The asymptotics of heavily burdened viscoelastic rods
Authors:
Stuart S. Antman and Süleyman Ulusoy
Journal:
Quart. Appl. Math. 70 (2012), 437467
MSC (2010):
Primary 35B25, 35C20, 35G31, 35K35, 35K70, 74C20, 74D10, 74K10
Published electronically:
May 25, 2012
MathSciNet review:
2986130
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
Abstract: This paper treats the spatial motion of a deformable nonlinearly viscoelastic rod carrying a heavy rigid body. The ratio of the inertia of the rod to that of the attached rigid body is characterized by a small parameter . The boundary conditions on the rod where it is attached to the rigid body are the ordinary differential equations of motion for the rigid body subject to the contact loads exerted on the rigid body by the rod. The entire system is thus governed by a quasilinear parabolichyperbolic system of partial differential equations coupled to the ordinary differential equations for the rigid body, with appearing in the coefficients of the acceleration terms of the rod. This paper gives a rigorous asymptotic expansion of the solutions of initialboundaryvalue problems for this system, consisting of a regular expansion and an initiallayer expansion. The leading term of the regular expansion satisfies the reduced problem, obtained by setting in the governing equations. The reduced problem is governed by a curious set of quasilinear functionaldifferential equations, the solutions of which exhibit a rich and interesting behavior. (In the absence of dissipation, which is needed for the justification of the asymptotic expansion, the leading term of the regular expansion satisfies a steadystate problem parametrized by time, which enters through the boundary conditions.) The remaining terms of the regular expansion satisfy linear problems. The leading term of the initiallayer expansion satisfies a quasilinear parabolic system, and the remaining terms satisfy linear parabolic systems. Thus the asymptotic expansion leads to greatly simplified equations. The regular and initiallayer corrections to the solution of the reduced problem show that it exhibits the main features of the solution to the whole system. The justification of the asymptotic expansion consists in estimating the error. For this purpose, a FaedoGalerkin method is used to obtain sharp estimates for the exponential decay in time of the terms of the initiallayer expansion (satisfying parabolic systems). (This method is far more efficient than the repeated use of the Maximum Principle à la S.N. Bernstein (see Wiegner, Math. Z. 188 (1984) 322) for treating the analogous scalar problem by Yip et al., J. Math. Pures Appl. 81 (2002) 283309. The Maximum Principle is not applicable to our parabolic systems. Even for such scalar problems, the FaedoGalerkin method as used here is far simpler and more efficient.) The main focus of this paper is on the derivation of these estimates. A significant part of the analysis is devoted to handling technical difficulties due to the peculiarities of the geometrically exact equations governing the spatial motion of viscoelastic rods with a general class of nonlinear constitutive equations of strainrate type invariant under rigid motions.
 1.
Stuart
S. Antman, The paradoxical asymptotic status of massless
springs, SIAM J. Appl. Math. 48 (1988), no. 6,
1319–1334. MR 968832
(90b:35194), 10.1137/0148081
 2.
Stuart
S. Antman, Nonlinear problems of elasticity, 2nd ed., Applied
Mathematical Sciences, vol. 107, Springer, New York, 2005. MR 2132247
(2006e:74001)
 3.
Stuart
S. Antman, Randall
S. Marlow, and Constantine
P. Vlahacos, The complicated dynamics of heavy rigid bodies
attached to deformable rods, Quart. Appl. Math. 56
(1998), no. 3, 431–460. MR 1637036
(99e:73068)
 4.
Stuart
S. Antman and Thomas
I. Seidman, The parabolichyperbolic system governing the spatial
motion of nonlinearly viscoelastic rods, Arch. Ration. Mech. Anal.
175 (2005), no. 1, 85–150. MR 2106258
(2006b:74015), 10.1007/s0020500403416
 5.
Stuart
S. Antman and J.
Patrick Wilber, The asymptotic problem for the
springlike motion of a heavy piston in a viscous gas, Quart. Appl. Math. 65 (2007), no. 3, 471–498. MR 2354883
(2009g:76128), 10.1090/S0033569X07010765
 6.
JeanPierre
Aubin, Un théorème de compacité, C. R.
Acad. Sci. Paris 256 (1963), 5042–5044 (French). MR 0152860
(27 #2832)
 7.
Millard
F. Beatty, Finite amplitude oscillations of a simple rubber support
system, Arch. Rational Mech. Anal. 83 (1983),
no. 3, 195–219. MR 701902
(84g:73039), 10.1007/BF00251508
 8.
C.
M. Dafermos and J.
A. Nohel, Energy methods for nonlinear hyperbolic Volterra
integrodifferential equations, Comm. Partial Differential Equations
4 (1979), no. 3, 219–278. MR 522712
(80b:45018), 10.1080/03605307908820094
 9.
