1 Introduction

Quasicrystals (QCs) as a new structure of solid matter were discovered first on April 8, 1982,and were first reported by Shechtman et al.^[1],who won the Nobel’s Prize in 2011. This discovery has brought a significant breakthrough for condensed matter physics in recent years, because QCs possess both quasi-periodic long-range translational symmetry and noncrystallographic rotational symmetry. According to the cut-and-projection method,a three-dimensional (3D) quasilattice can be obtained by the selected projection of the respective six-dimensional (6D) periodical lattice^{[2, 3]}. The one-dimensional (1D) (or two-dimensional (2D)) QCs are the ones in which the atomic structures of the materials are quasiperiodic in one direction or two directions,while are periodic in the other two directions or one direction. The 3D QCs show quasiperiodicity in all the three directions. Since the discovery of QCs,great progress has been made in the elastic theory for many years^{[4, 5, 6, 7, 8]}. To solve the boundary value problems of elasticity for QCs,the governing equations and the general solutions are of great importance. Liu et al.^[9] investigated systematically the governing equations of the plane elasticity problems for 1D QCs with all point groups,and obtained the general solutions. With the differential operator matrix method,Chen et al.^[10],Wang^[11],Wang and Pan^[12],and Gao et al.^[13] analyzed the general solutions of 3D elastic problems for 1D hexagonal QCs. For the plane elasticity problem of 2D QCs with noncrystal rotational symmetry,Liu et al.^[14] presented the general solutions of different point groups including dodecagonal,pentagonal,decagonal,and octagonal systems. By virtue of the operator method,Gao et al.^[15] studied a theory of general solutions of the plane problems for 2D octagonal QCs. Gao and Zhao^[16] obtained the general solutions of the 3D problems for 2D QCs by introducing the displacement functions and using the operator analysis technique. For the elastic problems of 3D QCs,Fan and Guo^[17] derived the final governing equation and the fundamental solution of plane elasticity of icosahedral QCs. Based on the stress potential function,Li and Fan^[18] presented the general solution of plane elasticity for icosahedral QCs. Gao and Zhao^[19] made a general treatment of 3D elasticity for QCs by the operator method. Gao^[20] further simplified the governing equation of cubic QCs by introducing a displacement function,and established the general solutions through an operator method.

QCs are sensitive to mechanical,thermal,electrical,magnetic,and optical effects. The physical properties of QCs have been investigated intensively^{[21, 22, 23, 24, 25, 26, 27, 28, 29]}. The independent and non-vanishing first-order piezoelectric,piezomagnetic,pyromagenetic,photoelastic,and magnetoelectric coefficients are obtained^{[30, 31, 32]}. The development of QCs,such as the material properties,the theories of elasticity,and some applications,has been addressed^{[33, 34, 35, 36]}.

Rao et al.^[32] studied the electric effects of QCs on the piezoelasticity in QCs. Altay and Dökmeci^[37] developed the 3D fundamental equations of piezoelasticity of QCs. As mentioned above,only the elastic problems of QCs have been concerned. Recently,Li et al.^[38] addressed the 3D general solutions to static problems of 1D hexagonal piezoelectric QCs by introducing two displacement functions and utilizing the rigorous operator theory. By introducing four potential functions,Zhang et al.^[39] obtained the general solutions of the plane problems in 1D orthorhombic QCs with the piezoelectric effect. However,the governing equations of the plane piezoelasticity theory of other 1D QCs with all point groups and the general solutions have not been done up to now. It is well-known that the governing equations and the general solutions play an important role in solving the boundary value problems of the piezoelasticity of QCs, because they not only have theoretical merits themselves,but also test the validity of various approximate methods such as the finite element method and the boundary element method. Meanwhile,they pave the way to the forthcoming study of dislocation,fracture,interface,and similar problems for the piezoelasticity of QCs. Therefore,it is the purpose of this work to investigate systematically the governing equations of the plane piezoelasticity of 1D QCs with all point groups. With the help of the decomposition and superposition principles,the general solutions are derived by the operator method and the complex variable function method.

