numero25

P. Lazzarin et alii, Frattura ed Integrità Strutturale, 25 (2013) 61-68; DOI: 10.3221/IGF-ESIS.25.10 61 Special Issue: Characterization of Crack Tip Stress Field Recent developments in multi-parametric three-dimensional stress field representation in plates weakened by cracks and notches P. Lazzarin, M. Zappalorto, F. Berto University of Padua, Department of management and engineering, Stradella San Nicola 3, 36100, Vicenza, Italy A BSTRACT . The paper deals with the three-dimensional nature and the multi-parametric representation of the stress field ahead of cracks and notches of different shape. Finite thickness plates are considered, under different loading conditions. Under certain hypotheses, the three-dimensional governing equations of elasticity can be reduced to a system where a bi-harmonic equation and a harmonic equation have to be simultaneously satisfied. The former provides the solution of the corresponding plane notch problem, the latter provides the solution of the corresponding out-of-plane shear notch problem. The analytical frame is applied to some notched and cracked geometries and its degree of accuracy is discussed comparing theoretical results and numerical data from 3D FE models. K EYWORDS . Three-dimensional elasticity; Stress fields; Antiplane solution; Plane solution. I NTRODUCTION pioneering analytical framework was proposed by Dougall [1] to evaluate the three-dimensional stress fields in plates; Dougall’s method was later applied by Green [2] to the stress analysis of a thick isotropic elastic plate weakened by a cylindrical hole. The same problem was discussed later also by Sternberg and Sadowsky [3] and by Folias and Wang [4]. As common denominator, these works have a sound discussion on the role played by the plate thickness on the hole tip stresses. As is well known, an approach providing a general solution for three-dimensional problems of elasticity was developed by Papkovich [5] and Neuber [6]. It is based on a general three-dimensional stress function. By using three harmonic functions, the fourth one being equal to zero, and different curvilinear coordinate systems, Neuber was able to provide some solutions for the three-dimensional problem, in particular those related to the axisymmetric hyperboloidal ligament and the axisymmetric ellipsoidal cavity in an infinite elastic body. Dealing with cracked components, Hartranft and Sih gave the approximate stress fields near the tip of a through crack in a thin elastic plate using a variational principle [7] and a refinement of the Reissner theory [8]; fundamental contributions are also the extension to the three-dimensional crack case [9] of Williams’ two dimensional eigenfunction series [10] as well as the application of the Papkovich-Neuber method to the same case [11]. Among the recent papers we remember here the work by Yang and Freund [12] who took advantage of the Kane and Mindlin hypothesis [13] to analyse the state of stress in a tensioned thin elastic plate containing through-cracks. By using the Fourier transform and the Wiener-Hopf technique they demonstrated the existence of a generalised plane strain field at the crack tip, and confirmed Hartranft and Sih’s previous findings [7]. The same result was found numerically also by Nakamura and Parks [14]. In parallel, discussing results from 3D finite element analyses on thin cracked elastic plates remotely subjected to mode II anti-symmetrical loading, Nakamura and Parks [15] found that the asymptotic stress fields were characterized by a combination of plane strain mode II and antiplane mode III singular fields. Their analyses put into light, for the first time, the existence of “local induced modes” due to three-dimensional effects. A