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

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] 𝜎_𝑖𝑗=𝐶_{𝑖𝑗𝑘𝑙}𝑢_𝑘,𝑙,𝜎_{𝑖𝑗,𝑗}=𝐶_{𝑖𝑗𝑘𝑙}𝑢_{𝑘,𝑙𝑗}=0, (1)(2) where 𝑢_𝑖 is displacement, 𝜎_𝑖𝑗 is stress, and 𝐶_{𝑖𝑗𝑘𝑙} is the elastic tensor of the anisotropy, in which 𝑖,𝑗,𝑘,𝑙=1,2,3 are denoted as coordinates 𝑥_𝑖(𝑖=1,2,3) of the three-dimensional problem.

In the plane problem of anisotropy, the displacements 𝐮_𝑘(𝑘=1,2,3) are assumed to be only associated with the coordinates 𝑥₁ and 𝑥₂. According to Stroh theory [8, 9], 𝐮_𝑘=𝐚_𝑘𝑔𝑧_𝑘, (3)

in which 𝑔 is the analytic function of 𝑧_𝑘, while 𝑧_𝑘=𝑥₁+𝑝_𝑘𝑥₂. 𝐚_𝑘 is the eigenvector about the eigenvalue 𝑝_𝑘 of the material.

Substituting (3) into (2), we have 𝐐+𝐑+𝐑^𝑇𝑝_𝑘+𝐓𝑝²_𝑘𝐚_𝑘𝑔^′′𝑧_𝑘=𝐃𝐚_𝑘𝑔^′′𝑧_𝑘=0, (4)

in which 𝐃=𝐐+𝐑+𝐑^𝑇𝑝_𝑘+𝐓𝑝²_𝑘. (5)

The material coefficient matrixes of 𝐐, 𝐑, and 𝐓 are defined as 𝑄_𝑖𝑘=𝐶_𝑖1𝑘1, and 𝑅_𝑖𝑘=𝐶_𝑖1𝑘2, 𝑇_𝑖𝑘=𝐶_𝑖2𝑘2. They can also be denoted as 𝐶𝐐=¹¹𝐶₁₆𝐶₁₅𝐶₆₁𝐶₆₆𝐶₆₅𝐶₅₁𝐶₅₆𝐶₅₅,𝐶𝐑=¹⁶𝐶₁₂𝐶₁₄𝐶₆₆𝐶₆₂𝐶₆₄𝐶₅₆𝐶₅₂𝐶₅₄,𝐶𝐓=⁶⁶𝐶₆₂𝐶₆₄𝐶₂₆𝐶₂₂𝐶₂₄𝐶₄₆𝐶₄₂𝐶₄₄. (6)

In order to obtain the nonzero solutions of (4), the determinant of the coefficient matrix 𝐃 must be zero: det𝐃=0. (7)

The solutions 𝑝_𝑘(𝑘=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 𝜙_𝑘, and we have 𝜙_𝑘=𝐛_𝑘𝑓𝑧_𝑘(),𝜎𝑘=1,2,3_𝑖1=−𝜕𝜙_𝑖,1,𝜎_𝑖2=𝜕𝜙_𝑖,2, (8)(9) where 𝑓 is the first derivative of 𝑔.

According to (1), we have 𝐛_𝑘=𝐑^𝑇+𝑝_𝑘𝐓𝐚_𝑘1=−𝑝_𝑘𝐐+𝑝_𝑘𝐑𝐚_𝑘. (10)

The general solutions of displacement in (1) can be denoted as 𝐮=𝐀⋅𝐠, (11)

in which 𝐀=[𝐚₁,𝐚₂,𝐚₃].

The general solutions Φ=[𝜙₁,𝜙₂,𝜙₃]^𝑇 in (8) of stress functions can be denoted as Φ=𝐁⋅𝐟, (12)

in which 𝐁=[𝐛₁,𝐛₂,𝐛₃].

Introducing the normalization condition 2𝐚^𝑇_𝑘𝐛_𝑘=1, (13)

the characteristic matrix of the material 𝐀=[𝐚₁,𝐚₂,𝐚₃] and 𝐁=[𝐛₁,𝐛₂,𝐛₃] can be determined.

