Decagonal quasicrystal plate with elliptic holes subjected to out-of-plane bending moments

Physics Letters A 378 (2014) 839–844

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Physics Letters A www.elsevier.com/locate/pla

Decagonal quasicrystal plate with elliptic holes subjected to out-of-plane bending moments Lian He Li a,b,c,∗ , Guan Ting Liu a a b c

College of Mathematics Science, Inner Mongolia Normal University, Hohhot 010022, China College of Physical Science and Technology, Inner Mongolia University, Hohhot 010021, China Inner Mongolia Key Lab of Nanoscience and Nanotechnology, Hohhot 010021, China

a r t i c l e

i n f o

Article history: Received 22 September 2013 Received in revised form 25 November 2013 Accepted 17 January 2014 Available online 22 January 2014 Communicated by A.R. Bishop Keywords: Decagonal quasicrystal Stroh-like formalism Elliptic hole Bending

a b s t r a c t In the present paper, we consider only the ideal elastic behavior, neglecting the dissipation associated with the atomic rearrangements. Under these conditions, the decagonal quasicrystal plate bending problems have been discussed. The Stroh-like formalism for the bending theory of decagonal quasicrystal plate is developed. The analytical solutions for problems of decagonal quasicrystal plate with elliptic hole subjected to out-of-plane bending moments are obtained directly by using the forms. The resultant bending moments around the hole boundaries are also given explicitly. When the phonon–phason coupling is absent, the results reduce to the corresponding solutions for the isotropic elastic plates. © 2014 Elsevier B.V. All rights reserved.

1. Introduction After the first discovery of a quasicrystalline phase (icosahedral structure) by Shechtman in 1984 [1], much research was performed on the electronic structure and the optic, magnetic, thermal and mechanical properties of quasicrystals [2]. Elasticity is one of the important properties of quasicrystals. The elastic behavior of quasicrystals is different from that of usual crystals. Based on Landau–Anderson symmetry-breaking, the phason as a new elementary excitation was introduced in addition to the well known phonon. Phonons are responsible for translations of particles while phasons are responsible for rearrangements of local atomic configurations [3–5]. The problems of quasicrystals containing holes and cracks have been studied extensively for two-dimensional deformations [6,7]. Many methods and techniques have been developed to solve problems of elasticity and defects in quasicrystals. Among them, the decomposition procedure, the Green function method and integral transformations have been particularly successful [8–11]. However, relatively little work involving the bending problems of quasicrystals has been done due to the mathematical complexity. Boundary conditions for plate bending in one-dimensional hexagonal

quasicrystals and two-dimensional dodecagonal quasi-crystal have been considered by Gao et al. [12,13]. Recently, the complex variable methods in quasicrystal elasticity have reached a big step by connecting Muskhelishvili method, Lekhnitskii formulation and Stroh formalism [14–16]. However, still very few contributions have been made to the plate bending problems. The Stroh formalism is an elegant and powerful tool for the study of two-dimensional deformation of quasicrystal materials, which has been applied successfully to solve the elliptical hole, the rigid-line inclusion problems and the interaction between defects [17]. Under the conditions (as already mentioned), Stroh-like formalism for the bending theory of quasicrystal plates is developed in this paper. In our formalism, the deflections, the moments and the transverse shear forces can all be expressed in complex matrix form. Based on the developed formalism, the analytical solutions for decagonal plates with elliptic hole subjected to out-ofplane bending moments are now obtained explicitly. The solutions for the crack problems are obtained by letting the minor axis of the ellipse approach to zero and the moment intensity factors of the cracks are also given. Furthermore, a numerical example is given for the reader to make a quantitative assessment. 2. Stroh-like formalism for the bending theory of decagonal quasicrystal plate

*

Corresponding author at: College of Mathematics Science, Inner Mongolia Normal University, Hohhot 010022, China. Tel.: +86 471 4392483; fax: +86 471 7383390. E-mail address: [email protected] (L.H. Li). 0375-9601/$ – see front matter © 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2014.01.024

