A Four-Variable Shear Deformation Theory for the Static Analysis of FG Sandwich Plates with Different Porosity Models
Abstract
:1. Introduction
2. Theoretical Model and Formulas
2.1. Material Properties and Porosity Models
2.1.1. FG with Even Porosities (Model A)
2.1.2. FG with Uneven Porosities (Model B)
2.1.3. FG with Linear-Uneven Porosities (Model C)
2.2. Displacement Field and Constitutive Equations
3. Governing Equations
4. Solution Method
5. Results and Discussion
5.1. Material Properties
5.2. Nondimensional Parameters
5.3. Comparison and Validation
5.4. Bending Analysis of the FG Porous Sandwich Plates
5.4.1. Sandwich Plates with Ceramic Core
5.4.2. Sandwich Plates with Metal Core
6. Conclusions
- Employing the four-variable shear deformation theory proved its soundness since it yielded similar results to those found in the literature.
- The current findings exhibit a strong similarity to the outcomes obtained through the Third Shear Deformation Theory (TSDT).
- Higher porosity leads to larger deflections. This can be referred to as the degradation in the modulus of elasticity as a consequence of high porosities, which leads to a reduction in the bending stiffness of the plate. The central deflection is further magnified at lower aspect ratios and at higher side-to-thickness ratios due to the reduction in the plate bending stiffness in these cases.
- At the same porosity coefficient (α), the central deflection is highest in the case of Model A, lower in Model B, and lowest in Model C. This is due to the gradual reduction in the modulus of elasticity in the porosity models from Model C down to Model A. In addition, Higher values of (defined in the volume fraction functions) result in higher central defections.
- When the core layer in Model B is made of metal instead of ceramic, the nondimensional central deflection increases linearly with the porosity coefficient α. The increase in the deflection is due to the reduction in the modulus of elasticity of as a consequence of increasing the porosity.
- The stress distributions differ according to the porosity models as well as the value of the porosity coefficient. In addition, the trend exhibited in the middle homogenous layer can be maintained or reversed once or twice in the external FG layers.
- The distributions of all stress types experience a jump at the interfaces between different layers in the case of sandwich plates with a metal core.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A
References
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Scheme | Method | |||||
---|---|---|---|---|---|---|
1-0-1 | CPT [14] | 1.16247 | 0.91865 | 0.35885 | 0.13590 | 0.05742 |
FSDT [14] | 1.19200 | 0.94473 | 0.37514 | 0.14592 | 0.06393 | |
SSDT [14] | 1.18808 | 0.94124 | 0.37297 | 0.14458 | 0.06305 | |
TSDT [14] | 1.18877 | 0.94186 | 0.37335 | 0.14481 | 0.06321 | |
Present | 1.18877 | 0.94186 | 0.37335 | 0.14481 | 0.06321 | |
2-1-2 | CPT [14] | 1.09971 | 0.86891 | 0.33942 | 0.12854 | 0.05431 |
FSDT [14] | 1.12611 | 0.89237 | 0.35408 | 0.13756 | 0.06017 | |
SSDT [14] | 1.12204 | 0.88876 | 0.35183 | 0.13617 | 0.05926 | |
TSDT [14] | 1.12293 | 0.88955 | 0.35231 | 0.13647 | 0.05946 | |
Present | 1.12293 | 0.88945 | 0.35231 | 0.13647 | 0.05946 | |
1-1-1 | CPT [14] | 1.03895 | 0.82090 | 0.32067 | 0.12144 | 0.05131 |
FSDT [14] | 1.06369 | 0.84289 | 0.33411 | 0.12989 | 0.05680 | |
SSDT [14] | 1.05989 | 0.83952 | 0.33230 | 0.12895 | 0.05619 | |
TSDT [14] | 1.06096 | 0.84046 | 0.33289 | 0.12895 | 0.05619 | |
Present | 1.06096 | 0.84046 | 0.33289 | 0.12895 | 0.05619 | |
2-2-1 | CPT [14] | 0.98512 | 0.77837 | 0.30405 | 0.11514 | 0.04085 |
