AXISYMMETRIC ELASTICITY PROBLEM OF CUBIC QUASICRYSTAL *

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1 Volume 9, Number April, 9-963//9/9- CHINESE PHYSICS c Chin. Phys. Soc. AXISYMMETRIC ELASTICITY PROBLEM OF CUBIC QUASICRYSTAL * Zhou Wang-min and Fan Tian-you Research Center of Materials Science, Beijing Institute of Technology, Beijing 8, China Received 8 September 999 A method for analyzing the elasticity problem of cubic quasicrystal is developed. The axisymmetric elasticity problem of cubic quasicrystal is reduced to a single higher-order partial differential equation by introducing a displacement function. As an example, the solutions of elastic field of cubic quasicrystal with a penny-shaped crack are obtained, and the stress intensity factor and strain energy release rate are determined. PACC: 6M; 67G I. INTRODUCTION Quasicrystal as a new structure of solid matter was discovered around 98 [,], it brings profound new ideas to the traditional condensed matter physics and encourages considerable theoretical and experimental studies on the physical and mechanical properties of the material, including the elasticity theory of the quasicrystal. [3,] Cubic quasicrystal is one of important threedimensional quasicrystals, Feng et al. [5] reported cubic quasicrystals with cubic symmetry3 observed in a VNiSi alloy cooled rapidly, Wang et al. [6] analyzed the structure of the material by using projective approach. Defects such as dislocation, disclination and stacking fault etc. in the material were observed soon after the discovery of the quasicrystal. [7,8] Crack is one of the important defects, its existence greatly influences the physical and mechanical properties of quasicrystals. The study on crack in quasicrystals is significant subjects. To describe the behaviour of defects in the material, elasticity theory of quasicrystals is necessary. In addition, appropriate mathematical methods are also needed for this purpose. The practice shows that the procedure of eliminating variables is powerful [9] which simplifies extremely the partial differential equations of elasticity of quasicrystals. Based on this, we further develop the Fourier analyses, and exact solutions for complicated defect problems of quasicrystals were found. In this paper, the formulation for the axisymmetric elasticity of cubic quasicrystals is suggested. By introducing a displacement function, the problem is reduced to solution of a single higher-order partial differential equation. As an example, the elastic field of a penny-shaped crack in the material is determined, as well as the stress intensity factor and the strain energy release rate, which provides some useful information for studying deformation and fracture of the quasicrystaline material. II. AXISYMMETRIC PROBLEM AND DIS- PACEMENT FUNCTION The stress-strain relationsthe generalized Hooke s law of cubic quasicrystal can be expressed as follows under an axisymmetric conditionabout z-axis symmetry by using cylindrical coordinatesr, θ, z. σ rr =C ε rr C ε θθ ε zz R E rr R E θθ E zz, a σ θθ =C ε θθ C ε rr ε zz R E θθ R E rr E zz, b σ zz =C ε zz C ε θθ ε rr R E zz R E θθ E rr, c σ zr =σ rz = C ε rz R E rz, H rr =R ε rr R ε θθ ε zz d K E rr K E θθ E zz, e H θθ =R ε θθ R ε rr ε zz K E θθ K E rr E zz, f H zz =R ε zz R ε θθ ε rr K E zz K E θθ E rr, g H zr =H rz = R ε rz K E rz, h where σ ij are the phonon stress components, H ij the phason stress components; ε ij the phonon strain components, E ij the phason strain components; C ij the elastic constants in phonon field, K ij the elastic con- * Project supported by the Research Fund of Doctoral Program of Higher Education and National Natural Science Foundation of China.

