Stress and strain recovery for functionally graded free-form and doubly-curved sandwich shells using higher-order equivalent single layer theory

Abstract We investigate recovery of through-the-thickness transverse normal and shear strains and stresses in statically deformed functionally graded (FG) doubly-curved sandwich shell structures and shells of revolution using the generalized zigzag displacement field and the Carrera Unified Formulation (CUF). Three different through-the-thickness distributions of the volume fractions of constituents and two different homogenization techniques are employed to deduce the effective moduli of linear elastic isotropic materials. The system of partial differential equations for different Higher-order Shear Deformation Theories (HSDTs) is numerically solved by using the Generalized Differential Quadrature (GDQ) method. Either the face sheets or the core is assumed to be made of a FGM. The through-the-thickness stress profiles are recovered by integrating along the thickness direction the 3-dimensional (3-D) equilibrium equations written in terms of stresses. The stresses are used to find the strains by using Hooke’s law. The computed displacements and the recovered through-the-thickness stresses and strains are found to compare well with those obtained by analyzing the corresponding 3-D problems with the finite element method and a commercial code. The stresses for the FG structures are found to be in-between those for the homogeneous structures made of the two constituents of the FGM.

[1]  N. Kuruoglu,et al.  Buckling and vibration of shear deformable functionally graded orthotropic cylindrical shells under external pressures , 2014 .

[2]  A. Fereidoon,et al.  Bending analysis of functionally graded annular sector plates by extended Kantorovich method , 2012 .

[3]  P. Weaver,et al.  Buckling analysis of variable angle tow, variable thickness panels with transverse shear effects , 2014 .

[4]  N. Rogacheva.  The Theory of Piezoelectric Shells and Plates , 1994 .

[5]  Alfred G. Striz,et al.  High-accuracy plane stress and plate elements in the quadrature element method , 1995 .

[6]  Moshe Eisenberger,et al.  Exact vibration frequencies of segmented axisymmetric shells , 2006 .

[7]  Arthur W. Leissa,et al.  Vibration of Plates , 2021, Solid Acoustic Waves and Vibration.

[8]  Guang Meng,et al.  A unified formulation for vibration analysis of functionally graded shells of revolution with arbitrary boundary conditions , 2013 .

[9]  W. Flügge.  Stresses in Shells , 1960 .

[10]  O. Anwar Bég,et al.  An efficient and simple higher order shear and normal deformation theory for functionally graded material (FGM) plates , 2014 .

[11]  J. R. Banerjee,et al.  Axiomatic/asymptotic PVD/RMVT-based shell theories for free vibrations of anisotropic shells using an advanced Ritz formulation and accurate curvature descriptions , 2014 .

[12]  S. Vel,et al.  Exact Solution for Thermoelastic Deformations of Functionally Graded Thick Rectangular Plates , 2002 .

[13]  J. N. Reddy,et al.  Winkler–Pasternak foundation effect on the static and dynamic analyses of laminated doubly-curved and degenerate shells and panels , 2014 .

[14]  Erasmo Carrera,et al.  Plates and Shells for Smart Structures: Classical and Advanced Theories for Modeling and Analysis , 2011 .

[15]  L. Dozio.  Exact free vibration analysis of Lévy FGM plates with higher-order shear and normal deformation theories , 2014 .

[17]  Chang Shu,et al.  Free vibration analysis of laminated composite cylindrical shells by DQM , 1997 .

[18]  Mohamad S. Qatu,et al.  Static and vibration analyses of thick deep laminated cylindrical shells using 3D and various shear deformation theories , 2012 .

[19]  Erasmo Viola,et al.  2-D solution for free vibrations of parabolic shells using generalized differential quadrature method , 2008 .

[20]  Zhu Su,et al.  A unified accurate solution for vibration analysis of arbitrary functionally graded spherical shell segments with general end restraints , 2014 .

[21]  M. M. Aghdam,et al.  A micromechanics based analysis of hollow fiber composites using DQEM , 2012 .

