The effect of the dynamic response of the composite plate depends on various parameters. These include the imperfection ratios whose values determine the predeflection shape of the plate. A plate subjected to mechanical and thermal loads increasing in time formulates a time-dependent complex problem, whose results are strongly related to the plate geometrical parameters. The annular and circular composite plates, which can be used in different kinds of industries, such as mechanical, civil engineering, and aerospace, can be subjected to the complex work conditions. This is the reason that the thermomechanical evaluation of plate sensitivity is still a current issue addressed in numerous works. Few works strictly focus on the issue of the imperfection of composite annular plates. Selected papers concerning the issue of buckling of mechanically or/and thermally loaded plates where the plate geometry has been taken into account are presented in the literature review.

The general solution and dynamic behavior of sandwich annular and circular plates are presented in works [1] and [2]. The thermal buckling effect of moderately thick functionally graded material (FGM) annular plate is presented in paper [3]. The thermoelastic problem showing the reactions of the imperfect, radially graded annular plate with a heated edge is presented in work [4]. The effect of temperature on the buckling behavior of the annular plate is presented in papers [5,6,7]. The elements of the critical state, like critical temperature and the corresponding modes, are examined for different materials and geometrical plate parameters. The FGM annular plates with imperfections are presented in paper [8]. Both buckling and dynamic postbuckling reactions are the main problems of consideration. The thermomechanical buckling of perforated, functionally graded annular sector plates under uniform temperature rises and radial, circumferential, or biaxial mechanical loads is investigated in work [9]. The final results present the effect of the sector geometry, direction of the mechanical loads, and the combination of the thermal and mechanical loads on the buckling loads and mode shapes. The viscoelastic FGM annular plates with different geometrical, material, and load parameters are presented in work [10]. The paper presents the unified dynamic analysis method for a viscoelastic FGM annular plate.

The novelties presented in this paper concern the numerical investigations which are focused on the evaluation of the reactions of the composite plate to the action of the temperature field or the participation of the temperature field in complex thermomechanical loading. The imperfection of the shape geometry of the plate's initial surface is the main analyzed element, which changes the dynamic response of the plate. The effect of various geometrical imperfect forms of the plate surface predeflection has been examined: waved circumferential predeflection, which corresponds to the plate buckling mode, complex initial shape, which is composed of rotational axisymmetrical predeflection, and the circumferential waved form for positive or mixed positive and negative numbers, which calibrate the grade of plate predeflection. The participation of various imperfections of the plate surface, which initiate the dynamic buckling phenomenon, complements the existing analyses and significantly enriches the cognition of the examined layered structure of the plate showing its resistance to the shape imperfections. The presented exemplary results of the numerous numerical analyses create both a practically important and scientifically interesting image of the buckling sensitivity of the structures to existing imperfections. Some numerical results are presented in work [11]. The additional and wider observations, numerical calculations, and results shown in the figures and the tables presented in this work make recognition of the formulated problem richer. To the best of the author's knowledge, the so-formulated thermomechanical problem and the idea to observe the plate buckling sensitivity on variously defined imperfections have not been sufficiently considered.

The method of the analytical and numerical solution to the problem as proposed in this paper refers to the solution of mechanically loaded annular plate presented in works [12], [13], [14]. The imperfection issue is analyzed in work [14] and particularly in work [15] for plates that are only loaded mechanically. Thermal and thermomechanical problems are examined for a layered, composite plate in work [16], [17]. The observations for a composite three-layered plate with a core layer made of viscoelastic material are presented in [18].

The three-layered annular plate composed of thin steel facings and a thicker foam core is the object of consideration. The plate cross section is symmetric. The analyzed forms of plate buckling can be axisymmetrical or asymmetrical. The plate is subjected to a complex thermomechanical state of loading. It is mechanically loaded by the compressing forces linearly increasing in time, which are uniformly distributed on the outer facings. The temperature field surrounds the plate's inner and outer perimeters. The temperature difference between the inner and outer plate edges creates the thermal gradient. The temperature difference can be fixed, constant in time, or it can change, dynamically increasing in time. Three models of loading, showing the temperature field effect on the dynamic plate reaction, are acceptable: thermal loading with the temperature difference between the plate edges increasing in time, both mechanical and thermal loading with mechanical and thermal loads increasing in time, and mechanical loading increasing in time, which is connected to the constant in time action of the temperature field. It should be emphasized that thermal loading is defined by the uncoupled temperature field, whose parameters are arbitrarily assumed.

