Several patterns of functional transformation and generalized variational principles with several arbitrary parameters

Abstract

In this paper, the functional transformations of variational principles in elasticity are classified as three patterns: pattern I (relaxation pattern) is a generalized equivalent pattern in which constraint conditions are transformed into natural conditions; pattern II (augmented pattern) is a generalized equivalent pattern in which augmented conditions are transformed into natural conditions; pattern III (equivalent pattern) is a pattern in which a nonconditional functional is transformed into an equivalent functional with several arbitrary parameters. Pattern I is the well-known pattern of Lagrange multipliers method: patterns II and III are new patterns proposed in this paper. On the basis of pattern III, generalized variational principles with several arbitrary parameters are formulated, and the general and simple forms of the functional are defined. Many existing functionals of variational principles in elasticity are special cases of this functional.