[22]  Romesh C. Batra,et al.  Three-dimensional analysis of transient thermal stresses in functionally graded plates , 2003 .

[23]  R. Batra.  Torsion of a Functionally Graded Cylinder , 2006 .

[24]  Guang Meng,et al.  A unified formulation for vibration analysis of composite laminated shells of revolution including shear deformation and rotary inertia , 2013 .

[25]  V. V. Novozhilov,et al.  Thin shell theory , 1964 .

[26]  Serge Abrate,et al.  Free vibration, buckling, and static deflections of functionally graded plates , 2006 .

[27]  Autar Kaw,et al.  Mechanics of Composite Materials , 1997 .

[28]  Erasmo Viola,et al.  Vibration Analysis of Damaged Circular Arches with Varying Cross-section , 2005 .

[29]  Nicholas Fantuzzi,et al.  Generalized differential quadrature finite element method for cracked composite structures of arbitrary shape , 2013 .

[30]  Philippe G. Ciarlet,et al.  Theory of Shells , 2000 .

[31]  Zhu Su,et al.  A unified solution for vibration analysis of functionally graded cylindrical, conical shells and annular plates with general boundary conditions , 2014 .

[32]  Erasmo Carrera,et al.  Static analysis of doubly-curved anisotropic shells and panels using CUF approach, differential geometry and differential quadrature method , 2014 .

[33]  I. I. Vorovich,et al.  Nonlinear Theory of Shallow Shells , 1999 .

[34]  Romesh C. Batra,et al.  Thickness shear vibrations of a circular cylindrical piezoelectric shell , 1995 .

[35]  C. Bert,et al.  The behavior of structures composed of composite materials , 1986 .

[36]  Francesco Tornabene,et al.  Free vibrations of laminated composite doubly-curved shells and panels of revolution via the GDQ method , 2011 .

[37]  P. M. Naghdi,et al.  THEORY OF ELASTIC THIN SHELLS , 1962 .

[38]  K. Tanaka,et al.  Average stress in matrix and average elastic energy of materials with misfitting inclusions , 1973 .

[39]  C. R. Calladine,et al.  Theory of Shell Structures , 1983 .

[40]  Reza Kolahchi,et al.  Nonlinear buckling response of embedded piezoelectric cylindrical shell reinforced with BNNT under electro–thermo-mechanical loadings using HDQM , 2013 .

[41]  K. Liew,et al.  A review of meshless methods for laminated and functionally graded plates and shells , 2011 .

[42]  Guoyong Jin,et al.  A general Fourier solution for the vibration analysis of composite laminated structure elements of revolution with general elastic restraints , 2014 .

[43]  Francesco Tornabene,et al.  Free vibration analysis of functionally graded conical, cylindrical shell and annular plate structures with a four-parameter power-law distribution , 2009 .

[44]  N. K. Srivastava.  Finite element analysis of shells of revolution , 1986 .

[45]  A. Leissa,et al.  Vibrations of continuous systems , 2011 .

[46]  Erasmo Carrera,et al.  Multilayered Shell Theories Accounting for Layerwise Mixed Description, Part 1: Governing Equations , 1999 .

[47]  N. Pagano,et al.  Exact Solutions for Rectangular Bidirectional Composites and Sandwich Plates , 1970 .

[48]  Alessandro Marzani,et al.  Critical Flow Speeds of Pipes Conveying Fluid Using the Generalized Differential Quadrature Method , 2010 .

[49]  Romesh C. Batra,et al.  Wave propagation in functionally graded materials by modified smoothed particle hydrodynamics (MSPH) method , 2007, J. Comput. Phys..

[50]  D. Chapelle,et al.  The Finite Element Analysis of Shells - Fundamentals , 2003 .

[51]  Guang Meng,et al.  Dynamic analysis of composite laminated and sandwich hollow bodies of revolution based on three-dimensional elasticity theory , 2014 .

[52]  Toshio Mura,et al.  Micromechanics of defects in solids , 1982 .

