The Temperature Field Effect on Dynamic Stability Response of Three-layered Annular Plates for Different Ratios of Imperfection

The main elements of the solution procedure are as follows: formulation of the system of the dynamic equilibrium equations of each plate layer, description of the transversally symmetrical deformation of the three-layered structure, formulation of the equations for angles describing the broken line in the plate cross section for the core layer in radial and circumferential directions, usage of the linear physical relation in the plate facings and the core, formulation of the sectional forces and moments in facings including the thermal elements, and determination of the resultant membrane forces including the assumed stress function.

The temperature distribution in the radial direction of the plate facings is expressed by the logarithmic equation [20] (3) TN=To+Ti−Tolnρilnρ {T_N} = {T_o} + {{{T_i} - {T_o}} \over {\ln {\rho _i}}}\ln \rho where T_i and T_o are the temperatures of the inner and outer plate perimeters, respectively (see Fig. 1), ρ=rro \rho = {r \over {{r_o}}} the dimensionless plate radius, ρi=riro {\rho _i} = {{{r_i}} \over {{r_o}}} the dimensionless inner plate radius, and r_o is the outer plate radius.

The basic equation describing the plate deflections in the dynamic problem takes the following form: (4) k1wd′rrrr+2k1rwd′rrr−k1r2wd′rr+k1r3wd′r+k1r4wd′θθθθ+2(k1+k2)r4wd′θθ++2k2r2wd′rrθθ−2k2r3wd′rθθ−G2H′h21r(γ′θ+δ+rδ′r+H′1rwd′θθ+H′wd′r+H′rwd′rr)=2h′r( 2r2Φ′θw′rθ−2rΦ′rθw′rθ+2r2w′θΦ′rθ−2r3w′θΦ′θ+w′rΦ′rr+Φ′rw′rr+ +1rΦ′θθw′rr+1rΦ′rrw′θθ )−Mwd'tt \matrix{ {{k_1}{w_{d'rrrr}} + {{2{k_1}} \over r}{w_{d'rrr}} - {{{k_1}} \over {{r^2}}}{w_{d'rr}} + {{{k_1}} \over {{r^3}}}{w_{d'r}} + {{{k_1}} \over {{r^4}}}{w_{d'\theta \theta \theta \theta }} + {{2\left( {{k_1} + {k_2}} \right)} \over {{r^4}}}{w_{d'\theta \theta }} + } \cr { + {{2{k_2}} \over {{r^2}}}{w_{d'rr\theta \theta }} - {{2{k_2}} \over {{r^3}}}{w_{d'r\theta \theta }} - {G_2}{{H'} \over {{h_2}}}{1 \over r}\left( {{\gamma _{'\theta }} + \delta + r{\delta _{'r}} + H'{1 \over r}{w_{d'\theta \theta }} + H'{w_{d'r}} + H'r{w_{d'rr}}} \right)} \cr { = {{2h'} \over r}\left( {{2 \over {{r^2}}}{\Phi _{'\theta }}{w_{'r\theta }} - {2 \over r}{\Phi _{'r\theta }}{w_{'r\theta }} + {2 \over {{r^2}}}{w_{'\theta }}{\Phi _{'r\theta }} - {2 \over {{r^3}}}{w_{'\theta }}{\Phi _{'\theta }} + {w_{'r}}{\Phi _{'rr}} + {\Phi _{'r}}{w_{'rr}} + } \right.} \cr {\left. { + {1 \over r}{\Phi _{'\theta \theta }}{w_{'rr}} + {1 \over r}{\Phi _{'rr}}{w_{'\theta \theta }}} \right) - {\rm{M}}{{\rm{w}}_{{\rm{d'tt}}}}} \cr } where w_d is the additional plate deflection, k₁ = 2D, k₂ = 4D_rθ + νk₁; D=Eh312(1−ν2) D = {{E{h^3}} \over {12\left( {1 - {\nu ^2}} \right)}} is the plate rigidity; Drθ=Gh312 {D_{r\theta }} = {{G{h^3}} \over {12}} is flexural rigidity of the facings; E and ν are the Young's modulus and Poisson's ratio of the facing material, respectively; h is the total plate thickness; δ and γ are the differences of radial and circumferential displacements of the points, respectively, in the middle surfaces of the facings δ = u₃ − u₁, γ = v₃ − v₁; H′ = h′ + h₂; Φ is the stress function; w is the plate total deflection; M = 2h′μ + h₂μ₂; μ and μ₂ are the facing and core mass density, respectively; h′ is the facings' thickness and h₂ is the core thickness.

