Stroh formalism for icosahedral quasicrystal and its application

Physics Letters A 376 (2012) 987–990

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

Stroh formalism for icosahedral quasicrystal and its application Lian He Li ∗ , Guan Ting Liu College of Mathematics Science, Inner Mongolia Normal University, Huhhot 010022, China

a r t i c l e

i n f o

Article history: Received 6 July 2011 Received in revised form 29 November 2011 Accepted 18 January 2012 Available online 21 January 2012 Communicated by A.R. Bishop

a b s t r a c t The Stroh formalism for two-dimensional deformations of the icosahedral quasicrystal is studied, for which there are six pairs of complex eigenvalues. The closed-form expressions for the elastic displacement and stress fields induced by a dislocation in an icosahedral quasicrystal are obtained using the extended Stroh formalism. The effect of phonon–phason coupling elastic constant on mechanical behavior are also discussed. © 2012 Elsevier B.V. All rights reserved.

Keywords: Icosahedral quasicrystal Plane elasticity Stroh formalism Dislocation

1. Introduction Quasicrystals are unique structures with long-range order but no periodicity. Since they were first proposed to exist [1], scientists have observed and studied more than 100 examples of these oddly ordered materials. In June 2009, 25 years after successfully creating artificial quasicrystals in the lab, researchers reported the existence of a fivefold, natural quasicrystal. The results suggest that quasicrystals can form and remain stable under geologic conditions, although there remain open questions as to how this mineral formed naturally [2]. The discovery could redefine the field of mineralogy and expand our understanding of how quasicrystals form, leading to new applications. The physical properties of quacisryatals have been the subject of numerous studies of solid-state physicists and materials scientists [3–5]. Among them, the theoretical research on elasticity and defects has been made great progress [6–8]. According to these theories, many methods were developed to solve the relevant elasticity and defects problems [9–12]. The icosahedral quasicrystals play a central role in the study of quasicrystalline solids, which are more complicated than oneand two-dimensional quasicrystals. Due to the complexity, no effective methods have been found so far to the elasticity and defect problem of icosahedral quasicrystals. Only a few results have been obtained under some approximate conditions [13]. It is well known that Stroh formalism [14,15] is an elegant and powerful tool for the study of two-dimensional deformation of anisotropic elastic materials, which has been applied successfully to some

*

Corresponding author. E-mail address: [email protected] (L.H. Li).

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areas, such as thermo-anisotropic elasticity, piezoelectric materials [16]. Recently, the Stroh formalism has been generalized successfully to one- and two-dimensional quasicrystals. Gao et al. [17] solved the elliptic hole and the rigid line inclusion problems of one-dimensional quasicrystals by the generalized Stroh formalism. Radi and Mariano [18] considered the straight cracks in two-dimensional quasicrystals by using Stroh formalism which is modified to account for a totally degenerate eigenvalue problem, a closed-form solution to the balance equations in terms of phonon and phason fields was provided under general loading conditions. Extensions of the Stroh formalism to icosahedral quasicrystals are investigated in this Letter. Explicit solutions for two-dimensional deformations of an infinite icosahedral quasicrystals subjected to a line dislocation are obtained. 2. Stroh formalism for two-dimensional deformation of icosahedral quasicrystals According to the elasticity theory of icosahedral quasicrystals [7], we have the generalized Hooke’s law

σi j = C i jkl εkl + R i jkl w kl H i j = R kli j εkl + K i jkl w kl

(1)

and the equilibrium equations (if the body force is neglected)

C i jkl uk, jl + R i jkl w k, jl = 0 R kli j uk, jl + K i jkl w k, jl = 0

(2)

in which a comma denotes differentiation, repeated indices imply summation, σi j , εi j , u i and C i jkl are stress, strain, displacement

