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

Dorota Pawlus

Dorota Pawlus

Faculty of Mechanical Engineering and Computer Science, University of Bielsko-BialaBielsko-Biala, Poland
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Pawlus, Dorota
  
28. Apr. 2023
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Artikel-Kategorie: Original Study
Online veröffentlicht: 28. Apr. 2023
Seitenbereich: 158 - 173
Eingereicht: 20. Sept. 2022
Akzeptiert: 05. Jan. 2023
DOI: https://doi.org/10.2478/sgem-2023-0005
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© 2023 Dorota Pawlus, published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.

Figure 1

Scheme of thermomechanical loading of a three-layered annular plate built of outer layers 1 and 3 and middle layer 2.
Scheme of thermomechanical loading of a three-layered annular plate built of outer layers 1 and 3 and middle layer 2.

Figure 2

Deflections of the axisymmetrical m = 0 plate model versus imperfection ratio ξ2 under a temperature field with a positive gradient and two rates a = 200 K/s and a = 800 K/s.
Deflections of the axisymmetrical m = 0 plate model versus imperfection ratio ξ2 under a temperature field with a positive gradient and two rates a = 200 K/s and a = 800 K/s.

Figure 3

Deflections of the asymmetrical m = 7 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.
Deflections of the asymmetrical m = 7 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.

Figure 4

Time histories of deflections and velocity of deflection for plate model m = 0 with the imperfection ratio ξ2 = 2 loaded thermally with a positive temperature gradient, with rate a = 200 K/s: a) FDM model, b) FEM model with critical deflection form.
Time histories of deflections and velocity of deflection for plate model m = 0 with the imperfection ratio ξ2 = 2 loaded thermally with a positive temperature gradient, with rate a = 200 K/s: a) FDM model, b) FEM model with critical deflection form.

Figure 5

Deflections of the asymmetrical m = 7 plate model with different imperfection ratios ξ2 under mechanical load and thermal load with a positive temperature gradient and various rates a.
Deflections of the asymmetrical m = 7 plate model with different imperfection ratios ξ2 under mechanical load and thermal load with a positive temperature gradient and various rates a.

Figure 6

Time histories of deflections for the FDM plate with ξ2 = 2 thermomechanically loaded with various rates a or fixed temperature ΔT = 800 K: a) axisymmetrical plate mode m = 0 and b) asymmetrical plate mode m = 7.
Time histories of deflections for the FDM plate with ξ2 = 2 thermomechanically loaded with various rates a or fixed temperature ΔT = 800 K: a) axisymmetrical plate mode m = 0 and b) asymmetrical plate mode m = 7.

Figure 7

Deflections of a) axisymmetrical plate mode m = 0 [11], b) asymmetrical plate mode m = 7 versus negative and positive imperfection ratios ξ2 under mechanical load and thermal load with a negative gradient.
Deflections of a) axisymmetrical plate mode m = 0 [11], b) asymmetrical plate mode m = 7 versus negative and positive imperfection ratios ξ2 under mechanical load and thermal load with a negative gradient.

Figure 8

Time histories of deflections and velocity of deflections for the axisymmetrical m = 0 FEM plate model thermomechanically loaded with a positive temperature gradient versus different imperfection ratios ξ2 and temperature growth loads a: a) ξ2 = 1, a = 200 K/s, b) ξ2 = 1, a = 800 K/s, c) ξ2 = 2, a = 200 K/s, d) ξ2 = 2, a = 800 K/s.
Time histories of deflections and velocity of deflections for the axisymmetrical m = 0 FEM plate model thermomechanically loaded with a positive temperature gradient versus different imperfection ratios ξ2 and temperature growth loads a: a) ξ2 = 1, a = 200 K/s, b) ξ2 = 1, a = 800 K/s, c) ξ2 = 2, a = 200 K/s, d) ξ2 = 2, a = 800 K/s.

Figure 9

Time histories of deflections and velocity of deflections for a) FDM plate model and b) FEM plate model m = 0, ξ2 = 1 loaded mechanically and thermally with a positive temperature gradient and rate a = 200 K/s.
Time histories of deflections and velocity of deflections for a) FDM plate model and b) FEM plate model m = 0, ξ2 = 1 loaded mechanically and thermally with a positive temperature gradient and rate a = 200 K/s.

Figure 10

Deflections of the FDM plate model ξ2 = 1 loaded thermally with a positive temperature gradient and rate a = 200 K/s versus calibrating number ξ1.
Deflections of the FDM plate model ξ2 = 1 loaded thermally with a positive temperature gradient and rate a = 200 K/s versus calibrating number ξ1.

Figure 11

Deflections of the FDM plate model ξ2 = 1 loaded thermomechanically with a positive temperature gradient and rate a = 200 K/s versus a) different values of calibrating number ξ1 and b) value of calibrating number ξ1 = 5 for axisymmetrical plate m = 0.
Deflections of the FDM plate model ξ2 = 1 loaded thermomechanically with a positive temperature gradient and rate a = 200 K/s versus a) different values of calibrating number ξ1 and b) value of calibrating number ξ1 = 5 for axisymmetrical plate m = 0.

Figure 12

Deflections of the FDM waved m = 7 plate model loaded thermally with a positive temperature gradient and rate a = 200 K/s for mixed values of imperfection ratios: a) only positive, b) positive and negative.
Deflections of the FDM waved m = 7 plate model loaded thermally with a positive temperature gradient and rate a = 200 K/s for mixed values of imperfection ratios: a) only positive, b) positive and negative.

Figure 13

Influence of imperfection ratios on the distribution of the FDM waved m = 7 plate model deflections in a radial direction caused by thermal loading with a positive temperature gradient and rate a = 200 K/s.
Influence of imperfection ratios on the distribution of the FDM waved m = 7 plate model deflections in a radial direction caused by thermal loading with a positive temperature gradient and rate a = 200 K/s.

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_

a (K/s)ΔT (K) pcrdyn (MPa)/DTb (K)
ξ2 = 2
Positive gradient Negative gradient
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
ΔT = 800 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_

Rate a (K/s) ΔTcrdyn (K)
ξ2
0.5 1 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_

Rate a (K/s) ΔTcrdyn (K)
ξ2
0.5 1 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_

Ratio a (K/s) ΔTcrdyn (K)
ξ2
0.5 1 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_

m ΔTcrdyn (K)
N = 11 N = 14 N = 17 N = 21 N = 26
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_

Number N 11 14 17 21 26
pcrdyn (MPa)/ΔTb (K) 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 ri, m 0.2
Outer radius ro, m 0.5
Facing thickness h′, mm 1
Core thickness h2, mm 5
Ratio of plate initial deflection ξ2 0.5, 1, 2
Material parameters
Steel facing Polyurethane foam of core
Young's modulus E, GPa 210 E2, MPa 13
Kirchhoff's modulus G, GPa 80 G2, MPa 5
Poisson's ratio ν 0.3 ν2 0.3
Mass density μ, kg/m3 7850 μ2, kg/m3 64
Linear expansion coefficient a, 1/K 1.2×10−5 a2, 1/K 7×10−5
Loading parameters
Rate of thermal loading growth a, K/s (TK7, 1/s)     200 (20), 800 (20)
Rate of mechanical loading growth s, MPa/s (K7, 1/s)     931 (20)
Constant temperature difference ΔT, K     800
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