Asymptotics and Numerics for Thin Ligaments

                  by Xanthippi Markenscoff
              UCSD, Dept of Applied Mechanics
                  and Engineering Sciences

               JTB 120, 3:20pm March 10, 1997

                          ABSTRACT

In narrow regions of materials, such as between holes, or
between holes and free boundaries, or between inclusions,
which we call thin ligaments, the stress amplifies
singularly as the ligament thickness vanishes.

The question that arises is how the stress relates
asymptotically to the ligament thickness as it becomes
vanishingly small. For many classical problems, such as a
hole near a boundary in tension, two nearby holes in
tension, two rigid fibers in shear, etc, due to mathematical
difficulties of the asymptotics of nonuniformly convergent
series, it was not possible to obtain the limiting behavior
of the stress and the question has remained open until
recently.

Actually, this is a case where the numerical procedure also
breaks down for very thin ligaments, because the number fo
terms required to be kept in the series for convergence
exceeds the capacity of the computers.

Singular asymptotics of series were developed and the
limiting behavior of the stress as a function of ligament
thickness was obtained. Comparisons with numerical
evaluation are presented. The relation of the full field
solution to the beam theory approximation of the thin
ligaments is discussed.

Several examples of practical importance exhibiting stress
amplification in thin ligaments under various loadings,
including tension, presssure, body forces and thermal
loadings are presented.

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This source can be found at http://www.math.utah.edu/research/