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This paper presents a new fractal finite element based method for continuum-based shape sensitivity analysis for a crack in a homogeneous, isotropic, and two-dimensional linear-elastic body subject to mixed-mode (modes I and II) loading conditions. The method is based on the material derivative concept of continuum mechanics, and direct differentiation. Unlike virtual crack extension techniques, no mesh perturbation is needed in the proposed method to calculate the sensitivity of stress-intensity factors. Since the governing variational equation is differentiated prior to the process of discretization, the resulting sensitivity equations predicts the first-order sensitivity of J-integral or mode-I and mode-II stress-intensity factors, KI and KII, more efficiently and accurately than the finite-difference methods. Unlike the integral based methods such as J-integral or M-integral no special finite elements and post-processing are needed to determine the first-order sensitivity of J-integral or KI and KII. Also a parametric study is carried out to examine the effects of the similarity ratio, the number of transformation terms, and the integration order on the quality of the numerical solutions. Four numerical examples which include both mode-I and mixed-mode problems, are presented to calculate the first-order derivative of the J-integral or stress-intensity factors. The results show that first-order sensitivities of J-integral or stress-intensity factors obtained using the proposed method are in excellent agreement with the reference solutions obtained using the finite-difference method for the structural and crack geometries considered in this study.

In recent years methods based on fractal geometry concepts to generate infinite number of finite elements around the crack tip to capture the crack-tip singularity have been developed or investigated to solve linear-elastic fracture-mechanics (LEFM) problems [1], [2], [3], [4] and [5]. The fractal finite element method (FFEM) is one such method developed for calculating the stress-intensity factors (SIFs) in linear-elastic crack problems. In its original form, the fractal two-level finite element method was first proposed by Leung and Su in 1993 [6]. Since its origin, it has been successfully applied to solve many kinds of crack problems under mode-I and mixed-mode loading conditions in 2D [7], [8], [9], [10], [11], [12], [13], [14], [15] and [16] and 3D [17]. Recently, this method has been found to be able to evaluate the coefficients of the higher-order terms of the crack-tip asymptotic fields [18]. Basically, FFEM separates a 2D or 3D cracked elastic body into a regular and a singular region (see Fig. 1), with the latter enclosing the crack tip. Both the regular and the singular regions are modeled by conventional, isoparametric finite elements. However, within the singular region an infinite number of elements are generated by a self-similar, fractal process to capture the singular behaviour at the crack tip. The nodal displacements in the singular region are transformed to a set of unknown coefficients using William’s analytical solution for the displacements near the crack tip [19]. Since the stiffness matrix of an isoparametric element depends only on its shape and not its actual dimensions, the above transformation can be performed at the element level and the results summed up as geometrical progression series to be assembled to the global stiffness matrix. The contributions of the infinite number of elements in the singular region are therefore fully accounted for while the number of degrees of freedom involved remains finite. Full-size image (25 K) Fig. 1. Cracked body domain with regular region, singular region, and fractal mesh. Figure options Compared with other numerical methods like finite element method (FEM), FFEM has several advantages. First, by using the concept of fractal geometry, infinite finite elements are generated virtually around the crack tip, and hence the effort for data preparation can be minimized. Second, based on the eigenfunction expansion of the displacement fields [19] and [20], the infinite finite elements that generate virtually by fractal geometry around the crack tip are transformed in an expeditious manner. This results in reducing the computational time and the memory requirement for fracture analysis of cracked structures. Third, no special finite elements and post-processing are needed to determine the SIFs. Finally, as the analytical solution is embodied in the transformation, the accuracy of the predicted SIFs is high. In addition to the SIFs, the derivatives of the SIFs are often required to predict the probability of fracture initiation and/or instability in cracked structures. Hence, sensitivity analysis of a crack-driving force plays an important role in many fracture- mechanics applications involving the stability and arrest of crack propagation, reliability analysis, parameter identification, or other considerations. For example, the first- and second-order reliability methods [21], frequently used in probabilistic fracture mechanics [22], [23], [24], [25], [26], [27] and [28], require the gradient and Hessian of the performance function with respect to random parameters. In linear-elastic fracture mechanics (LEFM), the performance function is built on SIF. Hence, both first- and/or second-order derivatives of J-integral or SIF are needed for probabilistic analysis. The calculation of these derivatives with respect to load and material parameters, which constitutes size-sensitivity analysis, is not unduly difficult. However, the evaluation of response derivatives with respect to crack size is a challenging task, since it requires shape sensitivity analysis. Using a brute-force type finite-difference method to calculate the shape sensitivities