The Parabolic-Hyperbolic System Governing the Spatial Motion of Nonlinearly Viscoelastic Rods
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- Antman, S. & Seidman, T. Arch. Rational Mech. Anal. (2005) 175: 85. doi:10.1007/s00205-004-0341-6
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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 , 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 the 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.