Analytical solutions for some defect problems in 1D hexagonal and 2D octagonal quasicrystals
|
|
- Randolph Holland
- 5 years ago
- Views:
Transcription
1 PRAMANA c Indian Academy of Sciences Vol. 70 No. 5 journal of May 008 physics pp 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 USA Department of Applied Mathematics University of Akron OH USA Corresponding author 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; 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
2 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
3 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 E 3 D 1 = e 1) 15 ε 31 + e ) 15 w E 1 D = e 1) 15 ε 3 + e ) 15 w 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
4 X Wang and E Pan σ σ 1 + σ 133 = ρü 1 σ 11 + σ + σ 33 = ρü σ σ 3 + σ 333 = ρü 3 H 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) 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
5 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) then eq. 5) can be written as 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
6 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
7 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
8 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) 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) φ e 1) 15 b e) π 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) 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) 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
9 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 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) 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
10 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 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
11 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 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 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 = ) 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
12 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 [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
13 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 C 1 )K + K 3 ) R ] 6 x 4 1 x 4R [ K 1 C C 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 C 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
14 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 C 1 )K + K 3 ) R ] 4Rp 4 [ j K1 C C 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
15 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 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
16 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
17 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 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
18 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
19 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
20 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 ) z 1 β 1j z 1 z jo ) B 1 + A 1 z1o z 1 z 1o A z z o ) 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
21 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 ) 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
22 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 ) [] P De and R A Pelcovits Phys. Rev. B ) [3] W G Yang R H Wang D H Ding and C Z Hu Phys. Rev. B ) [4] D H Ding W G Yang C Z Hu and R H Wang Phys. Rev. B ) [5] R H Wang W G Yang C Z Hu and D H Ding J. Phys. Condens. Matter ) [6] X Wang and Z Zhong Int. J. Eng. Sci ) [7] X Wang Int. J. Eng. Sci ) [8] W G Yang D H Ding R H Wang and C Z Hu Z. Phys. B ) [9] C L Li and L Y Liu Chin. Phys ) [10] K Rama Mohan Rao P Hemagiri Rao and B S K Chaitanya Pramana J. Phys ) [11] D H Ding R H Wang W G Yang and C Z Hu J. Phys. Condens. Matter ) [1] W G Yang M Feuerbacher N Tamura D H Ding R H Wang and K Urban Philos. Mag. A ) [13] X F Li T Y Fan and Y F Sun Philos. Mag. A ) [14] W M Zhou and T Y Fan Chin. Phys ) [15] G T Liu R P Guo and T Y Fan Chin. Phys ) [16] S F Li and P A Mataga J. Mech. Phys. Solids ) 93 Pramana J. Phys. Vol. 70 No. 5 May 008
23 Some defect problems in 1D hexagonal and D octagonal quasicrystals [17] X Wang and Z Zhong Mech. Res. Commun ) [18] C Hu D H Ding and R Wang Rep. Prog. Phys ) [19] X F Li and T Y Fan Chin. Phys. Lett ) [0] D L Clements Int. J. Eng. Sci ) [1] X Wang E Pan and W J Feng Euro. J. Mech. A/Solids ) Pramana J. Phys. Vol. 70 No. 5 May