[53]  C. Bert,et al.  Differential Quadrature Method in Computational Mechanics: A Review , 1996 .

[54]  E. Viola,et al.  General higher-order shear deformation theories for the free vibration analysis of completely doubly-curved laminated shells and panels , 2013 .

[55]  Maenghyo Cho,et al.  Efficient higher-order shell theory for laminated composites , 1995 .

[56]  J. G. Simmonds,et al.  The Nonlinear Theory of Elastic Shells , 1998 .

[57]  E. Carrera,et al.  Effects of thickness stretching in functionally graded plates and shells , 2011 .

[58]  J. F. Doyle.  Thin Plates and Shells , 2020, Encyclopedia of Continuum Mechanics.

[59]  Š. Markuš,et al.  The mechanics of vibrations of cylindrical shells , 1988 .

[60]  R. Bellman,et al.  DIFFERENTIAL QUADRATURE AND LONG-TERM INTEGRATION , 1971 .

[61]  S. Kitipornchai,et al.  AXISYMMETRIC NONLINEAR FREE VIBRATION OF SIZE-DEPENDENT FUNCTIONALLY GRADED ANNULAR MICROPLATES , 2013 .

[62]  A. Sofiyev.  On the dynamic buckling of truncated conical shells with functionally graded coatings subject to a time dependent axial load in the large deformation , 2014 .

[63]  Erasmo Carrera,et al.  Analysis of thick isotropic and cross-ply laminated plates by generalized differential quadrature method and a Unified Formulation , 2014 .

[64]  L. Donnell,et al.  Beams, plates and shells , 1976, Finite Element Analysis.

[65]  Y. Benveniste,et al.  A new approach to the application of Mori-Tanaka's theory in composite materials , 1987 .

[66]  Phillip L. Gould,et al.  Analysis of Shells and Plates , 1987 .

[67]  A. Kalnins,et al.  Thin elastic shells , 1967 .

[68]  Zhi Zong,et al.  Advanced Differential Quadrature Methods , 2009 .

[69]  M. A. McCarthy,et al.  Analysis of thick functionally graded plates by using higher-order shear and normal deformable plate theory and MLPG method with radial basis functions , 2007 .

[70]  Francesco Tornabene,et al.  Modellazione e soluzione di strutture a guscio in materiale anisotropo , 2007 .

[71]  Alfredo Liverani,et al.  General anisotropic doubly-curved shell theory: A differential quadrature solution for free vibrations of shells and panels of revolution with a free-form meridian , 2012 .

[72]  Daniel J. Inman,et al.  2-D differential quadrature solution for vibration analysis of functionally graded conical, cylindrical shell and annular plate structures , 2009 .

[73]  S. Timoshenko,et al.  THEORY OF PLATES AND SHELLS , 1959 .

[74]  Francesco Tornabene.  Meccanica delle Strutture a Guscio in Materiale Composito. Il metodo Generalizzato di Quadratura Differenziale , 2012 .

[75]  R. Nascimbene,et al.  Towards Non–Standard Numerical Modeling of Thin-Shell Structures: Geometrically Linear Formulation , 2014 .

[76]  R. Batra,et al.  Higher-Order Piezoelectric Plate Theory Derived from a Three-Dimensional Variational Principle , 2002 .

[77]  Hui-Shen Shen,et al.  Nonlinear bending analysis of shear deformable functionally graded plates subjected to thermo-mechanical loads under various boundary conditions , 2003 .

[78]  Moshe Eisenberger,et al.  Dynamic stiffness vibration analysis of thick spherical shell segments with variable thickness , 2010 .

[79]  Erasmo Viola,et al.  FREE VIBRATIONS OF FOUR-PARAMETER FUNCTIONALLY GRADED PARABOLIC PANELS AND SHELLS OF REVOLUTION , 2009 .

[80]  Bo Liu,et al.  Exact characteristic equations for free vibrations of thin orthotropic circular cylindrical shells , 2012 .

[81]  C. Shu.  Differential Quadrature and Its Application in Engineering , 2000 .

