Issue 41

F. Berto et alii, Frattura ed Integrità Strutturale, 41 (2017) 260-268; DOI: 10.3221/IGF-ESIS.41.35 261 I NTRODUCTION s a part of a more general two-dimensional study on sharp V-notches in plates, the specific case of a zero angle V- notch, or crack, was carried out by Williams [1] to supplement previous results by Inglis (1913), Griffith (1921), and Westergaard (1939). For the crack case Williams demonstrated that the stress function can be expressed as a series expansion, being the coefficients of each term undetermined and depending on the loading conditions. Williams also underlined that for practical cases when the plate has finite dimensions, higher order eigenfunctions should be used to determine a solution in the large. The stress associated with the symmetric and skew-symmetric singular terms in combination with the constant term were finally presented by Williams, neglecting the other terms proportional to r 1/2 , r , r 3/2 . Today, after Larsson and Carlsson and Rice, the constant stress term is commonly called T-stress [2, 3]. By using a boundary collocation technique, a procedure to determine mode I and mode II stress intensity factors was proposed by Gross and Mendelson [4]. The collocation technique was later used also by Carpenter [5] to determine the coefficients associated both to singular and non-singular stress terms. Carpenter presented some examples where the main aim was to handle the complex coefficients of higher order terms. However, after Williams up to now, classical theories of fracture mechanics generally assume that the near-crack-tip stresses can be characterized by the stress intensity factors but extensive studies in the last two decades have shown that also the T-stress is important for describing the states of stress and strain near the crack tip [6-10]. The role of the T -stress in brittle fracture for linear elastic materials has been emphasized by Ayatollahi et al. [6] who proposed a modified Erdogan-Sih’s criterion (Erdogan, Sih, 1963). Their generalized maximum tensile stress criterion is based on the mode I and II stress intensity factors, K I and K II , the T- stress and a fracture process zone parameter, r c.. Several closed form solutions of T-stress in plane elasticity crack problems in an infinite plate are investigated using the complex potential theory [7]. A rigorous derivation for T-stress in line crack problem is presented by Chen et al. (2010). Similar to the edge crack case, this paper provides the T-stress dependence on loading with the Dirac delta function property [11].Dealing with mixed mode loading a particular weight function method is used to determine the stress intensity factors (SIFs) and T -stresses for offset double edge cracked plates under mixed mode loading [12]. Beyond the T-stress, the problem related to the different role played by the higher order terms remain open. Ramesh et al. [13, 14] presented an over-deterministic least squares technique to evaluate the mixed-mode multi-parameter stress field by photoelasticity underlining the fact that the use of a multi-parametric representation is not just an a academic curiosity but a necessity in some cases of engineering interest. This fact is underlined also by Ayatollahi and Nejati [15] who provided a specific algorithm for a fast determinations of the unknown parameters. A method for the direct determination of SIF and higher order terms by a hybrid crack element has been developed in the past also by Xiao et al. [16] proving the versatility and accuracy of the element for pure mode I problems and for mode II and mixed mode cracks. An overview of a hybrid crack element and determination of its complete displacement field has been carried out by Xiao and Karihaloo [17]. A novel mathematical model of the stresses around the tip of a fatigue crack, which Includes the T-stress and considers the effects of plasticity through an analysis of their shielding effects on the applied elastic field was developed by Christopher et al. [8, 9]. The ability of the model to characterize plasticity-induced effects of cyclic loading and on the elastic stress fields was demonstrated using full field photoelasticity. The effects due to overloads have been also discussed [10]. The model can be seen as a modified linear elastic approach, to be applied outside the zone where nonlinear effects are prevailing. Two logarithmic terms are added to the Williams’ solution and three new stress intensity factors K F , K R and K S are proposed to quantify shielding effects ahead of the crack tip and on its back. The present Authors have been recently interested in the fatigue strength of welded lap joints and they have paid particular attention to thin plates characterized by a thickness equal or lower than 5 mm [18, 19]. In that study, the T- stress was considered together with K I and K II to describe the stress field. The closed form expressions involving only these three parameters was found to be inadequate to accurately describe the actual stress fields in thin welded lap joints as well as the mean value of the strain energy density on a control volume embracing the slit tip. Decreasing the plate thickness of the welded joint, the zone controlled by the first order terms became smaller than the characteristic control volume of the welded material. The effect of the higher order terms will be object of the present contribution and a demarcation line between linear elastic and elastic-plastic analyses will be drawn on the basis of the local strain energy density creating also a bridging with the recent model proposed by Christopher et al. [8, 9]. A