AXISYMMETRIC ELASTICITY PROBLEM OF CUBIC QUASICRYSTAL *
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
More informationMoving screw dislocations in piezoelectric bimaterials
phys stat sol (b) 38 No 1 10 16 (003) / DOI 10100/pssb00301805 Moving screw dislocations in piezoelectric bimaterials Xiang-Fa Wu *1 Yuris A Dzenis 1 and Wen-Sheng Zou 1 Department of Engineering Mechanics
More informationApplied Mathematics and Mechanics (English Edition)
Appl. Math. Mech. -Engl. Ed., 39(3), 335 352 (2018) Applied Mathematics and Mechanics (English Edition) https://doi.org/10.1007/s10483-018-2309-9 Static deformation of a multilayered one-dimensional hexagonal
More informationTianyou Fan. Mathematical Theory of Elasticity of Quasicrystals and Its Applications
Tianyou Fan Mathematical Theory of Elasticity of Quasicrystals and Its Applications Tianyou Fan Mathematical Theory of Elasticity of Quasicrystals and Its Applications With 82 figures Author Tianyou Fan
More informationA Piezoelectric Screw Dislocation Interacting with an Elliptical Piezoelectric Inhomogeneity Containing a Confocal Elliptical Rigid Core
Commun. Theor. Phys. 56 774 778 Vol. 56, No. 4, October 5, A Piezoelectric Screw Dislocation Interacting with an Elliptical Piezoelectric Inhomogeneity Containing a Confocal Elliptical Rigid Core JIANG
More informationElastic behaviour of an edge dislocation near a sharp crack emanating from a semi-elliptical blunt crack
Chin. Phys. B Vol. 19, No. 1 010 01610 Elastic behaviour of an edge dislocation near a sharp crack emanating from a semi-elliptical blunt crack Fang Qi-Hong 方棋洪, Song Hao-Peng 宋豪鹏, and Liu You-Wen 刘又文
More informationScrew Dislocation Interacting with Interfacial Edge-Cracks in Piezoelectric Bimaterial Strips
Freund Publishing House Ltd. International Journal of Nonlinear Sciences Numerical Simulation 5(4), 34-346, 4 Screw Dislocation Interacting with Interfacial Edge-Cracks in Piezoelectric Bimaterial Strips
More informationElastic Fields of Dislocations in Anisotropic Media
Elastic Fields of Dislocations in Anisotropic Media a talk given at the group meeting Jie Yin, David M. Barnett and Wei Cai November 13, 2008 1 Why I want to give this talk Show interesting features on
More informationHomework Problems. ( σ 11 + σ 22 ) 2. cos (θ /2), ( σ θθ σ rr ) 2. ( σ 22 σ 11 ) 2
Engineering Sciences 47: Fracture Mechanics J. R. Rice, 1991 Homework Problems 1) Assuming that the stress field near a crack tip in a linear elastic solid is singular in the form σ ij = rλ Σ ij (θ), it
More informationarxiv: v1 [cond-mat.mtrl-sci] 7 Nov 2016
Journal of Micromechanics and Molecular Physics c World cientific Publishing Company arxiv:1611.02146v1 [cond-mat.mtrl-sci] 7 Nov 2016 Eshelbian mechanics of novel materials: Quasicrystals Markus Lazar
More informationLecture 7: The Beam Element Equations.
4.1 Beam Stiffness. A Beam: A long slender structural component generally subjected to transverse loading that produces significant bending effects as opposed to twisting or axial effects. MECH 40: Finite
More informationRussia. St. Petersburg, , Russia
PACS 81.07.Bc, 61.72.Cc An asymptotic formula for displacement field in triangular lattice with vacancy V.A. Tsaplin 1, V.A. Kuzkin 1,2 vtsaplin@yandex.ru, kuzkinva@gmail.com 1 Institute for Problems in
More information17th European Conference on Fracture 2-5 September,2008, Brno, Czech Republic. Thermal Fracture of a FGM/Homogeneous Bimaterial with Defects
-5 September,8, Brno, Czech Republic Thermal Fracture of a FGM/Homogeneous Bimaterial with Defects Vera Petrova, a, Siegfried Schmauder,b Voronezh State University, University Sq., Voronezh 3946, Russia
More informationBasic Principles of Quasicrystallography
Chapter 2 Basic Principles of Quasicrystallography Conventional crystals are constructed by a periodic repetition of a single unit-cell. This periodic long-range order only agrees with two-, three-, four-
More informationSmall-Scale Effect on the Static Deflection of a Clamped Graphene Sheet
Copyright 05 Tech Science Press CMC, vol.8, no., pp.03-7, 05 Small-Scale Effect on the Static Deflection of a Clamped Graphene Sheet G. Q. Xie, J. P. Wang, Q. L. Zhang Abstract: Small-scale effect on the
More informationElectric field dependent sound velocity change in Ba 1 x Ca x TiO 3 ferroelectric perovskites
