Exact extraction of stress intensity factors via enriched numerical manifold method with composite patches

Abstract

Stress intensity factors (SIFs) determine whether cracks propagate and play vital role in the simulation of crack propagation. Therefore, the computation of SIFs is an important prerequisite for analysis of crack evolution via numerical methods. Nowadays, J-integral method widely is employed to obtain SIFs. However, it has a significant error in calculation of SIFs when the crack tips are close to the geometrical boundaries or the two crack tips are close to each other. Therefore, Numerical Manifold Method with Composite patches is proposed to obtain exact the value of SIFs. Then numerical examples are used to verify the proposed method. The compare of results from the proposed method and J-integral method demonstrate numerical manifold method with Composite patches can exactly extract SIFs.

Introduction

Material fracture widely used in various industrial applications, including chip manufacturing, petroleum and mineral resource extraction^1,2. It is a difficult and hot issue for scholars around the world. Numerical computation has become one of vital methods to study material fracture^3,4, including finite element method (FEM), discrete element method (DEM) and Numerical manifold method (NMM)⁵. The current numerical methods usually uses fracture parameters to obtain the stress field around crack tips⁴, then solve fracture parameters such as stress intensity factors⁶, energy release rate⁷ and maximum circumferential tension stress, and eventually compares fracture parameters and fracture criterion to determine whether crack propagate. It can be seen that accurately computation of fracture parameters is an important prerequisite for analysis of crack evolution via numerical methods. Among fracture parameters, SIFs is the most widely known and widely used parameter for engineers ⁸.

SIFs can be used to determine whether a crack has propagated. When the SIFs is greater than or equal to the critical SIFs, the crack will propagation. Otherwise, the crack remains in a stable state. Therefore, the exact computation of SIFs plays a vital role in determination of crack propagation⁹. Nowadays, there are two categories to obtain SIFs. The solution method of SIFs in the form of analytic solution is obtained through theoretical derivation. The closed solution is difficultly obtained for complex cracks, but is suitable for simple regular cracks^10,11. The other approach to obtain SIFs is the interaction integral method based on the J-integral. And path-independent J-integral method has been widely utilized the computations of SIFs^12,13. Two-dimensional problems were considered. The J-integral can be applied to elastic–plastic materials, if deformation theory of plasticity (i.e. non-linear elasticity) is assumed^14,15. In addition, a surface integral method was proposed and applied to three-dimensional problems¹⁶. As discussed later, the domain integral method is considered to be the most popular technique to evaluate the crack parameters in three-dimensional fracture mechanics problems¹⁷.

In the last two decades, a series of combinations of J-integral and numerical methods are proposed. Hamdi employed FEM to determine numerically J-integral values. These numerical values were compared with corresponding J-integral values using Single Edge Notched in Tension and Pure Shear specimens¹⁸. On the other hand, one of trends for the computation of SIFs is the application of artificial neural network^19,20. Mortazavi and Ince successfully applied an artificial neural network to obtain elastic–plastic J-integral values. The proposed method can shorten the computational cost²¹.

In addition, since NMM was proposed, a combinations of J-integral and NMM was applied to computation of SIFs^22,23. Zhang et al. extract SIFs via NMM²⁴.

Although J-integral method is widely used, it has a significant error in calculation of SIFs when the crack tip is close to the boundary or the two crack tips are close to each other. Therefore, this paper proposes a enriched numerical manifold method with composite patches, which can accurately extract SIFs and make up for the deficiency of J-integral method.

The basic of NMM

Mathematical cover and physical cover

In order to deal with discontinue problems in engineering, Shi proposed NMM. It does not introduce enriched functions such as XFEM, nor does it need element remesh such as FEM. But NMM deal with continuous and discontinuous problem via cutting instead of enriched functions. Figure 1 shows a trapezoidal problem domain, denoted as , surrounded by a thick solid line. In there is a discontinuous interface, denoted as .Triangular grids are used to cover the entire problem domain, and triangles with the same node are denoted as a mathematical patch. In Fig. 1M₁, M₂, M₃ are mathematical patch comprised of six triangle. In order to describe discontinuous interface, All mathematical patches are cut into physical patch one by one via interfaces such as crack and boundary. According to the spatial relationship between mathematical patch and interface, there are three situations for mathematical patch cutting. First, mathematical patch is cut by boundaries. M₂, are partitioned into two parts, denoted as P_2-1, P_2-2, by boundary. As shown in Fig. 2P_2-2 is within and is a physical patch, while P_2-1 is not considered. Second, mathematical patch is cut by cracks. M₃, are partitioned into two parts, denoted as P_3-1, P_3-2, by boundary. As shown in Fig. 2, both of them are within and are physical patch. Third, mathematical patch is not cut by any interfaces. M₁, are not cut by any interfaces and only produce a physical patch, denotes as P_1-1. By analogy, all mathematical patches and interfaces are subjected to Boolean operations to obtain physical patches, and all physical patches constitute the physical cover of the problem domain, as shown in Fig. 2. At this point, the physical cover coincides with the problem domain.

