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

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1 Mathematical Problems in Engineering Volume 2013, Article ID , 11 pages Research Article The Characteristic Solutions to the V-Notch Plane Problem of Anisotropy and the Associated Finite Element Method Ge Tian, 1 Xiang-Rong Fu, 1 Ming-Wu Yuan, 2 and Meng-Yan Song 1 1 Department of Civil Engineering, China Agricultural University, Beijing , China 2 Department of Mechanics and Engineering Science, Peking University, Beijing , China Correspondence should be addressed to Xiang-Rong Fu; fuxr@cau.edu.cn 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. 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, namedas thesubregionaccelerated 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 thecomplexvariables.theproposedatf-vnbasedonthe analytical trial functions, which contain the characteristic solutions of the V-notch problem, presents good performance in the benchmarks. ThoughmanyresearchersstudiedtheV-notchproblemof isotropy material [1 16], thereareseldom studiesabout thevnotch 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, (1) σ ij,j =C ijkl u k,lj =0, (2)

2 2 Mathematical Problems in Engineering 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 F F F E 13 F Table 2: Characteristic eigenvalues of V-notch problem. 2α ξ ξ 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) areassumedtobeonlyassociatedwiththe coordinates x 1 and x 2. According to Stroh theory [8, 9], u k = a k g (z k ), (3) in which g is the analytic function of z k, while z k =x 1 +p k x 2. a k is the eigenvector about the eigenvalue p k of the material. Substituting (3)into(2), we have [Q +(R + R T )p k + Tp 2 k ] a kg (z k )=Da k g (z k )=0, (4) in which D =[Q +(R + R T )p k + Tp 2 k ]. (5) In order to obtain the nonzero solutions of (4), the determinant of the coefficient matrix D must be zero: det D =0. (7) The solutions p k (k=1,2,3)of (7) are the eigenvalues of the material. Inthesameway,thestresses,whichsatisfiedequilibrium equations, can also be expressed by the characteristic solutions of stress functions φ k,andwehave φ k = b k f(z k ) (k =1,2,3), (8) σ i1 = φ i,1, σ i2 = φ i,2, (9) where f is the first derivative of g. According to (1), we have b k =(R T +p k T) a k = 1 p k (Q +p k R) a k. (10) The material coefficient matrixes of Q, R, and T are defined as Q ik =C i1k1,andr ik =C i1k2, T ik =C i2k2.they canalsobedenotedas The general solutions of displacement in (1) can be denoted as C 11 C 16 C 15 Q = [ C 61 C 66 C 65 ], [ C 51 C 56 C 55 ] C 16 C 12 C 14 R = [ C 66 C 62 C 64 ], [ C 56 C 52 C 54 ] C 66 C 62 C 64 T = [ C 26 C 22 C 24 ]. [ C 46 C 42 C 44 ] (6) u = A g, (11) in which A =[a 1, a 2, a 3 ]. The general solutions Φ = [φ 1,φ 2,φ 3 ] T in (8) ofstress functions can be denoted as in which B =[b 1, b 2, b 3 ]. Φ = B f, (12)

3 Mathematical Problems in Engineering 3 Table 3: Characteristic eigenvalues of V-notch problem when C 11 /C 22 =2. 2α θ 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

4 4 Mathematical Problems in Engineering Table 4: Characteristic eigenvalues of V-notch problem when C 11 /C 22 =8. 2α θ 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

5 Mathematical Problems in Engineering 5 Table 5: Characteristic eigenvalues of V-notch problem when C 11 /C 22 =32. 2α θ 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