Indian Journal of Pure & Applied Physics Vol. 49, February 2011, pp. 132-136 Electric field dependent sound velocity change in Ba 1 x Ca x TiO 3 ferroelectric perovskites Dushyant Pradeep, U C Naithani
More informationBeta-lattice multiresolution of quasicrystalline Bragg peaks
Beta-lattice multiresolution of quasicrystalline Bragg peaks A. Elkharrat, J.P. Gazeau, and F. Dénoyer CRC701-Workshop Aspects of Aperiodic Order 3-5 July 2008, University of Bielefeld Proceedings of SPIE
More information2 Basic Equations in Generalized Plane Strain
Boundary integral equations for plane orthotropic bodies and exterior regions G. Szeidl and J. Dudra University of Miskolc, Department of Mechanics 3515 Miskolc-Egyetemváros, Hungary Abstract Assuming
More informationIntroduction to the J-integral
Introduction to the J-integral Instructor: Ramsharan Rangarajan February 24, 2016 The purpose of this lecture is to briefly introduce the J-integral, which is widely used in fracture mechanics. To the
More informationRayleigh waves of arbitrary profile in anisotropic media
Rayleigh waves of arbitrary profile in anisotropic media D. A. Prikazchikov Dept. of Computational Mathematics and Mathematical Physics, The Bauman Moscow State Technical University, Moscow, Russia Abstract
More informationarxiv:physics/ v1 [physics.class-ph] 26 Sep 2003
arxiv:physics/0309112v1 [physics.class-ph] 26 Sep 2003 Electromagnetic vortex lines riding atop null solutions of the Maxwell equations Iwo Bialynicki-Birula Center for Theoretical Physics, Polish Academy
More informationA simple plane-strain solution for functionally graded multilayered isotropic cylinders
Structural Engineering and Mechanics, Vol. 24, o. 6 (2006) 000-000 1 A simple plane-strain solution for functionally graded multilayered isotropic cylinders E. Pan Department of Civil Engineering, The
More informationA rotating charged black hole solution in f (R) gravity
PRAMANA c Indian Academy of Sciences Vol. 78, No. 5 journal of May 01 physics pp. 697 703 A rotating charged black hole solution in f R) gravity ALEXIS LARRAÑAGA National Astronomical Observatory, National
More informationA half submerged metal sphere (UIC comprehensive
Problem 1. exam) A half submerged metal sphere (UIC comprehensive A very light neutral hollow metal spherical shell of mass m and radius a is slightly submerged by a distance b a below the surface of a
More informationReceiver. Johana Brokešová Charles University in Prague
Propagation of seismic waves - theoretical background Receiver Johana Brokešová Charles University in Prague Seismic waves = waves in elastic continuum a model of the medium through which the waves propagate
More informationMechanics of Earthquakes and Faulting
Mechanics of Earthquakes and Faulting www.geosc.psu.edu/courses/geosc508 Overview Milestones in continuum mechanics Concepts of modulus and stiffness. Stress-strain relations Elasticity Surface and body
More informationIntegral equations for crack systems in a slightly heterogeneous elastic medium
Boundary Elements and Other Mesh Reduction Methods XXXII 65 Integral equations for crack systems in a slightly heterogeneous elastic medium A. N. Galybin & S. M. Aizikovich Wessex Institute of Technology,
More informationImpact of size and temperature on thermal expansion of nanomaterials
PRAMANA c Indian Academy of Sciences Vol. 84, No. 4 journal of April 205 physics pp. 609 69 Impact of size and temperature on thermal expansion of nanomaterials MADAN SINGH, and MAHIPAL SINGH 2 Department
More informationarxiv:cond-mat/ v2 [cond-mat.stat-mech] 11 Sep 1997
Z. Phys. B 92 (1993) 307 LPSN-93-LT2 arxiv:cond-mat/9306013v2 [cond-mat.stat-mech] 11 Sep 1997 Surface magnetization of aperiodic Ising quantum chains 1. Introduction L Turban and B Berche Laboratoire
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING CAMBRIDGE, MASSACHUSETTS 02139
MASSACHUSETTS NSTTUTE OF TECHNOLOGY DEPARTMENT OF MATERALS SCENCE AND ENGNEERNG CAMBRDGE, MASSACHUSETTS 39 3. MECHANCAL PROPERTES OF MATERALS PROBLEM SET SOLUTONS Reading Ashby, M.F., 98, Tensors: Notes
More informationScrew dislocation interacting with interface and interfacial cracks in piezoelectric bimaterials
