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**The parabolic-hyperbolic system governing the spatial motion of nonlinearly viscoelastic rods.**
*(English)*
Zbl 1145.74020

Summary: This paper treats initial-boundary-value problems governing the motion in space of nonlinearly viscoelastic rods of strain-rate type. It introduces and exploits a set of physically natural constitutive hypotheses to prove that solutions exist for all time and depend continuously on the data. The equations are those for a very general properly invariant theory of rods that can suffer flexure, torsion, extension, and shear. In this theory, the contact forces and couples depend on strains measuring these effects and on the time derivatives of these strains.

The governing equations form an eighteenth-order quasilinear parabolic- hyperbolic system of partial differential equations in one space variable (the system consisting of two vectorial equations in Euclidean 3-space corresponding to the linear and angular momentum principles, each equation involving third-order derivatives). The existence theory for this system or even for its restrictced version governing planar motions has never been studied. Our work represents a major generalization of the treatment of purely longitudinal motions of [(*) the authors, J. Differ. Equations 124, No. 1, 132–185 (1996; Zbl 0844.35021)], governed by a scalar quasilinear third-order parabolic-hyperbolic equation. The paper [(*)] in turn generalizes an extensive body of work, which it cites.

Our system has a strong mechanism of internal friction embodied in the requirement that the consitutive function taking the strain rates to the contact forces and couples be uniformly monotone. As in [(*)], our system is singular in the sense that certain constitutive functions appearing in the principal part of the differential operator blow up as the strain variables approach a surface corresponding to a “total compression”.

We devote special attention to those inherent technical difficulties that follow from the underlying geometrical significance of the governing equations, from the requirement that the material properties be invariant under rigid motions, and from the consequent dependence on space and time of the natural vectorial basis for all geometrical and mechanical vector-valued functions. (None of these difficulties arises in [(*)].) In particular, for our model, the variables defining a configuration lie on a manifold, rather than merely in a vector space. These kinematical difficulties and the singular nature of the equations prevent our analysis from being a routine application of available techniques.

The foundation of our paper is the introduction of reasonable consitutive hypotheses that produce an a priori pointwise bound preventing a total compression and a priori pointwise bounds on the strains and strain rates. These bounds on the arguments of our consitutive functions allow us to use recent results on the extension of monotone operators to replace the original singular problem with an equivalent regular problem. This we analyze by using a modification of Faedo-Galerkin method, suitably adapted to the peculiarities of our parabolic-hyperbolic system, which stem from the underlying mechanics. Our consitutive hypotheses support bounds and consequent compactness properties for the Galerkin approximations so strong that these approximations are shown to converge to the solution of the initial-boundary-value problem without appeal to the theory of monotone operators to handle the weak convergence of composite functions.

The governing equations form an eighteenth-order quasilinear parabolic- hyperbolic system of partial differential equations in one space variable (the system consisting of two vectorial equations in Euclidean 3-space corresponding to the linear and angular momentum principles, each equation involving third-order derivatives). The existence theory for this system or even for its restrictced version governing planar motions has never been studied. Our work represents a major generalization of the treatment of purely longitudinal motions of [(*) the authors, J. Differ. Equations 124, No. 1, 132–185 (1996; Zbl 0844.35021)], governed by a scalar quasilinear third-order parabolic-hyperbolic equation. The paper [(*)] in turn generalizes an extensive body of work, which it cites.

Our system has a strong mechanism of internal friction embodied in the requirement that the consitutive function taking the strain rates to the contact forces and couples be uniformly monotone. As in [(*)], our system is singular in the sense that certain constitutive functions appearing in the principal part of the differential operator blow up as the strain variables approach a surface corresponding to a “total compression”.

We devote special attention to those inherent technical difficulties that follow from the underlying geometrical significance of the governing equations, from the requirement that the material properties be invariant under rigid motions, and from the consequent dependence on space and time of the natural vectorial basis for all geometrical and mechanical vector-valued functions. (None of these difficulties arises in [(*)].) In particular, for our model, the variables defining a configuration lie on a manifold, rather than merely in a vector space. These kinematical difficulties and the singular nature of the equations prevent our analysis from being a routine application of available techniques.

The foundation of our paper is the introduction of reasonable consitutive hypotheses that produce an a priori pointwise bound preventing a total compression and a priori pointwise bounds on the strains and strain rates. These bounds on the arguments of our consitutive functions allow us to use recent results on the extension of monotone operators to replace the original singular problem with an equivalent regular problem. This we analyze by using a modification of Faedo-Galerkin method, suitably adapted to the peculiarities of our parabolic-hyperbolic system, which stem from the underlying mechanics. Our consitutive hypotheses support bounds and consequent compactness properties for the Galerkin approximations so strong that these approximations are shown to converge to the solution of the initial-boundary-value problem without appeal to the theory of monotone operators to handle the weak convergence of composite functions.