FSDT [14] | 1.00911 | 0.79969 | 0.31738 | 0.12334 | 0.05398 | |
SSDT [14] | 1.00585 | 0.79679 | 0.34557 | 0.12222 | 0.05325 | |
TSDT [14] | 1.00694 | 0.79776 | 0.31617 | 0.12260 | 0.05325 | |
Present | 1.00694 | 0.79776 | 0.31617 | 0.12260 | 0.05349 | |
1-2-1 | CPT [14] | 0.94269 | 0.74489 | 0.29095 | 0.11018 | 0.04655 |
FSDT [14] | 0.96563 | 0.76524 | 0.30370 | 0.10803 | 0.05165 | |
SSDT [14] | 0.96248 | 0.76243 | 0.30195 | 0.11694 | 0.05094 | |
TSDT [14] | 0.96371 | 0.76353 | 0.30263 | 0.11737 | 0.05122 | |
Present | 0.96371 | 0.76353 | 0.30263 | 0.11737 | 0.05122 |
Perfect | Model A | Model B | Model C | ||
---|---|---|---|---|---|
10 | 0 | 0.33289 | -- | -- | -- |
0.1 | -- | 0.38414 | 0.35595 | 0.34794 | |
0.2 | -- | 0.45086 | 0.38155 | 0.36742 | |
0.3 | -- | 0.54184 | 0.41039 | 0.38677 | |
20 | 0 | 1.29489 | -- | -- | -- |
0.1 | -- | 1.49856 | 1.38653 | 1.36044 | |
0.2 | -- | 1.76382 | 1.48859 | 1.43190 | |
0.3 | -- | 2.12577 | 1.60341 | 1.50869 | |
50 | 0 | 8.02885 | -- | -- | -- |
0.1 | -- | 9.29942 | 8.60067 | 8.44051 | |
0.2 | -- | 10.95449 | 9.23779 | 8.88325 | |
0.3 | -- | 13.21322 | 9.95453 | 9.36211 |
Perfect | Model A | Model B | Model C | ||
---|---|---|---|---|---|
10 | 0 | 1.59370 | -- | -- | -- |
0.1 | -- | 0.14810 | 0.17069 | 0.16741 | |
0.2 | -- | 0.13692 | 0.18325 | 0.17624 | |
0.3 | -- | 0.12218 | 0.19741 | 0.18569 | |
20 | 0 | 1.58524 | -- | -- | -- |
0.1 | -- | 0.14827 | 0.16982 | 0.16666 | |
0.2 | -- | 0.13632 | 0.18239 | 0.17538 | |
0.3 | -- | 0.12169 | 0.19654 | 0.18483 | |
50 | 0 | 1.58287 | -- | -- | -- |
0.1 | -- | 0.14807 | 0.16957 | 0.16641 | |
0.2 | -- | 0.13615 | 0.18215 | 0.17515 | |
0.3 | -- | 0.12156 | 0.19629 | 0.18459 |
Perfect | Model A | Model B | Model C | ||
---|---|---|---|---|---|
10 | 0 | 0.27188 | -- | -- | -- |
0.1 | -- | 0.28065 | 0.27541 | 0.23777 | |
0.2 | -- | 0.29074 | 0.27915 | 0.28146 | |
0.3 | -- | 0.30260 | 0.28313 | 0.28661 | |
20 | 0 | 0.27197 | -- | -- | -- |
0.1 | -- | 0.28074 | 0.27507 | 0.26221 | |
0.2 | -- | 0.29083 | 0.27925 | 0.28155 | |
0.3 | -- | 0.30267 | 0.28323 | 0.28671 | |
50 | 0 | 0.27200 | -- | -- | -- |
0.1 | -- | 0.28077 | 0.27498 | 0.26998 | |
0.2 | -- | 0.29085 | 0.27928 | 0.28158 | |
0.3 | -- | 0.30270 | 0.28326 | 0.28674 |
Perfect | Model A | Model B | Model C | ||
---|---|---|---|---|---|
10 | 0 | 0.73130 | -- | -- | -- |
0.1 | -- | 0.76124 | 0.70389 | 0.73064 | |
0.2 | -- | 0.79066 | 0.66632 | 0.72685 | |
0.3 | -- | 0.81927 | 0.61660 | 0.72113 | |
20 | 0 | 1.46421 | -- | -- | -- |
0.1 | -- | 1.52380 | 1.40915 | 1.46221 | |
0.2 | -- | 1.58231 | 1.33397 | 1.45539 | |
0.3 | -- | 1.63915 | 1.23438 | 1.44396 | |
50 | 0 | 3.66166 | -- | -- | -- |
0.1 | -- | 3.81042 | 3.52382 | 3.65625 | |
0.2 | -- | 3.95646 | 3.33586 | 3.63964 | |
0.3 | -- | 4.09830 | 3.08676 | 3.61108 |
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Alghanmi, R.A.; Aljaghthami, R.H. A Four-Variable Shear Deformation Theory for the Static Analysis of FG Sandwich Plates with Different Porosity Models. Math. Comput. Appl. 2024, 29, 20. https://doi.org/10.3390/mca29020020
Alghanmi RA, Aljaghthami RH. A Four-Variable Shear Deformation Theory for the Static Analysis of FG Sandwich Plates with Different Porosity Models. Mathematical and Computational Applications. 2024; 29(2):20. https://doi.org/10.3390/mca29020020
Chicago/Turabian StyleAlghanmi, Rabab A., and Rawan H. Aljaghthami. 2024. "A Four-Variable Shear Deformation Theory for the Static Analysis of FG Sandwich Plates with Different Porosity Models" Mathematical and Computational Applications 29, no. 2: 20. https://doi.org/10.3390/mca29020020