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

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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)−σijϵij) or (B(σij)−σijϵij), 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 ΠHW (ui, ϵij, σij) is real three-field functional, and resolves the historic academic controversy on the issue of constructing a three-field functional.

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Acknowledgements

In the era of great discovery of the generalized variational principle, both Prof. Chien and Prof. Hu made original contribution. Their academic thoughts are very important to our understanding of the generalized variational principle. Now that both Prof. Chien and Prof. Hu have passed away, any progress on Chien’s question could serve as the best tribute to both of them. Therefore, it is my privilege to dedicate this paper to the memories of Prof. Chien and Prof. Hu for their great contribution to the establishment of the generalized variational principle. The author wishes to express his deep gratitude to Prof. Felippa for his private communications and for providing the copy of de Veubeke’s famous paper: Diffusion des inconnues hyperstatiques dans les voilures á longeron couples, Bull. Serv. Technique de L’Aéronautique No. 24, Imprimerie Marcel Hayez, Bruxelles, 56 (1951). Last but not least, the author wishes to express his appreciation to the anonymous reviewers for their high level comments.

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Project supported by Xi’an University of Architecture and Technology (No. 002/2040221134) ©Shanghai University 2022

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  1. School of Civil Engineering & Institute of Mechanics and Technology, Xi’an University of Architecture and Technology, Xi’an, 710055, China

    Bohua Sun

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  1. Bohua Sun
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Correspondence to Bohua Sun.

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Citation: SUN, B. H. On Chien’s question to the Hu-Washizu three-field functional and variational principle. Applied Mathematics and Mechanics (English Edition), 43(4), 537–546 (2022) https://doi.org/10.1007/s10483-022-2838-5

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Sun, B. On Chien’s question to the Hu-Washizu three-field functional and variational principle. Appl. Math. Mech.-Engl. Ed. 43, 537–546 (2022). https://doi.org/10.1007/s10483-022-2838-5

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  • DOI: https://doi.org/10.1007/s10483-022-2838-5

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