Some Remarks on Elastic Crack-tip Stress Fields

It is shown that if the displacement field and stress intensity factor are known as functions of crack length for any symmetrical load system acting on a linear elastic body in plane strain, then the stress intensity factor for any other symmetrical load system whatsoever on the same body may be directly determined. The result is closely related to Bueckner's [1] "weight function". through which the stress intensity factor is expressed as a sum of work-like products between applied forces and values of the weight function at their points of application. An example of the method is given wherein the solution for a crack in a remotely uniform stress field is used to generate the expression for the stress intensity factor due to an arbitrary traction distribution on the faces of a crack. A corresponding theory is developed in an appendix for three-dimensional crack problems, although this appears to be directly useful chiefly for problems in which there is axial symmetry. INTRODUCTION CONSIDER a two-dimensional linear elastic body containing a straight crack under conditions of plane strain or of generalized plane stress. Both the body and all applied load systems to be considered are assumed symmetrical about the crack line so that only the tensile opening mode of crack tip deformation may result. Two distinct load systems are shown in Fig. 1 and denoted by Q 1 and Ql' The displacement field and stress intensity factor are assumed known as a function of crack length 1 for one ofthe load systems, say Ql ' The principal result _of this study is in showing that this information is sufficient to determine the stress intensity factor for the other load system Q2' Of course, the 1 and 2 systems may represent any arbitrarily chosen load systems and thus it is being shown that if a solution for the displacement field and stress intensity factor is known for any particular load system} then this information is sufficient to determine the stress intensity factor for any other load system whatsoever. Bueckner [1] has presented a similar result, basing his argument on analytic function representations of elastic fields for isotropic materials. Here we see that this dependence between solutions for different load systems arises as a consequence of what is known on the relation between stress·intensity factors and strain energy variations [2, 3J and of the properties of point functions. To develop the argument, consider the following preliminary remarks: (a) Ql and Q2 are considered as "generalized forces" in the sense that the stress vector t on the boundary r and body force fwithin region A resulting from, say, load system 1 are expressible in the form t = Qlt(l) on rand f = Qlf(l) in A;