PSI - Issue 18

Johannes Scheel et al. / Procedia Structural Integrity 18 (2019) 268–273 J. Scheel et al. / Structural Integrity Procedia 00 (2019) 000–000

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the accurateness of all other approaches for model a) from Fig. 2, while still being equally accurate as the the CTEM for model b) and also being close to the results of the MCCI for model c). For a coarser mesh it tends to become more inaccurate than other approaches for models a) and c), while still being quite exact for the coarser two meshes of model b). Calculating the SIF using node C in connection with Eq. (13) also yields precise results, were for the three finer meshes (0.1 mm, 0.25 mm, 0.5 mm) the error is not exceeding five percent for model a) and c), and slightly exceeding five percent for the 0.1 mm mesh of model b). While being mostly less accurate than the MCCI, the calculation using node C is mostly more exact than the CTEM and, except for the finest mesh in model a), it is always more accurate than the DIM, which is based on an equivalent SIF relation but uses an extrapolation towards the crack tip. The more sophisticated EMCCI approach using nodes C and D and three eigenfunctions of the Williams series [Williams (1957)] is also reasonably accurate. For model b) it becomes the most accurate approach for the coarsest mesh, while it also hardly exceeds five percent relative error for all models and meshes, except for the coarsest mesh in model a). Finally, it has to be assessed whether the EMCCI approach is advantageous compared to classical methods of crack tip loading analysis. In all approaches, with a few exceptions, the relative error increases with a coarser mesh, but the higher order EMCCI is not as sensitive to this issue and partly even becomes more exact. It is favorable to use the higher order EMCCI approach, as it is in all cases (except for model a), mesh 2 mm) reasonably accurate, while being more precise for coarser meshes than the calculation of the SIF with Eqs. (13). As a very fine meshing (0.1 mm) is rarely chosen due to computational cost, a coarser mesh is more appropriate. Concerning the other methods, the higher order EMCCI is for a coarser mesh partly more or at least equivalently precise than the MCCI and CTEM and except for the finest mesh always more accurate than the DIM, while eluding the drawback of a somewhat arbitrary extrapolation. In this work a novel approach for crack tip loading analyses is presented and denoted as enriched modified crack closure integral. It is based on the ideas of the MCCI, however circumvents the use of nodal reaction forces on the crack ligament by inserting the analytical near tip stress field into the crack closure integral, while substituting the displacements by an interpolation function based on the Williams series. Di ff erent approaches exploit one to three series elements requiring di ff erent numbers of nodal displacements of the finite element solution. For di ff erent models the SIF are calculated using the EMCCI and other methods for comparison. Even though the EMCCI is a very convenient approach to calculate SIF, it yields results with an accurateness comparable to the classical MCCI and the CTEM, mostly exceeding the one of the DIM, without any extrapolation and search for a suitable node set required. While the DIM is the method of choice when implementation and meshing e ff ort as well as computational cost shall be low, the EMCCI holds all these advantages with equal or higher accuracy and even less implementation e ff ort. 4. Conclusion Rice, J. R., 1968. A path independent integral and the approximate analysis of strain concentration by notches and cracks. Journal of Applied Mechanics 35, 379–386. Cherepanov, G. P., 1967. Crack propagation in continuous media (translation from russian). Journal of Applied Mathematics and Mechanics 31 (3), 503–512. Strifors, H. C., 1974. A generalized force measure of conditions at crack tips. International Journal of Solids and Structures 10, 1389–1404. Judt, P. O., Ricoeur, A., Linek, G., 2015. Crack path prediction in rolled aluminum plates with fracture toughness orthotropy and experimental validation. Engineering Fracture Mechanics 138, 33–48. Barsoum, R. S., 1976. On the use of isoparametric finite elements in linear fracture mechanics. International Journal of Numerical Methods in Engineering 10, 25–37. Chan, S. K., Tuba, I. S.,Wilson, W. K., 1970. On the finite element method in linear fracture mechanics. Engineering fracture mechanics 2, 1–17. Irwin, G. R., 1958. Fracture, in “ Elasticity and Plasticity , Encyclopedia of Physics VI ”. In: Flu¨gge, S. (Eds.) . Springer, Berlin, Heidelberg, 558–590. Rybicki, E. F., Kanninen, M. F., 1977. A finite element calculation of stress intensity factors by a modified crack closure integral. Engineering fracture mechanics 9 (4), 931–938. Buchholz, F. G., 1984. Improved formulae for the finite element calculation of the strain energy release rate by modified crack closure integral method, in “ Accuracy, Reliability and Training in FEM Technology ”. In: Robinson, J. (Ed.). Robinson and Associates, Dorset, 650–659. Williams, M. L., 1957. On the stress distribution at the base of a stationary crack. Journal of Applied Mechanics 24, 109–114. References