1 Introduction

The problems of the inclusions embedded in an infinite homogeneous medium are a classic topic in elasticity. They are practically very important since inclusions have a significant effect on the mechanical behavior of the composites composed of matrices and reinforcing or functional inclusions^[1]. Inclusions can be regarded as rigid when their elastic constants are much larger than those of the matrices. Or,they may be treated as elastic when the differences of the elasticity between the inclusions and the matrices are not dramatically distinct. The problems are also common in mechanical engineering,where a circular hole in a plate may be filled or welded with different materials to enhance the whole integrity^[2]. A series of plane elasticity solutions and classical plate theory solutions for such problems have been obtained via the complex function theory. The readers are referred to Refs. [2]-[4] for instances.

Functionally graded materials (FGMs) are a new type of composites with continuously changing material properties along one or more directions. This special feature makes it possible to avoid the problems that are frequently encountered in conventional laminated materials, e.g.,stress concentration and interfacial debonding. Therefore, FGMs have shown significant application prospects in many fields. A few studies have been devoted to the static analysis of the FGM plates containing inclusions. Wang et al.^[5] investigated the problem of an FGM plane with a circular inclusion under a uniform antiplane eigenstrain when the shear modulus varies exponentially along one in-plane direction. Fang et al.^[6] considered the multiple scattering of electro-elastic waves and the dynamic stresses around a subsurface cylindrical inclusion in an FGM piezoelectric layer bonded on a homogeneous piezoelectric substrate. Yang and Gao^[7] studied the non-axisymmetric two-dimensional (2D) thermoelastic problem of an infinite matrix with an FGM-coated circular inclusion with the complex variable method. It is noted that the above-mentioned works all consider 2D problems only. There is an obvious lack of three-dimensional (3D) solutions in the literature.

In a series of papers,Spencer and his coworkers (see for example Mian and Spencer^[8]) have developed an elegant procedure for deriving the 3D analytical solutions of the equations of linear elasticity for the isotropic FGM plates with traction-free top and bottom surfaces. With the complex function theory,England^[9] extended the procedure of Mian and Spencer^[8] to an FGM plate bent by a pressure acting on its top surface,which satisfied the biharmonic equation or the higher-order ones. Hereinafter,this complex formulation will be referred to as the England-Spencer (plate) theory. With this theory,England^[10] considered the problem of an isotropic FGM annular plate containing a rigid inclusion,which was subject to some rigid-body motion. Yang et al.^{[11, 12]} extended the England-Spencer theory to transversely isotropic FGM plates,and obtained a series of 3D elasticity solutions for rectangular and annular FGM plates. Recently,their results were further utilized to develop a more general complex variable framework for the FGM plates with multiply connected regions^[13],and the 3D elasticity solutions for an infinite FGM plate with holes^[13] or an elastic circular inclusion^[14] were derived.

This work is a further extension of the previous studies of Refs. [13] and [14]. An infinite transversely isotropic FGM plate welded with a circular inclusion is considered,which may be subjected to the rotations or assigned with different material properties. In contrast to Yang et al.^[14] which focused on the axisymmetric case only,a general 3D situation is considered here, where the inclusion is assumed to be in more complex equilibrium states. 3D analytical solutions are derived and presented when the loads are applied at infinity on the cylindrical boundaries of the plate.

2 England-Spencer theory

We employ a system of rectangular Cartesian coordinates (x,y, and z). For a transversely isotropic FGM plate with the thickness h,the z-axis coincides with the material symmetry axis,and is perpendicular to the mid-plane (i.e.,the xy-plane) of the plate. Both the top and bottom surfaces of the plate are traction-free.

Using the (generalized) England-Spencer theory^[9],we can express the displacement field as follows:

where R₀, R₁, R₂, T₁, and T₂ are functions of z, u= u(x, y), v = v(x, y), and w = w(x, y) are the mid-plane displacements, and

The z-dependent functions R_j(j=0,1,2) and T_k(k=1,2) can be determined by the boundary conditions on the top and bottom surfaces of the plate,while the mid-plane displacements can be given in terms of four analytic functions as follows ^[11]:

where κ₁ and κ₂ are the constants defined in Ref. ^[11],the two analytic functions ø(ζ) and φ(ζ) are associated with the in-plane deformation,and the other two analytic functions α(ζ) and β(ζ) are associated with the bending deformation. The corresponding resultant forces are given by

where a₁,a₂,a₅,a₆,a₇,b₁,b₂,b₅, b₆,b₇,and b₈ are real constants.

The following transforms between the Cartesian coordinates and the cylindrical coordinates will be used later:

3 Infinite FGM plate welded with circular inclusion

Let us consider an infinite FGM plate welded with a circular inclusion of the radius a subjected to N_x^∞,N_y^∞,N_x,y^∞,M_x^∞,M_Y^∞,and M_x,y^∞ acting at infinity on the cylindrical boundaries of the plate (see Fig. 1). In the following,three cases regarding the state of the inclusion will be considered.