Substituting (6) into (5), we can get𝐶𝐃=¹¹+2𝑝𝐶₁₆+𝑝²𝐶₆₆𝐶₁₆+𝑝𝐶₆₆+𝐶₁₂+𝑝²𝐶₆₂𝐶₁₅+𝑝𝐶₆₅+𝐶₁₄+𝑝²𝐶₆₄𝐶₁₆+𝑝𝐶₆₆+𝐶₁₂+𝑝²𝐶₆₂𝐶₆₆+2𝑝𝐶₂₆+𝑝²𝐶₂₂𝐶₅₆+𝑝𝐶₅₂+𝐶₄₆+𝑝²𝐶₂₄𝐶₁₅+𝑝𝐶₆₅+𝐶₁₄+𝑝²𝐶₆₄𝐶₅₆+𝑝𝐶₅₂+𝐶₄₆+𝑝²𝐶₂₄𝐶₅₅+2𝑝𝐶₅₄+𝑝²𝐶₄₄. (14)

In the plane problem, we can define 𝐶₁₄=𝐶₁₅=𝐶₂₄=𝐶₂₅=𝐶₆₄=𝐶₆₅=0.

So matrix 𝐃 can be simplified as follows: 𝐶𝐃=¹¹+2𝑝𝐶₁₆+𝑝²𝐶₆₆𝐶₁₆+𝑝𝐶₆₆+𝐶₁₂+𝑝²𝐶₆₂0𝐶₁₆+𝑝𝐶₆₆+𝐶₁₂+𝑝²𝐶₆₂𝐶₆₆+2𝑝𝐶₂₆+𝑝²𝐶₂₂000𝐶₅₅+𝑝²𝐶₄₄. (15)

The characteristic matrix 𝐀 of the material can be written as 𝑘𝐀=¹𝐶₆₆+2𝑝₁𝐶₂₆+𝑝²₁𝐶₂₂𝑘₂𝐶₆₆+2𝑝₂𝐶₂₆+𝑝²₂𝐶₂₂0−𝑘₁𝐶₁₆+𝑝₁𝐶₆₆+𝐶₁₂+𝑝²₁𝐶₂₆−𝑘₂𝐶₁₆+𝑝₂𝐶₆₆+𝐶₁₂+𝑝²₂𝐶₂₆000𝑘₃. (16) The characteristic matrix 𝐁=[𝐛₁,𝐛₂,𝐛₃] can be calculated as 𝐛_𝟏=𝑘₁𝑝₁𝐶₁₆𝐶₂₆+𝑝₁𝐶₁₆𝐶₂₂+𝑝²₁𝐶₆₆𝐶₂₂−𝐶₆₆𝐶₁₂−𝑝₁𝐶₂₆𝐶₁₂−𝑝²₁𝐶²₂₆𝑘₁𝐶₆₆𝐶₁₂+𝑝₁𝐶₂₆𝐶₁₂+𝑝²₁𝐶²₂₆−𝐶₁₆𝐶₂₆−𝑝₁𝐶₁₆𝐶₂₂−𝑝²₁𝐶₆₆𝐶₂₂0,𝐛₂=𝑘₂𝑝₂𝐶₁₆𝐶₂₆+𝑝₂𝐶₁₆𝐶₂₂+𝑝²₂𝐶₆₆𝐶₂₂−𝐶₆₆𝐶₁₂−𝑝₂𝐶₂₆𝐶₁₂−𝑝²₂𝐶²₂₆𝑘₂𝐶₆₆𝐶₁₂+𝑝₂𝐶₂₆𝐶₁₂+𝑝²₂𝐶²₂₆−𝐶₁₆𝐶₂₆−𝑝₂𝐶₁₆𝐶₂₂−𝑝²₂𝐶₆₆𝐶₂₂0,𝐛₃=00𝑘₃𝑝₃𝐶₄₄. (17)

The constants 𝑘₁, 𝑘₂, and 𝑘₃ in (16) and (17) can be determined by (13).

It can be proved that according to the characteristic matrixes 𝐀 and 𝐁 of anisotropy defined in (16) and (17), we have 𝐀𝐀^𝑇+𝐀𝐀^𝑇=𝟎=𝐁𝐁^𝑇+𝐁𝐁^𝑇,𝐀^𝑇𝐁+𝐁^𝑇𝐀=𝐈=𝐀^𝑇𝐁+𝐁^𝑇𝐀. (18)(19)

3. The Boundary Characteristic Matrix of V-Notch

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

Figure 1 

The local coordinators.

