Contact problems exist in many fields,e.g.,civil engineering and mechanical engineering. The initial study of the subject began in 1881^[1]. All kinds of contact problems are described in monographs^{[2, 3, 4]}. Bagault et al.^{[5, 6]} investigated the anisotropy orientation effects on the contact solution by use of the semi-analytical methods. Mehmet^[7] and Alinia et al.^[8] studied the closed-form solution of a two-dimensional (2D) sliding frictional contact problem and a rolling contact problem of functionally graded materials by use of the singular integral equation approach, respectively. Chidlow et al.^[9] explored the adhesive and non-adhesive contact problems of functionally graded materials by use of the integral equation and Galerkin method. Due to the difficulties in mathematics,there have been analytical solutions for only some simple contact problems,and the closed-form solutions of complex contact problems are still hanging. Therefore,scholars turned to the help of numerical methods. Peric and Owen^[10], Shao et al.^[11],and Yan et al.^[12] successfully used the finite element theory to solve the contact problems for piezoelectric materials. Migorski^[13] investigated a class of inequality problems for the static frictional contact between a piezoelastic body and a foundation. Liu and Migorski^[14] considered an evolution problem to model the frictional skin effects in piezoelectricity. Migorski et al.^{[15, 16]} discussed a dynamic frictional problem of piezoelectric materials. Mikael and Mirceal^[17] studied a dynamic process of the frictionless contact problem between a piezoelectric body and an electrically conductive foundation by use of a penalized approach and a version of Newton's method. Matei and Ciurcea^[18] considered a static process of the unilateral frictionless contact mode under the small deformation hypothesis. The weak solvability of the considered model was proved based on a weak formulation with dual Lagrange multipliers.

As new materials,quasicrystals are different from crystals and non-crystals^[19]. The discovery of quasicrystals changes people's understanding in solid structures,and has been a significant breakthrough on condensed matter physics in recent years. To learn the physical properties and mechanical properties of quasicrystals more deeply,many mathematical elastic theory problems of quasicrystals have been studied. Fan^[20] and his research group introduced the basic concepts and frameworks of the mathematical elasticity theory of quasicrystals. At present,the defects of quasicrystals are mostly focused on^{[21, 22, 23, 24, 25, 26, 27]}.

Contact problems can be solved by the same method used for the duality problems on cracks^{[28, 29]}. Peng and Fan^[30] solved the contact and crack problem in one-dimensional (1D) hexagonal quasicrytal half-plane by use of the integral transformation and Fourier series methods. With the aid of the Fourier transform,Zhou^[31] and Yin et al.^[32] solved the smooth contact problems in decagonal and oetagonal quasicrystals, respectively. Within the framework of the Green function, Wu^[33] investigated the smooth contact problems in 1D and 2D quasicrytal half spaces. Gao and Ricoeur^[34] gave the perfect adhesive problem solutions in quasicrytal half-infinite spaces. Wang et al.^[35] solved two kinds of contact problems in decagonal quasicrystals by use of the complex variable function method. Zhou et al.^[36] explored the axisymmetry contact problem of cubic quasicrystals under the rigid cylindrical flat punch by use of the Hankel transform.

Three-dimensional (3D) icosahedral quasicrystals have been widely used in practice,though their structures are complex. Therefore,the research on the elastic theory of icosahedral quasicrystals plays a very important role in reality and practice. However,the contact problems for icosahedral quasicrystals have seldom been reported. Therefore,we focus on the study of the contact problems of 3D icosahedral quasicrystals in this paper.

The paper is organized as follows. In Section 2,the basic equations and the complex function expressions of the stress and displacement are given for icosahedral quasicrystals. Then,the frictional contact problem and the adhesive contact problem are described in Sections 3 and 4,respectively. By use of the classical contact theory method,the frictional contact problem and the adhesive contact problem are converted into the Riemann-Hilbert problems,and then the analytic expressions of the contact stress in the phonon field are obtained by solving the Riemann-Hilbert problems. According to the analytic expressions,the singularities are analyzed at the edge of the contact zone. In Section 5,some numerical examples are presented to analyze the change regulations of the contact stress. Meanwhile,the displacement-force relation is given for the adhesive problem. The summary and the related conclusions are given in Section 6.