The equations (1) and (2) express the thermomechanical loading quickly increasing in time:

The plate is loaded mechanically with the compressive forces uniformly distributed on the outer perimeter of the facings. The action of the outer forces on the lateral surfaces of both plate facings determines the compressive mechanical stress _{i}_{o}_{i}_{i}_{o}_{i}_{o}

The main assumptions which are adopted to describe the thermal environment are as follows:

axisymmetric and flat temperature field,

the lack of heat exchange between the plate surfaces,

the heat flow is only in a radial direction of the plate facings (see Eq. (3)), and

the material parameters are fixed and do not depend on temperature changes.

The main method of solution is based on an analytical and numerical analysis which uses the following approximation methods: orthogonalization and finite difference (the Finite Difference Model – FDM plate). The influence of various ratios of imperfections on plate stability response is expressed by the elements of assumed equation (9), which determine the shapes of plate predeflection. Equation (9) is composed of two terms: axisymmetrical and a term dependent on the number of circumferential waves. Assumed calibrating numbers change the participation of the mentioned two terms and make it possible to produce various forms of plate predeflection. The solution procedure is presented in the works [12,13,14,17] in detail.

Additionally, the finite element method has been used to evaluate selected examples of the plates being examined (the Finite Element Method - FEM plate). The ABAQUS system was used to conduct the calculations.

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]
_{i}_{o}_{o}

The basic equation describing the plate deflections in the dynamic problem takes the following form:
_{d}_{1} = 2_{2} = 4_{rθ} + νk_{1};
_{3}_{1}, _{3}_{1}; _{2}; _{2}_{2}; _{2} are the facing and core mass density, respectively; _{2} 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

The boundary conditions for the plate slidably clamped on both edges are as follows:
_{r}_{rθ}_{1}, _{2} 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
_{o}_{o} = w_{o}/h_{o}_{1}, _{2} are the calibrating numbers, ^{4}+_{1}^{2}+A_{2}^{2}_{3}_{4}, _{i}

The solution is based on shape functions for the additional plate deflection _{1} [21]
_{1} calibrating number equal to _{1} = 0.

Using the orthogonalization method after elimination of the angular variable _{i} – T_{o}_{K}_{V}, _{Y}_{z}_{YΔT} is a vector whose elements are expressed as the difference between the suitable element of vector _{Y}_{Y}_{N′ρ}_{YΔT const} is a vector whose elements are expressed as the difference between the suitable element of vector _{Y}_{Y}_{V}_{Z}_{D}_{G}_{U}_{GG}_{GD}_{GU}

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

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.

Table 1 presents the assumed material, geometrical, and loading parameters of the examined plate models.

Parameters of the plate model.

Geometrical parameters | |||

Inner radius _{i} | 0.2 | ||

Outer radius _{o} | 0.5 | ||

Facing thickness | 1 | ||

Core thickness _{2}, mm | 5 | ||

Ratio of plate initial deflection _{2} | 0.5, 1, 2 | ||

Material parameters | |||

Steel facing | Polyurethane foam of core | ||

Young's modulus | 210 | _{2}, MPa | 13 |

Kirchhoff's modulus | 80 | _{2}, MPa | 5 |

Poisson's ratio | 0.3 | _{2} | 0.3 |

Mass density ^{3} | 7850 | _{2}, kg/m^{3} | 64 |

Linear expansion coefficient a, 1/K | 1.2×10^{−5} | a_{2}, 1/K | 7×10^{−5} |

Loading parameters | |||

Rate of thermal loading growth | 200 (20), 800 (20) | ||

Rate of mechanical loading growth | 931 (20) | ||

Constant temperature difference | 800 |

The ratio _{2} of plate initial deflection is described in figures as

Two main models of plate loading are analyzed: a plate thermally loaded and a plate mechanically and thermally loaded. The critical dynamic temperature difference _{crdyn} is the main calculation result for plates loaded only thermally. Critical dynamic mechanical load _{crdyn} is the main result of the calculations of plates loaded mechanically and thermally. Additionally, to recognize the plate dynamic behavior, the time histories of plate deflections and the velocity of deflections are designed. The analyzed problem is the multiparameter task, for which the ratio of plate predeflection is the basic variable.

The first step of numerical analysis, which is performed with using the finite difference method, is the choice of the number _{crdyn} and the critical dynamic load _{crdyn} with relative temperature _{b}_{crdyn} of the FDM plate model with the imperfection ratio _{2} = 0.5. The plate is thermally loaded with a positive temperature gradient. Values of _{crdyn} show a tendency to increase with the number _{crdyn} for number _{crdyn}, which exists for _{crdyn} and corresponding temperature differences _{b}_{2} = 2 loaded mechanically and located in the thermal environment. The temperature field model is characterized by the rate _{crdyn} are comparable to the relative difference between the values calculated for a different number

The values of the dynamic, critical temperature differences _{crdyn} depending on the number _{2} = 0.5 subjected to a positive gradient of the temperature field.