[82]  E. Carrera.  Historical review of Zig-Zag theories for multilayered plates and shells , 2003 .

[83]  Abdullah H. Sofiyev,et al.  The influence of non-homogeneity on the frequency–amplitude characteristics of laminated orthotropic truncated conical shell , 2014 .

[84]  Mohamad S. Qatu,et al.  Vibration of Laminated Shells and Plates , 2004 .

[85]  Nicholas Fantuzzi,et al.  Strong formulation finite element method for arbitrarily shaped laminated plates - Part I. Theoretical analysis , 2014 .

[86]  Ali Fallah,et al.  Free vibration analysis of Mindlin plates partially resting on Pasternak foundation , 2013 .

[87]  Ahmed K. Noor,et al.  Predictor-corrector procedures for stress and free vibration analysis of multilayered composite plates and shells , 1990 .

[88]  R. Batra,et al.  Deflection relationships between the homogeneous Kirchhoff plate theory and different functionally graded plate theories , 2000 .

[89]  Gerald Wempner,et al.  Mechanics of Solids and Shells: Theories and Approximations , 2002 .

[90]  Elena Ferretti,et al.  Soft Core Plane State Structures Under Static Loads UsingGDQFEM and Cell Method , 2013 .

[91]  J. N. Reddy,et al.  Theory and analysis of elastic plates , 1999 .

[92]  Moshe Eisenberger,et al.  Exact vibration analysis of variable thickness thick annular isotropic and FGM plates , 2007 .

[93]  J. Douglas Faires,et al.  Numerical Analysis , 1981 .

[94]  Erasmo Viola,et al.  Free vibrations of three parameter functionally graded parabolic panels of revolution , 2009 .

[95]  Romesh C. Batra,et al.  Modified smoothed particle hydrodynamics method and its application to transient problems , 2004 .

[96]  Yoshinari Miyamoto,et al.  Functionally Graded Materials. , 1995 .

[97]  S. Vel,et al.  The generalized plane strain deformations of thick anisotropic composite laminated plates , 2000 .

[98]  Mohamad S. Qatu,et al.  Recent research advances on the dynamic analysis of composite shells: 2000-2009 , 2010 .

[99]  R. Batra,et al.  Static and dynamic deformations of thick functionally graded elastic plates by using higher-order shear and normal deformable plate theory and meshless local Petrov -Galerkin method , 2004 .

[100]  Alfredo Liverani,et al.  Static analysis of laminated composite curved shells and panels of revolution with a posteriori shear and normal stress recovery using generalized differential quadrature method , 2012 .

[101]  Nicholas Fantuzzi,et al.  Generalized Differential Quadrature Finite Element Method for vibration analysis of arbitrarily shaped membranes , 2014 .

[102]  Ashraf M. Zenkour,et al.  Analysis of Sandwich Plates by Generalized Differential Quadrature Method , 2013 .

[103]  A. Ceruti,et al.  Free-Form Laminated Doubly-Curved Shells and Panels of Revolution Resting on Winkler-Pasternak Elastic Foundations: A 2-D GDQ Solution for Static and Free Vibration Analysis , 2013 .

[104]  Nguyen Dinh Duc,et al.  Transient responses of functionally graded double curved shallow shells with temperature-dependent material properties in thermal environment , 2014 .

[105]  J. N. Reddy,et al.  FGM and Laminated Doubly-Curved and Degenerate Shells Resting on Nonlinear Elastic Foundations: A GDQ Solution for Static Analysis with a Posteriori Stress and Strain Recovery , 2013 .

[106]  R. Christensen,et al.  A HIGH-ORDER THEORY OF PLATE DEFORMATION, PART 1: HOMOGENEOUS PLATES , 1977 .

[107]  Alfred G. Striz,et al.  Static analysis of structures by the quadrature element method , 1994 .

[108]  Xinwei Wang,et al.  Accurate buckling loads of thin rectangular plates under parabolic edge compressions by the differential quadrature method , 2007 .

