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2. Modeling of heat conduction through rate equations

Author

Giorgi, Claudio, Morro, Angelo, and Zullo, Federico

Subjects
Mathematical Physics
Abstract

Starting from a classical thermodynamic approach, we derive rate-type equations to describe the behavior of heat flow in deformable media. Constitutive equations are defined in the material (Lagrangian) description where the standard time derivative satisfies the principle of objectivity. The statement of the Second Law is formulated in the classical form and the thermodynamic restrictions are then developed following the Coleman-Noll procedure. However, instead of the Clausius Duhem inequality we consider the corresponding equality where the entropy production rate is prescribed by a non-negative constitutive function. Both the free energy and the entropy production are assumed to depend on a common set of independent variables involving, in addition to temperature, both temperature gradient and heat-flux vector together with their time derivatives. This approach results in rate-type constitutive equations for the heat-flux vector that are intrinsically consistent with the Second Law and easily amenable to analysis. In addition to obtaining already known models (e.g. Cattaneo-Maxwell's, Jeffreys-like and Green-Naghdi's heat conductors), this scheme allows us to build new and more complex models of heat transport that may have applications in describing the thermal behavior in nanosystems. Indeed, when higher order time derivatives of the heat flux vector are involved, many different relaxation times occur within the rate equation., Comment: 32 pages, 2 tables, 1 appendix

Published
2024

3. Nonlinear and nonlocal models of heat conduction in continuum thermodynamics

Author

Giorgi, Claudio and Zullo, Federico

Subjects
Mathematical Physics
Abstract

The aim of this paper is to develop a general constitutive scheme within continuum thermodynamics to describe the behavior of heat flow in deformable media. Starting from a classical thermodynamic approach, the rate-type constitutive equations are defined in the material (Lagrangian) description where the standard time derivative satisfies the principle of objectivity. All constitutive functions are required to depend on a common set of independent variables and to be consistent with thermodynamics. The statement of the Second Law is formulated in a general nonlocal form, where the entropy production rate is prescribed by a non-negative constitutive function and the extra entropy flux obeys a no-flow boundary condition. The thermodynamic response is then developed based on Coleman-Noll procedure. In the local formulation, the free energy potential and the rate of entropy production function are assumed to depend on temperature, temperature gradient and heat-flux vector along with their time derivatives. This approach results in rate-type constitutive equations for the heat-flux vector that are intrinsically consistent with the Second Law and easily amenable to analysis. A huge class of linear and nonlinear models of the rate type are recovered (e.g., Cattaneo-Maxwell's, Jeffreys-like, Green-Naghdi's, Quintanilla's and Burgers-like heat conductors). In the (weakly) nonlocal formulation of the second law, both the entropy production rate and an entropy extra-flux vector are assumed to depend on temperature, temperature gradient and heat-flux vector along with their spatial gradients and time derivatives. Within this (classical) thermodynamic framework the nonlocal Guyer-Krumhansl model and some nonlinear generalizations devised by Cimmelli and Sellitto are obtained, Comment: 29 pages

Published
2023

4. Superconductivity and Superfluidity

Author

Morro, Angelo, Giorgi, Claudio, Bellomo, Nicola, Series Editor, Tezduyar, Tayfun E., Series Editor, Aoki, Kazuo, Editorial Board Member, Bazilevs, Yuri, Editorial Board Member, Chaplain, Mark, Editorial Board Member, Degond, Pierre, Editorial Board Member, Deutsch, Andreas, Editorial Board Member, Gibelli, Livio, Editorial Board Member, Herrero, Miguel Ángel, Editorial Board Member, Hughes, Thomas J.R., Editorial Board Member, Koumoutsakos, Petros, Editorial Board Member, Prosperetti, Andrea, Editorial Board Member, Rajagopal, K.R., Editorial Board Member, Takizawa, Kenji, Editorial Board Member, Tao, Youshan, Editorial Board Member, van Brummelen, Harald, Editorial Board Member, Morro, Angelo, and Giorgi, Claudio

