This paper presents the mathematical model consisting of conservation and balance laws (CBL) of classical continuum mechanics (CCM) and the constitutive theories derived using entropy inequality and representation theorem for thermoviscoelastic solids (TVES) matter without memory. The CBL and the constitutive theories take into account finite deformation and finite strain deformation physics. This mathematical model is thermodynamically and mathematically consistent and is ideally suited to study nonlinear dynamics of TVES and dynamic bifurcation and is used in the work presented in this paper. The finite element formulations are constructed for obtaining the solution of the initial value problems (IVPs) described by the mathematical models. Both space-time coupled as well as space-time decoupled finite element methods are considered for obtaining solutions of the IVPs. Space-time coupled finite element formulations based on space-time residual functional (STRF) that yield space-time variationally consistent space-time integral forms are considered. This approach ensures unconditional stability of the computations during the entire evolution. In the space-time decoupled finite element method based on Galerkin method with weak form for spatial discretization, the solutions of nonlinear ODEs in time resulting from the decoupling of space and time are obtained using Newmark linear acceleration method. Newton’s linear method is used to obtain converged solution for the nonlinear system of algebraic equations at each time step in the Newmark method. The different aspects of the deformation physics leading to the factors that influence nonlinear dynamic response and dynamic bifurcation are established using the proposed mathematical model, the solution method and their validity is demonstrated through model problem studies presented in this paper. Energy methods and superposition techniques in any form including those used in obtaining solutions are neither advocated nor used in the present work as these are not supported by calculus of variations and mathematical classification of differential operators appearing in nonlinear dynamics. The primary focus of the paper is to address various aspects of the deformation physics in nonlinear dynamics and their influence on dynamic bifurcation phenomenon using mathematical models strictly based on CBL of CCM using reliable unconditionally stable space-time coupled solution methods, which ensure solution accuracy or errors in the calculated solution are always identified. Many model problem studies are presented to
References
[1]
Surana, K.S. (1983) Geometrically Non-Linear Formulation for Two Dimensional Curved Beam Elements. Computers & Structures, 17, 105-114. https://doi.org/10.1016/0045-7949(83)90035-4
[2]
Mescall, J.F. (1965) Large Deflections of Spherical Shells under Concentrated Loads. Journal of Applied Mechanics, 32, 936-938. https://doi.org/10.1115/1.3627340
[3]
Surana, K.S. (1982) Geometrically Nonlinear Formulation for the Axisymmetric Shell Elements. International Journal for Numerical Methods in Engineering, 18, 477-502. https://doi.org/10.1002/nme.1620180402
[4]
Surana, K.S. (1983) Geometrically Nonlinear Formulation for the Curved Shell Elements. International Journal for Numerical Methods in Engineering, 19, 581-615. https://doi.org/10.1002/nme.1620190409
[5]
Ramm, E. (1977) A Plate/Shell Element for Large Deflections and Rotations. MIT Press, Cambridge, Chapter 10.
[6]
Surana, K.S. and Mathi, S.S.C. (2020) A Thermodynamically Consistent Non-Linear Mathematical Model for Thermoviscoelastic Plates/Shells with Finite Deformation and Finite Strain Based on Classical Continuum Mechanics. International Journal of Non-Linear Mechanics, 126, Article ID: 103565. https://doi.org/10.1016/j.ijnonlinmec.2020.103565
[7]
Zheng, Q.S. (1993) On the Representations for Isotropic Vector-Valued, Symmetric Tensor-Valued and Skew-Symmetric Tensor-Valued Functions. International Journal of Engineering Science, 31, 1013-1024. https://doi.org/10.1016/0020-7225(93)90109-8
[8]
Wang, C.C. (1969) On Representations for Isotropic Functions, Part I. Archive for Rational Mechanics and Analysis, 33, 249-267. https://doi.org/10.1007/BF00281278
[9]
Wang, C.C. (1969) On Representations for Isotropic Functions, Part II. Archive for Rational Mechanics and Analysis, 33, 268-287. https://doi.org/10.1007/BF00281279
[10]
Wang, C.C. (1970) A New Representation Theorem for Isotropic Functions, Part I and Part II. Archive for Rational Mechanics and Analysis, 36, 166-223. https://doi.org/10.1007/BF00272241
[11]
Wang, C.C. (1971) Corrigendum to “Representations for Isotropic Functions”. Archive for Rational Mechanics and Analysis, 43, 392-395. https://doi.org/10.1007/BF00252004
[12]
Smith, G.F. (1971) On Isotropic Functions of Symmetric Tensors, Skew-Symmetric Tensors and Vectors. International Journal of Engineering Science, 9, 899-916. https://doi.org/10.1016/0020-7225(71)90023-1
[13]
Spencer, A.J.M. and Rivlin, R.S. (1959) The Theory of Matrix Polynomials and Its Application to the Mechanics of Isotropic Continua. Archive for Rational Mechanics and Analysis, 2, 309-336. https://doi.org/10.1007/BF00277933
[14]
Spencer, A.J.M. (1971) Chapter 3. Theory of Invariants. In: Eringen, A.C., Ed., Mathematics, Academic Press, New York, 239-353. https://doi.org/10.1016/B978-0-12-240801-4.50008-X
Stoker, J.J. (1950) Nonlinear Vibrations in Mechanical and Electrical Systems. Interscience, New York.
[17]
Chattarjee, A. (2009) A Brief Introduction to Nonlinear Vibrations.
