2014, Vol. 35
The Chinese Meteorological Society
Article Information
- Yi-ming FU, Xue-fei SHAO, Yang CHEN 2014.
- Elasto-plastic buckling and post-buckling analysis of sandwich plates with functionally graded metal-metal face sheets and interfacial damage
- Appl. Math. Mech. -Engl. Ed., 35 (3) : 325–344
- http: //dx. doi. org/10.1007/s10483-014-1794-7
Article History
- Received 2013-03-02;
- in final form 2013-05-24
1 Introduction
In recent years,the development of the functionally graded material is at high speed for its designability and premium properties of resistance to the extreme environment.
Many scholars have studied the stability and the post-buckling behavior of the functionally graded structures so far. As we know,the material properties of the functionally graded plate are inhomogeneous through the thickness,which results in the tension-bend coupling deformation under in-plane compression load. Abrate[1] indicated that only when the load is acted on the physical neutral surface of plate does there exist bifurcational bucklings for the simply supported functionally graded plate under unidirectional or bi-directional compression load. Meanwhile,Zhang and Zhou[2] presented the expression of the physical neutral surface,in which the Poisson ratio was chosen as a constant. For accuracy,the Poisson ratio is considered as a function of position for functionally graded plate. As the physical neutral surface cannot be confirmed in advance,the in-plane load is hard to be subjected on the physical neutral surface, which is a difficulty for the application of this concept. In this paper,the buckling problem of a geometric symmetry functionally graded sandwich plate is investigated. Consequently, the difficulty can be avoided. According to the existing research results,Abrate[3] summarized the buckling behaviors of functionally graded plates. Using the semi-analytical method,Yang and Shen[4] researched the nonlinear response of the clamped functionally graded plate under transverse and in-plane loading. Considering the transverse shear deformation effect,Yang et al.[5] used the perturbation method combined with the one-dimensional differential quadrature method to analyze the post-buckling path of the functionally graded plate in various boundary conditions and initial geometry imperfections. Based on the higher shear deformation theory and considering the dependence of the temperature,Woo et al.[6] investigated the thermoelastic post-buckling property of the functionally graded plates and shells under the in-plane loading. Shukla et al.[7] used the rapid convergence Chebyshev polynomial method to study the postbuckling property of the clamped functionally graded plates with the mechanical and thermal loading. Considering the temperature dependency and the geometric nonlinearity,Reddy[8] proposed the theoretical formula and the finite element model for the nonlinear analysis of functionally graded plates on the basis of the higher shear deformation theory. Lee et al.[9] utilized the kth Ritz mesh-free method to analyze the post-buckling behavior of functionally graded plate under the compressive and the thermal loading. To investigate the thermal postbuckling of the functionally graded plate,Na and Kim[10] introduced the three-dimensional finite model.
As a new type composite material structure,functionally graded plates easily form imperfections in interface and within the functionally graded plates for the complex manufacture process and inside construction. When the imperfections reach a limit degree,they may assemble in the interface and interior,and then the interface and internal damage emerges. Therefore,in the present paper,the realization of the effect of the interface damage on mechanical properties of the sandwich plates with functionally graded metal-metal face sheets is significant. So far, there have been some research findings about normal layered structures with interfacial damage. Cheng et al.[11] firstly presented the interface damage model by modifying the original shear strip model. Afterwards,various interface theories and interface model were proposed[12, 13, 14, 15, 16, 17, 18, 19, 20]. Among that,the weak bonded interface theory was used widely in the complex interface damage problem for its clear sense and concise formulation.