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L.H. Li, G.T. Liu / Physics Letters A 376 (2012) 987–990

and elastic constants of phonon fields, respectively. H i j , w i j , w i and K i jkl are the stress, strain, displacement and elastic constants of phason fields. R i jkl are the phonon–phason coupling elastic constants. Consider the defect parallel to the fivefold direction of icosahedral quasicrystals, then all the field variables are independent of x3 [19]. Similar to reference [16], we introduce a generalized displacement vector u as

u = [u 1 , u 2 , u 3 , w 1 , w 2 , w 3 ]T = a f ( z)

(3)

where

z = x1 + px2 the superscript T represents the transpose, f ( z) is an analytic function, p is a complex number, and a is a constant six-element column vector. Eqs. (2) can be satisfied by (3) for arbitrary f ( z) if









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

(4)

where the 6 × 6 matrices Q, R and T are given by



C i1k1

R i1k1

R Ti1k1

K i1k1

C i2k2

R i2k2

R Ti2k2

K i2k2

Q=

 T=





R=

,  ,

C i1k2

R i2k1

R Ti1k2

K i1k2



i , k = 1, 2, 3

(5)

(6)

which gives twelve roots p α and p α (α = 1, 2, . . . , 6), where p α cannot be real because of the positive definiteness of the strain energy. The associated eigenvector a is determined from Eq. (4). Then the general solution of Eqs. (1) and (2) can be expressed as



u = 2 Re Af(z)

(7)

in which

f(z) =



a2

a3

f 1 ( z1 )

a4

a5

f 2 ( z2 )

a6 ]

f 3 ( z3 )

f 4 ( z4 )

f 5 ( z5 )

f 6 ( z6 )

T

(8)

and zα = x1 + p α x2 , without loss in generality we let Im p α > 0. Making use of Eqs. (1), (3) and (5), the phonon stress and phason stress can be rewritten as

σi1 = − pbi f  (z),

σi2 = bi f  (z)

H i1 = − pb i +3 f  ( z),

H i2 = b i +3 f  ( z)



(9)

(10)

Introducing the stress function φ , the phonon stress son stress H i j can be shown to be

σi2 = φi,1

H i1 = −φi +3,2 ,

H i2 = φi +3,1

σi j and pha-

i = 1, 2, 3

(11)

then the general solution for stress function φ is obtained as



φ = 2 Re Bf(z)

(12)

in which

B = [ b1

b2

b3

b4

b5

u(r, π ) − u(r, −π ) = bT∗ ,

φ(r, π ) − φ(r, −π ) = 0

(13)

Let

f α ( zα ) = qα ln zα ,

α = 1, 2, . . . , 6

(14)

where qα are arbitrary complex constants. Eqs. (7) and (12) can then be written as

u=

1

π





Im Aln zα q ,

φ=

1

π





Im Bln zα q

(15)

in which ln zα  is the diagonal matrix

ln zα  = diag[ln z1 , ln z2 , ln z3 , ln z4 , ln z5 , ln z6 ]   u

φ

b = RT + pT a

σi1 = −φi,2 ,

be infinitely long in the fivefold direction (x3 axis), b i and b⊥ i (i = 1, 2, 3) are Burgers components in parallel space b and perpendicular space b⊥ , respectively. Without loss in generality we take the plane x2 = 0, x1 < 0 to be the slip plane. Introducing a branch cut along the negative x1 axis we have r > 0, −π < θ < π (where x1 = r cos θ , x2 = r sin θ ). The boundary conditions for the problem at hand are

(16)

On substitution of (15) into (13) we have

i = 1, 2, 3

where, in matrix notation



In this section, consider a straight dislocation L (see Fig. 1)   [20] with an arbitrary Burgers vectors b∗ = b ⊕ b⊥ = (b1 , b2 , 

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

A = [ a1

3. Elastic field induced by a dislocation in icosahedral quasicrystals

⊥ ⊥ b3 , b⊥ 1 , b 2 , b 3 ) and the core at the origin, which is assumed to



for a nontrivial solution of a we must have



Fig. 1. The Burgers circuit λ surrounding the dislocation L .

b6 ]

Since the extended Stroh formalism for icosahedral quasicrystal materials preserves the most essential features of Stroh formalism, it will be a useful tool for the study of icosahedral quasicrystal anisotropic elasticity. As an illustration we will discuss two-dimensional deformations of an infinite icosahedral quasicrystal subject to a line dislocation.

