×

Dynamical problems for geometrically exact theories of nonlinearly viscoelastic rods. (English) Zbl 0845.73034

Summary: This paper surveys recent results and open problems for the equations of motion for geometrically exact theories of nonlinearly viscoelastic and elastic rods. These rods can deform in space by undergoing not only flexure and torsion, but also extension and shear. The paper begins with a derivation of the governing equations, which for viscoelastic rods form a quasilinear system of hyperbolic-parabolic partial differential equations of high order. It then derives the energy equation and discusses difficulties that can arise in getting useful energy estimates. The paper next treats constitutive assumptions precluding total compression. The paper then discusses the curious asymptotic problems that arise when the inertia of the rod is small relative to that of a rigid body attached to its end. The paper concludes with discussions of traveling waves and shock structure, Hopf bifurcation problems, and problems of control.

MSC:

74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74Hxx Dynamical problems in solid mechanics
74-02 Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] G. Andrews and J. M. Ball, Asymptotic behaviour and changes of phase in one-dimensional nonlinear viscoelasticity,J. Diff. Eqs. 44 (1982), 306–341. · Zbl 0501.35011
[2] S. S. Antman, The paradoxical asymptotic status of massless springs,SIAM J. Appl. Math. 48 (1988), 1319–1334. · Zbl 0661.73018
[3] S. S. Antman,Nonlinear Problems of Elasticity, Springer-Verlag, New York, 1995. · Zbl 0820.73002
[4] S. S. Antman and H. Koch, Self-sustained oscillations of nonlinearly viscoelastic bodies (Hopf bifurcation for a quasilinear hyperbolic-parabolic equation), to appear.
[5] S. S. Antman and C. S. Kenney, Large buckled states of nonlinearly elastic rods under torsion, thrust, and gravity,Arch. Rational Mech. Anal. 76 (1981), 289–338. · Zbl 0472.73036
[6] S. S. Antman and T.-P. Liu, Travelling waves in hyperelastic rods,Quart. Appl. Math. 36 (1979), 377–399. · Zbl 0408.73043
[7] S. S. Antman and R. Malek-Madani, Travelling waves in nonlinearly viscoelastic media and shock structure in elastic media,Quart. Appl. Math. 46 (1988), 77–93. · Zbl 0677.73022
[8] S. S. Antman and R. S. Marlow, Material constraints, Lagrange multipliers, and compatibility,Arch. Rational Mech. Anal. 116 (1991), 257–299. · Zbl 0769.73012
[9] 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., to appear. · Zbl 0960.74028
[10] S. S. Antman and T. I. Seidman, Quasilinear hyperbolic-parabolic equations of nonlinear viscoelasticity,J. Diff. Eqs., to appear. · Zbl 0844.35021
[11] S. S. Antman and T. I. Seidman, Large shearing motions of nonlinearly viscoelastic slabs,Bull. Tech. Univ. Istanbul 47 (1994), 41–56. · Zbl 0860.73018
[12] S. S. Antman and T. I. Seidman, Hyperbolic-parabolic systems governing the spatial motion of nonlinearly viscoelastic rods, to appear. · Zbl 1145.74020
[13] M. Beck, Die Knicklast des einseitig eingespannten tangential gedrückten Stabes,Z. Angew. Math. Phys. 3 (1952), 225–228. · Zbl 0046.17703
[14] J. Carr and M. Z. M. Malhardeen, Beck’s problem,SIAM J. Appl. Math. 37 (1979), 261–262. · Zbl 0426.35015
[15] M.-S. Chen, Hopf bifurcation in Beck’s problem,Nonlin. Anal. T. M. A. 11 (1987), 1061–1073. · Zbl 0657.35011
[16] A. Cimetière, G. Geymonat, H. Le Dret, A. Raoult, and Z. Tutek, Asymptotic theory and analysis for displacements and stress distribution in nonlinear elastic straight slender rods,J. Elasticity 19 (1988), 111–161. · Zbl 0653.73010
