As a fundamental component of an overall program in modeling smart material damping devices, we consider inactive host material models for moderate to highly filled rubbers undergoing uniaxial tensile deformations. Beginning from a neo-Hookean strain energy function formulation for nonlinear extension, we develop general constitutive models for both quasi-static and dynamic deformations of a viscoelastic rod. The constitutive laws are nonlinear and contain hysteresis through a Boltzmann superposition integral term. The resulting integropartial differential equations models are shown to be equivalent to the usual Lagrangian dynamic distributed parameter models coupled with linear ordinary differential equations for internal variables (internal strains). Comprehensive well-posedness results (existence, uniqueness and continuous dependence) are summarized in a discussion of theoretical aspects of the systems.
The models are validated with experiments designed and carried out explicitly for this study. In particular, quasi-static Instron experimental data are used in a least squares inverse problem formulation to estimate nonlinear elastic and nonlinear viscoelastic contributions to the general stress-strain constitutive laws proposed. It is shown that the models provide an excellent prediction of nested hysteresis loops manifested in the data. These models are then used as initial estimates in determining the nonlinear hysteretic constitutive laws for the dynamic experiments. It is shown that in cases of more highly filled rubbers, multiple internal variable models lead to best fits to the data.
Get full access to this article
View all access options for this article.
References
1.
1. Banks, H. T., R. H. Fabiano and Y. Wang, "Estimation of Boltzmann damping coefficients in beam models,"Proceedings of COMCON Workshop on Stabilization of Flexible Structures, A. V. Balakrishnan and J. P. Zolesio, eds., Optimization Software Inc., New York (1987), 13-35.
2.
2. Banks, H. T., R. H. Fabiano and Y. Wang, "Inverse problem techniques for beams with tip body and time hysteresis damping,"Mat. Aplic. e Comput. (1989), 8:101-118.
3.
3. Banks, H. T., R. H. Fabiano, Y. Wang, D. Inman and H. Cudney, "Spatial versus time hysteresis in damping mechanics,"Proc. 27th IEEE Conf: on Dec. and Control (1988), 1674-1677.
4.
4. Banks, H. T., D. S. Gilliam and V. I. Shubov, "Global solvability for damped abstract nonlinear hyperbolic systems,"Differential and Integral Equations (1997), 10:309-332.
5.
5. Banks, H. T. and N. Lybeck, "Modeling methodology for elastomer dynamics," in Systems and Control in the 21st Century, Birkhauser, Boston, 1996, 37-50.
6.
6. Banks, H. T., N. J. Lybeck, M. J. Gaitens, B. C. Muioz and L. C. Yanyo, "Computational methods for estimation in the modeling of nonlinear elastomers," CRSC-TR95-40, NCSU; Kybernetika (1996), 32:526-542.
7.
7. Banks, H. T., N. G. Medhin and Y. Zhang, "A mathematical framework for curved active constrained layer structures: well-posedness and approximation," CRSC-TR95-32, NCSU; Numerical Functional Analysis & Optimization (1996), 17:1-22.
8.
8. Banks, H. T. and G. A. Pintdr, "Approximation results for parameter estimation in nonlinear elastomers," CRSC-TR96-34, NCSU; Control and Estimation of Distributed Parameter Systems, F. Kappel et al., eds., Birkhauser, Boston, 1997.
9.
9. Banks, H. T., G. A. Pinter and L. K. Potter, "Existence of unique weak solutions to a dynamical system for nonlinear elastomers with hysteresis," CRSC-TR98-43, NCSU, Nov. 1998; Differential and Integral Equations, to appear.
10.
10. Banks, H. T., L. K. Potter and Y. Zhang, "Stress-strain laws for carbon black and silicon filled elastomers,"Proc. 36th IEEE Conf on Dec. and Control (1996), 3727-3732.
11.
11. Banks, H. T., R. C. Smith and Y. Wang, Smart Material Structures: Modeling, Estimation and Control, Masson, Paris and John Wiley and Sons, Chichester, 1996.
12.
12. Bensoussan, A., J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, New York, 1978.
13.
13. Bloch, R., W. V. Chang and N. W. Tschoegl, "The behavior of rubberlike materials in moderately large deformations,"Journal of Rheology (1978), 22:1-32.
14.
14. Bloch, R., W. V. Chang and N. W. Tschoegl, "On the theory of the viscoelastic behavior of soft polymers in moderately large deformations,"Rheologia Acta (1976), 15:367-378.
15.
15. Charlton, D. J., J. Yang and K. K. Teh, "Mechanical properties of solid polymers,"Rubber Chemistry & Technology (1994), 67:481-503.
16.
16. Christensen, R. M., Theory of Viscoelasticity: An Introduction, 2nd ed., Academic Press, New York, 1982.
17.
17. Codegone, M. and A. Negro, "Homogenization of the nonlinear quasistationary Maxwell equations with degenerated coefficients,"Applicable Analysis (1984), 18:159-173.