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Faculty Publications from the Department of Engineering Mechanics Mechanical & Materials Engineering, Department of 4-2003
More informationScienceDirect. Hamiltonian approach to piezoelectric fracture
Available online at www.sciencedirect.com ScienceDirect Procedia Materials Science 3 ( 014 ) 318 34 0th European Conference on Fracture (ECF0) Hamiltonian approach to piezoelectric fracture J.M. Nianga
More informationLinear Elastic Fracture Mechanics
Measure what is measurable, and make measurable what is not so. - Galileo GALILEI Linear Elastic Fracture Mechanics Krishnaswamy Ravi-Chandar Lecture presented at the University of Pierre and Marie Curie
More informationStress magnification due to stretching and bending of thin ligaments between voids
Arch Appl Mech 6) DOI 1.17/s419-6-8-6 ORIGINAL V. A. Lubarda X. Markenscoff Stress magnification due to stretching and bending of thin ligaments between voids Received: 8 December 5 Springer-Verlag 6 Abstract
More informationJournal of Applied Mathematics and Mechanics
Journal of Applied Mathematics and Mechanics Zeitschrift für Angewandte Mathematik und Mechanik Founded by Richard von Mises in 191 Antiplane shear deformations of an anisotropic elliptical inhomogeneity
More informationPROBLEM OF CRACK UNDER QUASIBRITTLE FRACTURE V.A. KOVTUNENKO
PROBLEM OF CRACK UNDER QUASIBRITTLE FRACTURE V.A. KOVTUNENKO Overview: 1. Motivation 1.1. Evolutionary problem of crack propagation 1.2. Stationary problem of crack equilibrium 1.3. Interaction (contact+cohesion)
More informationAn Interface Anticrack in a Periodic Two Layer Piezoelectric Space under Vertically Uniform Heat Flow
Mechanics and Mechanical Engineering Vol., No. 3 018 667 681 c Lodz University of Technology An Interface Anticrack in a Periodic Two Layer Piezoelectric Space under Vertically Uniform Heat Flow Andrzej
More informationEffect of interfacial dislocations on ferroelectric phase stability and domain morphology in a thin film a phase-field model
JOURNAL OF APPLIED PHYSICS VOLUME 94, NUMBER 4 15 AUGUST 2003 Effect of interfacial dislocations on ferroelectric phase stability and domain morphology in a thin film a phase-field model S. Y. Hu, Y. L.
More informationRayleigh waves in magneto-electro-elastic half planes
Acta Mech 202, 127 134 (2009) DOI 10.1007/s00707-008-0024-8 W. J. Feng E. Pan X. Wang J. Jin Rayleigh waves in magneto-electro-elastic half planes Received: 23 August 2007 / Revised: 17 February 2008 /
More information3 2 6 Solve the initial value problem u ( t) 3. a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1
Math Problem a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1 3 6 Solve the initial value problem u ( t) = Au( t) with u (0) =. 3 1 u 1 =, u 1 3 = b- True or false and why 1. if A is
More informationDYNAMIC WEIGHT FUNCTIONS FOR A MOVING CRACK II. SHEAR LOADING. University of Bath. Bath BA2 7AY, U.K. University of Cambridge
DYNAMIC WEIGHT FUNCTIONS FOR A MOVING CRACK II. SHEAR LOADING A.B. Movchan and J.R. Willis 2 School of Mathematical Sciences University of Bath Bath BA2 7AY, U.K. 2 University of Cambridge Department of
More informationA cylinder in a magnetic field (Jackson)
Problem 1. A cylinder in a magnetic field (Jackson) A very long hollow cylinder of inner radius a and outer radius b of permeability µ is placed in an initially uniform magnetic field B o at right angles
More informationDynamics of Glaciers
Dynamics of Glaciers McCarthy Summer School 01 Andy Aschwanden Arctic Region Supercomputing Center University of Alaska Fairbanks, USA June 01 Note: This script is largely based on the Physics of Glaciers
More informationStructural Dynamics Lecture 4. Outline of Lecture 4. Multi-Degree-of-Freedom Systems. Formulation of Equations of Motions. Undamped Eigenvibrations.
Outline of Multi-Degree-of-Freedom Systems Formulation of Equations of Motions. Newton s 2 nd Law Applied to Free Masses. D Alembert s Principle. Basic Equations of Motion for Forced Vibrations of Linear
More informationMathematical Model for Thermal Shock Problem of a Generalized Thermoelastic Layered Composite Material with Variable Thermal Conductivity
COMPUTATIONAL METHODS IN SCIENCE AND TECHNOLOGY ( 65-7 (6 Mathematical Model for Thermal Shock Problem of a Generalized Thermoelastic Layered Composite Material with Variable Thermal Conductivity H. M.