### MSC:

74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |

74D10 | Nonlinear constitutive equations for materials with memory |

35Q72 | Other PDE from mechanics (MSC2000) |

### Citations:

Zbl 0844.35021
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\textit{S. S. Antman} and \textit{T. I. Seidman}, Arch. Ration. Mech. Anal. 175, No. 1, 85--150 (2005; Zbl 1145.74020)

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### References:

[1] | Alexits, G.: Convergence Problems of Orthogonal Series. Pergamon, 1961 · Zbl 0098.27403 |

[2] | Andrews, G.: On the existence of solutions to the equation utt =uxxt +?(ux)x. J. Diff. Eqs. 35, 200-231 (1980) · Zbl 0415.35018 |

[3] | Andrews, G., Ball, J.M.: Asymptotic behaviour and changes of phase in one- dimensional nonlinear viscoelasticity. J. Diff. Eqs. 44, 306-341 (1982) · Zbl 0501.35011 |

[4] | Antman, S.S.: Ordinary differential equations of one-dimensional nonlinear elasticity II: Existence and regularity theory for conservative problems. Arch. Rational Mech. Anal. 61, 353-393 (1976) · Zbl 0354.73047 |

[5] | Antman, S.S.: Extensions of monotone mappings. C. R. Acad. Sci., Paris, Sér. I 323, 235-239 (1996) · Zbl 0889.47029 |

[6] | Antman, S.S.: Invariant dissipative mechanisms for the spatial motion of rods suggested by artificial viscosity. J. Elasticity 70, 55-64 (2003) · Zbl 1046.74025 |

[7] | Antman, S.S.: Nonlinear Problems of Elasticity. 2nd. edn., Springer, 2004 |

[8] | Antman, S.S., Kenney, C.S.: Large buckled states of nonlinearly elastic rods under torsion, thrust, and gravity. Arch. Rational Mech. Anal. 76, 289-338 (1981) · Zbl 0472.73036 |

[9] | Antman, S.S., Koch, H.: Self-sustained oscillations of nonlinearly viscoelastic layers. SIAM J. Appl. Math. 60, 1357-1387 (2000) · Zbl 0956.35009 |

[10] | Antman, S.S., Marlow, R.S., Vlahacos, C.P.: The complicated dynamics of heavy rigid bodies attached to light deformable rods. Quart. Appl. Math. 56, 431-460 (1998) · Zbl 0960.74028 |

[11] | Antman, S.S., Seidman, T.I.: Large shearing motions of nonlinearly viscoelastic slabs. Bull. Tech. Univ. Istanbul 47, 41-56 (1994) · Zbl 0860.73018 |

[12] | Antman, S.S., Seidman, T.I.: Quasilinear hyperbolic-parabolic equations of nonlinear viscoelasticity. J. Diff. Eqs. 124, 132-185 (1996) · Zbl 0844.35021 |

[13] | Aubin, J.P.: Un théorème de compacité. C. R. Acad. Sci., Paris 265, 5042-5045 (1963) · Zbl 0195.13002 |

[14] | Bethuel, F., Brezis, H., Hélein, F.: Ginzberg-Landau Vortices. Birkhäuser, 1993 |

[15] | Caflisch, R.E., Maddocks, J.H.: Nonlinear dynamical theory of the elastica. Proc. Roy. Soc. Edinburgh A 99, 1-23 (1984) · Zbl 0589.73057 |

[16] | Calderer, M.C.: The dynamic behavior of viscoelastic spherical shells. J. Diff. Eqs. 63, 289-305 (1986) · Zbl 0598.73098 |

[17] | Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. McGraw-Hill, 1955 · Zbl 0064.33002 |

[18] | Dafermos, C.M.: The mixed initial-boundary value problem for the equations of nonlinear 1-dimensional viscoelasticity. J. Diff. Eqs. 6, 71-86 (1969) · Zbl 0218.73054 |

[19] | Dafermos, C.M.: Global smooth solutions to the initial-boundary-value problem for the equations of one-dimensional nonlinear thermoviscoelasticity. SIAM J. Math. Anal. 13, 397-408 (1982) · Zbl 0489.73124 |

[20] | French, D.A., Jensen, S., Seidman, T.I.: A space-time finite element method for a class of nonlinear hyperbolic-parabolic equations. Appl. Num. Math. 31, 429-450 (1999) · Zbl 0937.74063 |

[21] | Hagan, R., Slemrod, M.: The viscosity-capillarity admissibility criterion for shocks and phase transitions. Arch. Rational Mech. Anal. 83, 333-361 (1983) · Zbl 0531.76069 |

