Analytical solutions for some defect problems in 1D hexagonal and 2D octagonal quasicrystals

PRAMANA c Indian Academy of Sciences Vol. 70 No. 5 journal of May 008 physics pp. 911 933 Analytical solutions for some defect problems in 1D hexagonal and D octagonal quasicrystals X WANG 1 and E PAN 1 1 Department of Civil Engineering University of Akron OH 4435-3905 USA Department of Applied Mathematics University of Akron OH 4435-3905 USA Corresponding author E-mail: xuwang@uakron.edu; pan@uakron.edu MS received 7 November 007; revised 4 December 007; accepted 10 December 007 Abstract. We study some typical defect problems in one-dimensional 1D) hexagonal and two-dimensional D) octagonal quasicrystals. The first part of this investigation addresses in detail a uniformly moving screw dislocation in a 1D hexagonal piezoelectric quasicrystal with point group 6mm. A general solution is derived in terms of two functions ϕ 1 ϕ which satisfy wave equations and another harmonic function ϕ 3. Elementary expressions for the phonon and phason displacements strains stresses electric potential electric fields and electric displacements induced by the moving screw dislocation are then arrived at by employing the obtained general solution. The derived solution is verified by comparison with existing solutions. Also obtained in this part of the investigation is the total energy of the moving screw dislocation. The second part of this investigation is devoted to the study of the interaction of a straight dislocation with a semi-infinite crack in an octagonal quasicrystal. Here the crack penetrates through the solid along the period direction and the dislocation line is parallel to the period direction. We first derive a general solution in terms of four analytic functions for plane strain problem in octagonal quasicrystals by means of differential operator theory and the complex variable method. All the phonon and phason displacements and stresses can be expressed in terms of the four analytic functions. Then we derive the exact solution for a straight dislocation near a semi-infinite crack in an octagonal quasicrystal and also present the phonon and phason stress intensity factors induced by the straight dislocation and remote loads. Keywords. Quasicrystal; dislocation; crack; stress intensity factors; piezoelectricity; general solution. PACS Nos 7.10.Fk; 61.7.Bb; 61.7.Nn; 6.40+i; 77.65.-j; 71.3.Ft 1. Introduction Quasicrystals which were first discovered in 1984 by Shechtman et al [1] possess a type of ordered structure characterized by crystallographically disallowed long-range orientational symmetry and by long-range quasiperiodic translational order. Since the discovery of quasicrystals the elastic theory of quasicrystals continues to attract investigators attention [ 7]. It has been experimentally verified 911

X Wang and E Pan that quasicrystals can really exist as stable phases then it is necessary to take quasicrystals as a thermodynamic system and to establish the corresponding thermodynamics of equilibrium properties. Yang et al [8] first generalized the thermodynamics of equilibrium thermal electrical magnetic and elastic properties to the case of quasicrystals. By group theory they derived physical property tensors for two-dimensional pentagonal octagonal decagonal and dodecagonal and three-dimensional icosahedral and cubic quasicrystals. Recently Li and Liu [9] derived physical property tensors for one-dimensional quasicrystals according to group representation theory. They presented particular matrix forms of the thermal expansion coefficient tensors and piezoelectric coefficient tensors under 31 point groups for the 1D quasicrystals. Most recently Rao et al [10] determined the maximum number of non-vanishing and independent second-order piezoelectric coefficients in pentagonal and icosahedral quasicrystals also by using group representation theory. With the development of the elasticity theory of quasicrystals theoretical investigations of dislocation and crack problems in quasicrystals also receive focused attention. Ding et al [11] obtained the displacement fields induced by a straight dislocation line along the period direction of decagonal quasicrystals. Yang et al [1] derived an analytic expression for the elastic displacement fields induced by a dislocation in an icosahedral quasicrystal. Li et al [13] considered an infinite decagonal quasicrystal containing a Griffith crack which penetrates through the solid along the period direction. Zhou and Fan [14] studied an octagonal quasicrystal weakened by a Griffith crack. Wang and Zhong [6] studied the interaction between a semi-infinite crack and a line dislocation in a decagonal quasicrystalline solid. Liu et al [15] addressed the interaction between a screw dislocation and a semi-infinite crack in one-dimensional quasicrystals. In the first part of this paper we investigate the problems of dislocation dynamics in a one-dimensional hexagonal piezoelectric quasicrystal with point group 6mm by employing the results of Li and Liu [9] as our starting step. A general solution is derived in terms of two functions ϕ 1 ϕ which satisfy wave equations and one harmonic function ϕ 3. Elementary expressions for the phonon and phason displacements strains stresses electric potential electric fields and electric displacements induced by a straight screw dislocation line parallel to the quasiperiodic axis moving along a period direction in this piezoelectric quasicrystal are then obtained by employing the obtained general solution. Also derived is the total energy of the moving dislocation. In the second part of this paper we address in detail the interaction problem between a straight dislocation and a semi-infinite crack in an octagonal quasicrystalline solid. We first present the general solution for plane strain problems in octagonal quasicrystals. All of the phonon and phason fields can be expressed in terms of four analytic functions. We then derive the field potentials for i) a straight dislocation in an infinite octagonal quasicrystal; ii) asymptotic fields around a semi-infinite crack in an octagonal quasicrystal; and iii) a straight dislocation near a semi-infinite crack in an octagonal quasicrystal. We also derive analytic expressions of the phonon and phason stress intensity factors induced by the straight dislocation. 91 Pramana J. Phys. Vol. 70 No. 5 May 008