_{crdyn} (K) | |||||
---|---|---|---|---|---|

0 | 128.6 | 130.0 | 130.1 | 131.6 | 131.5 |

1 | 131.9 | 133.7 | 133.7 | 134.2 | 134.7 |

2 | 133.5 | 135.5 | 135.5 | 137.2 | 137.0 |

3 | 126.4 | 129.3 | 131.2 | 130.9 | 132.4 |

4 | 117.5 | 120.7 | 122.1 | 123.5 | 124.8 |

5 | 108.7 | 112.3 | 114.9 | 115.9 | 117.1 |

6 | 105.7 | 108.9 | 110.4 | 112.8 | 113.8 |

7 | 103.8 | 106.8 | 108.8 | 109.5 | 111.7 |

8 | 103.7 | 107.9 | 110.3 | 112.8 | 116.4 |

The values of the dynamic, critical mechanical loads _{crdyn} with the corresponding temperature differences _{b}_{2} = 2 subjected to a mechanical load and increasing with the value

_{crdyn} (MPa)/Δ_{b} | 30.74/26.4 | 29.35/25.2 | 31.21/26.8 | 30.74/26.4 | 31.21/26.8 |

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 _{crdyn} (see Table 2). The plate models are loaded thermally with the rate of loading growth _{2} do not differ significantly. Tables 4 and 5 present the values of the critical temperature differences _{crdyn} for two FDM plate models _{crdyn} for the waved

Values of critical temperature differences _{crdyn} for the axisymmetrical _{2} under a temperature field with a positive gradient and two rates

_{crdyn} (K) | |||
---|---|---|---|

_{2} | |||

200 | 130.0 | 130.2 | 130.7 |

800 | 132.0 | 128.4 | 126.8 |

Values of critical temperature differences _{crdyn} for the asymmetrical _{2} under a temperature field with a positive gradient and two rates

_{crdyn} (K) | |||
---|---|---|---|

_{2} | |||

200 | 107.4 | 108.0 | 108.2 |

800 | 108.8 | 108.4 | 108.4 |

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 _{2} = 2 loaded thermally with _{crdyn} for the axisymmetric _{crdyn} for the FEM plate model are clearer than for the FDM plate mode. With an increase in imperfection ratio _{2} and temperature growth rate

Values of critical temperature differences _{crdyn} for the axisymmetrical _{2} under a temperature field with a positive gradient and two rates

_{crdyn} (K) | |||
---|---|---|---|

_{2} | |||

200 | 115.2 | 121.2 | 129.2 |

800 | 124.8 | 128.0 | 132.8 |

In summary, it can be observed that the analyzed plate modes _{2} for the asymmetric (_{crdyn}.

The two plate modes _{1} = 0 and various values of the ratio _{2} (see Eq. (9)). The plates are compressed on the outer edge with a growth rate equal to

Figure 5 shows the effect of the imperfection ratio _{2} and thermal loading growth _{2} = 0.5.

A comparison between the reactions of plate modes _{2} = 2 are subjected to three models of thermomechanical loads: without temperature field _{crdyn} and the corresponding temperature difference Δ_{b}_{crdyn} = 22.82 MPa and the corresponding temperatures Δ_{b}_{b}

Values of critical dynamic mechanical loads _{crdyn} and corresponding temperature differences _{b}_{2} = 2.

_{crdyn} (MPa)/D_{b} | ||
---|---|---|

_{2} = 2 | ||

0 | 35.8/0 | 35.8/0 |

200 | 34.47/7.4 | 37.26/8.0 |

800 | 27.12/23.2 | 42.39/36.4 |

Δ | 22.36/19.2 | 44.25/38.0 |

Additionally, the effect of the negative value of imperfection ratio _{2} has been examined. Selected results are presented in Fig. 7 for the FDM plate model. Plate reactions are for two plate modes _{1max} _{2} on the final results is observed. On comparing the results obtained for the asymmetric plate mode _{b}_{1max} =

Figure 8 shows the exemplary results presented for the axisymmetric (_{2} = 1 and _{2} = 2 loaded mechanically and thermally with a temperature difference increasing in time and expressed by a growth parameter _{cr} and dynamic load _{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 _{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 _{2} = 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 _{cr}, critical plate deflection, and, after calculation, dynamic load _{crdyn} as well as the corresponding difference in temperatures Δ_{b}

The obtained results can be summarized as follows:

The minimal value of _{crdyn} which is important in buckling analysis exists for the axisymmetrical _{2} = 2, whose thermal loading is characterized by a positive temperature gradient and a higher value of temperature difference between the plate edges.