2 No. Axisymmetric Elasticity Problem of Cubic Quasicrystal 95 stants in phason field, R ij the phonon-phason coupling elastic constants. The cubic quasicrystal is one of specific ones in which the phonon field u and phason field w have the same irreducible representation, which are different from those of the elasticity of other quasicrystal system studied by Bak [3] and Socolar et al. [] ε rr = u r, ε θθ = u r r, ε zz = u z z, ε rz = ε zr = ur E rr = w r, E θθ = w r r, E zz = w z z, E rz = E zr = For this reason, we denote its phason strain tensor by E ij rather than w ij in the following. In addition, the present paper is devoted to the study of the axisymmetric deformation of quasicrystals, in the case of symmetry with respect to z-axis, all the stress, strain and displacement components are independent of θ, and u θ = w θ =, so the strain field are defined as z u z wr z w z, a, b the rest strain components are vanishing. The equilibrium equations in the absence of body forces are σ rr σ rz z σ rr σ θθ r =, σ zr σ zz z σ zr r =, 3a H rr H rz z H rr H θθ r Substituting Eqs. and into Eqs.3 leads to =, H zr u C r u r r r u r C C u z w R r w r r C u z r u z R w z r u R r u r r r u r w K r w r r R u z r u z K w z r r w r C C w z R R H zz z z C R R w z u r z u r r z w r z r R R u z r w r R R w z K K H zr r u r z z R z R K K w z u r z u r r z w r z r =, 3b w r z =, u C z z w r w R z z z =, u r z z K u R z z w r z =, w r w K z z z =. These are named the equilibrium equations expressed by displacements. To simplify the equations, a new unknown function Fr, z is introduced, such as u r = [A z A r r z A 3 z u z = [B 3 B r r z B 3 r w r = [C z C r r z C 3 z w z = [D 3 D r r z D 3 r a b c d ] F, 5a 6 ] F, 5b z B z 6 ] F, 5c 6 z D z 6 ] F, 5d

3 96 Zhou Wang-min et al. Vol.9 where A i, B i, C i and D i are known constants composed of the elastic constants given in the Appendix, then Eqs. are automatically satisfied if Fr, z satisfies the following partial differential equation, [ 8 z 8 b 6 r z 6 c r z d r 3 z e r ] F =, 6 where b, c d and e are known constants composed of the elastic constants given in theappendix. Substituting Eqs.5 into Eqs.,the stress components σ ij and H ij can be expressed by Fr, z. Because these expressions are quite lengthy, they are omitted here. The axisymmetric elasticity problem of cubic quasicrystal u r = u z = w r = w z = σ rr = σ θθ = σ zz = r r σ rz =σ zr = H rr = H θθ = H zz = r r H rz =H zr = A ξ 6 ddz A ξ d3 B ξ 7 B ξ 5 d C ξ 6 ddz C ξ d3 D ξ 7 D ξ 5 d E ξ 7 ddz E ξ 5 d3 dz 3 A 3ξ d5 dz B 3ξ 3 d dz B ξ d6 dz 6 E 5 ξ 6 ddz E 6ξ d3 F ξ 7 ddz F ξ 5 d3 E 5 ξ 6 ddz E 6ξ d3 G ξ 7 ddz G ξ 5 d3 H ξ 8 H ξ 6 d I ξ 7 ddz I ξ 5 d3 I 5 ξ 6 ddz I 6ξ d3 J ξ 7 ddz J ξ 5 d3 I 5 ξ 6 ddz I 6ξ d3 K ξ 7 ddz K ξ 5 d3 dz 3 C 3ξ d5 is reduced to solving Eq.6. III. HANKEL TRANSFORM AND THE IN- TEGRAL EXPRESSIONS OF THE SOLU- TIONS Introducing the Hankel transform of zero order Fξ, z = rfr, zj ξrdr, where J ξr is the zero order Bessel function of first