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

Authors: Ge Tian; Xiang-Rong Fu; Ming-Wu Yuan; Meng-Yan Song
Journal: Mathematical Problems in Engineering
 (2013)

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## 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.


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## Body

## 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]
(1)σij=Cijkluk,l,(2)σij,j=Cijkluk,lj=0,
where  ui is displacement, σij is stress, and Cijkl is the elastic tensor of the anisotropy, in which i,j,k,l=1,2,3 are denoted as coordinates xi(i=1,2,3) of the three-dimensional problem.In the plane problem of anisotropy, the displacementsuk(k=1,2,3) are assumed to be only associated with the coordinates x1 and x2. According to Stroh theory [8, 9],
(3)uk=akg(zk),in whichg is the analytic function of zk, while zk=x1+pkx2. ak is the eigenvector about the eigenvalue pk of the material.Substituting (3) into (2), we have
(4)[Q+(R+RT)pk+Tpk2]akg′′(zk)=Dakg′′(zk)=0,in which(5)D=[Q+(R+RT)pk+Tpk2].The material coefficient matrixes ofQ, R, and T are defined as Qik=Ci1k1, and Rik=Ci1k2,  Tik=Ci2k2. They can also be denoted as
(6)Q=[C11C16C15C61C66C65C51C56C55],R=[C16C12C14C66C62C64C56C52C54],T=[C66C62C64C26C22C24C46C42C44].In order to obtain the nonzero solutions of (4), the determinant of the coefficient matrix D must be zero:
(7)detD=0.The solutionspk(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 functionsϕk, and we have
(8)ϕk=bkf(zk)(k=1,2,3),(9)σi1=-∂ϕi,1,σi2=∂ϕi,2,
where f is the first derivative of g.According to (1), we have
(10)bk=(RT+pkT)ak=-1pk(Q+pkR)ak.The general solutions of displacement in (1) can be denoted as
(11)u=A·g,in whichA=[a1,a2,a3].The general solutionsΦ=[ϕ1,ϕ2,ϕ3]T in (8) of stress functions can be denoted as
(12)Φ=B·f,in whichB=[b1,b2,b3].Introducing the normalization condition(13)2akTbk=1,the characteristic matrix of the materialA=[a1,a2,a3] and B=[b1,b2,b3] can be determined.Substituting (6) into (5), we can get(14)D=[C11+2pC16+p2C66C16+p(C66+C12)+p2C62C15+p(C65+C14)+p2C64C16+p(C66+C12)+p2C62C66+2pC26+p2C22C56+p(C52+C46)+p2C24C15+p(C65+C14)+p2C64C56+p(C52+C46)+p2C24C55+2pC54+p2C44].In the plane problem, we can defineC14=C15=C24=C25=C64=C65=0.So matrixD can be simplified as follows:
(15)D=[C11+2pC16+p2C66C16+p(C66+C12)+p2C620C16+p(C66+C12)+p2C62C66+2pC26+p2C22000C55+p2C44].The characteristic matrixA of the material can be written as
(16)A=[k1(C66+2p1C26+p12C22)k2(C66+2p2C26+p22C22)0-k1(C16+p1(C66+C12)+p12C26)-k2(C16+p2(C66+C12)+p22C26)000k3].
The characteristic matrix B=[b1,b2,b3] can be calculated as
(17)b1=[k1p1(C16C26+p1C16C22+p12C66C22-C66C12-p1C26C12-p12C262)k1(C66C12+p1C26C12+p12C262-C16C26-p1C16C22-p12C66C22)0],b2=[k2p2(C16C26+p2C16C22+p22C66C22-C66C12-p2C26C12-p22C262)k2(C66C12+p2C26C12+p22C262-C16C26-p2C16C22-p22C66C22)0],b3=[00k3p3C44].The constantsk1, k2, and k3 in (16) and (17)  can be determined by (13).It can be proved that according to the characteristic matrixesA and B of anisotropy defined in (16) and (17), we have
(18)AAT+A-A-T=0=BBT+B-B-T,(19)ATB+BTA=I=A-TB-+B-TA-.