6 6 Mathematical Problems in Engineering Introducing the normalization condition 2a T k b k =1, (13) the characteristic matrix of the material A =[a 1, a 2, a 3 ] and B =[b 1, b 2, b 3 ] can be determined. Substituting (6)into(5), we can get C 11 +2pC 16 +p 2 C 66 C 16 +p(c 66 +C 12 )+p 2 C 62 C 15 +p(c 65 +C 14 )+p 2 C 64 D = [ C 16 +p(c 66 +C 12 )+p 2 C 62 C 66 +2pC 26 +p 2 C 22 C 56 +p(c 52 +C 46 )+p 2 C 24 ]. (14) [ C 15 +p(c 65 +C 14 )+p 2 C 64 C 56 +p(c 52 +C 46 )+p 2 C 24 C 55 +2pC 54 +p 2 C 44 ] In the plane problem, we can define C 14 =C 15 =C 24 = C 25 =C 64 =C 65 =0. So matrix D can be simplified as follows: D =[ C 11 +2pC 16 +p 2 C 66 C 16 +p(c 66 +C 12 )+p 2 C 62 0 C 16 +p(c 66 +C 12 )+p 2 C 62 C 66 +2pC 26 +p 2 C 22 0 ]. 0 0 C 55 +p 2 C 44 (15) The characteristic matrix A of the material can be written as k 1 (C 66 +2p 1 C 26 +p 2 1 C 22) k 2 (C 66 +2p 2 C 26 +p 2 2 C 22) 0 k 1 (C 16 +p 1 (C 66 +C 12 )+p 2 1 C 26) k 2 (C 16 +p 2 (C 66 +C 12 )+p 2 2 C 26) 0 A=[ ]. 0 0 k 3 (16) The characteristic matrix B =[b 1, b 2, b 3 ] canbecalculatedas b 1 =[ k 1p 1 (C 16 C 26 +p 1 C 16 C 22 +p 2 1 C 66C 22 C 66 C 12 p 1 C 26 C 12 p 2 1 C2 26 ) ], k 1 (C 66 C 12 +p 1 C 26 C 12 +p 2 1 C2 26 C 16C 26 p 1 C 16 C 22 p 2 1 C 66C 22 ) 0 b 2 =[ k 2p 2 (C 16 C 26 +p 2 C 16 C 22 +p 2 2 C 66C 22 C 66 C 12 p 2 C 26 C 12 p 2 2 C2 26 ) k 2 (C 66 C 12 +p 2 C 26 C 12 +p 2 2 C2 26 C 16C 26 p 2 C 16 C 22 p 2 2 C 66C 22 ) ], 0 b 3 = [ 0 ]. [ k 3 p 3 C 44 ] 0 (17) The constants k 1, k 2,andk 3 in (16) and(17) canbe determined by (13). Itcanbeprovedthataccordingtothecharacteristic matrixes A and B ofanisotropydefinedin(16) and(17), we have AA T + A A T = 0 = BB T + B B T, (18) A T B + B T A = I = A T B + B T A. (19) 3. The Boundary Characteristic Matrix of V-Notch The plane polar coordinate system (r, θ) was defined in Figure1,andther-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 θ between the r-axis and the material principle axis of x 1. 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 cos 2 θ sin 2 θ 2sin θ cos θ T = [ sin 2 θ cos 2 θ 2sin θ cos θ ]. (21) [ sin θ cos θ sin θ cos θ cos 2 θ sin 2 θ] In the polar coordinate system, z k canbeexpressedas in which z k =rζ k (θ) (22) ζ k (θ) = cos (θ) +p k sin (θ). (23) r representstheradialdistancetotip. 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] f = c 2 (r r )λ Λ (θ, α, λ) q + c 2 (r r )λ Λ (θ, α, λ) q, (24) in which eigenvalues λ and parameter c are a complex constant, eigenvector q is undetermined complex vector, are r is reference length of the notch, and we have Λ (θ, α, λ) = diag [( ζ λ 1 (θ) ζ 1 (α) ), ( ζ λ 2 (θ) ζ 2 (α) ), ( ζ λ 3 (θ) ζ 3 (α) ) ] B 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 cλ (BΛ ( α, α, λ) + BΛ ( α, α, λ)) q =cλcq = 0, (27) in which the boundary characteristic matrix C is C = BΛ ( α, α, λ) + BΛ ( α, α, λ). (28)