Compared with the normal functionally graded material composed of metal and ceramic, the elasto-plastic deformation will occur in metal-metal functionally graded material under external force. At present,the study on the elasto-plastic of functionally graded structures is still in elementary stage and simulated by the finite element method. Mahmoud et al.[21] conducted thermal buckling analysis of two-dimensional functionally graded plate using the finite element method. Considering residual stress of the fabrication process,Shabana and Noda[22] used the finite element method to discuss thermo-elasto-plastic stress distribution in the functionally graded rectangular plate under thermal loading. Based on the small deformation assumption, Eraslan and Akis[23] obtained the analytical plane strain solution of the functionally graded elasto-plastic pressurized tube. The elasto-plastic impact problem of the functionally graded circular plates under low velocities was researched by Gunes et al.[24] using the finite element method. Based on the cohesive fracture model and dealing with elasto-plastic deformation by simple power hardening model,Jin and Pwaulino[25] analyzed the elasto-plastic crack propagation of the functionally graded material.
The elasto-plastic buckling and post-buckling analyses of the sandwich plates with functionally graded metal-metal face sheets with interfacial damage have rarely been presented so far. In the present paper,based on the elasto-plastic theory and the weak bonded theory,the increment constitutive equations of sandwich plates with functionally graded metal-metal face sheets can be derived. Then,considering the effect of geometric nonlinearity,the minimum potential energy principle is used to obtain the nonlinear increment differential equilibrium equations of the sandwich plates. Elasto-plastic post-buckling path of sandwich plates is researched by applying the finite difference method and the iterative method. When the geometric nonlinearity is ignored,the Galerkin and iterative methods can be utilized to analyze the elasto-plastic critical buckling load of the sandwich plates. To demonstrate the accuracy of the present results, the elasto-plastic critical buckling loads of isotropic plates are compared with the existing data from the related literature. Otherwise,another objective of this paper is to present the effects of the interfacial damage,the induced load ratio,the functionally graded index and the geometry parameters on the elasto-plastic post-buckling path and the elasto-plastic critical buckling load of the sandwich plates with functionally graded metal-metal face sheets and interfacial damage. 2 Basic equations of sandwich plates with functionally graded metal-metal face sheets with interface damage
Consider a metal plate with functionally graded coatings in the top and bottom surfaces with length a,width b,and thickness h,as shown in Fig. 1. The boundaries in the x- and y-directions are subjected to the in-plane uniform compress loads px,py,and py = mpx,herein, m indicates the induced load ratio. The reference plane (z = 0) of coordinate Oxyz is set on the top surface of the plate. The plate consists of 3 layers,in which Layer 1 and Layer 3 are functionally graded layers and Layer 2 is the metal layer. Surface 1 is between Layer 1 and Layer 2,and Surface 2 is between Layer 2 and Layer 3. Neglecting the effect of interlaminar extrusion,the damages caused by shear slip are considered only in the surfaces. Set the distance between the upper surface (z = 0) and the bottom surface of the ith layer as h(k)(k = 1,2,3), obviously,h = h(3). Assume that the thicknesses of Layer 1 and Layer 3,which both represent the functionally graded layers,are equal,that is,H = h(1) = h(3) − h(2).
 |
| Fig. 1. Geometric sketch of model |
As the functionally graded material is a specially heterogeneous material,it can be assumed that the material constants are inhomogeneous in the thickness,in other words,the material parameters are the functions in terms of the coordinate z. Then,the laminate approximation model,which equally divides the functionally graded layer into M sublaminates through the thickness,is adopted during analyzing the mechanical property of the functionally graded material. The thickness of each sublaminate,H/M,is small enough. Therefore,the sublaminates can be assumed as isotropic,that is to say the material parameters in the thickness are constant. For the purpose of uniform,in the lth sublaminate,set
where P indicates the material parameter of the functionally graded material,i.e.,the material parameters of each sublaminates are taken as the value of the middle plane of relevant sublaminates.
 |
| Fig. 2. Geometric sketch of model |
Based on the research results of Ref. [26],in the elasto-plastic deformation situation,we assume that (i) the stress spherical tensor causes plastic deformation,and the plastic strain can be compressed; (ii) the homogeneous volume expansion created by the positive stress do not affect the plastic deformation; (iii) the yield curved surface moves and extends along with the appearance of plastic strain in the stress space; (iv) the yield rule of material at an arbitrary point has the same format with the Mises yield rule. Based on the above assumption,the mixed hardening yield rule can be set as
where f(σij − bij) is the yield function,σij are the stress components,bij are the back stress
components which represent the motivation of the center of the yield curved surface caused by
kinematic hardening; κ(ζ),which indicates the size of the yield curved surface and the isotropic
hardening,is the hardening parameter and set as the equivalently positive stress ˜σ,herein,ζ is
the internal variable and chosen as the equivalently plastic strain 
.
