Investigating Electromechanical Buckling Response of FG-GPL-Reinforced Piezoelectric Doubly Curved Shallow Shells Embedded in an Elastic Substrate
Abstract
:1. Introduction
2. Problem Formulation
2.1. Shell Configuration
2.2. Displacement Field
2.3. Constitutive Relations
3. Governing Equations
4. Solution Procedure
5. Numerical Anaylsis and Discussions
6. Conclusions
- The GPL weight fraction and GPL distribution types significantly impact the stiffness as well as the dynamical characteristics of structures of the GPLs/piezoelectric nanocomposite doubly curved shallow shells. The GPLs improve high-strength and multifunctional nanocomposite materials. The results emphasize that the U-GPL type has the best mechanical characteristics, while the O-FG type has the weakest stiffness.
- An increase in the elastic stiffness and the aspect ratio leads to an increase in the critical buckling load.
- The sensitivity performance of the critical buckling load of GPLs/piezoelectric nanocomposite doubly curved shallow shells without elastic foundations is reduced by increasing the external electric voltage.
- The critical buckling loads noticeably depend on the dimensions of the shells. They increase as the shallowness ratio and the side-to-thickness ratio increase. Moreover, for small values of the shallowness ratio, the buckling load F may be independent of it.
- Increasing the graphene weight fraction enhances the plate stiffness and this leads to a noticeable increase in the critical buckling load.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
- Salehi-Khojin, A.; Jalili, N. Buckling of boron nitride nanotube reinforced piezoelectric polymeric composites subject to combined electro-thermo-mechanical loadings. Compos. Sci. Technol. 2008, 68, 1489–1501. [Google Scholar] [CrossRef]
- Sun, J.; Xu, X.; Lim, C.W.; Zhou, Z.; Xiao, S. Accurate thermo-electro-mechanical buckling of shear deformable piezoelectric fiber-reinforced composite cylindrical shells. Compos. Struct. 2016, 141, 221–231. [Google Scholar] [CrossRef]
- Tan, P.; Tong, L. Modeling for the electro-magneto-thermo-elastic properties of piezoelectric-magnetic fiber reinforced composites. Compos. Part A Appl. Sci. Manuf. 2002, 33, 631–645. [Google Scholar] [CrossRef]
- Potts, J.R.; Dreyer, D.R.; Bielawski, C.W.; Ruoff, R.S. Graphene-based polymer nanocomposites. Polymer 2011, 52, 5–25. [Google Scholar] [CrossRef] [Green Version]
- Abazid, M.A. The nonlocal strain gradient theory for hygrothermo-electromagnetic effects on buckling, vibration and wave propagation in piezoelectromagnetic nanoplates. Inter. J. Appl. Mech. 2019, 11, 1950067. [Google Scholar] [CrossRef]
- Sobhy, M. Piezoelectric bending of GPL-reinforced annular and circular sandwich nanoplates with FG porous core integrated with sensor and actuator using DQM. Arch. Civil Mech. Eng. 2021, 21, 78. [Google Scholar] [CrossRef]
- Polley, C.; Distler, T.; Detsch, R.; Lund, H.; Springer, A.; Boccaccini, A.R.; Seitz, H. 3D printing of piezoelectric barium titanate-hydroxyapatite scaffolds with interconnected porosity for bone tissue engineering. Materials 2020, 13, 1773. [Google Scholar] [CrossRef]