Equation (4) has been obtained after adding the summands of the dynamic equilibrium equations of forces in the transversal plate direction, which have been derived for each plate layer: the facings and the core. Then, the relations for the resultant radial and circumferential forces and the resultant membrane forces expressed by the introduced stress function Φ have been inserted.

The boundary conditions for the plate slidably clamped on both edges are as follows: (5) w|r=riro=0, w′r|r=riro=0, δ=γ|r=riro=0, δ′r|r=riro=0 w{|_{r = {r_i}\left( {{r_o}} \right)}} = 0,\;\;{w_{'r}}{|_{r = {r_i}\left( {{r_o}} \right)}} = 0,\;\;\delta = \gamma {|_{r = {r_i}\left( {{r_o}} \right)}} = 0,\;\;\;{\delta _{'r}}{|_{r = {r_i}\left( {{r_o}} \right)}} = 0 The initial and loading conditions are as follows: (6) wd/t=0=0,wd,t/t=0=0 {w_d}{/_{t = 0}} = 0,{w_{d,t}}{/_{t = 0}} = 0 The conditions associated with the mechanically loaded plate edges are the following: (7) σr|r=ri=−ptd1, σr|r=ro=−ptd2, τrθ|r=riro=0, {\sigma _r}{|_{r = {r_i}}} = - p\left( t \right){d_1},\;\;\;{\sigma _r}{|_{r = {r_o}}} = - p\left( t \right){d_2},\;\;\;{\tau _{r\theta }}{|_{r = {r_i}\left( {{r_o}} \right)}} = 0, where σ_r is the radial stress, τ_rθ the shear stress, and d₁, d₂ are the quantities equal to 0 or 1, determining the loading of the inner or/and outer plate perimeter.

Conditions for the plate edges subjected to only thermal loads are expressed by the equation (8) σr|r=ri−0, σr|r=ro=0 {\sigma _r}\left| {_{r = {r_i}}} \right. - 0,\;\;\;\;{\sigma _r}\left| {_{r = {r_o}}} \right. = 0 The form of plate imperfection ζ_o (ζ_o = w_o/h), important in the presented analysis, is expressed by the following equation presented in [21]: (9) ζo(ρ,θ)=ξ1(ρ)η(ρ)+ξ2(ρ)η(ρ)cos(mθ) {\zeta _o}\left( {\rho ,\theta } \right) = {\xi _1}\left( \rho \right)\eta \left( \rho \right) + {\xi _2}\left( \rho \right)\eta \left( \rho \right)cos \left( {m\theta } \right) where w_o is the plate initial deflection, m the number of initial circumferential waves, ξ₁, ξ₂ are the calibrating numbers, η(ρ) is a function: η(ρ)=ρ⁴+A₁ρ²+A₂ρ²lnr+A₃lnρ+A₄, A_i are the quantities fulfilling the conditions of clamped edges.

The solution is based on shape functions for the additional plate deflection ζ₁ [21] (10) ζ1(ρ,θ,t)=X1(ρ,t)cos(mθ), {\zeta _1}\left( {\rho ,\theta ,t} \right) = {X_1}\left( {\rho ,t} \right)cos \left( {m\theta } \right), stress function F [21] (11) F(ρ,θ,t)=Fa(ρ,t)+Fb(ρ,t)cos(mθ)+Fc(ρ,t)cos(2mθ), F\left( {\rho ,\theta ,t} \right) = {F_a}\left( {\rho ,t} \right) + {F_b}\left( {\rho ,t} \right)cos \left( {m\theta } \right) + {F_c}\left( {\rho ,t} \right)cos \left( {2m\theta } \right), and differences δ¯ \overline \delta , γ¯ \overline \gamma [14]: (12) δ¯(ρ,θ,t)=δ¯(ρ,t)cos(mθ), γ¯(ρ,θ,t)=γ¯(ρ,t)sin(mθ) \bar \delta \left( {\rho ,\theta ,t} \right) = \bar \delta \left( {\rho ,t} \right)cos \left( {m\theta } \right),\;\;\bar \gamma \left( {\rho ,\theta ,t} \right) = \bar \gamma \left( {\rho ,t} \right)sin \left( {m\theta } \right) where ζ1=wdh {\zeta _1} = {{{w_d}} \over h} , F=ΦEh2 F = {\Phi \over {E{h^2}}} , δ¯=δh \bar \delta = {\delta \over h} , γ¯=γh \bar \gamma = {\gamma \over h} , m are the number of circumferential waves, which is compatible with the initial number (see Eq. (9)) and characterizes the form of plate buckling – the mode for plates with ξ₁ calibrating number equal to ξ₁ = 0.