3.1 Rigid circular inclusion without or with small rotation about z-axis

When a rigid circular inclusion is welded to the plate,we take

where a,α_-2,α₀,β_-1,ø_-1,ψ_-1,and ψ_-3 are constants to be determined, while the other four constants,i.e.,ø₁^·,β₁^·,α₂,and ψ_-1,are determined by the following equations^[13]:

In this article,we express the complex constants as follows when necessary:

where superscripts "·" and "··" indicate the real and imaginary parts of a complex constant,respectively.

The expressions of the mid-plane displacements of the plate can be obtained by substituting Eq. (11) into Eqs. (3) and (4) as follows:

It can be found from Eq. (15) that

The following conditions should be satisfied at the welding interface (i.e.,r=a):

Substituting Eqs. (14)-(16) into Eq. (17) yields

It can be found from Eqs. (18)-(24) that

When the material of the plate degenerates to a homogeneous one,we have

If the material further becomes isotropic,we have

where G is the shear modulus,and v is Poisson's ratio.

When only a force N_x^∞=N is applied,we can obtain from Eq. (12) that

Substituting Eq. (27) into Eq. (25) and then into Eq. (14),we have

Take

Then,we will find that the expressions of the mid-plane displacements in Eqs. (28) and (29) are exactly the same as those in Ref. [2].

If the rigid inclusion is given a small rotation ω_z about the z-axis,the boundary conditions in Eq. (17) become

It can be found from Eq. (30) that all constants are the same as those in Eq. (25) except

3.2 Rigid circular inclusion with small rotations about x-and y-axes

If the rigid inclusion is given small rotations ω_x and ω_y about the x- and y-axes,respectively,then the resulting rigid-body displacement vector can be determined as follows:

Namely,

where

Equation (33) gives the rigid-body displacements corresponding to the rotations ω_x and ω_y. Therefore,the boundary conditions at r=a are

We now assume

where a,B,C,α₀ ,α_-1,α_-2, β_-1 ,ø_-1, ψ_-1,ψ_-2,and ψ_-3 are eleven constants to be determined.

Substituting Eq. (36) into Eqs. (3) and (4) and then into Eq. (35),we can obtain Eqs. (18)-(24) along with the following equations:

Thus,the constants α_-2,A,ψ_-1^·,ψ_-1^··,β_-1,α₀^·,ø_-1,and ψ_-3 can be determined from Eqs. (18)-(24), while B,C,α_-1,and ψ_-2 can be determined from Eqs. (37)-(40).

3.3 Elastic circular inclusion

In the following,the physical quantities of the elastic circular inclusion will be distinguished by the affix "0" (superscript or subscript) from those of the infinite plate. The solutions corresponding to the infinite FGM plate are given in Eq. (11). For the elastic circular inclusion,we assume

where α₂₀,β₁₀,β₃₀,ø₁₀ ,ø₃₀,and ψ₁₀ are six constants to be determined.

Substituting Eq. (41) into Eqs. (3)-(9) yields

There are thirteen unknown constants,i.e.,α₂₀,β₁₀,β₃₀,ø₁₀,ø₃₀,ψ₁₀ ; A,α₀,α_-2,β_-1,ø_-1,ψ_-1,and ψ_-3,which are determined by the following boundary conditions:

where

It can be found from Eq. (50) that

There are thirteen equations in Eqs. (51)-(63),in which the constants ø₁^·,β₁^·,ψ₁,and α₂ are known quantities as specified in Eq. (12). The above equations can be divided into two groups. The first group includes Eqs. (51),(53),(55),(58),and (61),which can be used to determine the constants ø₁₀^·,β₁₀^·,ψ_-1^·,a,and α₀^·. Since the constants α₀^·· and β₀^·· do not show up in the resultant forces and displacements,we have ψ_-1^··=0, and ø₁₀^··=0. Another group includes Eqs. (52),(54),(56),(57),(59), (60),(62),and (63),which can be used to obtain the constants ψ₁₀,α₂₀,ø₃₀,β₃₀,ø_-1,β_-1,α_-2,and ψ_-3. There are two limiting cases considering the elastic constants of the inclusion. The first one is when all elastic constants approach zero such that the problem becomes that of an infinite FGM plate containing a circular hole. The other case is when all elastic constants become so large that we have an infinite FGM plate welded with a fixed rigid circular inclusion that has been considered in Subsection 3.1. 4 Numerical example

For convenience,the following dimensionless quantities will be used:

Let a=1 m and h=0.2 m. The material parameters of the infinite plate and the elastic inclusion are assumed to be^[13]

where c₁₁^0(A1),c₁₃^0(A1),c₃₃^0(A1),and c₅₅^0(A1) are the elastic constants of Al at z=-h/2,and c₁₁^0(SiC),c₁₃^0(SiC),c₃₃^0(SiC),and c₅₅^0(SiC) are the elastic constants of SiC at z=h /2 in the power law model of FGMs,both of which are given in Table1. The parameter λ is the gradient index reflecting the degree of the material inhomogeneity. Obviously,λ =0 corresponds to homogeneous materials.

