Application of stress functions and its dual theory to finite element
应力函数及其对偶理论在有限元中的应用

Cosserat's continuum is a generalized model of the classical elasticity. Many important elastic problems can be taken as its special case subjected to some geometric/mechanical constrains. In some of these problems, there exist the C~1 continuity difficulty in finite element formulation when the elements are constructed in the displacement space. Using Cosserat's continuum, the present work discusses the reason of the appearance of the C~1 continuity difficulty. It is noted that when geometric or/and mechanical constraint(s) is(are) enforced upon Cosserat's model there must exist C~1 continuity requirement for either displacement field or stress function field. And the key point is that only one of these two fields has the C~1 continuity requirement and the other is free from this difficulty. So for some problems with C~1 continuity difficulty in displacement formulation, it is a natural approach to avoid this difficulty by using formulation in stress function space. Nevertheless, the finite element constructed in stress function space is not convenient to apply because stress functions have no explicit physical meaning and then it is difficult to appoint boundary condition for them. For this practical reason, the dual theory of stress functions is presented to provide an approach to transform an clement with stress functions as degree of freedom (DOF) to the element with ordinary displacement as DOF. Based on this dual theory, a general way to construct finite element using stress functions is discussed.