2 Basic equations for piezoelasticity of QCs

In a fixed rectangular coordinate system xi (i = 1,2,3),the basic equations for the piezoelasticity of QCs presented by Altay and D¨ okmeci^[37] are as follows. The equilibrium equations are

The gradient equations are The constitutive equations are

In the above equations,a comma denotes partial differentiation,and the repeated indices represent summation. σ_ij,ε_ij,and u_i are the stress,the strain,and the displacement of the phonon field,respectively. H_ij,ω_ij,and w_i are the stress,the strain,and the displacement of the phason field,respectively. D_i,E_i,and φ stand for the electric displacement,the electric field,and the electric potential,respectively. C_ijkl,K_ijkl,R_ijkl,e_ijk,and d_ijk stand for the phonon elastic,the phason elastic,and the phonon-phason coupling moduli,respectively. λ_ij stands for the dielectric permittivity. The following reciprocal symmetry conditions hold:

For stable materials,C_ijkl,K_ijkl,and λ_ij satisfy the positive-semidefinite conditions as follows:

for non-zero vector ηi and non-zero tensor η_ij.

For the piezoelasticity problems of 1D QCs,there are non-zero phonon displacements u_x, u_y,and u_z,phason displacement w_z (w_x = w_y = 0),and electric potential φ. Therefore,the corresponding strains and electric fields are

The above equation holds for the piezoelasticity of all 1D QCs. In this paper,we only consider the piezoelasticity of 1D QCs,because among various QCs,1D QCs are of particular interest for researchers after the success of Merlin et al.^[40] in growing model systems,in which quasiperiodicity was built up.

According to the symmetry operations of point groups^[41],for the piezoelasticity of monoclinic QCs with the point group m and the z-axis as a symmetry axis,we can obtain the constitutive equations as follows:

where short notations are used for the phonon elastic constant tensors,i.e.,

and C_ijkl is denoted as C_pq accordingly. There are 13 independent phonon elastic constants, i.e.,

4 independent phason elastic constants,i.e.,

8 phonon-phason coupling elastic constants,i.e.,

15 independent piezoelastic constants,i.e.,

and 4 independent dielectric permittivities,i.e.,κ₁₁,κ₂₂,κ₃₃,and κ₁₃. Thus,for the 1D monoclinic QCs,there are 44 non-zero material constants in total.

From Eq. (7),we can obtain the corresponding stress-strain relations as follows:

The corresponding equilibrium equations to Eq. (1) are

It is found from Eqs. (6),(8),and (9) that there are 29 equations and 29 field variables including 4 displacements,9 strains,9 stresses,3 electric fields,3 electric displacements,and one electric potential. Thus,the elastic equilibrium problem of piezoelasticity of 1D monoclinic QCs is more complicated than that of 3D classic elasticity,1D monoclinic QC elasticity and piezoelectric materials. We will present a rigorous treatment of the problem in this work.

3 Governing equations of plane piezoelasticity of QC systems

3.1 Monoclinic QC

If there is a straight dislocation or a Griffith crack along the direction of the atom quasiperiodic arrangement and the polarized direction of the electric field along the z-axis,the deformation is independent of the z-axis,i.e.,

Therefore,we have the following gradient equations and equilibrium equations in the absence of the body forces of phonon and phason fields and the electric density:

The constitutive equation (8) can be simplified as follows:

Substituting the gradient equations in Eq.(11) into the constitutive equations in Eq.(13), and then into the equilibrium equations in Eq.(12),we have the final governing equations in terms of the displacements and electric potential as follows:

This is a phonon-phason-electric coupling elasticity problem,involving the displacements u_x, u_y,u_z,w_z and the electric potential φ.

3.2 Orthorhombic QC

For the orthorhombic QC with the point group 2mm,the increase in the symmetric elements leads to

Therefore,the number of the non-zero independent electro-elastic constants of 1D orthorhombic QCs reduces to 28,i.e.,

for the phonon elastic constants,K₁,K₂,and K₃ for the phason elastic constants,R₁,R₂,R₃,R₄,R₅,and R₆ for the phonon-phason coupling elastic constants,

for the piezoelastic constants,and κ₁₁,κ₂₂,and κ₃₃ for the independent dielectric permittivities. With the superposition principle,we can decompose Eqs.(11)-(13) and (15) into the following two uncoupled problems.