## 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 Figure1, 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
(20)C′=TCTT,in whichT is the transformation matrix, and we have
(21)T=[cos2θsin2θ-2sinθcosθsin2θcos2θ2sinθcosθsinθcosθ-sinθcosθcos2θ-sin2θ].In the polar coordinate system,zk can be expressed as
(22)zk=rζk(θ)in which(23)ζk(θ)=cos(θ)+pksin(θ).r represents the radial distance to tip.The general stress solutionsf(zk) in (8) can be expressed by f(r,ζk(θ)). According to Stroh’s theory, vector f can be written as [10, 11]
(24)f=c2(rr^)λΛ(θ,α,λ)q+c-2(rr^)λ-Λ(θ,α,λ-)q-,in which eigenvaluesλ and parameter c are a complex constant, eigenvector q is undetermined complex vector, are r^ is reference length of the notch, and we have
(25)Λ(θ,α,λ)=diag[(ζ1(θ)ζ1(α))λ,(ζ2(θ)ζ2(α))λ,(ζ3(θ)ζ3(α))λ]BT.Taking account the stresstr in the boundaries of the V-notch, where θ=±α,
(26)tr=-∂Φ∂r.Generallytr is zero in the stress-free boundaries of the V-notch.Utilizing (18), tr is zero in the boundary θ=α.Considering the condition of(r/r^)λ-1≠0, in order to make tr equal to zero on the side θ=-α, according to (24), we have
(27)cλ(BΛ(-α,α,λ)+BΛ(-α,α,λ-)¯)q=cλCq=0,in which the boundary characteristic matrixC is
(28)C=BΛ(-α,α,λ)+BΛ(-α,α,λ-)¯.In order to obtain the nonzero solutions of (27), the determinant of the coefficient matrix C must be zero:
(29)Δ=|C|=0.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 Figure2, making a parabola g(z) which passes three points zi, zi-1, zi-2 on the complex function  f(z), we have
(30)g(z)=f(zi)+(z-zi)ai+f[zi,zi-1,zi-2](z-zi)2,
where f[zi,zi-1], f[zi,zi-1,zi-2] is the first and second difference of f(z).Figure 2
The schematic of Müller method.The root of theg(z)=0 is
(31)z*=zi+-ai±ai2-4f(zi)f[zi,zi-1,zi-2]2f[zi,zi-1,zi-2],in which(32)ai=f[zi,zi-1]+f[zi,zi-1,zi-2](zi-zi-1).In the next iteration step,zi+1=z* is the new initial value.To solve more than one root, there is an effective modification (RDM) to the functionf(z)(33)fn(z)=f(z)(z-z10)(z-z20)⋯(z-zn0),
where z10,z20,…,zn0 are series of characteristic roots which have been calculated.The Müller method is local convergence iteration method, if the root of equationf(z)=0 is confirmed in the interval of a and b. There is an effective method to accelerate the convergence of the Müller method
(34)f*(z)=f(z)(z-a)(b-z).It is called Shrink Boundary Method (SBM).Combining (33) with (34), we have the subregion accelerated müller (SRAM) method, whose iteration function is
(35)fn*(z)=f(z)(z-a)(b-z)(z-z10)⋯(z-zn0).Figure3 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 off(z) and f*(z) near the root.In Figure3, to the values of the tangent near root, f*(z) is larger than f(z) due to the SBM.Table1 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.
