Nomenclature

R,    radius of the circular plate;

t_c,    thickness of the core;

t_f,    thickness of the facings;

E₀,    Young's modulus in the middle plane of the core;

E₁,    Young's modulus at the top and bottom surfaces of the core;

E_f,    Young's modulus of the facings;

ν_c,    Poisson's ratio of the core;

ν_f,    Poisson's ratio of the facings;

N_{R, cr},   static critical intensity of the load;

U_ε,    elastic strain energy;

W,    work of the load;

w,    deflection;

u,    longitudinal displacement;

ψ₀, ψ₁,   dimensionless functions.

1 Introduction

The global buckling problems of layered structures have been studied since the mid-20th century. When designing layered structures, strength and stability conditions are active constraints. Stability conditions include global buckling problems. A basic theory of sandwich structures is described in the literature^[1-4]. Carrera^[5] formulated the zig-zag hypotheses for multilayered plates. Carrera and Brischetto^[6] presented a survey of the theories for the analysis of sandwich plates. Based on the extensive research, they concluded that the classical lamination theory (CLT) and the first-order shear deformation theory (FSDT) cannot be effectively used for the analysis of sandwich structures. Zenkour^[7] derived a refined trigonometric higher-order plate theory for functionally graded plates. Caliri et al.^[8] delivered a detailed review of the theories and respective solution methods devoted to laminated and sandwich structures. The solution methods based on the finite element method (FEM) were explained.

The buckling phenomenon was considered in various layer structures such as beams and rectangular, circular, and annular plates. Jasion et al.^[9] presented analytical, numerical, and experimental data of the global and local buckling-wrinkling of the face sheets of sandwich beams. Kedzia and Smyczynski^[10] studied buckling problems of a three-layer rectangular plate loaded by pockets filled with ferrofluids. Magnucki et al.^[11-13] presented analytical investigations of bending and buckling of rectangular plates made of a porous material and sandwich beams. Thai and Choi^[14] presented analytical solutions for bending, buckling, and vibration analyses of thick rectangular plates using two variable refined plate theories. Uysal and Güven^[15] investigated the buckling behaviors of adhesively bonded sandwich plates subject to the in-plane shear force and the in-plane normal compression force. Magnucka-Blandzi^[16-17] presented a mathematical model of a rectangular plate compressed in its middle plane and buckling and bending of the circular plate. Pawlus^[18-19] presented the computational results of critical load calculations for annular three-layer plates with a soft core. Alipour and Shariyat^[20] presented distributions of in-plane normal and transverse shear stresses for circular/annular sandwich plates with orthotropic composite face sheets and auxetic cores using a zigzag theory whose results are correct based on the three-dimensional theory of elasticity. Magnucki et al.^[21] analytically and experimentally studied pure bending of sandwich circular plates. Mao et al.^[22] investigated the load carrying capacity of circular sandwich plates subject to the transverse quasi-static point central loading. Mojahedin et al.^[23] analyzed the buckling of functionally graded circular plates made of porous materials using the higher-order shear deformation theory. Properties of the porous plate vary across its thickness. The boundary condition of the plate is assumed to be clamped.

The clamped sandwich isotropic circular plate under compression in its middle plane is analyzed. This plate (see Fig. 1) consists of two facings of the thickness t_f and the core of the thickness t_c. The middle plane of the plate is subject to the load intensity N_R. The radius of the plate equals R. The mechanical properties of the plate core vary along the thickness direction symmetrically with respect to the middle plane of the plate. As a result of the investigation, the mathematical model of the three-layer circular plate is formulated. The results obtained from both analytical and numerical studies using the FEM with the use of SOLIDWORKS system are compared.

2 Theoretical models of the circular plate 2.1 Mechanical properties of the plate

The mechanical properties of the core vary through the thickness in the z-direction (see Fig. 2), while the properties of the facings are constant. The middle surface of the plate is its symmetrical plane.

Young's modulus changes in accordance with the specific function,

(1)

where E₀ is Young's modulus in the middle plane, and E₁ is Young's modulus at the top and bottom surfaces of the core.

There are two particular cases as follows:

(ⅰ) if z= , then E_c=E₁;

(ⅱ) if z=0, then E_c=E₀.

Young's modulus in the facings E_f is constant. Poisson's ratios in the facings and in the core are also constant.