Fig. 1.

Mathematical cover and mathematical patches.

Fig. 2.

Physical cover and physical patches.

The composite patches

To exactly obtain the singularity of crack tip, the composite physical patches are proposed. It is make up of the physical patches where cover crack tips. As shown in Fig. 3, there are two crack tips, and . The crack tip is covered by three physical patches, P₄, P₅ and P₆. That is P₄, P₅ and P₆ form a composite physical patch, which is covered with red dotted lines. The crack tip is covered by three physical patches P₅, P₉ and P₁₀. That is P₄, P₅ and P₆ form a composite physical patch, surrounded by a red solid line.

Fig. 3.

The composite patches.

Weigh functions for physical patches and composite patches

For simplicity and clarity, renumber all physical patches sequentially, denoted as P_i, i = 1,2…m. m is the number of physical patches. Every physical patch has its own weight function, denoted as . And it need to satisfy the following conditions:

i.

ii.

iii.

here, z is an arbitrary point located in .

The weigh function of physical patches is shown in Fig. 4a. The weigh function for composite physical patches are shown in Fig. 4b. Every composite physical patch also has its own weight function. However, they are is significantly different. The shape of weight function of the physical patch likes a hat, which is also called the hat function, and its value is only 1 at the star point, while it of the composite physical patch likes a wheel table, which is called the platform function, that is, in the middle of the triangle area, its value is 1. It satisfies three conditions of weigh function.

Fig. 4.

Weigh functions for physical patches and composite patches. (a) weigh function for physical patches, (b) weigh function for composite patches.

Local approximation and global approximation

Physical patches and composite patches has own local approximation. In order to characterize the singularity, different local approximations are chosen for physical patches and composite physical patches, separately. On one hand, every physical patch has a local approximation function . can be either a polynomial function or other function that can reflect the local property of the system. Here B = {1} is chosen as local approximation for physical patch, namely

1

,meaning degree of freedom (DOF) vector of the displacement field of the physical patch.

Based on the concept of partition of unity, for an arbitrary point z in the ,

On the other hand, to characterize the singularity, different local approximations are chosen for composite physical patches. The enriched function are introduced as a enriched approximation for the composite physical patches. is denoted as enriched function, namely

2

here ,, , . r, θ represent the local polar coordinates established on the crack tip, as shown in Fig. 5.

Fig. 5.

Local polar coordinates established on the crack tip.

The enriched approximation of composite physical patch is

3

Here , , meaning DOF vector of the displacement field of the physical patch.

There are two types of cover of any point, one situation is only covered by physical patches, the other situation is covered by physical pieces and composite physical patch. Therefore, the global approximation can be represented by

4

n is the number of physical patch that can cover the point z. Here n = 3. That is

5

in Eq. (6) represents rigid body displacement. The other term,, describe the asymptotic displacement field at the crack tip.

The computation of SIFS via NMM with composite physical patches

According to Eq. (6), the asymptotic displacement field at the crack tip is obtained by the following equation²⁵:

6

are the enhanced displacement field of the problem domain, which are obtained by the enriched numerical manifold method based on external loads and boundary conditions.

In addition, taking a mode I and II crack as an example, according to the classical elastic theory²⁵, the asymptotic displacement field of the crack tip is as follows

7

8

here is passion ratio, .The relationships between them and I and II stress intensity factors, ,, are as follows:

9

The numerical examples

An uniformly tensioned infinite plate with a central crack

There is a central crack with a length of 2a in the infinite plate, and the angle between the crack and the horizontal direction is α, as shown in Fig. 6. The both ends of plate are subjected to uniform tensile stress, σ = 100Kpa. The material parameters are chosen as following: E = 210 MPa, passion’s ratio v = 0.26. The size of grid is 0.04a. And SIFs for various α are investigated.

Fig. 6.

Boundary conditions of an infinite plate with a central crack.

Table 1 shows the SIFs obtained from the proposed method and J-integral method. To obviously and clearly see the differences of from different methods, the results in Table 1 are employed to obtain Fig. 7. From Fig. 7 it can be found that both SIFs agree with the closed solution. But there are little errors. It can be seen that the results obtained by ENMM with composite patches can be used to the computation of SIFs. In addition, the solutions from the proposed method agree with the results from J-integral method and the analytical solutions²⁶. Although in this example the differences between the proposed method and J-integral method are far from obvious, the following examples can obtain the better conclusion.

Table 1.

Non dimensional SIFSs from different computational methods.

α	0°	15°	22.5°	45°	67.5°	90°
J-Integral	0	0.0901	0.1896	0.8001	1.5081	1.7942
ENMM	0	0.1106	0.2179	0.8228	1.4668	1.8224
Exact	0	0.1133	0.2199	0.8236	1.4680	1.8160

Fig. 7.