Problem I

It is similar to a plane strain problem for orthorhombic crystals. The solution of Problem I agrees well with that of the classical elasticity theory. Therefore,it is not given in this work.

Problem II

It is an anti-plane phonon-phason-electric coupling elasticity problem,involving only the displacements u_z and w_z and the electric potential φ.

For Problem II,the governing equations turn into

3.3 Tetragonal QC

For the tetragonal QC with the point group 4mm,besides Eq. (15),the number of new symmetrical elements increases,i.e.,

For Problem II,from Eq. (18),we can simplify the governing equations as follows: where ∇² is the Laplace operator defined by

3.4 Hexagonal QC

For the hexagonal QCs with the point group 6mm,we further have

For Problem II,the governing equations are the same as Eq. (20).

4 General solutions of QC systems

4.1 Monoclinic QC system

The governing equation (14) can be rewritten as the following matrix equation:

where V = (u_x,u_y,u_z,w_z,φ)^T,and D is a 5 × 5 differential operator matrix expressed by The elements in Eq. (23) are The general solutions of Eq. (22) can be obtained by the operator method developed by Gao and Zhao^[19],Wang and Wang^[42],and Wang and Shi^[43]. Due to the complexity,we do not give them here. Among various 1D QCs,the special QCs including orthorhombic QCs,tetragonal QCs,and hexagonal QCs are of particular interest for researchers. Therefore,we will give the general solutions of these special QCs by the operator method or the complex variable function method as follows.

4.2 Orthorhombic QC system

The solutions of Problem II are given as follows by using the operator method. The governing equation (18) is rewritten as the following matrix equation:

where U = (u_z,w_z,ϕ)^T,and A is a 3 × 3 differential operator matrix expressed by Let us introduce a 3 × 3 differential operator matrix B as the “adjoint matrix” of A such that where the components B_ij of B are “algebraic complement minors” of A,i.e., The determinant of A is defined by where Let us introduce a displacement function F,which satisfies where ∇_i² can be written as follows: In Eq. (32),s_i² (i = 1,2,3) are three characteristic roots of the following cubic algebra equation of s²: The three roots expressed by a,b,c,and d exist a real one among them. Assume that s₁² is the real root without loss of generality. Moreover,we further assume Re(s_i² ) > 0.

Therefore,the general solutions of Eq. (25) can be obtained as follows:

Take one of the general solutions of Eq. (34) as an example,i.e., or where From the work of Gao and Zhao^[16],it can be proved that the above-mentioned general solutions are complete in any limited domain in E^3[20]. According to the theorem,if the domain Ω is y-convex and F follows that then there exist displacement functions F_i (i = 1,2,3) in the three forms as follows:

Case 1

Case 2

Case 3

In the above equations,F_i (i = 1,2,3) satisfy the following second-order governing equations:

It is clearly seen that the complicated governing equations (18) can be simplified into several partial differential equations of lower order by the operator method. We will deduce three different forms of the general solutions of the anti-plane problem for the orthorhombic QC system by considering the different cases of three characteristic roots.

For Case 1,when

Eqs. (32),(39),and (36) yield where For Case 2,when

Eqs. (32),(36),and (40) lead to For Case 3,when

Eqs. (32),(36),and (41) result in 4.3 Tetragonal QC system

For Problem II,Eq. (20) is satisfied if u_z,w_z,and φ are harmonic functions. This can be achieved by letting u_z,w_z,and φ be the imaginary parts of the analytic functions U(z),W (z), and Φ(z),respectively,such that

where z = x + iy,and Im denotes the imaginary part of the complex function. The stresses of the phonon field,the phason field,and the electric displacements can then be expressed as follows: where Re is the real part of the complex function,and the prime indicates differentiation with respect to the complex variable z.

