Appl Math Optim 46:291–312 (2002)
2002 Springer-Verlag New York Inc.
A Nonlinear Elastic Beam System with Inelastic Contact Constraints
David L. Russell
and Luther W. White
Department of Mathematics,
Virginia Polytechnic Institute and State University,
Blacksburg, VA 24061-123, USA
Department of Mathematics, University of Oklahoma, Norman,
Norman, OK 73019, USA
Abstract. In this paper we study freely propagating inertial, i.e., unforced, waves,
in an elastic beam constrained so that all motion takes place above and on a ﬂat,
rigid support surface, subject to a gravitational force and a compressive longitudinal
load. Contact between the beam and the support surface is assumed to be completely
inelastic. A nonlinear beam model is used, incorporating a quartic extension of
the familiar quadratic potential energy functional for the standard Euler–Bernoulli
model. After brieﬂy reviewing the rationale for the model and some of its properties,
as developed in earlier articles, we present existence and uniqueness results for the
constrained system obtained with the use of a “penalty function” approach involving
the addition of a “uni-directional friction” dissipative term, active only when the
constraint is violated, to the unconstrained system.
Key Words. Nonlinear beam, Supported beam, Buckling, Traveling waves.
AMS Classiﬁcation. 35J25, 35J50, 35Q72, 49L10.
1. Introduction; Background
In a number of recent articles , , ,  the authors have studied static buckling
and free wave propagation phenomena in the context of a nonlinear beam model. The
model in question is similar to that originally introduced by Lagnese , and also to a
slightly different model treated by the present authors in the ﬁrst of these papers, but is
modiﬁed to the extent that the beam is supported on a ﬂat, rigid, inelastic surface, so that
the only permissible vertical displacements are positive. The beam is also subject to a