# On Symplectic Analysis for the Plane Elasticity Problem of Quasicrystals with Point Group 12 mm

Authors: Hua Wang; Jianrui Chen; Xiaoyu Zhang
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

The symplectic approach, the separation of variables based on Hamiltonian systems, for the plane elasticity problem of quasicrystals with point group 12 mm is developed. By introducing appropriate transformations, the basic equations of the problem are converted to two independent Hamiltonian dual equations, and the associated Hamiltonian operator matrices are obtained. The study of the operator matrices shows the feasibility of the method. Without any assumptions, the general solution is presented for the problem with mixed boundary conditions.


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## Body

## 1. Introduction

Quasicrystals (QCs), a new material and structure, were first discovered by the authors in [1] in 1984. QCs that exhibit excellent physical and mechanical properties, such as low friction, high hardness, and high wear resistance, have promising potential applications (cf. [2]). It is well known that the general solution of quasicrystal elasticity is very important, but it is difficult to be obtained because of the complexity of the basic governing equations. So far, many methods and techniques have been developed to seek for the general solution (see, e.g., [3–8]). However, some problems of quasicrystal elasticity have not been solved well due to the complicated assumptions of the solution, and the symplectic approach, developed by Zhong [9], may be helpful in those problems.The symplectic approach has advantages of avoiding the difficulty of solving high order differential equations and having no any further assumptions and has been applied into various research fields such as elasticity [10–12], viscoelasticity [13], fluid mechanics [14], piezoelectric material [15], and functionally graded effects [16]. In this method, one needs to transform the considered problem into Hamiltonian dual equations and then obtains the desired Hamiltonian operator matrix. Based on the eigenvalue analysis and eigenfunction expansion, the analytical solution of the problem can be explicitly presented. It should be noted that the feasibility of this method depends on the completeness of eigenfunction systems of the corresponding Hamiltonian operator matrices.To the best of the author’s knowledge, there are no reports of the method on the analysis of QCs. The objective of this paper is to propose the symplectic approach for the plane elasticity problem of quasicrystals with point group 12 mm. After derivation of two independent Hamiltonian dual equations of the problem, we prove the completeness of eigenfunction systems for the corresponding Hamiltonian operator matrices. Finally, we obtain the analytical solution with the use of the eigenfunction expansion.


## 2. Basic Equations and Their Hamiltonian Dual Equations

According to the quasicrystal elasticity theory, we have the deformation geometry equations of the plane elasticity problem of quasicrystals with point group 12 mm(1)ɛij=12(∂ui∂xj+∂uj∂xi),wij=∂wi∂xj,i,j=1,2,
the equilibrium equations
(2)∂σij∂xj+fi=0,∂Hij∂xj+gi=0,
and the generalized Hooke’s law
(3)σxx=C12(ɛxx+ɛyy)+2C66ɛxx,σyy=C12(ɛxx+ɛyy)+2C66ɛyy,σxy=σyx=2C66ɛxy,Hxx=K1wxx+K2wyy,Hyy=K1wyy+K2wxx,Hxy=(K1+K2+K3)wxy+K3wyx,Hyx=(K1+K2+K3)wyx+K3wxy.
Here ui and wi are the phonon and phason displacements, σij and ɛij are the phonon stresses and strains, Hij and wij are the phason stresses and strains, C12, C66, K1, K2, and K3 are the elastic constants, and fi and gi are the body and generalized body forces, respectively.Substituting (1) and (3) into (2), we get the displacement equilibrium equations
(4)C66∇2ux+(C12+C66)∂∂x(∂ux∂x+∂uy∂y)+f1=0,C66∇2uy+(C12+C66)∂∂y(∂ux∂x+∂uy∂y)+f2=0,K1∇2wx+(K2+K3)∂∂y(∂wx∂y+∂wy∂x)+g1=0,K1∇2wy+(K2+K3)∂∂x(∂wx∂y+∂wy∂x)+g2=0,
where ∇2=(∂2/∂x2)+(∂2/∂y2). Let
(5)q1=C66C12+2C66(∂ux∂y-∂uy∂x),q2=∂ux∂x+∂uy∂y,q3=K1+K2+K3K1(∂wx∂y+∂wy∂x),q4=-∂wx∂x+∂wy∂y.
Then (4) can be expressed in the following matrix forms:
(6)∂∂yZ1=H1Z1+F1,(7)∂∂yZ2=H2Z2+F2,
where Z1=(ux,uy,q1,q2)T, Z2=(wx,wy,q3,q4)T, F1=-(1/(C12+2C66))(0,0,f1,f2)T, F2=(-1/K1)(0,0,g1,g2)T,
(8)H1=(0∂∂xC12+2C66C660-∂∂x001000-∂∂x00∂∂x0),H2=(0-∂∂xK1K1+K2+K30∂∂x001000∂∂x00-∂∂x0).
Obviously, H1T=JH1J and H2T=JH2J, in which J=(0I2-I20) is the symplectic matrix with I2 being the 2×2 identity matrix. Thus, H1 and H2 are both Hamiltonian operator matrices and (6) and (7) are exactly the Hamiltonian dual equations for the plane elasticity problem of quasicrystals with point group 12 mm.We consider the problem satisfying the mixed boundary conditions(9)ux=0,σxy=0,forx=0,x=h,wx=0,Hxy=0,forx=0,x=h.From (9) and (3), we have
(10)q1=q3=0,forx=0,x=h.