More informationDRIVING FORCES. and INCLUSION BOUNDARIES with transformation strain. Xanthippi Markenscoff
DRIVING FORCES ON MOVING DEFECTS: DISLOCATIONS and INCLUSION BOUNDARIES with transformation strain Xanthippi Markenscoff UCSD Energy release rates (Driving forces) for moving defects: dislocations and
More informationShijiazhuang, P.R. China. Online Publication Date: 01 June 2008 PLEASE SCROLL DOWN FOR ARTICLE
This article was downloaded by:[feng, W. J.] On: 5 June 28 Access Details: [subscription number 793822887] Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 172954
More informationA short review of continuum mechanics
A short review of continuum mechanics Professor Anette M. Karlsson, Department of Mechanical ngineering, UD September, 006 This is a short and arbitrary review of continuum mechanics. Most of this material
More information(This is a sample cover image for this issue. The actual cover is not yet available at this time.)
(This is a sample cover image for this issue. The actual cover is not yet available at this time. This article appeared in a journal published by Elsevier. The attached copy is furnished to the author
More information16.21 Techniques of Structural Analysis and Design Spring 2005 Unit #8 Principle of Virtual Displacements
6. Techniques of Structural Analysis and Design Spring 005 Unit #8 rinciple of irtual Displacements Raúl Radovitzky March 3, 005 rinciple of irtual Displacements Consider a body in equilibrium. We know
More informationDepartment of Engineering Mechanics, SVL, Xi an Jiaotong University, Xi an
The statistical characteristics of static friction J. Wang, G. F. Wang*, and W. K. Yuan Department of Engineering Mechanics, SVL, Xi an Jiaotong University, Xi an 710049, China * E-mail: wanggf@mail.xjtu.edu.cn
More informationSchur decomposition in the scaled boundary finite element method in elastostatics
IOP Conference Series: Materials Science and Engineering Schur decomposition in the scaled boundary finite element method in elastostatics o cite this article: M Li et al 010 IOP Conf. Ser.: Mater. Sci.
More informationStatic Deformation Due To Long Tensile Fault Embedded in an Isotropic Half-Space in Welded Contact with an Orthotropic Half-Space
International Journal of cientific and Research Publications Volume Issue November IN 5-5 tatic Deformation Due To Long Tensile Fault Embedded in an Isotropic Half-pace in Welded Contact with an Orthotropic
More informationMechanics of Earthquakes and Faulting
Mechanics of Earthquakes and Faulting www.geosc.psu.edu/courses/geosc508 Surface and body forces Tensors, Mohr circles. Theoretical strength of materials Defects Stress concentrations Griffith failure
More informationSEMM Mechanics PhD Preliminary Exam Spring Consider a two-dimensional rigid motion, whose displacement field is given by
SEMM Mechanics PhD Preliminary Exam Spring 2014 1. Consider a two-dimensional rigid motion, whose displacement field is given by u(x) = [cos(β)x 1 + sin(β)x 2 X 1 ]e 1 + [ sin(β)x 1 + cos(β)x 2 X 2 ]e
More informationI. INTRODUCTION J. Acoust. Soc. Am. 109 (4), April /2001/109(4)/1398/5/$ Acoustical Society of America 1398
The explicit secular equation for surface acoustic waves in monoclinic elastic crystals Michel Destrade a) Mathematics, Texas A&M University, College Station, Texas 77843-3368 Received 11 December 2000;
More informationElectromagnetic Field Theory (EMT)
Electromagnetic Field Theory (EMT) Lecture # 9 1) Coulomb s Law and Field Intensity 2) Electric Fields Due to Continuous Charge Distributions Line Charge Surface Charge Volume Charge Coulomb's Law Coulomb's
More informationMechanics of Earthquakes and Faulting
Mechanics of Earthquakes and Faulting Lectures & 3, 9/31 Aug 017 www.geosc.psu.edu/courses/geosc508 Discussion of Handin, JGR, 1969 and Chapter 1 Scholz, 00. Stress analysis and Mohr Circles Coulomb Failure
More informationMechanics PhD Preliminary Spring 2017
Mechanics PhD Preliminary Spring 2017 1. (10 points) Consider a body Ω that is assembled by gluing together two separate bodies along a flat interface. The normal vector to the interface is given by n
More informationROTATING RING. Volume of small element = Rdθbt if weight density of ring = ρ weight of small element = ρrbtdθ. Figure 1 Rotating ring
ROTATIONAL STRESSES INTRODUCTION High centrifugal forces are developed in machine components rotating at a high angular speed of the order of 100 to 500 revolutions per second (rps). High centrifugal force
More informationIAP 2006: From nano to macro: Introduction to atomistic modeling techniques and application in a case study of modeling fracture of copper (1.
