Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 593640, 11 pages

http://dx.doi.org/10.1155/2013/593640

## The Characteristic Solutions to the V-Notch Plane Problem of Anisotropy and the Associated Finite Element Method

^{1}Department of Civil Engineering, China Agricultural University, Beijing 100083, China^{2}Department of Mechanics and Engineering Science, Peking University, Beijing 100871, China

Received 17 July 2013; Accepted 13 August 2013

Academic Editor: Song Cen

Copyright © 2013 Ge Tian et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper presents a novel way to calculate the characteristic solutions of the anisotropy V-notch plane problem. The material eigen equation of the anisotropy based on the Stroh theory and the boundary eigen equation of the V-notch plane problem are studied separately. A modified Müller method is utilized to calculate characteristic solutions of anisotropy V-notch plane problem, which are employed to formulate the analytical trial functions (ATF) in the associated finite element method. The numerical examples show that the proposed subregion accelerated Müller method is an efficient method to calculate the solutions of the equation involving the complex variables. The proposed element ATF-VN based on the analytical trial functions, which contain the characteristic solutions of the anisotropy V-notch problem, presents good performance in the benchmarks.

#### 1. Introduction

The characteristic solutions to the V-notch plane problem were studied by many researchers [1–12]. Fu et al. studied the finite element method based on the analytical trial functions of the V-notch plane problem of isotropy [2, 3]. Niu et al. studied the boundary element method based on the analytical solutions of the V-notch plane problem [4–6]. Ping et al. discussed the finite element of the V-notch plate problem [7].

This paper presents a novel way to calculate the characteristic solutions of the anisotropy V-notch plane problem and to study the finite element method associated with these solutions.

Firstly, the material eigen equation of the anisotropy based on the Stroh theory [1, 8–11] is studied, and the eigenvalues of the anisotropy are calculated.

Secondly, the calculated eigenvalues of material are employed to calculate the boundary eigenvalues of the V-notch plane problem.

Thirdly, the paper proposed a novel modified Müller method, named as the subregion accelerated müller (SRAM) method, which is utilized to calculate characteristic solutions of anisotropy V-notch plane problem.

At last, the calculated eigenvalues of the V-notch plane problem are employed to formulate the analytical trial functions (ATF) in the associated finite element method.

In the numerical examples, the proposed subregion accelerated Müller method [2, 3] is shown as an efficient method to calculate the solutions of the equation involving the complex variables. The proposed ATF-VN based on the analytical trial functions, which contain the characteristic solutions of the V-notch problem, presents good performance in the benchmarks.

Though many researchers studied the V-notch problem of isotropy material [1–16], there are seldom studies about the V-notch problem of anisotropy material. This paper presents a novel systematic strategy to calculate the V-notch problem of anisotropy material.

#### 2. The Material Characteristic Matrix of Anisotropic

The constitutive and equilibrium equations of anisotropy can be written as [1] where is displacement, is stress, and is the elastic tensor of the anisotropy, in which are denoted as coordinates of the three-dimensional problem.

In the plane problem of anisotropy, the displacements are assumed to be only associated with the coordinates and . According to Stroh theory [8, 9],

in which is the analytic function of , while . is the eigenvector about the eigenvalue of the material.

Substituting (3) into (2), we have

in which

The material coefficient matrixes of , **,** and are defined as , and , . They can also be denoted as

In order to obtain the nonzero solutions of (4), the determinant of the coefficient matrix must be zero:

The solutions of (7) are the eigenvalues of the material.

In the same way, the stresses, which satisfied equilibrium equations, can also be expressed by the characteristic solutions of stress functions , and we have where is the first derivative of .

According to (1), we have

The general solutions of displacement in (1) can be denoted as

in which .

The general solutions in (8) of stress functions can be denoted as

in which .

Introducing the normalization condition

the characteristic matrix of the material and can be determined.

Substituting (6) into (5), we can get

In the plane problem, we can define .

So matrix can be simplified as follows:

The characteristic matrix of the material can be written as The characteristic matrix can be calculated as

The constants , , and in (16) and (17) can be determined by (13).

It can be proved that according to the characteristic matrixes and of anisotropy defined in (16) and (17), we have

#### 3. The Boundary Characteristic Matrix of V-Notch

The plane polar coordinate system was defined in Figure 1, and the -axis divided the angle of the V-notch into two equal parts.

As showed in Figure 1, in the plane V-notch problem of anisotropic, the notch tip is defined as the origin, and there is an angle between the -axis and the material principle axis of . So the material matrix of anisotropic material can be redefined as

in which is the transformation matrix, and we have

In the polar coordinate system, can be expressed as

in which

represents the radial distance to tip.

The general stress solutions in (8) can be expressed by . According to Stroh’s theory, vector can be written as [10, 11]

in which eigenvalues and parameter are a complex constant, eigenvector is undetermined complex vector, are is reference length of the notch, and we have

Taking account the stress in the boundaries of the V-notch, where ,

Generally is zero in the stress-free boundaries of the V-notch.

Utilizing (18), is zero in the boundary .