## 3. Theoretical Analysis

In the following, we only discuss (6), and the analysis for (7) is similar.First, considering the homogeneous equation of (6),
(11)∂∂yZ1=H1Z1.
Applying the method of separation of variables to the above equation, we write the solution as
(12)Z1=X(x)Y(y),
in which Y(y)=eμy, and μ and X(x) are the eigenvalue of the Hamiltonian operator matrix H1 and its associated eigenvector, respectively. They are determined by the equation
(13)H1X(x)=μX(x).Solving (13) with the boundary conditions (9) and (10) at x=0,h, we obtain the eigenvalues of H1:
(14)μ0=0,μn=nπh,μ-n=-nπh,wwwwwwwwwwwwwwn=1,2,…,
and the associated eigenvectors of μ0, μn, and μ-n are
(15)X00=(0100),Xn0=(sin(μnx)-cos⁡(μnx)00),X-n0=(-sin(μnx)-cos⁡(μnx)00),
respectively. From
(16)H1Xn1(x)=μnXn1(x)+Xn0(x),
and the imposed boundary conditions, the first-order Jordan form eigenvectors of μ0, μn, and μ-n can be solved as
(17)X01=(0001),Xn1=(sin(μnx)(C12+3C66(C12+C66)μn-1)cos⁡(μnx)2C66C12+C66sin(μnx)2C66C12+C66cos⁡(μnx)),X-n1=(-sin(μnx)-(C12+3C66(C12+C66)μn+1)cos⁡(μnx)-2C66C12+C66sin(μnx)2C66C12+C66cos⁡(μnx)),
respectively. Besides, we can verify that there are no other high-order Jordan form eigenvectors in every chain.It is easy to prove that the above eigenvectors and Jordan form eigenvectors satisfy the symplectic conjugacy and orthogonality; that is,(18)∫0h‍X00TJX00dx=∫0h‍X00TJXn1dx=∫0h‍X01TJXn0dx=∫0h‍X01TJXn1dx=0,∫0h‍Xn0TJXn0dx=∫0h‍Xn0TJXn1dx=∫0h‍Xn0TJX-n0dx=∫0h‍Xn1TJX-n1dx=0,∫0h‍X00TJX01dx=h,∫0h‍Xn0TJX-n1dx=-2C66C12+C66h,∫0h‍X00TJX01dx=h,X01dxh,X01dx=h,n=±1,±2,….Next, we will prove the symplectic orthogonal expansion theorem, that is, the completeness theorem of the generalized eigenvector system (i.e., the collection of all the eigenvectors and Jordan form eigenvectors), which shows that the symplectic method can be adopted to solve the title problem.Theorem 1.
The generalized eigenvector system(19){X00,X01}∪{Xn0,Xn1∣n=±1,±2,…}
of the Hamiltonian operator matrix H1 is complete in the Hilbert space X; that is, there exist constant sequences {c00,c01}, {cni}n=1∞, and {c-ni}n=1∞(i=0,1) such that
(20)Φ=c00X00+c01X01+∑n=1+∞‍(cn0Xn0+cn1Xn1+c-n0X-n0+c-n1X-n1)
for each Φ=(ϕ1(x), ϕ2(x), ϕ3(x), ϕ4(x))T∈X, where X=L2[0,h]×L2[0,h]×L2[0,h]×L2[0,h].Proof.
For anyΦ∈V, in order to prove equality (20), we set
(21)c00=∫0h‍ΦTJX01dx∫0h‍X00TJX01dx=∫0h‍ϕ2dxh,c01=∫0h‍ΦTJX00dx∫0h‍X01TJX00dx=∫0h‍ϕ4dxh,cn0=∫0h‍ΦTJX-n1dx∫0h‍Xn0TJX-n1dx=∫0h‍ϕ1sin(μnx)dx-∫0h‍ϕ2cos⁡(μnx)dxh-(∫0h‍ϕ3sin⁡(μnx)dy+((C12+3C66)(C12+C66)μn+1)sin(μn)×∫0h‍ϕ4cos⁡⁡(μnx)dx)×(2C66C12+C66h)-1,cn1=∫0h‍ΦTJX-n0dx∫0h‍Xn1TJX-n0dx=∫0h‍ϕ3sin(μnx)dx+∫0h‍ϕ4cos⁡(μnx)dx(2C66/(C12+C66))h
by the symplectic orthogonal relationship (18). Then,
(22)c00X00+c01X01+∑n=1+∞‍(cn0Xn0+cn1Xn1+c-n0X-n0+c-n1X-n1)=(∑n=1+∞‍2h∫0h‍(ϕ1sinnπxhdx)sinnπxh1h∫0h‍ϕ2dx+∑n=1+∞‍2h(∫0h‍ϕ2cos⁡nπxhdx)cos⁡nπxh∑n=1+∞‍2h∫01‍(ϕ3sinnπxhdx)sinnπxh1h∫0h‍ϕ4dx+∑n=1+∞‍2h(∫0h‍ϕ4cos⁡nπxhdx)cos⁡nπxh),
in which the four components of the above expression are the corresponding Fourier series of ϕ1,ϕ2,ϕ3,ϕ4 associated with the orthogonal function system {sin⁡(nπx/h)}n=1+∞ or {cos⁡⁡(nπx/h)}n=0+∞ in L2[0,h]. Therefore, equality (20) is valid, which means that the generalized eigenvector system of H1 is complete in the Hilbert space X.Similarly, for the Hamiltonian operator matrixH2, we also have the following completeness theory.Theorem 2.
The generalized eigenvector system(23){X¯00,X¯01}∪{X¯n0,X¯n1∣n=±1,±2,…}
of the Hamiltonian operator matrix H2 is complete in the Hilbert space X, where
(24)X¯00=(0100),X¯n0=(sin(λnx)cos⁡(λnx)00),X¯01=(0001),X¯n1=(sin(λnx)(1+2K1+K2+K3λn(K2+K3))cos⁡(λnx)-(2K1K2+K3+2)sin(λnx)(2K1K2+K3+2)cos⁡(λnx))
are the associated eigenvectors and the first-order Jordan form eigenvectors of the eigenvalue λ0=0 and λn=nπ/h of H2.


