DOI 10.1007/s10704-006-6730-0 International Journal of Fracture (2006) 139:59–69 © Springer 2006 Frobenius’ method for curved cracks ROBERTO BALLARINI 1,∗ and PIERO VILLAGGIO 2 1 Department of Civil Engineering, Case Western Reserve University, Cleveland, OH 44106-7201, USA 2 Dipartimento di Ingegneria Strutturale, Universit´ a di Pisa, Pisa, Italy ∗ Author for Correspondence (E-mail: roberto.ballarini@case.edu) Received 10 November 2005; accepted in revised form 11 January 2006 Abstract. The distribution of stresses produced by an undulated crack in a plane elastic solid, and in particular, at its tips where stresses approach infinity, requires the solution of two coupled sin- gular integral equations. Except for simple crack geometries such as rectilinear and circular arcs in infinite plates, for which explicit analytic solutions have been obtained, the integral equations require numerical solutions. We propose a treatment of the integral equations by Frobenius’ method, which is particularly suitable for evaluating the stress intensity factors of slightly curved cracks. Key words: Curved crack, Frobenius’ method, stress intensity factors. 1. Introduction The stresses induced by a curved crack in an infinite elastic sheet subjected to a uni- form stress state at infinity can be obtained from the dislocation density, which sat- isfies two coupled singular integral equations. Explicit analytical solutions of these equations have been obtained only for rectilinear and circular cracks. But often the shape of the crack is wavy and hence numerical solution of the integral equations is required to determine the dislocation density, the stress field and the stress inten- sity factors. For finite length curved cracks Chen et al. (1991) proposed a method of solution of the integral equations that involves representing the regular part of the dislocation density as a linear combination of Chebyshev polynomials, whose coeffi- cients are determined by collocation. The same procedure was proposed by Dreilich and Gross (1985) for the specific case of a slightly curved crack. Brandinelli (1997) solved the same problems by approximating the dislocation density with Chebyshev polynomials. The numerical values of the stress intensity factors calculated for para- bolic and sinusoidal shaped cracks were, to within three significant figures, the same as those obtained by Savruk (1981) using a different method. An elegant treatment of the problem of slightly curved and kinked cracks involv- ing simultaneous analytical perturbation of the (given) boundary of the crack and of the unknown complex stress functions (Muskhelishvili, 1953) was presented by Cotterell and Rice (1980), who showed that satisfactory values of the stress intensity factors can be obtained even by arresting the perturbation procedure to first order. The integral equations of the curved crack can also be solved by the method of power series (Frobenius’ method) consisting of a double expansion, in increasing powers of a suitable curvilinear abscissa, of the equation of the crack and of the