kind. Performing the Hankel transform on Eq.6 yields [ ] d 8 d6 d d bξ cξ dξ6 dz8 dz6 dz dz eξ8 F =, 7 in which ξ represents the Hankel transform parameter. Using the relations among stress, displacement and function Fr, z, the stress and displacement components can be expressed by Fξ, z dz D 3ξ 3 d dz D ξ d6 dz 6 Fξ, zj ξrdξ, Fξ, zj ξrdξ, Fξ, zj ξrdξ, dz 3 E 3ξ 3 d5 E ξ d7 dz 3 E 7ξ d5 dz 3 F 3ξ 3 d5 F ξ d7 dz 3 E 7ξ d5 dz 3 G 3ξ 3 d5 G ξ d7 Fξ, zj ξrdξ, Fξ, zj ξrdξ Fξ, zj ξrdξ, Fξ, zj ξrdξ Fξ, zj ξrdξ, dz H 3ξ d dz H ξ d6 dz 6 dz 3 I 3ξ 3 d5 I ξ d7 dz 3 I 7ξ d5 dz 3 J 3ξ 3 d5 J ξ d7 dz 3 I 7ξ d5 Fξ, zj ξrdξ, Fξ, zj ξrdξ, Fξ, zj ξrdξ Fξ, zj ξrdξ, Fξ, zj ξrdξ Fξ, zj ξrdξ, dz 3 K 3ξ 3 d5 K ξ d7 L ξ 8 L ξ 6 d dz L 3ξ d dz L ξ d6 dz 6 Fξ, zj ξrdξ, Fξ, zj ξrdξ, 8a 8b 8c 8d 8e 8f 8g 8h 8i 8j 8k 8l

4 No. Axisymmetric Elasticity Problem of Cubic Quasicrystal 97 where E i, F i, G i, H i, I i, J i, K i and L i are known constants composed of the elastic constants shown in the Appendix, J ξr is the first order Bessel function of first kind. As a typical example of axisymmetric problem, the elastic problem of cubic quasicrystal with a pennyshaped crack under the action of axisymmetric stress is studied below. IV. SOLUTIONS OF A PENNY-SHAPED CRACK PROBLEM Assume a penny-shaped crack with radius a in the center of the cubic quasicrystal material, the size of the crack is much smaller than the solid, so that the size of the material can be considered as infinite, at the infinity, the quasicrystal material is subjected to a tension p in z-direction. The origin of the Coordinate system is at the center of the crack as shown in Fig. of the stress components are zero. So, it is enough to solve the boundary-value problems of the partial differential equation 6. In the Hankel transform domain, Eq.6 is reduced to an ordinary differential equation 7 with the follwing characteristic roots γ, = ±λ ξ, γ 3, = ±λ ξ, γ 5,6 = ±λ 3 ξ, γ 7,8 = ±λ ξ, where λ i i =,, 3, are known constants composed of the elastic constants given in the Appendix. We may assume λ i λ j i j and λ i > if λ i = λ j for some values of i and j, or λ i, or λ i are complex numbers for some values of i, for these cases the problem can be solved similarly. Only when λ i are pure imaginary numbers for some values of i, there is no physical meaning for solutions, from the condition -, we obtain Fξ, z =A ξe λξz A ξe λξz A 3 ξe λ3ξz A ξe λξz, where A i ξi =,, 3, are to be determined. Substituting Eq. into Eqs.8, and from the conditions - and -3, we obtain the dual integral equations Fig.. The penny-shaped crack in a cubic quasicrystal and the coordinate system. From the symmetry of the problem, it is enough to study the upper half-space z > or lower half-space z <. In this case, for studying the upper half-space, the boundary conditions of the problem are described by r z : σzz = p, Hzz =, 9a z =, r a : σ zz = σ rz = ; H zz = H rz =, 9b z =, r > a : σ rz =, u z = ; H rz =, w z =, 9c Using the principle of linear superposition, the