[109]  Alfredo Liverani,et al.  Laminated composite rectangular and annular plates: A GDQ solution for static analysis with a posteriori shear and normal stress recovery , 2012 .

[110]  Chang Shu,et al.  FREE VIBRATION ANALYSIS OF COMPOSITE LAMINATED CONICAL SHELLS BY GENERALIZED DIFFERENTIAL QUADRATURE , 1996 .

[111]  Z. Hashin.  Analysis of Composite Materials—A Survey , 1983 .

[112]  J. Reddy.  Mechanics of laminated composite plates and shells : theory and analysis , 1996 .

[113]  E. Viola,et al.  Static analysis of completely doubly-curved laminated shells and panels using general higher-order shear deformation theories , 2013 .

[114]  Chang Shu,et al.  An efficient approach for free vibration analysis of conical shells , 1996 .

[115]  Fiorenzo A. Fazzolari,et al.  A refined dynamic stiffness element for free vibration analysis of cross-ply laminated composite cylindrical and spherical shallow shells , 2014 .

[116]  Masoud Tahani,et al.  Transient dynamic and free vibration analysis of functionally graded truncated conical shells with non-uniform thickness subjected to mechanical shock loading , 2012 .

[117]  Nicholas Fantuzzi,et al.  Static analysis of functionally graded conical shells and panels using the generalized unconstrained third order theory coupled with the stress recovery , 2014 .

[118]  S. Vel,et al.  Three-dimensional exact solution for the vibration of functionally graded rectangular plates , 2004 .

[119]  Pizhong Qiao,et al.  On an exact bending curvature model for nonlinear free vibration analysis shear deformable anisotropic laminated beams , 2014 .

[120]  Trung Nguyen-Thoi,et al.  A cell-based smoothed discrete shear gap method (CS-FEM-DSG3) using layerwise theory based on the C0-HSDT for analyses of composite plates , 2014 .

[121]  Phillip L. Gould.  Thin Plates and Shells , 2013 .

[122]  S. A. Ambartsumyan,et al.  Theory of anisotropic shells , 1964 .

[123]  Erasmo Viola,et al.  Analytical and numerical results for vibration analysis of multi-stepped and multi-damaged circular arches , 2007 .

[124]  Alessandro Marzani,et al.  Nonconservative stability problems via generalized differential quadrature method , 2008 .

[125]  Nicholas Fantuzzi,et al.  Mechanics of Laminated Composite Doubly-Curved Shell Structures. The Generalized Differential Quadrature Method and the Strong Formulation Finite Element Method , 2014 .

[126]  S. Vel,et al.  Analytical Solution for Rectangular Thick Laminated Plates Subjected to Arbitrary Boundary Conditions , 1999 .

[127]  Erasmo Viola,et al.  Vibration analysis of conical shell structures using GDQ Method , 2006 .

[128]  Francesco Tornabene,et al.  Free vibrations of anisotropic doubly-curved shells and panels of revolution with a free-form meridian resting on Winkler–Pasternak elastic foundations , 2011 .

[129]  Erasmo Viola,et al.  Static analysis of functionally graded doubly-curved shells and panels of revolution , 2013 .

[130]  Abdullah H. Sofiyev,et al.  The combined influences of heterogeneity and elastic foundations on the nonlinear vibration of orthotropic truncated conical shells , 2014 .

[131]  Charles W. Bert,et al.  Free Vibration of Plates by the High Accuracy Quadrature Element Method , 1997 .

[132]  Nicholas Fantuzzi,et al.  Radial basis function method applied to doubly-curved laminated composite shells and panels with a General Higher-order Equivalent Single Layer formulation , 2013 .

[133]  J. N. Reddy,et al.  Energy principles and variational methods in applied mechanics , 2002 .

[134]  S. Hosseini-Hashemi,et al.  Accurate natural frequencies and critical speeds of a rotating functionally graded moderately thick cylindrical shell , 2013 .

[135]  S. K. Sahu,et al.  Effects of moisture on the frequencies of vibration of woven fibre composite doubly curved panels with strip delaminations , 2014 .

