The asymptotics of heavily burdened viscoelastic rods

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 $\varepsilon$. 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 parabolic-hyperbolic system of partial differential equations coupled to the ordinary differential equations for the rigid body, with $\varepsilon$ appearing in the coefficients of the acceleration terms of the rod. This paper gives a rigorous asymptotic expansion of the solutions of initial-boundary-value problems for this system, consisting of a regular expansion and an initial-layer expansion. The leading term of the regular expansion satisfies the reduced problem, obtained by setting $\varepsilon =0$ in the governing equations. The reduced problem is governed by a curious set of quasilinear functional-differential 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 steady-state 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 initial-layer 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 initial-layer 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 Faedo-Galerkin method is used to obtain sharp estimates for the exponential decay in time of the terms of the initial-layer 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) 3–22) for treating the analogous scalar problem by Yip et al., J. Math. Pures Appl. 81 (2002) 283–309. The Maximum Principle is not applicable to our parabolic systems. Even for such scalar problems, the Faedo-Galerkin 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 strain-rate type invariant under rigid motions.