18.
18. Defranceschi, A."An Introduction to Homogenization and G-Convergence,"Lecture Notes, School on Homogenization, ICTPTrieste, Sept. 6-8, 1993.
19.
19. Doi, M. and M. Edwards, The Theory of Polymer Dynamics, Oxford, New York, 1986.
20.
20. Findley, W. N., J. S. Lai and K. Onaran, Creep and Relaxation of Nonlinear Viscoelastic Materials, H. A. Lauwerier and W. T. Koiter, eds., North-Holland Series in Applied Mathematics and Mechanics, North-Holland Publishing Company, 1976.
21.
21. Ferry, J. D., Viscoelastic Properties of Polymers, John Wiley & Sons, Inc., New York, 1980.
22.
22. Friswall, M. I., D. J. Inman and M. J. Lam, "On the realisation of GHM models in viscoelasticity," preprint, 1997.
23.
23. Gandhi, M. V. and B. S. Thompson, Smart Materials and Structures, Chapman and Hall, London, 1992.
24.
24. Golla, D. F. and P. C. Hughes, "Dynamics of viscoelastic structures-A time-domain, finite element formulation,"ASME J. Applied Mechanics (1985), 52:897-905.
25.
25. Jikov, V. V., S. M. Kozlov and 0. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer Verlag, New York, 1994.
26.
26. Johnson, A. R., C. J. Quigley and J. L. Mead, "Large strain viscoelastic constitutive models for rubber, part I: Formulations,"Rubber Chemistry Technology (1994), 67:904-917.
27.
27. Johnson, A. R., A. Tessler and M. Dambach, "Dynamics of thick elastic beams,"J. Engr Materials and Tech. (1997), 119:273-278.
28.
28. Johnson, A. R. and R. G. Stacer, "Rubber viscoelasticity using the physically constrained systems' stretches as internal variables,"Rubber Chem. Tech. (1993), 66:567-577.
29.
29. Jolly, M. R., J. D. Carlson and B. C. Mufioz, "A model of the behavior of magnetorheological materials,"Smart Mater Struct. (1996), 5:607-614.
30.
30. Jolly, M. R., J. D. Carlson, B. C. Mutioz and T. A. Bullions, "The magnetoviscoelastic response of elastomer composites consisting of ferrous particles embedded in a polymer matrix,"Journal of Intelligent Material Systems and Structures (1996), 7:613-622.
31.
31. Kolsky, H., "The measurement of the material damping of high polymers over ten decades of frequency and its interpretation,"V. K. Kinra and A. Wolfenden, eds., Mechanics and Mechanisms of Material Damping, American Society for Testing and Materials, Philadelphia, 1992.
32.
32. Lesieutre, G. A."Modeling frequency-dependent longitudinal dynamic behavior of linear viscoelastic long fiber components,"J. Composite Materials (1994), 28:1770-1782.
33.
33. Lesieutre, G. A. and K. Govindswamy, "Finite element modeling of frequency-dependent and temperature-dependent dynamic behavior of viscoelastic materials in simple shear,"Int. J. Solids Structures (1996), 33:419-432.
34.
34. McTavish, D. J., P. C. Hughes, Y. Soucy and W. B. Graham, "Prediction and measurement of modal damping factors for viscoelastic space structures,"AIAA Journal (1992), 30:1392-1399.
35.
35. Mufioz, B. C. and M. R. Jolly, "Composites with field responsive rheology,"W. K. Brostow, ed., Performance of Plastics, Hansen Publishers, NY, 1997.
37. Rivera-Dominguez, A. and W. M. Jordan, "Predictive creep response of linear viscoelastic graphite/epoxy composites using the Laplace Transform method,"Journal of Materials Engineeirng and Performance (1992), 1:261-266.
38.
38. Rivlin, R. S., "Large elastic deformations of isotropic materials, I, II, III."Phil. Trans. Roy Soc. A (1948), 240:459-525.
39.
39. Schapery, R. A."On the characterization of nonlinear viscoelastic materials,"Polymer Engineering and Science (1969), 9:295-310.
40.
40. Simon, T. M., F. Reitich, M. R. Jolly, K. Ito and H. T. Banks, "Estimation of the effective permeability in magnetorheological fluids, CRSC-TR98-35, NCSU, Oct. 1998; J. Intelligent Mat. Systems and Structures, submitted.
41.
41. Stoer, J. and R. Bulirsch, Introduction to Numerical Analysis, Springer-Verlag, New York, 3rd ed. (1993).
42.
42. Ward, I. M., Mechanical Properties of Solid Polymers, John Wiley & Sons, New York, 2nd ed. (1983).
43.
43. Zhang, Y., Mathematical formulation of vibrations of a composite curved beam structure: Aluminum core material with viscoelastic layers, constraining layers and piezoceramic patches, Ph.D Thesis, NC State University, May 1997.
