On Analytic Constraints and Elements Methods in Modeling Stresses near the Tips of Cracks and V-Notches
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ON ANALYTIC CONSTRAINTS AND ELEMENTS METHODS IN MODELING STRESSES NEAR THE TIPS OF CRACKS AND V-NOTCHES A. Seweryn and A. Adamowicz The work deals with the application of the methods of analytic constraints and analytic elements in modeling stress fields near the tips of cracks and V-notches in elastic bodies. Both methods are based on the combined application of some analytic solutions for stress concentrators and the finite-element method. By using the proposed approach, we perform the numerical analysis of singular stresses formed in plates containing cracks or angular notches under the action of loads of various types.
Introduction The necessity of analysis of singular stress fields in structural elements of complex shapes caused a considerable progress of numerical methods, which led to modeling stress fields according to the analytic solutions and to determining the analytic parameters (e.g., the stress intensity factors). To this end, the methods presented in the monographs by Aliabadi and Rooke [1] and Atluri [2] are used. The finite- and boundary-elements methods are the methods most extensively applied in practice. The analytic solutions are used to model singular stress fields and calculate the related parameters. On their basis, special finite- or boundary-elements [3] methods for the determination of the generalized stress intensity factors are developed. The methods can be classified as follows: the direct methods in which the unknown values of analytic factors are obtained as a result of modeling of singular stress fields; they are in the vector-column of unknown nodal parameters together with the values of nodal displacements (the hybrid-elements methods [3], the analyticconstraints method [4], and the analytic-elements method); the asymptotic methods (the extrapolation method [5] and the method of changing of the notch shape [6]), and the energetic methods (the flexibility method [7], the virtual crack increment method [8, 9], and the invariant J-integrals method [10, 11] and H [12]). The present work is devoted to two direct methods for calculating analytic factors, namely, the analyticconstraints and analytic-elements methods (a new approach). The results obtained by using the indicated two methods for bodies with cracks and V-notches under different loading are also presented. Stress and Displacement Distributions Near the Crack Tip or a Sharp Notch For the linear problems of fracture mechanics, the general case of the fields of displacements and stresses at the crack tips can be obtained by using the superposition principle from the solutions of the following modes of crack deformation [13]: the in-plane opening mode (mode I), the in-plane shearing mode (mode II), and the antiplane shearing mode (mode III). Consider a crack of length 2l in an infinite sheet. Let ( r, ϑ ) be a polar coordinate system with origin at the crack tip (Fig. 1 for β = 0). We consider the first four terms of the asymptotic expansion of the fields of stresses and displacements near the crack tip. Bialystok University of
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