The drawbacks in standard format of quasicrystal linear elasticity have been discussed in Refs. [18–20]. In principle, a conservative

840

L.H. Li, G.T. Liu / Physics Letters A 378 (2014) 839–844

component of the inner self-action may exist, which make analysis more difficult. Following the point of view adopted in [21], here we adopt generalized elastic constitutive prescriptions [5] and the common assumption is accepted. The restriction of the analysis to the linear elastic case suggests to attribute an ideal limit character to the present results [21]. Assume that decagonal quasicrystal is periodic in z direction, and quasi-periodic in the x– y plane. According to the elastic theory of quasicrystals [5,6], we have the strain-displacement relations

εi j =

 ∂u j ∂ ui , + 2 ∂xj ∂ xi 1



wij =

h

M xx =

σxx z dz,

(1)

h

M xy = M yx =

N xx =

σxy = σ yx = 2C 66 εxy + R ( w yx − w xy )

H xx z dz,

N yy =

H y y z dz

− h2 h

2 N xy =

2 N yx =

H xy z dz,

⎫ ⎧ M xx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ M yy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ M xy ⎪ ⎬ ⎨

σ yz = σzy = 2C 44 ε yz σxz = σzx = 2C 44 εxz

N

H xx = K 1 w xx + K 2 w y y + R (εxx − ε y y )

xx ⎪ ⎪ N ⎪ y y ⎪ ⎪ ⎪ ⎪ ⎩ N xy

H y y = K 1 w y y + K 2 w xx + R (εxx − ε y y ) H xy = K 1 w xy − K 2 w yx − 2R εxy

N yx

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

H yx = K 1 w yx − K 2 w xy + 2R εxy H xz = K 3 w xz H yz = K 3 w yz

(3)

in which C 66 = (C 11 − C 12 )/2. σi j (σi j = σ ji ), εi j (εi j = ε ji ), u i and C i j are the stress, strain, displacement, and elastic constants of phonon fields, respectively. H i j (H i j = H ji ), w i j (w i j = w ji ), w i and K i are the stress, strain, displacement, and elastic constants of phason fields. R is the phonon–phason coupling elastic constant. The approximate theory of bending of decagonal quasicrystal plates (thin plates) is based on the Kirchhoff plate assumptions. It follows from the assumption that

∂w ∂y ∂v w y = −z ∂y

u y = −z

(4)

where w (x, y ) is the deflection of the middle plane, u (x, y ) and v (x, y ) are the generalized deflection of the middle plane. By Eqs. (1) and (4), we have

∂2 w , ∂ y2 ∂2v w y y = −z 2 ∂y ∂2v w yx = − z ∂ x∂ y

2

εxy = −z

H yx z dz

(8)

− h2

Substituting Eqs. (5) and (6) into Eqs. (3) then into Eqs. (7)–(8), the bending moments and the generalized bending moments can be expressed as

σzz = C 13 εxx + C 13 ε y y + C 33 εzz

ε y y = −z

h

2

− h2

σ y y = C 12 εxx + C 11 ε y y − R ( w xx + w y y )

∂2 w , ∂ x2 ∂ 2u w xx = − z 2 , ∂x ∂ 2u w xy = − z , ∂ x∂ y

(7)

h

σxx = C 11 εxx + C 12 ε y y + R ( w xx + w y y )

εxx = −z

σxy z dz

− h2

− h2

(2)

σ y y z dz

− h2

2

and the constitutive equations

∂w , ∂x ∂u w x = −z , ∂x

M yy =

h

∂ wi ∂xj

∂ Hij =0 ∂xj

u x = −z

2

− h2

the equilibrium equations

∂ σi j = 0, ∂xj

h

2

∂2 w ∂ x∂ y

(6)

When we cut this plate with certain surfaces parallel to the initial middle surface xy with height equal to the plate thickness h, then the bending moments M xx , M y y and M xy , the generalized bending moments N xx , N xy , N xy and N yx can be expressed as follows

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

∂2u ∂ x2 ∂2 v ∂ y2 ∂2u ∂ x∂ y ∂2 v ∂ x∂ y

(9)

