Variational Principles Of Continuum Mechanics Research Articles

SIMULATION OF THE STRESSED-DEFORMED STATE OF THE EXPLOSION-WELDING FIVELAYERS TITANO-STEEL COMPOSITE AFTER HOT ROLLING

Using experimental data, 3D modeling options were verified, which describe the strain during rolling of layers of a titanium-steel composite and based on the variational principles of continuum mechanics and the finite element method.

Exegesis of the Introduction and Sect. I from “Fundamentals of the Mechanics of Continua”** by E. Hellinger

The fundamental review article DIE ALLGEMEINEN ANSÄTZE DER MECHANIK DER KONTINUA in the Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, Bd. IV‐4, Hft. 5 (1913) by Ernst Hellinger has not been translated into English so far. We believe that such a circumstance is really deprecative, as the insight reached by Hellinger in the mathematical structure of continuum mechanics seems, in some aspects, unsurpassed even nowadays. Hellinger's scientific manuscripts do not fill more than one and half boxes in library storage , but their impact on mathematics and mechanical sciences is profound. Indeed, the Hellinger‐Reissner variational principle is still a fundamental tool in theoretical and numerical mechanics. The intent of this paper is threefold: i) to allow to those who cannot understand German to enjoy the reading of a crystal‐clear and still topical article whose content has some enlightening parts, ii) to show that only one century ago the principle of virtual work (or virtual velocities) was regarded as the central principle in continuum mechanics and that Hellinger did forecast already then the main lines of its development, iii) to discuss some technical and conceptual aspects of the variational principles in continuum mechanics which some authors consider still controversial.

Variational Principles of Continuum Mechanics with Engineering Applications. Volume 2: Introduction to Optimal Design Theory (Vadim Komkov)

Variational Principles of Continuum Mechanics with Engineering Applications. Volume 2: Introduction to Optimal Design Theory (Vadim Komkov)

Hamilton's principle for perfect fluids

Variational principles, invented in analytical mechanics, has become a tool in various fields of physics, engineering and applied mathematics. Very often, in an introductory course of analytical mechanics, there is not time for the discussion of variational principles in other disciplines. Perfect fluids represent a simple system for which the usefulness of Hamilton's principle can be easily demonstrated. Moreover, the physical notion of Lagrange multipliers and the role of constraints can readily be discussed for such a system. From the educational point of view this example can also be used as an introduction to the discussion of variational principles in continuum mechanics.

Spatial variational principles in continuum mechanics

Spatial variational principles in continuum mechanics

Variational principles in continuum mechanics

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.