On the Stress Distribution at the Base of a Stationary CrackBy M. L. WILLIAMS, PASADENA, CALIFAbstractIn an earlier paper it was suggested that a knowledge of the elastic-stress variation in the neighborhood of an angular corner of an infinite plate would perhaps be of value in analyzing the stress distribution at the base of a V-notch. As a part of a more general study, the specific case of a zero-angle notch, or crack, was carried out to supplement results obtained by other investigators. This paper includes remarks upon the anti-symmetric, as well as symmetric, stress distribution, and the circumferential distribution of distortion strain-energy density. For the case of a symmetrical loading about the crack, it is shown that the energy density is not a maximum along the direction of the crack but is one third higher at an angle cos-1 (1/3); i.e., approximately 70 deg. It is shown that at the base of the crack in the direction of its prolongation, the principal stresses are equal, thus tending toward a state of hydrostatic tension. As the distance from the point of the crack increases, the distortion strain energy increases, suggesting the possibility of yielding ahead of the crack as well as 70 deg to the sides. The maximum principal tension stress occurs on 60 deg rays. For the anti-symmetrical stress distribution the distortion strain energy is a relative maximum along the crack and 60 percent lower 85 deg to the sides.SummaryEarliest contributions Inglis 1 studied an internal crack using elliptical bounding surface. Griffith 2 set up an energy criterion for crack instability. Westergaard initially treated the crack problem in the same way to be exploited in this paper 3, as well as by the complex variable technique in a later paper 4. Hollister 5 examined the stress distribution at the base of crack photoelastically with first attempt to measure isochromatic-fringe patterns in this specific application apparently. Post has published some interesting results of his photoelastic observations for the case of an edge crackPurpose of this paperIt is the purpose of this paper to supplement the results of these investigators in certain respects which it is hoped will aid in the further understanding of the elastic-stress distribution at the base of a stationary crack.Formula derivation & Theoretical explanationFor the particular purpose of this paper it is desired to consider the case where the two radial edges of the plate are unloaded and the included angle approaches 2. The stress functions of the formFour boundary conditions can be satisfied byIt is a more convenient alternate form of expressing Equation 7 in terms of the bisector angle =-, the stress function (r, ) can be split into its even e(r, ) and odd 0(r, ) parts with respect to (Fig. 1)Upon writing out the first few termsFrom which the associated stresses may be found asThe total strain-energy densityand the strain energy due to distortion alone, i.e., the total strain energy less that due to change in volumeThe definition of the octahedral shearing, respectively, asThe displacements 7 are found to beIn the foregoing expressions it is seen that the term multiplied by the coefficient a1b1 represents a coupling between the symmetric and antisymmetric variations with respect to .Symmetrical Stress DistributionSpecializing the stresses and the other quantities to this case givesThere are certain other interesting features which, because of the relative simplicity of the previous expressions, become readily apparent.Antisymmetrical Stress DistributionWhen the stress distribution is antisymmetric, which is a case that does not seem previously to have been treated explicitly, such as may exist about a crack parallel to the neutral axis of a beam in pure bending, there resultsIt would be interesting to experiment photoelastically with the antisymmetric-loading condition, which is not too simple a matter, in an analysis similar to that of Post for the symmetrical case.Conclusion1. Even in the presence of a partial (two-dimensional) hydrostatic-stress field there would be a reduced tendency for yielding at the base of the crack.2. As the magnitude of the individual stresses increases as the inverse half power of the radius, the stress should become quite high with a tendency toward a cleavage failure.3. At the same time the elastic analysis indicates there should be large amount of distortion off to the sides of the crack and presumably some yielding should take place in these areas would tend toward a ductile failure.4. The character of failure actually occurring in a given specimen would depend upon the material.5. When the stress becomes nonhydrostatic as the distance from the point of the crack increases, by Equation 35, there may be reason therefore to suspect a yielding region ahead of the crack also, although not as severe.6. For concerning the direction of cracking or forking