a
b
Direct (31)
n
SBM (34)
n
RDM (33)
n
SRAM (35)
n
0.6
0.9
0.730900742
0
8
0.730900742
0
9
0.730900742
0
7
0.730900742
0
7
1.8
2.1
1.89349155
0
F
2.0748262
0.2294258
20
2.00933205
0
F
2.0748262
0.2294258
30
3
3.3
3.01938419
0
F
3.27976709
0.326690313
16
4.48290004
−0.38898376
23
3.27976709
−0.326690313
16
4.2
4.5
3.27976709
−0.32669031
19
4.48290004
0.388983761
16
4.48290004
0.388983761
9
4.48290004
−0.388983761
17
5.4
5.7
3.01938419
6.21124
E
-
13
F
3.27976709
−0.32669031
51
9.28902685
−0.5301614
25
5.68509375
−0.435407938
20
6.6
6.9
6.88672925
0.472546839
87
3.27976709
−0.32669031
45
9.28902685
0.530161398
31
6.88672925
−0.472546839
25


## 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(36)Π=ΠP-ΠC+HPC,
where ΠC is the complementary energy in the subregion-C that was defined by the stress field,  ΠP is the potential energy in the subregion-P that was defined by the displacement field, and HPC is the additional energy along the boundary between two subregions.The potential energyΠP can be denoted as
(37)ΠP=12{w}T[KO]{w}-{w}T{P}in which{w} is the nodal displacement vector, {P} is the equivalent nodal load vector, and  [KO] is the stiffness matrix of the potential energy in subregion-P (SRP).The complementary energyΠc can be denoted as
(38)Πc=12{β}T[V]{β}in which{β} is the undetermined parameters of the stress, and
(39)[V]=∬ΩC[S]T[D]-1[S]tdA,
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 energyHPC on the boundary Γ between two kinds of subregions can be written as
(40)HPC=∫Γ{T}T{u-}tdsin which{T} is the boundary force determined in SRC:
(41){T}=[L]{σ}=[L][S]{β},
where direction cosine matrix [L] can be written as
(42)[L]=[cosθcosγsinθsinγ-(sinθcosγ+cosθsinγ)sinθcosγ-cosθsinγcosθcosγ-sinθsinγ].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:
(43){u-}=[N-]{w-},
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
(44)HPC={β}T[H]{w-},in which(45)[H]=∫T[S]T[L]T[N-]tds.The total energyΠ can be denoted as
(46)Π=12{w}T[KO]{w}-{w}T{P}-12{β}T[V]{β}+{β}T[H]{w-}.Using the stationary conditions of the total energyΠ(47)δΠ=0the stiffness matrix of{w-} can be obtained as
(48)[K¯]=[H]T[V]-1[H].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 angleθ is 45°.The material parameters matrix of T700 is(49)C=[1.38×10114.14×10904.14×1099.79×1090004.47×109].The layer angleθ is 45°. According to (21) and (20)
(50)T=[cos245°sin245°-2sin45°cos45°sin245°cos245°2sin45°cos45°sin45°cos45°-sin45°cos45°cos245°-sin245°],C′=TCTT=[4.35×10103.45×10103.21×10103.45×10104.35×10103.21×10103.21×10103.21×10103.49×1010].According to (15), we can get the material eigenvalues from detD=0:
(51)p1=-0.934-0.358ip2=0.348-0.937ip3=i.According to (16) and (17), we can get material eigen matrixes A and B(52)A=[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+4.61×10-6i0002.68×10-6-2.68×10-6i]B=[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]Substituting matrixB 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.
2
α
190°
200°
210°
240°
270°
300°
330°
360°
ξ
1
0.93265944
0.87284430
0.81861307
0.67723697
0.55658245
0.50800061
0.50068400
0.50000000
ξ
2
1.86475055
1.74191187
1.62613243
1.28206691
0.88272695
0.65597019
0.56189499
0.50000000


### 6.2. Example 2

This example analyses the characteristic solutions of a V-notch and studies the influence of the material parameters, the angle of2α in the V-notch, and the layer angle of θ.Tables3, 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 whenC11/C22 = 2.
