Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed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)

How to cite: Sun, B. Thermodynamic Foundation of Generalized Variational Principle. Preprints 2020, 2020110168 (doi: 10.20944/preprints202011.0168.v1). Sun, B. Thermodynamic Foundation of Generalized Variational Principle. Preprints 2020, 2020110168 (doi: 10.20944/preprints202011.0168.v1).

## Abstract

One long-standing open question remains regarding the theory of the generalized variational principle, that is, why can the stress-strain relation still be derived from the generalized variational principle while the method of 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. In addition, that Chien's functional $\Pi_{Q}[u_i,\epsilon_{ij},\sigma_{ij},\lambda]$ is a much more general four-field functional and that the Hu-Washizu functional is its special case as $\lambda=0$ are confirmed.

## Subject Areas

variational principle; elasticity; Lagrangian multipliers; thermodynamics; entropy

Views 0