Using the orthogonalization method after elimination of the angular variable θ and approximation of the derivatives with respect to ρ by the central differences in the discrete points, the basic system of differential equations has been established: (13) PU+Q=U¨K {\boldsymbol {PU + Q = }}{{\boldsymbol {\ddot U}}_{\boldsymbol {K}}} (14) MYY=QYΔT {{\boldsymbol {M}}_Y}{\boldsymbol {Y}} = {{\boldsymbol {Q}}_Y}_{\Delta T} (15) MVV=QV {{\boldsymbol {M}}_V}{\boldsymbol {V}} = {{\boldsymbol {Q}}_V} (16) MZZ=QZ {{\boldsymbol {M}}_Z}{\boldsymbol {Z}} = {{\boldsymbol {Q}}_Z} (17) MDD+MUU+MGG=0 {{\boldsymbol {M}}_D}{\boldsymbol {D}} + {{\boldsymbol {M}}_U}{\boldsymbol {U}} + {{\boldsymbol {M}}_G}{\boldsymbol {G}} = 0 (18) MGGG+MGUU+MGDD=0 {{\boldsymbol {M}}_{GG}}{\boldsymbol {G}} + {{\boldsymbol {M}}_{GU}}{\boldsymbol {U}} + {{\boldsymbol {M}}_{GD}}{\boldsymbol {D}} = 0 For a mechanically loaded plate with a constant in time–temperature difference ΔT = T_i – T_o between the edges, Eq. (14) has been modified to the form (19) MYY=QYΔT const {{\boldsymbol {M}}_Y}{\boldsymbol {Y}} = {{\boldsymbol {Q}}_{Y\Delta T\;const}} where U, Y, Z, V are the vectors whose elements consist of the additional deflections and components of the stress function, respectively; Ü_K is a vector whose elements are expressed by the products of the derivative of the additional deflection with respect to time t and number K, equal to K=TK72⋅h′h⋅roh2M K = TK{7^2} \cdot {{h'} \over h} \cdot {r_o}{h_2}M ; Q, Q_V, Q_Y, Q_z, D, G are the vectors whose elements consist of the plate material parameters, geometrical dimensions, initial and additional deflections, number m, dimension radius ρ, and displacement differences δ; γ; Q_YΔT is a vector whose elements are expressed as the difference between the suitable element of vector Q_Y (vector Q_Y is used for plates which are not subjected to the temperature field) and the number ρ · S · T_N′ρ, which differs in time, S=rz2h2α S = {{r_z^2} \over {{h^2}}}\alpha ; Q_{YΔT const} is a vector whose elements are expressed as the difference between the suitable element of vector Q_Y and the number STi−Tolnρi S{{{T_i} - {T_o}} \over {\ln {\rho _i}}} , which is constant in time; and M_Y,M_V,M_Z,M_D,M_G,M_U,M_GG,M_GD,P,M_GU are the matrices whose elements consist of the plate material parameters, geometrical dimensions, number m, FDM parameter b (b – the interval in the finite difference method), and dimension radius ρ.

The Runge–Kutta's integration method for the initial state of the plate has been used in the solution of the presented system of equations.

The dimensionless time connected with mechanical loading (see Equation (1)) is expressed by t*=t×K7, where K7 is the number expressing the rate of mechanical loading growth but connected with thermal loading (see Equation (2)) and is expressed as t*=t×TK7, where TK7 is the number expressing the rate of thermal loading growth.

The plate model built with the use of the finite element method has been calculated in the ABAQUS system. The calculations were carried out at the Academic Computer Center CYFRONET-CRACOW (KBN/SGI_ORIGIN_2000/PLodzka/030/1999). The dynamic module is the main option, which was applied in the dynamic solution procedure [22]. The three-layered structure of the FEM plate model is composed of shell elements and solid ones to build the meshes of the plate facings and the core, respectively. 3D nine-node shell elements with six active degrees of freedom and 3D 27-node solid elements with three active degrees of freedom were used. The surface contact interaction with the TIE option was assumed to connect the surfaces of the facings and the core meshes.