IAP 2006: From nano to macro: Introduction to atomistic modeling techniques and application in a case study of modeling fracture of copper (1.978 PDF) http://web.mit.edu/mbuehler/www/teaching/iap2006/intro.htm
More informationFracture Mechanics, Damage and Fatigue Linear Elastic Fracture Mechanics - Energetic Approach
University of Liège Aerospace & Mechanical Engineering Fracture Mechanics, Damage and Fatigue Linear Elastic Fracture Mechanics - Energetic Approach Ludovic Noels Computational & Multiscale Mechanics of
More informationCRACK-TIP DIFFRACTION IN A TRANSVERSELY ISOTROPIC SOLID. A.N. Norris and J.D. Achenbach
CRACK-TIP DIFFRACTION IN A TRANSVERSELY ISOTROPIC SOLID A.N. Norris and J.D. Achenbach The Technological Institute Northwestern University Evanston, IL 60201 ABSTRACT Crack diffraction in a transversely
More informationCHARACTERISTIC WAVEFUNCTIONS OF ONE-DIMENSIONAL PERIODIC, QUASIPERIODIC AND RANDOM LATTICES
Modern Physics Letters B, Vol. 17, Nos. 27 & 28 (2003) 1461 1476 c World Scientific Publishing Company CHARACTERISTIC WAVEFUNCTIONS OF ONE-DIMENSIONAL PERIODIC, QUASIPERIODIC AND RANDOM LATTICES X. Q.
More informationFast multipole boundary element method for the analysis of plates with many holes
Arch. Mech., 59, 4 5, pp. 385 401, Warszawa 2007 Fast multipole boundary element method for the analysis of plates with many holes J. PTASZNY, P. FEDELIŃSKI Department of Strength of Materials and Computational
More informationLaminated Composite Plates and Shells
Jianqiao Ye Laminated Composite Plates and Shells 3D Modelling With 62 Figures Springer Table of Contents 1. Introduction to Composite Materials 1 1.1 Introduction 1 1.2 Classification of Composite Materials
More informationFundamentals of Linear Elasticity
Fundamentals of Linear Elasticity Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research of the Polish Academy
More informationPLEASE SCROLL DOWN FOR ARTICLE
This article was downloaded by:[youssef, Hamdy M.] On: 22 February 2008 Access Details: [subscription number 790771681] Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered
More informationInterfacial effects in electromagnetic coupling within piezoelectric phononic crystals
Acta Mech Sin (29) 25:95 99 DOI 1.17/s149-8-21-y RESEARCH PAPER Interfacial effects in electromagnetic coupling within pieoelectric phononic crystals F. J. Sabina A. B. Movchan Received: 14 July 28 / Accepted:
More informationCorrection of local-linear elasticity for nonlocal residuals: Application to Euler-Bernoulli beams
Correction of local-linear elasticity for nonlocal residuals: Application to Euler-Bernoulli beams Mohamed Shaat* Engineering and Manufacturing Technologies Department, DACC, New Mexico State University,
More informationExercise: concepts from chapter 8
Reading: Fundamentals of Structural Geology, Ch 8 1) The following exercises explore elementary concepts associated with a linear elastic material that is isotropic and homogeneous with respect to elastic
More informationNonlinear optical effects and carbon nanotubes. Abstract
Nonlinear optical effects and carbon nanotubes Chiyat Ben Yau Department of Physics, University of Cincinnati, OH 45221, USA (December 3, 2001) Abstract Materials with large second or third order optical
More informationElectrons and Phonons on the Square Fibonacci Tiling
Ferroelectrics, 305: 15 19, 2004 Copyright C Taylor & Francis Inc. ISSN: 0015-0193 print / 1563-5112 online DOI: 10.1080/00150190490462252 Electrons and Phonons on the Square Fibonacci Tiling RONI ILAN,
More informationExplicit models for elastic and piezoelastic surface waves
IMA Journal of Applied Mathematics (006) 71, 768 78 doi:10.1093/imamat/hxl01 Advance Access publication on August 4, 006 Explicit models for elastic and piezoelastic surface waves J. KAPLUNOV AND A. ZAKHAROV
More informationYou may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
