Abstract
The cohesive zone model (CZM) has been widely used for numerical simulations of interface crack growth. However, geometrical and material discontinuities decrease the accuracy and efficiency of the CZM when based on the conventional finite element method (CFEM). In order to promote the development of numerical simulation of interfacial crack growth, a new CZM, based on the wavelet finite element method (WFEM), is presented. Some fundamental issues regarding CZM of interface crack growth of double cantilever beam (DCB) testing were studied. The simulation results were compared with the experimental and simulation results of CFEM. It was found that the new CZM had higher accuracy and efficiency in the simulation of interface crack growth. At last, the impact of crack initiation length and elastic constants of material on interface crack growth was studied based on the new CZM. These results provided a basis for reasonable structure design of composite material in engineering.
1. Introduction
Interface in the composite materials is the common surface area of each connecting phase and it contributes to the transmission of mechanical property. Numerous studies have shown that failure often occurs along the interface; therefore, the fracture behavior of interface crack growth attracted considerable attention in recent years [1–4]. Numerical simulation is the main method for interface fracture analysis, and it is also the key to the development of interface fracture mechanics. At present, many scholars, both within the country and abroad, have carried out studies on the problem of interface crack growth, by using different numerical methods [5–9]. Although many numerical calculation methods for interface crack growth have already been reported, as indicated above, the number of papers that simulates interface crack growth based on WFEM is limited.
In most of the finite element applications of the CZM, it is natural that WFEM is a new numerical calculation method which has been gaining lot of interest in the last decade. It uses the scaling function or the wavelet function as interpolation function instead of the traditional polynomial, and its main characteristics are as follows: lower undetermined coefficient, higher approximation accuracy, strong localization performance, and multiresolution analysis. Regarding the problem of crack growth, in terms of not changing mesh dividing, the accuracy and efficiency of numerical calculation can be improved by increasing the node information, and it has better adaptive performance. Therefore, the method has an application prospect in the aspect of numerical simulation of crack growth. Zuo et al. [10] study static and free vibration problems of laminated composite plates adopting WFEM and higher-order plate theory, the effects of length-to-thickness ratios, layer numbers, and fiber orientations on the deflections and frequencies; the results showed that WFEM provides better results and that it is accurate and stable for free vibration analysis of laminated composite plates. Xiang et al. [11] constructed plane plate element using Hermite cubic spline wavelet for stress intensity factors (SIFs) evaluation in cracked plate structures; numerical simulation showed WFEM has better localization property. Chen et al. [12] proposed a second generation WFEM for diagnosing rotors with different crack location and size; the experimental results denote that the proposed method has higher identification precision. Chen et al. [13] proposed a novel numerical algorithm of crack fault prognosis based on WFEM and verified the effectiveness and accuracy of the proposed method through experiments. In short, WFEM has been successfully applied to various fields of engineering numerical analysis [14–19].
CZM is an effective approach for simulating fracture events. In recent years, CZM was studied and some excellent achievements have been obtained [20–23].
According to the current known researching achievement, the combination of WFEM with CZM for simulating interface crack growth has not been studied. In this paper, by combining the WFEM and cohesive model, the stiffness matrix of wavelet cohesive interface element was derived. The strain energy release rate (SERR) was calculated by using the virtual crack closure technique (VCCT) [24, 25], the process of crack growth was described by the nonlinear fracture criterion [26], and the experimental results and simulation results of CFEM were compared. Finally, the impact of initial length of interface crack and elastic modulus of material on interface crack growth was studied.
2. B-Spline Wavelet on the Interval
In order to avoid the numerical oscillation phenomenon of classical wavelet when solving boundary value problem, American scholars Chui and Quak [27] presented the B-spline wavelet on the interval (BSWI). The 0 scale th-order BSWI functions and wavelets were given by Goswami et al. [28]. For the needs of the following part, in this paper, the 0 scale 4th-order scaling functions and wavelets were given in the following:
The 0 scale 4th-order wavelet functions can be obtained by the following function:where is the corresponding coefficient to different , and it can be obtained by LUT [29].
In order to have at least one inner wavelet, the following condition must be satisfied:where and are the order and scale of BSWI, respectively.
Since the 0 scale th-order scaling and wavelet functions have been obtained, the corresponding scale th-order of BSWI () scaling functions and the corresponding wavelet functions can be evaluated by the following equations:where is the independent variable of scaling and wavelet functions on the interval . Therefore, the scaling functions can be obtained in vector form as follows:where are the scaling functions obtained from (4). The wavelet functions can also be obtained in vector form:where are the wavelet functions obtained from (5).
