Research Papers

A Nonlinear Finite Element Framework for Viscoelastic Beams Based on the High-Order Reddy Beam Theory

[+] Author and Article Information
G. S. Payette

Graduate Research Assistant

J. N. Reddy

Life Fellow ASME
e-mail: jnreddy@tamu.edu
Department of Mechanical Engineering,
Texas A&M University,
College Station, TX 77843-3123

1Corresponding author.

Contributed by the Materials Division of ASME for publication in the Journal of Engineering Materials and Technology. Manuscript received July 10, 2011; final manuscript received November 18, 2012; published online January 23, 2013. Assoc. Editor: Georges Cailletaud.

J. Eng. Mater. Technol 135(1), 011005 (Jan 23, 2013) (11 pages) Paper No: MATS-11-1147; doi: 10.1115/1.4023185 History: Received July 10, 2011; Revised November 18, 2012

A weak form Galerkin finite element model for the nonlinear quasi-static and fully transient analysis of initially straight viscoelastic beams is developed using the kinematic assumptions of the third-order Reddy beam theory. The formulation assumes linear viscoelastic material properties and is applicable to problems involving small strains and moderate rotations. The viscoelastic constitutive equations are efficiently discretized using the trapezoidal rule in conjunction with a two-point recurrence formula. Locking is avoided through the use of standard low-order reduced integration elements as well through the employment of a family of elements constructed using high-polynomial order Lagrange and Hermite interpolation functions.

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Flügge, W., 1975, Viscoelasticity, 2nd ed., Springer, Berlin/Heidelberg.
Christensen, R. M., 1982, Theory of Viscoelasticity, 2nd ed., Academic Press, New York.
Findley, W. N., Lai, J. S., and Onaran, K., 1976, Creep and Relaxation of Nonlinear Viscoelastic Materials, North-Holland Pub. Co., New York.
Reddy, J. N., 2008, An Introduction to Continuum Mechanics With Applications, Cambridge University Press, New York.
Chen, T.-M., 1995, “The Hybrid Laplace Transform/Finite Element Method Applied to the Quasi–Static and Dynamic Analysis of Viscoelastic Timoshenko Beams,” Int. J. Numer. Methods Eng., 38(3), pp. 509–522. [CrossRef]
Aköz, Y., and Kadioğlu, F., 1999, “The Mixed Finite Element Method for the Quasi-Static and Dynamic Analysis of Viscoelastic Timoshenko Beams,” Int. J. Numer. Methods Eng., 44(12), pp. 1909–1932. [CrossRef]
Temel, B., Calim, F. F., and Tütüncü, N., 2004, “Quasi-Static and Dynamic Response of Viscoelastic Helical Rods,” J. Sound Vib., 271(3–5), pp. 921–935. [CrossRef]
Chen, Q., and Chan, Y. W., 2000, “Integral Finite Element Method for Dynamical Analysis of Elastic-Viscoelastic Composite Structures,” Comput. Struct., 74(1), pp. 51–64. [CrossRef]
Trindade, M. A., Benjeddou, A., and Ohayon, R., 2001, “Finite Element Modelling of Hybrid Active-Passive Vibration Damping of Multilayer Piezoelectric Sandwich Beams—Part I: Formulation,” Int. J. Numer. Methods Eng., 51(7), pp. 835–854. [CrossRef]
Pálfalvi, A., 2008, “A Comparison of Finite Element Formulations for Dynamics of Viscoelastic Beams,” Finite Elem. Anal. Design, 44(14), pp. 814–818. [CrossRef]
McTavish, D. J., and Hughes, P. C., 1992, “Finite Element Modeling of Linear Viscoelastic Structures—The GHM Method,” Proceedings of the 33rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Dallas, TX, April 13–15, AIAA Paper No. 92-2380, pp. 1753–1763.
McTavish, D. J., and Hughes, P. C., 1993, “Modeling of Linear Viscoelastic Space Structures,” ASME J. Vib. Acoust., 115(1), pp. 103–110. [CrossRef]
Balamurugan, V., and Narayanan, S., 2002, “Finite Element Formulation and Active Vibration Control Study on Beams Using Smart Constrained Layer Damping (SCLD) Treatment,” J. Sound Vib., 249(2), pp. 227–250. [CrossRef]
Balamurugan, V., and Narayanan, S., 2002, “Active-Passive Hybrid Damping in Beams With Enhanced Smart Constrained Layer Treatment,” Eng. Struct., 24(3), pp. 355–363. [CrossRef]
Johnson, A. R., Tessler, A., and Dambach, M., 1997, “Dynamics of Thick Viscoelastic Beams,” J. Eng. Mater. Technol., 119(3), pp. 273–278. [CrossRef]
Austin, E. M., and Inman, D. J., 1998, “Modeling of Sandwich Structures,” Smart Structures and Materials 1998: Passive Damping and Isolation, Vol. 3327, No. 1, pp. 316–327.
Kennedy, T. C., 1998, “Nonlinear Viscoelastic Analysis of Composite Plates and Shells,” Compos. Struct., 41(3–4), pp. 265–272. [CrossRef]
Oliveira, B. F., and Creus, G. J., 2000, “Viscoelastic Failure Analysis of Composite Plates and Shells,” Compos. Struct., 49(4), pp. 369–384. [CrossRef]
Hammerand, D. C., and Kapania, R. K., 2000, “Geometrically Nonlinear Shell Element for Hygrothermorheologically Simple Linear Viscoelastic Composites,” AIAA J., 38, pp. 2305–2319. [CrossRef]
Payette, G. S., and Reddy, J. N., 2010, “Nonlinear Quasi-Static Finite Element Formulations for Viscoelastic Euler–Bernoulli and Timoshenko Beams,” Int. J. Numer. Methods Biomed. Eng., 26(12), pp. 1736–1755. [CrossRef]
Reddy, J. N., 1984, “A Simple Higher-Order Theory for Laminated Composite Plates,” ASME J. Appl. Mech., 51, pp. 745–752. [CrossRef]
Heyliger, P. R., and Reddy, J. N., 1988, “A Higher-Order Beam Finite Element for Bending and Vibration Problems,” J. Sound Vib., 126(2), pp. 309–326. [CrossRef]
Wang, C. M., Reddy, J. N., and Lee, K. H., 2000, Shear Deformable Beams and Plates. Relationships With Classical Solutions, Elsevier, Amesterdam.
Reddy, J. N., 2004, An Introduction to Nonlinear Finite Element Analysis, Oxford University Press, Oxford, UK.
Belytschko, T., Liu, W. K., and Moran, B., 2000, Nonlinear Finite Elements for Continua and Structures, John Wiley and Sons, Ltd, New York.
Reddy, J. N., 1999, Theory and Analysis of Elastic Plates, Taylor and Francis, Philadelphia.
Başar, Y., Ding, Y., and Schultz, R., 1993, “Refined Shear-Deformation Models for Composite Laminates With Finite Rotations,” Int. J. Solids Struct., 30(19), pp. 2611–2638. [CrossRef]
Cortés, F., and Elejabarrieta, M. J., 2007, “Finite Element Formulations for Transient Dynamic Analysis in Structural Systems With Viscoelastic Treatments Containing Fractional Derivative Models,” Int. J. Numer. Methods Eng., 69(10), pp. 2173–2195. [CrossRef]
Enelund, M., and Josefson, B. L., 1997, “Time-Domain Finite Element Analysis of Viscoelastic Structures With Fractional Derivatives Constitutive Relations,” AIAA J., 35(10), pp. 1630–1637. [CrossRef]
Escobedo-Torres, J., and Ricles, J. M., 1998, “The Fractional Order Elastic-Viscoelastic Equations of Motion: Formulation and Solution Methods,” J. Intell. Mater. Syst. Struct., 9(7), pp. 489–502. [CrossRef]
Galucio, A. C., Deü, J.-F., and Ohayon, R., 2004, “Finite Element Formulation of Viscoelastic Sandwich Beams Using Fractional Derivative Operators,” Comput. Mech., 33, pp. 282–291. [CrossRef]
Zheng-you, Z., Gen-guo, L., and Chang-jun, C., 2002, “Quasi-Static and Dynamical Analysis for Viscoelastic Timoshenko Beam With Fractional Derivative Constitutive Relation,” Appl. Math. Mech., 23, pp. 1–12. [CrossRef]
Taylor, R. L., Pister, K. S., and Goudreau, G. L., 1970, “Thermomechanical Analysis of Viscoelastic Solids,” Int. J. Numer. Methods Eng., 2(1), pp. 45–59. [CrossRef]
Simo, J. C., and Hughes, T. J. R., 1998, Computational Inelasticity, Springer-Verlag, Berlin.
Reddy, J. N., 1997, “On Locking-Free Shear Deformable Beam Finite Elements,” Comput. Methods Appl. Mech. Eng., 149(1–4), pp. 113–132. [CrossRef]
Reddy, J. N., 2002, Energy Principles and Variational Methods in Applied Mechanics, 2nd ed., John Wiley and Sons, Ltd, New York.
Hamming, R., 1987, Numerical Methods for Scientists and Engineers, 2nd ed., Dover Publications, Mineola, NY.
Karniadakis, G. E., and Sherwin, S. J., 1999, Spectral/hp Element Methods for CFD, Oxford University Press, Oxford, UK.
Lai, J., and Bakker, A., 1996, “3-D Schapery Representation for Non-Linear Viscoelasticity and Finite Element Implementation,” Comput. Mech., 18, pp. 182–191. [CrossRef]
Van Krevelen, D. W., 1990, Properties of Polymers, 3rd ed., Elsevier, Amsterdam.
Newmark, N. M., 1959, “A Method of Computation for Structural Dynamics,” J. Eng. Mech., 85, pp. 67–94.
Reddy, J. N., 2006, An Introduction to the Finite Element Method, 3rd ed., McGraw-Hill, New York.


