On Chien’s question to the Hu-Washizu three-field functional and variational principle

doi:10.1007/s10483-022-2838-5

摘要/Abstract

摘要： There is an open question, namely Chien’s question, in construction of a generalized functional in elasticity, i.e., why the stress-strain relation can still be derived from the Hu-Washizu generalized variational principle while the Lagrangian multiplier method is applied in vain? This study shows that the generalized variational principle can only be understood and implemented correctly within the framework of thermodynamics. This investigation finds that as long as the functional has one of the combinations (A(εij)-σ_ijε_ij) or (B(σ_ij)-σ_ijε_ij), its corresponding variational principle will produce the stress-strain relation without the need to introduce extra constraints by the Lagrangian multiplier method. This research proves that the Hu-Washizu functional Π_HW(u_ij, ε_ij; σ_ij) is real three-field functional, and resolves the historic academic controversy on the issue of constructing a three-field functional.

关键词: variational principle, elasticity, Lagrangian multiplier, thermodynamics, entropy

Abstract: There is an open question, namely Chien’s question, in construction of a generalized functional in elasticity, i.e., why the stress-strain relation can still be derived from the Hu-Washizu generalized variational principle while the Lagrangian multiplier method is applied in vain? This study shows that the generalized variational principle can only be understood and implemented correctly within the framework of thermodynamics. This investigation finds that as long as the functional has one of the combinations (A(εij)-σ_ijε_ij) or (B(σ_ij)-σ_ijε_ij), its corresponding variational principle will produce the stress-strain relation without the need to introduce extra constraints by the Lagrangian multiplier method. This research proves that the Hu-Washizu functional Π_HW(u_ij, ε_ij; σ_ij) is real three-field functional, and resolves the historic academic controversy on the issue of constructing a three-field functional.

Key words: variational principle, elasticity, Lagrangian multiplier, thermodynamics, entropy

中图分类号:

Bohua SUN. On Chien’s question to the Hu-Washizu three-field functional and variational principle[J]. Applied Mathematics and Mechanics (English Edition), 2022, 43(4): 537-546.

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