Isogeometric analysis (IGA) based on nonuniform rational B-splines (NURBS) is applied for static and free vibration analysis of laminated composite plates by using the third order shear deformation theory (TSDT). TSDT requires C1-continuity of generalized displacements and NURBS basis functions are well-suited for this requirement. Due to the noninterpolatory nature of NURBS basis functions, a penalty method is applied to enforce the essential boundary conditions. The validity and accuracy of the present method are demonstrated through a series of numerical examples of isotropic and laminated composite plates with different shapes, boundary conditions, fiber orientations, lay-up numbers, and so forth. The obtained numerical results are compared with either the analytical solutions or other available numerical methods. 1. Introduction The requirements for higher stiffness and strength-to-weight ratio, better wear resistance, and greater health characteristics over conventional materials have resulted in the wide application of composite laminates in the area of aerospace, automotive, and underwater structures. Therefore, static and dynamic analysis of both thin and thick laminates becomes an important research topic. Based on different assumptions for displacement fields, several theories for analysis of composite laminates have been developed. The classical plate theory (CPT) relying on the Kirchhoff-Love assumptions [1, 2] can only be applied if plates are thin, because transverse shear deformations are not considered. The first order shear deformation theory (FSDT) which accounts for transverse shear effects [3, 4] can be applied for both moderately thick and thin plates. However, in FSDT, the transverse shear strain is assumed to be constant along the thickness direction, and, thus a shear correction factor (SCF) has to be considered. The SCF depends on many factors, such as material coefficients, stacking scheme, plate geometry and boundary conditions; therefore the evaluation of the SCF remains a subject of research [5]. In addition, with constant shear strain assumption, transverse shear stresses obtained may not be zero on the plate surface, which contradicts the traction free boundary conditions. To overcome the limitations of FSDT, a more accurate high order shear deformation theory, termed the third order shear deformation theory (TSDT) [6, 7], has been developed to formulate the mechanical behavior of composite laminates. TSDT introduces the cubic variation of the displacement such that more accurate interlaminar stress distributions can be
References
[1]
E. Reissner and Y. Stavsky, “Bending and stretching of certain types of heterogeneous aeolotropic elastic plates,” Journal of Applied Mechanics, vol. 28, pp. 402–408, 1961.
[2]
S. B. Dong, K. S. Pister, and R. L. Taylor, “On the theory of laminated anisotropic shells and plates,” Journal of the Aerospace Sciences, vol. 29, no. 8, pp. 969–975, 1962.
[3]
E. Reissner, “The effect of transverse shear deformation on the bending of elastic plates,” Journal of Applied Mechanics, vol. 12, no. 2, pp. 69–77, 1945.
[4]
J. N. Reddy, Energy Principles and Variational Methods in Applied Mechanics, John Wiley & Sons, New York, NY, USA, 2002.
[5]
P. Madabhusi-Raman and J. F. Davalos, “Static shear correction factor for laminated rectangular beams,” Composites Part B, vol. 27, no. 3-4, pp. 285–293, 1996.
[6]
J. N. Reddy, “A refined nonlinear theory of plates with transverse shear deformation,” The International Journal of Solids and Structures, vol. 20, no. 9-10, pp. 881–896, 1984.
[7]
J. N. Reddy, “A Simple higher-order theory for laminated composite plates,” Journal of Applied Mechanics, vol. 51, no. 4, pp. 745–752, 1984.
[8]
K. Y. Dai, G. R. Liu, K. M. Lim, and X. L. Chen, “A mesh-free method for static and free vibration analysis of shear deformable laminated composite plates,” Journal of Sound and Vibration, vol. 269, no. 3–5, pp. 633–652, 2004.
[9]
L. Liu, L. P. Chua, and D. N. Ghista, “Mesh-free radial basis function method for static, free vibration and buckling analysis of shear deformable composite laminates,” Composite Structures, vol. 78, no. 1, pp. 58–69, 2007.
[10]
A. J. M. Ferreira, C. M. C. Roque, A. M. A. Neves, R. M. N. Jorge, C. M. M. Soares, and J. N. Reddy, “Buckling analysis of isotropic and laminated plates by radial basis functions according to a higher-order shear deformation theory,” Thin-Walled Structures, vol. 49, no. 7, pp. 804–811, 2011.
[11]
A. J. M. Ferreira, C. M. C. Roque, and P. A. L. S. Martins, “Analysis of composite plates using higher-order shear deformation theory and a finite point formulation based on the multiquadric radial basis function method,” Composites Part B, vol. 34, no. 7, pp. 627–636, 2003.
[12]
J. A. Cottrell, T. J. Hughes, and Y. Bazilevs, Isogeometric Analysis: Toward Integration of CAD and FEA, John Wiley & Sons, Chichester, UK, 2009.