Avner
Friedman, Partial differential equations of parabolic type,
PrenticeHall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
(31 #6062)
 10.
G.
H. Handelman and J.
B. Keller, Small vibrations of a slightly stiff pendulum,
Proc. 4th U.S. Nat. Congr. Appl. Mech. (Univ. California, Berkeley, Calif.,
1962) Amer. Soc. Mech. Engrs., New York, 1962, pp. 195–202. MR 0153150
(27 #3119)
 11.
P.
S. Krishnaprasad and J.
E. Marsden, Hamiltonian structures and stability for rigid bodies
with flexible attachments, Arch. Rational Mech. Anal.
98 (1987), no. 1, 71–93. MR 866725
(87m:58084), 10.1007/BF00279963
 12.
O.
A. Ladyženskaya, Smešannaya zadača dlya
giperboličeskogo uravneniya, Gosudarstv. Izdat. Tehn.Teor.
Lit., Moscow, 1953 (Russian). MR 0071631
(17,160c)
 13.
O.
A. Ladyženskaja, V.
A. Solonnikov, and N.
N. Ural′ceva, Linear and quasilinear equations of parabolic
type, Translated from the Russian by S. Smith. Translations of
Mathematical Monographs, Vol. 23, American Mathematical Society,
Providence, R.I., 1968 (Russian). MR 0241822
(39 #3159b)
 14.
J.L.
Lions, Quelques méthodes de résolution des
problèmes aux limites non linéaires, Dunod;
GauthierVillars, Paris, 1969 (French). MR 0259693
(41 #4326)
 15.
Donald
R. Smith, Singularperturbation theory, Cambridge University
Press, Cambridge, 1985. An introduction with applications. MR 812466
(87d:34001)
 16.
S.
L. Sobolev, Applications of functional analysis in mathematical
physics, Translated from the Russian by F. E. Browder. Translations of
Mathematical Monographs, Vol. 7, American Mathematical Society, Providence,
R.I., 1963. MR
0165337 (29 #2624)
 17.
Michael
Wiegner, On the asymptotic behaviour of solutions of nonlinear
parabolic equations, Math. Z. 188 (1984), no. 1,
3–22. MR
767358 (86b:35017), 10.1007/BF01163868
 18.
J.
Patrick Wilber, Absorbing balls for equations modeling nonuniform
deformable bodies with heavy rigid attachments, J. Dynam. Differential
Equations 14 (2002), no. 4, 855–887. MR 1940106
(2003j:37139), 10.1023/A:1020716727905
 19.
J.
Patrick Wilber, Invariant manifolds describing the dynamics of a
hyperbolicparabolic equation from nonlinear viscoelasticity, Dyn.
Syst. 21 (2006), no. 4, 465–489. MR 2273689
(2007m:37197), 10.1080/14689360600821828
 20.
J.
Patrick Wilber and Stuart
S. Antman, Global attractors for degenerate partial differential
equations from nonlinear viscoelasticity, Phys. D 150
(2001), no. 34, 177–206. MR 1820734
(2001m:74013), 10.1016/S01672789(00)002207
 21.
Shui
Cheung Yip, Stuart
S. Antman, and Michael
Wiegner, The motion of a particle on a light viscoelastic bar:
asymptotic analysis of its quasilinear parabolichyperbolic equation,
J. Math. Pures Appl. (9) 81 (2002), no. 4,
283–309. MR 1967351
(2004k:35378), 10.1016/S00217824(01)012272
 22.
W. Weaver, Jr., S.P. Timoshenko, and D.H. Young, Vibration Problems in Engineering, 5th edition, Wiley, 1990.
 23.
Eberhard
Zeidler, Nonlinear functional analysis and its applications.
II/B, SpringerVerlag, New York, 1990. Nonlinear monotone operators;
Translated from the German by the author and Leo F. Boron. MR 1033498
(91b:47002)
 1.
 S.S. Antman, The paradoxical asymptotic status of massless springs, SIAM J. Appl. Math. 48 (1988) 13191334. MR 968832 (90b:35194)
 2.
 S.S. Antman, Nonlinear Problems of Elasticity, 2nd edition, Springer, 2005. MR 2132247 (2006e:74001)
 3.
 S.S. Antman, R.S. Marlow, and C.P. Vlahacos, The complicated dynamics of heavy rigid bodies attached to light deformable rods, Quart. Appl. Math. 56 (1998) 431460. MR 1637036 (99e:73068)
 4.
 S.S. Antman and T.I. Seidman, The parabolichyperbolic system governing the spatial motion of nonlinearly viscoelastic rods, Arch. Rational Mech. Anal. 175 (2005) 85150. MR 2106258 (2006b:74015)
 5.