=

1

π



Im

Aln zα BT Bln zα BT



bT∗

(17)

Assuming that there is no coupling between the phonon and phason strain (i.e. R = 0), the analytic expressions for both the phonon and phason displacements are obtained (see Appendix A), which agree with Ref. [19]. The explicit solutions for stresses (R = 0) are given in Appendix B, which shows that the phonon stresses are in accordance with the classical solutions of isotropic materials. In order to have a clear and direct picture on how the phonon– phason coupling elastic constants R affect the displacements, here, the icosahedral Al–Pd–Mn is used as an example to numerically demonstrate the displacements of phonon and phason fields. The material properties [21] are given by

λ = 74.9 GPa, K 1 = 125 MPa,

μ = 72.4 GPa K 2 = −50 MPa,

R 2 = 0.1μ K 1

Other parameters are taken to have the values below 



b1 = 2 × 10−9 m,

b2 = 1.5 × 10−9 m,

−9 m, b⊥ 1 = 2 × 10

−9 b⊥ m, 2 = 6.3 × 10



b3 = 1.6 × 10−9 m −9 b⊥ m 3 = 10.7 × 10

L.H. Li, G.T. Liu / Physics Letters A 376 (2012) 987–990

Fig. 4. The normalized displacements w x versus the angular θ for R = 0.

Fig. 2. The normalized displacements u x versus the angular θ for R = 0.

1

Fig. 3. The normalized displacements u x versus the angular θ for R = (0.1μ K 1 ) 2 .

The displacements are normalized by 1 m. The normalized displacements u x and w x versus the angular θ under the above parameters and R = 0 are depicted in Figs. 2–5, respectively, where r is fixed at r = 15 × 10−9 m. The above figures show the phonon and phason displacements are significantly affected by the phonon–phason coupling elastic constants R. The results corresponding to R = 0 to be about ten 1

989

times larger than the ones for R = (μ K 1 ) 2 . It has more favorable effects on phason displacements than phonon displacements. Therefore the phonon–phason coupling effect on the displacements is negligible. Mariano [22] pointed out that the nature of the influence of phason activity on a macroscopic discontinuity surface endowed with its own surface energy, and is generated by the variation of the energy with respect to variations of the material metric, the latter variations induced by the defects themselves.

1

Fig. 5. The normalized displacements w x versus the angular θ for R = (0.1μ K 1 ) 2 .

4. Conclusion and discussion A theoretical analysis is carried out for the generalized twodimensional problem of a line dislocation in icosahedral quasicrystals by using the extended Stroh formalism. The influences of the phonon–phason coupling elastic constants R on the displacements are numerically demonstrated. The results are reduced to the classical ones when the phason field is neglected. These results show that the extended Stroh formalism are powerful for the complicated boundary value problem of icosahedral quasicrystals. Acknowledgements The authors are very grateful to the reviewers for their helpful suggestions. This work is supported by the National Natural Science Foundation of China (Grant Nos. 11026175 and 10761005), the Inner Mongolia Natural Science Foundation (Grant

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L.H. Li, G.T. Liu / Physics Letters A 376 (2012) 987–990

Nos. 2009MS0102 and 2009BS0104), the Natural Science Foundation of Inner Mongolia Department of Public Education (Grant No. NJzy08024) and the Natural Science Foundation of Inner Mongolia Normal University (Grant No. QN07034). Appendix A



H 11 = −

− H 12 =

The phonon and phason displacement fields (R = 0) can be calculated using (17) as



