18 th International Conference on composite materials CALCULATION OF STRESS INTENSITY FACTORS WITH THE MODIFIED VIRTUAL CRACK CLOSURE TECHNIQUE Zhou Hongliang 1,* 1 Institute of Structural Mechanics, Chinese Academy of Engineering Physics, MianYang ， China * Corresponding author (zhouhongliang1986@gmail.com.cn) Keywords : stress intensity factors (SIFs); strain energy release rate (SERR); crack closure technique (VCCT); finite element (FE) crack element is developed. The interface crack Abstract: element can be implemented easily in the commercial The modified equations of VCCT with different software ABAQUS TM by the user subroutine UEL. element lengths in front of the crack tip and behind Several examples are analyzed to demonstrate the are given out in the paper. In order to avoid the accuracy of the present method with agreement with complex post proceeding to extract fracture analytical solutions. parameters such as SERR and SIFs, the interface by Krueger [3]; it is not dependent on the hypothesis 1 Introduction of the singularity of the stress field at the crack tip. Delamination is one of the most common failure The relationship between node opening mode of composite structure [1-2], and fracture displacements at the crap tip before crack growth and mechanics is one effective tool to characterize the after are established by taking into account the shape onset and growth of delamination [3]. As the basic functions of the elements or approximated by simple fracture parameters, SERR ( G ) and SIF ( K ) need to linear interpolation. In the paper, a unit form of corrected formulas of be calculated. Up to now, the two kinds of numerical VCCT with different element lengths in front of the approaches are widely used: one is called direct method, such as stress extrapolation method and crack tip and behind is established to the two dimensional crack problems, and two typical displacement extrapolation method; the other is indirect method, including J integral 、 interaction modified equations are represented. The SERR is calculated by the VCCT interface element proposed integral 、 VCCT and et al [4-5]. by Xie [9], and then the SIFs can be obtained. The In contrast with other methods, VCCT has many interface element is implemented by the commercial merits such as simplicity 、 convenience 、 high software ABAQUS TM with the user subroutine UEL accuracy 、 no sensitivity to mesh size 、 explicit [10]. Two classical examples (center crack and separation of fracture modes and et al, therefore, it is slanted crack) are computed, and the accuracy and widely investigated by scientists and engineers [2,6]. affectivity of the modified equations presented are In previous research, the equations have been derived validated by the excellent agreement with the under the assumption that the element lengths in front analytical results. Compared with the traditional of the crack tip and behind are identical. However, modified equations, the modified equations presented once automatic mesh generators are used to create in the paper can get more accurate results with the complex models, especially in the situation of grid same mesh. Furthermore, the modified VCCT can be transition, the ideal case of identical element length simplily implemented in the engineering analysis of can no longer be assumed and corrections are complex structure. required. 2 Modified equations The study about 2D-VCCT with different element lengths in front of the crack tip and behind is very As shown in Fig.1, when the element lengths in front limited. Based on the assumption that the stress of the crack tip and behind are different, it has two distribution is same at the crack tip before the crack Δ < Δ Δ < Δ ≤ Δ + Δ cases: a c and c a c d , the growth and after, a kind of modified equations is Δ > Δ + Δ a c d case is not suggested. The modified derived by Rybichi and Kanninen [7]. A equations are presented by Rybichi and Kanninen[7] mathematical explanation to the corrections is made on the assumption that the stress distribution at the by Xie and Wass [8], and using this mathematical crack tip is same before and after the crack growth: explanation, the VCCT calculation formulas of kinking crack are obtained. In the comprehensive survey, another approach to corrections is illustrated
the nodal forces at the crack tip are invariable. Δ Δ F v F u Δ σ ( ) ( ) = = y 1 34 x 1 34 G lim G lim ∫ a ; ( 1 ) ( 2 ) ( 1 ) ( 2 ) Δ = Δ x v x d x F v I Δ Δ II Δ Δ Δ → Δ → yy y 1 a 0 2 D a c a 0 2 D a c ' 1 , 1 0 (1) (4) The quantitative relation of the opening Therefore, for the VCCT with different element displacements at the nodes behind the crack tip length in front of the crack tip and behind, the most before and after the crack growth is establish by Δ v important work is to calculate with the opening ' ' 34 Krueger[3], then another kind of modified equations displacements supplied by finite element analysis are obtained: (FEA). Δ Δ F v F u Δ = y 1 34 = x 1 34 v lim 2 lim 2 In general, there are three methods to calculate , G ; G ' ' 34 Δ Δ I II Δ → Δ → D c D c a 0 a 0 Δ Δ v v the first method to get is using by linearly ' ' 34 34 (2) interpolating: The two cases of different mesh at the crack tip are Δ a not considered in the above two kinds of modified Δ = Δ v v ' ' 34 34 Δ in front of the crack Δ equations, the element length a c tip is not involved in the modified equations given by (5) Krueger. Take the mode I as example, the modified The modified equations of VCCT derived by the equations of VCCT are derived in the paper. one-point interpolation method are the same as that given by Krueger. The second method is based on the = basic formula v B r of the linear elastic fracture mechanics, and then the following relation can be obtained: Δ a Δ = Δ v v ' ' 34 34 Δ c (6) The modified equations of VCCT derived by the Δ < Δ (a) a c basic formula method are the same as that given by Rybichi and Kanninen. The third method to Δ Δ Δ v v v get is using and by linearly ' ' 34 34 56 interpolating: Δ − Δ ( ) a c Δ = Δ − Δ + Δ v v v v ' ' 34 34 Δ 56 34 d (7) Then the corresponding modified equations of VCCT are: Δ < Δ ≤ Δ + Δ d (b) c a c ⎡ Δ − Δ ⎤ ( ) F a c = y 1 Δ − Δ + Δ Fig.1. the FE model at the crack tip in local coordinates G lim 2 v v v ⎢ ⎥ 34 Δ Δ I ⎣ 56 34 ⎦ Δ → 0 D a d Similar to VCCT equations with the same element a length in front of the crack tip and behind, the general (8) VCCT equations with different element length in The method is called two-point interpolation method. front of the crack tip and behind are as: The difference between the two cases of different Δ Δ element lengths in front of the crack tip and behind is F v F u ' ' ' ' = y 1 34 = x 1 34 G lim G lim ; not considered in the above three methods. In general, Δ Δ I II Δ → Δ → 2 D a 2 D a a 0 a 0 Δ < Δ , the result of equation (5) is for the case a c (3) lower, and the results of equations (6) and (7) are Δ v is the opening displacement of nodes higher; it is on the contrary for the where ' ' 34 Δ < Δ ≤ Δ + Δ Δ case c a c d . behind the crack tip with the distance . a Therefore, the mean form of the two methods is used The element length in front of the crack tip is the Δ virtual extension with a given FE mesh, according to to calculate v in the paper. By averaging the ' ' 34 the mathematical explanation presented by Raju[11],
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