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.     

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.     

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 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     

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.     

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)     

On the solutions of stochastic initial-value problems in continuum mechanics

This work deals with the analysis of a class of initial-value problems for nonlinear hyperbolic equations with stochastic process coefficients and four time-space independent variables. A perturbation procedure is developed, leading to a sequence of stochastic, partial differential equations which are solved by application of the Adomian decomposition method. A pertinent application in mechanics is also considered, and quantitative results are presented and discussed.     

On the direct connection of rheological elements in nonlinear continuum mechanics

The structure of complicated phenomenological material models at finite strains is often exemplified with the help of rheological elements. Thereby, simple material behaviour, i.e. elasticity or viscous and plastic flow, are composes by components. In our approach, we directly apply this concept to obtain material models at finite strains. Towards this end, the thermodynamically consistent material behaviour of single elements is defined first. Subsequently, the elements are connected by evaluation of stress equilibria equations formulated on interconnecting configurations. The basic equations of this concept are presented using the example of nonlinear viscoelasticity of Maxwell type. The model results from a series connection of an elastic and a viscous element, whereas both are formulated in a thermodynamically consistent way within the framework on nonlinear continuum mechanics. Furthermore, an approach of numerical implementation using the stress equilibria is suggested. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

On the solvability of a variational problem about phase transitions in continuum mechanics

We study a variational problem about phase transitions in continuum mechanics under the condition that the surface tension coefficient vanishes. A homogeneous isotropic two-phase elastic medium occupies a ball-shaped domain, the zero displacement field is fixed on the boundary of this domain, and a spherically symmetric force field acts on the medium. The solvability of this problem is established. As is shown, if a force field is nonzero almost everywhere, then the problem has only spherically symmetric solutions. Bibliography: 9 titles. Translated from Problemy Matematicheskogo Analiza, No. 38, December 2008, pp. 61–71.     

On the set of solutions to a variational phase transition problem of continuum mechanics

A variational problem describing phase transitions is considered. It is shown that a multi-vaued function associating with the set of parameters the set of solutions to the problem is continuous and has compact values. Bibliography: 5 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 35, 2007, pp. 111–119     

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)     

The symmetry principle in continuum mechanics

For problems of the mechanics of an anisotropic inhomogeneous continuum, theorems are given concerning the uninterrupted symmetrical and antisymmetrical analytical continuation of the solution through the plane part of the boundary surface of the medium. Theorems are given for two types of mechanics problem; in the first of these both symmetrical and antisymmetrical continuations of the solution are allowed, while in the second only symmetrical continuation of the solution is allowed. Problems of the first type include problems which are reduced to linear thermoelastic dynamic differential equations of motion of an inhomogeneous anisotropic medium possessing a plane of elastic symmetry, to linear thermoelastic dynamic differential equations of motion of an inhomogeneous Cosserat medium, to non-linear differential equations describing the static elastoplastic stress state of a plate, etc. The second type includes problems which are reduced to non-linear differential equations describing geometrically non-linear strains of shells, to Navier–Stokes equations, etc. These theorems extend the principle of mirror reflection (the Riemann–Schwartz principle of symmetry) to linear and non-linear equations of continuum mechanics. The uninterrupted continuation of the solutions is used to solve problems of the equilibrium state of bodies of complex shape.     

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.     

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.     

Well-posedness of mathematical models of continuum mechanics and thermodynamics

Mathematical models of nonequilibrium thermodynamics related to the concepts of stability and well-posedness are considered.Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 3, Partial Differential Equations, 2003.