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

in which the matrix D (see Appendix A) is bending stiffness matrix. We consider now the case when the plate is subjected to bending only by forces and moments distributed along the edge. The force and moment equilibrium equations of the plate can be expressed as

∂ M xx ∂ M xy ∂Qx ∂Q y + = 0, + − Qx = 0 ∂x ∂y ∂x ∂y ∂ M yx ∂ M y y + − Qy =0 ∂x ∂y ∂ N yx ∂ N y y ∂ N xx ∂ N xy + = 0, + =0 ∂x ∂y ∂x ∂y

(10) (11)

in which Q x and Q y are the transverse shear forces defined by h

h

2 Qx = − h2

(5)

= −D

⎧ ∂2 w ⎫ ⎪ ⎪ ⎪ ∂ x2 ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ∂ w ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ∂y ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ∂ w ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎨ ∂ x∂ y ⎪ ⎬

2

σxz dz,

Qy =

σ yz dz

− h2

The equilibrium equations (10) and (11) can be satisfied automatically if we introduce stress functions ψ1 (x, y ), ψ2 (x, y ), φ1 (x, y ) and φ2 (x, y ) such that

∂ψ1 ∂ψ2 , M yy = , ∂y ∂x ∂φ1 ∂φ2 N xx = − , N yy = ∂y ∂x ∂φ1 ∂φ2 N xy = , N yx = − ∂x ∂y M xx = −

M xy =

1



∂ψ1 ∂ψ2 − 2 ∂x ∂y



(12)

L.H. Li, G.T. Liu / Physics Letters A 378 (2014) 839–844

Then the curvatures κ1x , κ1 y and κ1xy , the generalized curvatures κ2x , κ2 y , κ2xy and κ2 yx can be expressed in terms of the stress functions as

⎧ ⎫ κ1x ⎪ ⎪ ⎪ ⎪ ⎪ κ1 y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ κ1xy ⎪ ⎬

κ

2x ⎪ ⎪ ⎪ κ2 y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ κ ⎪ ⎪ 2xy ⎩ ⎭

=−

κ2 yx

⎧ ∂2 w ⎫ ⎪ ⎪ ⎪ ∂ x2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂2 w ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ∂ y ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ∂ w ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ∂ x ∂ y ⎬ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

∂2u ∂ x2 ∂2 v ∂ y2 ∂2u ∂ x∂ y ∂2 v ∂ x∂ y

⎧ ⎫ 1 − ∂ψ ∂y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ψ 2 ⎪ ⎪ ⎪ ⎪ ∂ x ⎪ ⎪ ⎪ ⎪ 1 ( ∂ψ1 − ∂ψ2 ) ⎪ ⎪ ⎪ ⎪ ⎪ ∂y ⎪ ⎨ 2 ∂x ⎬ ∂φ1 −1 =D − ∂y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂φ2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂x ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂φ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ x ⎩ ⎭ ⎪ ⎪ ∂φ2 ⎪ − ⎪ ∂y ⎭

(13)

D ∗33 ∂ 2 ψ1





Q + p R + RT + p 2 T b = 0

(17)

where

⎡ − 1 D∗

Q= (14)

4 33 ⎢ − 12 D ∗23 ⎢ ⎣ − 1 D∗ 2 63 − 12 D ∗53

⎡ 1 D∗ 2

13

⎢ D ∗12 R=⎢ ⎣ 1 D∗

2 34 1 ∗ D 2 37

+ D ∗13

⎡ −D∗ 11 ⎢ − 12 D ∗31 ⎢ T=⎣ − D ∗41 − D ∗71

− 12 D ∗32 − 12 D ∗36 − D ∗22 − D ∗26 − D ∗62 − D ∗66 ∗ − D 52 − D ∗56 1 ∗ D D D∗ ⎤ 4 33 1 ∗ D 2 23 1 ∗ D 2 36 1 ∗ D 2 35