2
α
θ
10°
20°
30°
40°
50°
60°
70°
80°
ξ
1
ξ
2
ξ
1
ξ
2
ξ
1
ξ
2
ξ
1
ξ
2
ξ
1
ξ
2
ξ
1
ξ
2
ξ
1
ξ
2
ξ
1
ξ
2
190°
0.9128563
1.8249625
0.9113253
1.8218496
0.9086988
1.8165083
0.9049076
1.8087957
0.8999710
1.7987498
0.8941736
1.7869486
0.8883017
1.7749937
0.8837307
1.7656869
200°
0.8397233
1.6748301
0.8371422
1.6693737
0.8327417
1.6600658
0.8264549
1.6467584
0.8183932
1.6296842
0.8091217
1.6100415
0.7999652
1.5906445
0.7930116
1.5759189
210°
0.7777646
1.5429819
0.7744839
1.5356576
0.7689332
1.5232544
0.7611038
1.5057428
0.7512566
1.4837038
0.7402290
1.4590221
0.7296815
1.4354289
0.7219154
1.4180723
240°
0.6420085
1.2222747
0.6381055
1.2111531
0.6317307
1.1929553
0.6232522
1.1687083
0.6134684
1.1406951
0.6036664
1.1126176
0.5953715
1.0888543
0.5898835
1.0731292
270°
0.5608742
0.9750736
0.5578813
0.9625454
0.5532895
0.9432595
0.5477408
0.9198607
0.5420796
0.8958804
0.5371148
0.8747394
0.5333855
0.8587638
0.5311212
0.8490085
300°
0.5183809
0.7767464
0.5169893
0.7660143
0.5150402
0.7508982
0.5129504
0.7345539
0.5110686
0.7196690
0.5095865
0.7077811
0.5085560
0.6993955
0.5079587
0.6944767
330°
0.5023271
0.6186594
0.5020954
0.6130523
0.5017991
0.6057997
0.5015112
0.5986297
0.5012728
0.5925552
0.5010958
0.5879303
0.5009771
0.5847549
0.5009095
0.5829168
360°
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000Table 4
Characteristic eigenvalues of V-notch problem whenC11/C22 = 8.
2α
θ
10°
20°
30°
40°
50°
60°
70°
80°
ξ
1
ξ
2
ξ
1
ξ
2
ξ
1
ξ
2
ξ
1
ξ
2
ξ
1
ξ
2
ξ
1
ξ
2
ξ
1
ξ
2
ξ
1
ξ
2
190°
0.9300319
1.8597539
0.9295683
1.8557960
0.9252591
1.8500525
0.9204215
1.8402016
0.9122592
1.8235521
0.8983587
1.7951506
0.8753016
1.7480014
0.8442929
1.6846406
200°
0.8682780
1.7344681
0.8656370
1.7288880
0.8606216
1.7182604
0.8520654
1.7000556
0.8377687
1.6694968
0.8140318
1.6185886
0.7772342
1.5397211
0.7346126
1.4488744
210°
0.8144412
1.6229892
0.8108555
1.6149757
0.8040481
1.5996819
0.7924631
1.5734684
0.7732964
1.5297892
0.7425320
1.4594576
0.6993252
1.3612058
0.6581962
1.2687287
240°
0.6868593
1.3415308
0.6811489
1.3251930
0.6703026
1.2937616
0.6521517
1.2404467
0.6245291
1.1589485
0.5906116
1.0602030
0.5627811
0.9806798
0.5479446
0.9383353
270°
0.5969306
1.1069820
0.5898858
1.0778538
0.5769633
1.0235558
0.5581487
0.9442040
0.5384141
0.8624828
0.5247156
0.8064764
0.5174707
0.7762371
0.5141545
0.7618821
300°
0.5367164
0.8897706
0.5303145
0.8434998
0.5210972
0.7768489
0.5128778
0.7183591
0.5078752
0.6825068
0.5052792
0.6629644
0.5039741
0.6524804
0.5033598
0.6472710
330°
0.5056623
0.6756323
0.5037204
0.6363077
0.5021314
0.6041338
0.5012388
0.5853208
0.5007847
0.5749354
0.5005517
0.5690795
0.5004307
0.5657721
0.5003720
0.5640673
360°
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000Table 5
Characteristic eigenvalues of V-notch problem whenC11/C22 = 32.
