Preprint Article Version 2 Preserved in Portico This version is not peer-reviewed

# Thermodynamic Foundation of Generalized Variational Principle

Version 1 : Received: 3 November 2020 / Approved: 3 November 2020 / Online: 3 November 2020 (15:28:25 CET)
Version 2 : Received: 2 August 2021 / Approved: 3 August 2021 / Online: 3 August 2021 (13:01:53 CEST)

How to cite: Sun, B. Thermodynamic Foundation of Generalized Variational Principle. Preprints 2020, 2020110168 (doi: 10.20944/preprints202011.0168.v2). Sun, B. Thermodynamic Foundation of Generalized Variational Principle. Preprints 2020, 2020110168 (doi: 10.20944/preprints202011.0168.v2).

## Abstract

One open question remains regarding the theory of the generalized variational principle, that is, why the stress-strain relation still be derived from the 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. As long as the functional has one of the combination $A(\epsilon_{ij})-\sigma_{ij}\epsilon_{ij}$ or $B(\sigma_{ij})-\sigma_{ij}\epsilon_{ij}$, its corresponding variational principle will produce the stress-strain relation without the need to introduce extra constraints by the Lagrangian multiplier method. It is proved herein that the Hu-Washizu functional $\Pi_{HW}[u_i,\epsilon_{ij},\sigma_{ij}]$ and Hu-Washizu variational principle comprise a real three-field functional.

## Keywords

variational principle; elasticity; Lagrangian multipliers; thermodynamics; entropy

Comment 1
Commenter: Bohua Sun
Commenter's Conflict of Interests: Author
Comment: some typos are corrected
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