MATHEMATICAL TRIPOS Part III Thursday 1 June 2006 1.30 to 4.30 PAPER 76 NONLINEAR CONTINUUM MECHANICS Attempt FOUR questions. There are SIX questions in total. The questions carry equal weight. STATIONERY
More informationNonlinear stability of steady flow of Giesekus viscoelastic fluid
Nonlinear stability of steady flow of Giesekus viscoelastic fluid Mark Dostalík, V. Průša, K. Tůma August 9, 2018 Faculty of Mathematics and Physics, Charles University Table of contents 1. Introduction
More informationElectro-elastic fields of a plane thermal inclusion in isotropic dielectrics with polarization gradient
Arch. Mech., 56, 1, pp. 33 57, Warszawa 24 Electro-elastic fields of a plane thermal inclusion in isotropic dielectrics with polarization gradient J. P. NOWACKI The Polish-Japanese Institute of Information
More informationTransactions on Engineering Sciences vol 6, 1994 WIT Press, ISSN
A computational method for the analysis of viscoelastic structures containing defects G. Ghazlan," C. Petit," S. Caperaa* " Civil Engineering Laboratory, University of Limoges, 19300 Egletons, France &
More informationNatural Boundary Element Method for Stress Field in Rock Surrounding a Roadway with Weak Local Support
Copyright 011 Tech Science Press CMES, vol.71, no., pp.93-109, 011 Natural Boundary Element Method for Stress Field in Rock Surrounding a Roadway with Weak Local Support Shuncai Li 1,,3, Zhengzhu Dong
More informationModeling of Variable Lamé s Modulii for a FGM Generalized Thermoelastic Half Space
75 Modeling of Variable Lamé s Modulii for a FGM Generalized Thermoelastic Half Space Abstract In this work we consider a problem in the contet of the generalized theory of thermoelasticity for a half
More informationSquare-Triangle-Rhombus Random Tiling
Square-Triangle-Rhombus Random Tiling Maria Tsarenko in collaboration with Jan de Gier Department of Mathematics and Statistics The University of Melbourne ANZAMP Meeting, Lorne December 4, 202 Crystals
More informationMott metal-insulator transition on compressible lattices
Mott metal-insulator transition on compressible lattices Markus Garst Universität zu Köln T p in collaboration with : Mario Zacharias (Köln) Lorenz Bartosch (Frankfurt) T c Mott insulator p c T metal pressure
More informationAccuracy and transferability of GAP models for tungsten
Accuracy and transferability of GAP models for tungsten Wojciech J. Szlachta Albert P. Bartók Gábor Csányi Engineering Laboratory University of Cambridge 5 November 214 Motivation Number of atoms 1 1 2
More informationON THE SUM OF POWERS OF TWO. 1. Introduction
t m Mathematical Publications DOI: 0.55/tmmp-06-008 Tatra Mt. Math. Publ. 67 (06, 4 46 ON THE SUM OF POWERS OF TWO k-fibonacci NUMBERS WHICH BELONGS TO THE SEQUENCE OF k-lucas NUMBERS Pavel Trojovský ABSTRACT.
More informationTIME HARMONIC BEHAVIOUR OF A CRACKED PIEZOELECTRIC SOLID BY BIEM. Marin Marinov, Tsviatko Rangelov
Serdica J. Computing 6 (2012), 185 194 TIME HARMONIC BEHAVIOUR OF A CRACKED PIEZOELECTRIC SOLID BY BIEM Marin Marinov, Tsviatko Rangelov Abstract. Time harmonic behaviour of a cracked piezoelectric finite
More informationS.P. Timoshenko Institute of Mechanics, National Academy of Sciences, Department of Fracture Mechanics, Kyiv, Ukraine
CALCULATION OF THE PREFRACTURE ZONE AT THE CRACK TIP ON THE INTERFACE OF MEDIA A. Kaminsky a L. Kipnis b M. Dudik b G. Khazin b and A. Bykovtscev c a S.P. Timoshenko Institute of Mechanics National Academy
More informationRational Form Solitary Wave Solutions and Doubly Periodic Wave Solutions to (1+1)-Dimensional Dispersive Long Wave Equation
Commun. Theor. Phys. (Beijing, China) 43 (005) pp. 975 98 c International Academic Publishers Vol. 43, No. 6, June 15, 005 Rational Form Solitary Wave Solutions and Doubly Periodic Wave Solutions to (1+1)-Dimensional
More informationMechanical Properties of Polymer Rubber Materials Based on a New Constitutive Model