## 4. General Solution

By completeness Theorem1, the general solution of the inhomogeneous equation (6) is represented in the form
(25)Z1(x,y)=X00Y00(y)+X01Y01(y)+∑n=1+∞‍(Xn0Yn0(y)+Xn1Yn1(y)+X-n0Y-n0(y)wwwww+X-n1Y-n1(y)).
The vector F1 can also be expanded as
(26)F1=X00F00(y)+X01F01(y)+∑n=1+∞‍(Xn0Fn0(y)+Xn1Fn1(y)+X-n0F-n0(y)+∑n=1+∞‍e++X-n1F-n1(y)).
Multiplying both sides of (26) by X01TJ, X00TJ, X-n1TJ, X-n0TJ, Xn1TJ, and Xn0TJ and then integrating by x from 0 to h, respectively, we have
(27)F00(y)=∫0h‍X01TJF1dx∫0h‍X01TJX00dx=0,F01(y)=∫0h‍X00TJF1dx∫0h‍X00TJX01dx=-∫0h‍f2dx(C12+2C66)h,Fn0(y)=∫0h‍X-n1TJF1dx∫0h‍X-n1TJXn0dx=(∫0h‍f1sin(μnx)dx+(C12+3C66(C12+C66)μn+1)×∫0h‍f2cos⁡⁡(μnx)dx)×(2C66(C12+2C66)C12+C66h)-1,Fn1(y)=∫0h‍X-n0TJF1dx∫0h‍X-n0TJXn1dx=-∫0h‍f1sin⁡(μnx)dx-∫0h‍f2cos⁡⁡(μnx)dx(2C66(C12+2C66)/(C12+C66))h∫0h‍X-n0TJF1x∫0h‍X-n0TJXn1x∫0h‍X-n0TJF1dx∫0h‍X-n0TJXn1dxn=±1,±2,….
Then, substituting (25) and (26) into (6) yields
(28)dY01(y)dy=F01(y),dY00(y)dy=Y01(y),dYn1(y)dy=μnYn1(y)+Fn1(y),dYn0(y)dy=μnYn0(y)+Yn1(y)+Fn0(y).
Thus, we obtain
(29)Y01(y)=c01+∫0y‍F01(ξ)dξ,Y00(y)=c00+c01y+∫0y‍∫0τ‍F01(ξ)dξdτ,Yn1(y)=cn1eμny+∫0y‍Fn1(ξ)eμn(y-ξ)dξ,Yn0(y)=(cn0+cn1y)eμny+∫0y‍Fn0(ξ)eμn(y-ξ)dξ+∫0y‍∫0τ‍Fn1(ξ)eμn(y-ξ)dξdτ,
where c01, c00, cn1, and cn0 are unknown constants to be determined by imposing the remaining boundary conditions at y. Substituting (29) into (25), we have the analytical solutions ux and uy of (4) given by