It is found that the solution to Problem II of the hexagonal QC system is the same as that of the tetragonal QC system.

5 Fracture mechanics of Griffith crack

To our interest,the phonon-phason-electric coupling anti-plane elasticity problem described by Eq. (47) may bring new insight into the piezoelasticity scope of QCs.

Consider a mode III fracture problem,for which a Griffith crack with the length 2a is embedded in an infinite 1D hexagonal QC subjected to far-field phonon,phason,and electrical loads (see Fig. 1). The boundary conditions on the upper and lower surfaces of the crack are free of the surface traction and the surface charge,i.e.,

Then,with Eq. (47),we can obtain the stresses and the electric displacements as follows:

where Re is the real part of the complex function,and the prime indicates differentiation with respect to the complex variable z. If the medium is loaded uniformly at infinity,we take a semi-inverse method by assuming U(z),W(z),and Φ(z) to be

It can be seen that Eqs. (47) and (49) are satisfied. The unknown real constants A,B and C will be determined from the far-field loading conditions. Substituting Eq. (51) into Eqs.(47) and (50) yields

where r and θ are the coordinates defined in Fig. 2.

By applying the far-field loading conditions,the constants A,B,and C are obtained for the following possible boundary conditions at infinity:

Case 1 σ_yz^∞ = τ^∞,H_yz^∞ = H^∞,D_y^∞ = D^∞ as x² + y² → ∞

Case 2 σ_yz^∞ = τ^∞,H_yz^∞ = H^∞,E_y^∞ = E^∞ as x² + y² → ∞

Case 3 σ_yz^∞ = τ^∞,ω_yz^∞ = ω^∞,D_y^∞ = D^∞ as x² + y² → ∞

Case 4 σ_yz^∞ = τ^∞,ω_yz^∞ = ω^∞,E_y^∞ = E^∞ as x² + y² → ∞

Case 5 γ_yz^∞ = 2ε_yz^∞ = γ^∞,H_yz^∞ = H^∞,D^∞_y = D^∞ as x² + y² → ∞

Case 6 γ_yz^∞ = 2ε_yz^∞ = γ^∞,H_yz^∞ = H^∞,E^∞_y = E^∞ as x² + y² → ∞

Case 7 γ_yz^∞ = 2ε_yz^∞ = γ^∞,ω_yz^∞ = ω^∞,D^∞_y = D^∞ as x² + y² → ∞

Case 8 γ_yz^∞ = 2ε_yz^∞ = γ^∞,ω_yz^∞ = ω^∞,E^∞_y = E^∞ as x² + y² → ∞

Evaluating the solution (53) near the right crack tip and extending the traditional concept of stress intensity factors to other field variables,we have

where K_‖^S and K_‖^III denote the strain factor and the stress intensity factor of the phonon field,respectively. K_⊥^S and K_⊥^III stand for the strain factor and the stress intensity factor of the phason field,respectively. K_E and K_D are the electric field factor and the electric displacement intensity factor,respectively. For this problem,these field intensity factors have the followin forms:

For this particular problem,the field variables have the same crack-tip behavior as the classical mode III fracture problem. If all electrical quantities are made to vanish,the present solutions can be reduced to the solutions of 1D hexagonal QCs^[36]. If all the phason field quantities vanish,the present solutions can be reduced to the solutions of piezoelectric materials^[44]. It can be seen from Eqs. (62) and (63) that the stresses of the phonon field,the phason field, and the electric displacement are uncoupled with each other. Further,the field intensity factors corresponding to the field variables used at infinity are independent of the material constants, and are uncoupled with each other. Therefore,we will consider the energy release rate in the characterizing defects subjected to more than one field loading. For this,we firstly derive a conservative integral. Mariano and Planas^[45] pointed out that the phason tractions at the surface of the QC are null because we do not know any loading device that is able to use non-zero phason tractions at the external boundary of a quasicrystalline body. In this work,we only consider the phason tractions similar to the phonon tractions from the theoretical point of view.

For an antiplane shear crack in other 1D QCs,the solutions can be obtained to utilize the rigorous operator theory^{[38, 39]}.