Mechanical Properties of Polymer Rubber Materials Based on a New Constitutive Model Mechanical Properties of Polymer Rubber Materials Based on a New Constitutive Model J.B. Sang*, L.F. Sun, S.F. Xing,
More informationGeometry-dependent MITC method for a 2-node iso-beam element
Structural Engineering and Mechanics, Vol. 9, No. (8) 3-3 Geometry-dependent MITC method for a -node iso-beam element Phill-Seung Lee Samsung Heavy Industries, Seocho, Seoul 37-857, Korea Hyu-Chun Noh
More informationCRACK-TIP DRIVING FORCE The model evaluates the eect of inhomogeneities by nding the dierence between the J-integral on two contours - one close to th
ICF 100244OR Inhomogeneity eects on crack growth N. K. Simha 1,F.D.Fischer 2 &O.Kolednik 3 1 Department ofmechanical Engineering, University of Miami, P.O. Box 248294, Coral Gables, FL 33124-0624, USA
More informationROTATIONAL STABILITY OF A CHARGED DIELEC- TRIC RIGID BODY IN A UNIFORM MAGNETIC FIELD
Progress In Electromagnetics Research Letters, Vol. 11, 103 11, 009 ROTATIONAL STABILITY OF A CHARGED DIELEC- TRIC RIGID BODY IN A UNIFORM MAGNETIC FIELD G.-Q. Zhou Department of Physics Wuhan University
More informationLECTURE 3: Quantization and QFT
LECTURE 3: Quantization and QFT Robert Oeckl IQG-FAU & CCM-UNAM IQG FAU Erlangen-Nürnberg 14 November 2013 Outline 1 Classical field theory 2 Schrödinger-Feynman quantization 3 Klein-Gordon Theory Classical
More informationResearch Article Analysis of Mode I Periodic Parallel Cracks-Tip Stress Field in an Infinite Orthotropic Plate
Mathematical Problems in Engineering Volume 3, Article ID 47, 8 pages http://dx.doi.org/.55/3/47 Research Article Analysis of Mode I Periodic Parallel Cracks-Tip Stress Field in an Infinite Orthotropic
More informationModelling and design of complete photonic band gaps in two-dimensional photonic crystals
PRAMANA c Indian Academy of Sciences Vol. 70, No. 1 journal of January 2008 physics pp. 153 161 Modelling and design of complete photonic band gaps in two-dimensional photonic crystals YOGITA KALRA and
More informationRadiation energy flux of Dirac field of static spherically symmetric black holes
Radiation energy flux of Dirac field of static spherically symmetric black holes Meng Qing-Miao( 孟庆苗 ), Jiang Ji-Jian( 蒋继建 ), Li Zhong-Rang( 李中让 ), and Wang Shuai( 王帅 ) Department of Physics, Heze University,
More informationVector Spaces. Vector space, ν, over the field of complex numbers, C, is a set of elements a, b,..., satisfying the following axioms.
Vector Spaces Vector space, ν, over the field of complex numbers, C, is a set of elements a, b,..., satisfying the following axioms. For each two vectors a, b ν there exists a summation procedure: a +
More informationFirst-Order Solutions for the Buckling Loads of Euler-Bernoulli Weakened Columns
First-Order Solutions for the Buckling Loads of Euler-Bernoulli Weakened Columns J. A. Loya ; G. Vadillo 2 ; and J. Fernández-Sáez 3 Abstract: In this work, closed-form expressions for the buckling loads
More informationModule 3 : Equilibrium of rods and plates Lecture 15 : Torsion of rods. The Lecture Contains: Torsion of Rods. Torsional Energy
The Lecture Contains: Torsion of Rods Torsional Energy This lecture is adopted from the following book 1. Theory of Elasticity, 3 rd edition by Landau and Lifshitz. Course of Theoretical Physics, vol-7
More informationTheoretische Physik 2: Elektrodynamik (Prof. A-S. Smith) Home assignment 11
WiSe 22..23 Prof. Dr. A-S. Smith Dipl.-Phys. Matthias Saba am Lehrstuhl für Theoretische Physik I Department für Physik Friedrich-Alexander-Universität Erlangen-Nürnberg Problem. Theoretische Physik 2:
More informationarxiv:cond-mat/ v1 1 Apr 1999
Dimer-Type Correlations and Band Crossings in Fibonacci Lattices Ignacio G. Cuesta 1 and Indubala I. Satija 2 Department of Physics, George Mason University, Fairfax, VA 22030 (April 21, 2001) arxiv:cond-mat/9904022
More information