To satisfy the assumptions (ii)-(iv),the mixed hardening yield rule is written as
in which 
(i,j no sum),Σij are the yield stress components of materials,
is the
function of equivalent plastic strain,and can be obtained by the uniaxial traction test curve
σ-ε. In this paper,the formulations of f,Σij,,and are set according to Ref. [26].
Assure that the plastic dissipation function satisfies the mixed hardening yield rule. According to the orthogonality law,the plastic strain components can be written as
where λp is the factor determined by the continuity of the yield curved surface,and
The plastic strain can be decomposed into the isotropic hardening plastic strain increments dεp(i)ij and the kinematic hardening plastic strain increments dεp(i)ij. Meanwhile,
where α is the mixed hardening parameter,which can be determined by the test. The range of α is (-1,1). When α = 1,the material is isotropically hardened. The back stress can be expressed as the linear function with respect to the plastic strain,i.e.,
in which c is the proportion constant. From (3) and (5),(6) can be rewritten as
2.1.2 Elasto-plastic constitutive relations
Assume that the strain increment consists of two parts
The increment constitutive relation of the material can be written as
in which Qe indicates the elastic stiffness tensor of material. Form (3) and (8),(9) can be expressed as
According to the consistency condition,(2) can be derived by setting 
.
λp can be obtained by (4),(7),(8),(10),and (11) as
where Xij and S are determined by the parameters Σkl,Qeijkl,c,α,and stresses 
.
Substituting (12) into (10),the incremental plastic constitutive equations of the functionally graded material are
in which
By introducing the plastic state factor 
,(13) can be rewritten as
When Fp= 0 (i.e.,the material reaches the plastic yield limit) and 
> 0 (i.e.,
loading),set = 1 which means that the material is in the plastic loading state. If Fp< 0 (i.e.,
the material is in the plastic state),set = 0 which means that the material is in the elastic
loading state. Other if Fp= 0 and 6 0 (i.e.,unloading),set = 0 which means the
elastic unloading.
The above mixed hardening model is usually applied to normal orthotropic material. If the material is isotropic,the elastic stiffness Qe should be set as
2.2 Elasto-plastic constitutive equations of sandwich plates with functionally graded metal-metal face sheets
By applying a simple power law distribution,the elastic modulus E(k)(z) (k = 1,2,3) of the plate can be expressed as
where Ea and Eb are the elastic moduli of the two constituents,and n is the functionally graded index. For convenience,take the Poison ratio of the two constituents as constant ν.
By analyzing the laminate approximation model of the functionally graded coatings,the elastic modulus of the lth sublaminate can be expressed as
The values of other material parameters of the lth sublaminate of functionally graded coatings follow the similar law in the above equation.
Assume that Qe(k) denote elastic stiffness tensor of the kth layer of the plate. For simplicity, the double subscripts 11,22,33,13,23,12 of Qe(k)ijkl are substituted by 1,2,3,4,5,6,respectively, i.e.,
As the effect of interlaminar extrusion is ignored (σz = 0),the elastic stiffness parameters of the plate are defined as
The yield stresses of plate are given as
in which Σ(k)ij are the yield stress components in the arbitrary point of the plate,Σ11a and Σ11b are the yield stress components of the a and b consistent material,in the x-coordinate,Σ12a and Σ12b are the pure shear stress components of the a and b consistent materials through the xy-plane.