The SIFs from J-integral method and the proposed method.

An uniformly tensioned infinite plate with a boundary crack

Figure 8 shows a length of 2 h infinite plate, subjected to uniform tensional stress σ on both ends of the plate. And there is a length of a crack closer to the boundary of plate. The computational parameters are chose as following: E = 210 kPa, passion’s ratio v = 0.26, σ = 200Kpa. Two types of grid size , = 0.02 h. and = 0.06 h., were considered. And SIFs for various of a are investigated. Figure 9 shows the SIFs obtained from the proposed method and J-integral method using different grid size.

Fig. 8.

Boundary conditions of an infinite plate with a central crack.

Fig. 9.

The SIFs from J-integral method and the proposed method for different grid size. (a) =0.06 h, (b) =0.02 h.

From the Fig. 9a, when a/b is between 0.25 and 0.75, the results from J-integral method agree well with the closed solutions²⁶. However, when a/b approaches 0 or 1, the result of the J-integral deviates significantly from the exact value, and the error becomes larger and larger. As can be seen from Fig. 9, when a/b approaches 0, the crack is very short and the crack tip is closer to the bottom edge. When a/b approaches 1, the crack tip is closer to the top edge. Compared with Fig. 9b, even if the grid is refined, the error generated by the J-integral in calculating the SIF of the boundary crack will neither disappear nor decrease. It can be seen that the error of the J-integral in calculating the SIF of boundary cracks has no correlation with the mesh size. In other words, when the crack tip is close to the geometric boundaries, the error of the J-integral is large. This is because when the crack is very close to the geometric boundary, it is difficult to obtain a large enough integral region. This is the root cause of the large error in the J-integral calculation of boundary cracks.

In addition, only when the annular integration area is large enough, is the SIFs obtained by the J-integral method highly accurate. When a/b is between 0.25 and 0.75, the results from J-integral method agree well with the exact values. This is why the results of J-integral agree well with the exact values when a/b is between 0.25 and 0.75.

As you can see from the Fig. 9a, the results of the proposed method match the closed solutions well, regardless of what a/b is. In other words, even when the crack tip is close to the boundary, the proposed method can obtain high precision calculation results. Compared with J-integral method, however, the proposed method does not have its deficiency. And the proposed methods can make up for the deficiency of J-integral method well.

In addition, although the results of the proposed method match the closed solutions well, there are little errors. This is because of the error of the numerical method itself. As the grid gets smaller, the errors of the proposed method gets smaller and smaller.

An uniformly tensioned infinite plate with merging cracks

The other numerical example are investigated for demonstration the advantages of proposed method in treating merging cracks. Figure 10 shows a rectangle plate subjected uniform tension at its two ends. There is a pair of collinear cracks at half height of the plate. The length of the two cracks is a and the distance between them is 2(b-a). And SIFs obtained from J-integral method and the proposed method for various of a/b are investigated. The computational parameters are chosen as following: E = 210 kPa, passion’s ratio v = 0.26. The size of grid is 0.04 h.

Fig. 10.

Boundary conditions of an infinite plate with merging cracks.

When a/b is less than 0.7, the results of the J-integral method agree well with the closed solution²⁷, but when a/b is greater than 0.7, the error is obvious, and as a/b approaches 1, the error tends to be amplified. Combined with Fig. 10, it can be seen that when a/b is less than 0.7, the distance of two crack tips are far, and as a/b approaches 1, the crack tips are closer and closer until the two cracks converge. It can be seen that the error of J-integral is large when calculating the merging crack. This may be because the stress field near the crack tip is more singular when the two crack tips are close together. This results in a large error in the calculated stress. And the J-integral method uses the stress near the crack tip to obtain SIFs.

However, the proposed method does not have the above disadvantages. As you can see from Fig. 11, the proposed method can accurately extract SIFs regardless of the a/b. In other words, when the two crack tips approach each other, the proposed method is able to obtain an accurate SIFs, which can make up for the deficiency of J-integration method.

Fig. 11.

The SIFs from J-integral method and the proposed method.

Conclusions

It can be seen that accurate calculation of SIFs is the key to numerical simulation of crack propagation. Although the J-integral method is the main method to obtain SIFs, there are shortages.

1. The SIFs error obtained by the J-integral method in the calculation of edge cracks and merging cracks is large. On one hand, because it is difficult to obtain a large enough integral region due to the influence of the boundary, thus affecting the accuracy of the J-integral method for boundary crack. On the other hand, For merging method, J-integral method is difficult to calculate SIFs precisely, which may be because the stress field singularity increases when the two crack tips are close to each other, resulting in the stress field calculation is not accurate enough.

2. The proposed method can obtain SIFs with higher precision and wider application, especially in the case of boundary cracks and merging cracks, and it make up for the shortages of J-integral.

Author contributions

Liang Yang and Manying Wu wrote the main manuscript text and Yan Bi prepared programs. All authors reviewed the manuscript.

Data availability

The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.