[13]
T. J. R. Hughes, J. A. Cottrell, and Y. Bazilevs, “Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement,” Computer Methods in Applied Mechanics and Engineering, vol. 194, no. 39–41, pp. 4135–4195, 2005.
[14]
Y. Bazilevs and T. J. R. Hughes, “NURBS-based isogeometric analysis for the computation of flows about rotating components,” Computational Mechanics, vol. 43, no. 1, pp. 143–150, 2008.
[15]
Y. Bazilevs, V. M. Calo, T. J. R. Hughes, and Y. Zhang, “Isogeometric fluid-structure interaction: theory, algorithms, and computations,” Computational Mechanics, vol. 43, no. 1, pp. 3–37, 2008.
[16]
Y. Bazilevs, V. M. Calo, Y. Zhang, and T. J. R. Hughes, “Isogeometric fluid-structure interaction analysis with applications to arterial blood flow,” Computational Mechanics, vol. 38, no. 4-5, pp. 310–322, 2006.
[17]
X. Qian, “Full analytical sensitivities in NURBS based isogeometric shape optimization,” Computer Methods in Applied Mechanics and Engineering, vol. 199, no. 29–32, pp. 2059–2071, 2010.
[18]
W. A. Wall, M. A. Frenzel, and C. Cyron, “Isogeometric structural shape optimization,” Computer Methods in Applied Mechanics and Engineering, vol. 197, no. 33–40, pp. 2976–2988, 2008.
[19]
D. J. Benson, Y. Bazilevs, M. C. Hsu, and T. J. R. Hughes, “Isogeometric shell analysis: the Reissner-Mindlin shell,” Computer Methods in Applied Mechanics and Engineering, vol. 199, no. 5–8, pp. 276–289, 2010.
[20]
J. Kiendl, Y. Bazilevs, M.-C. Hsu, R. Wüchner, and K.-U. Bletzinger, “The bending strip method for isogeometric analysis of Kirchhoff-Love shell structures comprised of multiple patches,” Computer Methods in Applied Mechanics and Engineering, vol. 199, no. 37–40, pp. 2403–2416, 2010.
[21]
J. Kiendl, K.-U. Bletzinger, J. Linhard, and R. Wüchner, “Isogeometric shell analysis with Kirchhoff-Love elements,” Computer Methods in Applied Mechanics and Engineering, vol. 198, no. 49–52, pp. 3902–3914, 2009.
[22]
T. Elguedj, Y. Bazilevs, V. M. Calo, and T. J. R. Hughes, “ and projection methods for nearly incompressible linear and non-linear elasticity and plasticity using higher-order NURBS elements,” Computer Methods in Applied Mechanics and Engineering, vol. 197, no. 33–40, pp. 2732–2762, 2008.
[23]
R. L. Taylor, “Isogeometric analysis of nearly incompressible solids,” The International Journal for Numerical Methods in Engineering, vol. 87, no. 1–5, pp. 273–288, 2011.
[24]
J. A. Cottrell, A. Reali, Y. Bazilevs, and T. J. R. Hughes, “Isogeometric analysis of structural vibrations,” Computer Methods in Applied Mechanics and Engineering, vol. 195, no. 41–43, pp. 5257–5296, 2006.
[25]
S. Shojaee, N. Valizadeh, E. Izadpanah, T. Bui, and T.-V. Vu, “Free vibration and buckling analysis of laminated composite plates using the NURBS-based isogeometric finite element method,” Composite Structures, vol. 94, no. 5, pp. 1677–1693, 2012.
[26]
C. H. Thai, H. Nguyen-Xuan, N. Nguyen-Thanh, T. H. Le, T. Nguyen-Thoi, and T. Rabczuk, “Static, free vibration, and buckling analysis of laminated composite Reissner-Mindlin plates using NURBS-based isogeometric approach,” International Journal for Numerical Methods in Engineering, vol. 91, no. 6, pp. 571–603, 2012.
[27]
L. V. Tran, A. J. M. Ferreira, and H. Nguyen-Xuan, “Isogeometric analysis of functionally graded plates using higher-order shear deformation theory,” Composites Part B, vol. 51, pp. 368–383, 2013.
[28]
L. A. Piegl and W. Tiller, The NURBS Book, Springer, New York, NY, USA, 2nd edition, 1997.
[29]
G. R. Liu, Mesh Free Methods: Moving Beyond the Finite Element Method, CRC Press, Boca Raton, Fla, USA, 2nd edition, 2009.
[30]
J. N. Reddy, Mechanics of Laminated Composite Plates: Theory and Analysis, CRC press, Boca Raton, Fla, USA, 1997.
[31]
S. Shojaee, E. Izadpanah, N. Valizadeh, and J. Kiendl, “Free vibration analysis of thin plates by using a NURBS-based isogeometric approach,” Finite Elements in Analysis and Design, vol. 61, pp. 23–34, 2012.
