Justification and Imfroveme~ of Kienzler and Herrma~s Estimate of Stress Intensity Factors of Cracked Beam

Kienzler and Hernnan recently showed that good and remarkably simple approximations of the stress intensity factors of cracks in beams can be obtained by bending theory estimations of the strain energy release as the crack is widened into a band of finite width. Their method has been based on equating this energy release to the energy release rate due to crack length extension, which has been postulated as a hypothesis. This note presents a justification of this hypothesis and further shows a possibility of improvement by introduction of an additional factor. The improvement is demonstrated by numerical comparison with the exact solution of a cracked beam. The afo~mention~ factor, however, can be dete~in~ only by optimum fitting of the exact solution. INTRODUCTION A REMARKABLY simple method for close approximation of stress intensity factor 4 in cracked or notched beams was recently discovered by Kienzler and Herrmann[l] (see also Herrmann and Sosa[2]). The method was derived from a certain postulated hypothesis regarding the energy release as the crack is widened into a fracture band. The purpose of this note is to present a justification of this hypothesis and also show a different derivation of this new method. This derivation is simpler and at the same time indicates that the hypothesis used by Kienzler and Herrmann is not exact but merely a good approximation. The present method avoids sophisticated elegant concepts, such as material forces, which were introduced by Kienxler and Hemnann, but are not used in the present derivation. They do not seem to be necessary for obtaining the result. KIENZLER AND HERRMANN’S METHOD We may illustrate this new method by considering a cracked beam that is subjected to bending moment M. The beam has bending stiffness EI,, and the notched cross-section has bending stiffness ,??I,, where I, = bH3/12 and I, = b(H ~)~/12; I,, I2 = moments of inertia, H = beam depth, a = notch depth, b = beam thickness. Kienxler and Hemnann[l] consider the energy release AU of the beam as the notch thickness b is widened from zero to A/I. From the theory of bending one has AU = M2(1/EI, l/El,) Ahb/2, and so au --.-i= dh where U = strain energy of the beam. To calculate the energy releases rate, and from that the stresses intensity factor, Herrmann and Kienxler write (for 6 = I): au au aa”2ah (2) where SJiaa = -bG, and G is the energy release rate of the beam per unit advance of the crack (and per unit length of the crack front edge). The stress intensity factor is K, = (EG)“‘. According to Kienxler and Herrmann’s[l] Figs 2-4, the values of K, calculated in this manner compare quite well with accurate solutions from handbooks. Equation (2) represents a crucial step. This step, aowever, has not been justified theoretically. It has been postulated as a hypothesis. JUSTIFICATION AND IMPROVEMENT Formation of a crack in an untracked body may be imagined to completely relieve the strain energy from the triangular areas 021 and 023 which are limited by the so-called “stress effusion lines” (see e.g. Knot#3]), as shown in Fig. l(a), (b). When the crack length a increases by Au, the additional energy release, therefore, comes from the strips 2683 and 2641 of width a (Fig. Ia) from which a~/au, G and X;: can be calculated. For bodies with an initially homo~n~us stress field, this method gives correct formulas for X,, except that the value of the proper slope k can be determined only by comparison