Figure 2 shows the variations of the elastic constant c₁₁ along the thickness direction of the plate for different values of λ. Obviously,λ = 0 corresponds to the case of a homogeneous material Al,while λ > 0 corresponds to an FGM with the material properties changing continuously from Al at z=-h /2 to SiC at z=h/2. It can also be found that the whole rigidity of the plate in the thickness increases with λ.

4.1 Rigid circular inclusion with small rotations about x-,y-,and z-axes

An infinite FGM plate welded with a rigid circular inclusion subjected to the load N_x^∞ and the small rotations ω_x,ω_y and ω_z of one unit are considered.

Figure 3 presents the dimensionless resultant force 10N_θ versus λ for an FGM plate welded with a rigid inclusion subjected to N_x^∞ and ω_z. It is clear that the dimensionless resultant force first increases and then decreases with λ. Near λ =1,there is an inflection point. The closer the point is to the inclusion,the more evident the phenomenon is,and of course,the larger the dimensionless resultant force is.

Figure 4 shows the dimensionless resultant force 10N_θ along the r-direction for an FGM plate welded with a rigid inclusion subjected to N_x^∞ and ω_x. It is observed that the distribution curve for a homogeneous plate is significantly different from that for an FGM plate. In the vicinity of the inclusion,the absolute values of the dimensionless resultant force first increase and then decrease with λ. Far away from the inclusion,the dimensionless resultant force vanishes.

Figure 5 depicts the dimensionless resultant force 10^-9N_θ along the r-direction for an FGM plate welded with a rigid inclusion subjected to N_x^∞ and ω_y. It is shown that the distribution curves are similar to those shown in Fig. 4. However,the magnitude of the dimensionless resultant force N_θ for an FGM plate is very large in Fig. 5.

4.2 Elastic circular inclusion

In this case,an infinite FGM plate welded with an elastic circular inclusion subjected to N_x^∞ and M_x^∞ at infinity is considered.

Figures 6 and 7 plot the distributions of the dimensionless resultant force N_θ at r=a/2 along the circumferential direction. In Fig. 6,the material of the inclusion keeps homogeneous (λ =0) while the material of the infinite plate is functionally graded with different λ. The distributions of N_θ for λ =0 and λ=10 are similar,which are contrary to the case of λ =4. The absolute extremum of N_θ at θ=π/2 or θ= 3π/2 decreases with the increase in the rigidity of the plate. In Fig. 7,the material of the infinite plate keeps homogeneous (λ =0) while the material of the inclusion is functionally graded with different λ. It is seen that the distributions of N_θ for the three λ have the same periodicity. The absolute extremum of N_θ at θ=π/2 or θ =3π/2 first increases,and then decreases.

Figures 8 and 9 show the variations of the dimensionless resultant force M_θ with λ at different locations when θ=π/2. The material of the infinite plate is homogeneous (λ =0) in Fig. 8,while the material of the inclusion is homogeneous (λ =0) in Fig. 9. In Fig. 8,the distributions of M_θ inside and outside the inclusion are quite different. The M_θ inside the inclusion is bigger than that outside the inclusion,which approaches 0.5 at infinity,just as expected. In Fig. 9,the variations of M_θ with λ are all similar at different locations. In particular,the M_θ near the inclusion dramatically changes in the range from λ =0 to λ =2,while it approaches 0.5 again far away from the inclusion.

5 Conclusions

The equilibrium problem of the FGM plates welded with a circular inclusion is studied by the generalized England-Spencer theory. The 3D analytical solutions are found for three cases of the welded inclusion,i.e.,a rigid circular inclusion fixed in the space,a rigid circular inclusion with small rotations about the three coordinate axes,and an elastic circular inclusion,when the plate is subjected to the loads applied at infinity on the cylindrical boundaries of the plate. Since no transverse force acts on the top or bottom surfaces of the plate,the general solution utilized in the paper (i.e.,Eqs. (3) and (4)) is expressed by four analytic functions only,enabling an easy determination of the displacements and resultant forces for the considered problem. The material properties of the plate can vary arbitrarily in a continuous fashion along the thickness direction. The numerical results show that the material inhomogeneity has an obvious effect on the responses of the elastic inclusion and infinite plate,especially when the rigidity of the inclusion and that of the plate are different. Therefore,the mechanical behavior of the FGM plates reinforced by an elastic inclusion can be optimized in engineering applications.

We finally remark that the present analytical solutions exactly satisfy the 3D equilibrium equations and the traction boundary conditions on the top and bottom surfaces of the plate. However,the boundary conditions on the cylindrical boundaries of the plate can only be satisfied in the Saint-Venant sense. Therefore,these solutions can be used in cases of thin or moderately thick FGM plates.