3D Size-Dependent Dynamic Instability Analysis of FG Cylindrical Microshells Subjected to Combinations of Periodic Axial Compression and External Pressure Using a Hermitian C2 Finite Layer Method Based on the Consistent Couple Stress Theory
Abstract
1. Introduction
2. The Initial Stresses Induced at the Pre-Instability State
3. The CCST for Elastic Bodies
4. The System Equations at the Incremental Perturbation State
4.1. The CCST-Based Hermitian C2 FLM
4.1.1. Generalized Kinematics Models
4.1.2. Hamilton’s Principle
4.1.3. Layer Element Equations and Structural Equations
4.2. Bolotin’s Method
4.2.1. The Principal Instability Regions
4.2.2. The Secondary Instability Regions
4.3. Reduced Cases
5. Numerical Examples
5.1. Static Buckling
5.2. Dynamic Instability
6. Concluding Remarks
- The CCST-based Hermite C2 FLM for analyzing FG cylindrical microshells can be reduced to those for analyzing FG cylindrical macroshells by setting a zero value to the material length-scale parameter. The static buckling and dynamic instability analyses of the reduced model showed that the CCST-based Hermite C2 FLM was validated by comparing the solutions it produced with the solutions obtained using the 3D elasticity theory and the 2D advanced shear deformation shell theories reported in the literature.
- The implementation of Bolotin’s method in the numerical examples showed that convergent solutions were obtained when two terms of the trigonometric functions were used. When arranging by descending order of bandwidth between the upper and lower bounds of excitation frequency, we obtain the following list of instability regions: the first principal instability region, the first secondary instability region, the second principal instability region, and the second secondary instability region.
- The magnitude of the excitation frequency and its bandwidth between the upper and lower bounds of various instability regions increased when the value of the material length-scale parameter increased, which indicated that an increase in the value of the material length-scale parameter caused the microshell to stiffen, in turn increasing the magnitude of the excitation frequency and its bandwidth for the instability region.
- The magnitude of the excitation frequency and its bandwidth between the upper and lower bounds of various instability regions decreased when the value of the inhomogeneity index increased, which indicates that an increase in the value of caused the microshell to soften, in turn decreasing the magnitude of the excitation frequency and its bandwidth for the instability region.
- The magnitude of the excitation frequency and its bandwidth between the upper and lower bounds of various instability regions decreased when the L/R ratio or the R/h ratio increased, which indicates that an increase in either the L/R ratio or the R/h ratio results in a decrease in the overall stiffness of the microshells, in turn decreasing the magnitude of the excitation frequency and its bandwidth for the instability region.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A. The Detailed Expressions of Relevant Tensors in Equations (37)–(39)