2α
θ
10°
20°
30°
40°
50°
60°
70°
80°
ξ
1
ξ
2
ξ
1
ξ
2
ξ
1
ξ
2
ξ
1
ξ
2
ξ
1
ξ
2
ξ
1
ξ
2
ξ
1
ξ
2
ξ
1
ξ
2
190°
0.9383544
1.8765620
0.9375017
1.8748302
0.9358295
1.8714294
0.9328018
1.8652572
0.9271630
1.8537207
0.9156380
1.8300284
0.8883484
1.7736743
0.8177936
1.6281756
200°
0.8837766
1.7665839
0.8822227
1.7633014
0.8791667
1.7568144
0.8736061
1.7449163
0.8631675
1.7223247
0.8416098
1.6750388
0.7909583
1.5632250
0.6867746
1.3375999
210°
0.8350726
1.6673544
0.8329023
1.6624997
0.8286071
1.6527989
0.8207038
1.6346735
0.8055930
1.5992951
0.7736118
1.5229344
0.7007226
1.3501031
0.6040837
1.1317090
240°
0.7160093
1.4159385
0.7120920
1.4046503
0.7040059
1.3806731
0.6879708
1.3312028
0.6538841
1.2229404
0.5878978
1.0238868
0.5382270
0.8890728
0.5231031
0.8477859
270°
0.6264983
1.2064383
0.6203072
1.1806093
0.6063290
1.1193694
0.5758734
1.0000000
0.5336490
0.8148305
0.5145313
0.7463110
0.5084959
0.7230788
0.5063534
0.7139696
300°
0.5585617
1.0047370
0.5486459
0.9336876
0.5277819
0.7857225
0.5105224
0.6807309
0.5046508
0.6461264
0.5026413
0.6325957
0.5018261
0.6262824
0.5014813
0.6233311
330°
0.5118620
0.7488441
0.5050930
0.6348976
0.5017876
0.5848921
0.5007722
0.5681839
0.5004135
0.5610688
0.5002638
0.5575139
0.5001946
0.5556253
0.5001632
0.5546819
360°
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000Figures4, 5, 6, and 7 show the first and the second items of the eigenvalues with different 2α and θ.Figure 4
The first eigenvalue whileC11/C22=8.Figure 5
The second eigenvalue whileC11/C22=8.Figure 6
The first eigenvalue while2α=270°.Figure 7
The second eigenvalue while2α=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 Figure8. 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.
2α
330°
350°
ξ
1
0.50145301
0.50005299
η
1
0
0
ξ
2
0.59819181
0.52935471
η
2
0
0
ξ
3
1
1
η
3
0
0
ξ
4
1.20295709
1.05884289
η
4
0
0
ξ
5
1.49037781
1.49972777
η
5
0
0
ξ
6
1.83893404
1.5886091
η
6
0
0
ξ
7
1.948556
1.99910697
η
7
0
0
ξ
8
2.44049194
2.11882272
η
8
0.11420683
0
ξ
9
2.98700495
2.49797992
η
9
0.16674099
0
ξ
10
3.53334548
2.64969856
η
10
0.20371063
0
ξ
11
4.07955465
2.99614098
η
11
0.2330165
0
ξ
12
4.62566114
3.18153251
η
12
0.25754031
0
ξ
13
5.17168563
3.71477243
η
13
0.27872957
0
ξ
14
6.26354629
3.98901973
η
14
0.31420638
0Table 7
The values ofKII calculated by ATF-VN.
N items of the ATF
4
6
8
10
12
14
Reference [12]
350°
0.41956745
0.40362374
0.4003461
0.39869345
0.39876045
0.39877248
0.401
330°
0.15386682
0.12706163
0.12363552
0.12187221
0.12195786
0.12195786
/Figure 8
V-notch specimen under the antisymmetric load.


## 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 angleθ is 45°.The material parameters matrix of T700 is(49)C=[1.38×10114.14×10904.14×1099.79×1090004.47×109].The layer angleθ is 45°. According to (21) and (20)
(50)T=[cos245°sin245°-2sin45°cos45°sin245°cos245°2sin45°cos45°sin45°cos45°-sin45°cos45°cos245°-sin245°],C′=TCTT=[4.35×10103.45×10103.21×10103.45×10104.35×10103.21×10103.21×10103.21×10103.49×1010].According to (15), we can get the material eigenvalues from detD=0:
(51)p1=-0.934-0.358ip2=0.348-0.937ip3=i.According to (16) and (17), we can get material eigen matrixes A and B(52)A=[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+4.61×10-6i0002.68×10-6-2.68×10-6i]B=[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]Substituting matrixB 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.
2
α
190°
200°
210°
240°
270°
300°
330°
360°
ξ
1
0.93265944
0.87284430
0.81861307
0.67723697
0.55658245
0.50800061
0.50068400
0.50000000
ξ
2
1.86475055
1.74191187
1.62613243
1.28206691
0.88272695
0.65597019
0.56189499
0.50000000


## 6.2. Example 2

This example analyses the characteristic solutions of a V-notch and studies the influence of the material parameters, the angle of2α in the V-notch, and the layer angle of θ.Tables3, 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 whenC11/C22 = 2.
