Matched asymptotic expansions in nonlinear fracture mechanics—III. In-plane loading of an elastic perfectly-plastic symmetric specimen

Abstract

T he matched expansion method, introduced by the authors in two earlier papers (1976) devoted to mode III loading, is applied to the practically important case of mode I loading of a symmetric specimen. The method allows the linear elastic far-field to be considered separately from the elasto-plastic near-tip field, except for coupling through a set of parameters that are determined explicitly in the matching. The effects of the plasticity are thus found, once and for all, from the solution of a set of standard elasto-plastic problems for a semi-infinite crack in an infinite body, whose properties may be tabulated. The solution for any particular specimen geometry and loading then follows from a small set of linear elastic solutions for the specimen, which define, through coefficients γ ij appearing in their near-tip expansions, all the parameters in the “inner” and “outer” solutions. The effects of plasticity appear in these parameters only through a set of constants C ti that define the far-field expansions of the “inner” (near-tip) solutions: they are material constants, depending upon the constitutive relation for the material, but not upon specimen geometry and loading. The J-integral, being obtainable from the far-field, is expressed as an explicit asymptotic series in the loading parameter ε, whose coefficients are given as functions of the “elastic” parameters γ ij and the material constants C i. It is demonstrated that a plastic-zone correction term, r y, can be chosen to yield a two-term asymptotic expansion for J; the value of r y depends upon the yielding model only through the constant C 1. The Dugdale (1960) model of yielding is treated, as a simple example for which all calculations can be performed analytically, and for which exact solutions are available for comparison. Finally, the near-tip solutions are constructed for a material obeying the Mises yield criterion and associated flow-rule, using a specially developed finite element program. The first eight of the constants C i are tabulated, which suffice to define the J-integral up to terms of order ε 6 (where ε is a loading parameter) and some representative near-tip features are displayed graphically. The computed value of C 1 shows that the conventionally adopted value for the plastic-zone correction r y is too large by a factor of roughly 2.8, if it is to yield a genuine asymptotic estimate for J. As an example, the “elastic” parameters γ ij are found, from a boundary collocation program, for a centre-cracked square plate subjected to tensile loading; and a plot of J versus load, and the plastic-zone shape at a particular load level, are displayed.