6 Conservative integral

Let us define an energy function F defined by

Substituting Eq. (3) into Eq. (64),we have If the QC material is homogeneous and is free of any defects,then Eq. (65) becomes With the help of Eq. (3),Eq. (66) can be further written as Now,let us consider the following integral: where I_p denotes the pth component of the conservation integral,and δjp is the Kronecker delta.

With Eq. (1),we can rewrite Eq. (68) as follows:

Moreover,according to the divergence theorem,Eq. (68) turns into

where n_j denotes the n_j-direction component of the unit normal vector n of a closed surface S (n directs towards the environment). It is found from Eqs. (68) and (70) that I_p = 0. Thus, for any closed surface S in a homogeneous material,the following relation holds: Therefore,Eq. (71) is called the path-independent integral for the linear piezoelasticity of QCs. The x-component of Ip is the J-integral of fracture mechanics of the piezoelasticity of QCs.

It is seen from Eq. (65) that the energy function F is not positive definite. We further define an internal energy density W that represents the internal energy per unit volume as follows:

7 Energy release rate

The path-independent integral derived earlier can be used to obtain the energy release rate for the mode III piezoelasticity fracture problem of QCs. Denote J to be the x-component of the conservation integral I_p. Then,the path-independent integral takes the following form:

Using the solution obtained previously,the J-integral can be obtained by evaluating Eq. (73) on a vanishingly small contour at a crack tip as follows: This result can also be obtained by considering the virtual crack closure integral as follows: It is clear that due to the linear piezoelasticity of QCs,for the purely elastic and piezoelectroelastic cases,the value of the J-integral is identical to the energy release rate G,i.e., Substituting Eq.(63) into Eq.(75),the energy release rate can be expressed by the field intensity factors as follows: which indicates that the energy release rate depends not only on the field intensity factors but also on the material constants. Thus,the energy release rate can be used as the fracture criterion for the piezoelasticity of QCs. If there is no applied electric loading at infinity,i.e., D^∞ = 0 or E^∞ = 0,Eq.(77) reduces to the results of 1D hexagonal QCs^[36]. If there is no applied phason field at infinity,i.e.,H_zy^∞ = 0 or w_z^∞ = 0,Eq.(77) reduces to the results of piezoelectric materials^[44].

8 Conclusions

Based on the fundamental equations of piezoelasticity of QCs,the governing equations of plane piezoelasticity problems for 1D QCs with all point groups are investigated systematically with the symmetry operations of point groups. The equilibrium problem of piezoelasticity of 1D QCs is more complicated than that of 3D classical elasticity,1D QC elasticity,and purely piezoelectric materials. When the electric field is neglected,the obtained governing equations in this paper are identical to the governing equations of the plane elasticity problems for the corresponding QCs^[9]. If the phason field is not considered,the present governing equations can be reduced to the results of piezoelectric materials^[36]. The general solutions of the plane piezoelasticity problems for 1D QCs with all point groups are derived by the operator method and the complex variable function method. For some special QC systems such as orthorhombic QCs,tetragonal QCs,and hexagonal QCs,the plane piezoelasticity problem can be decomposed into two uncoupled problems,i.e.,the classical plane strain elasticity problem of conventional crystals and the phonon-phason-electric coupling anti-plane elasticity problem of 1D QCs.

As an application,a mode III piezoelasticity fracture is formulated,and the solutions of the phonon,phason,and electric fields near the crack tip are obtained by the semi-inverse method for the case of out-of-plane mechanical and in-plane electrical loadings. The stresses of the phonon field,the phason field,and the electric displacements at the crack tip show traditional square root singularities. Among the phonon field,the phason field,and the electrical field,any applied load alone cannot cause the singularities for the other two fields. The path-independent integral derived from the conservation integral equals the energy release rate,which can be used as the fracture criterion. The present results can be reduced to the earlier theories of elasticity of QCs and piezoelectric materials,which pave the way to the forthcoming study of dislocation, fracture,interface,and similar problems of both elasticity and piezoelasticity of QCs.