- Abouelregal, A.E.; Ahmad, H.; Yao, S.W. Functionally graded piezoelectric medium exposed to a movable heat flow based on a heat equation with a memory-dependent derivative. Materials 2020, 13, 3953. [Google Scholar] [CrossRef] [PubMed]
- Toron, B.; Szperlich, P.; Koziol, M. SbSI composites based on epoxy resin and cellulose for energy harvesting and sensors—The influence of SBSI nanowires conglomeration on piezoelectric properties. Materials 2020, 13, 902. [Google Scholar] [CrossRef] [Green Version]
- Wu, C.C.M.; Kahn, M.; Moy, W. Piezoelectric ceramics with functional gradients: A new application in material design. J. Am. Ceram. Soc. 1996, 79, 809–812. [Google Scholar] [CrossRef]
- El Harti, K.; Rahmoune, M.; Sanbi, M.; Saadani, R. Bentaleb, M.; Rahmoune, M. Dynamic control of euler bernoulli FG porous beam under thermal loading with bonded piezoelectric materials. Ferroelectrics 2020, 558, 104–116. [Google Scholar] [CrossRef]
- Mallek, H.; Jrad, H.; Wali, M.; Dammak, F. Nonlinear dynamic analysis of piezoelectric-bonded FG-CNTR composite structures using an improved FSDT theory. Eng. Comp. 2021, 37, 1389–1407. [Google Scholar] [CrossRef]
- Sobhy, M.; Al Mukahal, F.H.H. Magnetic control of vibrational behavior of smart FG sandwich plates with honeycomb core via a quasi-3D plate theory. Adv. Eng. Mater. 2023. [Google Scholar] [CrossRef]
- Garg, A.; Chalak, H.D.; Belarbi, M.O.; Zenkour, A.M. Hygro-thermo-mechanical based bending analysis of symmetric and unsymmetric power-law, exponential and sigmoidal FG sandwich beams. Mech. Advan Mater. Struct. 2021, 29, 4523–4545. [Google Scholar] [CrossRef]
- Meyer, J.C.; Geim, A.K.; Katsnelson, M.I.; Novoselov, K.S.; Booth, T.J.; Roth, S. The structure of suspended graphene sheets. Nature 2007, 446, 60–63. [Google Scholar] [CrossRef] [Green Version]
- Rafiee, M.A.; Rafiee, J.; Yu, Z.Z.; Koratkar, N. Buckling resistant graphene nanocomposites. Appl. Phys. Lett. 2009, 95, 223103. [Google Scholar] [CrossRef]
- Sobhy, M.; Abazid, M.A. Dynamic and instability analyses of FG graphene-reinforced sandwich deep curved nanobeams with viscoelastic core under magnetic field effect. Compo Part B Eng. 2019, 174, 106966. [Google Scholar] [CrossRef]
- Abazid, M.A. 2D magnetic field effect on the thermal buckling of metal foam nanoplates reinforced with FG-GPLs lying on pasternak foundation in humid environment. Euro. Phys. J. Plus 2020, 135, 910. [Google Scholar] [CrossRef]
- Abbasipour, M.; Khajavi, R.; Yousefi, A.A.; Yazdanshenas, M.E.; Razaghian, F. The piezoelectric response of electrospun PVDF nanofibers with graphene oxide, graphene, and halloysite nanofillers: A comparative study. J. Mater. Sci. Mater. Elect. 2017, 28, 15942–15952. [Google Scholar] [CrossRef]
- Liao, Y.; Li, Z.; Xia, W. Size-dependent structural behaviors of crumpled graphene sheets. Carbon 2021, 174, 148–157. [Google Scholar] [CrossRef]
- Maity, N.; Mandal, A.; Nandi, A.K. Hierarchical nanostructured polyaniline functionalized graphene/poly (vinylidene fluoride) composites for improved dielectric performances. Polymer 2016, 103, 83–97. [Google Scholar] [CrossRef]
