In this study, a three-dimensional discontinuous Galerkin isogeometric analysis framework is presented for the analysis of composite laminates. Non-uniform rational B-splines are employed as basis functions for both geometric and computational implementations. From a practical point of view, modeling with multiple non-uniform rational B-spline patches is required in many different applications due to the complexity of computational domains. Then, a special numerical technique is necessary to couple different non-uniform rational B-spline patches to carry out the isogeometric analysis. In this study, therefore, one of the discontinuous Galerkin methods, namely, symmetric interior penalty Galerkin formulation is utilized to deal with multi-patch isogeometric analysis applications. In order to show the applicability of the proposed framework, composite laminates under sinusoidally distributed load with different stacking sequences are studied in the numerical examples. The predicted results are compared with those obtained by the three-dimensional elasticity solutions and various numerical models available in the literature.
ShenoiRAWellicomeJFDyerI.Composite materials in maritime structures: Volume 1: Fundamental aspects. Cambridge: Cambridge University Press, 1993.
2.
ShenoiRAWellicomeJF.Composite materials in maritime structures: Volume 2: Practical considerations. Cambridge: Cambridge University Press, 1993.
3.
YildizdagMETranCABarchiesiE, et al. A multi-disciplinary approach for mechanical metamaterial synthesis: a hierarchical modular multiscale cellular structure paradigm. In: AltenbachHÖchsnerA (eds) State of the art and future trends in material modeling. Cham: Springer, 2019. pp.485–505.
4.
dell’IsolaFSeppecherPAlibertJJ, et al. Pantographic metamaterials: an example of mathematically driven design and of its technological challenges. Continuum Mech Therm2018; 31: 851–884.
5.
dell’IsolaFSeppecherPSpagnuoloM, et al. Advances in pantographic structures: design, manufacturing, models, experiments and image analyses. Continuum Mech Therm2019; 31(4): 1231–1282.
6.
VangelatosZKomvopoulosKGrigoropoulosC.Vacancies for controlling the behavior of microstructured three-dimensional mechanical metamaterials. Math Mech Solids2019; 24(2): 511–524.
7.
GiorgioIRizziNLAndreausU, et al. A two-dimensional continuum model of pantographic sheets moving in a 3d space and accounting for the offset and relative rotations of the fibers. Math Mech Complex Syst2019; 7(4): 311–325.
8.
PaganoNJ.Exact solutions for rectangular bidirectional composites and sandwich plates. J Compos Mater1970; 4(1): 20–34.
9.
SrinivasSRaoCJRaoA.An exact analysis for vibration of simply-supported homogeneous and laminated thick rectangular plates. J Sound Vib1970; 12(2): 187–199.
10.
SrinivasSRaoA.Bending, vibration and buckling of simply supported thick orthotropic rectangular plates and laminates. Int J Solids Struct1970; 6(11): 1463–1481.
11.
VelSSBatraR.Analytical solution for rectangular thick laminated plates subjected to arbitrary boundary conditions. AIAA J1999; 37(11): 1464–1473.
12.
ReissnerEStavskyY.Bending and stretching of certain types of heterogeneous aeolotropic elastic plates. J Appl Mech1961; 28: 402–408.
13.
WhitneyJ.The effect of transverse shear deformation on the bending of laminated plates. J Compos Mater1969; 3(3): 534–547.
14.
ReissnerE.A consistent treatment of transverse shear deformations in laminated anisotropic plates. AIAA J1972; 10(5): 716–718.
15.
LibrescuL.On the theory of anisotropic elastic shells and plates. Int J Solids Struct1967; 3(1): 53–68.
16.
LevinsonM.An accurate, simple theory of the statics and dynamics of elastic plates. Mech Res Commun1980; 7(6): 343–350.
17.
BhimaraddiAStevensL.A higher order theory for free vibration of orthotropic, homogeneous, and laminated rectangular plates. J Appl Mech1984; 51: 195–198.
18.
ReddyJN.A simple higher-order theory for laminated composite plates. J Appl Mech1984; 51: 742–752.
19.
RenJ.A new theory of laminated plate. Compos Sci Technol1986; 26(3): 225–239.
20.
KantTPandyaB.A simple finite element formulation of a higher-order theory for unsymmetrically laminated composite plates. Compos Struct1988; 9(3): 215–246.
21.
MohanPRNaganarayanaBPrathapG.Consistent and variationally correct finite elements for higher-order laminated plate theory. Compos Struct1994; 29(4): 445–456.
22.
MantariJOktemASoaresCG.A new higher order shear deformation theory for sandwich and composite laminated plates. Compos Part B: Eng2012; 43(3): 1489–1499.
23.
ReddyJ.Mechanics of laminated plates: Theory and analysis. Boca Raton, FL: CRC Press, 1997.
24.
ReddyJN.Mechanics of laminated composite plates and shells: Theory and analysis. Boca Raton, FL: CRC press, 2003.
25.
SolleroPAliabadiM.Anisotropic analysis of cracks in composite laminates using the dual boundary element method. Compos Struct1995; 31(3): 229–233.
26.
Dos ReisAAlbuquerqueÉLTorsaniFL, et al. Computation of moments and stresses in laminated composite plates by the boundary element method. Eng Anal Bound Elem2011; 35(1): 105–113.
27.