[17] C. M. Dafermos, The mixed initial-boundary value problem for the equations of nonlinear one-dimensional viscoelasticity,J. Diff. Eqs. 6 (1969), 71–86. · Zbl 0218.73054
[18] L. Euler,Additamentum I de curvis elasticis, methodus inveniendi lineas curvas maximi minimivi proprietate gaudentes, Bousquent, Lausanne, 1744, inOpera Omnia I, Vol. 24, 231–297.
[19] R. L. Fosdick and R. D. James, The elastica and the problem of pure bending for a nonconvex stored energy function,J. Elasticity 11 (1981), 165–186. · Zbl 0481.73018
[20] A. E. Green and P. M. Naghdi, Electromagnetic effects in the theory of rods,Phil. Trans. R. Soc. Lond. A 314 (1985), 311–352. · Zbl 0568.73108
[21] J. M. Greenberg, R. C. MacCamy, and V. J. Mizel, On the existence, uniqueness, and stability of solutions of the equation {\(\sigma\)}’(u x )u xx +{\(\lambda\)}u txt ={\(\rho\)}0 u tt ,J. Math. Mech. 17 (1968), 707–728.
[22] R. D. James, The equilibrium and post-buckling behavior of an elastic curve governed by a non-convex energy,J. Elasticity 11 (1981), 239–269. · Zbl 0514.73029
[23] Ya. I. Kanel’, On a model system of equations of one-dimensional gas motion (in Russian),Diff. Urav. 4 (1969), 721–734; English transl:Diff. Eqs. 4 (1969), 374–380.
[24] A. J. Karwowski, Asymptotic models for a long, elastic cylinder,J. Elasticity 24 (1990), 229–287. · Zbl 0733.73038
[25] H. Koch and S. S. Antman, Hopf bifurcation for fully nonlinear hyperbolic-parabolic equations. Applications to nonlinear viscoelasticity, in preparation. · Zbl 1049.35145
[26] R. C. MacCamy, Existence, uniqueness and stability ofu tt =[{\(\sigma\)}(u x )+{\(\lambda\)}(u x )(u xt)],Indiana Univ. Math. J. 20 (1970), 231–238. · Zbl 0204.10903
[27] A. Mielke, Saint-Venant’s problem and semi-inverse solutions in nonlinear elasticity,Arch. Rational Mech. Anal. 102 (1988), 205–229; Corrigendum, ibid.110 (1990), 351–352. · Zbl 0651.73006
[28] A. Mielke, Normal hyperbolicity of center manifolds and Saint-Venant’s principle,Arch. Rational Mech. Anal. 110 (1990), 353–372. · Zbl 0706.73016
[29] E. L. Nikolai, On the stability of the straight equilibrium form of a compressed and twisted rod (in Russian),Izv. Leningr. Politekh. Inst. 31 (1928), 357–387; reprinted in E. L. Nikolai,Works on Mechanics (in Russian), G.I.T.T.L., 1955.
[30] R. L. Pego, Phase transitions in one-dimensional nonlinear viscoelasticity: Admissibility and stability,Arch. Rational Mech. Anal. 97 (1987), 353–394. · Zbl 0648.73017
[31] H. T. Savage and M. L. Spano, Theory and application of highly magnetoelastic Metglas 2605SC,J. Appl. Phys. (1982), 8092–8097.
[32] J. C. Simo, A finite strain beam formulation. The three-dimensional dynamical problem. Part I,Comp. Meths. Appl. Mech. Eng. 49 (1985), 55–70. · Zbl 0583.73037
[33] J. C. Simo and L. Vu-Quoc, Three-dimensional finite strain rod model. Part I: Computational aspects,Comp. Meths. Appl. Mech. Eng. 58 (1986), 79–116. · Zbl 0608.73070
[34] J. C. Simo and L. Vu-Quoc, On the dynamics of flexible beams under large overall motions–The plane case: Part I, Part II,J. Appl. Mech. 53 (1986), 849–863. · Zbl 0607.73057
[35] J. C. Simo and L. Vu-Quoc, On the dynamics in space of rods undergoing large motions–A geometrically exact approach.Comp. Meths. Appl. Mech. Eng. 66 (1988), 125–161. · Zbl 0618.73100
[36] J. C. Simo and L. Vu-Quoc, A geometrically exact rod model incorporating shear and torsion-warping deformation,Int. J. Solids Structures 27 (1991), 371–393. · Zbl 0731.73029
[37] J. Smoller,Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983. · Zbl 0508.35002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.