[32]
S. Long and S. Atluri, “A meshless local petrov-galerkin method for solving the bending problem of a thin plate,” Computer Modeling in Engineering and Sciences, vol. 3, no. 1, pp. 53–64, 2002.
[33]
A. Embar, J. Dolbow, and I. Harari, “Imposing dirichlet boundary conditions with Nitsche's method and spline-based finite elements,” The International Journal for Numerical Methods in Engineering, vol. 83, no. 7, pp. 877–898, 2010.
[34]
D. Wang and J. Xuan, “An improved NURBS-based isogeometric analysis with enhanced treatment of essential boundary conditions,” Computer Methods in Applied Mechanics and Engineering, vol. 199, no. 37–40, pp. 2425–2436, 2010.
[35]
Z. Xu, Elasticity, High Education Press, Beijing, China, 4th edition, 2006.
[36]
N. J. Pagano and H. J. Hatfield, “Elastic behavior of multilayered bidirectional composites,” The American Institute of Aeronautics and Astronautics Journal, vol. 10, no. 7, pp. 931–933, 1972.
[37]
N. Pagano, “Exact solutions for rectangular bidirectional composites and sandwich plates,” Journal of Composite Materials, vol. 4, no. 1, pp. 20–34, 1970.
[38]
G. Akhras, M. S. Cheung, and W. Li, “Static and vibration analysis of anisotropic composite laminates by finite strip method,” The International Journal of Solids and Structures, vol. 30, no. 22, pp. 3129–3137, 1993.
[39]
S. Srinivas, “A refined analysis of composite laminates,” Journal of Sound and Vibration, vol. 30, no. 4, pp. 495–507, 1973.
[40]
B. N. Pandya and T. Kant, “Higher-order shear deformable theories for flexure of sandwich plates—finite element evaluations,” The International Journal of Solids and Structures, vol. 24, no. 12, pp. 1267–1286, 1988.
[41]
A. J. M. Ferreira, MATLAB Codes for Finite Element Analysis: Solids and Structures, Springer, London, UK, 2008.
[42]
A. J. M. Ferreira, R. C. Batra, C. M. C. Roque, L. F. Qian, and R. M. N. Jorge, “Natural frequencies of functionally graded plates by a meshless method,” Composite Structures, vol. 75, no. 1–4, pp. 593–600, 2006.
[43]
C. M. C. Roque, D. Cunha, C. Shu, and A. J. M. Ferreira, “A local radial basis functions—finite differences technique for the analysis of composite plates,” Engineering Analysis with Boundary Elements, vol. 35, no. 3, pp. 363–374, 2011.
[44]
T. Q. Bui and M. N. Nguyen, “A moving Kriging interpolation-based meshfree method for free vibration analysis of Kirchhoff plates,” Computers and Structures, vol. 89, no. 3-4, pp. 380–394, 2011.
[45]
G. R. Liu and X. L. Chen, “A mesh-free method for static and free vibration analyses of thin plates of complicated shape,” Journal of Sound and Vibration, vol. 241, no. 5, pp. 839–855, 2001.
[46]
X. Y. Cui, G. R. Liu, G. Y. Li, and G. Y. Zhang, “A thin plate formulation without rotation DOFs based on the radial point interpolation method and triangular cells,” The International Journal for Numerical Methods in Engineering, vol. 85, no. 8, pp. 958–986, 2011.
[47]
X. L. Chen, G. R. Liu, and S. P. Lim, “An element free Galerkin method for the free vibration analysis of composite laminates of complicated shape,” Composite Structures, vol. 59, no. 2, pp. 279–289, 2003.
[48]
F. Auricchio, L. B. da Veiga, A. Buffa, C. Lovadina, A. Reali, and G. Sangalli, “A fully “locking-free” isogeometric approach for plane linear elasticity problems: a stream function formulation,” Computer Methods in Applied Mechanics and Engineering, vol. 197, no. 1–4, pp. 160–172, 2007.
[49]
R. Bouclier, T. Elguedj, and A. Combescure, “Locking free isogeometric formulations of curved thick beams,” Computer Methods in Applied Mechanics and Engineering, vol. 245-246, pp. 144–162, 2012.
[50]
R. Echter and M. Bischoff, “Numerical efficiency, locking and unlocking of NURBS finite elements,” Computer Methods in Applied Mechanics and Engineering, vol. 199, no. 5–8, pp. 374–382, 2010.
[51]
M.-C. Hsu, I. Akkerman, and Y. Bazilevs, “High-performance computing of wind turbine aerodynamics using isogeometric analysis,” Computers and Fluids, vol. 49, no. 1, pp. 93–100, 2011.
