Deriving generalized variational principles in general mechanics by using Lagrangian multiplier method

1999 ◽  
Vol 42 (12) ◽  
pp. 1332-1339 ◽  
Author(s):  
Lifu Liang
Keyword(s):  
Variational Principles ◽  
Multiplier Method ◽  
Lagrangian Multiplier ◽  
General Mechanics ◽  
Generalized Variational Principles ◽  
Lagrangian Multiplier Method

Lagrangian Multiplier Method

2014 ◽  
pp. 33-37 ◽  
Author(s):  
Sinniah Ilanko ◽  
Luis E. Monterrubio ◽  
Yusuke Mochida
Keyword(s):  
Multiplier Method ◽  
Lagrangian Multiplier ◽  
Lagrangian Multiplier Method

scholarly journals Thermodynamic Foundation of Generalized Variational Principle

2021 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Variational Principle ◽  
Multiplier Method ◽  
Generalized Variational Principle ◽  
Lagrangian Multiplier ◽  
Stress Strain ◽  
Strain Relation ◽  
Stress Strain Relation ◽  
Lagrangian Multiplier Method ◽  
Open Question

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.

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