A Partial Report on the Controversies About the Principle of VirtualWork: From Archytas of Tarentum to Lagrange, Piola, Mindlin and Toupin

Barchiesi, Emilio; Ciallella, Alessandro; Scerrato, Daria

doi:10.1007/978-3-030-80550-0_5

Cited by 7 publications

(6 citation statements)

References 146 publications

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“…This is true also for its application to second-gradient continua. Indeed the so-called “capillary fluids” were the first continua of this type to be described ([32,37,38], more historical remarks can be found in previous studies [39–42]). We have argued that D’Alembert–Lagrange postulation scheme is more suitable than Cauchy’s postulation scheme for introducing generalized continuum models (see among others [43–49]).…”

Section: Discussionmentioning

confidence: 99%

Second-gradient continua: From Lagrangian to Eulerian and back

Dell’Isola

Eugster

Fedele

et al. 2022

Mathematics and Mechanics of Solids

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In this paper, we represent second-gradient internal work functionals in Lagrangian (referential) and Eulerian (spatial) descriptions, and we deduce the corresponding expressions for the Piola transformations of stress and double-stress tensors and of external forces and double-forces. We also derive, in both the Eulerian and Lagrangian description, the expression of surface and edge contact interactions (which include forces and double-forces) for second-gradient continua in terms of the normal and the curvature of contact boundary surfaces and edge shapes.

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Section: Discussionmentioning

confidence: 99%

Second-gradient continua: From Lagrangian to Eulerian and back

Dell’Isola

Eugster

Fedele

et al. 2022

Mathematics and Mechanics of Solids

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“…Variational methods of modeling the processes of thermodynamics in complex environments are preferable because they allow you to propose the most complete and fully thermodynamically consistent models that are invariant with respect to coordinate transformations. In this regard, we also note papers [20][21][22], where variational models were used to describe irreversible processes and where the fundamental nature of the principle of possible displacements, which is valid for reversible and irreversible processes, was emphasized.…”

Section: Introductionmentioning

confidence: 99%

“…For modeling reversible thermodynamic deformation processes, variational methods are very effective [11,[20][21][22]27,28], etc. They involve the introduction of thermodynamic potentials for models of media of varying complexity and the use of variational principles.…”

Section: Introductionmentioning

confidence: 99%

Symmetry Properties of Models for Reversible and Irreversible Thermodynamic Processes

Lurie,

Belov,

Matevossian

2023

Symmetry

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The problem of formulating variational models for irreversible processes of media deformation is considered in this paper. For reversible processes, the introduction of variational models actually comes down to defining functionals with a given list of arguments of various tensor dimensions. For irreversible processes, an algorithm based on the principle of stationarity of the functional is incorrect. In this paper, to formulate a variational model of irreversible deformation processes with an expanded range of coupled effects, an approach is developed based on the idea of the introduction of the non-integrable variational forms that clearly separate dissipative processes from reversible deformation processes. The fundamental nature of the properties of symmetry and anti-symmetry of tensors of physical properties in relation to multi-indices characterizing independent arguments of bilinear forms in the variational formulation of models of thermomechanical processes has been established. For reversible processes, physical property tensors must necessarily be symmetric with respect to multi-indices. On the contrary, for irreversible thermomechanical processes, the tensors of physical properties that determine non-integrable variational forms must be antisymmetric with respect to the permutation of multi-indices. As a result, an algorithm for obtaining variational models of dissipative irreversible processes is proposed. This algorithm is based on determining the required number of dissipative channels and adding them to the known model of a reversible process. Dissipation channels are introduced as non-integrable variational forms that are linear in the variations of the arguments. The hydrodynamic models of Darcy, Navier–Stokes, and Brinkman are considered, each of which is determined by a different set of dissipation channels. As another example, a variational model of heat transfer processes is presented. The equations of heat conduction laws are obtained as compatibility equations by excluding the introduced thermal potential from the constitutive equations for temperature and heat flux. The Fourier and Maxwell–Cattaneo equations and the generalized heat conduction laws of Gaer–Krumhansl and Jeffrey are formulated.

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“…These last authors became representatives of two diverse approaches: Piola based the postulation of mechanics on the principle of virtual work, while Cauchy on the balance law of forces and of moments of forces. The querelle concerning the most appropriate and effective way to formulate new models in continuum mechanics began immediately after the publication of Piola's works [5][6][7]. Among the others, Ernst Hellinger in his famous article for the Encyklopa¨die der mathematische dated 1913 (see [8][9][10]), established continuum mechanics on the basis of the principle of virtual work: his list of open problems included the generalization of Piola's approach to higher gradient continua.…”

Section: Introductionmentioning

confidence: 99%

Third-gradient continua: nonstandard equilibrium equations and selection of work conjugate variables

Fedele

2022

Mathematics and Mechanics of Solids

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This paper outlines the variational derivation of the Lagrangian equilibrium equations for the third-gradient materials, stemming from the minimization of the total potential energy functional, and the selection of suitable dual variables to represent the inner work in the Eulerian configuration. Volume, face, edge and wedge contributions were provided through integration by parts of the inner virtual work and by repeated applications of the divergence theorem extended to embedded submanifolds with codimension one and two. Detailed expressions were provided for the contact pressures and the edge loading, revealing the complex dependence on the face normals and on the mean curvature. Relationships were specified among the Lagrangian (hyper-)stress tensors of rank lower or equal to four, and their Eulerian counterparts.

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A Partial Report on the Controversies About the Principle of VirtualWork: From Archytas of Tarentum to Lagrange, Piola, Mindlin and Toupin

Cited by 7 publications

References 146 publications

Second-gradient continua: From Lagrangian to Eulerian and back

Second-gradient continua: From Lagrangian to Eulerian and back

Symmetry Properties of Models for Reversible and Irreversible Thermodynamic Processes

Third-gradient continua: nonstandard equilibrium equations and selection of work conjugate variables

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