Some defect problems in 1D hexagonal and D octagonal quasicrystals. A uniformly moving screw dislocation in a 1D hexagonal piezoelectric quasicrystal with point group 6 mm.1 Basic formulations The generalized Hooke s law for 1D hexagonal piezoelectric quasicrystal with point group 6 mm whose period plane is the x 1 x )-plane and whose quasiperiodic direction is the x 3 -axis is given by [59] σ 11 = c 11 ε 11 + c 1 ε + c 13 ε 33 + R 1 w 33 e 1) 31 E 3 σ = c 1 ε 11 + c ε + c 13 ε 33 + R 1 w 33 e 1) 31 E 3 σ 33 = c 13 ε 11 + c 13 ε + c 33 ε 33 + R w 33 e 1) 33 E 3 σ 3 = σ 3 = c 44 ε 3 + R 3 w 3 e 1) 15 E σ 13 = σ 31 = c 44 ε 31 + R 3 w 31 e 1) 15 E 1 σ 1 = σ 1 = c 66 ε 1 H 33 = R 1 ε 11 + ε ) + R ε 33 + K 1 w 33 e ) 33 E 3 H 31 = R 3 ε 31 + K w 31 e ) 15 E 1 H 3 = R 3 ε 3 + K w 3 e ) 15 E D 3 = e 1) 31 ε 11 + ε ) + e 1) 33 ε 33 + e ) 33 w 33+ 33 E 3 D 1 = e 1) 15 ε 31 + e ) 15 w 31+ 11 E 1 D = e 1) 15 ε 3 + e ) 15 w 3+ 11 E 1) where σ ij and H ij are the phonon and phason stress components D i are the electric displacements; ε ij and w 3j are the phonon and phason strains E i are the electric fields; c 11 c 1 c 13 c 33 c 44 c 66 are six elastic constants in the phonon field and c 66 = c 11 c 1 )/; K 1 and K are two elastic constants in the phason field; R 1 R R 3 are three phonon phason coupling elastic constants; e 1) ij and e ) ij are piezoelectric coefficients and 11 and 33 are two dielectric coefficients. The strain displacement and electric field electric potential relations are given by ε 11 = u 11 ε = u ε 33 = u 33 ε 1 = 1 u 1 + u 1 ) ε 31 = 1 u 13 + u 31 ) ε 3 = 1 u 3 + u 3 ) w 31 = w 31 w 3 = w 3 w 33 = w 33 E 1 = φ 1 E = φ E 3 = φ 3 ) where u i i = 1 3) are three phonon displacement components w 3 is the phason displacement φ is the electric potential; a comma followed by ii = 1 3) denotes partial derivative with respect to the ith spatial coordinate. In the absence of body force and electric charge density the equations of motion and the charge equilibrium equation are [416] Pramana J. Phys. Vol. 70 No. 5 May 008 913

X Wang and E Pan σ 111 + σ 1 + σ 133 = ρü 1 σ 11 + σ + σ 33 = ρü σ 311 + σ 3 + σ 333 = ρü 3 H 311 + H 3 + H 333 = ρẅ 3 D 11 + D + D 33 = 0 3) where the superdot means the differentiation with respect to time and ρ is the mass density of the piezoelectric quasicrystalline solid. For the anti-plane shear problem in which the non-trivial displacements u 3 w 3 and the electric potential φ are independent of x 3 the equations of motion and the charge equilibrium equation can be expressed in terms of u 3 w 3 and φ as c 44 u 3 + R 3 w 3 + e 1) 15 φ = ρü 3 R 3 u 3 + K w 3 + e ) 15 φ = ρẅ 3 e 1) 15 u 3 + e ) 15 w 3 11 φ = 0 4) where = + is the two-dimensional Laplace operator. x 1 x It follows from 4) 3 that φ can be expressed in terms of u 3 and w 3 as φ = e1) 15 u 3 + e) 15 w 3. 5) 11 11 Inserting the above into 4) 1 and eliminating φ we arrive at c 44 u 3 + R 3 w 3 = ρü 3 R 3 u 3 + K w 3 = ρẅ 3 6) where c 44 = c 44 + e 1) 15 / 11 is the piezoelectrically stiffened elastic constant in the phonon field K = K + e ) 15 / 11 is the piezoelectrically stiffened elastic constant in the phason field and R 3 = R 3 + e 1) 15 e) 15 / 11 is the piezoelectrically stiffened phonon phason coupling elastic constant. Next we introduce two new functions ϕ 1 and ϕ given by u 3 = αϕ 1 R 3 ϕ w 3 = R 3 ϕ 1 + αϕ 7) where α = 1 [ ] c 44 K + c 44 K ) + 4 R 3. 8) Consequently eq. 6) can be rewritten in the following canonical form: where ϕ 1 = 1 s 1 ϕ 1 t ϕ = 1 ϕ s t 9) 914 Pramana J. Phys. Vol. 70 No. 5 May 008

Some defect problems in 1D hexagonal and D octagonal quasicrystals c 44 + K ) + c 44 K ) + 4 R 3 s 1 = ρ c 44 + K ) c 44 K ) + 4 R 3 s = ρ are two wave speeds under anti-plane shear conditions. Meanwhile if we introduce a third function ϕ 3 given by φ = ϕ 3 + e1) 15 u 3 + e) 15 w 3 = ϕ 3 + e1) 15 α + e) 11 11 then eq. 5) can be written as 11 15 R 3 10) ϕ 1 + e) 15 α e1) 15 R 3 11 ϕ 11) ϕ 3 = 0. 1) The nontrivial stresses and electric displacements can be expressed in terms of three new functions ϕ 1 ϕ and ϕ 3 as σ 13 = σ 31 = c 44 α + R 3)ϕ 11 + R 3 α c 44 )ϕ 1 + e 1) 15 ϕ 31 σ 3 = σ 3 = c 44 α + R 3)ϕ 1 + R 3 α c 44 )ϕ + e 1) 15 ϕ 3 H 31 = R 3 α + K )ϕ 11 + K α R 3)ϕ 1 + e ) 15 ϕ 31 H 3 = R 3 α + K )ϕ 1 + K α R 3)ϕ + e ) 15 ϕ 3 D 1 = 11 ϕ 31 D = 11 ϕ 3 13) Consequently eqs 7) and 11) for the displacements and electric potential and eq. 13) for stresses and electric displacements give a general solution for a kind of elasticity dynamic problems in a 1D hexagonal piezoelectric quasicrystal with point group 6mm where the dislocation line is parallel to the quasiperiodic axis. The three unknown functions ϕ 1 ϕ and ϕ 3 can be determined from the appropriate boundary or initial-boundary conditions.. Electroelastic fields induced by a moving piezoelectric screw dislocation Now consider a straight screw dislocation line parallel to the quasiperiodic x 3 axis. The screw dislocation suffers a finite discontinuity in the displacements and in the electric potential across the slip plane. We further assume that the screw dislocation moves at a constant velocity V along the x 1 -axis. We choose a new coordinate system x y) which moves together at the same velocity V as the dislocation and the moving and the fixed coordinate systems coincide at t = 0. Make the following transformation to transform the fixed coordinate system x 1 x ) to the moving coordinate system x y): Pramana J. Phys. Vol. 70 No. 5 May 008 915

