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 引用本文: 黄若煜, 吴长春. 应力函数及其对偶理论在有限元中的应用[J]. 力学学报, 2004, 36(4): 419-426.
Application of stress functions and its dual theory to finite element[J]. Chinese Journal of Theoretical and Applied Mechanics, 2004, 36(4): 419-426.
 Citation: Application of stress functions and its dual theory to finite element[J]. Chinese Journal of Theoretical and Applied Mechanics, 2004, 36(4): 419-426.

## Application of stress functions and its dual theory to finite element

• 摘要: 借助于Cosserat连续介质模型，探讨了应力函数和位移对避免有限元C^1连续性困难的互补性作用. 通过对应力函数对偶理论的深入分析，为将应力函数列式得到的余能单元转化为具有一般位移自由度的势能单元提供了严格的理论基础，在此基础上，给出应用应力函数构造有限元的一般方法.

Abstract: 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.

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