Appendix B. The Detailed Expressions of Matrices
References
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(l/h) | Theories | L/R = 1 | L/R = 3 | ||||||
---|---|---|---|---|---|---|---|---|---|
R/h = 300 | R/h = 500 | R/h = 1000 | R/h = 3000 | R/h = 300 | R/h = 500 | R/h = 1000 | R/h = 3000 | ||
0 (0) | Two-node Hermitian C2 () | 1273.7314 | 349.4958 | 60.6715 | 3.8187 | 413.8408 | 115.0668 | 19.7346 | 1.2594 |
Two-node Hermitian C2 () | 1273.7314 | 349.4958 | 60.6715 | 3.8188 | 413.8408 | 115.0668 | 19.7345 | 1.2592 | |
Vodenitcharova and Ansourian [44] | 1269.6 | 348.43 | 60.488 | 3.8100 | 407.19 | NA | NA | 1.2510 | |
Shen [45] | 1272.6 | 348.59 | 60.536 | 3.8144 | 402.60 | NA | NA | 1.2511 | |
Sofiyev [46] | 1273.5 | 349.45 | 60.599 | 3.8153 | 412.62 | NA | NA | 1.2562 | |
Khazaeinejad et al. [47] | 1273.1 | 349.39 | 60.595 | 3.8151 | 412.57 | NA | NA | 1.2564 | |
Mehralian et al. [48] | 1273.2 | 349.40 | 60.597 | 3.8152 | 412.58 | NA | NA | 1.2561 | |
(1, 11) | (1, 13) | (1, 15) | (1, 20) | (1, 7) | (1, 8) | (1, 9) | (1, 12) | ||
0.5 | Two-node Hermitian C2 () | 2219.8700 | 608.4825 | 104.7762 | 6.5912 | 700.4623 | 194.2209 | 33.9787 | 2.1644 |
(0.25) | Two-node Hermitian C2 () | 2219.8700 | 608.4824 | 104.7761 | 6.5912 | 700.4624 | 194.2211 | 33.9785 | 2.1645 |
(1, 10) | (1, 12) | (1, 14) | (1, 18) | (1, 6) | (1, 7) | (1, 8) | (1, 11) | ||
1.0 | Two-node Hermitian C2 () | 4578.1057 | 1247.0089 | 214.3564 | 13.3529 | 1467.7834 | 393.3222 | 69.1036 | 4.4041 |
(0.5) | Two-node Hermitian C2 () | 4578.1057 | 1247.0088 | 214.3564 | 13.3530 | 1467.7834 | 393.3222 | 69.1036 | 4.4047 |
(1, 9) | (1, 10) | (1, 12) | (1, 16) | (1, 5) | (1, 6) | (1, 7) | (1, 10) |
2R/h | Numerical Results (Kim and Kim [50]) | Analytical Results (Kim and Kim [50]) | MCST-Based LCST Results (Mehralian and Beni [49]) | CCST-Based FCLMs | ||
---|---|---|---|---|---|---|
Hermitian C2 | Relative Errors | |||||
800 | 0 | 1.5013 | 1.5141 | 1.5131 | 1.48742 | 1.73% |
1.0 | NA | NA | 3.4099 | 3.34969 | 1.80% | |
900 | 0 | 1.3586 | 1.3459 | 1.3450 | 1.33319 | 0.89% |
1.0 | NA | NA | 3.0464 | 3.01984 | 0.88% | |
1000 | 0 | 1.2111 | 1.2113 | 1.2105 | 1.19073 | 1.66% |
1.0 | NA | NA | 2.7497 | 2.71544 | 1.26% | |
1100 | 0 | 1.1021 | 1.1012 | 1.1005 | 1.08321 | 1.60% |
1.0 | NA | NA | 2.4842 | 2.44885 | 1.44% | |
1200 | 0 | 1.0170 | 1.0094 | 1.0087 | 1.00143 | 0.72% |
1.0 | NA | NA | 2.2783 | 2.24608 | 1.43% | |
1300 | 0 | 0.9365 | 0.9318 | 0.9311 | 0.92267 | 0.91% |
1.0 | NA | NA | 2.1164 | 2.08826 | 1.35% | |
1400 | 0 | 0.8654 | 0.8652 | 0.8646 | 0.85258 | 1.41% |
1.0 | NA | NA | 1.9674 | 1.95456 | 0.66% | |
1500 | 0 | 0.8075 | 0.8075 | 0.8069 | 0.79602 | 1.37% |
1.0 | NA | NA | 1.8309 | 1.81369 | 0.95% |
Theories | R/h = 10 | R/h = 100 | R/h = 500 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
0 (0) | Two-node Hermitian C2 FCLM | 0 | 153.078 | 145.158 | 1.356 | 0.383 | 0.377 | 0.020 | 0.0065 | 0.0065 | 7.7299 × 10−4 |
Khazaeinejad et al. [47] | 150.244 | 146.626 | NA | 0.370 | 0.366 | NA | 0.006 | 0.006 | NA | ||