X Wang and E Pan x = x 1 V t y = x 14) then eqs 9) and 1) can be transformed to the following equations in the moving coordinates β1 ϕ 1 x + ϕ 1 y = 0 ϕ β x + ϕ y = 0 ϕ 3 x + ϕ 3 = 0 15) y where β 1 = 1 V /s 1 β = 1 V /s. 16) The general solution of eq. 15) can be immediately arrived at ϕ 1 = Im f 1 z 1 ) ϕ = Im f z ) ϕ 3 = Im f 3 z) 17) where z 1 = x + iβ 1 y z = x + iβ y z = x + iy. The screw dislocation investigated here is defined as du 3 = b dw 3 = d dφ = φ 18) or equivalently dϕ 1 = αb + R 3 d α + R 3 dϕ = αd R 3 b α + R 3 dϕ 3 = φ e1) 15 b + e) 15 d 11 19) where b is the phonon displacement jump across the slip plane d is the phason displacement jump across the slip plane φ is the electric potential jump across the slip plane. Consequently the three analytic functions f 1 z 1 ) f z ) and f 3 z) take the forms f 1 z 1 ) = f z ) = αb + R 3 d πα + R 3 ) ln z 1 αd R 3 b πα + R 3 ) ln z f 3 z) = 11 φ e 1) 15 b e) 15 d π 11 ln z. 0) In view of eqs 7) 11) 17) and 0) the phonon and phason displacements and electric potential can be expressed in terms of the mechanical and electric dislocations as follows: 916 Pramana J. Phys. Vol. 70 No. 5 May 008

Some defect problems in 1D hexagonal and D octagonal quasicrystals u 3 = ααb + R 3 d) πα + R 3 ) tan1 β 1 x x 1 V t + R 3 R 3 b αd) πα + R 3 ) tan1 β x x 1 V t w 3 = R 3 αb + R 3 d) πα + R 3 ) tan1 β 1 x x 1 V t + ααd R 3 b) πα + R 3 ) tan1 β x x 1 V t φ = 11 φ e 1) 15 b e) 15 d π 11 tan 1 x x 1 V t + e1) 15 α + e) 15 R 3 )αb + R 3 d) π 11 α + R 3 ) tan 1 β 1 x x 1 V t + e) 15 α e1) 15 R 3 )αd R 3 b) π 11 α + R 3 ) tan 1 β x x 1 V t 1) Similarly the phonon and phason strains are given by γ 31 = ε 31 = ααb + R 3 d) πα + R 3 ) β 1 x x 1 V t) + β 1 x R 3 R 3 b αd) β x πα + R 3 ) x 1 V t) + β x γ 3 = ε 3 = ααb + R 3 d) πα + R 3 ) β 1 x 1 V t) x 1 V t) + β 1 x + R 3 R 3 b αd) β x 1 V t) πα + R 3 ) x 1 V t) + β ) x the electric fields by w 31 = R 3 αb + R 3 d) πα + R 3 ) β 1 x x 1 V t) + β 1 x ααd R 3 b) β x πα + R 3 ) x 1 V t) + β x w 3 = R 3 αb + R 3 d) πα + R 3 ) β 1 x 1 V t) x 1 V t) + β 1 x + ααd R 3 b) β x 1 V t) πα + R 3 ) x 1 V t) + β 3) x E 1 = 11 φ e 1) 15 b e) 15 d x π 11 x 1 V t) + x + e1) 15 α + e) R 15 3 )αb + R 3 d) β 1 x π 11 α + R 3 ) x 1 V t) + β1 x + e) 15 α e1) R 15 3 )αd R 3 b) β x π 11 α + R 3 ) x 1 V t) + β x Pramana J. Phys. Vol. 70 No. 5 May 008 917

X Wang and E Pan E = 11 φ e 1) 15 b e) 15 d x 1 V t) π 11 x 1 V t) + x e1) 15 α + e) R 15 3 )αb + R 3 d) β 1 x 1 V t) π 11 α + R 3 ) x 1 V t) + β1 x e) 15 α e1) R 15 3 )αd R 3 b) β x 1 V t) π 11 α + R 3 ) x 1 V t) + β 4) x the phonon and phason stresses by σ 13 = σ 31 = c 44α + R 3)αb + R 3 d) πα + R 3 ) β 1 x x 1 V t) + β 1 x R 3 α c 44 )αd R 3 b) πα + R 3 ) β x x 1 V t) + β x e1) 15 11 φ e 1) 15 b e) 15 d) x π 11 x 1 V t) + x σ 3 = σ 3 = c 44α + R 3)αb + R 3 d) πα + R 3 ) β 1 x 1 V t) x 1 V t) + β 1 x + R 3 α c 44 )αd R 3 b) πα + R 3 ) β x 1 V t) x 1 V t) + β x + e1) 15 11 φ e 1) 15 b e) π 11 15 d) x 1 V t x 1 V t) + x 5) H 31 = R 3 α + K )αb + R 3 d) πα + R 3 ) β 1 x x 1 V t) + β 1 x K α R 3)αd R 3 b) πα + R 3 ) β x x 1 V t) + β x e) 15 11 φ e 1) 15 b e) 15 d) x π 11 x 1 V t) + x H 3 = R 3 α + K )αb + R 3 d) β 1 x 1 V t) πα + R 3 ) x 1 V t) + β1 x + K α R 3)αd R 3 b) πα + R 3 ) β x 1 V t) x 1 V t) + β x + e) 15 11 φ e 1) 15 b e) π 11 and the electric displacements are given by 15 d) x 1 V t x 1 V t) + x 6) 918 Pramana J. Phys. Vol. 70 No. 5 May 008