16

65

− D ∗55

15

D ∗24

D ∗27 ⎥

D ∗67

D ∗57

D ∗46

− 12 D ∗35 ⎤ − D ∗25 ⎥ ⎥ −D∗ ⎦

⎥

D ∗45 ⎦

− 12 D ∗13 − D ∗14 − D ∗17 ⎤ 1 ∗ 1 ∗ − 4 D 33 − 2 D 34 − 12 D ∗37 ⎥ ⎥ − 12 D ∗43 − D ∗44 − D ∗47 ⎦ − 12 D ∗73 − D ∗74 − D ∗77

(18)

and the superscript T represents the transpose. The explicit expressions of the matrices Q , R and T are given in Appendix B. For a nontrivial solution of b we must have

 

Q + p R + RT + p 2 T = 0

2 2 D ∗33 ∂ 2 ψ2 ∂ 2 ψ2 ∗ ∂ ψ2 ∗ ∂ φ1 − D 22 + D − − D 23 26 ∂ x∂ y 4 ∂ y2 ∂ x2 ∂ x2  2  ∗ ∗ 2 2 D D ∂ φ1 ∂ φ2 ∂ φ1 + D ∗24 + 36 − 34 − D ∗25 2 2 ∂ x∂ y 2 ∂ y2 ∂x  2  ∗ ∗ 2 D D ∂ φ2 ∂ φ2 + D ∗27 + 35 − 37 =0 2 ∂ x∂ y 2 ∂ y2   2 2 D ∗ ∂ 2 ψ1 D ∗34 ∂ 2 ψ1 ∗ ∗ ∂ ψ1 ∗ ∂ ψ2 − 63 + D + − D − D 16 41 62 2 ∂ x2 2 ∂ x∂ y ∂ y2 ∂ x2  2  ∗ ∗ 2 2 2 D D ∂ ψ2 ∂ ψ2 ∗ ∂ φ1 ∗ ∂ φ1 + D ∗42 + 63 − 43 − D + 2D 66 46 2 ∂ x∂ y 2 ∂ y2 ∂ x∂ y ∂ x2 2 2 2 2

∂ φ2 ∂ φ1 ∂ φ2 ∂ φ2 − D ∗44 − D ∗65 2 + D ∗45 + D ∗67 − D ∗47 =0 ∂ x∂ y ∂ y2 ∂x ∂ y2   2 D ∗ ∂ 2 ψ1 D ∗37 ∂ 2 ψ1 ∗ ∗ ∂ ψ1 − 53 + D + − D 15 17 2 ∂ x2 2 ∂ x∂ y ∂ y2  2  ∗ ∗ 2 D 53 ∂ ψ2 D 73 ∂ 2 ψ2 ∂ ψ2 ∗ − D ∗52 + D + − 72 2 ∂ x∂ y 2 ∂ y2 ∂ x2 2 2

∂ φ1 ∂ φ1 ∂ 2 φ1 − D ∗56 2 + D ∗45 + D ∗67 − D ∗47 ∂ x∂ y ∂x ∂ y2

∗

∂ 2 φ2 ∂ 2 φ2 ∂ 2 φ2 − D 55 2 + 2D ∗57 − D ∗77 =0 ∂ x∂ y ∂x ∂ y2

z = x + py



2 D ∗32 ∂ 2 ψ2 ∂ 2 ψ1 ∗ ∂ ψ1 − D − 11 4 ∂ x2 ∂ x∂ y 2 ∂ x2 ∂ y2  2  ∗ ∗ 2 D D ∂ ψ2 D ∗36 ∂ 2 φ1 ∂ ψ2 + D ∗12 + 33 − 13 − 4 ∂ x∂ y 2 ∂ y2 2 ∂ x2  2  ∗ ∗ 2 D D 35 ∂ 2 φ2 ∂ φ1 ∂ φ1 + D ∗16 + 34 − D ∗14 − 2 ∂ x∂ y 2 ∂ x2 ∂ y2  2  ∗ 2 D ∂ φ2 ∂ φ2 + D ∗15 + 37 − D ∗17 =0 2 ∂ x∂ y ∂ y2   D ∗ ∂ 2 ψ1 D ∗33 ∂ 2 ψ1 D ∗31 ∂ 2 ψ1 ∗ − 23 + D + − 21 2 ∂ x2 4 ∂ x∂ y 2 ∂ y2

