Elasticity_ Barber

512 33 Variational methods
of the given geometry in the laboratory, then there exists one and only one
solution.
To examine this question from a mathematical point of view, suppose
provisionally that the solution is non-unique, so that there exist two distinct
stress fields σ 1 , σ 2 , both satisfying the field equations and the same boundary
conditions. We can then construct a new stress field ∆σ =σ 1 − σ 2 by taking
the difference between these fields, which is a form of linear superposition.
The new field ∆σ clearly involves no external loading, since the same external
loads were included in each of the constituent solutions ex hyp. We therefore
conclude from (33.1) that the corresponding total strain energy U associated
with the field ∆σ must be zero. However, U can also be written as a volume
integral of the strain energy density U 0 as in (33.5), and U 0 must be everywhere
positive or zero. The only way these two results can be reconciled is if U 0 is zero
everywhere, implying that the stress is everywhere zero, from (33.3). Thus,
the difference field ∆σ is null, the two solutions must be identical contra hyp.
and only one solution can exist to a given elasticity problem.
The question of existence of solution is much more challenging and will not
be pursued here. A short list of early but seminal contributions to the subject
is given by Sokolnikoff 6 who states that “the matter of existence of solutions
has been satisfactorily resolved for domains of great generality.” More recently,
interest in more general continuum theories including non-linear elasticity has
led to the development of new methodologies in the context of functional
analysis 7 .
33.8.1 Singularities
In §11.2.1, we argued that stress singularities are acceptable in the mathematical
solution of an elasticity problem if and only if the strain energy in a small
region surrounding the singularity is bounded. In the two-dimensional case
(line singularity), this leads to equation (11.37) and hence to the conclusion
that the stresses can vary with r a as r →0 only if a>−1. We can now see the
reason for this restriction, since if there were any points in the body where
the strain energy was not integrable, the total strain energy (33.5) would be
ill-defined and the above proof of uniqueness would fail.
If these restrictions on the permissible strength of singularities are not imposed,
the uniqueness theorem fails and it is quite easy to generate examples in
which a given set of boundary conditions permits multiple solutions. Consider
the flat punch problem of §12.5.2 for which the contact pressure distribution
is given by equation (12.47). If we differentiate the stress and displacement
fields with respect to x, we shall generate a field in which the contact traction
is
6 I.S.Sokolnikoff, Mathematical Theory of Elasticity, McGraw-Hill, New York,
2nd.ed. 1956, §27.
7 See for example, J.E.Marsden and T.J.R.Hughes, Mathematical Foundations of
Elasticity, Prentice-Hall, Englewood Cliffs, 1983, Chapter 6.