Geometrical Obstacles for the PDEs of Solid Mechanics

Stuart S. Antman
University of Maryland

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.