∗

(16)

f ( z) is an analytic function, p is a complex number, and b is a constant four-element column vector. Similar to the generalized Stroh formalism for two-dimensional problems of quasicrystal, Eqs. (15) can be satisfied by Eq. (16) for arbitrary f ( z) if

Substituting Eq. (13) into Eqs. (14), the governing equations for the plate bending problem can then be obtained as

−

trix D−1 . The governing equations (15) are a set of homogeneous second-order differential equations. Without loss of generality we choose

where

ness D. The curvatures expressed by the stress functions should satisfied the following compatibility relations,

∂ κ1 y ∂ κ1xy = ∂x 2∂ y ∂ κ2 y ∂ κ2 yx = ∂x ∂y

where D ∗i j is the value in the ith row and jth column of the ma-

Φ = b f ( z)

in which the matrix D−1 is the inverse matrix of bending stiff-

∂ κ1xy ∂ κ1x = , ∂y 2∂ x ∂ κ2xy ∂ κ2x = , ∂y ∂x

841

(15)

(19)

which gives eight roots p α and p α (α = 1, 2, 3, 4). By Eqs. (13) and (16), we introduce the slope vector as

β = a f ( z)

(20)

where

β1 = −

∂w , ∂x

β2 = −

∂w , ∂y

β3 = −

∂u , ∂x

β4 = −

∂v ∂y

and

1 a = − (Q + pR)b p

(21)

The general solutions (16) and (20) for the plate bending problems can now be written as

  β = 2 Re Af( z) , where

⎧ ⎫ β ⎪ ⎨ 1⎪ ⎬ β2 β= , ⎪ ⎩ β3 ⎪ ⎭ β4 A = [ a1

a2

  Φ = 2 Re Bf( z) ⎧ ⎫ ψ ⎪ ⎬ ⎨ 1⎪ ψ2 Φ= , ⎪ ⎭ ⎩ φ1 ⎪ φ2

a3

a4 ] ,

(22)

⎧⎡ ⎤⎫ f (z ) ⎪ ⎨ 1 1 ⎪ ⎬ ⎢ f (z ) ⎥ f( z) = ⎣ 2 2 ⎦ ⎪ ⎩ f 3 ( z3 ) ⎪ ⎭ f 4 ( z4 )

B = [ b1

b2

b3

b4 ]

and zα = x + p α y, α = 1, 2, 3, 4. The appropriate form of f( z) in Eqs. (22) depends on the boundary conditions of the problems considered.

842

L.H. Li, G.T. Liu / Physics Letters A 378 (2014) 839–844

i.e., ζα−1  = diag[ζ1−1 , ζ2−1 , ζ3−1 , ζ4−1 ]. Therefore, the solution for the present problem can be expressed as

    β = β ∞ + 2 Re A ζα−1 q , ∞

    Φ = Φ ∞ + 2 Re B ζα−1 q

(27)

, Φ ∞ are given in (24) and (25), respectively. Knowing

where β the value ζα = e i ϕ along the hole boundary and the boundary condition (24) will then provide us



1 ∞ q = − B−1 am∞ 2 − ibm1 2

Fig. 1. A decagonal quasicrystal plate weakened by an elliptical hole subjected to out-of-plane bending moments.

3. Elliptical holes Consider an unbounded decagonal quasicrystal plate weakened by an elliptical hole subjected to out-of-plane bending moments ˆ xx , M ˆ y y and generalized bending moments Nˆ xx , Nˆ y y at infinity, M see Fig. 1. There is no load around the edge of the elliptical hole. The contour of the elliptical hole is represented by

x = a cos ϕ ,

y = b sin ϕ

where 2a, 2b are the major and minor axes of the ellipse and ϕ is a real parameter. The boundary conditions of this problem can be expressed as

ˆ xx , M xx = M ˆ xx , N xx = N

ˆ yy M yy = M

(28)