[18]
Cammarano, A., Hill, T.L., Neild, S.A. and Wagg, D.J. (2014) Bifurcations of Backbone Curves for Systems of Coupled Nonlinear Two-Mass Oscillators. Nonlinear Dynamics, 77, 311-320. https://doi.org/10.1007/s11071-014-1295-3
[19]
Kovacic, I. and Rand, R. (2013) Straight-Line Backbone Curve. Communications in Nonlinear Science and Numerical Simulation, 18, 2281-2288. https://doi.org/10.1016/j.cnsns.2012.11.031
[20]
Londoo, J.M., Cooper, J.E. and Neild, S.A. (2017) Identification of Systems Containing Nonlinear Stiffnesses Using Backbone Curves. Mechanical Systems and Signal Processing, 84, 116-135. https://doi.org/10.1016/j.ymssp.2016.02.008
[21]
Peter, S., Riethmüller, R. and Leine, R. (2016) Tracking of Backbone Curves of Nonlinear Systems Using Phaselocked-Loops. In: Kerschen, G., Ed., Nonlinear Dynamics, Volume 1, Springer International Publishing, Cham, 107-120. https://doi.org/10.1007/978-3-319-29739-2_11
[22]
Renson, L., Gonzalez-Buelga, A., Barton, D.A.W. and Neild, S.A. (2016) Robust Identification of Backbone Curves Using Control-Based Continuation. Journal of Sound and Vibration, 367, 145-158. https://doi.org/10.1016/j.jsv.2015.12.035
[23]
Amabili, M., Balasubramanian, P. and Ferrari, G. (2016) Travelling Wave and Non-Stationary Response in Nonlinear Vibrations of Water-Filled Circular Cylindrical Shells: Experiments and Simulations. Journal of Sound and Vibration, 381, 220-245. https://doi.org/10.1016/j.jsv.2016.06.026
[24]
Alijani, F., Amabili, M., Balasubramanian, P., Carra, S., Ferrari, G. and Garziera, R. (2016) Damping for Large Amplitude Vibrations of Plates and Curved Panels, Part 1: Modeling and Experiments. International Journal of Non-Linear Mechanics, 85, 23-40. https://doi.org/10.1016/j.ijnonlinmec.2016.05.003
[25]
Amabili, M., Alijani, F. and Delannoy, J. (2016) Damping for Large-Amplitude Vibrations of Plates and Curved Panels, Part 2: Identification and Comparisons. International Journal of Non-Linear Mechanics, 85, 226-240. https://doi.org/10.1016/j.ijnonlinmec.2016.05.004
[26]
Amabili, M. (2018) Nonlinear Damping in Large-Amplitude Vibrations: Modelling and Experiments. Nonlinear Dynamics, 93, 5-18. https://doi.org/10.1007/s11071-017-3889-z
[27]
Breunung, T. and Haller, G. (2018) Explicit Backbone Curves from Spectral Submanifolds of Forced-Damped Nonlinear Mechanical Systems. Proceedings of the Royal Society A, 474, Article ID: 20180083. https://doi.org/10.1098/rspa.2018.0083
[28]
Ghorbel, B., Zghal, A., Abdennadher, M., Walha, L. and Haddar, M. (2020) Analysis of Strongly Nonlinear Systems by Using HBM-AFT Method and Its Comparison with the Five-Order Runge-Kutta Method: Application to Duffing Oscillator and Disc Brake Model. International Journal of Applied and Computational Mathematics, 6, Article No. 50. https://doi.org/10.1007/s40819-020-0803-z
[29]
Balasubramanian, P., Ferrari, G. and Amabili, M. (2017) A Tool to Identify Damping during Large Amplitude Vibrations of Viscoelastic Structures. Proceedings of the ASME 2017 International Mechanical Engineering Congress and Exposition. Volume 4B: Dynamics, Vibration, and Control, Tampa, 3-9 November 2017, V04BT05A006.1. https://doi.org/10.1115/IMECE2017-70773
[30]
Amabili, M. (2008) Nonlinear Vibrations and Stability of Shells and Plates. Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9780511619694
[31]
Balasubramanian, P., Ferrari, G., Amabili, M. and Del Prado, Z.G.J.N. (2017) Experimental and Theoretical Study on Large Amplitude Vibrations of Clamped Rubber Plates. International Journal of Non-Linear Mechanics, 94, 36-45. https://doi.org/10.1016/j.ijnonlinmec.2016.12.006
[32]
Surana, K.S. and Reddy, J.N. (2017) The Finite Element Method for Initial Value Problems. CRC/Taylor and Francis, Boca Raton. https://doi.org/10.1201/b22512
[33]
Surana, K.S. and Mathi, S.S.C. (2020) A Thermodynamically Consistent Formulation for Dynamic Response of Thermoviscoelastic Plate/Shell Based on Classical Continuum Mechanics (ccm). International Journal of Structural Stability and Dynamics, 20, Article ID: 2050043. https://doi.org/10.1142/S0219455420430129
[34]
Surana, K.S., Mysore, D. and Carranza, C.H. (2020) A Thermodynamically Consistent Formulation for Bending of Thermoviscoelastic Beams for Small Deformation, Small Strain Based on Classical Continuum Mechanics. Mechanics of Advanced Materials and Structures, 27, 1120-1140. https://doi.org/10.1080/15376494.2020.1725987