Phan-DaoHNguyen-XuanHThai-HoangC, et al. An edge-based smoothed finite element method for analysis of laminated composite plates. Int J Comp Method2013; 10(1): 1340005.
28.
HerathMTNatarajanSPrustyBG, et al. Smoothed finite element and genetic algorithm based optimization for shape adaptive composite marine propellers. Compos Struct2014; 109: 189–197.
29.
KefalAHasimKAYildizM.A novel isogeometric beam element based on mixed form of refined zigzag theory for thick sandwich and multilayered composite beams. Compos Part B: Eng2019; 167: 100–121.
CazzaniAMalagùMTurcoE.Isogeometric analysis: a powerful numerical tool for the elastic analysis of historical masonry arches. Continuum Mech Therm2016; 28(1–2): 139–156.
32.
CazzaniAStochinoFTurcoE.An analytical assessment of finite element and isogeometric analyses of the whole spectrum of Timoshenko beams. Zamm: J Appl Math Mech/Z Angew Math Me2016; 96(10): 1220–1244.
33.
CazzaniAMalagùMTurcoE, et al. Constitutive models for strongly curved beams in the frame of isogeometric analysis. Math Mech Solids2016; 21(2): 182–209.
34.
CazzaniASerraMStochinoF, et al. A refined assumed strain finite element model for statics and dynamics of laminated plates. Continuum Mech Therm. Epub ahead of print 3 September 2018. DOI: 10.1007/s00161-018-0707-x.
35.
CottrellJARealiABazilevsY, et al. Isogeometric analysis of structural vibrations. Comput Method Appl M2006; 195(41–43): 5257–5296.
36.
YildizdagMEArdicITDemirtasM, et al. Hydroelastic vibration analysis of plates partially submerged in fluid with an isogeometric FE-BE approach. Ocean Eng2019; 172: 316–329.
GómezHCaloVMBazilevsY, et al. Isogeometric analysis of the Cahn–Hilliard phase-field model. Comput Method Appl M2008; 197(49–50): 4333–4352.
39.
ThaiCHFerreiraACarreraE, et al. Isogeometric analysis of laminated composite and sandwich plates using a layerwise deformation theory. Compos Struct2013; 104: 196–214.
40.
ThaiCHFerreiraABordasSPA, et al. Isogeometric analysis of laminated composite and sandwich plates using a new inverse trigonometric shear deformation theory. Eur J Mech A: Solid2014; 43: 89–108.
41.
ThaiCHNguyen-XuanHBordasSPA, et al. Isogeometric analysis of laminated composite plates using the higher-order shear deformation theory. Mech Adv Mater Struct2015; 22(6): 451–469.
42.
ThaiCHFerreiraAWahabMA, et al. A generalized layerwise higher-order shear deformation theory for laminated composite and sandwich plates based on isogeometric analysis. Acta Mech2016; 227(5): 1225–1250.
43.
FaroughiSShafeiESchillingerD.Isogeometric stress, vibration and stability analysis of in-plane laminated composite structures. Int J Struct Stab Dy2018; 18(5): 1850070.
44.
HoferCToulopoulosI.Discontinuous Galerkin isogeometric analysis of elliptic problems on segmentations with non-matching interfaces. Comput Math Appl2016; 72(7): 1811–1827.
45.
LangerUMooreSE. Discontinuous Galerkin isogeometric analysis of elliptic PDEs on surfaces. In: DickopfTGanderMHalpernL, et al. (eds) Domain decomposition methods in science and engineering XXII. Cham: Springer, 2016, pp.319–326.
46.
YildizdagMEDemirtasMErginA.Multipatch discontinuous Galerkin isogeometric analysis of composite laminates. Continuum Mech Therm. Epub ahead of print 19 July 2018. DOI: 10.1007/s00161-018-0696-9.
DornischWVitucciGKlinkelS.The weak substitution method–an application of the mortar method for patch coupling in NURBS-based isogeometric analysis. Int J Numer Meth Eng2015; 103(3): 205–234.
NguyenVPKerfridenPBrinoM, et al. Nitsche’s method for two and three dimensional NURBS patch coupling. Comput Mech2014; 53(6): 1163–1182.
51.
GuoYRuessM.Nitsche’s method for a coupling of isogeometric thin shells and blended shell structures. Comput Method Appl M2015; 284: 881–905.
52.
Nguyen-ThanhNZhouKZhuangX, et al. Isogeometric analysis of large-deformation thin shells using RHT-splines for multiple-patch coupling. Comput Method Appl M2017; 316: 1157–1178.
53.
CuomoMGrecoL.An implicit strong G1-conforming formulation for the analysis of the Kirchhoff plate model. Continuum Mech Therm. Epub ahead of print 7 August 2018. DOI: 10.1007/s00161-018-0701-3.
54.
GrecoLCuomoMContrafattoL.A quadrilateral G1-conforming finite element for the Kirchhoff plate model. Comput Method Appl M2019; 346: 913–951.
55.
PieglLTillerW.The NURBS book. Berlin: Springer Science & Business Media, 2012.
56.
CottrellJAHughesTJBazilevsY.Isogeometric analysis: Toward integration of CAD and FEA. Chichester: John Wiley & Sons, 2009.
57.
RiviereB.Discontinuous Galerkin methods for solving elliptic and parabolic equations: Theory and implementation. Philadelphia, PA: SIAM, 2008.
