### Geometrical Obstacles for the PDEs of Solid Mechanics

Stuart S. Antman

University of Maryland

Mathematics

Problems in the classical field theories of physics, namely, solid
and fluid mechanics, thermodynamics, electricity and magnetism, and
relativity, are typically governed by (nonlinear) partial differential
equations or by more complicated equations. Each of these theories
has a rich and subtle structure. Perhaps the conceptually simplest
theory is that of continuum mechanics, which describes the motion of
deformable bodies in three-dimensional Euclidean space under the
action of forces. The aims of this lecture are to show (i) that the
requirement that the motion occur naturally in the familiar Euclidean
space is a source of severe analytical difficulties, i.e.,
three-dimensional geometry is intrinsically hard, (ii) how some of
these difficulties can be overcome for relatively simple (vectorial)
nonlinear problems of viscoelasticity, and (iii) that failure to
respect the underlying geometry can lead to serious errors in the
numerical treatment of the governing equations.