The first step of numerical analysis, which is performed with using the finite difference method, is the choice of the number N of discrete points. Tables 1 and 2 present the results of the critical dynamic temperature ΔT_crdyn and the critical dynamic load p_crdyn with relative temperature ΔT_b for the FDM plate model with a different number N of discrete points equal to N = 11, 14, 17, 21, 26. Table 2 presents the critical dynamic temperature differences ΔT_crdyn of the FDM plate model with the imperfection ratio ξ₂ = 0.5. The plate is thermally loaded with a positive temperature gradient. Values of ΔT_crdyn show a tendency to increase with the number N of discrete points. Also, the relative difference between the values of ΔT_crdyn for number N = 26 and 21 increases with the number m. Its value for the minimal value of ΔT_crdyn, which exists for m = 7, is less than 5%, which is treated as a technical error. This is the way the number N = 14 of the discrete points has been chosen in the FDM numerical calculations. Table 3 presents the critical dynamic mechanical load p_crdyn and corresponding temperature differences ΔT_b for the axisymmetric (m = 0) FDM plate model with the imperfection ratio ξ₂ = 2 loaded mechanically and located in the thermal environment. The temperature field model is characterized by the rate a of a temperature linear growth equal to a = 800 K/s and a positive temperature gradient. The values of critical load p_crdyn are comparable to the relative difference between the values calculated for a different number N, which is about 5% of the technical error.

In summary, it can be noticed that values of the critical temperature difference and the critical load are converged. It confirms that the solution process is correct and the numerical calculations are accurate.

Figures 2 and 3 show a comparison of the thermal reaction of two FDM plate models: axisymmetrical m = 0 and circumferentially waved m = 7. Number m = 7 of plate buckling waves in a circumferential direction corresponds to the minimal value of the critical temperature difference between the plate edges ΔT_crdyn (see Table 2). The plate models are loaded thermally with the rate of loading growth a = 200 K/s and a = 800 K/s. A temperature field model with a positive gradient is considered. Figures 2 and 3 show the strong plate reaction on the temperature growth with a high value of the rate a, a = 800 K/s. Then, both the plate modes m = 0 and m = 7 lose dynamic stability quickly. However, the temperature difference, which exists between the plate edges at the critical buckling moment for the two examined rates a, and the imperfection ratios ξ₂ do not differ significantly. Tables 4 and 5 present the values of the critical temperature differences ΔT_crdyn for two FDM plate models m = 0 and m = 7, respectively. The values of ΔT_crdyn for the waved m = 7 plate model are smaller than those observed for the axisymmetrical m = 0 one. It shows that a full analysis, which includes the asymmetric plate modes, is required to recognize the different plate thermal reactions. Imperfection ratios do not have a significant meaning here.

Figure 2

Figure 3

Exemplary time histories of plate deflection and velocity of deflection for the FDM and FEM plate models are shown in Figure 4. Additionally, the axisymmetric m = 0 form of plate buckling is shown (see Fig. 4b). The FEM results are for the plate with the imperfection ratio ξ₂ = 2 loaded thermally with a = 200 K/s. On comparing the results shown in Fig. 4a and b, a good compatibility of responses of the FDM and FEM plate models is observed. Table 6 presents the values of the critical temperature differences ΔT_crdyn for the axisymmetric m = 0 FEM plate model loaded with two rates a = 200 K/s and a = 800 K/s. The differences between the critical values ΔT_crdyn for the FEM plate model are clearer than for the FDM plate mode. With an increase in imperfection ratio ξ₂ and temperature growth rate a, loss of dynamic stability occurs for a higher value of the temperature field.

Figure 4

In summary, it can be observed that the analyzed plate modes m = 0 and m = 7 and plate cases loaded with different rates a influence the values of the critical temperature differences. The imperfection ratio ξ₂ for the asymmetric (m = 7) plate mode does not have a significant effect on the critical temperature difference ΔT_crdyn.

