 
Summary: Scuola Normale Superiore  Pisa September 1988
A Variational Definition of the Strain Energy
for an Elastic String
E. Acerbi, G. Buttazzo, D. Percivale
1. Introduction
The aim of this paper is to deduce the constitutive equations of an elastic one
dimensional string from the stressstrain relations of nonlinear threedimensional elas
ticity, by passing to the limit when the other dimensions go to zero. We will use the
variational point of view.
Denote by the reference configuration of the string:
= (x1, x2, x3) : 0 x1 1, x2 = x3 = 0 ,
and by the "thick" elastic body
= (x1, x2, x3) : 0 x1 1, x2
2 + x2
3 2
.
We assume the stored strain energy, associated to a displacement field u, to be given
by a functional of the form
f(Du) dx,