 S.S. Antman and J.P. Wilber, The asymptotic problem for the springlike motion of a heavy piston in a viscous gas, Quart. Appl. Math. 65 (2007) 471498. MR 2354883 (2009g:76128)
 6.
 J.P. Aubin, Un théorème de compacité, C. R. Acad. Sci., Paris 265 (1963) 50425045. MR 0152860 (27:2832)
 7.
 M.F. Beatty, Finite amplitude oscillations of a simple rubber support system, Arch. Rational Mech. Anal. 83 (1983) 195219. MR 701902 (84g:73039)
 8.
 C. M. Dafermos and J. A. Nohel, Energy methods for nonlinear hyperbolic differential equations, Comm. P.D.E.s 4 (1979) 219278. MR 522712 (80b:45018)
 9.
 A. Friedman, Partial Differential Equations of Parabolic Type, PrenticeHall, 1964. MR 0181836 (31:6062)
 10.
 G.H. Handelman and J.B. Keller, Small vibrations of a slightly stiff pendulum. Proc. 4th U.S. Nat. Congr. Appl. Mech., Vol. 1, Amer. Soc. Mech. Engrs., 1962, pp. 195202. MR 0153150 (27:3119)
 11.
 P. Krishnaprasad and J.E. Marsden, Hamiltonian structures and stability for rigid bodies with flexible attachments, Arch. Rational Mech. Anal. 98 (1987) 7193. MR 866725 (87m:58084)
 12.
 O.A. Ladyženskaja, Mixed Problems for Hyperbolic Equations (in Russian), GITTL, Moscow, 1953. MR 0071631 (17:160c)
 13.
 O.A. Ladyženskaja, V.A. Solonnikov, and N.N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, vol. 23, Amer. Math. Soc., 1967. MR 0241822 (39:3159b)
 14.
 J.L.Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, GauthierVillars, 1969. MR 0259693 (41:4326)
 15.
 D.R. Smith, Singular Perturbation Theory, Cambridge, 1985. MR 812466 (87d:34001)
 16.
 S.L. Sobolev, Applications of Functional Analysis in Mathematical Physics, Amer. Math. Soc., 1963. MR 0165337 (29:2624)
 17.
 M. Wiegner, On the asymptotic behaviour of solutions of nonlinear parabolic equations, Math. Z. 188 (1984) 322. MR 767358 (86b:35017)
 18.
 J.P. Wilber, Absorbing balls for equations modeling nonuniform deformable bodies with heavy rigid attachments, J. Dyn. Diff. Eqs. 19 (2002) 855887. MR 1940106 (2003j:37139)
 19.
 J.P. Wilber, Invariant manifolds describing the dynamics of a hyperbolicparabolic equation from nonlinear elasticity, Dyn. Systems 21 (2006) 465490. MR 2273689 (2007m:37197)
 20.
 J.P. Wilber and S.S. Antman, Global attractors for a degenerate partial differential equation from nonlinear viscoelasticity, Physica D 150 (2001) 179208. MR 1820734 (2001m:74013)
 21.
 S.C. Yip, S.S. Antman, and M. Wiegner, The motion of a particle on a light viscoelastic bar: Asymptotic analysis of the quasilinear parabolichyperbolic equation, J. Math. Pures Appl. 81 (2002) 283309. MR 1967351 (2004k:35378)
 22.
 W. Weaver, Jr., S.P. Timoshenko, and D.H. Young, Vibration Problems in Engineering, 5th edition, Wiley, 1990.
 23.
 E. Zeidler, Nonlinear Functional Analysis and its Applications, Vol. IIB, SpringerVerlag, New York, 1990. MR 1033498 (91b:47002)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC (2010):
35B25,
35C20,
35G31,
35K35,
35K70,
74C20,
74D10,
74K10
Retrieve articles in all journals
with MSC (2010):
35B25,
35C20,
35G31,
35K35,
35K70,
74C20,
74D10,
74K10
Additional Information
Stuart S. Antman
Affiliation:
Department of Mathematics, Institute for Physical Science and Technology, and Institute for Systems Research, University of Maryland, College Park, Maryland 20742
Email:
ssa@math.umd.edu
Süleyman Ulusoy
Affiliation:
Faculty of Education, Zirve University, K$𝜄$z$𝜄$lhisar Campus, 27260 Gaziantep, Turkey
Email:
suleyman.ulusoy@zirve.edu.tr
DOI:
http://dx.doi.org/10.1090/S0033569X2012013250
PII:
S 0033569X(2012)013250
Received by editor(s):
January 2, 2012
Published electronically:
May 25, 2012
Dedicated:
This paper is dedicated to Costas Dafermos on the occasion of his 70th birthday.
Article copyright:
© Copyright 2012
Brown University
The copyright for this article reverts to public domain 28 years after publication.