2
α
θ
10°
20°
30°
40°
50°
60°
70°
80°
ξ
1
ξ
2
ξ
1
ξ
2
ξ
1
ξ
2
ξ
1
ξ
2
ξ
1
ξ
2
ξ
1
ξ
2
ξ
1
ξ
2
ξ
1
ξ
2
190°
0.9128563
1.8249625
0.9113253
1.8218496
0.9086988
1.8165083
0.9049076
1.8087957
0.8999710
1.7987498
0.8941736
1.7869486
0.8883017
1.7749937
0.8837307
1.7656869
200°
0.8397233
1.6748301
0.8371422
1.6693737
0.8327417
1.6600658
0.8264549
1.6467584
0.8183932
1.6296842
0.8091217
1.6100415
0.7999652
1.5906445
0.7930116
1.5759189
210°
0.7777646
1.5429819
0.7744839
1.5356576
0.7689332
1.5232544
0.7611038
1.5057428
0.7512566
1.4837038
0.7402290
1.4590221
0.7296815
1.4354289
0.7219154
1.4180723
240°
0.6420085
1.2222747
0.6381055
1.2111531
0.6317307
1.1929553
0.6232522
1.1687083
0.6134684
1.1406951
0.6036664
1.1126176
0.5953715
1.0888543
0.5898835
1.0731292
270°
0.5608742
0.9750736
0.5578813
0.9625454
0.5532895
0.9432595
0.5477408
0.9198607
0.5420796
0.8958804
0.5371148
0.8747394
0.5333855
0.8587638
0.5311212
0.8490085
300°
0.5183809
0.7767464
0.5169893
0.7660143
0.5150402
0.7508982
0.5129504
0.7345539
0.5110686
0.7196690
0.5095865
0.7077811
0.5085560
0.6993955
0.5079587
0.6944767
330°
0.5023271
0.6186594
0.5020954
0.6130523
0.5017991
0.6057997
0.5015112
0.5986297
0.5012728
0.5925552
0.5010958
0.5879303
0.5009771
0.5847549
0.5009095
0.5829168
360°
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000Table 4
Characteristic eigenvalues of V-notch problem whenC11/C22 = 8.
2α
θ
10°
20°
30°
40°
50°
60°
70°
80°
ξ
1
ξ
2
ξ
1
ξ
2
ξ
1
ξ
2
ξ
1
ξ
2
ξ
1
ξ
2
ξ
1
ξ
2
ξ
1
ξ
2
ξ
1
ξ
2
190°
0.9300319
1.8597539
0.9295683
1.8557960
0.9252591
1.8500525
0.9204215
1.8402016
0.9122592
1.8235521
0.8983587
1.7951506
0.8753016
1.7480014
0.8442929
1.6846406
200°
0.8682780
1.7344681
0.8656370
1.7288880
0.8606216
1.7182604
0.8520654
1.7000556
0.8377687
1.6694968
0.8140318
1.6185886
0.7772342
1.5397211
0.7346126
1.4488744
210°
0.8144412
1.6229892
0.8108555
1.6149757
0.8040481
1.5996819
0.7924631
1.5734684
0.7732964
1.5297892
0.7425320
1.4594576
0.6993252
1.3612058
0.6581962
1.2687287
240°
0.6868593
1.3415308
0.6811489
1.3251930
0.6703026
1.2937616
0.6521517
1.2404467
0.6245291
1.1589485
0.5906116
1.0602030
0.5627811
0.9806798
0.5479446
0.9383353
270°
0.5969306
1.1069820
0.5898858
1.0778538
0.5769633
1.0235558
0.5581487
0.9442040
0.5384141
0.8624828
0.5247156
0.8064764
0.5174707
0.7762371
0.5141545
0.7618821
300°
0.5367164
0.8897706
0.5303145
0.8434998
0.5210972
0.7768489
0.5128778
0.7183591
0.5078752
0.6825068
0.5052792
0.6629644
0.5039741
0.6524804
0.5033598
0.6472710
330°
0.5056623
0.6756323
0.5037204
0.6363077
0.5021314
0.6041338
0.5012388
0.5853208
0.5007847
0.5749354
0.5005517
0.5690795
0.5004307
0.5657721
0.5003720
0.5640673
360°
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000Table 5
Characteristic eigenvalues of V-notch problem whenC11/C22 = 32.
