Dynamics of a viscoelastic spherical shell with a nonconvex strain energy function
Authors:
Roger Fosdick, Yohannes Ketema and Jang-Horng Yu
Journal:
Quart. Appl. Math. 56 (1998), 221-244
MSC:
Primary 73F15; Secondary 73G25, 73K12, 73K15
DOI:
https://doi.org/10.1090/qam/1622558
MathSciNet review:
MR1622558
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Abstract: We study the radial motion of an incompressible viscoelastic spherical shell with a nonconvex strain energy function that models a material that can undergo a phase transition. In addition to the classical Newtonian viscosity for viscoelastic materials, we consider a material with two microstructural coefficients that are supposed to sense local configurational changes that take place during a deformation. Conditions necessary to show the effect of the nonconvexity of the strain energy function during a phase transition of the material, are determined, and the resulting dynamics is analyzed. It is shown that, though small periodic vibrations are possible, the system can easily revert into a mode of large amplitude motion as a result of small external excitation. Such motion may be transient to periodic motion or to chaotic motion. Boundaries in parameter space for the occurrence of this type of motion are determined and examples are given.
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R. Abeyaratne and J. K. Knowles, Dynamics of propagating phase boundaries: Thermoelastic solids with heat conduction, Archive for Rational Mechanics and Analysis 126(3), 203–230 (1994)
M. Baker and J. L. Ericksen, Inequalities restricting the form of the stress-deformation relations for isotropic elastic solids and Reiner-Rivlin fluids, J. Wash. Acad. Sci. 44, 33–35 (1954)
C. Chu and R. D. James, Biaxial loading experiments on Cu-Al-Ni single crystals, AMD-Vol. 181, ASME, 61–69 (1993)
Thomas S. Parker and Leon O. Chua, Chaos: A tutorial for engineers, Proceedings of the IEEE 75(8), 982–1008 (1987)
P. M. Culkowski and H. Reismann, The spherical sandwich shell under axisymmetric static and dynamic loading, Journal of Sound and Vibration 14, 229–240 (1971)
J. E. Dunn and R. L. Fosdick, Thermodynamics, stability, and boundedness of fluids of complexity 2 and fluids of second grade, Archive for Rational Mechanics and Analysis 56(3), 191–252 (1974)
F. Falk, Model free energy, mechanics, and thermodynamics of shape memory alloys, Acta Metallurgica 28, 1773–1780 (1980)
R. L. Fosdick and G. P. MacSithigh, Minimization in incompressible nonlinear elastic theory, Journal of Elasticity 16, 267–301 (1986)
R. L. Fosdick, W. H. Warner, and J. H. Yu, Steady, structured shock wave in a viscoelastic solid of differential type, International Journal of Engineering Science 28(6), 469–483 (1990)
B. P. Gautham and N. Ganesan, Vibration and damping characteristics of spherical shells with a viscoelastic core, Journal of Sound and Vibration 170(3), 289–301 (1994)
H. Goldstein, Classical Mechanics, second edition, Addison-Wesley Publishing Company, Reading, MA, 1980
J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcation of vector fields, Applied Mathematical Sciences, Vol. 42, Springer-Verlag, Berlin, 1983
Z. Guo and R. Solecki, Free and forced finite-amplitude oscillations of an elastic thick-walled hollow sphere made of incompressible material, Archiwum Mechaniki Stosowanej 3(15), 427–433 (1963)
Y. Ketema, A physical interpretation of Melnikov’s method, International Journal of Bifurcation and Chaos 2(1), 1–9 (1991)
J. K. Knowles and M. T. Jakub, Finite dynamic deformation of an incompressible elastic medium containing a spherical cavity, Archive for Rational Mechanics and Analysis 18, 367–378 (1965)
P. H. Leo, T. W. Shield, and O. P. Bruno, Transient heat transfer effects of the pseudoelastic behavior of shape-memory wires, Acta Metall. Mater., 2477–2485 (1993)
A. Okazaki, Y. Urata, and A. Tatemichi, Damping properties of a three layered shallow spherical shell with a constrained viscoelastic layer, Japan Society of Mechanical Engineers International Journal, Ser. I, 33(2), 145–151 (1990)
C. Truesdell and W. Noll, The non-linear field theories of mechanics, Handbook of Physics, III/3, Springer-Verlag, New York, 1965
J. Yu, Ph.D. Thesis: Nonlinear oscillations of viscoelastic cylindrical and spherical shells, University of Minnesota, 1994
R. L. Fosdick and J. H. Yu, Thermodynamics, stability and nonlinear oscillations of viscoelastic solids. Part I: Differential type solids of second grade, Internat. J. Non-Linear Mech. 31 (4), 495–516 (1996)
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Article copyright:
© Copyright 1998
American Mathematical Society