تجزیه و تحلیل حساسیت شکل از سیستم ترک چندگانه مبتنی بر روش المان محدود فراکتال

This paper presents fractal finite element based continuum shape sensitivity analysis for a multiple crack system in a homogeneous, isotropic, and two dimensional linear-elastic body subjected to mixed-mode (modes I and II) loading conditions. The salient feature of this method is that the stress intensity factors and their derivatives for the multiple crack system can be obtained efficiently since it only requires an evaluation of the same set of fractal finite element matrix equations with a different fictitious load. Three numerical examples are presented to calculate the first-order derivative of the stress intensity factors or energy release rates.

Methods based on fractal geometry concepts such as fractal finite element method (FFEM), to generate infinite number of finite elements around the crack tip to capture the crack tip singularity have been successfully adopted to solve many kinds of crack problems under mode-I and mixed mode loading conditions [1], [2], [3], [4], [5], [6], [7] and [8]. 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 [9] and [10], 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 stress intensity factors (SIFs) and their derivatives. Finally, as the analytical solution is embodied in the transformation, the accuracy of the predicted SIFs and their derivatives is high. In addition to the SIFs, the derivatives of the SIFs are often required for reliable analysis of crack growth behavior under LEFM conditions. Hence, sensitivity analysis of a crack-driving force plays an important role in many fracture mechanics applications which includes the prediction of stability and arrest of a single crack [11], the growth pattern analysis of a system of interacting cracks [12] and [13], configurational stability analysis of evolving cracks [14], size effect model [15], stability analysis of crack path [16] and probabilistic fracture mechanics analysis [17] and [18]. Hwang et al. [19] and [20] clearly outlined the potential applications of the derivatives of the energy release rate and the SIFs. The first- and second-order reliability methods [21], frequently used in probabilistic fracture mechanics [22], [23] and [24], 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 SIFs. Hence, both first- and/or second-order derivatives of the SIFs or energy release rates are needed for probabilistic analysis. 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 fracture mechanics applications is to evaluate the rates of SIFs or energy release rates accurately and efficiently. As a result, analytical methods based on virtual crack extension [19], [20], [25], [26], [27] and [28] and continuum shape sensitivity theory [29], [30], [31] and [32] have emerged. Rao and Rahman [33] and [34] developed a sensitivity analysis method for a crack in an isotropic, linear-elastic functionally graded materials under mode-I and mixed mode loading conditions. However, all of the above methods have been developed in conjunction with FEM. Recently, Reddy and Rao [35] developed FFEM based method for continuum-based shape sensitivity analysis to calculate the first-order derivative of the J-integral or the SIFs for a single crack in a homogeneous, isotropic, and two dimensional linear-elastic body subjected to mixed-mode loading condition. Subsequently, based on the first-order sensitivities computed using continuum shape sensitivity analysis of mixed-mode fracture in conjunction with FFEM [35], Reddy and Rao [36] employed FORM to conduct probabilistic fracture mechanics analysis. However, estimation of the derivative of the SIFs at one crack tip due to the extension of any other crack tip, which is very important for the growth pattern analysis of multiple crack system, is not considered in the above papers. In this paper the potential of FFEM based method is explored, for predicting the first-order derivative of the SIFs or energy release rates at one crack tip due to the extension of any other crack tip, for a multiple crack system in a homogeneous, isotropic, and two dimensional linear-elastic structure subjected 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 SIFs or energy release rates at one crack tip due to the extension of any other crack tip, using FFEM based method. The predicted numerical results from this method are compared with those obtained using the finite-difference methods.

Fractal finite element based method is presented for predicting the first-order sensitivity of the SIFs or energy release rates for a multiple crack system crack in a homogeneous, isotropic, and two dimensional linear-elastic body subjected 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 energy release rates 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, and the number of transformation terms on the quality of the numerical solutions. Three examples are presented to calculate the first-order derivative of SIFs or energy release rates. Results show that first-order sensitivities of SIFs or energy release rates 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.