A boundary element implementation for fracture mechanics problems using generalised Westergaard stress functions
DOI:
https://doi.org/10.13052/17797179.2018.1499188Keywords:
Fracture mechanics, generalised Westergaard functions, boundary elementsAbstract
In the traditional boundary element methods, the numerical modelling of cracks is usually carried out by means of a hypersingular fundamental solution, which involves a 1=r2 kernel for two-dimensional problems. A more natural procedure should make use of fundamental solutions that represent the square root singularity of the gradient field around the crack tip (a Green’s function). Such a representation has been already accomplished in a variationally based framework that also addresses a convenient means of evaluating results at internal points. This paper proposes a procedure for the numerical simulation of two-dimensional problems with a fundamental solution that can be in part or for the whole structure based on generalised Westergaard stress functions. Problems of general topology can be modelled, such as in the case of unbounded and multiply-connected domains. The formulation is naturally applicable to notches and generally curved cracks. It also provides an easy means of evaluating stress intensity factors, when particularly applied to fracture mechanics. The main features of the theory are briefly presented in the paper, together with several validating examples and some convergence assessments.
Downloads
References
Anderson, T. L. (1995). Fracture mechanics: Fundamentals and applications. 2nd edition. C.
R. C. Press. Boca Raton, New York
Ang, W. T., & Telles, J. C. F. (2004). A numerical Green’s function for multiple cracks in
anisotropic bodies. Journal of Engineering Mathematics, 49(3), 197–207.
Barenblatt, G. I. (1962).The mathematical theory of equilibrium cracks in brittle fracture.
In Advances in applied mechanics. VII, 55–129. Academic Press: Amsterdam.
Brebbia, C. A., Telles, J. C. F., & Wrobel, L. C. (1984). Boundary element techniques:
Theory and applications in engineering. Berlin: Prentice Hall.
Brown, W. F., & Strawley, J. E. (1966). Plane strain crack toughness testing of high strength
metallic materials. In ASTM STP 410 (pp. 5). American Society For Testing And
Materials: Philadelphia, PA.
Cardoso, M. L. (2017). A boundary element implementation for fracture mechanics problems
using generalized westergaard stress functions. MSc. thesis (in Portuguese),
Pontifical Catholic University of Rio de Janeiro, Brazil.
Crouch, S. L., & Starfield, A. M. (1983). Boundary element methods in solid mechanics.
London: George Allen & Unwin.
Dugdale, D. S. (1960). Yielding in steel sheets containing slits. Journal of the Mechanics and
Physics of Solids, 8, 100–104.
Dumont, N. A. (1989). The hybrid boundary element method: An alliance between
mechanical consistency and simplicity. Applied Mechanics Reviews, 42(11–2), S54–
S63.
Dumont, N. A. (1994). On the efficient numerical evaluation of integrals with complex
singularity poles. Engineering Analysis with Boundary Elements, 13, 155–168.
Dumont, N. A. (2014). The hypersingular boundary element method revisited. In C. A.
Brebbia & A. H.-D. Cheng (eds), Boundary elements and other mesh reduction methods
XXXVII, WIT transactions on modelling and simulation (Vol. 57, pp. 27–39).
Dumont, N. A. (2018). The collocation boundary element method revisited: Perfect code
for 2D poblems. International Journal Comparative Meth and Experiments
Measurement, 6(6), 965–975.
Dumont, N. A., & Aguilar, C. A. (2012). The best of two worlds: The expedite boundary
element method. Engineering Structures, 43, 235–244.
N. A. DUMONT ET AL.
Dumont, N. A., Chaves, R. P., & Paulino, G. H. (2004). The hybrid boundary element
method applied to problems of potential of functionally graded materials. International
Journal of Computational Engineering Science (IJCES), 5(4), 863–891.
Dumont, N. A., & De Oliveira, R. (2001). From frequency-dependent mass and stiffness
matrices to the dynamic response of elastic systems. International Journal of Solids and
Structures, 38(10–13), 1813–1830.
Dumont, N. A., & Huaman, D. (2009). Hybrid finite/boundary element formulation for
strain gradient elasticity problems. In E. J. Sapountzakis & M. H. Aliabadi, eds.,
Advances in Boundary Element Techniques X, Proceedings of the 10th International
Conference, pp. 295–300, EC, Ltd., UK.
Dumont, N. A., & Lopes, A. A. O. (2003). On the explicit evaluation of stress intensity
factors in the hybrid boundary element method. Fatigue & Fracture of Engineering
Materials & Structures, 26, 151–165.
Dumont, N. A., & Mamani, E. Y. (2011a). Use of generalized Westergaard stress functions
as fundamental solutions. In E. L. Albuquerque & M. H. Aliabadi (eds), Advances in
boundary element techniques XII (pp. 170–175). UK: EC, Ltd.
