An element-free Galerkin method for three-dimensional fracture mechanics

The application of a coupled ®nite element± element-free Galerkin (EFG) method to problems in threedimensional fracture is presented. The EFG method is based on moving least square (MLS) approximations and uses only a set of nodal points and a CAD-like description of the body to formulate the discrete model. The EFG method is coupled with the ®nite element method which allows for the use of the EFG method in the crack region and the ®nite element method to model the remainder of the problem. Domain integral methods are used to evaluate stress intensity factors along the 3D crack front. Both planar and volume representations of the domain integrals are considered. The former require derivatives of stress and strain which are readily obtainable in the EFG method due to the C continuity of the MLS approximations used here. Applications of the method to the determination of stress intensity factors along planar cracks in 3D are presented. 1 Introduction The modeling of fracture and failure remains one of the most challenging problems in mechanics. Failure modeling is an essential ingredient in the lifetime prediction of critical components in structures such as aircraft, automobiles and pressure vessels. It also plays an important role in the development of advanced materials such as composites and in understanding their durability and integrity. Finite element methods based on singular elements (Henshell and Shaw 1975; Barsoum 1977) as well as enriched elements (Benzley 1974; Gifford and Hilton 1978) are reasonably effective in the analysis of stationary cracks. However, the modeling of growing cracks (Gray et al. (1994)) presents automatic mesh generation dif®culties and, in certain cases, manual intervention is required. No general purpose computational method currently exists which can handle crack growth in complex 3D bodies for arbitrary constitutive response without recourse to extensive remeshing. In this paper, a coupled ®nite element±element-free Galerkin (EFG) method for problems in 3D fracture is presented. EFG methods (Belytschko et al. 1994; Lu et al. 1994) require only nodal data ± no element structure is needed for the construction of the approximation. The approximating functions in EFG are moving least square approximants (MLS). They are not interpolants because the approximation does not pass through the data; this is often referred to as failure to satisfy the Kronecker delta property. As a consequence, essential boundary conditions cannot be speci®ed directly. Typically, Lagrange multiplier methods or modi®ed variational forms are used to implement the essential boundary conditions in the EFG method (Belytschko et al. 1994; Lu et al. 1994). A coupled ®nite element±element-free Galerkin method has been developed by Belytschko, Organ, and Krongauz (1995) and applied to fracture problems in two dimensions. This allows for the use of the EFG method in the crack region and the ®nite element method to handle complex geometries and essential boundary conditions. In the following section, the governing equations for elastostatics are given. In Sect. 3, the element free Galerkin method together with its coupling to the ®nite element method is described. The enriched basis method for enhancement of the crack tip ®eld is presented and the visibility and diffraction methods for construction of shape functions in the presence of the crack discontinuity are extended to three dimensions. In Sect. 4, domain integral methods for evaluation of stress intensity factors along a 3D crack front are described. Results for several benchmark problems in 3D fracture mechanics are presented in Sect. 5 and compared with solutions from the literature. Some ®nal remarks are given in Sect. 6. 2 Element-free Galerkin method 2.1 Governing equations and weak form We consider small displacement elastostatics, which is governed by the equation of equilibrium: $ r‡ b ˆ 0 in X …2:1† where r ˆ C : ; ˆ $su : …2:2† Computational Mechanics 20 (1997) 170±175 Ó Springer-Verlag 1997