7 Mathematical Problems in Engineering 7 In order to obtain the nonzero solutions of (27), the determinant of the coefficient matrix C must be zero: Δ = C = 0. (29) x 2 α x 1 Equation (29) is the boundary characteristic equation. Its solution λ is complex; u and φ in (11) and(12) arealso 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),wehave g (z) =f(z i )+(z z i )a i +f[z i,z i 1,z i 2 ](z z i ) 2, (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). The root of the g(z) = 0 is in which z =z i + a i ± ai 2 4f(z i )f[z i,z i 1,z i 2 ], (31) 2f [z i,z i 1,z i 2 ] a i =f[z i,z i 1 ] +f[z i,z i 1,z i 2 ](z 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 (z) f n (z) = (z z1 0 (33) )(z z0 2 ) (z z0 n ), where z 0 1,z0 2,...,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 f (z) (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 z1 0 (35) ) (z z0 n ). 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. In Figure 3, to the values of the tangent near root, f (z) is larger than f(z) due to the SBM. λ i+1 α Figure 1: The local coordinators. y=f(λ) λ i 1 λ i λ i 2 θ r y=g(λ) Figure 2: The schematic of Müller method. 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 somesubregions.sramcangetridoftheinfluenceofthe 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 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 = 1 2 {w}t [K 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 canbedenotedas Π c = 1 2 {β}t [V] {β} (38)

8 8 Mathematical Problems in Engineering f(x), f (z) z f(z) f (z) Figure 3: Values of f(z) and f (z) near the root. ξ θ=0 θ=10 θ=20 θ=30 θ=40 2α ( ) θ=50 θ=60 θ=70 θ=80 Figure 5: The second eigenvalue while C 11 /C 22 =8. ξ θ=0 θ=10 θ=20 θ=30 θ=40 2α ( ) θ=50 θ=60 θ=70 θ=80 Figure 4: The first eigenvalue while C 11 /C 22 =8. in which {β} is the undetermined parameters of the stress, and [V] = Ω C [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 twokindsofsubregionscanbewrittenas H PC = {T} T {u} tds (40) Γ in which {T} is the boundary force determined in SRC: {T} = [L] {σ} = [L][S] {β}, (41) Table 6: The eigenvalues of the V-notch specimen. 2α ξ η ξ η ξ η ξ η ξ η ξ η ξ η ξ η ξ η ξ η ξ η ξ η ξ η ξ η

9 Mathematical Problems in Engineering 9 Table 7: The values of K II calculated by ATF-VN. N items of the ATF Reference [12] / where direction cosine matrix [L] can be written as cos θ cos γ sin θ sin γ (sin θ cos γ+cos θ sin γ) [L] =[ sin θ cos γ cos θ sin γ cos θ cos γ sin θ sin γ ]. (42) γ istheanglebetweenthenormaldirectionofγ 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 in which H PC ={β} T [H] {w}, (44) [H] = [S] T [L] T [N]tds. (45) T The total energy Π canbedenotedas Π= 1 2 {w}t [K O ] {w} {w} T {P} 1 2 {β}t [V] {β}+{β} T [H] {w}. Using the stationary conditions of the total energy Π (46) 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 = [ ]. (49) [ ] The layer angle θ is 45. According to (21)and(20) cos 2 45 sin sin 45 cos 45 T = [ sin 2 45 cos sin 45 cos 45 ], [ sin 45 cos 45 sin 45 cos 45 cos 2 45 sin 2 45 ] C = TCT T = [ ]. [ ] (50) According to (15), we can get the material eigenvalues from det D =0: δπ = 0 (47) the stiffness matrix of {w} canbeobtainedas [K] =[H] T [V] 1 [H]. (48) p 1 = i p 2 = i p 3 =i. (51) The generalized element defined in (48)is namedas ATF- VN. According to (16) and(17), we can get material eigen matrixes A and B i i 0 A = [ i i 0 ] [ i] i i 0 B = [ i i 0 ] [ i] (52)