With the explicit solutions found in (27) and (28), all the physical responses can be obtained. In practical applications, one is usually interested in the moments around the hole boundary since most of the critical stress occurs along the hole boundary. The moment of phonon fields around the hole boundary in the normal– tangent coordinate (n–s coordinate) can be obtained as

∂ψ1 ∂ψ2 cos θ − sin θ ∂n ∂n ˆ xx + M ˆ yy =M

Ms = −

(29)

The n and s denote, respectively, the directions normal and tangent to the boundary and θ denotes the angle directed clockwise from the positive x-axis to the tangent direction of s. By Eq. (29), we know that the expression for the moment of phonon fields is identical to the well-known results of the classical elasticity theory [22]. The moment of phason fields around the hole boundary can also be given as

∂φ1 ∂φ2 cos θ − sin θ ∂n ∂n



ˆ xx + 1 − 2/e sin2 θ cos2 θ M ˆ yy = d1 1 + 2e cos2 θ sin2 θ M  

1 ˆ xx + ed1 (θ) + d2 (θ) + (1 + e ) cos2 θ N 2d2  

1 ˆ yy − d1 (θ) + 1/ed2 (θ) + (1 + 1/e ) sin2 θ N (30)

Ns = − at infinity

ˆ yy N yy = N

(23)

The boundary conditions (23) can now be expressed by the stress function vector Φ as ∞ Φ = Φ ∞ = xm∞ at infinity 2 − ym1

Φ = 0 along the hole boundary

2d2

(24)

The slope vector β ∞ associated with Φ ∞ can be obtained from the constitutive laws given in (13), which leads to

in which e = b/a, d1 , d2 , d1 (θ) and d2 (θ) can be found in Appendix C. From Eq. (30), we see that the moment of phason fields depends on the loading, the ratios b/a as well as the material properties of the plate. Consider a decagonal quasicrystal plate weakened by an elliptic ˆ xx at infinity. hole subjected to out-of-plane bending moments M The material properties of the decagonal quasicrystal are [23]

∞ β ∞ = xκ ∞ 1 + yκ 2

C 11 = 23.43 GPa,

in which T ˆ ˆ m∞ 1 = { M xx , 0, N xx , 0} ,

ˆ ˆ T m∞ 2 = {0, M y y , 0, N y y }

(25)

K 2 = 2.4 GPa,

where

⎧ ∞ ⎫ κ1x ⎪ ⎪ ⎪ ⎬ ⎨ κ ∞ /2 ⎪ ∞ ∞ 1 yx κ∞ 1 = ∞ ⎪ = −Tm1 + Rm2 ⎪ κ ⎪ ⎪ 2x ⎭ ⎩ ∞ κ2 yx /2 ⎧ ∞ ⎫ κ1xy /2 ⎪ ⎪ ⎪ ⎬ ⎨ κ∞ ⎪ ∞ 1y κ∞ = Rm∞ 2 = 1 − Qm2 ∞ ⎪ κ /2 ⎪ ⎪ ⎪ 2xy ⎭ ⎩

C 12 = 5.741 GPa,

K 1 = 12.2 GPa

R = −0.11 GPa

The effects of ratios b/a on the phason moment distributions around the hole boundary are depicted in Fig. 2. From Fig. 2 we see that for hole problems, the maximum value of N s increases when the ratio b/a increases. 4. Moment intensity factors for crack problems

κ2∞y

Similar to the corresponding two-dimensional problems, the complex function vector f( z) of (22) is selected to be

As a special case of the results, the solutions for an infinite decagonal quasicrystal plate containing a crack of length 2a can be obtained by letting b = 0 in (28). Consider an infinite plate conˆ xx , M ˆ yy taining a center crack subjected to bending moments M

(26)

ˆ xx , Nˆ y y at infinity. The stress and generalized bending moments N function vector Φ and slope vector β of this problem can therefore be obtained as

where q is a constant vector to be determined by the boundary condition, the angular bracket   stands for the diagonal matrix,