Some defect problems in 1D hexagonal and D octagonal quasicrystals D 1 = 11 φ e 1) 15 b e) π 15 d D = 11 φ e 1) 15 b e) π 15 d x x 1 V t) + x x 1 V t x 1 V t) + x. 7) It can be easily checked that when R 3 = e ) 15 = 0 the results derived here can just reduce to those obtained by Wang and Zhong [17] for a screw dislocation moving in piezoelectric crystals..3 Energy of the moving piezoelectric screw dislocation The total energy W per unit length on the dislocation line of the moving screw dislocation is composed of the kinetic energy W k and the potential energy W p which are given by the following integrals: W k = ρ u 3 + ẇ 3)dx 1 dx 8) W p = 1 σ 3j u 3j + H 3j w 3j + D j φ j )dx 1 dx 9) where the integration should be taken over the circular annulus r 0 r R 0 with r 0 being the radius of the screw dislocation core. The specific expressions of W k and W p are finally given by where W k = k k 4π ln R 0 r 0 ρv k k = α + R 3 ) W p = k p 4π ln R 0 r 0 30) [ αb + R 3 d) + αd R ] 3 b) 31) β 1 β k p = β 1 + 1 β 1 ) c44 α + K R 3 + α R 3) α + R 3 ) αb + R 3 d) + β + 1 ) c44 R 3 + K α α R 3) β α + R αd R 3 3 b) ) 11 φ e1) 15 b + e) 11 15 d Consequently the total energy is given by W = k k + k p 4π ). 3) ln R 0 r 0. 33) Pramana J. Phys. Vol. 70 No. 5 May 008 919

X Wang and E Pan It can be observed from the above that the total energy W becomes infinite when V mins 1 s. Thus mins 1 s is the limit of the velocity of the screw dislocation. In addition when V mins 1 s the total energy can be written as follows: W = W 0 + 1 m 0V 34) where W 0 is the potential energy per unit length of a stationary screw dislocation i.e. W 0 = c 44b + K d 11 φ + R 3 bd + e 1) 15 4π b φ + e) 15 d φ) ln R 0 r 0 35) and m 0 = ρb +d ) 4π ln R0 r 0 is the static mass of the dislocation per unit length. 3. Interaction of a straight dislocation with a semi-infinite crack in an octagonal quasicrystal 3.1 General solution The generalized Hooke s law for octagonal quasicrystalline materials with point groups 8mm 8 8m and 8/mmm whose period direction is the x 3 -axis and whose quasiperiodic plane is the x 1 x )-plane is given by [18] σ 11 = C 11 ε 11 + C 1 ε + C 13 ε 33 + Rw 11 + w ) σ = C 1 ε 11 + C 11 ε + C 13 ε 33 Rw 11 + w ) σ 33 = C 13 ε 11 + C 13 ε + C 33 ε 33 σ 1 = σ 1 = C 11 C 1 )ε 1 Rw 1 w 1 ) σ 3 = σ 3 = C 44 ε 3 σ 13 = σ 31 = C 44 ε 13 H 11 = Rε 11 ε ) + K 1 w 11 + K w H = Rε 11 ε ) + K w 11 + K 1 w H 1 = Rε 1 + K 1 + K + K 3 )w 1 + K 3 w 1 H 1 = Rε 1 + K 3 w 1 + K 1 + K + K 3 )w 1 H 13 = K 4 w 13 H 3 = K 4 w 3. 36) where σ ij and H ij are phonon and phason stress components; ε ij and w ij are phonon and phason strains; C 11 C 1 C 13 C 33 C 44 are five elastic constants in the phonon field K 1 K K 3 K 4 are four elastic constants in the phason field R is the phonon phason coupling constant. 90 Pramana J. Phys. Vol. 70 No. 5 May 008

Some defect problems in 1D hexagonal and D octagonal quasicrystals The phonon and phason strains ε ij and w ij are related to the phonon and phason displacements u i and w i through the following relationship: ε ij = 0.5u ij + u ji ) w ij = w ij. 37) In the absence of body forces the static equilibrium equations for the quasicrystal are given by [18] σ ijj = 0 H ijj = 0. 38) For the plane strain problems in which the displacement components u 1 u w 1 w are independent of x 3 and furthermore u 3 = 0 the equations of motion can be expressed in terms of the displacement components u 1 u w 1 w as follows: C 11 u 111 + C 11 C 1 )u 1 + C 11 + C 1 )u 1 +Rw 111 w 1 + w 1 ) = 0 C 11 + C 1 )u 11 + C 11 C 1 )u 11 + C 11 u +Rw 11 w w 11 ) = 0 K 1 w 111 + K 1 + K + K 3 )w 1 + K + K 3 )w 1 +Ru 111 u 1 u 1 ) = 0 K 1 + K + K 3 )w 11 + K 1 w + K + K 3 )w 11 +Ru 11 u + u 11 ) = 0. 39) The above set of equations can be equivalently written in the following matrix form: u 1 u L w = 0 4 1 40) 1 w where the components of the 4 4 symmetric differential operator L are given by L 11 = C 11 x 1 + C 11 C 1 ) x L 1 = L 1 = C 11 + C 1 ) x 1 x ) L 13 = L 31 = R x 1 x L 14 = L 41 = 4R x 1 x L = C 11 C 1 ) x + C 11 1 x L 3 = L 3 = 4R x 1 x ) L 4 = L 4 = R x 1 x L 33 = K 1 x +K 1 +K +K 3 ) 1 x L 34 = L 43 = K +K 3 ) x 1 x Pramana J. Phys. Vol. 70 No. 5 May 008 91