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 𝜃 between the 𝑟-axis and the material principle axis of 𝑥₁. So the material matrix of anisotropic material can be redefined as 𝐂^′=𝐓𝐂𝐓^𝑇, (20)

in which 𝐓 is the transformation matrix, and we have 𝐓=cos²𝜃sin²𝜃−2sin𝜃cos𝜃sin²𝜃cos²𝜃2sin𝜃cos𝜃sin𝜃cos𝜃−sin𝜃cos𝜃cos²𝜃−sin²𝜃. (21)

In the polar coordinate system, 𝑧_𝑘 can be expressed as 𝑧_𝑘=𝑟𝜁_𝑘(𝜃) (22)

in which 𝜁_𝑘(𝜃)=cos(𝜃)+𝑝_𝑘sin(𝜃). (23)

𝑟 represents the radial distance to tip.

The general stress solutions 𝑓(𝑧_𝑘) in (8) can be expressed by 𝑓(𝑟,𝜁_𝑘(𝜃)). According to Stroh’s theory, vector 𝐟 can be written as [10, 11] 𝑐𝐟=2𝑟𝑟̂^𝜆Λ(𝜃,𝛼,𝜆)𝐪+𝑐2𝑟𝑟̂^𝜆Λ𝜃,𝛼,𝜆𝐪, (24)

in which eigenvalues 𝜆 and parameter 𝑐 are a complex constant, eigenvector 𝐪 is undetermined complex vector, are 𝑟̂ is reference length of the notch, and we have Λ()𝜁𝜃,𝛼,𝜆=diag¹(𝜃)𝜁₁(𝛼)^𝜆,𝜁₂(𝜃)𝜁₂(𝛼)^𝜆,𝜁₃(𝜃)𝜁₃(𝛼)^𝜆𝐁^𝑇. (25)

Taking account the stress 𝐭_𝑟 in the boundaries of the V-notch, where 𝜃=±𝛼, 𝐭_𝑟=−𝜕Φ.𝜕𝑟 (26)

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

Utilizing (18), 𝐭_𝑟 is zero in the boundary 𝜃=𝛼.

Considering the condition of (𝑟/𝑟)̂^𝜆−1≠0, in order to make 𝐭_𝑟 equal to zero on the side 𝜃=−𝛼, according to (24), we have 𝑐𝜆𝐁Λ(−𝛼,𝛼,𝜆)+𝐁Λ−𝛼,𝛼,𝜆𝐪=𝑐𝜆𝐂𝐪=𝟎, (27)

in which the boundary characteristic matrix 𝐂 is 𝐂=𝐁Λ(−𝛼,𝛼,𝜆)+𝐁Λ−𝛼,𝛼,𝜆. (28)

In order to obtain the nonzero solutions of (27), the determinant of the coefficient matrix 𝐂 must be zero: Δ=|𝐂|=0. (29)

Equation (29) is the boundary characteristic equation. Its solution 𝜆 is complex; 𝐮 and 𝜙 in (11) and (12) are also complex. If 𝜆 is the root of the equation, 𝜆 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 𝑔(𝑧) which passes three points 𝑧_𝑖, 𝑧_𝑖−1, 𝑧_𝑖−2 on the complex function 𝑓(𝑧), we have 𝑔(𝑧)=𝑓𝑧_𝑖+𝑧−𝑧_𝑖𝑎_𝑖+𝑓𝑧_𝑖,𝑧_𝑖−1,𝑧_𝑖−2𝑧−𝑧_𝑖², (30) where 𝑓[𝑧_𝑖,𝑧_𝑖−1], 𝑓[𝑧_𝑖,𝑧_𝑖−1,𝑧_𝑖−2] is the first and second difference of 𝑓(𝑧).

Figure 2 

The schematic of Müller method.

The root of the 𝑔(𝑧)=0 is 𝑧^∗=𝑧_𝑖+−𝑎_𝑖±𝑎²_𝑖−4𝑓𝑧_𝑖𝑓𝑧_𝑖,𝑧_𝑖−1,𝑧_𝑖−22𝑓𝑧_𝑖,𝑧_𝑖−1,𝑧_𝑖−2, (31)

in which 𝑎_𝑖=𝑓𝑧_𝑖,𝑧_𝑖−1+𝑓𝑧_𝑖,𝑧_𝑖−1,𝑧_𝑖−2𝑧_𝑖−𝑧_𝑖−1. (32)

In the next iteration step, 𝑧_𝑖+1=𝑧^∗ is the new initial value.