is often computationally expensive, in that numerous repetitions of deterministic FEM or FFEM analysis may be required for a complete reliability analysis. Furthermore, if the finite-difference perturbations are too large relative to finite element meshes, the approximations can be inaccurate, whereas if the perturbations are too small, numerical truncation errors may become significant. Therefore, an important requirement of some fracture-mechanics applications is to evaluate the rates of SIF accurately and efficiently. Consequently, analytical methods based on virtual crack extension [29], [30], [31], [32], [33] and [34] and continuum shape sensitivity theory [35], [36], [37], [38], [39] and [40] have emerged. In 1988, Lin and Abel [29] introduced a virtual crack extension technique that employs a variational formulation and a FEM to calculate the first-order derivative of mode-I SIF for a structure containing a single crack. This method maintains all of the advantages of similar virtual crack extension techniques introduced by deLorenzi [30] and [31], Haber and Koh [32], and Barbero and Reddy [33], but adds a capability to calculate the derivatives of the SIF. Subsequently, Hwang et al. [34], [35], [36] and [37] generalized this method to calculate both first- and second-order derivatives for structures with multiple crack systems, axisymmetric stress states, and crack-face and thermal loading. However, this method requires mesh perturbation – a fundamental requirement of all virtual crack extension techniques. For second-order derivatives, the number of elements affected by mesh perturbation surrounding the crack tip has a significant effect on solution accuracy [34], [35], [36] and [37]. Recently, Feijo’o et al. [38] applied the concepts of continuum shape sensitivity theory [39] to calculate the first-order derivative of the potential energy. Since the energy release rate (ERR) is the first-order derivative of potential energy, the ERR or SIF can be calculated using this approach, without any mesh perturbation. Later, Taroco [40] extended this approach to formulate the second-order sensitivity of potential energy to predict the first-order derivative of the ERR. However, this presents a formidable task, since it involves calculation of second-order stress and strain sensitivities. To overcome this difficulty, Chen et al. [41] and [42] invoked the domain integral representation of the J-integral and used the material derivative concept of continuum mechanics to obtain first-order sensitivity of the J-integral for linear-elastic cracked structures. Since this method requires only the first-order sensitivity of a displacement field, it is simpler and more efficient than existing methods. Subsequently, Chen et al. [43] extended their continuum shape sensitivity method for mixed-mode loading conditions. Recently, Rao and Rahman [44] and [45] developed a sensitivity analysis method for a crack in an isotropic, linear-elastic functionally graded materials under mode-I and mixed-mode loading conditions. All of these methods, however, have been developed only in conjunction with FEM. Consequently, there is a need to develop FFEM based sensitivity equations for SIFs so that subsequent stochastic FFEM analysis of cracks can be efficiently performed. This paper presents a new FFEM based method for predicting the first-order sensitivity of J-integral or mode-I and mode-II stress-intensity factors, KI and KII, respectively, for a crack in a homogeneous, isotropic, and two-dimensional linear-elastic structure subject to mixed-mode (modes I and II) loading conditions. The method is based on the material derivative concept of continuum mechanics, and direct differentiation. Numerical examples are presented to calculate the first-order derivative of the J-integral or SIFs, using the proposed method. The predicted numerical results from this method are compared with those obtained using the finite-difference methods.

A new fractal finite element based method is presented for predicting the first-order sensitivity of J-integral or mode-I and mode-II stress-intensity factors for a crack in a homogeneous, isotropic, and two-dimensional linear-elastic body subject to mixed-mode (modes I and II) loading conditions. The method is based on the material derivative concept of continuum mechanics, and direct differentiation. Unlike virtual crack extension techniques, no mesh perturbation is needed in the proposed method to calculate the sensitivity of stress-intensity factors. Since the governing variational equation is differentiated prior to the process of discretization, the resulting sensitivity equations predicts the first-order sensitivity of J-integral or mode-I and mode-II stress-intensity factors, KI and KII, more efficiently and accurately than the finite-difference methods. Unlike the integral based methods such as J-integral or M-integral no special finite elements and post-processing are needed to determine the first-order sensitivity of J-integral or KI and KII. Also a parametric study is carried out to examine the effects of the similarity ratio, the number of transformation terms, and the integration order on the quality of the numerical solutions. It is concluded that the similarity ratio greater than 0.4, and the number of transformation terms greater than 6 should be used to create fractal mesh, and that reduced integration may be used without producing significant errors in the sensitivity solutions. Four examples are presented to calculate the first-order derivative of J-integral or stress-intensity factors. Results show that first-order sensitivities of J-integral or stress-intensity factors obtained using the proposed method are in excellent agreement with the reference solutions obtained from the finite-difference methods for the structural and crack geometries considered in this study.