X Wang and E Pan L 44 = K 1 + K + K 3 ) x + K 1 1 x. 41) Now we introduce a displacement function F which satisfies the following equation: L F = 0 4) where L is the determinant of the differential operator matrix L. Omitting the intermediate procedures the displacement function F finally satisfies the following partial differential equation [14]: where F 4ε Λ Λ F + 4εΛ Λ Λ Λ F = 0 43) and = x + 1 x Λ = x 1 x R C 11 + C 1 )K + K 3 ) ε = [C 11 C 1 )K 1 + K + K 3 ) R ] [C 11 K 1 R ] 0 < ε < 1. 44) Applying the differential operator theory one general solution to eq. 40) can now be expressed as u 1 = L 11F u = L 1F w 1 = L 13F w = L 14F 45) where L 11 L 1 L 13 and L 14 are the algebraic cofactors of L. The specific expressions of L 11 L 1 L 13 and L 14 are given below: L 11 = 4K 1 [C 11 C 1 )K 1 + K + K 3 ) R ] 6 x 6 1 +8[K 1 C 11 C 1 )K 1 + K + K 3 ) 6 3R K 1 + 3K + 3K 3 )] x 4 1 x [ +4 K 1 5C 11 C 1 )K 1 + K + K 3 ) ] 6R 6 K 1 K K 3 ) x 1 x4 +8C 11 K 1 R )K 1 + K + K 3 ) 6 x 6 46) 9 Pramana J. Phys. Vol. 70 No. 5 May 008

Some defect problems in 1D hexagonal and D octagonal quasicrystals L 1 = 4 [ K 1 C 11 + C 1 )K 1 + K + K 3 ) + 6R K + K 3 ) ] ) [ 6 x 5 1 x + 6 x 1 x 5 8 K 1 C 11 + C 1 )K 1 + K + K 3 ) ] 10R 6 K + K 3 ) x 3 47) 1 x3 L 13 = 4R [ C 11 C 1 )K 1 + K + K 3 ) R ] 6 x 6 1 8R [ K 1 C 11 + C 1 ) + C 11 + 3C 1 )K + K 3 ) R ] 6 x 4 1 x 4R [ K 1 C 11 + 3C 1 ) 7C 11 + C 1 )K + K 3 ) + R ] 6 x 1 x4 +8RC 11 K 1 R ) 6 x 6 48) L 14 = 4R [ K 1 3C 1 C 11 ) + C 11 C 1 )K + K 3 ) + 4R ] 6 x 5 1 x +8R [ C 11 + 3C 1 )K + K 3 ) K 1 3C 11 C 1 ) + 4R ] 6 x 3 1 x3 4R [ K 1 5C 11 + C 1 ) + 7C 11 + C 1 )K + K 3 ) 4R ] 6 x 1 x 5. 49) Here it shall be mentioned that the displacement function F is derived by means of the differential operator theory. It is observed that the method given here for the derivation of F is more straightforward than that presented in [14] and [19]. In addition it is not difficult to understand the fact that the displacement function F satisfies exactly the same partial differential equation as that derived by Zhou and Fan [14]. The general solution to eq. 43) can be expressed as F = Re f 1 z 1 ) + f z ) + f 3 z 3 ) + f 4 z 4 ) 50) where z j = x 1 + p j x Imp j > 0 j = 1 4) and p j j = 1 4) are explicitly given by p 1 = [1 + ε) 1/ + ε 1/4 ] [ε 1/4 + i1 ε) 1/ ] p = [1 + ε) 1/ + ε 1/4 ] [ε 1/4 + i1 ε) 1/ ] p 3 = [1 + ε) 1/ ε 1/4 ] [ε 1/4 + i1 ε) 1/ ] p 4 = [1 + ε) 1/ ε 1/4 ] [ε 1/4 + i1 ε) 1/ ]. 51) Consequently the phonon and phason displacements can also be expressed in terms of f j z j ) j = 1 4) as follows: Pramana J. Phys. Vol. 70 No. 5 May 008 93

X Wang and E Pan u 1 = Re δ 11 f 6) 1 z 1) + δ 1 f 6) z ) + δ 13 f 6) 3 z 3) + δ 14 f 6) 4 z 4) u = Re δ 1 f 6) 1 z 1) + δ f 6) z ) + δ 3 f 6) 3 z 3) + δ 4 f 6) 4 z 4) w 1 = Re δ 31 f 6) 1 z 1) + δ 3 f 6) z ) + δ 33 f 6) 3 z 3) + δ 34 f 6) 4 z 4) w = Re δ 41 f 6) 1 z 1) + δ 4 f 6) z ) + δ 43 f 6) 3 z 3) + δ 44 f 6) 4 z 4) 5) where f j) z) = j f/ z j and the constants δ ij are given by [ δ 1j = 4K 1 C11 C 1 )K 1 + K + K 3 ) R ] +8p [ j K1 C 11 C 1 )K 1 + K + K 3 ) 3R K 1 + 3K + 3K 3 ) ] +4p 4 [ j K1 5C 11 C 1 )K 1 + K + K 3 ) 6R K 1 K K 3 ) ] +8p 6 jc 11 K 1 R )K 1 + K + K 3 ) δ j = 4p j 1 + p 4 j) [ K 1 C 11 + C 1 )K 1 + K + K 3 ) + 6R K + K 3 ) ] 8p 3 [ j K1 C 11 + C 1 )K 1 + K + K 3 ) 10R K + K 3 ) ] δ 3j = 4R [ C 11 C 1 )K 1 + K + K 3 ) R ] 8Rp [ j K1 C 11 + C 1 ) + C 11 + 3C 1 )K + K 3 ) R ] 4Rp 4 [ j K1 C 11 + 3C 1 ) 7C 11 + C 1 )K + K 3 ) + R ] +8Rp 6 jc 11 K 1 R ) [ δ 4j = 4Rp j K1 3C 1 C 11 ) + C 11 C 1 )K + K 3 ) + 4R ] +8Rp 3 [ j C11 + 3C 1 )K + K 3 ) K 1 3C 11 C 1 ) + 4R ] 4Rp 5 [ j K1 5C 11 + C 1 ) + 7C 11 + C 1 )K + K 3 ) 4R ] j = 1 4) 53) The phonon and phason stress components can also be expressed in terms of the four analytic functions f j z j ) j = 1 4) as σ 1 = σ 1 = Re γ 11 f 7) 1 z 1) + γ 1 f 7) z ) + γ 13 f 7) 3 z 3) + γ 14 f 7) 4 z 4) σ = Re γ 1 f 7) 1 z 1) + γ f 7) z ) + γ 3 f 7) 3 z 3) + γ 4 f 7) 4 z 4) H 1 = Re γ 31 f 7) 1 z 1) + γ 3 f 7) z ) + γ 33 f 7) 3 z 3) + γ 34 f 7) 4 z 4) H = Re γ 41 f 7) 1 z 1) + γ 4 f 7) z ) + γ 43 f 7) 3 z 3) + γ 44 f 7) 4 z 4) 54) σ 11 = Re λ 11 f 7) 1 z 1) + λ 1 f 7) z ) + λ 13 f 7) 3 z 3) + λ 14 f 7) 4 z 4) σ 33 = Re λ 1 f 7) 1 z 1) + λ f 7) z ) + λ 3 f 7) 3 z 3) + λ 4 f 7) 4 z 4) H 11 = Re λ 31 f 7) 1 z 1) + λ 3 f 7) z ) + λ 33 f 7) 3 z 3) + λ 34 f 7) 4 z 4) H 1 = Re λ 41 f 7) 1 z 1) + λ 4 f 7) z ) + λ 43 f 7) 3 z 3) + λ 44 f 7) 4 z 4) 55) 94 Pramana J. Phys. Vol. 70 No. 5 May 008

