Abstract
The present contribution reviews a recently proposed method to rapidly estimate the averaged SED at the tip of short as well as long cracks under in-plane I+II and long cracks under out-of-plane I+III mixed mode loadings. Short cracks are distinguished from long cracks by considering that the stress fields within the control volume of short cracks are no longer governed solely by the stress intensity factors (SIFs), but the contribution of higher order terms, and primarily the T-stress, becomes significant to estimate the averaged SED. According to the proposed method, the averaged SED is calculated using the linear elastic nodal stresses evaluated by FEM either at the crack tip, to account for the SIFs contribution, and at selected FE nodes of the crack free edges, to include the T-stress contribution. The advantage of the proposed approach is two-fold: coarse FE meshes can be adopted; moreover, geometrical modelling the control volume is no longer necessary. To validate the approach, cracked plates subjected to in-plane I+II mixed mode loading as well as bars weakened by circumferential outer cracks subjected to out-of-plane mixed mode I+III loading have been analysed. A comparison between approximate values of the averaged SED according to the nodal stress approach and those derived directly from the FE strain energy adopting very refined FE meshes has been successfully performed.
Highlights
Dealing with cracks under in-plane mixed mode I+II loading conditions, according to Williams [1], the local stress fields expressed in terms of Cartesian stress components as functions of the polar coordinates (r,θ), with origin at the crack tip (Fig. 1a), can be written in the following form: σσxyxy = τxy cos θ 2 1 2 sin θ sin 3 2 θ KI 2πr + θ -2sin K II 2πr 1 2 θ 2 sin
The present contribution reviews the nodal stress approach and its validation, which is based on the FE analyses of cracked plates subjected to in-plane I+II mixed mode loading as well as bars weakened by circumferential outer cracks subjected to out-of-plane mixed mode I+III loading, while varying (i) the crack lengths, (ii) the mode mixity and (iii) the finite element size adopted in the numerical analyses
To verify the applicability of Eqn (16) to the considered small cracks subjected to mixed mode I+II loading, the ratio between the Nodal T-stress according to Eqn (16) and the exact T-stress is shown in Fig. 5b for a mode mixity ratio MM = 0.50, the results for other MM values being identical
Summary
Dealing with cracks under in-plane mixed mode I+II loading conditions, according to Williams [1], the local stress fields expressed in terms of Cartesian stress components as functions of the polar coordinates (r,θ), with origin at the crack tip (Fig. 1a), can be written in the following form: σσxyxy. Out-of-plane I+III mixed mode loading Dealing with the out-of-plane I+III mixed mode crack problem of Fig. 1b, exact values of the averaged SED, WFEM (Eqn (8)), have been evaluated for the geometrical and loading cases reported in Tab. 1, by adopting the direct approach with very refined meshes. These deviations are due to the contribution of the T-stress, and of further higher order non-singular terms, O(r1/2) in Eqns. While Eqns. (3)-(5) require to process a number of stress-distance numerical results, the PSM requires a single stress value to evaluate the SIFs
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