numero25

P. Lazzarin et alii, Frattura ed Integrità Strutturale, 25 (2013) 61-68; DOI: 10.3221/IGF-ESIS.25.10 63 2 2 2 2 2 xx yy xy x y y x                  (5) At the same time, accounting for the generalised Hooke law for stresses and strains, the in-plane compatibility equation can be written in the form [22]: 4 2 0 zz        (6) where the equality to zero is guaranteed by the third of Beltrami-Mitchell’s equations. As a result, the three dimensional notch problem can be converted into a bi-harmonic problem and a harmonic problem being valid the following differential equation system: 4 2 0 0 w         (7a-b) Here w and  are implicitly defined according to Eqs. (2) and (5), respectively. Note that Eq. (7a) is the common bi- harmonic equation governing the solution of the plane problem; Eq. (7b) is, instead, the harmonic equation governing the antiplane elasticity problem. It was demonstrated [22, 23] that this new frame is effective not only in the presence of pointed notches (sharp cracks or re-entrant corners) but also in the case of sharply radiused notches. The range of applicability does mainly depend on the notch tip radius. One should note that, in order to guarantee all the fundamental equations of 3D elasticity, Eq. (7a) and Eq. (7b) must be simultaneously satisfied. This allows to explain an important feature of three-dimensional stress distributions in plates: far applied loads resulting in local skew-symmetric   function (mode II stress components) inherently provoke a local skew- symmetric w function (mode III stress components). Viceversa , far applied loads resulting in local skew-symmetric w  function (twisting) inherently provoke a local skew-symmetric   function (mode II stress components). The same holds valid, obviously, for symmetric loading conditions. Far applied loads resulting in a local symmetric   function (mode I stress components) results in a local symmetric w function, which is non-singular also in the presence of a singularity point. This justifies by a theoretical point of view the mutual interaction between loading modes shown numerically in refs. [19-21] and allows to understand why Nisitani and Chen [26] discussed, in principle, about four distinct loading modes. It is also worth mentioning that the proposed solution is not valid on the free surfaces of plates, where some edge effects, like the corner point singularities noted by Benthem [27], might arise. It is valid up to a small distance from the free-surfaces. In the next sections two examples are provided with the aim to discuss the degree of accuracy of the newly developed three-dimensional theory. T HE SPATIAL RECTANGULAR HOLE PROBLEM he plane problem of a plate weakened by a rectangular hole has been analysed by Savin [28]. The mapping function approximating the rectangular hole was constructed on the basis of the Schwarz-Christoffel transformation and the complex functions used to determine the stress state according to the Kolosof- Muskelisvily method were given explicitely. In this section Savin’s two dimensional analysis is extended to the three- dimensional case. For more details, see Ref. [23]. In particular, a rectangular hole in a finite thickness plate loaded in tension is considered (figure 1a). The rectangle corner can be regarded as a pointed V-notch of which the bisector is 45° inclined with respect to the loading direction. As a consequence, the uniaxial remote applied tension induces local in-plane mixed mode stresses (mode I plus mode II), which can be described, according to Eq. (7a), using Williams’ plane solution for re-entrant corners. For a fixed value of the through-the-thickness coordinate, z, taking advantage of a polar reference system centred at the notch tip (see figure 1b), the in-plane stresses must vary according to Williams’ singularity degrees, 1-  1 for mode I and 1-  2 for mode II; in parallel, the stress field intensities can be quantified by the corresponding Notch Stress Intensity Factors (NSIFs) [29]. However, the three-dimensional nature of the problem induces, besides the in-plane stresses, the out-of-plane shear stress components,  zr and  z  which do not belong, by nature, to the Williams solution. These stress components can be obtained by the following w function [26]: T