The singular edge-based smoothed finite element method for stationary dynamic crack problems in 2D elastic solids P. Liu a , T.Q. Bui b,⇑ , Ch. Zhang b , T.T. Yu a , G.R. Liu c , M.V. Golub d a Department of Mechanical Engineering, Hohai University, Nanjing 210098, PR China b Department of Civil Engineering, University of Siegen, Paul-Bonatz-Strasse 9-11, D-57076 Siegen, Germany c University of Cincinnati, Cincinnati, OH 45221-0070, USA d Institute for Mathematics, Mechanics and Informatics, Kuban State University, Krasnodar 350040, Russia article info Article history: Received 13 December 2011 Received in revised form 7 March 2012 Accepted 12 April 2012 Available online 24 April 2012 Keywords: Singular ES-FEM Strain smoothing method Dynamic fracture mechanics Dynamic stress intensity factors abstract In this paper, the recently developed singular edge-based smoothed finite element method (sES-FEM) is further developed for dynamic crack analysis in two-dimensional elastic solids. The objective of this work is to provide an efficient and accurate numerical simulation tool for the dynamic fracture behaviors of linear elastic solids in the framework of the strain smoothing approaches. Following this approach, the strains are smoothed and the system stiffness matrix is thus performed using the strain smoothing tech- nique over the smoothing domains associated with the element edges. In order to accurately capture the singular fields at the crack-tip, a two-layer singular 5-node crack-tip element is employed. The governing dynamic equations are transformed into a weakened weak (W2) form, which is then discretized into a sES-FEM system of time-dependent equations to be solved by the unconditionally stable implicit New- mark time integration method. To analyze the fracture behaviors of linear elastic solids, mixed-mode dynamic stress intensity factors (DSIFs) are evaluated using the domain forms of the interaction integrals in terms of the smoothing technique. Four test examples including pure mode-I and mixed-modes are studied to validate the accuracy of the proposed method. The computed results for the normalized DSIFs are compared with analytical and other numerical reference solutions in a wide range of benchmark dynamic crack problems which shows high accuracy of the sES-FEM. Ó 2012 Elsevier B.V. All rights reserved. 1. Introduction Of great importance in modeling fracture mechanics problems is to accurately describe the singular fields near the crack-tip, and in contrast to the static loading conditions, numerical simula- tions of the dynamic fracture problems remain a challenging task for many practical engineering applications. Understanding the dy- namic fracture behaviors of a cracked body through the dynamic stress intensity factors (DSIFs) is essential and an accurate compu- tation of the DSIFs thus plays a crucial role in practices. In general, the dynamic fracture mechanics studies cases in which the inertial effects are taken into account and the responses are often caused by the time-dependent loads. Because of the mathematical difficul- ties, analytical approaches are often not feasible to solve general dynamic fracture problems in practices, numerical methods are thus required. Many different methods have been introduced in the last decades to model such dynamic crack problems and among them, the boundary element methods (BEM) [1] and the finite ele- ment methods (FEM) [2–4] are frequently applied by using special singular elements. In recent years, the strain smoothing technique has proposed by Chen et al. [5,6] to stabilize the solutions of the nodal integrated Galerkin mesh-free methods. The smoothing technique is later incorporated into the FEM using cell-based smoothing domains, which forms the smoothed finite element method (SFEM) [7]. Soon after, different versions of the SFEM have been formulated by Liu’s group including the n-side polygonal smoothed finite element method (nSFEM) [8], the node-based smoothed finite element method (NS-FEM) [9], the edge-based smoothed finite element method (ES-FEM) [10], and so on. In particular, the NS-FEM faces to the problem of the temporal instability, and the ES-FEM is thus devoted to overcome such difficulty by creating the smoothing do- mains constructed based on the element edges instead. The advan- tages by using the ES-FEM lie in the fact that it is much more accurate than the linear FEM using the same mesh and even more accurate than the FEM using quadrilateral elements. More recently, the ES-FEM has been improved significantly by introducing a spe- cial singular triangular element associated with the smoothing do- mains (further on named as sES-FEM), which has been recently applied to accurately modeling the singular stress field near the 0045-7825/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cma.2012.04.008 ⇑ Corresponding author. Tel.: +49 2717402836; fax: +49 2717404074. E-mail addresses: [email protected] (T.Q. Bui), [email protected] (Ch. Zhang). Comput. Methods Appl. Mech. Engrg. 233–236 (2012) 68–80 Contents lists available at SciVerse ScienceDirect Comput. Methods Appl. Mech. Engrg. journal homepage: www.elsevier.com/locate/cma
The singular edge-based smoothed finite element method for stationary dynamic crack problems in 2D elastic solids
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