The two plate modes m = 0 and m = 7 present the plate reaction on the action of the mechanical load and temperature field. The selected modes enable showing the behavior of the axisymmetric m = 0 and asymmetric plates m = 7, whose value of the critical load is minimal [12,13,14]. Both thermomechanical responses of the FDM and FEM plate modes are shown in Figs 5–9. Plate imperfection is expressed by the ratio ξ₁ = 0 and various values of the ratio ξ₂ (see Eq. (9)). The plates are compressed on the outer edge with a growth rate equal to s = 931 MPa/s (K7 = 20 1/s).

Figure 5

Figure 6

Figure 7

Figure 8

Figure 9

Figure 5 shows the effect of the imperfection ratio ξ₂ and thermal loading growth a. The presented results are for the asymmetric m = 7 FDM plate mode. The analyzed value of the temperature loading growth a does not change the dynamic response of the plate subjected to only the mechanical load a = 0. Very small differences are observed for a plate with the value ξ₂ = 0.5.

A comparison between the reactions of plate modes m = 0 and m = 7 is shown in Fig. 6. Plates with the imperfection ratio ξ₂ = 2 are subjected to three models of thermomechanical loads: without temperature field a = 0, with increasing in time temperature difference between the plate edges a = 200 K/s and a = 800 K/s, and the model with a fixed value of the temperature field ΔT = 800 K (marked in the figures as DT). The results presented for a greater value of the rate a = 800 K/s or for the plate located in a thermal environment with a high, constant temperature difference equal to ΔT = 800 K can be the exemplary dynamic response of the plate under the impact of thermal loading, which exists during the aerodynamic or laser heating mentioned in work [8]. Additionally, two temperature gradients are taken into account: positive with a higher value of temperature on the inner plate edge and negative with a higher value of temperature on the outer edge. Plate mode m = 0 responds differently to temperature field profiles. The detailed values are presented in Table 7. Critical dynamic load p_crdyn and the corresponding temperature difference ΔT_b, when a loss of plate stability occurs, decrease with the growth of the rate a for a temperature field with a positive gradient and increase when the plate is subjected to a negative temperature gradient. Such differences are not observed for the plate mode m = 7 (see Fig. 6b). Practically, for all examined cases, the same values of p_crdyn = 22.82 MPa and the corresponding temperatures ΔT_b = 4.9 K or ΔT_b = 19.6 K for the field characterized by the rate a = 200 or 800 K/s, or ΔT = 800 K, respectively, are observed.

Additionally, the effect of the negative value of imperfection ratio ξ₂ has been examined. Selected results are presented in Fig. 7 for the FDM plate model. Plate reactions are for two plate modes m = 0 and m = 7. The plate is compressed on the outer edge and subjected to an increase in the time–temperature field with a negative gradient. The character of curves ζ_1max = f(t*) is similar for the examined plate cases. The lack of influence of the negative ratio ξ₂ on the final results is observed. On comparing the results obtained for the asymmetric plate mode m = 7 with the axisymmetric one m = 0 [11], differences are found in the values of critical parameters: time, deflection, load with the corresponding temperature ΔT_b and also in the supercritical part of curves ζ_1max = f(t*), where for axisymmetrical plate m = 0, the vibrations are initiated.

Figure 8 shows the exemplary results presented for the axisymmetric (m = 0) FEM plate model with two imperfection ratios ξ₂ = 1 and ξ₂ = 2 loaded mechanically and thermally with a temperature difference increasing in time and expressed by a growth parameter a = 200 K/s and a = 800 K/s. The temperature field has a positive gradient. Time histories of deflections (red lines) and velocity of deflections (blue lines) with critical parameters, time t_cr and dynamic load p_crdyn, are presented. The nature of the curves is similar. Increase in the temperature difference shortens the time to plate stability loss and decreases the critical dynamic loads p_crdyn. This observation is of importance in evaluation of the plate buckling phenomenon.

Additionally, the correspondence of results between the two plate models is shown in Fig. 9. The plate is loaded mechanically and thermally with a positive gradient and a temperature growth rate a = 200 K/s. The plate imperfection ratio ξ₂ = 2. The lines of time histories of deflections and velocity of deflections are similar. The additional black lines indicate the points with a maximum value of the speed of plate deflections. According to the assumed criterion [19], they enable expression of the critical time t_cr, critical plate deflection, and, after calculation, dynamic load p_crdyn as well as the corresponding difference in temperatures ΔT_b. The presented results are an example confirming the proper calculation process and the method of plate modeling.

