On the principle of virtual powers in continuum mechanics

Abstract We give a general formulation of the Principle of virtual powers in Continuum Mechanics from a distributional point of view, and study some of its relevant consequences in the field of balance equations. Keywords: Virtual Powers, Contact Interactions, Balance Equations Mathematics Subject Classification (2000): 74A10, 74G70, 74A30, 74A60     

Application of the symmetry principle for biharmonic functions to a problem in fracture mechanics

It is shown that a well-known series expansion of the stress function around a tip of a crack in an elastic plate, converges on a two-sheet Riemann surface. Explicit expressions for its coefficients, the stress intensity factors, are obtained. More generally, a new series expansion around the whole crack is found and investigated.     

Variational formulations in continuum mechanics

Variational formulations of the governing equations are of great interest in continuum mechanics: on the one hand a deeper theoretical insight in the respective system is possible, on the other hand variational formulations give rise for the development of semi-analytical and numerical methods like Ritz' direct method, especially FEM. Despite these benefits, there are many problems in continuum mechanics for which a variational principle is not available. The reason for this is that in contrast to conservative Newtonian mechanics, where the Lagrangian is given as difference between kinetic and potential energy, no generally valid construction rule for the Lagrangian has been established in the past. In this paper a construction rule is developed, on the Galilei-invariance of the system, leading to a general scheme for Lagrangians the individual analytical form of which is determined by an inverse treatment of Noether's theorem. This procedure is demonstrated for an elastically deforming body. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Potential methods in continuum mechanics

This article is a survey of the applications of the potential methods to the problems of continuum mechanics. A historical review, new results, and prospects for further development are given.See Burchuladze and Gegelia [1] where the reader can find rather complete information on the development of the potential method in elasticity theory.     

Relative continuum mechanics in general relativity

Summary An axiomatic approach to the study of relative continuum mechanics in curved space-time is proposed. The explicit assumptions are: a) existence of the energy-momentum tensor T^ij, satisfying the equations of motion T^ij _‖j=0, and b) existence of the congruence of stream-lines of the given continuum. The argument relies on a relativistic extension of the Lagrangian viewpoint, and involves the analysis of the relative dynamical behaviour of an arbitrary infinitestimal globule Δ of continuum in a given frame of reference [Γ]. The plan is fulfilled in two steps:1) geometrical theory of the Lagrangian viewpoint, valid for any type of continua satisfying the stated requirements;2) physical applications, illustrating the general theory in the case of an energy-momentum tensor of the form T^ij=μ ₀ VⁱV^j—S^ij. Entrata in Redazione il 22 novembre 1977. Lavoro eseguito nell'ambito dell'attività del Gruppo Nazionale per la Fisica Matematica del C.N.R.     

Beltrami's continuum mechanics in non-Euclidean spaces

Eugenio Beltrami investigated if ambient space, where physical phenomena occur, is Euclidean: such a subject was trendy in the mathematical scientific community at the end of the 19th century. Beltrami, along with Riemann, was interested also on how non-Euclidean geometry affects elasticity and electro-magnetism. Beltrami opened new paths for mathematical physics; his pupils Ernesto Cesaro, Ernesto Padova, Carlo Somigliana also studied elasticity in non-Euclidean spaces. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Relative continuum mechanics in general relativity

Summary An axiomatic approach to relative continuum kinematics in General Relativity is proposed. In the space-time manifold ϑ₄, we consider a physical frame of reference [Γ], and a congruence ξ of world-lines, interpreted as the congruence of stream-lines of an ideal physical system ℬ. The kinematical and geometrical properties of ξ relative to [Γ] are then analysed on the basis of the following scheme: a) definition and group theoretical interpretation of the class C(ξ) of connection vectors associated with ξ; b) spatial resolution of C(ξ), and construction of the velocity gradient of ξ relative to [Γ]; c) analysis of the time-dependent metric concepts associated with ξ in the frame of reference [Γ] (spatial distances between neighbouring world-lines, spatial volumes, Lorentz contractions, etc.).

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Sunto. Si propone un approccio assiomatico allo studio della cinematica relativa dei continui in Relatività Generale. Nello spazio-tempo ϑ₄ è assegnato un sistema di riferimento [Γ], e una congruenza ξ di linee di universo, pensata come la congruenza di linee di corrente di un sistema fisico ideals ℬ. Vengono quindi analizzate le proprietà cinematiche e geometriche di ξ relative al riferimento [Γ], sulla base del seguente schema: a) definizione e caratterizzazione in termini gruppali della classe C(ξ) dei vettori di connessione associati alla congruenza ξ; b) risoluzione spaziale di C(ξ), e costruzione del gradiente di velocità di ξ relativo a [Γ]; c) analisi delle proprietà metriche dipendenti dal tempo della congruenza ξ nel riferimento [Γ] (distanze spaziali tra linee di universo contigue, volumi spaziali, contrazioni di Lorentz, etc.).

