In the present work, the doubly curved spherical and cylindrical laminated composite shells are modelled and analysed in the framework of non-polynomial axiomatic approach. The inverse hyperbolic shear deformation theory originally developed for the laminated composite plates is extended to model the deformation characteristics of laminated composite shells. The mathematical formulation is developed under the assumption of linear structural kinematics and linear-elastic-orthotropic material behaviour. The governing equations of the model are obtained using the principle of virtual work and solved in exact manner for simply supported boundary conditions following the Navier solution methodology. The bending response of thick and thin spherical and cylindrical shells subjected to different types of transverse loads such as point load, uniform load and sinusoidal load is analysed in the framework of developed methodology. The obtained results due to inverse hyperbolic shear deformation theory are compared with other shell theories and on the basis of this comparison, the validity and applicability of the inverse hyperbolic shear deformation theory for doubly curved spherical and cylindrical shells is ensured.
LoveAEH.The small free vibrations and deformation of a thin elastic shell. Phil Trans Royal Soc London1888;
179: 491–546.
2.
HildebrandFBReissnerEThomasGB. Notes on the foundations of small displacements of orthotropic shells. Technical Note NACA, Report no. 1833, 1949.
3.
NaghdiPM.On the theory of thin elastic shells. Quart Appl Math1957;
14: 369–380.
4.
ReissnerE.On a variational theorem in elasticity. J Math Phys1950;
29: 90–95.
5.
DongSBTsoFKW.On a laminated orthotropic shell theory including transverse shear deformation. ASME J Appl Mech1972;
39: 1091–1097.
6.
WhitneyJMSunCT.A refined theory for laminated anisotropic cylindrical shells. ASME J Appl Mech1974;
41: 471–476.
7.
KantTRameshCK. Analysis of thick orthotropic shells. In: Proceedings of IASS world congress on space enclosures, Montreal, Canada, 4–9 July 1976, pp. 401–409.
8.
WideraGEOLoganDL.Refined theories for non-homogeneous anisotropic cylindrical Shells. ASCE J Eng Mech1980;
106: 1053–1090.
9.
ReddyJN.Exact solutions of moderately thick laminated shells. ASCE J Eng Mech1984;
110: 794–809.
10.
KantT. A higher-order general shell theory. In: Proceedings of 32nd congress of the Indian society of theoretical and applied mechanics, IIT-Bombay, Mumbai, India, 17–20 December 1987.
11.
ReddyJNLiuCF.A higher order shear deformation theory of laminated elastic shells. Int J Eng Sci1985;
23: 319–330.
12.
BhimaraddiA.Three-dimensional elasticity solution for static response of orthotropic doubly curved shallow shells on rectangular platform. Compos Struct1993;
24: 67–77.
13.
WuCPTarnJQChiSM.Three-dimensional analysis of doubly curved laminated shells. J Eng Mech1996;
122: 391–401.
14.
BhimaraddiAStevensLK.On the higher order theories in plates and shells. Int J Solid Struct1986;
6: 35–50.
15.
KrishnamurthyAVReddyTSR.A higher order theory for laminated composite cylindrical shells. J Aero Soc India1986,
38: 161–171.
16.
NoorAKBurtonWS.Assessment of computational models for multilayered composite shells. Appl Mech Rev1990;
43: 67–97.
17.
BuiTQDoTVTonLHT, et al.
On the high temperature mechanical behaviors analysis of heated functionally graded plates using FEM and a new third-order shear deformation plate theory. Compos Part B: Eng2016;
92: 218–241.
18.
YinSYuTBuiTQ, et al.
In-plane material inhomogeneity of functionally graded plates: a higher-order shear deformation plate isogeometric analysis. Compos Part B: Eng2016;
106: 273–284.
19.
Van DoTNguyenDKDucND, et al.
Analysis of bi-directional functionally graded plates by FEM and a new third-order shear deformation plate theory. Thin-Walled Struct2017;
119: 687–699.
20.
Sanders Jr JL. An improved first-approximation theory for thin shells. NASA TR-R24, 1959.
21.
TouratierM.An efficient standard plate theory. Int J Eng Sci1991;
29: 901–916.
22.
SoldatosKP.A transverse shear deformation theory for homogeneous monoclinic plates. Acta Mech1992;
94: 195–220.
23.
KaramaMAfaqKSMistouS.A new theory for laminated composite plates. Proc IMechE, Part L: J Mater Des Appl2009;
223: 53–62.
24.
MantariJLOktemASSoaresCG.Static and dynamic analysis of laminated composite and sandwich plates and shells by using a new higher-order shear deformation theory. Compos Struct2011;
94: 37–49.
25.
MantariJLOktemASSoaresCG.A new trigonometric deformation theory for isotropic, laminated composite and sandwich plates. Int J Solids Struct2012;
49: 43–53.
26.
MantariJLSoaresCG.Bending analysis of thick exponentially graded plates using a new trigonometric higher order shear deformation theory. Compos Struct2012;
94: 1991–2000.
27.
GroverNMaitiDKSinghBN.A new inverse hyperbolic shear deformation theory for static and buckling analysis of laminated composite and sandwich plates. Compos Struct2013;
95: 667–675.
28.