Assume that Q(k) indicate elasto-plastic stiffness tensors of the kth layer of the plate (Q(k) =
Qe(k) − Qp(k),where Qp(k) represent plastic stiffness tensors),σ(k) are stresses,and ε(k)
are strains. Simplify the double subscript of Q(k). From (14),the elasto-plastic stress-strain
relations of the kth layer are derived as
where dσ(k) and dε(k) are the incremental stresses and strains,respectively. Qp(k)ij (k = 1,3) depend on the current stresses,the loading history,and the coordinate z. Layer 2 represents the metal layer. Hence,Qp(2)ij are independent of the coordinate z. (18) can be transformed to the total strain-stress relation under the proportional loading condition as follows:
2.3 Geometric relation of sandwich plates with functionally graded metal-metal face sheets with interfacial damage
Use u to indicate the displacement vector of the arbitrary point in the plate,whose expressions are written as[12, 27]
where u(k) represent the displacement vectors of the kth layer,and H(z − h(k)) is the Heaviside step function and can be defined as
Expand (20) by Taylor series as
in which
If the displacement components through the x-,y-,and z-coordinates of vector u are given as u1,u2,and u3,the displacement components of the arbitrary point can be approximately express as
where (u1,u2,u3),(ψ1,ψ2),(ϕ1,ϕ2),(η1,η2),(∆U(k),∆V(k)),and (Ω1(k),Ω2(k)) correspond to U(0),(U(0))(1),(U(0))(2),(U(0))(3),(U(k))(0),and (v(k))(1) in (22),u,v,and w are the displacements in the upper surface z = 0,w is the initial deflection,∆U(k) and ∆V(k) are the relative displacements in the kth surface,ψ1,ψ2,φ1,φ2,η1,and η2 are the high order shear functions,Ω1(k) and Ω2(k) are the surface functions.
On the basis of the von Karman nonlinear plate theory,the strains in the arbitrary point of the plate are written as
At first,the high order shear functions in the displacement field are determined. In the uppper and bottom surfaces,z = 0,z = h,σxz|z=0 = σxz|z=h = 0,and σyz|z=0 = σyz|z=h= 0. Then,applying (19),(23),and (24),we can get
After that,according to the continuity of stresses in the interface,the surface shape functions Ω1(k) and Ω2(k) (k = 1,2) in the displacement field are confirmed. In the kth surface,the interlayer shear stresses continue,that is,
Then,combining (19),(23),and (25) gives
(27) contains 4 linear algebraic equations related to Ω1(k) and Ω2(k). Solving these equations, we can get
where a(k)1i and a(k)2i are constants determined by Q(k)44,Q(k)55,h(k),and h. The detailed expressions of a(k)1i and a(k)2i can be given during the specific solution. Substituting (28) into (23), the displacement expressions of the sandwich plates with functionally graded metal-metal face sheets are derived as
in which
where δij are the Kronecker functions.
For the interfacial damage,the relations between the relative displacement and the interfacial stresses use the following linear elastic model:
where ∆U(k),∆V(k),and ∆W(k) indicate the relative displacements in the kth interface,σz(k), σxz(k) ,and σyz(k) are the stresses in the kth interface. Rx(k) and Ry(k) are the damage components related to the interfacial shear slips,Rz(k) is the damage components related to transverse opening. When Rx(k) = 0,Ry(k) = 0,and Rz(k) = 0,the interface is perfectly bonded. Rx(k) → ∞,R(k)y → ∞,and Rz(k) → ∞ indicate that the interface is completely opened. R is chosen as the different value with respect to the corresponding interfacial conditions. In the present paper, set Rz(k) = 0 and Rx(k),Ry(k) ≠ 0. Combining (19),(24),(29),and (30),the displacement field of the sandwich plates with functionally graded metal-metal face sheets with interfacial damage is expressed as
where
Substituting (31) into (24) gives
in which the quadratic terms of differential of initial displacement w are small quantities,which can be ignored.