Considering the condition of , in order to make equal to zero on the side , according to (24), we have

in which the boundary characteristic matrix is

In order to obtain the nonzero solutions of (27), the determinant of the coefficient matrix must be zero:

Equation (29) is the boundary characteristic equation. Its solution is complex; and in (11) and (12) are also complex. If is the root of the equation, is also the root of the equation.

#### 4. The Müller Method Accelerated by Subregion

The Müller method is an effective method in solving the zero point calculation. As showed in Figure 2, making a parabola which passes three points , , on the complex function , we have where , is the first and second difference of .

The root of the is

in which

In the next iteration step, is the new initial value.

To solve more than one root, there is an effective modification (RDM) to the function where are series of characteristic roots which have been calculated.

The Müller method is local convergence iteration method, if the root of equation is confirmed in the interval of and . There is an effective method to accelerate the convergence of the Müller method

It is called Shrink Boundary Method (SBM).

Combining (33) with (34), we have the subregion accelerated müller (SRAM) method, whose iteration function is

Figure 3 shows the values of and near the root, where the subregion of the iteration is and , , .

In Figure 3, to the values of the tangent near root, is larger than due to the SBM.

Table 1 shows the results of the eigenvalues to in four different Müller methods (Direct, SBM, RDM, and SRAM). The value in Table 1 presents the times of iteration ( means fail to converge). SRAM shows very good performance in the calculation of the eigenvalues. The SBM and the SRAM can reach the convergence results in every subregion, but the direct Müller method and the RDM fail to converge in some subregions. SRAM can get rid of the influence of the eigenvalues calculated before the step and converge faster than other methods in most cases.

#### 5. The Element ATF-VN Based on the Analytical Trial Functions

According to the subregion mixed energy principle, the total energy can be written as where is the complementary energy in the subregion- that was defined by the stress field, is the potential energy in the subregion- that was defined by the displacement field, and is the additional energy along the boundary between two subregions.

The potential energy can be denoted as

in which is the nodal displacement vector, is the equivalent nodal load vector, and is the stiffness matrix of the potential energy in subregion- (SRP).

The complementary energy can be denoted as

in which is the undetermined parameters of the stress, and where is elastic coefficient matrix, is thickness, and is the stress trial functions of the complementary energy subregion- (SRC), which is defined in (9).

The additional energy on the boundary between two kinds of subregions can be written as

in which is the boundary force determined in SRC: where direction cosine matrix can be written as

is the angle between the normal direction of and *-*axis.

In (41), is the displacement on the boundary, which is defined in SRP: where is the nodal displacements on the boundary between two subregions and is the shape function defined in SRP.

Equation (40) can also be expressed as

in which

The total energy can be denoted as

Using the stationary conditions of the total energy

the stiffness matrix of can be obtained as

The generalized element defined in (48) is named as ATF-VN.

#### 6. Numerical Examples

##### 6.1. Example 1

This example shows the calculation of the characteristic solutions of a V-notch slab, which is made of anisotropic material T700, and its layer angle is 45°.

The material parameters matrix of T700 is

The layer angle is 45°. According to (21) and (20)

According to (15), we can get the material eigenvalues from :

According to (16) and (17), we can get material eigen matrixes and

Substituting matrix into (28) and (29), with the application of the modified Müller method SRAM, the characteristic eigenvalues of different V-notch angles are showed in Table 2. and are the first and second eigenvalues, respectively.

##### 6.2. Example 2

This example analyses the characteristic solutions of a V-notch and studies the influence of the material parameters, the angle of in the V-notch, and the layer angle of .

Tables 3, 4, and 5 study the first and the second characteristic eigenvalues ( and ) of three kinds of material (), eight types of V-notch ( = 190°, 200°, 210°, 240°, 270°, 300°, 330°, 360°), and eight types of layer angle ( = 10°, 20°, 30°, 40°, 50°, 60°, 70°, 80°).

Figures 4, 5, 6, and 7 show the first and the second items of the eigenvalues with different and .

##### 6.3. Example 3

This example employs the proposed element ATF-VN to calculate the tip area of the V-notch, which is subjected to the antisymmetric load as shown in the Figure 8. , , , elastic constant is , Poisson’s ratio is , the load sum is , and the values of angle are 330° and 350°. Table 6 gives the eigenvalues of the V-notch problem. Table 7 gives the values of KII calculated by ATF-VN involving different number () of the items of the analytical trial functions.

#### 7. Conclusion

In this paper, the eigenvalues of anisotropic material in plane V-notch problem are analyzed. The material characteristic matrix of anisotropic and boundary characteristic equations of plane problems with notch is derived. The eigenvalues of the V-notch anisotropic plane problem are calculated by the SRAM method. Numerical examples show that the presented SRAM method has advantages of fast convergence and high accuracy and is easy to implement. The proposed element ATF-VN based on the analytical trial functions provides good performance in the calculation of the stress field near the tip of the V-notch.

#### Conflict of Interests

The authors do not have any conflict of interests with the content of the paper.

#### Acknowledgments

This project was supported by the National Natural Science Foundation of China (no. 11272340), the National Basic Research Programs of China (no. 2010CB731503).

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