1





r





+ (λ + 2μ)b1 θ   + (λ + μ)b1 sin θ cos θ + (λ + μ)b2 sin2 θ   1 r   −μb1 ln + (λ + 2μ)b2 θ u2 = 2π (λ + 2μ) r0   − (λ + μ)b2 sin θ cos θ + (λ + μ)b1 sin2 θ u1 =

μb2 ln

2π (λ + 2μ)

+ − H 22 = −



u3 = w1 =

b3

2π

+ w2 =

w3 =

K2

K 22

θ+

8K

b⊥ 3

 

4π K2

ln

r

sin 4θ

−

⊥ 

K 22 b2

K 4π

ln

r r0



+ 2 sin2 θ cos2 θ

3

K 1 K 2 b⊥ 1 2π

− 2 sin2 θ cos2 θ +

b⊥ 2 2π

 K2 θ − 2 sin 4θ 8K

K 1 K 2 b⊥ 2 K

2π

sin2 θ +

b⊥ 3 2π

θ

in which r0 is the radius of the dislocation core and K = K 12 − K 1 K 2 − K 22 . Appendix B The phonon and phason stress fields(R = 0) can be calculated using (17) as





2 2 μ(λ + μ)  x2 (3x21 + x22 )  x1 (x2 − x1 ) σ11 = − b1 + b 2 π (λ + 2μ) r4 r4  2 2 2 2  μ(λ + μ)  x1 (x1 − x2 )  x2 (x1 − x2 ) σ12 = σ21 = b1 + b 2 π (λ + 2μ) r4 r4



σ22 =

2 2 μ(λ + μ)  x2 (x21 − x22 )  x1 (x1 + 3x2 ) b1 + b2 4 4 π (λ + 2μ) r r

σ31 = − σ32 =

μb3 x2 2π r 2

μb3 x1 2π r 2

H 32 = −

K 1 K 22 b⊥ 2

−

π r4 r6  K 22 x1 (3x42 − 6x21 x22 − x41 )

+

r6

K

x2 (x41

− 6x21 x22 r6

+ x42 )

−

4K π 4 2 2 4 K 1 K 22 b⊥ 1 x1 (x1 − 6x1 x2 + x2 ) 4K π  K 1 b⊥ 2 4π

−

2x2 r2

2 K 2 b⊥ 3 x1 (x1

2 K 2 b⊥ 3 x1 x2

+

2 K 2 b⊥ 3 x1 x2

π

r4

r6  K 22 x2 (5x41 − 2x21 x22 + x42 ) r6

K

− x22 )

r4

2π 4 2 2 4 K 1 K 22 b⊥ 1 x2 (x1 − 6x1 x2 + x2 ) 4K π  K 1 b⊥ 2x1 2 4π

r2

−

r6  K 22 x1 (5x42 − 2x21 x22 + x41 ) r6

K

2 2 K 2 b⊥ 3 x2 (x1 − x2 )

2π r4 ⊥ 2 2 2 K 2 b1 x1 x2 K 2 b⊥ 2 x1 (x2 − x1 )

π

r4

2 K 2 b⊥ 1 x1 x2

π

r4

+ +

2π

r4

2 K 2 b⊥ 2 x2 (x2

2π

−

r4

x21 )

− +

K b⊥ 3 x2 2K 1 π r 2 K b⊥ 3 x1 2K 1 π r 2

References

sin2 θ cos θ sin θ −

− H 31 = −



r0

b⊥

K 1 2π K



sin 2θ

K 1 4π

b⊥ 1

−

+

θ

2π  b⊥ 1

r2



r6

K

4 2 2 4 K 1 K 22 b⊥ 2 x1 (x1 − 6x1 x2 + x2 )

4π

−

K 22 x2 (3x41 − 6x21 x22 − x42 )

+

r2

4π

4K π  K 1 b⊥ 2x1 1

H 21 = −

r0

K 1 b⊥ 2x2 1



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