Variational Principles in Continuum Mechanics

1983 ◽  
pp. 139-214
Author(s):  
John T. Oden ◽  
Junuthula N. Reddy
Keyword(s):  
Continuum Mechanics ◽  
Variational Principles

Variational Principles of Continuum Mechanics

2003 ◽  
pp. 283-295
Author(s):  
Lothar Gaul ◽  
Martin Kögl ◽  
Marcus Wagner
Keyword(s):  
Continuum Mechanics ◽  
Variational Principles

On dual approximation principles and optimization in continuum mechanics

1969 ◽  
Vol 265 (1162) ◽  
pp. 319-351 ◽  
Keyword(s):  
Mathematical Programming ◽  
Boundary Value Problems ◽  
Continuum Mechanics ◽  
Variational Principles ◽  
Diffusion Theory ◽  
Boundary Value ◽  
Uniqueness Theorems ◽  
Systematic Method ◽  
And Diffusion ◽  
Dual Functions

A unified expression of some of the boundary value problems of continuum mechanics is developed. A central role is given to the notion of a Legendre dual transformation in displaying the simple analytical structure of each problem considered. A systematic method of deriving reciprocal variational principles is described. General boundary value problems governed by inequalities as well as equations are then considered. Convexity of the dual functions related by the Legendre transformation is shown to be the basis of uniqueness theorems and extremum principles. Attention is drawn to the relevance of the literature on mathematical programming theory. M any examples are given, involving new or recent results in elasticity, plasticity, fluid mechanics and diffusion theory.

Variational principles in continuum mechanics

1968 ◽  
Vol 305 (1480) ◽  
pp. 1-25 ◽  
Keyword(s):  
Variational Principle ◽  
Continuum Mechanics ◽  
General Problem ◽  
Water Waves ◽  
Variational Principles ◽  
Differential Forms ◽  
Lagrangian Description ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Vector Potentials

Variational principles for problems in fluid dynamics, plasma dynamics and elasticity are discussed in the context of the general problem of finding a variational principle for a given system of equations. In continuum mechanics, the difficulties arise when the Eulerian description is used; the extension of Hamilton’s principle is straightforward in the Lagrangian description. It is found that the solution to these difficulties is to represent the Eulerian velocity v by expressions of the type v = ∇ X + λ∇ μ introduced by Clebsch (1859) for the case of isentropic fluid flow. The relation with Hamilton’s principle is elucidated following work by Lin (1963). It is also shown that the potential representation of electromagnetic fields and the variational principle for Maxwell’s equations can be fitted into the same overall scheme. The equations for water waves, waves in rotating and stratified fluids, Rossby waves, and plasma waves are given particular attention since the need for variational formulations of these equations has arisen in recent work on wave propagation (Whitham 1967). The idea of solving some of the equations by ‘potential representations’ (such as the Clebsch representation in continuum mechanics and the scalar and vector potentials in electromagnetism), and then finding a variational principle for the remaining equations, seems to be the crucial one for the general problem. An analogy with Pfaff’s problem in differential forms is given to support this idea.

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