10 10 Mathematical Problems in Engineering ξ ξ θ ( ) C 11 /C 22 =2 C 11 /C 22 =8 C 11 /C 22 =32 Figure 6: The first eigenvalue while 2α = θ ( ) C 11 /C 22 =2 C 11 /C 22 =8 C 11 /C 22 =32 Figure 7: The second eigenvalue while 2α = 270. Tables 3, 4, and5 study the first and the second characteristic eigenvalues (ξ 1 and ξ 2 )ofthreekindsofmaterial (C 11 /C 22 =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 ). Figures 4, 5, 6,and7 show the first and the second items of the eigenvalues with different 2α and θ 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,elasticconstantise = , Poisson s ratio is μ = 0.3, the load sum is P=1,andthevaluesofangle 2α are 330 and 350. Table 6 givestheeigenvaluesofthevnotch problem. Table 7 gives the values of KII calculated by ATF-VN involving different number (N) 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 accuracyandiseasytoimplement.theproposedelement 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 P h The authors do not have any conflict of interests with the contentofthepaper. 2α Acknowledgments P a w h This project was supported by the National Natural Science Foundation of China (no ), the National Basic Research Programs of China (no. 2010CB731503). Figure 8: V-notch specimen under the antisymmetric load. 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. ξ 1 and ξ 2 are the first and second eigenvalues, respectively 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. References [1] M. Z. Wang, Advanced Elasticity, Peking University Press, Beijing, China, 2002, (Chinese). [2]X.Fu,S.Cen,andY.Long, TheAnalyticalTrialFunction Method (ATFM) for finite element analysis of plane crack/notch problems, Key Engineering Materials, vol , pp , [3]X.FuandY.Long, Analysisofplanenotchproblemswith analytical trial functions method, Engineering Mechanics, vol. 20,no.4,pp.33 38,2003(Chinese). [4] Z.Niu,C.Cheng,J.Ye,andN.Recho, Anewboundaryelement approach of modeling singular stress fields of plane V-notch problems, International Solids and Structures, vol. 46, no.16,pp ,2009.

11 Mathematical Problems in Engineering 11 [5]Z.Niu,D.Ge,C.Cheng,J.Ye,andN.Recho, Evaluationof the stress singularities of plane V-notches in bonded dissimilar materials, Applied Mathematical Modelling, vol. 33, no. 3, pp , [6]C.Cheng,Z.Niu,H.Zhou,andN.Recho, Evaluationof multiple stress singularity orders of a V-notch by the boundary element method, Engineering Analysis with Boundary Elements, vol. 33, no. 10, pp , [7] X. C. Ping, M. C. Chen, and J. L. Xie, Finite element analyses of singular stresses in tips of V-notched anisotropic plates, in Proceedings of the International Conference on Mechanical Engineering and Mechanics, vol. 1-2, pp , Wuxi, China, [8] A. N. Stroh, Dislocations and cracks in anisotropic elasticity, vol. 3, pp , [9] A.N.Stroh, Steadystateproblemsinanisotropicelasticity, vol. 41, pp , [10] K. Wu and F. Chang, Near-tip fields in a notched body with dislocations and body forces, Applied Mechanics,vol. 60, no. 4, pp , [11] C. Dongye and T. C. T. Ting, Explicit expressions of Barnett- Lothe tensors and their associated tensors for orthotropic materials, Quarterly of Applied Mathematics, vol.47,no.4,pp , [12] B. Gross and A. Mendelson, Plane elastostatic analysis of V- notched plates, International Fracture Mechanics, vol. 8, no. 3, pp , [13] A. Carpinteri, M. Paggi, and N. Pugno, Numerical evaluation of generalized stress-intensity factors in multi-layered composites, International Solids and Structures, vol. 43, no. 3-4, pp , [14] D. H. Chen, Stress intensity factors for V-notched strip under tension or in-plane bending, International Fracture, vol.70,no.1,pp.81 97,1995. [15] M.-C. Chen and K. Y. Sze, A novel hybrid finite element analysis of bimaterial wedge problems, Engineering Fracture Mechanics,vol.68,no.13,pp ,2001. [16] M. Chen and X. Ping, Finite element analysis of piezoelectric corner configurations and cracks accounting for different electrical permeabilities, Engineering Fracture Mechanics, vol.74, no. 9, pp , 2007.

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