problem is reduced to a superposition of two problems, one of them has the boundary conditions r z : σ ij = H ij = ; a z =, r a : σ zz = p, σ rz = ; H zz = H rz = ; b z =, r > a : σ rz =, u z = ; H rz =, w z =. c The other is that the quasicrystal material without crack is subjected to a tension p at infinity, the solution of the problem is known, i.e:, σ zz = p the rest A i ξξ 8 J ξrdξ = M i p, r a, a A i ξξ 7 J ξrdξ =, r > a, b where M i i =,, 3, are known constants composed of the elastic constants listed in the Appendix. Solving the dual integral equations, we obtain A i ξ = a M i pπaξ / ξ 7 J 3/ aξ, 3 where J 3/ aξ is the Bessel function with order 3/ of the first kind. The problem is then solved. Substituting Eqs. and 3 into Eqs.8, the analytical expressions of all the stress and displacement components can be evaluated. Thus stress intensity factor K I, strain energy W I and strain energy release rate G I can be defined as follows [5] K I = lim πr aσzz r,, r a W I = a G I = πa πrσ zz r, u z r, dr, 5 W I a. 6

5 98 Zhou Wang-min et al. Vol.9 Calculation shown that K I = p a π, W I = Mp a 3, G I = 3Mp a π, 7 where M is known constant composed of the elastic constants shown in the Appendix V. CONCLUSION AND DISCUSSION It has been shown that the basic equation system of elasticity for the quasicrystal is much more complicated than conventional crystals. The solution in straightforward manner is not available. In the work of our group, including the present paper, we have developed a procedure, which is named the eliminating variable procedure, by introducung the displacement function or stress function to simplify the original basic equations to a single higher-order equation, In this way, the complicated boundary-value problem of elasticity of cubic quasicrystal is solved, and the exact analytic solution for a penny-shaped crack is achived. The experiment [6] indicated that quasicrystal is quite brittle, and in Ref.[6] was reported the measurment of fracture toughness of quasicrystal Al 65 Cu Co 5, It is well-known that the failure of brittle solid is always connected with crack that was observed by Griffith [7]. The present work attempts to extend the classical Griffith work to quasicrystal. However, at present, due to lack of experimental results for penny-shaped crack in cubic quasicrystal, the exact analytic solution of this paper provides a complete theoretical predication for experiment investigation and an assessment for numerical analysis. The formulation and mathematical method developed in this study can also be used for reference for other problems of elasticity of the quasicrystal. APPENDIX The expressions of the constants in the text are as follows: A = R b K b C K R, A = R b 3 K b C K R R R b B = K K b C C K K R R, { R R b B = 3 K K b C C K K R R,, A 3 = R b 5 K b 6, c b c b C K R [C C K K R R ] R R b B 3 = 5 K K b 6 C C K K R R, c b 3 c b C K R [C C K K R R ], c B = b 5 c b 6 C K R [C C K K R R ], C = C b R b C K R, C = C b 3 R b C K R, C 3 = C b 5 R b 6, C C b D = R R b C C K K R R, C C b D = 3 R R b C C K K R R, c 3 b c b C K R [C C K K R R ], C C b D 3 = 5 R R b 6 C C K K R R, c 3 b c b 3 C K R [C C K K R R ], c D = 3 b 6 c b 5 C K R [C C K K R R ], },