- Olabi, A.G.; Abdelkareem, M.A.; Wilberforce, T.; Sayed, E.T. Application of graphene in energy storage device—A review. Renew. Sustain. Energy Rev. 2021, 135, 110026. [Google Scholar] [CrossRef]
- Mao, J.J.; Zhang, W. Linear and nonlinear free and forced vibrations of graphene reinforced piezoelectric composite plate under external voltage excitation. Compos. Struct. 2018, 203, 551–565. [Google Scholar] [CrossRef]
- Mao, J.J.; Zhang, W. Buckling and post-buckling analyses of functionally graded graphene reinforced piezoelectric plate subjected to electric potential and axial forces. Compos. Struct. 2019, 216, 392–405. [Google Scholar] [CrossRef]
- Sobhani, E.; Avcar, M. Natural frequency analysis of imperfect GNPRN conical shell, cylindrical shell, and annular plate structures resting on Winkler-Pasternak Foundations under arbitrary boundary conditions. Eng. Anal. Bound. Elem. 2022, 144, 145–164. [Google Scholar] [CrossRef]
- Salmani, R.; Gholami, R.; Ansari, R.; Fakhraie, M. Analytical investigation on the nonlinear postbuckling of functionally graded porous cylindrical shells reinforced with graphene nanoplatelets. Euro. Phys. J. Plus 2021, 136, 53. [Google Scholar] [CrossRef]
- Wang, M.; Xu, Y.-G.; Qiao, P.; Li, Z.-M. Buckling and free vibration analysis of shear deformable graphene-reinforced composite laminated plates. Compos. Struct. 2022, 280, 114854. [Google Scholar] [CrossRef]
- Wang, Y.; Feng, C.; Santiuste, C.; Zhao, Z.; Yang, J. Buckling and postbuckling of dielectric composite beam reinforced with graphene platelets (gpls). Aerosp. Sci. Technol. 2019, 91, 208–218. [Google Scholar] [CrossRef]
- Wang, Y.; Feng, C.; Yang, J.; Zhou, D.; Liu, W. Static response of functionally graded graphene platelet–reinforced composite plate with dielectric property. J. Intell. Mater. Syst. Struct. 2020, 31, 2211–2228. [Google Scholar] [CrossRef]
- Duc, N.D.; Quan, T.Q.; Laut, V.D. Nonlinear dynamic analysis and vibration of shear deformable piezoelectric fgm double curved shallow shells under damping-thermo-electro-mechanical loads. Compos. Struct. 2015, 125, 29–40. [Google Scholar] [CrossRef]
- Kiani, Y.; Akbarzadeh, A.H.; Chen, Z.T.; Eslami, M.R. Static and dynamic analysis of an fgm doubly curved panel resting on the pasternak-type elastic foundation. Compos. Struct. 2012, 94, 2474–2484. [Google Scholar] [CrossRef]
- Amabili, M. A new third-order shear deformation theory with non-linearities in shear for static and dynamic analysis of laminated doubly curved shells. Compos. Struct. 2015, 128, 260–273. [Google Scholar] [CrossRef] [Green Version]
- Sobhy, M. Magneto-electro-thermal bending of fg-graphene reinforced polymer doubly-curved shallow shells with piezoelectromagnetic faces. Compos. Struct. 2018, 203, 844–860. [Google Scholar] [CrossRef]
- Esmaeili, H.R.; Kiani, Y. Vibrations of graphene platelet reinforced composite doubly curved shells subjected to thermal shock. Mech. Based Des. Struct. Mach. 2022. [Google Scholar] [CrossRef]
- Ngoc, V.; Hoangc, V.; Tien, N.D.; Ninh, D.G.; Thang, V.T.; Truong, D.V. Nonlinear dynamics of functionally graded graphene nanoplatelet reinforced polymer doubly-curved shallow shells resting on elastic foundation using a micromechanical model. J. Sandw. Struct. Mater. 2021, 23, 3250–3279. [Google Scholar]
