Effects of arbitrary holes/voids on crack growth using local mesh refinement adaptive XIGA
Introduction
Defects, e.g., inner holes or void, are often produced in engineering materials and structures. In addition, cracks generally exist in the service life of structures. Owing to high stress concentrations, the existence of crack and hole significantly affects the mechanical behavior and performance of structures. Hence, it is necessary and important to thoroughly understand the crack-hole interaction for evaluating the safety of structures.
Some researchers have examined the crack-hole interaction using theoretical approaches, e.g., see Refs. [1], [2], [3]. Due to the obvious limitations of analytical methods to general problem with complex geometry or loading conditions, numerical methods are more feasible. Ooi et al. [4] and Dai et al. [5] investigated the effect of existing hole on the crack propagation path by using polygon-based scaled boundary finite elements. The singular stress fields around the crack tip are analytically described in the solutions of stresses and displacements, accurate crack path can be predicted by using relatively coarse meshes, re-meshing for the evolution of crack is however rigorously required. Newman [6] solved the hole-crack-hole problem with the collocation method. Rashid [7] and Bouchard et al. [8] analyzed the influence of holes on crack growth path through the finite element method (FEM). In Ref. [8], an advanced re-meshing technique was proposed, and a fully automatic remesher was developed to deal with multiple boundaries and materials. Daux et al. [9] evaluated the stress intensity factors (SIFs) of a plate with two cracks emanating from a hole by using the extended finite element method (XFEM). Jiang et al. [10] used the XFEM to numerically investigate the effects of voids, inclusions, and minor cracks on major crack growth. To improve the accuracy, a fine mesh around the cracks and holes is required. However, the coarse mesh is applied far way the cracks and holes to save computational cost. Adaptive local refinement technique can be a highly effective approach to reduce the computational time. In recent years, some works e.g., see Refs. [11], [12], [13], have been developed for crack propagation in structures with holes using the adaptive local refinement technique.
The XFEM, an effective numerical method for modeling discontinuity, allows one to represent internal geometries or physical interfaces (such as cracks, voids, and inclusions) independently of the computational mesh. Zhou et al. [14] developed the XFEM coupled with the hierarchical mesh adaptation method and direct method for modeling crack propagation, and the method is simple and can obtain accurate SIFs with coarse meshes. Zhou et al. [15] introduced the node-scheme method into the XFEM for modeling frictional contact crack problem, and the locking phenomenon in the numerical simulation of the frictional contact problem can be solved. Zhou and Chen [16], [17] investigated the brittle failure mechanism of rock slopes with non-persistent en-echelon joints and the propagation of complex branched cracks using the XFEM. In spite of the success of XFEM for modeling various discontinuities, especially for cracks, there are several inherent disadvantages including only -continuity between elements, discretization errors in complex geometry, and time-consuming mesh generation. The extended isogeometric analysis (XIGA) [18] adopts the spline basis functions used in CAD as the shape functions of XFEM, and it can effectively overcome the existing drawbacks by the XFEM. The XIGA owns some excellent features including higher-order continuity, smooth stresses, exact geometry, higher-order convergence rate, high accuracy and without traditional meshing, besides the features of the XFEM [19]. The XIGA has been successfully applied in various discontinuities, for instance, see Refs. [20], [19], [21], [22], [23], [24] and references therein.
Non-uniform radial B-splines (NURBS) basis functions have been applied in XIGA, whereas the implementation of adaptive local refinement with NURBS is difficult due to its tensor-product structure. To conduct local refinement, some splines with local refinement ability have been developed and available in the literature, e.g., Refs. [25], [26], [27], [28], [29], [30], [31]. The method based on the locally refined (LR) B-splines has the very versatile and flexible local refinement ability. In our previous works, we modeled multiple inclusions [32], cracks [33], [34], [35] and voids [36] by using adaptive XIGA based on LR B-splines. Our study shows that the adaptive XIGA based on LR B-splines can effectively analyze the discontinuity, and can improve the accuracy with a low cost. The main objective of this study is to examine the effects of holes on crack growth path and fracture parameters in solids. An effective computational approach for modeling mechanical behavior of cracked structures with holes is developed, and then the crack-hole interactions are investigated using the developed adaptive XIGA based on LR B-splines. In this study, the geometry of hole or crack is described with the NURBS curve, so the arbitrary shaped hole or crack can be accurately presented.
The structure of this paper is given as follows. Problem statement and fundamental equations are given in Section 2. Section 3 presents the XIGA discretization for cracks, holes and voids. Information about modeling crack growth is presented in Section 4. The adaptive procedure is discussed in Section 5. Section 6 illustrates numerical implementation procedure. Section 7 provides several numerical examples to verify the performance and effectiveness of the developed method. Some concluding remarks are given in Section 8.
Section snippets
Problem description
Consider a linear and elastic body with traction-free holes and cracks occupying bounded by . , where is the displacement boundary, is the traction boundary, and represent the boundaries of cracks and holes, respectively. The strong form of the boundary value problem (BVP) can be expressed aswhere is the unit outward normal vector on and are the body force, displacement vector and stress tensor, respectively. and
A brief on LR B-splines
The LR B-splines, which are called as weighted B-splines, have the local refinement ability, they are thus used for the adaptive IGA/XIGA [37], [38], [39], [33], [34], [36]. Here, the LR B-splines are briefly introduced [30], [40].
and are two local knot vectors, p is the order of basis function in -direction, and q is the order of basis function in -direction, and the corresponding B-spline basis functions are denoted as and , respectively. The bivariate
Computation of stress intensity factors
For the sake of completeness, the interaction integral technique to evaluate the stress intensity factors (SIFs) is briefly presented [45]. Consider two states, the first state is the actual state and the second state is an auxiliary state . The interaction integral can be given as follows [34]where q is the weight function defined as , and A is the region determined by a circle with
Recovery technique
Similar to the displacement approximation, the approximation of smoothed or recovered stress field can be expressed aswhere and are the recovered stress and enrichment variable vectors at control points.The crack tip enrichment functions are extracted from the asymptotic stress field at crack tip, and the detailed expression can be found in Ref. [47]
Numerical implementation procedure
The main steps of simulating the hole-crack interaction using adaptive XIGA based on LR B-splines are presented in this section. There are two main steps for modeling hole-crack interaction and three steps for modeling crack propagation.
- 1.
Initialization
- (i)
Level set functions: Define the level set functions for hole boundaries and initial cracks.
- (ii)
Data input: Input problem descriptions and all the parameters, and construct the initial mesh for representing the physical domain without considering the
- (i)
Numerical examples and results
In this section, five examples are examined to investigate the crack-hole interaction using the adaptive XIGA. Unless mentioned otherwise, Young’s modulus units and Poisson’s ratio are used. In all the numerical examples, the cubic LR B-splines are used, and the interaction integral radius are taken to compute the SIFs, where is the area of crack tip element.
Conclusions
In this paper, we have presented numerical investigation of the crack-hole interaction using the adaptive XIGA based on LR B-splines. In XIGA, the computational mesh is independent of the geometry of cracks and holes, and the re-meshing is avoided when crack grows. An a posteriori error estimator based on stress recovery technique and the structured mesh refinement strategy are adopted to guide the adaptive refinement. The interaction integral and the maximum circumferential stress criterion
CRediT authorship contribution statement
Weihua Fang: Conceptualization, Methodology, Software, Investigation, Formal analysis, Writing - original draft. Xin Chen: Software, Data curation, Visualization, Investigation, Validation, Writing - original draft. Tiantang Yu: Supervision, Writing - review & editing. Tinh Quoc Bui: Supervision, Writing - review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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