GroverNSinghBNMaitiDK.Free vibration and buckling characteristics of laminated composite and sandwich plates implementing a secant function based shear deformation theory.Proc IMechE, Part C: J Mechanical Engineering Science2014;
229: 391–406.
29.
MantariJLBonillaEMSoaresCG.A new tangential-exponential higher order shear deformation theory for advanced composite plates. Compos Part B: Eng2014;
60: 319–328.
30.
JoshanYSGroverNSinghBN.A new non-polynomial four variable shear deformation theory in axiomatic formulation for hygro-thermo-mechanical analysis of laminated composite plates. Compos Struct2017;
182: 685–693.
31.
JoshanYSSantapuriSGroverN.Analysis of laminated piezoelectric composite plates using an inverse hyperbolic coupled plate theory. Appl Math Model2020;
82: 359–378.
32.
JoshanYSSharmaLKGroverN, et al.A couple stress model in non-polynomial framework to examine structural responses of laminated composite micro-plates: an analytical solution. Compos Struct2021;
261: 113533.
33.
GiuntaGBiscaniFBelouettarSCarreraE.Hierarchical modelling of doubly curved laminated composite shells under distributed and localised loadings. Compos Part B: Eng2011;
42: 682–691.
ChaudhuriRAKabirHRH.Fourier solution to higher order theory based laminated shell boundary-value problem. AIAA J1995;
33: 1681–1688.
38.
KhareRKKantTGargAK.Closed-form thermo-mechanical solutions of higher-order theories of cross-ply laminated shallow shells. Compos Struct2003;
59: 313–340.
39.
OktemSChaudhuriAR.Levy type Fourier analysis of thick cross-ply doubly curved panels. Compos Struct2007;
80: 445–488.
40.
PandaSKSinghBN.Non-linear free vibration analysis of laminated composite cylindrical/hyperboloid shell panels based on higher-order shear deformation theory using non-linear finite-element method. Proc IMechE, Part G: J Aerospace Engineering2008;
222: 993–1006.
41.
SahuSKRathMKSahooR.Parametric instability of laminated composite doubly curved shell panels subjected to hygrothermal environment. Adv Mater Res2012;
383: 3212–3216.
42.
OktemASSoaresCG.Higher order theory-based Fourier analysis of cross-ply plates and doubly curved panels. J Compos Mater2011;
46: 2675–2694.
43.
KhaliliSMRTafazoliSFardKM.Free vibrations of laminated composite shells with uniformly distributed attached mass using higher order shell theory including stiffness effect. J Sound Vib2011;
330: 6355–6371.
44.
GargAKKhareRKKantT.Higher-order closed-form solutions for free vibration of laminated composite and sandwich shells. J Sandw Struct Mater2006;
8: 205–235.
45.
MantariJLOktemASSoaresCG.A new higher order shear deformation theory for sandwich and composite laminated plates. Compos Part B: Eng2012;
43: 1489–1499.
46.
KumarABhargavaPChakrabartiA.Vibration of laminated composite skew hypar shells using higher order theory. Thin-Walled Struct2013;
63: 82–90.
47.
TornabeneFFantuzziNBacciocchiM.On the mechanics of laminated doubly-curved shells subjected to point and line loads. Int J Eng Sci2006;
109: 115–164.
48.
SinghVKMahapatraTRPandaSK.Nonlinear flexural analysis of single/doubly curved smart composite shell panels integrated with PFRC actuator. Euro J Mech A Solids2016;
60: 300–314.
49.
SadamotoSOzdemirMTanakaS, et al.
An effective meshfree reproducing kernel method for buckling analysis of cylindrical shells with and without cutouts. Comput Mech2017;
59: 919–932.
50.
SadamotoSTanakaSTaniguchiK, et al.
Buckling analysis of stiffened plate structures by an improved meshfree flat shell formulation. Thin-Walled Struct2017;
117: 303–313.
51.
YoshidaKSadamotoSSetoyamaY, et al.
Meshfree flat-shell formulation for evaluating linear buckling loads and mode shapes of structural plate. J Marine Sci Technol2017;
22: 501–512.
52.
OzdemirMTanakaSSadamotoS, et al.
Numerical buckling analysis for flat and cylindrical shells including through crack employing effective reproducing kernel meshfree modelling. Eng Anal Bound Elem2018;
97: 55–66.
53.
TanakaSDaiMJSadamotoS, et al.
Stress resultant intensity factors evaluation of cracked folded structures by 6DOFs flat shell meshfree modelling.Thin-Walled Struct2019;
144: 106285.
54.
ShahPHBatraRC.Stress singularities and transverse stresses near edges of doubly curved laminated shells using TSNDT and stress recovery scheme. Eur J Mech A Solids2017;
63: 68–83.
ShokriehMMParkestaniAN.Post buckling analysis of shallow composite shells based on the third order shear deformation theory. Aero Sci Technol2017;
66: 332–341.
57.
AmabiliM.Nonlinear vibrations and stability of laminated shells using a modified first order shear deformation theory. Eur J Mech A Solids2018;
68: 75–87.
58.
ReddyJN.Mechanics of laminated composite plates and shells. 2nd ed. Boca Raton, FL:
CRC Press, 2004.
59.
PaganoNJ.Exact solutions for rectangular bidirectional composite and sandwich plates. J Compos Mater1970;
4: 20–34.
