numero25

P. Lazzarin et alii, Frattura ed Integrità Strutturale, 25 (2013) 61-68; DOI: 10.3221/IGF-ESIS.25.10 62 In the same period Pook analysed the stress and displacement distributions in three-dimensional plates weakened by cracks and narrow notches [16, 17]. From the results of 3D finite element analyses Pook [17] noted that in a plate under a nominal, far applied, Mode II loading there was a zone, close the notch tip (the notch tip radius being small but different from zero), where Mode III displacements were induced. The Kane and Mindlin theory was applied by Kotousov and Lew [18] to analyse the stress singularities related to angular corners in plates of arbitrary thickness subjected to in-plane loadings and various boundary conditions. The new singular stress states were referred to as the out-of-plane singularities and the corresponding fracture mode as the out-of-plane singular mode or K O - mode. The intensity of local out-of-plane stress fields at the tip of pointed V-notches in 3D plates under remote mode II loading was widely documented in refs. [19-21] on the basis of detailed three-dimensional FE analyses. Cracks, blunt cracks as well as a variety of notches, sharp and blunt, were considered in these contributions. Also the role played by higher order terms of the stress field was analysed in detail with reference to the crack case. In the presence of a notch tip radius equal to zero, the out-of-plane stress singularity arising at a small distance from the free surfaces, prior to these ones, was found to match that of the sharp V-notch problem under pure antiplane elasticity. When the V-notch opening angle is large enough to make non singular the mode II stress distribution, the induced Mode III stress field remains singular. The local interaction between loading modes has been recently justified analytically by Lazzarin and Zappalorto [22] for pointed and sharply radiused notches in finite thick plates. On the basis of a generalised plane strain assumption, it was demonstrated that the governing equations of three-dimensional elasticity can be reduced in complexity solving, in combination, a bi-harmonic equation and a harmonic equation. The former provides the solution to the corresponding plane notch problem, the latter that of the antiplane elasticity problem for the same notch geometry. The two equations have to be simultaneously satisfied in a 3D problem, thus justifying theoretically the mutual interaction between mode II and mode III. This result was supported by a number of finite element analyses carried out on mode II loaded thick plates with sharp and blunt notches [22, 23]. As a result, some previous closed-form solutions [24,25] obtained for axi-symmetric bodies under torsion can be used to analyse the out-of-plane stress distributions in 3D plates. Starting from the analytical frame provided in [22], the aim of the present work is to present the 3D stress fields in some cases of practical interest. In the first case the stress fields close to a rectangular hole in a plate of finite thickness under tension is investigated. The second example deals with a finite thickness plate weakened by a crack. Two forces are applied to generate Mode III loading conditions on the plate and the automatically generated coupled Mode II is investigated paying also attention to the scale effect governing the induced mode . A NEW FRAME FOR THE ANALYSIS OF THE THREE - DIMENSIONAL STRESS FIELD new approach to the analysis of the three-dimensional notch problems has been recently proposed in Ref. [22]. According to a generalized plane strain hypothesis, the displacement components are given by: ( , ) ( , ) ( , ) x y z u u x y u v x y u bz w x y     (1) where z is the through-the-thickness coordinate and b is a constant term. Doing so, the normal strains  xx ,  yy ,  zz as well as  xy are independent of z. Taking advantage of the stress-strain relationships it was demonstrated that also the stress components  xx ,  yy ,  xy and  zz are independent of z, whereas the out-of-plane shear stress components depend on z according to the following expressions: yz xz w w G bz G bz y x           (2) Then the equilibrium equation in the z direction simply gives [22]: 2 0 w   (3) where 2  denotes the two-dimensional Laplacian operator. Differently, invoking Eq. (3), the equilibrium equations in the x and y directions give: 2 2 2 2 2 2 0 yy xy xx y x x y              (4) Eq. (4) is automatically satisfied by the classic Airy stress function  x,y  : A