Influence of load biaxiality on the fracture characteristics of two collinear cracks

Abstract

A theoretical investigation has been performed to determine the influence of load biaxiality on quantities pertinent to the brittle (elastic) fracture of an infinite sheet containing a pair of collinear cracks. It is shown how the biaxiality of the applied load affects the stress, the displacements and the maximum shear near the ends of the cracks, as well as the angle of initial crack extension and the displacement of the crack borders. The influence of load biaxiality on the rate at which the elastic strain energy of the entire body is altered with change in the length of the cracks is also demonstrated, and is used in conjunction with Griffith's crack instability hypothesis to show the effect of load biaxiality on the fracture load. The analysis indicates that loads applied parallel to the cracks influence the value of the critical (fracture) tensile load applied perpendicular to them, and that the Poisson ratio of the material determines the characteristics of this influence. Nomenclature (x,y)(r,O) = Z tjk, ejk = T~, uk, bk = txx , txy , tyy = l=(b-a) = U = W = P,V = F = R = A k = ~, ~, e~, ~o = F(~,p), E(~,p) = V(p), E(p) = x(p) = KI, K2 = E = k = o % = ~L = p y = 0o = rectangular and polar coordinates, respectively.x + iy, complex variable. components of the stress and strain tensors, respectively. components of the surface traction, displacement and body force vectors, respectively. rectangular stress components. rectangular displacements components. crack dimension. elastic strain energy per unit thickness. work of forces applied to the body. the total potential energy of the system and the elastic potential energy, respectively. surface energy. bound region of the x-y plane. closed boundary curves of R. sectionally holomorphic functions of the complex variable z. elliptic integrals of the first and second kinds, respectively. complete elliptic integrals of the first and second kinds.

On the fundamental energy argument of elastic fracture mechanics

“…In a paper in 1965, Swedlow [ 10] reproduces Wolf's result, (20), and is similarly in error. Also in a paper in 1965, Spencer [11] undertakes a correction to Griffith's 1920 result for G~, for either fixed grips or fixed loads, by truly assuring that these boundary conditions are adhered to (he uses polar coordinates rather than elliptical as in the Appendix).…”

“…6. This results in the following for the functional release rate (20) to recover Ge of (18); therefore, the derivation underlying (20) shares the same error as Griffith's first analysis, and (20) may be discounted.…”

“…Griffith was the first to recognize this in his seminal paper [1] in 1920. Since [1], there have been a number of related contributions which are primarily concerned with aspects of such an energy analysis: Wolf [2] in 1923; Griffith [3] in 1924; Sneddon [4] in 1946; Eshelby [5,6,7] in 1951, 1956 and 1959; Irwin [8] in 1957; Sanders [9] in 1960; Swedlow [10] and Spencer [11] in 1965; Cherepanov [12] in 1967; Sih and Liebowitz [13] in 1967; Rice [14] in 1968; Bilby and Eshelby [15] in 1968; Eshelby [16] in 1969; Eftis, Subramonian and Liebowitz [17] in 1977; Cherepanov [18] in 1979; Eftis and Jones [19] in 1982; Eftis [20] in 1984; Eftis [21,22] in 1987; Tyson and Roy [23] in 1989; and Eftis, Jones and Liebowitz [24] in 1990. A review of the above listed papers can possibly lead to confusion with respect to some facets of the fundamental energy argument of fracture mechanics.…”

On the fracture stress for the inclined crack under biaxial load

“…Both types of calculation have been previously shown and discussed for the horizontal crack geometry [3,4,5].…”

“…Attention in each of these studies was confined to analysis of the local region enclosing the crack-tip. Calculation of the critical (fracture) stress under biaxial applied load, in which the Griffith crack instability hypothesis was utilized, then followed for the horizontal single crack and for horizontally oriented collinear cracks [3,4]. A similar type of calculation was also performed for the so-called 'shear panel' [5].…”