[22] | Kanel?, Ya. I.: On a model system of equations of one-dimensional gas motion (in Russian). Diff. Urav. 4, 721-734 (1969); English translation: Diff. Eqs. 4, 374-380 (1969) |

[23] | Koch, H., Antman, S.S.: Stability and Hopf bifurcation for fully nonlinear parabolic-hyperbolic equations. SIAM J. Math. Anal. 32, 360-384 (2001) · Zbl 1049.35145 |

[24] | Ladyzenskaja, O.A., Solonnikov, V.A., Ural?ceva, N.: Linear and Quasi-Linear Equations of Parabolic Type. Amer. Math. Soc., 1968 |

[25] | Leslie, F.M., Stewart, I.W. (eds): Mathematical models of liquid crystals. Euro. J. Appl. Math. 8, 251-310 (1997) |

[26] | Lions, J.L.: Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires. Dunod, Gauthier-Villars, 1969 |

[27] | MacCamy, R.C.: Existence, uniqueness and stability of Indiana Univ. Math. J. 20, 231-238 (1970) · Zbl 0204.10903 |

[28] | Pego, R.L.: Phase transitions in one-dimensional nonlinear viscoelasticity: Admissibility and stability. Arch. Rational Mech. Anal. 97, 353-394 (1987) · Zbl 0648.73017 |

[29] | Seidman, T.I.: The transient semiconductor problem with generation terms, II. In: Nonlinear Semigroups, Partial Differential Equations, and Attractors. T. E. Gill, W. W. Zachary (eds), 1394 Springer Lect. Notes Math. 1989, pp. 185-198 |

[30] | Seidman, T.I., Antman, S.S.: Optimal control of a nonlinearly viscoelastic rod. In: Control of Nonlinear Distributed Parameter Systems. G. Chen, I. Lasiecka, J. Zhou (eds), Marcel Dekker, 2001, pp. 273-283 · Zbl 0976.49006 |

[31] | Seidman, T.I., Antman, S.S.: Optimal control of the spatial motion of a viscoelastic rod. Dynamics of Continuous, Discrete, and Impulsive Systems 10, 679-691 (2003) · Zbl 1033.49003 |

[32] | Seidman, T.I., Wolfe, P.: Equilibrium states of an elastic conducting rod in a magnetic field. Arch. Rational Mech. Anal. 102, 307-329 (1988) · Zbl 0668.73071 |

[33] | Simo, J.C., Vu-Quoc, L.: On the dynamics in space of rods undergoing large motions?-A geometrically exact approach. Comp. Meths. Appl. Mech. Engg. 66, 125-161 (1988) · Zbl 0618.73100 |

[34] | Simon, J.: Compact sets in the space Lp(0,T;B). Ann. Mat. Pura Appl. 146, 65-96 (1987) · Zbl 0629.46031 |

[35] | Slemrod, M.: Dynamics of first order phase transitions, Phase Transformations and Instabilities in Solids. M. E. Gurtin(ed), Academic Pr., 1984, 163-203 |

[36] | Watson, S.: Unique global solvability for initial-boundary-value problems in one-dimensional nonlinear thermoviscoelasticity. Arch. Rational Mech. Anal. 153, 1-37 (2000) · Zbl 0996.74032 |

[37] | Whittaker, E.T.: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. 4th ed., Cambridge Univ. Press, 1937 · JFM 63.1286.03 |

[38] | Yip, S.C., Antman, S.S., Wiegner, M.: The motion of a particle on a light viscoelastic bar: Asymptotic analysis of the quasilinear parabolic-hyperbolic equation. J. Math. Pures Appl. 81, 283-309 (2002) · Zbl 1134.35397 |

[39] | Zeidler, E.: Nonlinear Functional Analysis and it Applications, Vol. II/B, Nonlinear Monotone Operators. Springer, 1990 · Zbl 0684.47029 |

[40] | Zheng, S: Nonlinear Parabolic Equations and Hyperbolic-Parabolic Coupled Systems. Longman, 1995 · Zbl 0835.35003 |

[41] | Ziegler, H.: An Introduction to Thermomechanics. North-Holland, 1977 · Zbl 0358.73001 |

[42] | Ziegler, H., Wehrli, C.: The derivation of constitutive relations from the free energy and the dissipative function. In: Advances in Applied Mechanics, vol. 25, T. Y. Wu, J. W. Hutchinson (eds), Academic Press, 1987, pp. 183-238 · Zbl 0719.73001 |

[43] | Zygmund, A.: Trigonometric Series. 2nd edn., Cambridge Univ. Press, 1959 · Zbl 0085.05601 |

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