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Entrata in Redazione il 18 maggio 1977.

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Lavoro eseguito nell'ambito dell'attività del Gruppo Nazionale per la Fisica Matematica del C.N.R.     

On tensor rates in continuum mechanics

Zusammenfassung Es wird gezeigt, dass Tensorgeschwindigkeiten materiell objektiv sind, wenn sie von den Geschwindigkeiten ihrer Abbildung in einer festen Konfiguration abgeleitet sind. Höhere Geschwindigkeiten werden formuliert, und eine höhere Jaumanngeschwindigkeit wird definiert, die gegenüber der vonPrager kürzlich eingeführten Definition gewisse Vorteile aufweist.Bekannte Stabilitätskriterien für Körper, deren Randbedingungen gegebene Kräfte enthalten, werden auf Grund des Druckerschen Postulats bestätigt, und neue Kriterien werden für Körper unter anderen Randbedingungen entwickelt.Allgemeine bewegliche räumliche Koordinatsysteme werden schliesslich eingeführt, und die zugehörigen Tensorgeschwindigkeitskomponenten werden mit Rücksicht auf grösste Einfachheit abgeleitet.

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This work was sponsored by the National Science Foundation.     

The relation between director and multipolar theories in continuum mechanics

Résumé On décrit la théorie thérmo-mécanique d'un milieu continu dans laquelle la déformation est décrite par plusieurs champs directeurs. Il est demontré qu'une des formes de la théorie est equivalente à une théorie de directeurs. Une théorie dans laquelle la déformation est décrite par plusieurs champs de déplacement est discutée et il est demontré qu'elle est equivalente à une théorie de directeurs sous une condition restrictive.     

Conservation laws and constitutive equations in continuum mechanics

The authors examine one of the general problems of continuum mechanics, namely the problem of the constitutive equations.Moscow. Translated from Mekhanika Polimerov, Vol. 5, No. 1, pp. 14–21, January–February, 1969.     

Noncommutative mechanics and exotic Galilean symmetry

In terms of the Lagrange-Souriau 2-form formalism, we describe a wide set of Hamiltonian dynamical systems with first-order Lagrangians. We consider a wide class of systems derived in different phenomenological contexts. The noncommutativity of the particle position coordinates are a natural consequence. We present various explicit examples.     

On the geometric character of stress in continuum mechanics

This paper shows that the stress field in the classical theory of continuum mechanics may be taken to be a covector-valued differential two-form. The balance laws and other fundamental laws of continuum mechanics may be neatly rewritten in terms of this geometric stress. A geometrically attractive and covariant derivation of the balance laws from the principle of energy balance in terms of this stress is presented. Jerrold G. Marsden: Research partially supported by the California Institute of Technology and NSF-ITR Grant ACI-0204932.     

On the theorem of power expended in continuum mechanics

We show that the theorem of power expended may be claimed for unbounded bodies, under mild hypotheses on the behavior at infinity of the velocity, the density, and the stress fields.     

The limiting behavior of the value-function for variational problems arising in continuum mechanics

In this paper we study the limiting behavior of the value-function for one-dimensional second order variational problems arising in continuum mechanics. The study of this behavior is based on the relation between variational problems on bounded large intervals and a limiting problem on [0,∞)$[0,\infty )$.     

Generalized symmetry and Neumann’s principle

Neumann’s principle is examined in the context of the recent generalizations relating to symmetry and physical properties of crystals. It is shown that the difficulties encountered in applying the principle to transport properties are only apparent and that it is not necessary to impose restrictions on the principle. A proper interpretation of the symmetry of crystals resolves the difficulties and brings out clearly the axiomatic nature of this principle and its universal applicability.     

On the treatment of active behaviour in continuum muscle mechanics

Two approaches of including active contractile behaviour of muscle tissue written in a continuum-mechanical formulation are presented. One approach relies on the addition of active and passive stress contributions, while the other approach is based on a multiplicative decomposition of the deformation gradient tensor. Both formulations can be stated in a thermodynamically consistent manner, each with different constraints, and both models can reproduce experimental data of passive and fully active muscle. Different behaviours are observed when comparing the active muscle models at submaximal stimulation rates. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonequilibrium statistical mechanics of a solid immersed in a continuum

We study actions of the symmetric group S₄ on K3 surfaces X that satisfy the following condition: there exists an equivariant birational contraction \(\bar r:X \to \bar X\) to a K3 surface \(\bar X\) with ADE singularities such that the quotient space \(\bar X\)/S₄ is isomorphic to P². We prove that up to smooth equivariant deformations there exist exactly 15 such actions of the group S₄ on K3 surfaces, and that these actions are realized as actions of the Galois groups on the Galoisations \(\bar X\) of the dualizing coverings of the plane which are associated with plane rational quartics without A₄, A₆, and E₆ singularities as their singular points.