Similar to (33),the corresponding incremental geometric relations are taken as
2.4 Nonlinear differential equilibrium equations of sandwich plates with functionally graded metal-metal face sheets with interfacial damage
The nonlinear differential equilibrium equations of the sandwich plates with functionally graded metal-metal face sheets with damage can be deduced by the variation principle. Energy of the whole system can be expressed as
where V represents the volume of the plate,Ti are the surface loads subjected to the system,and A is the surface area where the loads are subjected to. Based on the minimum potential energy principle δ∏ = 0,through several variation procedures,the nonlinear differential equilibrium equations of the sandwich plates with functionally graded metal-metal face sheets with damage can be derived by considering the arbitraries of δu,δv,δw,δϕ1 and δϕ2.
The corresponding incremental nonlinear differential equilibrium equations can be written as
in which the superscript “c” indicates the value in terms of the classical plate and shell theory, the superscript “a” represents the component caused by the interfacial damage. Meanwhile,
Substitute (18) into (34) and set
For the post-buckling problem of the sandwich plates with functionally graded metal-metal face sheets with damage,assume the damages in every interface are uniformed. Then,hij,x = 0 and hij,y = 0. Omit the effect of motive hardening (bij = 0) by assuming that the unloading does not appear during the buckling procedure. Before buckling,Nxc = −px = −hσ11,Nyc =−mpx = −hmσ11,and Nxyc = 0. For simplicity,the following dimensionless parameters are introduced:
From (36)-(40),it can be obtained the non-dimensional incremental elasto-plastic nonlinear differential equilibrium equations of the sandwich plates with functionally graded metal-metal face sheets with interfacial damage.
Assume that the boundaries are simply supported. Then,the boundary conditions are set as
3 Solution methodology 3.1 Elasto-plastic critical buckling load of sandwich plates with functionally graded metal-metal face sheets with interfacial damage
In the process of the solution of elasto-plastic critical buckling load,the nonlinear terms in incremental elasto-plastic nonlinear differential equilibrium equations can be ignored. To satisfy the boundary conditions in (41),set the dimensionless displacement functions as
Firstly,use the Galerkin method to obtain the algebraic equations in terms of dU,dV,dW, dΦ1,and dΦ2. Then,by setting the parameter determinant of the equation set as 0,the critical buckling load pcr of the plate can be derived.
kling load pcr of the plate can be derived.
During the solve procedure,it is important to judge whether the sandwich plates with
functionally graded metal-metal face sheets is elastic buckling or elasto-plastic buckling. The
existence of the plastic can cause the decline of the stiffness. Therefore,the elasto-plastic critical
buckling load is lower than the elastic critical buckling load. In view of the foregoing,in the
beginning of the solution,assume that the plate is at the elastic state,the elastic constitutive
relations are employed to obtain the critical buckling load pcr,and the corresponding elastic
buckling stress is σcr = pcr/h. Substitute the current stresses to the yield criterion (2),if the
yield criterion Fp< 0,the buckling appears before reaching the plastic yield limit,i.e.,the plate
occurs elastic buckling,and pcr is the elastic critical buckling load of the plate. Otherwise,the
yield criterion in terms of current stresses,Fp> 0,which means that buckling occurs after
arriving in the plastic yield limit,i.e.,elasto-plastic buckling occurs in the plate. The critical
buckling load ppcr should be obtained by the elasto-plastic constitutive relations. As elastoplastic constitutive equations are relevant to the current stresses and the loading history,the
iteration method is needed to solve the problem. In this paper,set the equivalent plastic
strain as the internal variable and calculate the external load corresponding to . If the
derived external load and the elasto-plastic critical buckling are equal,the external load can be
considered as the elasto-plastic critical buckling load. The detailed process of the computation
are described as follows:
(i) Calculate the critical buckling load by using the elastic constitutive equations
When = 0,the plate is in the elastic state. Set = 0 in (14). Then,the elastic critical
buckling stress σcr is obtained by the incremental elasto-plastic nonlinear differential equilibrium
equations. If Fp< 0,the buckling of the plate happens in the elastic state,and pcr is the elastic
critical load. Otherwise,the buckling of the plate happens in the plastic state.
