Application of stress functions and its dual theory to finite element

Cosserat's continuum is a generalized model of the classical elasticity. Many important elastic problems can be taken as itsspecial case subjected to some geometric/mechanical constrains. In some ofthese problems, there exist the C^1 continuity difficulty in finiteelement formulation when the elements are constructed in the displacementspace. Using Cosserat's continuum, the present work discusses the reason ofthe appearance of the C^1 continuity difficulty. It is noted that whengeometric or/and mechanical constraint(s) is(are) enforced upon Cosserat'smodel there must exist C^1 continuity requirement for eitherdisplacement field or stress function field. And the key point is that onlyone of these two fields has the C^1 continuity requirement and the otheris free from this difficulty. So for some problems with C^1 continuitydifficulty in displacement formulation, it is a natural approach to avoidthis difficulty by using formulation in stress function space. Nevertheless,the finite element constructed in stress function space is not convenient toapply because stress functions have no explicit physical meaning and then itis difficult to appoint boundary condition for them. For this practicalreason, the dual theory of stress functions is presented to provide anapproach to transform an element with stress functions as degree of freedom(DOF) to the element with ordinary displacement as DOF. Based on this dualtheory, a general way to construct finite element using stress functions isdiscussed.