Based on the quasicrystals elasticity theory,the generalized Hooke's law can be expressed as follows^[20]:

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The deformation equations are^[20]

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If the body force is neglected,the equilibrium equations can be written as follows^[20]:

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where $\sigma_{ij}$ and $H_{ij}$ are the stress components of the phonon field and the phason field,respectively. $\varepsilon _{ij}$ and $w_{ij}$ are the strain components of the phonon field and the phason field,respectively. $u_i$ and $w_i$ are the displacement components of the phonon field and the phason field,respectively. $C_{ijkl}$ $(i,j,k,l = 1,2,3)$ and $K_{ijkl}$ $(i,j,k,l = 1,2,3)$ are the elastic constants of the phonon field and the phason field, respectively. $R_{ijkl}$ $(i,j,k,l = 1,2,3)$ are the coupling elastic constants of the phonon-phason field. $ {C_{ijkl}} = \lambda {\delta_{ij}}{\delta _{kl}} + \mu ({\delta _{ik}}{\delta _{jl}} + {\delta_{il}}{\delta _{jk}})$ $(i,j,k,l = 1,2,3)$,where $\lambda $ and $\mu$ are the Lame constants,and $\delta _{ij}$ are the Kronecker symbols.

Since all the field variables are independent of $x_3$,we have

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Under the approximation $ \big(\frac{R}{\mu}\big)^2 \ll 1,$ the solution of Eqs. (1)-(3) can be converted into solving the function $G(x_1,x_2)$ satisfying

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where

Equation (5) is called the final governing equation of icosahedral quasicrystals. Applying the properties of the operator theory,the basic solution of Eq. (5) can be written as follows:

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where ${g_i}(z)\;(i = 1,2,\cdots,6)$ are arbitrary analytic functions about $z$. $z = {x_1} + {\rm{i}}{x_2}$,and $\bar z = {x_1}-{\rm{i}}{x_2}$,where ${\rm{i = }}\sqrt {-1} $ is the imaginary unit.

According to Ref. ^[26],the complex variable functions of the stress and displacement can be expressed as follows:

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where

In the above equations,$g_i^{(n)}(z)$ is the $n$th derivatives of the function.

As shown in Fig. 1,the upper half space of the 3D icosahedral quasicrystals is indented by a single rigid punch. The width of the punch is $2a$. In this case,the boundary conditions of the frictional problem can be derived as follows:

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Fig. 1 Upper half space of icosahedral quasicrystals indented by single rigid punch

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where $\rho > 0$ is the static frictional coefficient.

Obviously,$\Psi '(z) = \Theta '(z) = 0$ meets the phason field boundary conditions on the $x_1$-axis. On the contact surface,the ratio relationship between the shear stress and the normal stress in the phonon field can be expressed by $\Gamma '(z)$ and $\Omega '(z)$ as follows:

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The left-side of Eq. (10) is analytic on the upper half plane and tends to 0 at infinity,and the right-side of Eq. (10) is analytic on the lower half plane and also tends to 0 at infinity. Therefore, it is also established if $| {{x_1}} |> a$. Thus,we can derive

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On the contact surface,the normal displacement condition in the phonon field can be expressed by $\Gamma '(z)$ and $\Omega '(z)$ as follows:

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Substituting Eq. (11) into Eq. (12),then eliminating ${\Gamma ^\prime }^{\rm{ + }}({x_1})$ and ${\overline{\Omega} '^-}({x_1})$, we have

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where

Outside of the contact surface,the normal stress in the phason field can be expressed as follows:

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As before,substituting Eq.(11) into Eq.(12),and eliminating ${\Gamma ^\prime }^{\rm{ + }}({x_1})$ and ${\overline{\Omega} '^-}({x_1})$,respectively,we have

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where

Combining Eqs. (11) and (15),we have

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Let

Introduce the functions $H_1(z)$ as follows:

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where ${H_1}(z)$ satisfies the relation on the $x_1$-axis as follows:

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Obviously,Eq. (18) is a Riemann-Hilbert problem,whose solution is

