Stress intensity factors for an elliptical crack approaching the surface of a semi-infinite solid

Abstract

A B S T R A C TStress intensity factors for an embedded elliptical crack approaching the free surface of the semi-infinite solid that is subjected to uniform tension perpendicular to the plane of crack are presented in a nondimensional form for various crack aspect ratios and crack distances from the free surface. Stress intensity factors are determined numerically using an alternating technique with two solutions. The first solution involves an elliptical crack in a solid and subjected to normal loading expressible in a polynomial of x and y. The second solution involves stresses in the half space due to prescribed normal and shear stresses on the surface. Effect of the Poisson's ratio on these stress intensity factors is also investigated. Stress intensity factors for a semi-elliptical surface crack in a finite thickness plate are then estimated in a nondimensional form for various crack aspect ratios and crack depth to plate thickness ratios. Nomenclature~b suitable harmonic stress function = ~f= o ~ffi 0 ~ij; i+j < 3 2,/a, v ellipsoidal coordinates a semi-major axis of ellipse b semi-minor axis of ellipse h distance from the free surface to the center of ellipse or plate thickness X 2 y2 z2 co(s) 1 a2+s bZ+s s Q(s) s(a 2 + s)(b 2 + s) r/ Poisson's ratio G shear modulus u~ incomplete elliptic integral of the first l~ind K(k) complete elliptic integral of the first kind E(ul) incomplete elliptic integral of the second kind E(k) complete elliptic integral of the second kind snu. cnul, dnux, dcut, cdut, ndu~, ncul, sdu~ = Jacobian elliptic functions k, k' modulus and complimentary modulus of Jacobian elliptic functions, respectively 0 K~e with k 2 = (1 -b2/a 2)angle in the parametric equations of ellipse, x = a cos 0; y = b sin 0 90 °-0 i.e., x = a sin t, y = b cos fl opening mode stress intensity factor stress intensity factor for an elliptical crack embedded in an elastic solid

Alternating method applied to edge and surface crack problems

“…Sin [22] has discussed this point in detail in connection with the finite thickness crack problem. This difficulty is clearly evidenced by the variation among results [6,9,10] published on the semi-circular and semi-el 1iptical surface crack problems. In [6,9], the maximum value of the stress-intensity factor was found to be on the free surface, whereas in [10] the maximum occurred at the utmost interior point of the crack front.…”

“…The trend of the curves differs substantially from those obtained in [6] using basically the same numerical procedure. The results of [6], shown as a dashed line in Figure 3 As a result of the difficulty, the elliptical crack problem was restricted in [9] to be of the form p(x.y) = a OQ + a 01 y + a 2Q x 2 + a 02 y 2 + a 21 x 2 y + a Q3 y 3 It is conceivable that in some applications the alternating method will involve three or more sequences of solutions. In general, one sequence would be expected to be necessary for each part of the boundary of the material.…”

“…-stress intensity factor k(0) -dimensionl ess stress intensity factor, lv ; k (n-l) ( | )ks(e) + k (n) (e ), contribu tion of n iteration to k(e) ki n) (6) -k(9) produced by 0_ = p n ti n) (x,y) on n crack z u IM k q (9) -k(9) produced by singular stress (Section 3 One of the requisites for solving any crack problem is to handle the stress singularities at the crack tips properly.…”

“…A generalization of [8] was later made by Shah and Kobayashi [9] in which the embedded crack is elliptical in shape with the minor axis perpendicular to the surface.…”

Evaluating the Integrity of Pressure Pipelines by Fracture Mechanics

“…In order to account for the finite thickness of a body and the plastic zone at the crack tip, correction factors were introduced for the "front" surface and the "rear" surface of the body and for the plastic region at the crack tip (Shah & Kobayashi, 1973). However, solutions by different authors often showed rather considerable disagreement.…”