis it was not only shown that J is path independent but also that J is equivalent to the energy release rate. This latter interpretation established the correlation with the K concept of LEFM via the energy release rate G and, therefore, the potential of J as a fracture parameter became evident. Begley and Landes[4,5] used J as a fracture criterion parameter and proposed a multispecimen procedure to measure the material fracture toughness J,,. To avoid principal difficulties arising from a multi-specimen technique, Rice et aZ.[6] proposed estimation techniques for J from a single load-displacement record. By plotting J from their original results vs crack length change, Au, Begley and Landes[l derived the J integral resistance curve as a characterization of the material resistance against ductile tearing after initiation. Clarke et a1.[8] simplified this approach and proposed the partial unloading technique where in the course of a single-specimen experiment, the amount of crack extension is determined from the change of the elastic compliance during repeated small unloadings. The application of J-resistance curves to the evaluation of crack stability under ductile conditions was established by Paris et u1.[9] and Hutchinson and Paris[lO] who introduced a tearing instability theory. The acceptance of the J-integral concept was promoted by the fact that J was easily determined in the course of numerical analyses, especially by the finite element method. In two-dimensional cases J may be computed directly from the contour integral since all quantities entering J are available. More efficient and easily extended to three-dimensional situations is the stiffness derivative method introduced by Parks[l 1,121. DeLorenzi[l3] presented a node shifting and releasing technique to model crack extension for plane problems. In a plastic fracture handbook Kumar et al.[l4] compiled a series of approximate solutions for J and J(Aa) based on finite element results. A major concern regarding the application of J to materials described by an incremental or flow theory of plasticity (as is the case for most of the structural materials like e.g. steels or aluminium alloys) came from the fact that the theoretical basis of J was within a deformation theory of plasticity. For proportional loading up to initiation it could be argued that the stressstrain curves are identical for nonlinear elastic and incrementally plastic material laws and therefore the near-tip stress and strain field is the same. The J integral evaluated on the basis of deformation theory, was considered a valid fracture parameter, provided that J uniquely defines the stress and strain field in the vicinity of the crack. For plane situations and for certain approximations to the stress-strain curve, asymptotic solutions for the stress and strain field in the vicinity of a crack, the so called HRR-field have been derived by Hutchinson[lS] and Rice and Rosengren[l6]. In the case of crack extension this assumption is not valid, but the possible error was considered tolerable if the relative amount of crack extension stayed within certain limits and if it was ensured that the