To solve more than one root, there is an effective modification (RDM) to the function 𝑓(𝑧)𝑓_𝑛(𝑧)=𝑓(𝑧)𝑧−𝑧⁰₁𝑧−𝑧⁰₂⋯𝑧−𝑧⁰_𝑛, (33) where 𝑧⁰₁,𝑧⁰₂,…,𝑧⁰_𝑛 are series of characteristic roots which have been calculated.

The Müller method is local convergence iteration method, if the root of equation 𝑓(𝑧)=0 is confirmed in the interval of 𝑎 and 𝑏. There is an effective method to accelerate the convergence of the Müller method 𝑓^∗(𝑧)=𝑓(𝑧)().𝑧−𝑎)(𝑏−𝑧 (34)

It is called Shrink Boundary Method (SBM).

Combining (33) with (34), we have the subregion accelerated müller (SRAM) method, whose iteration function is 𝑓^∗_𝑛(𝑧)=𝑓(𝑧)(𝑧−𝑎)(𝑏−𝑧)𝑧−𝑧⁰₁⋯𝑧−𝑧⁰_𝑛. (35)

Figure 3 shows the values of 𝑓(𝑧) and 𝑓^∗(𝑧) near the root, where the subregion of the iteration is 𝑎=0.6 and 𝑏=0.9, 𝑓(𝑧)=𝑧⋅sin(𝐴)−sin(𝑧⋅𝐴), 𝐴=300^∘.

Figure 3 

Values of 𝑓(𝑧) and 𝑓∗(𝑧) near the root.

In Figure 3, to the values of the tangent near root, 𝑓^∗(𝑧) is larger than 𝑓(𝑧) due to the SBM.

Table 1 shows the results of the eigenvalues to 𝑓(𝑧) in four different Müller methods (Direct, SBM, RDM, and SRAM). The value 𝑛 in Table 1 presents the times of iteration (𝐹 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.

Table 1 

The comparison of the convergence properties of several iterative methods.

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 Π=Π_𝑃−Π_𝐶+𝐻_𝑃𝐶, (36) where Π_𝐶 is the complementary energy in the subregion-𝐶 that was defined by the stress field, Π_𝑃 is the potential energy in the subregion-𝑃 that was defined by the displacement field, and 𝐻_𝑃𝐶 is the additional energy along the boundary between two subregions.

The potential energy Π_𝑃 can be denoted as Π_𝑃=12{𝐰}^𝑇𝐊_𝐎{𝐰}−{𝐰}^𝑇{𝐏} (37)

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-𝑃 (SRP).

The complementary energy Π_𝑐 can be denoted as Π_𝑐=12{𝜷}^𝑇[𝐕]{𝜷} (38)

in which {𝜷} is the undetermined parameters of the stress, and [𝐕]=_Ω𝐶[𝐒]^𝑇[𝐃]⁻¹[𝐒]𝑡𝑑𝐴, (39) where [𝐃] is elastic coefficient matrix, 𝑡 is thickness, and [𝐒] is the stress trial functions of the complementary energy subregion-𝐶 (SRC), which is defined in (9).

The additional energy 𝐻_𝑃𝐶 on the boundary Γ between two kinds of subregions can be written as 𝐻_𝑃𝐶=_Γ{𝐓}^𝑇{𝐮}𝑡𝑑𝑠 (40)

in which {𝐓} is the boundary force determined in SRC: {𝐓}=[𝐋]{𝝈}=[𝐋][𝐒]{𝜷}, (41) where direction cosine matrix [𝐋] can be written as [𝐋]=cos𝜃cos𝛾sin𝜃sin𝛾−sin𝜃cos𝛾+cos𝜃sin𝛾sin𝜃cos𝛾−cos𝜃sin𝛾cos𝜃cos𝛾−sin𝜃sin𝛾. (42)

𝛾 is the angle between the normal direction of Γ and 𝑟-axis.

