Extended iso geometry analysis of crack propagation

August 27 th , 2015) ABSTRACT: The purpose of this paper is simulating the crack propagation in steel structures with isogeometry analysis (IGA). In this method, CAD model is integrated into the CAE model by using non uniform rational B-Splines (NURBS) function. Crack propagation in isotroptic linear elastic material will be presented. The numerical example is a rectangular plate assumed to be plane strain condition with an edge crack under uniform shear loading. The obtained results are investigated and compared with analytical method and reference solutions. Very good agreements on the solutions are found. It is showed that isogometry analysis is better than standard finite element method in modeling and simulating. Consequently, isogometry analysis is an effective numerical method in future, especially when


INTRO DUCTIO N
In simulating the crack growth problems with arbitrary paths, the FEM has encountered many difficulties because the finite element mesh must be re-meshing after each increment of growthing cracks. To overcome these difficulties, the extened finite element method (Moes et al.1999) was developed to solve crack growth problems. XFEM is developed based on Partition of Unity Finite Element Method (PUFEM) [1]. Belytschko và Black (1999) [2] introduced a minimal remeshing method for crack propagation problems. Moës (1999) [3] improved this method. Dolbow (1999) [4] applied XFEM to solve crack problem in shell structures.
In recent years, Isogeometric Analysis -IGA has been successfully developed by Hughes at Institute for Computational Engineering and Sciences [5,6], The University of Texas (USA). The main idea of this method is the use of NURBS basis functions to build CAD geometry for modeling, the concept is similar to the finite element method (FEM). The difference is in FEM, Lagrange shape functions is used while IGA using NURBS shape functions to approximate the problem domain.There are several articles have demonstrated the approximate model discontinuities by using NURBS is better FEM shape function [7,8] The combination XFEM and IGA opens a modern approaching in the field of computational fracture mechanics, that is Extended Isogeometric Analysis -XIGA. XIGA inherited the advantages of XFEM and IGA, fully capable of solving some complex crack propagation problem without re-meshing. On the other hand, the complex geometry of objects can be modeled with a few of elements, so the calculation time can be reduced significantly.

B-Spli ne basi s func ti ons
A knot vector, defined by Ξ . B-Spline basis functions, according to [9], are constructed from a given knot vector. Generally, the B-Spline basis functions of order p = 0 are defined by:

B-Spli ne c ur ve
Given n basis functions corresponding to the knot vector   the B-Spline surface is thus defined by: ,

NURB S ge ometry
NURBS basis is then given as follows: The NURBS curve is then defined as in the same manner as the B-spline curves: The NURBS surfaces are defined as: Where , , ( , ) The mapping from the parametric domain  to the physical domain  is given by: In an isoparametric formulation, the displacement field is approximated by the same shape functions: where i u denotes the value of the displacement field at the control point i B .

Exte nde d i sogome try anal ysi s (XIGA)
General form of the XFEM for modeling the crack is given by where u h (x) is the approximated function of the displacement field; N is the shape function computed at the control points; u, b, c respectively, are the unknown degrees of freedom corresponding to the sets named as I, J and K. I is the set of total nodes in the problem domain, whereas J is the set of nodes enriched by the sign function S(x) (12) wherw (x,t) is the level set function where (r,) are the local polar coordinates defined at the crack tip.
In NURBS-based XFEM, the sets I, J and K are also associated with control points, correspondingly.

The le vel set me thod
According to [10,11], the level set method is use to detect the discontinuous surfaces. As sketched in Figure 5, the crack is considered to be the zero level set of .
Within the framework of crack growth problems, the level set must be updated appropriately, but only nodes locally close to the crack are updated. In addition, it is assumed that once a part of a crack has formed, that part will be fixed (Stolarska et al. (2001) [5]).
The evolution of the crack is modeled by appropriately updating the functions level set  and  then reconstructing the  function. In each step, the incremental length and the angle of the propagating direction, c are known. The displacement of the crack tip is given by the vector   , 1 ,n , tip n tip where xi tip,n+1 is the current crack tip (step n+1) and xi tip,n = (xi, yi) is the crack tip at step n.
Let the values of  and  at step n be  n và  n .
The updated values of  and ,  n+1 and  n+at n+1 are determined by the following algorithm [5]: (1) i n+1 is updated at each step.
(2) F is not necessarily orthogonal to the zero level set i n . Thus For the XIGA, the values at control points are also stored. The values at other points within a given element are approximated from the values of the control points that support the given element as follows: where Ncp is the number of control points which support the element. The other level set functions are also approximated by using the same form of (17).
For the NURBS-based XFEM, an element is determined to be split element, i.e. discontinuous-enriched, or tip element, i.e. tipenriched, from the nodal values of  and . The enrichment has to be chosen for the control points. The values of enrichment function in (11) are computed at the control points.

Maxi mum c i rc umfe re nti al stre ss cr iteri on
The maximum circumferential stress criterion states that the crack will propagate from its tip in a direction assigning by an angle c where the circumferential stress  is maximum The stress intensity factors are computed using the interaction integral [3].

Inte rac ti on i nte gr al
Interaction integral for states 1 and 2 is given as follow (19) With W (1,2) is the interaction strain energy

NUMERICAL EXAM PLES 3.1. Edge c rack unde r uni for m she ar
A rectangular plate assumed to be plane strain condition with an edge crack under uniform shear loading  = 1 N/cm 2 as depicted in figure 6.
The geometrical parameters are chosen as follows:: a/W = 0.5; h/W = 0.5; L/W = 16/7, W = 7 cm. while the material parameters involve Young's modulus E = 3x10 7 N/cm 2 and Poisson's ratio  = 0.25. Relative error is given by: The mixed mode stress intensity factors computed for the first-, second-and third-order NURBSbased XFEM are presented in Table 1 with a uniform mesh of 21x4.
As observed in Table 1, the stress intensity factors of mode I are more accurate when the order of the NURBS functions gets higher, but this behavior does not apply for mode-II. The computational time is increased rapidly when the order of the NURBS functions increases. In practice, the order of the NURBS functions may be chosen in a way dependent on the problems of interest, but second-order could yield a good solution.

Edge cr ack pr opagati on under uni for m she ar
A rectangular plate assumed to be plane strain condition with an edge crack under uniform shear loading  = 1 N/m 2 . The geometrical parameters are chosen as depicted in Fig. 8. The unit of length is the metre (m) while the material parameters involve Young's modulus E = 30x10 6 N/m 2 and Poisson's ratio  = 0.25.
We check the accuracy of the XIGA by comparing the obtained solutions with those given in previous work [12] (Scale boundary finite element method -SBFEM). The compared results show a good agreement between two methods. Additionally, The crack paths obtained by two method are shown in Figs. 9, respectively, which shows a good agreement as expected.