Some defect problems in 1D hexagonal and D octagonal quasicrystals where the constants γ ij and λ ij are related to δ ij through γ 1j = C 11 C 1 p j δ 1j + δ j ) Rp j δ 3j δ 4j ) γ j = C 1 δ 1j + C 11 p j δ j Rδ 3j + p j δ 4j ) γ 3j = Rp j δ 1j + δ j ) + K 1 + K + K 3 )p j δ 3j + K 3 δ 4j γ 4j = Rδ 1j p j δ j ) + K δ 3j + K 1 p j δ 4j 56) λ 1j = C 11 δ 1j + C 1 p j δ j + Rδ 3j + p j δ 4j ) λ j = C 13 δ 1j + p j δ j ) λ 3j = Rδ 1j p j δ j ) + K 1 δ 3j + K p j δ 4j λ 4j = Rp j δ 1j + δ j ) + K 3 p j δ 3j + K 1 + K + K 3 )δ 4j. 57) Now introduce four stress functions Φ 1 Φ Ψ 1 Ψ which are related to the phonon and phason stresses through the following equations: σ 11 = Φ 1 x σ 1 = Φ 1 x 1 σ 1 = Φ x σ = Φ x 1 H 11 = Ψ 1 x H 1 = Ψ 1 x 1 H 1 = Ψ x H = Ψ x 1. 58) Apparently the introduced stress functions automatically satisfy the equilibrium equations 38). It follows from eqs 54) and 58) that the four stress functions Φ 1 Φ Ψ 1 Ψ can be expressed in terms of the four analytic functions f j z j ) j = 1 4) as Φ 1 = Re γ 11 f 6) 1 z 1) + γ 1 f 6) z ) + γ 13 f 6) 3 z 3) + γ 14 f 6) 4 z 4) Φ = Re γ 1 f 6) 1 z 1) + γ f 6) z ) + γ 3 f 6) 3 z 3) + γ 4 f 6) 4 z 4) Ψ 1 = Re γ 31 f 6) 1 z 1) + γ 3 f 6) z ) + γ 33 f 6) 3 z 3) + γ 34 f 6) 4 z 4) Ψ = Re γ 41 f 6) 1 z 1) + γ 4 f 6) z ) + γ 43 f 6) 3 z 3) + γ 44 f 6) 4 z 4). 59) 3. Field potentials 3..1. A straight dislocation in an infinite octagonal quasicrystal. Consider the Burgers vector of a straight dislocation which is infinitely long in the period direction with the core at the origin in an infinite octagonal quasicrystal b d = b 1 b 0 d 1 d ) where Pramana J. Phys. Vol. 70 No. 5 May 008 95

X Wang and E Pan du 1 = b 1 du = b dw 1 = d 1 dw = d 60) for any loop C surrounding the dislocation line. We can assume that in this case the four analytic functions f j z j ) j = 1 4) take the following forms: f 6) 1 z 1) = A 1 ln z 1 f 6) z ) = A ln z f 6) 3 z 3) = A 3 ln z 3 f 6 4 z 4) = A 4 ln z 4 61) where A j j = 1 4) are complex constants to be determined. In view of eq. 60) we can arrive at the following set of linear algebraic equations: Im δ 11 A 1 + δ 1 A + δ 13 A 3 + δ 14 A 4 = b 1 π Im δ 1 A 1 + δ A + δ 3 A 3 + δ 4 A 4 = b π Im δ 31 A 1 + δ 3 A + δ 33 A 3 + δ 34 A 4 = d 1 π Im δ 41 A 1 + δ 4 A + δ 43 A 3 + δ 44 A 4 = d π Im γ 11 A 1 + γ 1 A + γ 13 A 3 + γ 14 A 4 = 0 Im γ 1 A 1 + γ A + γ 3 A 3 + γ 4 A 4 = 0 Im γ 31 A 1 + γ 3 A + γ 33 A 3 + γ 34 A 4 = 0 Im γ 41 A 1 + γ 4 A + γ 43 A 3 + γ 44 A 4 = 0. 6a) 6b) Then the four unknown constants A j j = 1 4) and their conjugates can be uniquely determined from the above set of equations. 3.. Asymptotic fields around a semi-infinite crack. Here we consider a semiinfinite crack which lies on the negative real axis. The crack penetrates through the solid along the period direction. We assume that in this case the four analytic functions f j z j ) j = 1 4) take the following form: f 6) 1 z 1) = B 1 z1 f 6) z ) = B z f 6) 3 z 3) = B 3 z3 f 6) 4 z 4) = B 4 z4 63) where B j j = 1 4) are complex constants to be determined. The satisfaction of traction-free boundary conditions σ 1 = σ = H 1 = H = 0 on the crack surfaces x 1 < 0 x = 0 will result in Im γ 11 B 1 + γ 1 B + γ 13 B 3 + γ 14 B 4 = 0 Im γ 1 B 1 + γ B + γ 3 B 3 + γ 4 B 4 = 0 Im γ 31 B 1 + γ 3 B + γ 33 B 3 + γ 34 B 4 = 0 Im γ 41 B 1 + γ 4 B + γ 43 B 3 + γ 44 B 4 = 0. 64) 96 Pramana J. Phys. Vol. 70 No. 5 May 008