6 No. Axisymmetric Elasticity Problem of Cubic Quasicrystal 99 where where c =R R R K R K K K C K R R, c =R R C K R R K K C R C R, c 3 =R R C K R R C C R K R K, c =R R C R C R C C C K R R, R K R K b = C C K K R R, b = R R R K C C C C K K R R, b 3 = K R R R K K C C K K R R K R R R K K R c K c 3 [C C K K R R ], b = R R R K C C C C K K R R C K K R R R R c K c [C C K K R R ], R c b 5 = K c 3 [C C K K R R ], R c b 6 = K c [C C K K R R ], b = C C K K R R C K R a b 5 a 5 b a 6 b 3 a 3 b 6, c = C C K K R R C K R a b 6 a 6 b a 3 b a b 3 a 5 b a b 5, d = C C K K R R C K R a b 3 a 3 b a b a b, e = C C K K R R C K R a b a b, a = R R R C K K C C K K R R, C R C R a = C C K K R R, a 3 = R R R C K K C C K K R R K C C R R R C c R c 3 [C C K K R R ], a = C R R R C C C C K K R R C R R R C C C c R c [C C K K R R ], C c a 5 = R c 3 [C C K K R R ], C c a 6 = R c [C C K K R R ], C K R E = C C K K R R,

7 3 Zhou Wang-min et al. Vol.9 E =b C R C R b 3 [R R R C K K ]b C C K K R R C c R c 3 b C c R c b [C C K K R R ], E 3 = b 6 C R C R b 5 [R R R C K K ]b 6 C C K K R R C c R c 3 b C c R c b 3 [C C K K R R ], C c E = R c 3 b 6 C c R c b 5 [C C K K R R ], E 5 = C R C R b R R R C K C K b, E 6 = C R C R b 3 R R R C K C K b, E 7 = C R C R b 5 R R R C K C K b 6, C K R F =b C C K K R R C R C R b R R C K b, F = C R C R b 3 C K R R b, C R C R b 3 [R R R C K K ]b C C K K R R C c R c 3 b C c R c b [C C K K R R ], F 3 = C R C R b 5 C K R R b 6, C R C R b 5 [R R R C K K ]b 6 C C K K R R C c R c 3 b C c R c b 3 [C C K K R R ], F =E, G = C R C R b R R C K b C K R [C R R R C C ]b [C K K R R R ]b C C K K R R, G = C R C R b 3 C K R R b C K R [R C C C R R ]b 3 [R R R C K K ]b C C K K R R, C c R c 3 b C c R c b C K R [C C K K R R ], G 3 = C R C R b 5 C K R R b 6 C K R [R C C C R R ]b 5 [R R R C K K ]b 6 C C K K R R, C c R c 3 b C c R c b 3 C K R [C C K K R R ], C c G = R c 3 b 6 C c R c b 5 [C C K K R R ],

8 No. Axisymmetric Elasticity Problem of Cubic Quasicrystal 3 H = [R C C C R R ]b [R R R C K K ]b C C K K R R, H = C R C R b R R C K b C K R C R C R b 3 [C K K R R R ]b C C K K R R, C c R c b C c R c 3 b [C C K K R R ], H 3 = C R C R b 3 R R C K b C K R C R C R b 5 [C K K R R R ]b 6 C C K K R R, C c R c b 3 C c R c 3 b [C C K K R R ], H = C R C R b 5 R R C K b 6 C c R c b 5 C c R c 3 b 6 [C C K K R R ], I =b R K R K b [R R R K C C ]b C C K K R R, I = b 3 R K R K b [R R R K C C ]b 3 C C K K R R, R c K c b R c K c 3 b C K R [C C K K R R ], I 3 =b 5 R K R K b 6 [R R R K C C ]b 5 C C K K R R, R c K c b 3 R c K c 3 b [C C K K R R ], R c I = K c b 5 R c K c 3 b 6 [C C K K R R ], I 5 = b R R C K b R K R K b, I 6 =b 3 R R C K b 3 R K R K b, I 7 = b 5 R R C K b 5 R K R K b 6, J = C K R R b R K R K b C K R [R R R K C C ]b R K R K b C C K K R R, J = R R C K b 3 R K R K b C K R [K C C R R R ]b 3 R K R K b C C K K R R, K c 3 R c b K c R c b C K R [C C K K R R ], J 3 = R R C K b 5 R K R K b 6 C K R [K C C R R R ]b 5 R K R K b 6 C C K K R R,