Grahic Jump Location
Fig. 1

Deformation of a beam structure according to the third-order Reddy beam theory: (a) undeformed configuration and (b) deformed configuration

Grahic Jump Location
Fig. 2

Interpolation functions for a high-order RBT finite element, where n = 6 and i = 1,…, n: (a) Lagrange interpolation functions ψi(1), (b) Hermite interpolation functions ψ∧2i-1(2), and (c) Hermite interpolation functions ψ∧2i(2)

Grahic Jump Location
Fig. 3

Maximum vertical deflection w0(L/2,t) of various viscoelastic beams, each subjected to a uniform vertically distributed load q0. 2 RBT-6 elements employed in each finite element discretization: (a) hinged–hinged beam configuration and (b) pinned–pinned and clamped–clamped beam configurations.

Grahic Jump Location
Fig. 4

Maximum vertical deflection w0(L/2,t) of a hinged–hinged viscoelastic beam subjected to a time-dependent transverse load q(t)

Grahic Jump Location
Fig. 5

A comparison of the time-dependent vertical response w0(L/2,t) (with units of mm) of hinged–hinged beams due to a suddenly applied transverse load q(t). Results are for both viscoelastic as well as elastic beams.

Grahic Jump Location
Fig. 6

A comparison of the time-dependent vertical response w0(L/2,t) (with units of mm) of hinged–hinged viscoelastic and elastic beams due to a periodic concentrated load F(t) (where η=2.744×108Ns/m2)




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