Some defect problems in 1D hexagonal and D octagonal quasicrystals According to the following stress intensity factors on the crack tip [6] K I = lim πx1 σ K II = lim πx1 σ 1 x 1 0x 0 x 1 0x 0 T I = lim πx1 H T II = lim πx1 H 1. 65) x 1 0x 0 x 1 0x 0 Then we arrive at another set of linear algebraic equations Re γ 11 B 1 + γ 1 B + γ 13 B 3 + γ 14 B 4 = K II π Re γ 1 B 1 + γ B + γ 3 B 3 + γ 4 B 4 = Re γ 31 B 1 + γ 3 B + γ 33 B 3 + γ 34 B 4 = Re γ 41 B 1 + γ 4 B + γ 43 B 3 + γ 44 B 4 = K I π T II π T I π. 66) Consequently the four unknowns B j j = 1 4) can be uniquely determined from eqs 64) and 66) as B 1 B B = 1 3 π B 4 γ 11 γ 1 γ 13 γ 14 γ 1 γ γ 3 γ 4 γ 31 γ 3 γ 33 γ 34 γ 41 γ 4 γ 43 γ 44 1 K II K I T II T I. 67) Substituting eq. 63) into eqs 54) and 55) we can obtain the asymptotic phonon and phason stress fields around the semi-infinite crack in an octagonal quasicrystal. Apparently all the phonon and phason stress components σ ij and H ij exhibit the classical inverse square root singularity near the crack tip. 3..3 A straight dislocation near a semi-infinite crack in an octagonal quasicrystal. As shown in figure 1 we consider an octagonal quasicrystal containing a straight dislocation near a semi-infinite crack. The semi-infinite crack also lies on the negative real axis and the dislocation is located at x 1 = x 1o x = x o. The satisfaction of continuity condition of tractions across the total real axis < x 1 < + x = 0 will result in the following equations: γ 1j f 7) j z) = γ j f 7) j z) = γ 3j f 7) j z) = γ 4j f 7) j z) = γ 1j f 7) j z) γ j f 7) j z) γ 3j f 7) j z) γ 4j f 7) j z) γ1j A j γ ) 1jA j z z jo z z jo γj A j γ ) ja j z z jo z z jo γ3j A j γ ) 3jA j z z jo z z jo γ4j A j γ ) 4jA j 68) z z jo z z jo Pramana J. Phys. Vol. 70 No. 5 May 008 97

X Wang and E Pan Figure 1. A line dislocation near a semi-infinite crack in an octagonal quasicrystal. where z jo = x 1o + p j x o j = 1 4) and A j j = 1 4) are determined from eqs 6a) and 6b). Here we have replaced the complex variables z j j = 1 4) by the common complex variable z = x 1 + ix due to the fact that z 1 = z = z 3 = z 4 = z on the real axis [01]. When the analysis is finished the complex variable z = x 1 + ix shall be changed back to the corresponding variables z j j = 1 4). The traction-free boundary conditions σ 1 = σ = H 1 = H = 0 on the surfaces of the semi-infinite crack x 1 < 0 x = 0 can be expressed as γ 1j f 7) j γ j f 7) j γ 3j f 7) j γ 4j f 7) j x + 1 ) + x + 1 ) + x + 1 ) + x + 1 ) + γ 1j f 7) j x 1 ) = 0 γ j f 7) j x 1 ) = 0 γ 3j f 7) j x 1 ) = 0 Substitution of eq. 68) into eq. 69) will result in γ 1j [ f 7) j x + 1 ) + f 7) j x 1 ) ] = γ j [ f 7) j x + 1 ) + f 7) j x 1 ) ] = γ 3j [ f 7) j x + 1 ) + f 7) j x 1 ) ] = γ 4j f 7) j x 1 ) = 0 x 1 < 0). 69) γ1j A j γj A j γ3j A j γ 1jA j γ ja j γ 3jA j ) ) ) 98 Pramana J. Phys. Vol. 70 No. 5 May 008

Some defect problems in 1D hexagonal and D octagonal quasicrystals γ 4j [ f 7) j x + 1 ) + f 7) j x 1 ) ] = γ4j A j γ 4jA j ) x 1 < 0). In order to simplify the analysis we introduce four new analytic functions defined by h 1 z) = f 7) 1 z) A 1 z z 1o h z) = f 7) z) A z z o 70) h 3 z) = f 7) 3 z) A 3 z z 3o h 4 z) = f 7) 4 z) A 4 z z 4o. 71) Consequently eq. 70) can be rewritten as [ γ 1j hj x + 1 ) + h jx 1 )] = [ γ j hj x + 1 ) + h jx 1 )] = [ γ 3j hj x + 1 ) + h jx 1 )] = [ γ 4j hj x + 1 ) + h jx 1 )] = or equivalently where h 1 x + 1 ) + h 1x 1 ) = A 1 x 1 z 1o + h x + 1 ) + h x 1 ) = A x 1 z o + h 3 x + 1 ) + h 3x 1 ) = A 3 x 1 z 3o + h 4 x + 1 ) + h 4x 1 ) = A 4 x 1 z 4o + β 1j β j β = A j 3j β 4j γ1j A j + γj A j + γ3j A j + γ4j A j + γ 11 γ 1 γ 13 γ 14 γ 1 γ γ 3 γ 4 γ 31 γ 3 γ 33 γ 34 γ 41 γ 4 γ 43 γ 44 β 1j β j β 3j γ 1jA j γ ja j ) ) γ ) 3jA j ) γ 4jA j x 1 < 0). 7) β 4j x 1 < 0) 73) γ 1j γ j γ j = 1 4). 74) 3j γ 4j 1 Pramana J. Phys. Vol. 70 No. 5 May 008 99