(ii) Calculate the critical buckling load according to the elasto-plastic constitutive equations
In the buckling process,the material properties of the metal layer are the same in every point,
and so are the sublaminates of the functionally graded coatings. For convenience,assume that
the upper and bottom functionally graded layers are both divided to M laminates. The plastic
status of each sublaminate is judged by the yield criterion (2). If the sublaminate accesses to the plastic status,set = 1. Otherwise,in this sublaminate = 0. For the ith (i = 1,2,· · · )
iteration step,set
where d is the chosen incremental equivalent plastic strain.
Take the hardening parameter κ(ζ) as
in which Σ11 is determined by (17). According to the incremental elasto-plastic nonlinear
differential equilibrium equations,the critical buckling load (σcr)i in terms of is obtained.
Then,compare (σcr)i with 
. If < (σcr)i,substitute i by i + 1,and repeat the procedure
(ii).
(iii) If > (σcr)i,set
Take hardening parameter κ(ζ) as
Calculate the critical buckling stress 
by the incremental elasto-plastic nonlinear differential equilibrium equations,and then compare with 
. If | − | < δ (δ is
a given small quantity),the critical elasto-plastic buckling load is pcr = h · .
Otherwise,if < ,
,and if > ,
. Then,
repeat the preceding step.
3.2 Solution of elasto-plastic post-buckling of sandwich plates with functionally
graded metal-metal face sheets with interfacial damage
Assume the initial deflection of the functionally graded plate as
where W0 is the dimensionless amplitude of the initial deflection.
During the solve procedure,from the elasto-plastic nonlinear equilibrium equations expressed
by the displacement components,it is found that the results are influenced by the displacement
increment,the present displacement,and initial deflection. For every loading step,the elastoplastic deformation of arbitrary point is different. Therefore,the analytical result cannot be
obtained from the governing equations. For this reason,in the present paper,disperse the
displacement functions in the space field firstly. Divide this region into m1 × m2 elements. The
whole problem is solved by the iterative method. Divide the loading p into N uniform parts.
Thus,the increment of every step is 
. In the Jth step,the nonlinear terms in the
equations and the boundary conditions are linearized,and the process can be written as
in which (y)JP is the average value of the former two steps,and the second extrapolation method can be used to solve the initial iteration step,that is,
For different iteration steps,the parameters A,B,and C can be set as
The iteration process goes until the difference between the results of two adjacent iteration steps is lower than 0.1%,which means that the Jth step is convergent,and then the (J + 1)th step can start. 4 Numerical examples
To verify the validity of the present method,the equations in this paper are degenerated to those about the elasto-plastic buckling problem of the isotropic plate. Set E1 = E2 = 73.7739 GPa,G12 = 27.944 7 GPa,ν12 = 0.32,Σ11 = Σ22 = 423.338 MPa,Σ12 = 200 MPa, n = 0,and m = 0. Ignoring the interfacial damage,the comparisons of the critical buckling loads of the elasto-plastic isotropic plate in terms of different geometric parameters are given in Table 1. From the table,it can be found that the results of this paper agree well with those of Ref. [28]. Thus,the analytical method in the present paper is reliable.
 |
The following numerical examples investigate the functionally graded material composed of Copper (Cu) and Wolfram (W). The material parameters of the Cu and W are set as
In the numerical examples,H = h(1) = h(3) − h(2) = 0.1,m1 = m2 = 5,and W0 = 0.1. In all figures,W0 is the dimensionless additional deflection of the center point of the plate,and p indicates the dimensionless external load. Other parameters will be given in the specific example. 4.1 Effects of relevant parameters on elasto-plastic critical buckling load for sandwich plates with functionally graded metal-metal face sheets with interfacial damage
The following numerical examples investigate the effects of geometric parameters λ1 and λ2,interfacial damages R(k)x and R(k)y (k = 1,2),induced load ratio m,and functionally graded index n on the elasto-plastic critical buckling load of the sandwich plates with functionally graded metal-metal face sheets.