(30)ux=∑n=1+∞‍[∫0τ‍(cn0+cn1+cn1y)eμny-(c-n0+c-n1+c-n1y)e-μny+∫0y‍((Fn0(ξ)+Fn1(ξ))eμn(y-ξ)+∫0y‍w-(F-n0(ξ)+F-n1(ξ))e-μn(y-ξ))dξ+∫0y‍∫0τ‍(Fn1(ξ)eμn(y-ξ)+∫0y‍∫0τ‍e-F-n1(ξ)e-μn(y-ξ))dξdτ∫0τ‍]sinμnx,uy=c00+c01y+∫0y‍∫0τ‍F01(ξ)dξdτ-∑n=1+∞‍[(cn0-(C12+3C66(C12+C66)μn-1)cn1+cn1y)eμny-w∑n=1+∞‍+(c-n0+(C12+3C66(C12+C66)μn+1)c-n1+c-n1y)e-μny-w∑n=1+∞‍+∫0y‍((Fn0(ξ)-(C12+3C66(C12+C66)μn-1)-w∑n=1+∞‍+=∫0y‍w×Fn1(ξ)Fn0(ξ)-(C12+3C66(C12+C66)μn-1))eμn(y-ξ)-w∑n=1+∞‍+∫0y‍w+(F-n0(ξ)+(C12+3C66(C12+C66)μn+1)-w∑n=1+∞‍+=∫0y‍ww×F-n1(ξ)(C12+3C66(C12+C66)μn+1))e-μn(y-ξ)(C12+3C66(C12+C66)μn+1))dξ-w∑n=1+∞‍+∫0y‍∫0τ‍(Fn1(ξ)eμn(y-ξ)+F-n1(ξ)e-μn(y-ξ))dξdτ]-w∑n=1+∞‍-w∑n=1+∞‍×cos⁡μnx.According to the above procedure for (7), the analytical solutions wx and wy of (4) can be obtained:
(31)wx=∑n=1+∞‍[(∫0τ‍F¯-n1(ξ)e-λn(y-ξ))(dn0+dn1+dn1y)eμny=∑n=1+∞‍w-(d-n0+d-n1+d-n1y)e-λny=∑n=1+∞‍w+∫0y‍((F¯n0(ξ)+F¯n1(ξ))eλn(y-ξ)=∑n=1+∞‍w+∫0y‍w-(F¯-n0(ξ)+F¯-n1(ξ))e-λn(y-ξ))dξ=∑n=1+∞‍w+∫0y‍∫0τ‍(F¯n1(ξ)eλn(y-ξ)=∑n=1+∞‍+∫0y‍wwww-F¯-n1(ξ)e-λn(y-ξ))dξdτ(∫0τ‍F¯-n1(ξ)e-λn(y-ξ))]sinλnx,wy=d00+d01y+∫0y‍∫0τ‍F¯01(ξ)dξdτ+∑n=1+∞‍[(dn0+(1+2K1+K2+K3λn(K2+K3))dn1+dn1y)eλny+∑n=1+∞‍e+(d-n0+(1-2K1+K2+K3λn(K2+K3))d-n1+∑n=1+∞‍e++d-n1y(1-2K1+K2+K3λn(K2+K3)))e-λny+∑n=1+∞‍e+∫0y‍((F¯n0(ξ)+(1+2K1+K2+K3λn(K2+K3))F¯n1(ξ))+∑n=1+∞‍e+w∫0y‍×eλn(y-ξ)+∑n=1+∞‍e+∑n=1+∞‍+(F¯-n0(ξ)+(1-2K1+K2+K3λn(K2+K3))F¯-n1(ξ))+∑n=1+∞‍e+∑n=1+∞‍+∑n=1+∞‍×e-λn(y-ξ))dξ+∑n=1+∞‍e+∫0y‍∫0τ‍(F¯n1(ξ)eλn(y-ξ)+∑n=1+∞‍+∫0y‍weee+F¯-n1(ξ)e-λn(y-ξ))dξdτ(1+2K1+K2+K3λn(K2+K3))]cos⁡λnx,
where d01, d00, dn1, and dn0 are unknown constants to be determined by imposing the remaining boundary conditions at y and
(32)F¯01=∫0h‍g2dxK1h,F¯n1=-∫0h‍g1sinλnxdx+∫0h‍g2cos⁡λnxdx((2K1+K2+K3)/(K2+K3))h,F¯n0=((1-2K1+K2+K3λn(K2+K3))∫0h‍g1sinλnxdy==-(1-2K1+K2+K3λn(K2+K3))∫0h‍g2cos⁡λnxdx)×(2K1+K2+K3K2+K3h)-1.