[136]  A. Leissa,et al.  Vibration of shells , 1973 .

[137]  Paul Seide,et al.  An approximate semi-analytical method for prediction of interlaminar shear stresses in an arbitrarily laminated thick plate , 1987 .

[138]  Francesco Tornabene,et al.  2-D GDQ solution for free vibrations of anisotropic doubly-curved shells and panels of revolution , 2011 .

[139]  Romesh C. Batra,et al.  Analysis of cylindrical bending thermoelastic deformations of functionally graded plates by a meshless local Petrov–Galerkin method , 2004 .

[140]  Guoyong Jin,et al.  Three-dimensional free vibration analysis of thick cylindrical shells with general end conditions and resting on elastic foundations , 2014 .

[141]  Romesh C. Batra,et al.  Free vibration of three-layer circular cylindrical shells with functionally graded middle layer , 2010 .

[142]  Romesh C. Batra,et al.  Symmetric smoothed particle hydrodynamics (SSPH) method and its application to elastic problems , 2009 .

[143]  Elena Ferretti,et al.  GDQFEM Numerical Simulations of Continuous Mediawith Cracks and Discontinuities , 2013 .

[144]  Hongguang Li,et al.  A variational formulation for dynamic analysis of composite laminated beams based on a general higher-order shear deformation theory , 2013 .

[145]  A. Saada.  Elasticity : theory and applications , 1993 .

[146]  Nicholas Fantuzzi,et al.  Numerical investigation of functionally graded cylindrical shells and panels using the generalized unconstrained third order theory coupled with the stress recovery , 2012 .

[147]  W. Soedel.  Vibrations of shells and plates , 1981 .

[148]  A. Ceruti,et al.  Mixed Static and Dynamic Optimization of Four-Parameter Functionally Graded Completely Doubly Curved and Degenerate Shells and Panels Using GDQ Method , 2013 .

[149]  Erasmo Viola,et al.  Vibration analysis of spherical structural elements using the GDQ method , 2007, Comput. Math. Appl..

[150]  Ahmed K. Noor,et al.  Accurate determination of transverse normal stresses in hybrid laminated panels subjected to electro-thermo-mechanical loadings , 2000 .

[151]  Erasmo Carrera,et al.  Multilayered Shell Theories Accounting for Layerwise Mixed Description, Part 2: Numerical Evaluations , 1999 .

[152]  Alfredo Liverani,et al.  FGM and laminated doubly curved shells and panels of revolution with a free-form meridian: A 2-D GDQ solution for free vibrations , 2011 .

[153]  H. Tzou,et al.  Piezoelectric Shell Vibration Theory , 1993 .

[154]  Zhu Su,et al.  A unified approach for the vibration analysis of moderately thick composite laminated cylindrical shells with arbitrary boundary conditions , 2013 .

[156]  Reza Masoomi,et al.  Deflection and stress analysis of thin FGM skew plates on Winkler foundation with various boundary conditions using extended Kantorovich method , 2013 .

[157]  E. Viola,et al.  On Static Analysis of Composite Plane State Structures viaGDQFEM and Cell Method , 2013 .

[158]  E. Viola,et al.  General higher-order equivalent single layer theory for free vibrations of doubly-curved laminated composite shells and panels , 2013 .

[159]  Zhu Su,et al.  Free vibration analysis of laminated composite shallow shells with general elastic boundaries , 2013 .

[160]  Nicholas Fantuzzi,et al.  Generalized Differential Quadrature Finite Element Method applied to Advanced Structural Mechanics , 2013 .

[161]  Serge Abrate,et al.  FUNCTIONALLY GRADED PLATES BEHAVE LIKE HOMOGENEOUS PLATES , 2008 .

[162]  S. G. Lekhnit︠s︡kiĭ.  Theory of elasticity of an anisotropic body , 1981 .

[163]  Erasmo Viola,et al.  Free vibration analysis of functionally graded panels and shells of revolution , 2009 .