- Karimiasl, M.; Ebrahimi, F.; Akgöz, B. Buckling and post-buckling responses of smart doubly curved composite shallow shells embedded in sma fiber under hygro-thermal loading. Compos. Struct. 2019, 223, 110988. [Google Scholar] [CrossRef]
- Salehipour, H.; Emadi, S.; Tayebikhorami, S.; Shahmohammadi, M.A. A semi-analytical solution for dynamic stability analysis of nanocomposite/fibre-reinforced doubly-curved panels resting on the elastic foundation in thermal environment. Eur. Phys. J. Plus 2022, 137, 2. [Google Scholar] [CrossRef]
- Feng, S.; Kitipornchai, C.; Yang, J. Nonlinear free vibration of functionally graded polymer composite beams reinforced with graphene nanoplatelets (gpls). Eng. Struct. 2017, 140, 110–119. [Google Scholar] [CrossRef]
- Song, S.; Kitipornchai, M.; Yang, J. Free and forced vibrations of functionally graded polymer composite plates reinforced with graphene nanoplatelets. Compos. Struct. 2017, 159, 579–588. [Google Scholar] [CrossRef]
- Halpin, J.C.; Kardos, J.L. The halpin-tsai equations: A review. Polym. Eng. Sci. 1976, 16, 344–352. [Google Scholar]
- Sobhy, M.; Zenkour, A.M. Vibration analysis of functionally graded graphene platelet-reinforced composite doubly-curved shallow shells on elastic foundations. Steel Compos. Struct. 2019, 33, 195–208. [Google Scholar]
- Al Mukahal, F.H.H.; Sobhy, M. Wave propagation and free vibration of fg graphene platelets sandwich curved beam with auxetic core resting on viscoelastic foundation via dqm. Arch. Civ. Mech. Eng. 2022, 22, 12. [Google Scholar] [CrossRef]
- Sobhy, M. Analytical buckling temperature prediction of fg piezoelectric sandwich plates with lightweight core. Mater. Res. Express 2021, 8, 095704. [Google Scholar] [CrossRef]
- Sobhy, M. Stability analysis of smart FG sandwich plates with auxetic core. Int. J. Appl. Mech. 2021, 13, 2150093. [Google Scholar] [CrossRef]
- Oktem, A.S.; Mantari, J.L.; Soares, C.G. Static response of functionally graded plates and doubly-curved shells based on a higher order shear deformation theory. Eur. J. -Mech.-A/Solids 2012, 36, 163–172. [Google Scholar] [CrossRef]
- Reddy, J.N.; Liu, C.F. A higher-order shear deformation theory of laminated elastic shells. Int. J. Eng. Sci. 1985, 23, 319–330. [Google Scholar] [CrossRef]
- Zhang, S.; Xia, R.; Lebrun, L.; Anderson, D.; Shrout, T.R. Piezoelectric materials for high power, high temperature applications. Mater. Lett. 2005, 59, 3471–3475. [Google Scholar] [CrossRef]
- Meyers, C.A.; Hyer, M.W. Thermal buckling and postbuckling of symmetrically laminated composite plates. J. Therm. Stress. 1991, 14, 519–540. [Google Scholar] [CrossRef]
- Shariat, B.A.S.; Eslami, M.R. Buckling of thick functionally graded plates under mechanical and thermal loads. Compos. Struct. 2007, 78, 433–439. [Google Scholar] [CrossRef]
- Sobhy, M. Size-dependent hygro-thermal buckling of porous fgm sandwich microplates and microbeams using a novel four-variable shear deformation theory. Int. J. Appl. Mech. 2020, 12, 2050017. [Google Scholar] [CrossRef]