Dumont, N. A., & Mamani, E. Y. (2011b). Generalized Westergaard stress functions as
fundamental solutions. Computer Modeling in Engineering & Sciences, 78(2), 109–150.
Dumont, N. A., & Mamani, E. Y. (2013). Simulation of plastic zone propagation around
crack tips using the hybrid boundary element method. In Z. J. G. N. Del Prado ed,
CILAMCE – XXXIV iberian latin-american congress on computational methods in
engineering 20 pp on CD. Pirenópolis, Brazil
Dumont, N. A., Mamani, E. Y., & Cardoso, M. L. (2017). A boundary element implementation
for fracture mechanics problems using generalized Westergaard stress functions.
In L. Marin & M. H. Aliabadi (eds), Advances in boundary element & meshless
techniques XVIII (pp. 120–125). UK: LC, Ltd.
Eftis, J., & Liebowitz, H. (1972). On the modified Westergaard equations for certain plane
crack problems. International Journal of Fracture Mechanics, 8(4), 383–392.
Gupta, M., Alderliesten, R. C., & Benedictus, R. (2015). A review of T-stress and its effects
in fracture mechanics. Engineering Fracture Mechanics, 134, 218–241.
Harmain, G. A., & Provan, J. W. (1997). Fatigue crack-tip plasticity revisited – The issue of
shape addressed. Theory and Application Fracture Mechanics, 26, 63–79.
Irwin, G. R. (1957). Analysis of stresses and strain near the end of a crack traversing a
plate. Journal of Applied Mechanics, 24, 361–364.
Jing, P., Khraishi, T., & Gorbatikh, L. (2003). Closed-form solutions for the mode II crack
tip plastic zone shape. International Journal of Fracture, 122, 137–142.
Lopes, A. A. O. (1998). The hybrid boundary element method applied to fracture mechanics
problems. MSc. thesis (in Portuguese), Pontifical Catholic University of Rio de Janeiro,
Brazil.
Lopes, A. A. O. (2002). Evaluation of stress intensity factors with the hybrid boundary
element method. PhD. thesis (in Portuguese), Pontifical Catholic University of Rio de
Janeiro, Brazil.
Mamani, E. Y. (2011). The hybrid boundary element method based on generalized
Westergaard stress functions. MSc. thesis (in Portuguese), Pontifical Catholic
University of Rio de Janeiro, Brazil.
Mamani, E. Y. (2015). Crack modeling using generalized Westergaard stress functions in the
hybrid boundary element method. PhD. thesis (in Portuguese), Pontifical Catholic
University of Rio de Janeiro, Brazil.
Mamani, E. Y., & Dumont, N. A. (2015). Use of improved Westergaard stress functions to
adequately simulate the stress field around crack tips. In N. A. Dumont ed, CILAMCE –
EUROPEAN JOURNAL OF COMPUTATIONAL MECHANICS 423
XXXVI iberian latin-american congress on computational methods in engineering (17).
ISSN: 2178-4949. doi:10.20906/CPS/CILAMCE2015-0840http://www.swge.inf.br/pro
ceedings/CILAMCE2015/ Rio de Janeiro, Brazil
Pian, T. H. H. (1964). Derivation of element stiffness matrices by assumed stress distribution.
AIAA Journal, 2, 1333–1336.
Sneddon, I. N. (1946). The distribution of stress in the neighborhood of a crack in an
elastic solid. Proceedings of the Royal Society of London A, 187, 229–260.
Sousa, R. A., Castro, J. T. P., Lopes, A. A. O., & Martha, L. F. (2013). On improved crack
tip plastic zone estimates based on t-stress and on complete stress fields. Fatigue &
Fracture of Engineering Materials & Structures, 36(1), 25–38.
Tada, H., Ernst, H., & Paris, P. (1993). Westergaard stress functions for displacementprescribed
crack problems - I. International Journal of Fracture, 61, 39–53.
Tada, H., Ernst, H., & Paris, P. (1994). Westergaard stress functions for displacementprescribed
crack problems - II. International Journal of Fracture, 67, 151–167.
Telles, J. C. F., Castor, G. S., & Guimares, S. (1995). A numerical Green’s function
approach for boundary elements applied to fracture mechanics. International Journal
for Numerical Methods in Engineering, 38, 3259–3274.
Westergaard, H. M. (1939). Bearing pressures and cracks. Journal of Applied Mechanics, 6,
–53.
Williams, M. L. (1957). On the stress distribution at the base of a stationary crack. Journal
of Applied Mechanics, 24, 109–114.
Xin, G., Hangong, W., Xingwu, K., & Liangzhou, J. (2010). Analytic solutions to crack tip
plastic zone under various loading conditions. European Journal of Mechanics A/Solids,
, 738–745.
Zhang, Y., Qiang, H., & Yang, Y. (2007). Unified solutions to mixed mode crack tip under
small scale yielding. Chinese Journal of Mechanical Engineering, 43(2), 50–54
