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![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.](https://sciendo-parsed.s3.eu-central-1.amazonaws.com/64737a3e4e662f30ba53f8da/j_sgem-2023-0005_fig_007.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Content-Sha256=UNSIGNED-PAYLOAD&X-Amz-Credential=ASIA6AP2G7AKFPEESX7D%2F20260201%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20260201T092035Z&X-Amz-Expires=3600&X-Amz-Security-Token=IQoJb3JpZ2luX2VjEP3%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FwEaDGV1LWNlbnRyYWwtMSJGMEQCIB5m%2Bb075uLP13en3Y8G274QXNZ95t732P0zOdGuOVGwAiBAE43t56HGHnOdCi%2BQVKOOoNSglQ5nBryIiaoL37aGBCrGBQjG%2F%2F%2F%2F%2F%2F%2F%2F%2F%2F8BEAIaDDk2MzEzNDI4OTk0MCIMe%2BjQO%2FtzopqHrGjTKpoFhLwufuEx1FgnDI9dJikf0GH9o17paaMf6GNpyOFNzF8WJyQB2aF52Q0IR%2BeZmfbNpXOax0%2FOF6q3kStL%2B2eC89rwzEu6RE3Zle%2FIf8vmNQaQMzXEY3XLnl92FDIhsP0IePh1%2B%2B8sGWrpXtfFmXcgS%2F1j5cON8hR54%2BqywJBoiEnqC7OXru%2BkdUY9U%2FHp3Z7cWGvOi81oWZx7ooiPAeTE3DL79AtSJVj%2Bz%2BIkMqmU94O3VRAft%2BvNJUSKA0QknYo2OhlgFjzf0bUtk2%2BsGzmSgINFykY9XZ7Dzx08Kck8jhHp5ZPoBNbBdHLwU5Y3dg1bUVerK9MyJdWOi3Bgmckz%2BbevuEtubAVQxiKFszxl%2Bz%2BLzbF51HOgNNt6Ppz78esy8zNCLG450pJWjfDrkNOyAOsJZeim4ARn8HINu7GrYZgtm5x%2FYesNXSK6PakNYFyeYcd%2FfLWRqcBjqCP5KyMNKYAnBNdq%2FYG%2BPLu2ndVLJeTRPggE36z0y4cQEZf7mKe39JeIodEwSO9BqLZPO07hosAkaQBfhSJVnvGXsac8VLUx6fmUEDy5s7sZ%2B5gQ16N%2B8Z7ujqDrQxetR0%2FAdUALT6%2Fc5PpEsatvuCJVdFdPdjcYUBRR3YvsQcTdfYaUIogUcTXL7I9GX3Imv%2FIbMvp0mFHfq19me9JLp168w3%2FDpwR0cDnz3MmhV0u0REp10k0HWz1Y5I2UUyUnl5e%2FQXt1wWcFBHZsGbb8nRwjwyMcj5WNO9cgFpJ%2BbvNSItJxlP9kHtDnVi5sILuFN%2FZ24hiHvcJxUkprluYQ1kHSp4kW53rRj1C%2FEk5k3%2Bxzg%2F2KSIJ3vetHl5nyEG1ZGZfcyv%2FEo4FWIj%2F1Jd610IUk%2FOsi9WG%2FeBsnLgxjKt%2F1MOG5%2B8sGOrIBjL6eKwxlLj32nGRQJRTcjcNa3vk8oK6EWSoHaVcUy2K0IAyYsCn%2BINM7CQPgDbjshrVe7c323vT529dgXJ7lUqZIegAlEE%2Bdw9Md7fHDmMwVg9duplRPbvtNoIaZAMTmgevN3mFZosgNuVLxqrUs3Ci5%2FbonwmWocuB5vMQiuuLTYnRzv7yAcgQ8YVWPVbcF3gPAnsD9Jw4ODE3VwWCSWBV5utaxxwBql5WsD8dmNNXb0g%3D%3D&X-Amz-Signature=bb5c0da0e74cad782fd86eff69d9bacea9e1594db6843c01ed71fd3fa2cb1178&X-Amz-SignedHeaders=host&x-amz-checksum-mode=ENABLED&x-id=GetObject)
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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) | 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 | ||