- Matsunaga, H. Vibration and stability of thick simply supported shallow shells subjected to in-plane stresses. J. Sound Vib. 1999, 225, 41–60. [Google Scholar] [CrossRef]
- Matsunaga, H. Free vibration and stability of functionally graded shallow shells according to a 2D higher-order deformation theory. Compos. Struct. 2008, 84, 132–146. [Google Scholar] [CrossRef]
Load Type | Ref. [51] | Present | ||
---|---|---|---|---|
Uniaxial | 0 | 0 | 3.7412 | 3.7866 |
0.2 | 0.2 | 4.1630 | 4.2350 | |
0.2 | 0 | 3.8391 | 3.8987 | |
0.2 | −0.2 | 3.7100 | 3.7866 | |
Biaxial | 0 | 0 | 1.8706 | 1.8933 |
0.2 | 0.2 | 2.0815 | 2.1175 | |
0.2 | 0 | 1.9195 | 1.9493 | |
0.2 | −0.2 | 1.8550 | 1.8933 |
Type | Ref. [52] | Present | ||||
---|---|---|---|---|---|---|
CST | FSDT | HST | ||||
2 | Plate | 0 | 0.64040 | 0.34790 | 0.35810 | 0.37947 |
Sph. | 0.5 | 0.64460 | 0.36270 | 0.36790 | 0.40480 | |
1 | 0.65560 | 0.40230 | 0.39480 | 0.48079 | ||
Cyl. | 0.5 | 0.61430 | 0.34440 | 0.35310 | 0.38580 | |
1 | 0.56850 | 0.33560 | 0.34030 | 0.40480 | ||
Hyp. | 0.5 | 0.56440 | 0.32200 | 0.33110 | 0.37947 | |
5 | Plate | 0 | 0.13570 | 0.11300 | 0.11400 | 0.11808 |
Sph. | 0.5 | 0.15610 | 0.13430 | 0.13410 | 0.14341 | |
1 | 0.21190 | 0.19270 | 0.18950 | 0.21940 | ||
Cyl. | 0.5 | 0.13890 | 0.11680 | 0.11740 | 0.12441 | |
1 | 0.14760 | 0.12710 | 0.12690 | 0.14341 | ||
Hyp. | 0.5 | 0.12840 | 0.10700 | 0.10790 | 0.11808 | |
10 | Plate | 0 | 0.03557 | 0.03372 | 0.03381 | 0.03423 |
Sph. | 0.5 | 0.05921 | 0.05774 | 0.05720 | 0.05956 | |
1 | 0.12420 | 0.12260 | 0.12160 | 0.13555 | ||
Cyl. | 0.5 | 0.04104 | 0.03924 | 0.03924 | 0.04056 | |
1 | 0.05625 | 0.05457 | 0.05435 | 0.05956 | ||
Hyp. | 0.5 | 0.03381 | 0.03205 | 0.03214 | 0.03423 |
Uniaxial | Biaxial | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
U-GPLs | X-FG | V-FG | O-FG | U-GPLs | X-FG | V-FG | O-FG | |||
5 | 0.5 | 1 | 6.4683 | 6.1167 | 5.9348 | 5.8803 | 1.9903 | 1.8821 | 1.8261 | 1.0691 |
0.5 | −1 | 6.2759 | 6.0204 | 5.8394 | 5.7840 | 1.9311 | 1.8524 | 1.7968 | 1.0516 | |
0.5 | 0 | 6.3507 | 6.0578 | 5.8764 | 5.8215 | 1.9541 | 1.8640 | 1.8081 | 1.0585 | |
0 | 0 | 6.2425 | 6.0037 | 5.8232 | 5.7673 | 1.9208 | 1.8473 | 1.7918 | 1.0486 | |
10 | 0.5 | 1 | 10.1233 | 8.0287 | 7.7431 | 7.6875 | 3.1149 | 2.4704 | 2.3825 | 1.3977 |
0.5 | −1 | 6.6440 | 6.2881 | 6.0095 | 5.9469 | 2.0443 | 1.9348 | 1.8491 | 1.0813 | |
0.5 | 0 | 6.9433 | 6.4378 | 6.1582 | 6.0966 | 2.1364 | 1.9809 | 1.8948 | 1.1085 | |
0 | 0 | 6.5104 | 6.2212 | 5.9437 | 5.8801 | 2.0032 | 1.9142 | 1.8288 | 1.0691 | |
20 | 0.5 | 1 | 21.7943 | 13.9199 | 13.5845 | 13.5413 | 6.7059 | 4.2831 | 4.1798 | 2.4621 |
0.5 | −1 | 7.8771 | 6.9575 | 6.6422 | 6.5788 | 2.4237 | 2.1408 | 2.0438 | 1.1961 | |
0.5 | 0 | 9.0743 | 7.5564 | 7.2385 | 7.1777 | 2.7921 | 2.3250 | 2.2272 | 1.3050 | |
0 | 0 | 7.3426 | 6.6901 | 6.3773 | 6.3114 | 2.2593 | 2.0585 | 1.9622 | 1.1475 | |
30 | 0.5 | 1 | 26.9927 | 16.5649 | 16.2176 | 16.1786 | 8.3054 | 5.0969 | 4.9900 | 2.9416 |
0.5 | −1 | 9.9048 | 8.0162 | 7.6922 | 7.6299 | 3.0476 | 2.4665 | 2.3668 | 1.3873 | |
0.5 | 0 | 12.5985 | 9.3638 | 9.0350 | 8.9775 | 3.8765 | 2.8812 | 2.7800 | 1.6323 | |