Figure 3 shows the effects of the divided number M and functionally graded index n on the elastic critical buckling load of sandwich plates with functionally graded metal-metal face sheets under the two-way compressive load. Herein,λ1 = 1,λ2 = 0.05,m = 1,R(k)x = R(k)y = 5 (k = 1,2),and n = 2,3,4. It can be seen that the elastic critical buckling load increases as the functionally grade index increases. Since the content of W,which possesses higher stiffness, increases with the increase of the functionally grade index,the whole stiffness of the plate increases. The laminate approximation model proposed in the present paper will get closer to the real model as the divided number increases. From Fig. 3,it can be found that when the divided number is 8-12,the accuracy of the results is enough. Based on the conclusion,the divided number is M = 8 in this paper without special explanation.
 |
| Fig. 3. Effects of divided number M and functionally graded index n on elastic critical buckling load of sandwich plates with functionally graded metal-metal face sheets under two-way compressive load |
Figure 4 shows the effects of the aspect ratio λ1 on the elastic and elasto-plastic critical buckling loads of sandwich plates with functionally graded metal-metal face sheets under the two-way compressive load,in which R(k)x = R(k)y = 0 (k = 1,2),λ2 = 0.05,m = 1,and n = 2. It can be seen that the difference between the elasto-plastic and elastic critical buckling loads decreases as the aspect ratio of the plate increases. It means that the possibility of the elastoplastic buckling increases as the aspect ratio of the plate increases. The elasto-plastic critical buckling loads are always lower than the elastic critical buckling loads.
 |
| Fig. 4. Effects of aspect ratio λ1 on elastic and elasto-plastic critical buckling loads of sandwich plates with functionally graded metal-metal face sheets |
Figure 5 shows the effects of the induced load ratio m on the elastic and elasto-plastic critical buckling load of sandwich plates with functionally graded metal-metal face sheets,where λ1 = 1, λ2 = 0.05,n = 2,and R(k)x = R(k)y = 5 (k = 1,2). It can be seen that the elastic and elastoplastic critical buckling load of sandwich plates with functionally graded metal-metal face sheets decline with the increase of the induced load ratio,and the instability of the plate occurs more easily.
 |
| Fig. 5. Effects of induced load ratio m on elastic and elasto-plastic critical buckling loads of sandwich plates with functionally graded metal-metal face sheets |
Figure 6 depicts the effect of the interfacial damage R on the elastic and elasto-plastic critical buckling loads of sandwich plates with functionally graded metal-metal face sheets under the two-way compressive load,where λ1 = 1,λ2 = 0.05,n = 2,m = 1,and R(k)x = R(k)y = R (k = 1,2). It can be seen that the elastic and elasto-plastic critical buckling loads of sandwich plates with functionally graded metal-metal face sheets decline with the increase of interfacial damage. However,the influence of interfacial damage on the elasto-plastic behavior is larger than the elastic. When the interface damage reaches some degree,the elasto-plastic critical buckling load varies slightly.
 |
| Fig. 6. Effects of interfacical damage R on elastic and elasto-plastic critical buckling loads of sandwich plates with functionally graded metal-metal face sheets |
The following numerical examples investigate the effects of geometric parameters λ1 and λ2,interfacial damages R(k)x and R(k)y (k = 1,2),induced load ratio m,and functionally graded index n on the elasto-plastic post-buckling path of the sandwich plates with functionally graded metal-metal face sheets. The initial deflection is set as W0 = 0.1. Other parameters will be given in the specific example.
