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ENGINEERING COMPUTATIONAL TECHNOLOGY
Edited by: B.H.V. Topping, Z. Bittnar
Combining SGBEM and FEM for Modeling 3D Cracks
G.P. Nikishkov+ and S.N. Atluri*
+Department of Computer Software, The University of Aizu, Aizu-Wakamatsu City, Fukushima, Japan
G.P. Nikishkov, S.N. Atluri, "Combining SGBEM and FEM for Modeling 3D Cracks", in B.H.V. Topping, Z. Bittnar, (Editors), "Engineering Computational Technology", Saxe-Coburg Publications, Stirlingshire, UK, Chapter 8, pp 167-192, 2002. doi:10.4203/csets.8.8
Keywords: crack, FEM, SGBEM, elastic-plastic, fracture mechanics, fatigue crack growth.
In this paper further development of the SGBEM-FEM alternating method  suitable for the solution of elastic and elastic-plastic three-dimensional fracture mechanics problems is presented. The crack is modeled by the symmetric Galerkin boundary element method (SGBEM), as a distribution of displacement discontinuities in an infinite medium. The finite element method (FEM) is used for stress analysis of the uncracked finite body. The solution for the structural component with the crack is obtained in an iterative procedure, which alternates between FEM solution for the uncracked body, and the SGBEM solution for the crack in an infinite body.
Using jointly the symmetric Galerkin boundary element method for modeling an arbitrary non-planar crack in an infinite body, and the finite element method for an uncracked finite body, in fracture mechanics problems, allows us to employ advantages of both methods. The finite element method is a robust method for elastic and elastic-plastic problems. It can easily incorporate various types of boundary conditions. The boundary element method is most suitable for modeling cracks in infinite bodies. The displacement discontinuity approach provides for a simple modeling of the crack. Only one surface of the crack should be discretized.
The superposition principle can be used for combining the SGBEM and the FEM. According to the superposition principle, the solution for a finite body with a crack can be obtained as a superposition of two solutions:
For a correct superposition corresponding to the solution for a finite body with a crack, fictitious forces on the boundary of the finite element model should be found in order to compensate for the stresses caused by the presence of a crack in an infinite body. While this can be done with a direct procedure, the alternating method  provides for a more efficient solution, without assembling the joint SGBEM-FEM matrix.
The SGBEM-FEM alternating method alternates between the finite element solution for an uncracked body and the boundary element solution for a crack in an infinite body. The iterative procedure leads to the correct tractions at the crack surface thus making possible to compute correct values of the stress intensity factors at the crack front. In the elastic alternating procedure, elastic material relations are used in both SGBEM and FEM solutions.
Elastic-plastic alternating method based on combination of FEM solution for the uncracked body and an analytical solution for the crack has been proposed in Reference . Here this algorithm is generalized for the SGBEM-FEM elastic-plastic alternating method. In the elastic-plastic solution procedure the stresses at the finite element integration points are determined with the use of elastic-plastic constitutive material equations. The residual vector at nodes of the finite element model is calculated as a volume integral. The iteration procedure is terminated when the residual vector becomes small enough in comparison to the applied load.
Brick-type 20-node finite elements and quadrilateral 8-node boundary elements are used for the solution of 3D fracture mechanics problems. Quarter-point boundary elements with proper modeling of stress singularity are placed at the crack front. The elastic stress intensity factors , and are determined by employing asymptotic formulae for displacements in the vicinity of the crack front. Modeling of fatigue crack growth is performed with the use of the -integral as a criterion for the amount and direction of crack growth. The crack-front advancement is done in finite steps by adding extra element layers to the existing crack model.
The object-oriented approach and Java programming language have been used for the development of the SGBEM-FEM alternating code for crack analysis. Developed classes are organized with the use of Java packages. Java 3D application programming interface is employed for visualization of models and results.
Efficiency of the SGBEM-FEM alternating method is demonstrated on several examples. The stress intensity factors are calculated for inclined semielliptical surface crack in a tensile plate. Then this surface crack is used for modeling non-planar fatigue crack growth under cyclic loading. Finally, elastic-plastic stress intensity factor is determined for a semielliptical surface crack in a plate under tensile loading.
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