In (41), {𝐮} is the displacement on the boundary, which is defined in SRP: {𝐮}=𝐍{𝐰}, (43) 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 𝐻_𝑃𝐶={𝜷}^𝑇[𝐇]{𝐰}, (44)

in which [𝐇]=_𝑇[𝐒]^𝑇[𝐋]^𝑇𝐍𝑡𝑑𝑠. (45)

The total energy Π can be denoted as 1Π=2{𝐰}^𝑇𝐊_𝐎{𝐰}−{𝐰}^𝑇{𝐏}−12{𝜷}^𝑇[𝐕]{𝜷}+{𝜷}^𝑇[𝐇]{𝐰}. (46)

Using the stationary conditions of the total energy Π𝛿Π=0 (47)

the stiffness matrix of {𝐰} can be obtained as 𝐊=[𝐇]^𝑇[𝐕]⁻¹[𝐇]. (48)

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 𝜃 is 45°.

The material parameters matrix of T700 is 𝐂=1.38×10¹¹4.14×10⁹04.14×10⁹9.79×10⁹0004.47×10⁹. (49)

The layer angle 𝜃 is 45°. According to (21) and (20) 𝐓=cos²45^∘sin²45^∘−2sin45^∘cos45^∘sin²45^∘cos²45^∘2sin45^∘cos45^∘sin45^∘cos45^∘−sin45^∘cos45^∘cos²45^∘−sin²45^∘,𝐂^′=𝐓𝐂𝐓^𝑇=4.35×10¹⁰3.45×10¹⁰3.21×10¹⁰3.45×10¹⁰4.35×10¹⁰3.21×10¹⁰3.21×10¹⁰3.21×10¹⁰3.49×10¹⁰. (50)

According to (15), we can get the material eigenvalues from det𝐃=0: 𝑝₁𝑝=−0.934−0.358𝑖₂=0.348−0.937𝑖𝑝₃=𝑖. (51)

According to (16) and (17), we can get material eigen matrixes 𝐀 and 𝐁𝐀=1.90×10⁻⁶+2.25×10⁻⁶𝑖−4.61×10⁻⁶−4.22×10⁻⁶𝑖02.65×10⁻⁶+1.09×10⁻⁶𝑖4.22×10⁻⁶+4.61×10⁻⁶𝑖0002.68×10⁻⁶−2.68×10⁻⁶𝑖𝐁=6.71×10⁴−4.61×10⁴𝑖−8.90×10³+4.96×10⁴𝑖04.61×10⁴−6.71×10⁴𝑖4.96×10⁴−8.90×10³𝑖0009.34×10⁴+9.34×10⁴𝑖 (52)

Substituting matrix 𝐁 into (28) and (29), with the application of the modified Müller method SRAM, the characteristic eigenvalues of different V-notch angles 2𝛼 are showed in Table 2. 𝜉₁ and 𝜉₂ are the first and second eigenvalues, respectively.

Table 2 

Characteristic eigenvalues of V-notch problem.

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 2𝛼 in the V-notch, and the layer angle of 𝜃.

Tables 3, 4, and 5 study the first and the second characteristic eigenvalues (𝜉₁ and 𝜉₂) of three kinds of material (𝐶₁₁/𝐶₂₂=2,8,32), eight types of V-notch (2𝛼 = 190°, 200°, 210°, 240°, 270°, 300°, 330°, 360°), and eight types of layer angle (𝜃 = 10°, 20°, 30°, 40°, 50°, 60°, 70°, 80°).

Table 3 

Characteristic eigenvalues of V-notch problem when 𝐶₁₁/𝐶₂₂ = 2.

Table 4 

Characteristic eigenvalues of V-notch problem when 𝐶₁₁/𝐶₂₂ = 8.

Table 5 

Characteristic eigenvalues of V-notch problem when 𝐶₁₁/𝐶₂₂ = 32.

Figures 4, 5, 6, and 7 show the first and the second items of the eigenvalues with different 2𝛼 and 𝜃.

Figure 4 

The first eigenvalue while 𝐶11/𝐶22=8.

Figure 5 

The second eigenvalue while 𝐶11/𝐶22=8.

Figure 6 

The first eigenvalue while 2𝛼=270∘.

Figure 7 

The second eigenvalue while 2𝛼=270∘.

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. 𝑎=1, 𝑤=3, ℎ=1, elastic constant is 𝐸=0.21×10⁷, Poisson’s ratio is 𝜇=0.3, the load sum is 𝑃=1, and the values of angle 2𝛼 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 (𝑁) of the items of the analytical trial functions.

Table 6 

The eigenvalues of the V-notch specimen.

Table 7 

The values of 𝐾_II calculated by ATF-VN.

Figure 8 

V-notch specimen under the antisymmetric load.

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).