Figure 7 shows the effects of the aspect ratio λ1 on the elasto-plastic post-buckling deformation of the middle point of sandwich plates with functionally graded metal-metal face sheets under the two-way compressive load,in which R(k)x = R(k)y = 0 (k = 1,2),λ2 = 0.05,m = 1, and n = 2. It can be seen that the elasto-plastic post-buckling has a maximum load compared with the normal elastic post-buckling. If the loads in the boundary exceed the maximum loading,the plate will be unstable. Meanwhile,the maximum loading decreases as the aspect ratio increases.
 |
| Fig. 7. Effects of aspect ratio λ1(a/b) on elasto-plastic post-buckling deformation of middle point for sandwich plates with functionally graded metalmetal face sheets under two-way compressive load |
Figure 8 depicts the effects of the functionally graded index n on the elasto-plastic postbuckling deformation of the middle point of sandwich plates with functionally graded metalmetal face sheets under the two-way compressive load. Herein,λ1 = 1,λ2 = 0.05,m = 1, R(k)x = R(k)y = 5 (k = 1,2),and n = 2,3,4. It can be seen that the maximum load in elastoplastic post-buckling increases as the functionally graded index increases because of the content of W,which possesses higher stiffness,and increases with the functionally graded index.
 |
| Fig. 8. Effects of functionally graded index n on elasto-plastic post-buckling deformation of middle point for sandwich plates with functionally graded metalmetal face sheets under two-way compressive load |
Figure 9 shows the effects of the induced load ratio m on the elasto-plastic post-buckling deformation of the middle point of sandwich plates with functionally graded metal-metal face sheets,where λ1 = 1,λ2 = 0.05,n = 2,and R(k)x = R(k)y = 5 (k = 1,2). It can be seen that, the elasto-plastic maximum loads of sandwich plates with functionally graded metal-metal face sheets decline as the induced load ratio increases.
 |
| Fig. 9. Effects of induced load ratio m on elasto-plastic post-buckling deformation of middle point of sandwich plates with functionally graded metal-metal face sheets |
Figure 10 depicts the effects of the interfacial damage R on the elasto-plastic post-buckling deformation of the middle point of sandwich plates with functionally graded metal-metal face sheets under the two-way compressive load,where λ1 = 1,λ2 = 0.05,n = 2,m = 1,and R(k)x = R(k)y = R (k = 1,2). It can be seen that the maximum load of elasto-plastic postbuckling of sandwich plates with functionally graded metal-metal face sheets declines as the interfacial damage increases. That is because the interfacial damage weakens the whole stiffness of the plate. With the increase of the deflections of the middle point,the effect of the damage on the rigidity of the plate rises.
 |
| Fig. 10. Effects of interfacial damage R on elasto-plastic post-buckling deformation of middle point of sandwich plates with functionally graded metal-metal face sheets under twoway compressive load |
In this paper,based on the elasto-plastic theory,the mixed hardening constitutive model of anisotropic functionally graded material is established. According to the interfacial weak bonded theory,the Heaviside step function is introduced to the displacement field to reflect the interfacial damage. The continuity of the interfacial stresses and the stress boundary condition is used to determine the interfacial shape function. The nonlinear differential equilibrium equations of the sandwich plates with functionally graded metal-metal face sheets are obtained by the minimum potential energy. Use the Galerkin method,the finite difference method,and the iteration method to solve the derived equation set. To describe elasto-plastic deformation growth of the functionally graded material,functionally graded layers,whose material property depends on the coordinate z,are divided into several laminates. In the present paper,numerical examples can derive the following conclusions:
The W-Cu functionally graded coating/W plate is decomposed to certain laminates,which can satisfy the accuracy by choosing appropriate numbers of layers. With the increases of the aspect ratio and induced load ratio,the elastic and elasto-plastic critical buckling loads decrease. The change of the functionally graded index in functionally graded layer causes the change of the whole stiffness,which considerably affects the values of the elastic and elastoplastic critical buckling loads. Meanwhile,the interfacial damage obviously reduces the elastic and elasto-plastic critical buckling loads. The elasto-plastic post-buckling possesses a maximum load which is different from the elastic one. In addition,the existence of the plastic deformation weakens the stiffness of the plate. As the aspect ratio and induced load ratio increase,the elasto-plastic maximum load lessens. The interfacial damage obviously reduces the value of elasto-plastic post-buckling maximum load of the plate along with the increase of the external load.
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