Method of High-Order Lagrange Multiplier and Generalized Variational Principles of Elasticity with More General Forms of Functionals

It is known^[1] that the minimum principles of potential energy and complementary energy are the conditional variation principles under respective conditions of constraints. By means of the method of La-grange multipliers, we are able to reduce the functionals of conditional variation principles to new functionals of non-conditional variation principles. This method can be described as follows:Multiply undetermined Lagrange multipliers by various constraints, and add these products to the original functionals.Considering these undetermined Lagrange multipliers and the original variables in these new functionals as independent variables of variation,we can see that the stationary conditions of these functionals give these unceter -mined Lagrange multipliers in terms of original variables. The substitutions of Ihese results for Lagrange multipliers into the above functionals lead to the functionals of these non-conditional variation principles.However, in certain cases, some of the undetermined Lagrange multipliers may turn out to be zero during variation.This is a critical state of variation. In this critical state,the corresponding variational constraint cannot be eliminated by means of the simple Lagrange multiplier method. This is indeed the case when one tries to eliminate the constraint condition of stress-strain relation in the variational principle of minimum complementary energy by the me-thod of Lagrange multiplier. By means of Lagrange multiplier method one can only derive, from minimum complementary energy principle,the Hellinger-Reissner principle^[2,3], in which only two types of independent variables, stresses and displacements, exist in the new functional. The strain-stress relation remains to be a constraint,from which one derives the strain from the given stress. Thus the Hellinger-Reissner principle remains to be a conditional variation with one constraint uneliminated.In ordinary Lagrange multiplier method, only the linear terms of constraint conditions are taken into consideration. It is impossible to incorporate this condition of constraint into functional whenever the corresponding Lagrange multiplier turns out to be zero. Hence, we extend the Lagrange multiplier method by considering not only the linear term, but also the high-order terms,such as thequa-dratic terms of constraint in the Taylor's series expansion.We call this method the high order Lagrange multiplier method.With this method we find the more general form of functional of the generalized variational principle ever known to us from the Hellinger-Reissner principle. In particular, this more general form of functional can be all known functionals of existing generalized variational principles in elasticity. Similarly, we can also find the more general form of functional from He-Washizu principle^[4,5].It is also shown that there are equivalent theorem and related equivalent relation between these two general forms of functionals in elasticity.