Abstract
Goal-oriented a posteriori error estimators are presented in this contribution for the error obtained while approximately evaluating the J-integral, i.e. the material force acting at the crack tip, in nonlinear elastic fracture mechanics using the finite element method. The error estimators rest upon the strategy of solving an auxiliary dual problem and can be classified as equilibrated residual error estimators based on the solutions of Neumann boundary value problems on the element level. Finally, an illustrative numerical example is presented.
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References
Ainsworth, M. and Oden, J. (2000). A posteriori error estimation in finite element analysis. John Wiley & Sons, New York.
Becker, R. and Rannacher, R. (1996). A feed-back approach to error control in finite element methods: Basic analysis and examples. East-West J. Numer. Math., 4:237–264.
Eriksson, K., Estep, D., Hansbo, P., and Johnson, C. (1995). Introduction to adaptive methods for differential equations. Acta Numer., pages 106–158.
Eshelby, J. (1951). The force on an elastic singularity. Philos. Trans. Roy. Soc. London Ser. A, 244:87–112.
Larsson, F., Hansbo, P., and Runesson, K. (2002). On the computation of goal-oriented a posteriori error measures in nonlinear elasticity. Int. J. Numer. Methods Engrg., 55:879–894.
Maugin, G. (1993). Material Inhomogeneities in Elasticity. Chapman & Hall, London.
Prudhomme, S. and Oden, J. (1999). On goal-oriented error estimation for local elliptic problems: application to the control of pointwise errors. Comput. Methods Appl. Mech. Engrg., 176:313–331.
Rice, J. (1968). A path independent integral and the approximate analysis of strain concentration by notches and cracks. J. Appl. Mech., 35:379–386.
Rüter, M. and Stein, E. (2003). Goal-oriented a posteriori error estimates in linear elastic fracture mechanics. Submitted to Int. J. Numer. Methods Engrg.
Rüter, M. and Stein, E. (2003). On the duality of global finite element discretization error-control in small strain Newtonian and Eshelbian mechanics. Technische Mechanik, 23:265–282.
Shih, C., Moran, B., and Nakamura, T. (1986). Energy release rate along a three-dimensional crack front in a thermally stressed body. Intern. J. of Fract., 30:79–102.
Steinmann, P. (2000). Application of material forces to hyperelastostatic fracture mechanics. I. Continuum mechanical setting. Int. J. Solids Structures, 37:7371–7391.
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Rüter, M., Stein, E. (2005). Error-Controlled Adaptive Finite Element Methods in Nonlinear Elastic Fracture Mechanics. In: Steinmann, P., Maugin, G.A. (eds) Mechanics of Material Forces. Advances in Mechanics and Mathematics, vol 11. Springer, Boston, MA. https://doi.org/10.1007/0-387-26261-X_9
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DOI: https://doi.org/10.1007/0-387-26261-X_9
Publisher Name: Springer, Boston, MA
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