2α
θ
10°
20°
30°
40°
50°
60°
70°
80°
ξ
1
ξ
2
ξ
1
ξ
2
ξ
1
ξ
2
ξ
1
ξ
2
ξ
1
ξ
2
ξ
1
ξ
2
ξ
1
ξ
2
ξ
1
ξ
2
190°
0.9383544
1.8765620
0.9375017
1.8748302
0.9358295
1.8714294
0.9328018
1.8652572
0.9271630
1.8537207
0.9156380
1.8300284
0.8883484
1.7736743
0.8177936
1.6281756
200°
0.8837766
1.7665839
0.8822227
1.7633014
0.8791667
1.7568144
0.8736061
1.7449163
0.8631675
1.7223247
0.8416098
1.6750388
0.7909583
1.5632250
0.6867746
1.3375999
210°
0.8350726
1.6673544
0.8329023
1.6624997
0.8286071
1.6527989
0.8207038
1.6346735
0.8055930
1.5992951
0.7736118
1.5229344
0.7007226
1.3501031
0.6040837
1.1317090
240°
0.7160093
1.4159385
0.7120920
1.4046503
0.7040059
1.3806731
0.6879708
1.3312028
0.6538841
1.2229404
0.5878978
1.0238868
0.5382270
0.8890728
0.5231031
0.8477859
270°
0.6264983
1.2064383
0.6203072
1.1806093
0.6063290
1.1193694
0.5758734
1.0000000
0.5336490
0.8148305
0.5145313
0.7463110
0.5084959
0.7230788
0.5063534
0.7139696
300°
0.5585617
1.0047370
0.5486459
0.9336876
0.5277819
0.7857225
0.5105224
0.6807309
0.5046508
0.6461264
0.5026413
0.6325957
0.5018261
0.6262824
0.5014813
0.6233311
330°
0.5118620
0.7488441
0.5050930
0.6348976
0.5017876
0.5848921
0.5007722
0.5681839
0.5004135
0.5610688
0.5002638
0.5575139
0.5001946
0.5556253
0.5001632
0.5546819
360°
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000
0.5000000Figures4, 5, 6, and 7 show the first and the second items of the eigenvalues with different 2α and θ.Figure 4
The first eigenvalue whileC11/C22=8.Figure 5
The second eigenvalue whileC11/C22=8.Figure 6
The first eigenvalue while2α=270°.Figure 7
The second eigenvalue while2α=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 Figure8. 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.
2α
330°
350°
ξ
1
0.50145301
0.50005299
η
1
0
0
ξ
2
0.59819181
0.52935471
η
2
0
0
ξ
3
1
1
η
3
0
0
ξ
4
1.20295709
1.05884289
η
4
0
0
ξ
5
1.49037781
1.49972777
η
5
0
0
ξ
6
1.83893404
1.5886091
η
6
0
0
ξ
7
1.948556
1.99910697
η
7
0
0
ξ
8
2.44049194
2.11882272
η
8
0.11420683
0
ξ
9
2.98700495
2.49797992
η
9
0.16674099
0
ξ
10
3.53334548
2.64969856
η
10
0.20371063
0
ξ
11
4.07955465
2.99614098
η
11
0.2330165
0
ξ
12
4.62566114
3.18153251
η
12
0.25754031
0
ξ
13
5.17168563
3.71477243
η
13
0.27872957
0
ξ
14
6.26354629
3.98901973
η
14
0.31420638
0Table 7
The values ofKII calculated by ATF-VN.
N items of the ATF
4
6
8
10
12
14
Reference [12]
350°
0.41956745
0.40362374
0.4003461
0.39869345
0.39876045
0.39877248
0.401
330°
0.15386682
0.12706163
0.12363552
0.12187221
0.12195786
0.12195786
/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.



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*Source: 593640-2013-09-26.xml*