The obtained results can be summarized as follows: –

The minimal value of p_crdyn which is important in buckling analysis exists for the axisymmetrical m = 0 plate model with a greater value of imperfection ratio ξ₂ = 2, whose thermal loading is characterized by a positive temperature gradient and a higher value of temperature difference between the plate edges.

Figures 10 and 11 show the FDM plate model response to thermal and thermomechanical loading. The assumed plate imperfection is composed of two forms: the axisymmetrical one expressed by the calibrating number ξ₁ and the form which is dependent on the number m of circumferential waves m = 0 or m ≠ 0 calibrated by the number ξ₂. Mathematically, this complex form of plate imperfection is described by Equation (9). The form consists of two terms expressed by the aforementioned calibrating numbers ξ₁ (ksi1) and ξ₂ (ksi2). The plate thermally loaded is subjected to a positive temperature gradient. In the case of complex thermomechanical loading, the plate is mechanically compressed with the forces on the outer edge and subjected to the thermal load with a positive gradient, too. Two exemplary forms of plate buckling are examined: axisymmetrical m = 0 and waved with the number m = 7 of circumferential buckling waves. The value of the number ξ₂ = 1.

Figure 10

Figure 11

Figure 10 shows the plate's behavior which was invoked by thermal loading only. The increase in the axisymmetrical form of imperfection expressed by calibrating the number ξ₁ = 5, 10 decreases the values of critical temperature differences by more than 10%. Additionally, critical plate deflections are observed to be increasing. Very small differences in the values of critical parameters exist for the plate with several buckling waves (here, the analyzed number is m = 7).

The effect of mechanical loads significantly changes the nature of plate responses. The effect of the additional axisymmetric imperfection shape is presented in Figure 11a. It should be underlined that the main axisymmetrical imperfection is expressed by the second term of Equation (9) for a non-zero value of calibrating number ξ₂. Axisymmetrical (m = 0) plates with strong imperfection (ξ₁ = 10) lose dynamic stability quicker for smaller values of critical dynamic loads p_crdyn and corresponding temperature difference between the plate edges ΔT_b. In the case of the axisymmetrical plate m = 0 with the value ξ₁ = 5, the area of a possible loss of the plate dynamic stability can be expressed, which is between two rectangular gray points shown in Figure 11a. In Figure 11b, the curves of the time history of deflection and time history of the velocity of deflection present the range of the plate stability loss between the points with maximum values of the velocity of deflection. An opposite observation can be made for the waved form of plate buckling (m = 7), where the increase in value ξ₁ of additional imperfection shape increases both the values of critical dynamic loads p_crdyn and the corresponding temperature ΔT_b. Small fluctuations of critical deflections are observed particularly for the analyzed plate with the number m = 7.

The effect of the mixed participation of the values of imperfection ratios on the run of deflection curves ζ_1max = f(t*) is shown in Figure 12. The results are for the positive numbers of both ratios ξ₁ and ξ₂ or for the positive and negative numbers of ratios ξ₁, ξ₂. The results are presented for the case of the FDM plate model where its form of the loss of stability corresponds to m = 7 circumferential waves. The plate is subjected to only thermal loads with a positive gradient. The results show the meaning of the higher positive or negative value of ratio ξ₂, where axisymmetric predeflection also exists (ξ₁≠0), and the meaning of the higher values of ratio ξ₁ (ξ₁ = 5 or ξ₁ = −5) for the positive value of the ratio ξ₂. For assumed values of ξ₁, additional waved shape of predeflection is expressed by the ratio x₂. This complex form of predeflection influences the positive or negative run of plate deflection curve ζ_1max = f(t*).

Figure 12

The waved form of plate deflection in a radial direction obtained for selected numbers of ratios ξ₁, ξ₂ (ξ₁ = 0, ξ₂ = 1), (ξ₁ = 1, ξ₂ = 1), (ξ₁ = 1, ξ₂ = −1) is shown in Figure 13.

Figure 13

The presented results can be summarized as follows: –

The effect of the additional axisymmetric form of plate predeflection expressed by the calibrating number ξ₁ causes little change to the values of critical temperature differences for the axisymmetrical m = 0 form of plate buckling. The increase in the value of calibrating number ξ₁ decreases the value of critical temperature differences.