PSI - Issue 45

Koji Fujimoto et al. / Procedia Structural Integrity 45 (2023) 74–81 Koji Fujimoto / Structural Integrity Procedia 00 (2019) 000 – 000 where the point symmetry of the model with respect to the point ( /2, /2) is considered. Further, on the crack surface AB is expressed as = 2 ( + 1) ∫ [ 1 − − ( − + ){( − + ) 2 − 2 } {( − + ) 2 + 2 } 2 ] 1 ( ) − − 2 ( + 1) ∫ {3( − + ) 2 + 2 } {( − + ) 2 + 2 } 2 2 ( ) − ∓ 4 + 1 2 ( ) (− < < ) ∙∙∙∙∙(30) where the signs – and + in the right hand side of the equation (30) correspond to the upper and the lower surfaces of the crack, respectively. In most cases, a large number of the collocation points is not necessary for obtaining converged numerical solutions. When both / and / are small, a large number of the collocation points is necessary for convergence; however, we can certainly obtain the converged values of the T -stresses and the stress intensity factors by setting the number of the collocation points sufficiently large. Examples of converged values of T -stresses and stress intensity factors at the crack tip B are shown in Fig. 6. 4. Conclusions The method for evaluating T -stresses of cracks in two-dimensional elasticity was developed using the method of continuously distributed dislocations model. The procedure for obtaining T -stress by solving a singular integral equation in which dislocation density is unknown was shown by extending the numerical integration method for the solution of singular integral equations proposed by Theocaris and Ioakimidis (1977). The weight function of the dislocation density was approximated as the expansion of Chebyshev polynomials of the first kind and the coefficients of the polynomial were determined by letting boundary conditions be satisfied at collocation points. Further, some crack problems were solved by the method and T -stresses were obtained together with stress intensity factors. As a result, it has been clarified that the convergence of calculated T -stresses and stress intensity factors with the increase of the number of collocation points were excellent. The method of continuously distributed dislocations model lacks versatility; however, this method can be expected to provide highly accurate solutions for problems to which this method can be applied. Note that although engineering significance of T -stress (its effect on the stability of crack propagating direction, the size of plastic zone at crack tip, etc.) is very important and interesting for engineers, that was not mentioned in this report. References Cotterell, B., 1966, Notes on the Paths and Stability of Cracks, International Journal of Fracture Mechanics, Vol. 2, No. 3, pp. 526-533. Dundurs, J., 1969, Elastic Interaction of Dislocations with Inhomogeneities, in “ Mathematical Theory of Dislocations ”, Mura, T. (Ed.), The American Society of Mechanical Engineers, New York, pp. 70-115. Fujimoto, K., Nagata, H., 2022, T-Stress Evaluation of Mode I Crack Close to Bimaterial Interface Using Continuous Dislocations Model, Advances in Applied Plasma Science, Vol.13, pp. 31-32. Gupta, M., Alderliesten, R.C., Benedictus, R., 2015, A Review of T-stress and Its Effects in Fracture Mechanics, Engineering Fracture Mechanics, Vol. 134, pp. 218-241. Isida, M., Noguchi, H., 1983, Transactions of the JSME, Ser. A, The Japan Society of Mechanical Engineers, Tokyo, Vol. 49, No. 437, pp. 36-45 (in Japanese). Theocaris, P.S. and Ioakimidis, N.I., 1977, Numerical Integration Methods for the Solution of Singular Integral Equations, Quarterly of Applied Mathematics, Vol. 35, pp. 173-183. 81 8