9 3 Zhou Wang-min et al. Vol.9 K c 3 R c b K c R c b 3 C K R [C C K K R R ], K c J = 3 R c b 6 K c R c b 5 [C C K K R R ], K = C K R R b R K R K b C K R [R R R K C C ]b [R K K K R R ]b C C K K R R, K = R R C K b 3 R K R K b C K R [K C C R R R ]b 3 [K R R R K K ]b C C K K R R, R c K c 3 b R c K c b [C C K K R R ], K 3 = R R C K b 5 R K R K b 6 C K R [K C C R R R ]b 5 [K R R R K K ]b 6 C C K K R R, R c K c 3 b R c K c b 3 [C C K K R R ], R c K = K c 3 b 6 R c K c b 5 [C C K K R R ], L = [K C C R R R ]b R K R K b C C K K R R, L = C K R R b R K R K b C K R [R R R K C C ]b 3 R K R K b C C K K R R, R c K c b R c K c 3 b [C C K K R R ], L 3 = C K R R b 3 R K R K b C K R [R R R K C C ]b 5 R K R K b 6 C C K K R R, R c K c b 3 R c K c 3 b [C C K K R R ], L = C K R R b 5 R K R K b 6 C K R R c K c b 5 R c K c 3 b 6 C K R [C C K K R R ]. λ = [ b b c t λ = [ b b c t λ 3 = [ b b c t λ = [ b b c t b b c t t ] / t e ξ, b b c t t ] / t e ξ, b b c t t ] / t e ξ, b b c t t ] / t e ξ,

10 No. Axisymmetric Elasticity Problem of Cubic Quasicrystal 33 where, t = q q /3 p3 q q /3 7 p3, 7 p = bd e c 3, q = 7 c3 c 3 bd e b e ce d. M i = i, i =,, 3,, in which a a a 3 a a = a a a 3 a a 3 a a 3 a 3 a 33 a, = a 3 a 33 a 3 3 a a a a 3 a a 3 a a a a a 3 = a 3 a 3 a 3 a a a, a a 3 = a 3 a 3 a 33 a a a, 3 a, a 3 a = a 3 a 33 a 3 a a 3 a, a i = G λ i G λ 3 i G 3λ 5 i G λ 7 i, a i = K λ i K λ 3 i K 3λ 5 i K λ 7 i, a 3i = H H λ i H 3λ i H λ 6 i, a i = L L λ i L 3λ i L λ 6 i, M =, in which 3 a a a 3 a = a 3 a 3 a 33 a 3 a a a 3 a, a 5i = B B λ i B 3 λ i B λ 6 i. a 5 a 5 a 53 a 5 REFERENCES [] D. Shechtman, I. Blech, D. Gratias and J. W. Cahn, Phys. Rev. Lett., 5398, 95. [] H. Q. Ye, D. N. Wang and K. H. Kuo, Ultramicroscopy, 6985, 73. [3] D. H. Ding, W. G. Yang, C. Z. Hu and R. H. Wang, Phys. Rev., B8993, 73. [] W. G. Yang, R. H. Wang, D. H. Ding and C. Z. Hu, Phys. Rev., B8993, [5] Y. C. Feng. et al, J. Phys.: Condensed Matter, 989, [6] Y. C. Feng. et al, J. Phys: Condensed Matter, 99, 979. [7] R. H. Wang et al, Acta Crystal., A599, 366. [8] P. De and R. A. Pelcovits, Phys. Rev., B35987, 869. [9] P. De and R. A. Pelcovits, Phys. Rev., B36987, 93. [] X. F. Li and T. Y. Fan, Chin. Phys. Lett., 5998, 78. [] X. F. Li, X. Y. Dun, T. Y. Fan and Y. F. Sun, J. Phys. Condens. Matter, 999, 73. [] T. Y. Fan, X. F. Li and Y. F. Sun, Acta Physica Sinnca Overseas Edition, 8999, 88. [3] W. M. Zhou and T. Y. Fan, Dislocation in two-dimensional octagonal quasicrystal and exact solutions, submitted to Int. J. Mod. Phys., 999. [] P. Bak, Phys. Rev., B3985, 576. [5] J. E. S. Socolar et al, Phys. Rev., B3987, 335. [6] T. Y. Fan, Foundation of Fracture Mechanics, Jiangsu Science and Technology Press, Nanjing, China 978, in Chinese. [7] X. M. Meng, B. Y. Tong and Y. K. Wu, Acta Metallugia Sinica, 399, 6, in Chinese. [8] A. A. Grriffith, Phil. Trans, Roy. Soc., A9, 63.