There is a difference between the responses of plates subjected to the temperature field increasing in time and constant in time.

The direction of temperature gradient affects the values of critical loads _{crdyn} and the corresponding values of temperature difference Δ_{b}_{crdyn} decrease and this is the opposite for the negative gradient.

A greater value of imperfection ratio _{2} decreases the critical value of loads: _{crdyn} and Δ_{b}

The direction of the plate's initial deflection does not affect the plate response.

The results obtained for the FDM and FEM plate models are comparable.

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 _{1} and the form which is dependent on the number _{2}. Mathematically, this complex form of plate imperfection is described by Equation (9). The form consists of two terms expressed by the aforementioned calibrating numbers _{1} (_{2} (_{2} = 1.

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 _{1} = 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

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 _{2}. Axisymmetrical (_{1} = 10) lose dynamic stability quicker for smaller values of critical dynamic loads _{crdyn} and corresponding temperature difference between the plate edges Δ_{b}_{1} = 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 (_{1} of additional imperfection shape increases both the values of critical dynamic loads _{crdyn} and the corresponding temperature Δ_{b}

The effect of the mixed participation of the values of imperfection ratios on the run of deflection curves _{1max} = _{1} and _{2} or for the positive and negative numbers of ratios _{1}, _{2}. The results are presented for the case of the FDM plate model where its form of the loss of stability corresponds to _{2}, where axisymmetric predeflection also exists (_{1}≠0), and the meaning of the higher values of ratio _{1} (_{1} = 5 or _{1} = −5) for the positive value of the ratio _{2}. For assumed values of _{1}, additional waved shape of predeflection is expressed by the ratio x_{2}. This complex form of predeflection influences the positive or negative run of plate deflection curve _{1max} =

The waved form of plate deflection in a radial direction obtained for selected numbers of ratios _{1}, _{2} (_{1} = 0, _{2} = 1), (_{1} = 1, _{2} = 1), (_{1} = 1, _{2} = −1) is shown in 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 _{1} causes little change to the values of critical temperature differences for the axisymmetrical _{1} decreases the value of critical temperature differences.

The imperfection of plates working in a thermal environment and subjected to mechanical loads has little effect on the critical values, which is dependent on the form of plate buckling.

The results show that plate dynamic buckling can be expressed by a range of values of critical parameters.

Values of ratios _{1}, _{2} determining the form of plate predeflection have an effect on plate deflections. The effect of ratio _{2} which determines the waved form of plate predeflection is dominant. The participation of the axisymmetrical term of Equation (9) exists for higher values of _{1}.

The paper presents the effect of the temperature field on the stability reaction of a plate with various forms of imperfection. A three-layered annular plate with thin steel facings and a thicker foam core was examined. Different shapes and ratios of imperfection were considered. The thermal effect on the plate response was analyzed for plates subjected to only the thermal environment or plates both mechanically loaded and surrounded by a temperature field that increased in time or was fixed in time. Different elements which characterize the field of assumed loading were taken into account. They are as follows: the temperature gradient direction, the dynamic effect expressed by the rate of temperature difference growth, the effect of the mechanical loading growth, the sensitivity of the examined plate to negative imperfections, and the participation of the axisymmetrical term in complex Equation (9), which expresses the form of imperfection. Vibrations are an additional element which characterizes the dynamic response of the plate. In the undertaken analysis, vibrations were observed in the overcritical region of plate work under the increasing load in time. Particularly, the plate oscillations existed for plates with the axisymmetrical buckling mode

The results presented herein show a different effect of imperfection parameters on the values of critical temperature differences for plates located in a temperature field. It depends on the participation of two assumed ratios _{1} and _{2}, which describe the shape of plate predeflection. A rather minimal effect of a single imperfection ratio _{2} on the final results has been observed. However, the complicated shape of plate predeflection, which is expressed by the values of imperfection ratios _{1} and _{2}, has an effect on plate thermal dynamic response. Two terms of the plate form of predeflection (see Eq. (9)) calibrated by the ratios, axisymmetrical _{1} (_{1}≠0) and waved _{2}, reveal the plate structure sensitivity. However, the participation of numbers that define plate imperfection is not unambiguous and is difficult to predict in the evaluation of the process of plate buckling behavior. Effective analytical and numerical dynamic analysis is of importance here. The plate reaction to thermomechanical loading depends on many elements, like the parameters of mechanical load, the profile of the temperature field, and dynamic rates of mechanical and thermal growth. The results show the possibility to design the conditions and structure parameters of the composite plate to use it more effectively. Further analyses can focus on the evaluation of both structural heterogeneities connected with the oriented material properties and the imposed form of plate predeflection to obtain the expected plate reactions on mechanical and thermal loads. Such investigations can be helpful in the plate design process.

