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

This paper presents a novel way to calculate the characteristic solutions of the anisotropyV-notch plane problem.Thematerial eigen equation of the anisotropy based on the Stroh theory and the boundary eigen equation of the V-notch plane problem are studied separately. AmodifiedMüllermethod is utilized to calculate characteristic solutions of anisotropyV-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.


Introduction
The characteristic solutions to the V-notch plane problem were studied by many researchers [1][2][3][4][5][6][7][8][9][10][11][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][5][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][9][10][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 Vnotch 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.

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 , , , = 1, 2, 3 are denoted as coordinates ( = 1, 2, 3) of the three-dimensional problem.
In the plane problem of anisotropy, the displacements u ( = 1, 2, 3) are assumed to be only associated with the coordinates 1 and 2 . According to Stroh theory [8,9], in which is the analytic function of , while = 1 + 2 . a is the eigenvector about the eigenvalue of the material.
Substituting (3) into (2), we have in which The material coefficient matrixes of Q, R, and T are defined as = 1 1 , and = 1 2 , = 2 2 . They can also be denoted as .
In order to obtain the nonzero solutions of (4), the determinant of the coefficient matrix D must be zero: The solutions ( = 1, 2, 3) 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 A = [a 1 , a 2 , a 3 ].
The general solutions Φ = [ 1 , 2 , 3 ] in (8) of stress functions can be denoted as      Introducing the normalization condition .
In the plane problem, we can define 14  ] .
The characteristic matrix A of the material can be written as The .
It can be proved that according to the characteristic matrixes A and B of anisotropy defined in (16) and (17), we have

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 1 . So the material matrix of anisotropic material can be redefined as in which T is the transformation matrix, and we have . (21) 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 f can be written as [10,11] in which eigenvalues and parameter are a complex constant, eigenvector q is undetermined complex vector, arê is reference length of the notch, and we have Taking account the stress t in the boundaries of the Vnotch, where = ± , Generally t is zero in the stress-free boundaries of the Vnotch.
Utilizing (18), t is zero in the boundary = .
Considering the condition of ( /̂) −1 ̸ = 0, in order to make t equal to zero on the side = − , according to (24), we have Mathematical In order to obtain the nonzero solutions of (27), the determinant of the coefficient matrix C must be zero: Equation (29) is the boundary characteristic equation. Its solution is complex; u and in (11) and (12) are also complex. If is the root of the equation, is also the root of the equation.

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 , −1 , −2 on the complex function ( ), we have is the first and second difference of ( ).

The root of the ( ) = 0 is
in which In the next iteration step, +1 = * is the new initial value.
To solve more than one root, there is an effective modification (RDM) to the function ( ) where 0 1 , 0 2 , . . . , 0 are series of characteristic roots which have been calculated.
The Müller method is local convergence iteration method, if the root of equation ( ) = 0 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 * ( ) = ( ) .
(35) Figure 3 shows the values of ( ) and * ( ) near the root, where the subregion of the iteration is = 0.6 and = 0.9, ( ) = ⋅ sin( ) − sin( ⋅ ), = 300 ∘ . 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.

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 Π is the complementary energy in the subregionthat was defined by the stress field, Π 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 Π can be denoted as in which {w} is the nodal displacement vector, {P} is the equivalent nodal load vector, and [K O ] is the stiffness matrix of the potential energy in subregion-(SRP). The complementary energy Π can be denoted as  in which { } is the undetermined parameters of the stress, and where [D] is elastic coefficient matrix, is thickness, and [S] is the stress trial functions of the complementary energy subregion-(SRC), which is defined in (9). The additional energy on the boundary Γ between two kinds of subregions can be written as in which {T} is the boundary force determined in SRC:  Figure 5: The second eigenvalue while 11 / 22 = 8.
is the angle between the normal direction of Γ and -axis.
In (41), {u} is the displacement on the boundary, which is defined in SRP: where {w} is the nodal displacements on the boundary between two subregions and {N} is the shape function defined in SRP. Equation (40) can also be expressed as in which The total energy Π can be denoted as Using the stationary conditions of the total energy Π Π = 0 (47) the stiffness matrix of {w} can be obtained as The generalized element defined in (48) is named as ATF-VN.

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 layer angle is 45 ∘ . According to (21) According to (15), we can get the material eigenvalues from det D = 0: According to (16) and (17)    Substituting matrix B into (28) and (29), with the application of the modified Müller method SRAM, the characteristic eigenvalues of different V-notch angles 2 are showed in Table 2. 1 and 2 are the first and second eigenvalues, respectively.
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. = 1, = 3, ℎ = 1, elastic constant is = 0.21×10 7 , Poisson's ratio is = 0.3, the load sum is = 1, and the values of angle 2 are 330 ∘ and 350 ∘ . Table 6 gives the eigenvalues of the Vnotch problem. Table 7 gives the values of KII calculated by ATF-VN involving different number ( ) of the items of the analytical trial functions.

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.