2.2 The field of displacements

The field of displacements for the global buckling of the sandwich circular plate is shown in Fig. 3, where the displacements u(r, z) and u_f(r) are presented. The adapted Mindlin-Reissner hypothesis for three-layer plates is presented in Fig. 3(a), while the proposed nonlinear one is shown in Fig. 3(b). These hypotheses include the shear effect and generalize the classical Kirchhoff-Love hypothesis which neglects the shear effect.

According to Fig. 3, the displacements in the case of the nonlinear hypothesis are as follows:

(ⅰ) for the upper layer, ,

(ⅱ) for the core, ,

(ⅲ) for the lower layer, ,

where ψ₀(r)=u_f(r)/t_c.

The proposed nonlinear hypothesis is a generalization of the Mindlin-Reissner hypothesis. In the particular case when the function ψ₁(r)≡0, the above field of displacements describes the adapted Mindlin-Reissner hypothesis. If the functions ψ₀(r)≡0 and ψ₁(r)≡0, the above field of displacements describes the Kirchhoff-Love hypothesis. If these functions are not equal to zero, then one obtains the nonlinear hypothesis.

Based on the displacements, the strains and stresses are determined.

The strains (ε is the normal strain, and γ is the shear strain) are as follows:

(ⅰ) for the upper layer (the upper facing),

(ⅱ) for the core,

(ⅲ) for the lower layer (the lower facing),

The stresses (σ is the normal stress, and τ is the shear stress) are as follows:

(ⅰ) for the facings,

(ⅱ) for the core,

where ν_f is Poisson's ratio of the facings, and ν_c is Poisson's ratio of the core.

3 Equilibrium equations

Based on the principle of the total potential energy,

(2)

the system of equilibrium equations has been obtained, where the work of the load is

(3)

and the elastic strain energy is

(4)

where U_ε^(uf) is the elastic strain energy of the upper facing,

U_ε^(c) is the elastic strain energy of the core,

and U_ε^(lf) is the elastic strain energy of the lower facing,

Then, the system of equilibrium equations is as follows:

(5)

where

4 Analytical solutions

The system (5) has three unknown functions (w is the deflection, and ψ₀ and ψ₁ are dimensionless functions that determine the normal deformations in the middle plane of the plate). Then, the following formulas for these functions are assumed:

(6)

which identically satisfy the boundary conditions when the plate is clamped at its edge for r=R,

(7)

and the condition in the middle of the plate for r=0,

(8)

where w_a is the amplitude of deflection, and ψ_a0 and ψ_a1 are amplitudes of dimensionless functions, respectively. By substituting (6) to the system (5) and using the Bubnov-Galerkin method, the system (5) takes the form of

(9)

The functions ψ_a0 and ψ_a1 are obtained from the second and third equations of the system (9),

(10)

where

(11)

After substituting (10) into the first equation of the system (9), the intensity of critical loads is obtained as

(12)

which is related to the proposed nonlinear hypothesis.

5 Numerical calculations

Critical loads are calculated numerically for a family of the plates. Numerical calculations are conducted using the FEM with SOLIDWORKS. The plate is clamped at the outer edge and loaded, as shown in Fig. 1. A quarter of the circular plate is modeled because of its symmetry (see Fig. 4). The model consists of three layers, two facings and the core. The core is divided into 9 layers (see Fig. 5(b)) with the same thicknesses and different mechanical properties, according to (1). The integrated system component for analyzing composites has been used in calculations. In consequence, all the layers have been modeled as shell elements, and the tie conditions between the layers have been imposed automatically. Figure 5(b) shows the division of the plate core into layers. The properties of the plate core have been changed according to the relationship shown in Fig. 2. The finite element mesh of the plate is presented in Fig. 5(a). The buckling analysis of the plate under compression has been performed with a view to obtain the intensity of critical loads. A quarter of the buckled plate is presented in Fig. 6 (the first mode is the global buckling mode).

The analytical and numerical calculations are carried out for a family of plates with the following parameters: t_f = 1 mm, t_c=18 mm, 900 mm < R < 2 250 mm, E_f=65 600 MPa, E₀=500 MPa, and ν_f=ν_c=0.3.