In order to construct the two-dimensional wavelet plane plate element, the two-dimensional BSWI scaling function is required. Through the tensor product, the two-dimensional scaling functions arewhere is the one row vector combined with the scaling functions for at the scale and is another row vector combined with the scale th-order scaling functions. is the Kronecker symbol.
The two-dimensional wavelet functions arewhere and are the vector forms of scaling functions and and are wavelet functions.
3. Wavelet Cohesive Zone Model
The CZM is a simplified model of the interface layer. The interface characteristics are described by the relationship between the interface traction and the relative displacement of upper and lower interface. The CZM is based on elastic-plastic fracture mechanics and is an effective approach for simulating fractures, which can potentially avoid the stress singularity at the crack tip in linear elastic fracture mechanics, and the crack interface stress and fracture energy can be therefore obtained. As the research and development of CZM evolves, two main CZMs have been proposed and have already been used for the simulating of ductile and two-phase material interface crack growth [30–32].
3.1. The Bilinear CZM
The relationship between normal cohesive traction and normal displacement jump can be expressed as
The normal critical fracture energy values are computed aswhere is the normal cohesive stiffness , is the maximum normal cohesive traction, is the normal displacement jump at the maximum normal cohesive traction, and is the normal displacement jump at the completion of debonding. is the interface damage parameter and is an irreversible quantity. The value of ranges from 0 to 1; when , the interface is completely out of bond and is defined aswhere is the normal displacement jump at the completion of debonding.
Figure 1 shows the relationship between interface normal cohesive traction and normal displacement jump . It clearly shows that, with the increase of normal displacement jump , when the interface layer is detached, the interface cohesive traction is increased, reaching a maximum value and then decreasing to zero in the end.
When the normal displacement , then . The interface has been completely detached and has lost the ability of load transferring; thus, the damage process of interface layer can be simulated by this curve.
3.2. Wavelet CZM Interface Element
With the aim to converge all difficulties in CZM numerical analyses, a method of combining of WFEM with CZM has been proposed, where wavelet CZM interface element is used to simulate the interface separation process. In this paper, as an example, two-dimensional wavelet interface elements were used to deduce the wavelet interface element stiffness matrix. By considering the scale th order of the BSWI scaling function as the interpolation function, an interface element was constructed, and its structure is presented in Figure 2. The element contains nodes, and each node has two degrees of freedom, which includes horizontal and vertical direction. In applications, the upper nodes of interface initially coincide with the lower nodes, which mean that the initial thickness of interface element is 0.
The definition of the wavelet interface element nodes displacement in the standard solution domain is as follows:
The continuous displacement field of upper and lower interface of wavelet interface element iswhere “+”, “−“ represent the upper and lower displacement of wavelet interface element, respectively.
When using the interval B-spline wavelet scaling function of the scale th order as an interpolation function, the displacement interpolation function iswhere is the tangential coordinates along with the wavelet interface element in the standard solution domain, the wavelet coefficient , and the wavelets scaling function can be obtained in vector form .
After matrix conversion, the displacement interpolation function represented by the nodal displacement array is where the converted matrix . The normal displacement field iswhere
The wavelet interface element stiffness matrix is where additionallyis the interface elasticity matrix; and are the tangential and normal cohesive traction, respectively; and are the tangential and normal displacement jump, respectively.
4. Numerical Examples
4.1. Interface Crack Growth Problem of DCB Testing
DCB model is made of a steel plate of length mm and width mm and crack length mm at the free end of DCB; the SERRs are J/m2, J/m2, and J/m2, and the properties of both DCB materials and interfaces are presented in Table 1.
By taking advantage of two BSWI43 wavelet plane plate elements to model the upper and lower solid beam, respectively, and one wavelet cohesive interface element to model the interface for the DCB, a wavelet finite element model of crack growth was established, as demonstrated in Figure 3. The process of interface crack growth was simulated by the nonlinear fracture criterion.
When material 1 and material 2 are the same material, Table 2 shows the relative error of the WFEM calculation results compared to the CFEM calculation results, and it indicated that the WFEM can obtain higher accuracy with less elements and nodes and that it is suitable to solve the problem of interface crack growth.
