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

and Applied Analysis 3 and the associated eigenvectors of μ 0 , μ n , and μ −n are X 0 0 = ( 0 1 0 0 ) , X 0 n = ( sin (μ n x) − cos (μ n x) 0 0 ) , X 0 −n = ( − sin (μ n x) − cos (μ n x) 0 0 ) , (15)


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][4][5][6][7][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][11][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.

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 Abstract and Applied Analysis the equilibrium equations and the generalized Hooke's law Here where Then ( 4) can be expressed in the following matrix forms: where Obviously, ) is the symplectic matrix with  2 being the 2×2 identity matrix.Thus,  1 and  2 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 From ( 9) and (3), we have

Theoretical Analysis
In the following, we only discuss (6), and the analysis for ( 7) is similar.First, considering the homogeneous equation of ( 6), Applying the method of separation of variables to the above equation, we write the solution as in which () =   , and  and () are the eigenvalue of the Hamiltonian operator matrix  1 and its associated eigenvector, respectively.They are determined by the equation Solving (13) with the boundary conditions ( 9) and ( 10) at  = 0, ℎ, we obtain the eigenvalues of  1 : and the associated eigenvectors of  0 ,   , and  − are respectively.From and the imposed boundary conditions, the first-order Jordan form eigenvectors of  0 ,   , and  − can be solved as 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, 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.
Similarly, for the Hamiltonian operator matrix  2 , we also have the following completeness theory.
are the associated eigenvectors and the first-order Jordan form eigenvectors of the eigenvalue  0 = 0 and   = /ℎ of  2 .

General Solution
By completeness Theorem 1, the general solution of the inhomogeneous equation ( 6) is represented in the form The vector  1 can also be expanded as Multiplying both sides of ( 26  Then, substituting (25) and ( 26) into (6) yields Thus, we obtain where  1 0 ,  0 0 ,  1  , and  0  are unknown constants to be determined by imposing the remaining boundary conditions at . Substituting (29) into (25), we have the analytical solutions   and   of (4) given by According to the above procedure for (7), the analytical solutions   and   of (4) can be obtained: where  1 0 ,  0 0 ,  1  , and  0  are unknown constants to be determined by imposing the remaining boundary conditions at  and

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.

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.
and   are the phonon and phason displacements,   and   are the phonon stresses and strains,   and   are the phason stresses and strains,  12 ,  66 ,  1 ,  2 , and  3 are the elastic constants, and   and   are the body and generalized body forces, respectively.

Table 1 :
The computed results.