Two-node Hermitian C2 FCLM | 0.5 | 100.745 | 95.414 | 0.925 | 0.253 | 0.250 | 0.014 | 0.0044 | 0.0044 | 5.2023 × 10−4 | |
Khazaeinejad et al. [47] | 97.934 | 95.575 | NA | 0.244 | 0.241 | NA | 0.004 | 0.004 | NA | ||
Two-node Hermitian C2 FCLM | 1 | 78.106 | 73.924 | 0.730 | 0.197 | 0.194 | 0.011 | 0.0035 | 0.0034 | 4.0765 × 10−4 | |
Khazaeinejad et al. [47] | 75.514 | 73.696 | NA | 0.189 | 0.187 | NA | 0.003 | 0.003 | NA | ||
Two-node Hermitian C2 FCLM | 5 | 50.960 | 48.204 | 0.437 | 0.126 | 0.124 | 0.006 | 0.0021 | 0.0021 | 2.5206 × 10−4 | |
Khazaeinejad et al. [47] | 49.357 | 48.169 | NA | 0.121 | 0.120 | NA | 0.002 | 0.002 | NA | ||
(1, 2) | (1, 2) | (2, 2) | (1, 3) | (1, 3) | (1, 2) | (1, 4) | (1, 4) | (1, 4) | |||
0.5 (0.25) | Two-node Hermitian C2 FCLM | 0 | 305.332 | 289.533 | 1.483 | 0.719 | 0.708 | 0.023 | 0.0113 | 0.0112 | 0.0011 |
Two-node Hermitian C2 FCLM | 0.5 | 210.961 | 199.796 | 1.101 | 0.498 | 0.491 | 0.016 | 0.0079 | 0.0078 | 7.7691 × 10−4 | |
Two-node Hermitian C2 FCLM | 1 | 167.457 | 158.489 | 0.905 | 0.396 | 0.390 | 0.013 | 0.0063 | 0.0062 | 6.2731 × 10−4 | |
(1, 2) | (1, 2) | (1, 1) | (1, 3) | (1, 3) | (1, 2) | (1, 4) | (1, 4) | (1, 3) | |||
1.0 (0.5) | Two-node Hermitian C2 FCLM | 0 | 757.117 | 717.920 | 1.486 | 1.728 | 1.647 | 0.031 | 0.0255 | 0.0253 | 0.0016 |
Two-node Hermitian C2 FCLM | 0.5 | 537.975 | 509.495 | 1.103 | 1.232 | 1.190 | 0.022 | 0.0182 | 0.0181 | 0.0011 | |
Two-node Hermitian C2 FCLM | 1 | 432.510 | 409.343 | 0.907 | 0.993 | 0.965 | 0.018 | 0.0147 | 0.0146 | 9.1795 × 10−4 | |
(1, 2) | (1, 2) | (1, 1) | (1, 3) | (1, 2) | (1, 2) | (1, 4) | (1, 4) | (1, 3) |
First Principal Instability Region | K-Term Approximations | |||||||
---|---|---|---|---|---|---|---|---|
K = 1 | K = 2 | K = 3 | ||||||
Cosine Terms | Sine Terms | Cosine Terms | Sine Terms | Cosine Terms | Sine Terms | |||
(1, 2) | 0 | Hermitian C2 FLM solutions | 0.64712 | 0.64712 | 0.64712 | 0.64712 | 0.64712 | 0.64712 |
Ng et al.’s solutions [51] | 0.6485 | 0.6485 | NA | NA | NA | NA | ||
Sofiyev’s solutions [28] | 0.6519 | 0.6519 | NA | NA | NA | NA | ||
0.1 | Hermitian C2 FLM solutions | 0.64518 | 0.64907 | 0.64518 | 0.64907 | 0.64518 | 0.64907 | |
Ng et al.’s solutions [51] | 0.6481 | 0.6488 | NA | NA | NA | NA | ||
Sofiyev’s solutions [28] | 0.6478 | 0.6560 | NA | NA | NA | NA | ||
0.3 | Hermitian C2 FLM solutions | 0.64126 | 0.65293 | 0.64127 | 0.65295 | 0.64127 | 0.65295 | |
Ng et al.’s solutions [51] | 0.6475 | 0.6494 | NA | NA | NA | NA | ||
Sofiyev’s solutions [28] | 0.6396 | 0.6640 | NA | NA | NA | NA | ||
0.5 | Hermitian C2 FLM solutions | 0.63732 | 0.65678 | 0.63736 | 0.65681 | 0.63736 | 0.65681 | |
Ng et al.’s solutions [51] | 0.6468 | 0.6500 | NA | NA | NA | NA | ||
Sofiyev’s solutions [28] | 0.6312 | 0.6720 | NA | NA | NA | NA | ||
(1, 3) | 0 | Hermitian C2 FLM solutions | 0.37843 | 0.37843 | 0.37843 | 0.37843 | 0.37843 | 0.37843 |
Ng et al.’s solutions [51] | 0.3778 | 0.3778 | NA | NA | NA | NA | ||
Sofiyev’s solutions [28] | 0.3719 | 0.3719 | NA | NA | NA | NA | ||