Figure 4 presents the changing trends of the interface reaction forces as a function of time, for different elastic moduli. As presented in Figure 4, for different elastic moduli, the interface reaction forces experienced a hardening process of gradually reaching their maximum value. The interface cracks were propagated, during the hardening process, which required the increase of external force of the interface, which, subsequently, resulted into crack growth.
In order to study the impact of elastic modulus ratio on interface crack growth, it is supposed that the interface strength is lower than that of material 1 and material 2. The interface crack was extended along the predefined interface, and the elastic modulus of material 1 was 135 MPa, while the elastic modulus of material 2 was altered, without changing other parameters. Regarding the wavelet interface element, the variation rule of stress at the crack tip was obtained in different elastic modulus ratio, as presented in Figure 5. It can be concluded that the stress at the crack tip increases with the increase of elastic modulus ratio; the higher the elastic modulus ratio is the faster the interface crack grows.
Figure 6 shows the effect of different initial crack length values on the interface crack growth. As can be observed from Figure 6, the longer the initial crack length of DCB, the more pronounced the cohesive traction of DCB will be. This indicates that the longer the initial crack length is, the easier the interface crack grows.
4.2. Interface Crack Growth Problem of Nonuniform Material DCB
As presented in Figure 7, the geometrical dimensions and materials parameters of nonuniform material DCB are beam length mm, beam height mm, the initial crack length mm, elastic modulus Pa, and Poisson ratio . The corresponding interface parameters are maximum normal traction MPa and normal separation displacement mm.
Table 3 presents the interface stress at the crack tip of DCB under different nonuniform material parameters . It can be observed by comparing the results of WFEM (484 DOFs) with CFEM (38400 DOFs) that WFEM and CFEM are in good agreement. Therefore, good computational accuracy and efficiency were further demonstrated. The relationship between the corresponding laws of interface stress as a function of time, under different parameters , is demonstrated in Figure 8. The curves show that the cohesive traction increases and the interface damage initiated when reaching the highest point. Then, it was successively decreased to an approximately constant value, which is consistent with interface damage process. Also, it was demonstrated that the interface damage initiated earlier, when the nonuniform parameter was at higher values; therefore, the uniform changing of material for DCB is beneficial for preventing crack growth.
5. Conclusions
The new CZM based on the WFEM was constructed and the corresponding wavelet interface element stiffness matrix was obtained by (17). The relative error of the WFEM, when 3 elements and 360 nodes were employed with the experimental mean, was 5.58%, while that of the CFEM, when 1540 elements and 5500 nodes were employed with the experimental mean, was 10.85%, which indicated that the WFEM can obtain higher accuracy with less elements and nodes and it is suitable to solve the problem of interface crack growth.
The elastic modulus has a great impact on interface crack growth. The higher the elastic modulus of the material, the lower the required energy consumption of interface crack growth, and the faster the crack growth.
The study on the impact of the elastic modulus ratio of DCB on interface crack growth shows that, with the increase of the elastic modulus ratio of DCB, the stress concentration in interface crack tip increases, and interface crack is easier to extend. Therefore, interface crack growth can be postponed by adjusting the elastic modulus ratio. Also, the material parameters for DCB changes uniformly, which can prevent interface cracks from growing.
The study results of interface crack growth based on the initial length of interface crack show that cohesive traction decreases with the increase of initial length of interface crack; therefore, the interface cracks get easier growth.
Abbreviations
| : | Scale |
| : | Order |
| : | Scaling function |
| : | Wavelet function |
| scale th order of BSWI | |
| : | Tangential coordinate, independent variable of wavelet scaling functions on the interval |
| : | Maximum normal cohesive traction, MPa |
| : | Normal displacement jump at maximum normal cohesive traction, m |
| : | Interface damage parameter |
| : | Normal displacement jump at the completion of debonding, m |
| : | Interface normal cohesive traction |
| : | Normal displacement jump, m |
| : | Scaling functions vector |
| : | Wavelet functions vector |
| : | Kronecker symbol |
| : | Interface element nodes displacement |
| : | Continuous displacement field of upper and lower interface |
| : | Converted matrix |
| : | Interface element stiffness matrix |
| : | Normal critical fracture energy |
| : | Tangential critical fracture energy. |
Competing Interests
The authors declare that they have no competing interests.
Acknowledgments
The work was financially supported by the National Natural Science Foundation of China (Grant no. 11502051), Major Project of Chinese Heilongjiang Province (Grant no. GA13A402), and Education Department Foundation of Chinese Heilongjiang Province (Grant no. 12541091).