It has been shown in a recent work [1] that the elastic T-term at the tip of a mixed mode crack can be determined by the so-called second order weight functions through a work-conjugate integral that is akin to that of the Bueckner-Rice weight function method for evaluating stress intensity factors. In this paper, the development of the second order weight functions is reviewed. These second order weight functions are determined using a unified finite element method introduced in [2]. The finite element procedure handles both traction and displacement boundaries and it permits the Bueckner-Rice weight functions and the second order weight functions for the elastic T-term to be determined in one single finite element run. The accuracy of the computed weight functions is assessed by comparing the computed results with special closed form solutions. The numerical values of the elastic T-term for single edge notch specimens under tension, pure bending and three-point bend are given. The corresponding second order weight functions are tabulated.

AN EXACT asymptotic analysis is presented of the stress and deformation fields near the tip of a quasistatically advancing plane strain tensile crack in an elastic-ideally plastic solid. In contrast to previous approximate analyses, no assumptions which reduce the yield condition, n priori, to the form of constant inplane principal shear stress near the crack tip are made, and the analysis is valid for genera1 Poisson ratio 1~. Specific results are given for v = 0.3 and 0.5, the latter duplicating solutions in previous work by L. I. Slepyan, Y.-C. Gao and the present authors. The crack tip field is shown to divide into five angular sectors of four different types; in the order in which these sweep across a point in the vicinity of the advancing crack, they are: two plastic sectors which can be described asymptotically (i.e., as r + 0, where r is distance from the crack tip) in slip-line terminology as 'constant stress' and 'centered fan' sectors, respectively; a plastic sector of nonconstant stress which cannot be described asymptotically in terms ofslip lines; an elastic unloading sector; and a trailing plastic sector of the same type as that directly preceding the elastic sector. Further, these four different sector types constitute the full set of asymptotically possible solutions at the crack tip. As is known from prior work, the plastic strain accumulated by a material point passing through such a moving 'centered fan'sector is O(ln r) as r + 0; it is proved in the present work that the plastic strain accumulated by a material point passing through the 'constant stress' sector ahead of a growing crack must be 6ess sinyular than In r as r -+ 0. As suggested also in earlier studies, the rate of increase of opening gap 6 at a point currently at a distance r behind, but very near, the crack tip is given for crack advance under contained yielding by 1where a is crack length, o. is tensile yield strength, E is Young's modulus, J is the value ofthe i-integrai taken in surrounding elastic material, and the parameters c( and R are undetermin~ by the asymptotic analysis. The exact sohttion for v = 0.3 gives g = 5.462, which agrees very closely with estimates obtained from finite element solutions. An approximate analysis based on use of slip line representations in all plastic sectors is outlined in the Appendix.

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In an extension of e a rlie r studies by Rice and Sorensen, we discuss the e la s tic-p la s tic stre ss and deformation field s at the tip of a crack which grows in an ideally p la s tic solid under plane stra in , small scale yielding conditions. The re su lts of an asymptotic analysis suggest the existence of a crack tip stress state sim ilar to th at of the cla ssic a l Prandtl fie ld , but containing a zone of e la s tic unloading between the centered fan region and the tr a ilin g constant stress p la s tic region. The near tip expression for the ra te of open ing displacement 6 at distance r from the growing tip is found to have the same form suggested by Rice and Sorensen, 6 = a j / c + 3(a /E) a £n(R/r) o o but now the presence of the e la s tic wedge causes B to have the revised value of 5.08 (for Poisson ra tio v = 0.3),. Here, a = crack length , oq = yield strength, E = e la s tic modulus, and J denotes the f a r-fie ld value, namely (1-v2)K2/E for the small scale yielding conditions considered. The parameters a and R cannot be determined from the asymptotic analysis, but comparisons with f in ite element solutions suggest th a t, a t least for small amounts of growth, *Presented at ASTM 12th Annual Symposium on Fracture Mechanics, Washington University, St. Louis, 21-23 May 1979; submitted to ASTM for publication. a is approximately the same for stationary and growing cracks, and R scales approximately with the size of the p lastic zone, being about 15% to 30% larger. For large scale yielding i t is argued that a similar form applies with possible variations in a and 6 , at least in cases which maintain tria x ia l constraint at the crack tip , but in the fully yielded case R is expected to be proportional to the dimension of the uncracked ligament. The model crack growth criterion of Rice and Sorensen, requiring a c ritic a l 6 at some fixed r from the tip , is reexamined in light of the more accurate solution. The resu lts suggest that the J versus Aa relation describing growth will be dependent on the extent of yielding, although i t is suggested that th is dependency might be small for highly ductile materials, provided that a similar tria x ia l constraint is maintained in a ll cases.