The values of the intensity of critical loads obtained both analytically and numerically are presented in Table 1 for different values of the parameter e₁₀=E₁/E₀ when λ=50, where analytical N_{R, cr}ⁿ, N_{R, cr}^M-R, and N_{R, cr}^K-L represent results of the nonlinear hypothesis, the adapted Mindlin-Reissner hypothesis, and the Kirchhoff-Love hypothesis, respectively, N_{R, cr}^FEM is the numerical result, and λ=R/t_c.

In addition, the relative difference (R_d) between these values is obtained as

The values of the intensity of critical loads obtained both analytically and numerically for different values of the parameter e₁₀=E₁/E₀ when λ=75 are presented in Table 2.

The values with the bold font (see Tables 1 and 2) correspond to the core with the constant mechanical properties (e₁₀=1 means E₁=E₀, then, Young's modulus is constant in the plate core). If e₁₀≠ 1, then the values of the intensity of critical loads correspond to the plate core with variable mechanical properties. The expression (12) which is a function of the intensity of critical loads obtained analytically is a solid line presented in Fig. 7, whereas the values obtained numerically (given in Tables 1 and 2) are marked by dots. It can be observed that, if the parameter e₁₀ increases, then the intensity of critical loads also increases. The largest difference between the intensity of critical loads, given by analytical and FEM calculations for e₁₀ =2 (see Tables 1 and 2), does not exceed 1.9% in the case of the nonlinear hypothesis. For the Kirchhoff-Love hypothesis, the difference is about 2.8%. The relative differences between the analytical and numerical results increase (see Tables 1 and 2 and Fig. 7) with the increase in e₁₀.

The values of the intensity of critical loads obtained with these methods when λ=100 and the relative difference between these results are presented in Table 3.

The values with the bold font (see Table 3) also correspond to the core with constant mechanical properties (e₁₀=1). It can also be observed that the intensity of critical loads increases for the increasing parameter e₁₀.

The values of the intensity of critical loads obtained with these methods when λ=125 are presented in Table 4.

The obtained results when λ=100 and λ=125 are illustrated in Fig. 8. The analytical results (the expression (12) is a function of the intensity of critical loads) and the numerical ones (the values of the intensity of critical loads for selected parameters e₁₀) are presented in Fig. 8 when λ=100 and λ=125. The solid lines correspond to the analytical solutions, while the dots correspond to the numerical ones. It can be noticed that the intensity of critical loads increases for the growing parameter e₁₀. The largest difference between the intensity of critical loads given by analytical and FEM calculations is about 2% in the case of the nonlinear hypothesis and occurs for e₁₀ =2 (see Tables 3 and 4). The difference between the analytical and numerical results is larger for larger values of λ (see Tables 1-4). The relative differences with the analytical and numerical results are almost the same when λ=125, while for smaller values of λ, these differences increase (see Tables 1 and 2 and Fig. 7) for the increasing value e₁₀. For the constant mechanical properties of the core (e₁₀=1 that means E₁=E₀), the difference between the numerical and analytical results is the smallest when λ=50 (about 1% in the case of the nonlinear hypothesis) and grows for increasing λ (about 2% in the case of the nonlinear hypothesis when λ= 125).

6 Conclusions

The mathematical and numerical models of sandwich circular plates are presented, which makes it possible to investigate the influence of the material properties of the core on the critical load intensity. Numerical calculations for a family of plates are performed not only using the analytical formula obtained in the paper but also with the FEM method. Based on the analysis, the following conclusions can be drawn:

(Ⅰ) If the parameter e₁₀ increases, then the intensity of critical loads increases.

(Ⅱ) If the radius of the plate R increases (if the parameter λ increases), then the intensity of critical loads decreases.

(Ⅲ) There are small discrepancies between the analytical and numerical results.

(Ⅳ) The largest difference between the results obtained numerically and analytically is about 2%.

(Ⅴ) The smallest change of critical load intensity induced by varying the parameter e₁₀ occurs for the largest λ (the largest plate radius).

The choice of the hypothesis does not significantly affect the intensity of critical loads in the case of symmetrical sandwich structures.

The proposed nonlinear hypothesis generalizing the Mindlin-Reissner hypothesis takes into account the shearing effect. Thus, significant differences will be noticeable in the distribution of shear stresses for symmetric sandwich structures.

Acknowledgements The research was carried out within the statutory activities of the Ministry of Science and Higher Education.