Keywords: scaled boundary finite element method, fracture mechanics, singularity order, classical laminate theory.

This paper deals with a new formulation of the scaled boundary finite element method (SBFEM) for the analysis of circular, arbitrarily laminated plates in the framework of classical laminate theory. The SBFEM is a new, semi-anayltical analysis technique, which was developed by Wolf and Song in 2000 [1,2]. Essential for the SBFEM is, that a domain is described by the mapping of its boundary with respect to a scaling centre, if the whole boundary is visible from the scaling centre. The geometry of the domain is described by the so-called scaled boundary coordinates, where a scaling coordinate runs from the scaling centre to the boundary and the other circumferential coordinate describes a length along the boundary. The governing partial differential equations are transformed into scaled boundary coordinates. The displacements are approximated as products of the displacement shape functions and unknown functions of the scaling coordinate. Because of the kinematic assumptions of the classical laminate theory, the displacement shape functions for the description of the out of plane displacement have to fulfill continuity requirements for the first derivatives of the displacement, in order that the plate does not kink. Next, the governing partial differential equations are reduced to a set of ordinary differential equations, which are called the scaled boundary equations in displacements, applying a discrete form of the Kantorovich reduction method. The scaled boundary equations in displacements are solved using the corresponding eigenvalue problem. Finally, the element stiffness matrix of the domain is derived for laminates with arbirtrary layup, using the homogenous solution of the ODE-system. If kinematic boundary conditions are respected, the displacements and rotations on the boundaries can be determined and the solution all over the domain can be expanded in the form of a power series.

In the case of crack problems, the scaling centre is selected at the crack tip and the SBFEM enables the effective and precise calculation of singularity orders, solving the corresponding eigenvalue problem [3]. Although the method has been applied successfully to many problems of continuum mechanics, its application to plate bending problems is new. This paper presents a new formulation of the method for the static analysis of circular composite laminates using the kinematics of classical laminate theory. The scaled boundary finite element equations in the displacements are derived and the computation of the element stiffness matrices for bounded and unbounded media are presented selecting appropriate subsets of the general solution of the ODE-system. Numerical examples show the performance and efficiency of the method.

1

J.P. Wolf, C. Song, "The scaled boundary finite-element method - a primer: Derivations", Computers and Structures, 78, 191-210, 2000. doi:10.1016/S0045-7949(00)00099-7

2

J.P. Wolf, C. Song, "The scaled boundary finite-element method - a primer: Solution procedures", Computers and Structures, 78, 211-225, 2000.

3

C. Song, "Evaluation of power-logarithmic singularities, T-stresses and higher order terms of in-plane singular stress fields at cracks and multi-material corners", Engineering Fracture Mechanics, 72, 1498-1530, 2005. doi:10.1016/j.engfracmech.2004.11.002

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