(31)





f( z) = ζα−1 q,

ζα =

zα +



2 − a2 − p 2 b 2 zα α

a − ip α b

    β = β ∞ − a Re A ζα−1 B−1 m∞ 2     Φ = Φ ∞ − a Re B ζα−1 B−1 m∞ 2

L.H. Li, G.T. Liu / Physics Letters A 378 (2014) 839–844

843

and Program of Higher-level Talents of Inner Mongolia University (Grant No. 125125). Appendix A

⎡

C 11 ⎢ C 12 ⎢ 0 h3 ⎢ ⎢ D= ⎢ R 12 ⎢ ⎢ R ⎣ 0 0

C 12 C 11 0 −R −R 0 0

0 0 C 66 0 0 −R R

R −R 0 K1 K2 0 0

R −R 0 K2 K1 0 0

0 0 −R 0 0 K1 −K2

⎤

0 0 ⎥ ⎥ R ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ −K2 ⎦ K1

Appendix B

⎡ Fig. 2. Phason moments around the elliptical hole in decagonal quasicrystal plate.

12 ⎢ ⎢

Q=

− K 1 +2 K 2 0

⎢ ch3 ⎣

1 a

zα +



⎡ 2 − a2 zα

R=

6

lim h2 r →0

√

=

h2

⎢ ⎣

2π rM y (r , θ)

ˆ yy π aM

0

−R

⎡ T=

θ =0

6√

ch3

)+2R 2 − C 12 ( KC111++KC212

12 ⎢ ⎢

)−2R − C 11 ( KC111++KC212

⎢ ch3 ⎣

(32)

where r is the distance ahead of the crack tip, h is the thickness of the plate, and θ = 0 is the direction along the crack.

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11072104, 1272053 and 11262017), the Key Project of Chinese Ministry of Education (Grant No. 212029), the Inner Mongolia Natural Science Foundation (Grant No. 2013MS0114), the Natural Science Foundation of Inner Mongolia Department of Public Education (Grant No. NJZZ13037), Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region (Grant No. NJYT-13-B07)

−R 0 2 − 2(CK661K−1K−2R )

⎤

K1+K2 2

0

0

R

0

R

0

0

2(C 66 K 2 − R 2 ) K1−K2

2 − 2(CK661K−2K−2R )

−R

0

R

− K 1 +2 K 2

0

R

0

0

R

− 2(CK661K−1K−2R

⎥ ⎥ ⎥ ⎦

⎥ ⎥ ⎥ ⎦

0

⎤

0

⎥ ⎥ ⎥ ⎦

R 2

)

0

0

− 2(CK661K−1K−2R

2

)

Appendix C

d1 =

Acknowledgements

0

⎤

in which c = 2(C 66 ( K 1 + K 2 ) − 2R 2 ).

5. Conclusion and discussion Under the conditions (as already mentioned), a Stroh-like formalism for the bending theory of quasicrystal plates is developed. Based on this formalism, the explicit closed form solutions for a decagonal quasicrystal plate containing an elliptical hole subjected to out-of-plane bending moments are obtained. The solutions for the crack problems are given by letting the minor axis of the ellipse approach to zero and the phonon moment intensity factors of the cracks are also obtained. One should note that the Stroh-like formalism introduced in this paper considers only the quasicrystal plate bending problems. Due to the possible coupling between bending and in-plane deformation, this formalism may not be used to solve the problem for the general laminated composites.

0

0

2

0

−R 2 − 2(CK661K−1K−2R )

−R 0

12 ⎢ ⎢

2

0

0

For the crack problems, it is interesting to know the phonon moment intensity factor defined by

KI =

)−2R − C 11 ( KC111++KC212

−R

where

ζα =

0

2C 66 ( K 1 − K 2 ) R K 1 (C 66 ( K 1 + K 2 ) − 2R 2 )

2 d2 = 2C 11 K 1 ( K 1 + K 2 )2



+ R 2 16K 1 R 2 + C 12 K 12 + 6K 2 K 1 + K 22

− C 11 2C 12 K 1 ( K 1 + K 2 )2 + 17K 12 + 6K 1 K 2 + K 22 R 2

d1 (θ) = cos 6θ + cos θ,

d2 (θ) = cos 6θ − cos θ

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