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where the integral path $L$ represents the punch indentation on the contact surface,$F_1$ and $F_2$ are the horizontal phonon field force and the vertical phonon field force under the punch, respectively,and

The solution of Eq. (19) is satisfied with the equilibrium condition of the phonon field force,which is under the punch

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For the flat punch,$f'({x_1}) = 0$,and the solution of Eq.(19) can be rewritten as

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Combining Eqs. (17) and (21) yields

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Thus,the phonon field contact stress under the punch can be derived as follows:

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According to Eq. (23),it can be obtained that the contact stress shows the power type singularities $-\frac{1}{2} \pm \beta$ at the edge of the contact zone,where $\beta$ is determined by the static frictional factor,the elastic constants,and the coupling constant.

The hypothesis is the same as before. For the adhesive contact problem,the boundary conditions can be written as follows:

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Then,$\Psi '(z) = \Theta '(z) = 0$ is satisfied by the phason field boundary condition completely on the $x_1$-axis. On the contact surface,the displacement condition in the phone field can be expressed by $\Gamma '(z)$ and $\Omega '(z)$ as follows:

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Outside the contact surface,the condition of the normal stress in the phason field can be indicated as follows:

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where ${\eta _4} = 3(1-{\rm{i}}).$

Combining Eqs. (25) and (26),we have

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where

Let ${\tau _2} ={{{\eta _3}}}/{{{\eta _4}}}.$ Introduce $H_2(z)$ as follows:

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Then,$H_2(z)$ can be satisfied with the relation on the $x_1$-axis as follows:

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Equation (29) is also a Riemann-Hilbert problem. For the flat punch, its solution is

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where $P_0$ is the indentation force (vertical direction) in the phone field under the punch,and

Therefore,the contact stress in the phonon field under the punch can be expressed as follows:

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Obviously,for the adhesive contact problem,the contact stress shows the oscillatory singularities $-\frac 12 \pm {\rm{i}}\varepsilon $ at the edge of the contact zone,where $\varepsilon $ is determined by the elastic constants and the coupling coefficient.

Then,the displacement-force relation is considered for the adhesive problem of icosahedral quasicrystals. With the help of Eq. (8),we have

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Combining Eqs. (28) and (30),we have

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Integrating both sides of Eq. (32) around the indentation under the punch,we have^[4]

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Substituting Eq. (34) into Eq. (32) and then separating the real part yield

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Equation (35) is the expression of the force-displacement relation for the adhesive contact problem of icosahedral quasicrystals,from which we can find that the indentation force $P_0$ is proportional to the contact displacement $u_2$.

In this section,numerical examples are used to analyze the contact stress distribution on the contact surface and at the edge of the contact zone when the upper half space surface of the icosahedral quasicrystals is indented by a single rigid punch. In the calculation,the elastic constants are given as follows^[20]: $\lambda = 74.9$,$\mu = 72.9 {\rm{GPa}}$,${K_1} = 72$,${K_2} = -73 {\rm{GPa}}$,and $\frac{R}{\mu} = 0.02.$

Figure 2 shows the relation between the contact stress and the static frictional factor in the frictional problem. It indicates that the contact stress is 0 outside the contact surface. On the contact surface,when the width of the punch $a$ is given,the phonon field contact stress $\sigma _{22}$ increases with the increase in the static friction coefficient $\rho$. When the static friction coefficient $\rho$ is given,the contact stress $\sigma _{22}$ decreases with the increase in $a$. The conclusions are the same as those of classical materials. It shows that quasicrystals have some characteristics identical to other materials,and the solutions are consistent with the reality. Therefore,the correctness of the results in the problem is verified.

Fig. 2 Relation between contact stress and static frictional factor (frictional contact problem) with different ρ

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Figure 3 shows the relation between the contact stress and the coupling coefficient in the frictional problem. In Fig. 3,the phonon field contact stress $\sigma _{22}$ increases with the increase in the coupling coefficient $R$. It indicates that the phonon-phason coupling constant $R$ has a significant effect on the contact stress intensity,while has little impact on the contact stress distribution regulation. When $R = 0$,the quasicrystal materials degenerate into elastic materials,and the distribution of the contact stress is the same as that of the classical elastic materials. If $R \ne 0$,due to the effect of the phason field,the value of the contact stress is larger and the stress concentration is stronger than those of the classical elastic materials. It is consistent with the mechanical property of quasicrystals^[20]. Otherwise,according to the value of the contact stress,the value of the coupling coefficient $R$ can be determined. It can be used in indention experiments to ascertain the value of $R$. It also provides an idea to research the prosperity of quasicrystals by qualitative and quantitative analysis methods.