#### Values of critical dynamic mechanical loads pcrdyn and corresponding temperature differences ΔTb for the axisymmetrical m = 0 FDM plate model thermomechanically loaded and imperfected with ratio ξ2 = 2.

_{crdyn} (MPa)/D_{b} |
||
---|---|---|

_{2} = 2 |
||

0 | 35.8/0 | 35.8/0 |

200 | 34.47/7.4 | 37.26/8.0 |

800 | 27.12/23.2 | 42.39/36.4 |

Δ |
22.36/19.2 | 44.25/38.0 |

#### Values of critical temperature differences ΔTcrdyn for the axisymmetrical m = 0 FDM plate model versus the imperfection ratio ξ2 under a temperature field with a positive gradient and two rates a = 200 K/s and a = 800 K/s.

_{crdyn} (K) |
|||
---|---|---|---|

_{2} |
|||

200 | 130.0 | 130.2 | 130.7 |

800 | 132.0 | 128.4 | 126.8 |

#### Values of critical temperature differences ΔTcrdyn for the axisymmetrical m = 0 FEM plate model versus the imperfection ratio ξ2 under a temperature field with a positive gradient and two rates a = 200 K/s and a = 800 K/s.

_{crdyn} (K) |
|||
---|---|---|---|

_{2} |
|||

200 | 115.2 | 121.2 | 129.2 |

800 | 124.8 | 128.0 | 132.8 |

#### Values of critical temperature differences ΔTcrdyn for the asymmetrical m = 7 FDM plate model versus the imperfection rate ξ2 under a temperature field with a positive gradient and two rates a = 200 K/s and a = 800 K/s.

_{crdyn} (K) |
|||
---|---|---|---|

_{2} |
|||

200 | 107.4 | 108.0 | 108.2 |

800 | 108.8 | 108.4 | 108.4 |

#### The values of the dynamic, critical temperature differences ΔTcrdyn depending on the number N of discrete points for the FDM plate model with the imperfection ratio ξ2 = 0.5 subjected to a positive gradient of the temperature field.

_{crdyn} (K) |
|||||
---|---|---|---|---|---|

0 | 128.6 | 130.0 | 130.1 | 131.6 | 131.5 |

1 | 131.9 | 133.7 | 133.7 | 134.2 | 134.7 |

2 | 133.5 | 135.5 | 135.5 | 137.2 | 137.0 |

3 | 126.4 | 129.3 | 131.2 | 130.9 | 132.4 |

4 | 117.5 | 120.7 | 122.1 | 123.5 | 124.8 |

5 | 108.7 | 112.3 | 114.9 | 115.9 | 117.1 |

6 | 105.7 | 108.9 | 110.4 | 112.8 | 113.8 |

7 | 103.8 | 106.8 | 108.8 | 109.5 | 111.7 |

8 | 103.7 | 107.9 | 110.3 | 112.8 | 116.4 |

#### The values of the dynamic, critical mechanical loads pcrdyn with the corresponding temperature differences ΔTb for the axisymmetric FDM plate model (m = 0) with the imperfection ratio ξ2 = 2 subjected to a mechanical load and increasing with the value a = 800 K/s temperature field with a positive gradient.

_{crdyn} (MPa)/Δ_{b} |
30.74/26.4 | 29.35/25.2 | 31.21/26.8 | 30.74/26.4 | 31.21/26.8 |

#### Parameters of the plate model.

Geometrical parameters | |||

Inner radius _{i} |
0.2 | ||

Outer radius _{o} |
0.5 | ||

Facing thickness |
1 | ||

Core thickness _{2}, mm |
5 | ||

Ratio of plate initial deflection _{2} |
0.5, 1, 2 | ||

Material parameters | |||

Steel facing | Polyurethane foam of core | ||

Young's modulus |
210 | _{2}, MPa |
13 |

Kirchhoff's modulus |
80 | _{2}, MPa |
5 |

Poisson's ratio |
0.3 | _{2} |
0.3 |

Mass density ^{3} |
7850 | _{2}, kg/m^{3} |
64 |

Linear expansion coefficient a, 1/K | 1.2×10^{−5} |
a_{2}, 1/K |
7×10^{−5} |

Loading parameters | |||

Rate of thermal loading growth |
200 (20), 800 (20) | ||

Rate of mechanical loading growth |
931 (20) | ||

Constant temperature difference |
800 |

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