0 | 0 | 8.7023 | 7.4147 | 7.0946 | 7.0283 | 2.6776 | 2.2814 | 2.1829 | 1.2779 | |
40 | 0.5 | 1 | 36.2940 | 21.2738 | 20.9154 | 20.8847 | 11.1674 | 6.5458 | 6.4355 | 3.7972 |
0.5 | −1 | 12.7400 | 9.4903 | 9.1618 | 9.1012 | 3.9200 | 2.9201 | 2.8190 | 1.6548 | |
0.5 | 0 | 17.5287 | 11.8859 | 11.5502 | 11.4969 | 5.3934 | 3.6572 | 3.5539 | 2.0903 | |
0 | 0 | 10.6021 | 8.4208 | 8.0981 | 8.0317 | 3.2622 | 2.5910 | 2.4917 | 1.4603 |
Uniaxial | Biaxial | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
U-GPLs | X-FG | V-FG | O-FG | U-GPLs | X-FG | V-FG | O-FG | |||
0.1 | 0.5 | 1 | 7.4137 | 6.6731 | 6.3923 | 6.3319 | 2.2811 | 2.0533 | 1.9669 | 1.9483 |
0.5 | −1 | 6.6440 | 6.2881 | 6.0095 | 5.9469 | 2.0443 | 1.9348 | 1.8491 | 1.8298 | |
0.5 | 0 | 6.9433 | 6.4378 | 6.1582 | 6.0966 | 2.1364 | 1.9809 | 1.8948 | 1.8759 | |
0 | 0 | 6.5104 | 6.2212 | 5.9437 | 5.8801 | 2.0032 | 1.9142 | 1.8288 | 1.8093 | |
0.2 | 0.5 | 1 | 9.0369 | 7.6087 | 7.0641 | 6.9572 | 2.7806 | 2.3411 | 2.1736 | 2.1407 |
0.5 | −1 | 7.5612 | 6.8805 | 6.3438 | 6.2290 | 2.3265 | 2.1171 | 1.9519 | 1.9166 | |
0.5 | 0 | 8.1351 | 7.1637 | 6.6233 | 6.5122 | 2.5031 | 2.2042 | 2.0379 | 2.0037 | |
0 | 0 | 7.3050 | 6.7541 | 6.2212 | 6.1026 | 2.2477 | 2.0782 | 1.9142 | 1.8777 | |
0.3 | 0.5 | 1 | 10.5266 | 8.4577 | 7.6661 | 7.5214 | 3.2389 | 2.6024 | 2.3588 | 2.3143 |
0.5 | −1 | 8.3942 | 7.4157 | 6.6401 | 6.4794 | 2.5828 | 2.2818 | 2.0431 | 1.9937 | |
0.5 | 0 | 9.2235 | 7.8210 | 7.0379 | 6.8846 | 2.8380 | 2.4065 | 2.1655 | 2.1183 | |
0 | 0 | 8.0240 | 7.2348 | 6.4669 | 6.2985 | 2.4689 | 2.2261 | 1.9898 | 1.9380 | |
0.4 | 0.5 | 1 | 11.9005 | 9.2328 | 8.2095 | 8.0336 | 3.6617 | 2.8409 | 2.5260 | 2.4719 |
0.5 | −1 | 9.1558 | 7.9025 | 6.9049 | 6.7033 | 2.8172 | 2.4315 | 2.1246 | 2.0626 | |
0.5 | 0 | 10.2232 | 8.4198 | 7.4103 | 7.2207 | 3.1456 | 2.5907 | 2.2801 | 2.2217 | |
0 | 0 | 8.6793 | 7.6715 | 6.6861 | 6.4724 | 2.6706 | 2.3605 | 2.0573 | 1.9915 | |
0.5 | 0.5 | 1 | 13.1732 | 9.9438 | 8.7028 | 8.5011 | 4.0533 | 3.0596 | 2.6778 | 2.6157 |
0.5 | −1 | 9.8561 | 8.3477 | 7.1431 | 6.9050 | 3.0327 | 2.5685 | 2.1979 | 2.1246 | |
0.5 | 0 | 11.1461 | 8.9684 | 7.7470 | 7.5257 | 3.4296 | 2.7595 | 2.3837 | 2.3156 | |
0 | 0 | 9.2803 | 8.0706 | 6.8832 | 6.6279 | 2.8555 | 2.4833 | 2.1179 | 2.0394 |
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Al Mukahal, F.H.H.; Abazid, M.A.; Sobhy, M. Investigating Electromechanical Buckling Response of FG-GPL-Reinforced Piezoelectric Doubly Curved Shallow Shells Embedded in an Elastic Substrate. Materials 2023, 16, 2975. https://doi.org/10.3390/ma16082975
Al Mukahal FHH, Abazid MA, Sobhy M. Investigating Electromechanical Buckling Response of FG-GPL-Reinforced Piezoelectric Doubly Curved Shallow Shells Embedded in an Elastic Substrate. Materials. 2023; 16(8):2975. https://doi.org/10.3390/ma16082975
Chicago/Turabian StyleAl Mukahal, Fatemah H. H., Mohammad Alakel Abazid, and Mohammed Sobhy. 2023. "Investigating Electromechanical Buckling Response of FG-GPL-Reinforced Piezoelectric Doubly Curved Shallow Shells Embedded in an Elastic Substrate" Materials 16, no. 8: 2975. https://doi.org/10.3390/ma16082975