5.1. Conclusions
The NURBS-based XFEM has been successfully applied in numerically modeling of crack growth in isotropic materials. The obtained results are in good agreement with the reference solutions, including analytical, experimental and numerical solutions available in the literatures.
In this work, the crack geometry is modeled as the zero level set of function ψ(x), which is defined as the signed distance from a point x to the crack surface, and function φ(x), which is defined as the signed distance from a point x to the line contains crack tip and orthogonal to the crack surface. The level sets are updated accordingly to the evolution of the crack and they are also used as criteria to identify the split and tip elements. These criteria are however insufficient for any configuration, a double check thus must be taken into account to ensure all the split elements and the true tip elements are detected completely and correctly.
One important issue in the proposed approach is the quadrature of the weak form. In this work, the technique of sub-division of elements is applied to numerical integration to deal with the singularities issue. In this technique, certain elements in the surroundings of the discontinuity surface, i.e. crack surface, are sub-divided to be implemented for the purpose of the numerical integration. Due to this technique, many more Gaussian quadrature points are added systematically in order to increase the accuracy of numerical integration. However, it leads to an issue of computational cost. Nevertheless, simpler and more effective methods would be desirable.
Crack growth is modeled by an incremental scheme. In each step of crack propagation, the crack geometry is updated by updating the level sets. Next, the asymptotic stress fields are evaluated.
Finally, the propagating direction is estimated based on a specified appropriate criterion, such as the maximum circumferential stress criterion or the maximum principal stress criterion. The incremental length of crack in each step is pre-defined.
The accuracy of the stress intensity factors, which are key factors in determining the propagating angle using the maximum circumferential stress criterion, is illustrated through a number of benchmark examples. The obtained results show a very good agreement with analytical solution and alternative numerical approaches.
The crack path is calculated based on the averaged stress criterion and the maximum circumferential stress criterion. The averaged stress criterion works well for the problem of the edge crack under uniform tension and the problem of three-point bending. However, it is not able to predict a reasonable crack path for the L-shaped panel problem. The nearly equal biaxial tensile stress state might be a possible reason. Hence, the averaged stress criterion needs further improvement to be able to model such problem. On the other hand, the maximum circumferential stress criterion works well for all the considered benchmarks and it thus seem to be more reliable than the averaged stress criterion. However, it contains some inherent disadvantages. The key in the maximum circumferential stress criterion is the stress intensity factors and most importantly, the ratio between mode II and mode I. When this ratio becomes large, the crack direction is unstable and crack path oscillates [Belytschko and Fleming, 1999], as observed in the three-point bending test. Since determining the propagating direction accurately plays an important role in crack modeling, further attempt should be invested in finding a better criterion is accurate and can be generally applicable.
It has been demonstrated through the numerical results that the NURBS-based XFEM can yield good solutions for numerically modeling of crack growth. The use of NURBS functions in isogeometric analysis as basis functions allows the ability to control the polynomial order and the continuity of such basis functions, which is a property not known by the standard FEM. Finite element mesh is structured and can be automatically refined by knot insertion. However, global refinement is somehow a disadvantage of NURBS functions. Local refinement is available in the recently introduced T-Spline ([Sederberg et al. (2003)] and [Sederberg et al. (2004)]). T-Spline is also more accurate than the NURBS in modeling complicated geometries that require the use of multiple patches [Bazilevs et al. (2008)]. Hence, the coupling of T-Spline and XFEM would be an interesting subject to be studied in our future research works.
5.2. Future works
The current work has many possible extensions. Many advanced aspects of the XFEM such as blending functions and corrected XFEM have not been yet considered to be incorporated into the present model. Finding an effective model, i.e. simple and accurate, for the numerical integration of singularities is also an interesting issue worth working on. These aspects shall be carefully investigated in future works. Besides that, finding an appropriate criterion in order to determine the growth direction of crack that can be generally applicable is also important. Furthermore, the NURBS-based XFEM formulation can be adopted in solving other types of discontinuous problems, such as voids and inclusions.
From the isogeometric analysis point of view, T-Spline has been proven to have many advantages compared to NURBS (see [Sederberg et al. (2003)], [Sederberg et al. (2004)] and [Bazilevs et al. (2008)]), such as the ability of local refinement, reduction of degrees of freedom, better accuracy in multiple-patches modeling, which in turn has great advantages in dealing with complicated geometries. Hence, the T-Spline could be possible to be considered as a better tool than the NURBS to be incorporated into the XFEM formulation to numerically model discontinuous problems. Like NURBS, the T-Spline basis function does not possess the Kronecker delta property. Hence, imposing of the essential boundary conditions is also an important task.
Moreover, this thesis is restricted to linear elastic fracture mechanics; future works can be open to take the effect of plastic zone around the crack tip into account, or the dynamic effect and so on.
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