## 5. Numerical Calculations

Compared with [17], the present paper is devoted to the symplectic analysis of the plane elasticity problem of quasicrystals. To guarantee the feasibility of our method, we also prove the completeness for the eigenfunction system of the associated Hamiltonian operator matrices. Note that the completeness does not always hold for the Hamiltonian operator matrices.In order to determine the unknown constantscnk and dnk of the analytical solution in (30) and (31), we consider the boundary conditions at y=0, y=l given by
(33)ux=sin4πhx,uy=0,fory=0,y=l,wx=sin4πhx,wy=0,fory=0,y=l.In the following, leth=5, l=1, and we take the constants C12=0.5714, C66=0.88445, K1=1.22, K2=0.24, and K3=0.6. The computed results are listed in Table 1 for illustrating previous main results, and the data is the same as that of using the treatment in [17].Table 1
The computed results.
(x,y)
(0.1464, 0.8536)
(1.4, 0.55)
(2.1, 0.34)
(3.42, 0.89)
(4.7, 0.27)
u
x
0.204
−0.08131
−0.237
0.4822
−0.2427
u
y
−0.06272
−0.01618
−0.02746
−0.0399
−0.04801
w
x
0.2965
−0.243
−0.583
0.6355
−0.4975
w
y
−0.02621
−0.006761
−0.01148
−0.01667
−0.02006


## 6. Conclusions

The symplectic approach is established for the plane elasticity problem of quasicrystals with point group 12 mm satisfying the mixed boundary conditions. The corresponding Hamiltonian operator matrix plays an important role in this method, whose eigenvalues and eigenfunctions need to be obtained. Through calculations, the eigenfunction system is symplectic orthogonal. Based on this, we further verify the feasibility of this approach. Then the exact analytical solution is given with the use of the symplectic eigenfunction method. We can know that the method is totally rational and gives us a systematic way to solve physical problems. In addition, this approach is expected to apply to other quasicrystal problems.



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