Fig. 3 Relation between contact stress and coupling constant (frictional contact problem) with different R/μ

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Figure 4 shows the relation between the contact stress and the coupling constant in the adhesive problem. From Fig. 4,it can be seen that the phonon field contact stress $\sigma _{22}$ increases with the increase in the coupling coefficient $R$. The coupling constant $R$ has obviously effects on the contact stress magnitude, while has little impact on the contact stress distribution regulation. The adhesive contact problem is the limited case of the frictional contact problem. Therefore,the contact stress distribution regulation is similar between the two kinds of contact problems. The difference is just in the magnitude of the stress. The above conclusions can be obtained from the comparison of Figs. 3 and 4. It can also be used to verify the correctness of the conclusions.

Fig. 4 Relation between contact stress and coupling constant (adhesive contact problem) with different R/μ

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As shown in Figs. 2-4,the contact stress exhibits singularities, and reaches the maximum value at the edge of the contact zone,and is therefore reasonable.

Figure 5 shows the relation between the indentation force and the contact displacement in the adhesive problem. It can be seen that the indentation force in the phone field is directly proportional to the contact displacement. The effects of the coupling constant $R$ on the indentation force are barely perceptible,because it is much smaller than the phonon elastic constants. The conclusion is consistent with that in Ref. ^[33]. In Ref. ^[33],the force-displacement relation is a curve,while it is straight in the present paper. The difference is caused by the shape of the punch^[33]. This shows that the variation tendency of the force with the displacement is similar.

Fig. 5 Force-displacement relation (adhesive contact problem) with different R/μ

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By use of the complex variable function method,two kinds of contact problems,i.e.,the frictional contact problem and the adhesive contact problem,in 3D icosahedral quasicrystals are discussed. Based on the complex variable function expressions of the stress and displacement in icosahedral quasicrystals,these two kinds of contact problems are converted into Riemann-Hilbert problems,and the analytic expressions of the phonon field contact stress under the action of a single rigid flat punch are obtained by solving the Riemann-Hilbert problems.

The analytic expressions of the contact stress show that:

(i) For the frictional contact problem,the contact stress exhibits power singularities at the edge of the contact zone.

(ii) For the adhesive contact problem,the contact stress exhibits oscillatory singularities at the edge of the contact zone.

The numerical examples show that:

(i) For these two kinds of contact problems,the contact stress exhibits singularities and reaches the maximum value at the edge of the contact zone. The effects of the phonon-phason coupling constant $R$ on the contact stress magnitude are obvious, while the effects of the phonon-phason coupling constant $R$ on the contact stress distribution regulation are little. When $R=0$,the results are consistent with the classical solutions in the contact problems of elastic materials,which can verify the correctness of the conclusions.

(ii) For the adhesive contact problem,the indentation force has positive correlation with the contact displacement,and the coupling constant effects are barely perceptible.

The paper only discusses two kinds of contact problems in 3D icosahedral quasicrystals under the action of single rigid flat punches. The solving processes of different shape punches are the same as those of the rigid flat punch. If changing a single punch into multiple punches or a moving punch,the contact problems can be solved by the same method.

For some quasicrystals,e.g.,2D octagonal quasicrystals and 3D dicubic quasicrystals,the final governing equations are not regular,and their analytic solutions are not easy or impossible to be obtained. In this case,the contact problems seem not to be perfectly solved by complex variable methods. Therefore,we can consider to find their weak solutions and take the advantages of the methods for the solutions of the contact problems for piezoelectric and functionally graded materials. This maybe another way to investigate the contact problems in quasicrystals.

Acknowledgements The authors are grateful to the valuable suggestions of Professor Xing LI for the paper.