The scaled boundary finite element method (SBFEM) is extended for shape sensitivity analysis of stress intensity factors (SIFs) with respect to the crack geometry. The procedure combines the finite element formulations with the boundary discretization. The original equations of the SBFEM are reformulated as functions of the so-called scaling centre, synonymous to the crack tip. The variation in crack geometry is modelled without remeshing. Sensitivity is analyzed by direct differentiation. Following the computation of the displacement field sensitivity, the SIF sensitivity is evaluated directly from the present SIF definition. Numerical examples are presented.
Stress intensity factors (SIFs) are used to represent the strength of the singularity near the tip of a crack due to remotely applied loads. They are utilized in failure criterion of cracked structural components. While there are many analytical methods of evaluating the SIFs, like explicit handbook equations (e.g. [1]), recent times have seen numerical techniques to be more favourable due to its computational ease and superiority in handling problems with more complex geometries and boundary conditions. The displacement/stress extrapolation and J-integral techniques, based on the finite/boundary element method (FEM/BEM) results, are examples. Some areas of fracture mechanics, however, also require the variation or derivative of the SIFs with respect to the crack geometry, i.e. shape sensitivity. Its significance is well recognized in, for example, the prediction of stability and arrest of a single crack [2] and [3], universal size effect modelling [4] and [5] and configurational stability analysis of evolving cracks [6] and [7]. In particular, shape sensitivity plays a major role in the reliability analysis of cracked structures with uncertainties in the crack geometry [8].
Over the years, various numerical approaches to shape sensitivity analysis have been investigated. The use of predefined equations of the SIFs (e.g. [1]) is an option, but are limited owing to complexities in loading, material behaviour and crack geometry. Direct application of the FEM [9], [10] and [11] in combination with a finite difference (FD) approximation is also possible, at the computational expense of numerous deterministic analyses and excessive remeshing, especially near the crack tip region. Applying the BEM incurs a simpler boundary mesh [12], [13] and [14], but still requires remeshing as the crack surface forms part of the boundary. Analytical techniques built on the FEM have also emerged. The virtual crack extension (VCE) technique, proposed by [15], is one of the earliest examples. A fundamental requirement of the technique is mesh perturbation. Hwang et al. [16], [17] and [18] and Hwang and Ingraffea [19] applied the VCE technique to obtain first/second-order derivatives of multiple crack systems, axisymmetric stress states and crack-face and thermal loading cases. Latter techniques include those based on continuum shape sensitivity theory, which introduces a velocity field expression [20] to simulate the shape variation or change. Chen et al. [21] and [22] used this to determine the first-order derivative of the J-integral. Rao and Rahman [23] and [24] carried on, but also investigated the effects of functionally graded materials. Reddy and Rao [25], [26] and [27] and Rao and Reddy [28] then developed a continuum shape sensitivity based approach using the well known fractal finite element method (FFEM).
The scaled boundary finite element method (SBFEM) was introduced by [29]. It possesses many characteristics that can simplify shape sensitivity analysis. Unlike the BEM, no fundamental solution is required and unlike the FEM, only the boundary need be meshed. The stress singularity at a crack tip is expressed analytically. Special elements or numerical techniques are not required at the crack tip for fracture analysis, and the crack surface remains meshless. High accuracy and efficiency in evaluating the SIFs have been vastly demonstrated in [30], [31], [32], [33] and [34]. Only Chowdhury et al. [35] and [36] have previously explored the possible capabilities of the SBFEM for shape sensitivity analysis.
The purpose of this paper is to continue the works of [35] and [36] and analytically extend the SBFEM for shape sensitivity analysis of the SIFs. An efficient, accurate and simple procedure is introduced to compute the first-order derivative of the SIFs with respect to the crack geometry. The procedure is primarily based on placing the so-called scaling centre of the SBFEM at the crack tip, and reformulating the governing equations as a function of this scaling centre. The variation in crack geometry is modelled by changing the scaling centre while leaving the domain mesh intact. Direct differentiation with respect to the crack geometry is applied. The sensitivity of the stiffness matrix and displacement field is obtained by solving efficient algebraic equations. No mesh perturbation or complex velocity field is needed. Following the normalization of the stress modes corresponding to square-root singularities, the SIF sensitivity is computed without requiring the stress field sensitivity. Existing numerical methods based on the energy-derived J-integral expression, however, require the second-order sensitivity of the energy release rate [22].
The remaining parts of the paper are organized as follows. A summary of the SBFEM is presented in Section 2, with the original equations reformulated. A stress mode normalization technique is also employed, leading to an alternate SIF formulation. The SBFEM shape sensitivity analysis is developed in Section 3. In Section 4, numerical examples and discussions are presented. Conclusions are stated in Section 5.