0.1 | Hermitian C2 FLM solutions | 0.37509 | 0.38174 | 0.37509 | 0.38175 | 0.37509 | 0.38175 | |
Ng et al.’s solutions [51] | 0.3771 | 0.3784 | NA | NA | NA | NA | ||
Sofiyev’s solutions [28] | 0.3696 | 0.3742 | NA | NA | NA | NA | ||
0.3 | Hermitian C2 FLM solutions | 0.36831 | 0.38828 | 0.36838 | 0.38834 | 0.36838 | 0.38834 | |
Ng et al.’s solutions [51] | 0.3757 | 0.3796 | NA | NA | NA | NA | ||
Sofiyev’s solutions [28] | 0.3649 | 0.3788 | NA | NA | NA | NA | ||
0.5 | Hermitian C2 FLM solutions | 0.36141 | 0.39471 | 0.36162 | 0.39487 | 0.36162 | 0.39487 | |
Ng et al.’s solutions [51] | 0.3743 | 0.3807 | NA | NA | NA | NA | ||
Sofiyev’s solutions [28] | 0.3601 | 0.3834 | NA | NA | NA | NA | ||
(1, 4) | 0 | Hermitian C2 FLM solutions | 0.24507 | 0.24507 | 0.24507 | 0.24507 | 0.24507 | 0.24507 |
Ng et al.’s solutions [51] | 0.2473 | 0.2473 | NA | NA | NA | NA | ||
Sofiyev’s solutions [28] | 0.2471 | 0.2471 | NA | NA | NA | NA | ||
0.1 | Hermitian C2 FLM solutions | 0.23988 | 0.25015 | 0.23991 | 0.25018 | 0.23991 | 0.25018 | |
Ng et al.’s solutions [51] | 0.2462 | 0.2481 | NA | NA | NA | NA | ||
Sofiyev’s solutions [28] | 0.2456 | 0.2487 | NA | NA | NA | NA | ||
0.3 | Hermitian C2 FLM solutions | 0.22914 | 0.26002 | 0.22944 | 0.26022 | 0.22944 | 0.26022 | |
Ng et al.’s solutions [51] | 0.2441 | 0.2500 | NA | NA | NA | NA | ||
Sofiyev’s solutions [28] | 0.2425 | 0.2517 | NA | NA | NA | NA | ||
0.5 | Hermitian C2 FLM solutions | 0.21788 | 0.26953 | 0.21886 | 0.27002 | 0.21886 | 0.27002 | |
Ng et al.’s solutions [51] | 0.2420 | 0.2518 | NA | NA | NA | NA | ||
Sofiyev’s solutions [28] | 0.2393 | 0.2547 | NA | NA | NA | NA |
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Wu, C.-P.; Wu, M.-L.; Hsu, H.-T. 3D Size-Dependent Dynamic Instability Analysis of FG Cylindrical Microshells Subjected to Combinations of Periodic Axial Compression and External Pressure Using a Hermitian C2 Finite Layer Method Based on the Consistent Couple Stress Theory. Materials 2024, 17, 810. https://doi.org/10.3390/ma17040810
Wu C-P, Wu M-L, Hsu H-T. 3D Size-Dependent Dynamic Instability Analysis of FG Cylindrical Microshells Subjected to Combinations of Periodic Axial Compression and External Pressure Using a Hermitian C2 Finite Layer Method Based on the Consistent Couple Stress Theory. Materials. 2024; 17(4):810. https://doi.org/10.3390/ma17040810
Chicago/Turabian StyleWu, Chih-Ping, Meng-Luen Wu, and Hao-Ting Hsu. 2024. "3D Size-Dependent Dynamic Instability Analysis of FG Cylindrical Microshells Subjected to Combinations of Periodic Axial Compression and External Pressure Using a Hermitian C2 Finite Layer Method Based on the Consistent Couple Stress Theory" Materials 17, no. 4: 810. https://doi.org/10.3390/ma17040810
APA StyleWu, C.-P., Wu, M.-L., & Hsu, H.-T. (2024). 3D Size-Dependent Dynamic Instability Analysis of FG Cylindrical Microshells Subjected to Combinations of Periodic Axial Compression and External Pressure Using a Hermitian C2 Finite Layer Method Based on the Consistent Couple Stress Theory. Materials, 17(4), 810. https://doi.org/10.3390/ma17040810