X Wang and E Pan The Riemann Hilbert problems eq. 73)) can be easily solved as A 1 h 1 z) = z z 1o ) + β 1j z z jo ) + 1 z B 1 + A 1 z1o z z 1o β 1j zjo z z jo A h z) = z z o ) + β j z z o ) + 1 z B + A zo z z o β j zjo z z jo A 3 h 3 z) = z z 3o ) + β 3j z z 3o ) + 1 z B 3 + A 3 z3o z z 3o β 3j zjo z z jo A 4 h 4 z) = z z 4o ) + β 4j z z 4o ) + 1 z B 4 + A 4 z4o z z 4o β 4j zjo 75) z z jo where B j j = 1 4) are given by eq. 67). In view of eqs 71) and 75) the explicit expressions for the four analytic functions f j z j ) j = 1 4) are given by f 7) 1 z 1) = f 7) z ) = f 7) 3 z 3) = A 1 z 1 z 1o ) + + 1 z 1 β 1j z 1 z jo ) B 1 + A 1 z1o z 1 z 1o A z z o ) + + 1 z β 1j zjo z 1 z jo β j z z o ) B + A zo z z o A 3 z 3 z 3o ) + β j zjo z z jo β 3j z 3 z 3o ) 930 Pramana J. Phys. Vol. 70 No. 5 May 008

Some defect problems in 1D hexagonal and D octagonal quasicrystals f 7) 4 z 4) = + 1 B 3 + A 3 z3o z 3 z 3 z 3o A 4 z 4 z 4o ) + + 1 z β 3j zjo z 3 z jo β 4j z 4 z 4o ) B 4 + A 4 z4o z 4 z 4o β 4j zjo. 76) z 4 z jo The local phonon and phason stress intensity factors induced by the straight dislocation and remote loads are determined by k I = lim πx1 σ k II = lim πx1 σ 1 x 1 0x 0 x 1 0x 0 t I = lim πx1 H t II = lim πx1 H 1. 77) x 1 0x 0 x 1 0x 0 The local phonon and phason stress intensity factors induced by the straight dislocation and remote loads can be finally derived to be k II = K II γ 1j A j πre zjo k I = K I γ j A j πre zjo t II = T II γ 3j A j πre zjo t I = T I γ 4j A j πre zjo. 78) Particularly when the dislocation lies on the positive real axis i.e. x 1o > 0 and y 1o = 0 then the stress intensity factors induced by dislocation are given below in view of eqs 6b) and 78): π π k II = K II γ 1j A j k I = K I γ j A j x 1o x 1o π π t II = T II γ 3j A j t I = T I γ 4j A j. 79) x 1o x 1o Pramana J. Phys. Vol. 70 No. 5 May 008 931

4. Conclusions X Wang and E Pan In a moving screw dislocation in a 1D hexagonal piezoelectric quasicrystal with point group 6mm is analyzed in detail within the framework of Landau theory. We first present a general solution in terms of two functions ϕ 1 ϕ which satisfy wave equations and another harmonic function ϕ 3. Based on the obtained general solution analytic expressions for all the field variables such as displacements electric potential strains stresses electric fields and electric displacements are found. We also present the total energy of the moving screw dislocation. The obtained solutions are verified by comparison with existing ones. It shall be pointed out that the derived solutions are valid when V < mins 1 s. The obtained general solution can also be conveniently utilized to analyze a Yoffe-type moving crack in a 1D hexagonal piezoelectric quasicrystal with point group 6mm. In 3 we have investigated the interaction of a straight dislocation with a semiinfinite crack in an octagonal quasicrystalline solid. The shielding or anti-shielding effect on the crack tip due to the neighboring straight dislocation can be easily observed from the expressions 78) and 79) for the local stress intensity factors induced by the straight dislocation. Acknowledgements The authors are indebted to the referee for his/her very helpful comments and suggestions. This work is supported in part by AFRL/ARL. References [1] D Shechtman I Blech D Gratias and J W Cahn Phys. Rev. Lett. 53 1951 1984) [] P De and R A Pelcovits Phys. Rev. B35 8609 1987) [3] W G Yang R H Wang D H Ding and C Z Hu Phys. Rev. B48 6999 1993) [4] D H Ding W G Yang C Z Hu and R H Wang Phys. Rev. B48 7003 1993) [5] R H Wang W G Yang C Z Hu and D H Ding J. Phys. Condens. Matter 9 411 1997) [6] X Wang and Z Zhong Int. J. Eng. Sci. 4 51 004) [7] X Wang Int. J. Eng. Sci. 4 1911 004) [8] W G Yang D H Ding R H Wang and C Z Hu Z. Phys. B100 447 1996) [9] C L Li and L Y Liu Chin. Phys. 13 94 004) [10] K Rama Mohan Rao P Hemagiri Rao and B S K Chaitanya Pramana J. Phys. 68 481 007) [11] D H Ding R H Wang W G Yang and C Z Hu J. Phys. Condens. Matter 7 543 1995) [1] W G Yang M Feuerbacher N Tamura D H Ding R H Wang and K Urban Philos. Mag. A77 1481 1998) [13] X F Li T Y Fan and Y F Sun Philos. Mag. A79 1943 1999) [14] W M Zhou and T Y Fan Chin. Phys. 10 743 001) [15] G T Liu R P Guo and T Y Fan Chin. Phys. 1 1149 003) [16] S F Li and P A Mataga J. Mech. Phys. Solids 44 1799 1996) 93 Pramana J. Phys. Vol. 70 No. 5 May 008

Some defect problems in 1D hexagonal and D octagonal quasicrystals [17] X Wang and Z Zhong Mech. Res. Commun. 9 45 00) [18] C Hu D H Ding and R Wang Rep. Prog. Phys. 63 1 000) [19] X F Li and T Y Fan Chin. Phys. Lett. 15 78 1998) [0] D L Clements Int. J. Eng. Sci. 9 57 1971) [1] X Wang E Pan and W J Feng Euro. J. Mech. A/Solids 6 901 007) Pramana J. Phys. Vol. 70 No. 5 May 008 933