Published
2023
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5. Ferroics

Author

Morro, Angelo, Giorgi, Claudio, Bellomo, Nicola, Series Editor, Tezduyar, Tayfun E., Series Editor, Aoki, Kazuo, Editorial Board Member, Bazilevs, Yuri, Editorial Board Member, Chaplain, Mark, Editorial Board Member, Degond, Pierre, Editorial Board Member, Deutsch, Andreas, Editorial Board Member, Gibelli, Livio, Editorial Board Member, Herrero, Miguel Ángel, Editorial Board Member, Hughes, Thomas J.R., Editorial Board Member, Koumoutsakos, Petros, Editorial Board Member, Prosperetti, Andrea, Editorial Board Member, Rajagopal, K.R., Editorial Board Member, Takizawa, Kenji, Editorial Board Member, Tao, Youshan, Editorial Board Member, van Brummelen, Harald, Editorial Board Member, Morro, Angelo, and Giorgi, Claudio

Published
2023
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6. Materials with Memory

Author

Morro, Angelo, Giorgi, Claudio, Bellomo, Nicola, Series Editor, Tezduyar, Tayfun E., Series Editor, Aoki, Kazuo, Editorial Board Member, Bazilevs, Yuri, Editorial Board Member, Chaplain, Mark, Editorial Board Member, Degond, Pierre, Editorial Board Member, Deutsch, Andreas, Editorial Board Member, Gibelli, Livio, Editorial Board Member, Herrero, Miguel Ángel, Editorial Board Member, Hughes, Thomas J.R., Editorial Board Member, Koumoutsakos, Petros, Editorial Board Member, Prosperetti, Andrea, Editorial Board Member, Rajagopal, K.R., Editorial Board Member, Takizawa, Kenji, Editorial Board Member, Tao, Youshan, Editorial Board Member, van Brummelen, Harald, Editorial Board Member, Morro, Angelo, and Giorgi, Claudio

Published
2023
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7. Phase Transitions

Author

Morro, Angelo, Giorgi, Claudio, Bellomo, Nicola, Series Editor, Tezduyar, Tayfun E., Series Editor, Aoki, Kazuo, Editorial Board Member, Bazilevs, Yuri, Editorial Board Member, Chaplain, Mark, Editorial Board Member, Degond, Pierre, Editorial Board Member, Deutsch, Andreas, Editorial Board Member, Gibelli, Livio, Editorial Board Member, Herrero, Miguel Ángel, Editorial Board Member, Hughes, Thomas J.R., Editorial Board Member, Koumoutsakos, Petros, Editorial Board Member, Prosperetti, Andrea, Editorial Board Member, Rajagopal, K.R., Editorial Board Member, Takizawa, Kenji, Editorial Board Member, Tao, Youshan, Editorial Board Member, van Brummelen, Harald, Editorial Board Member, Morro, Angelo, and Giorgi, Claudio

Published
2023
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8. Electromagnetism of Continuous Media

Author

Morro, Angelo, Giorgi, Claudio, Bellomo, Nicola, Series Editor, Tezduyar, Tayfun E., Series Editor, Aoki, Kazuo, Editorial Board Member, Bazilevs, Yuri, Editorial Board Member, Chaplain, Mark, Editorial Board Member, Degond, Pierre, Editorial Board Member, Deutsch, Andreas, Editorial Board Member, Gibelli, Livio, Editorial Board Member, Herrero, Miguel Ángel, Editorial Board Member, Hughes, Thomas J.R., Editorial Board Member, Koumoutsakos, Petros, Editorial Board Member, Prosperetti, Andrea, Editorial Board Member, Rajagopal, K.R., Editorial Board Member, Takizawa, Kenji, Editorial Board Member, Tao, Youshan, Editorial Board Member, van Brummelen, Harald, Editorial Board Member, Morro, Angelo, and Giorgi, Claudio

Published
2023
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9. Mixtures

Author

Morro, Angelo, Giorgi, Claudio, Bellomo, Nicola, Series Editor, Tezduyar, Tayfun E., Series Editor, Aoki, Kazuo, Editorial Board Member, Bazilevs, Yuri, Editorial Board Member, Chaplain, Mark, Editorial Board Member, Degond, Pierre, Editorial Board Member, Deutsch, Andreas, Editorial Board Member, Gibelli, Livio, Editorial Board Member, Herrero, Miguel Ángel, Editorial Board Member, Hughes, Thomas J.R., Editorial Board Member, Koumoutsakos, Petros, Editorial Board Member, Prosperetti, Andrea, Editorial Board Member, Rajagopal, K.R., Editorial Board Member, Takizawa, Kenji, Editorial Board Member, Tao, Youshan, Editorial Board Member, van Brummelen, Harald, Editorial Board Member, Morro, Angelo, and Giorgi, Claudio

Published
2023
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10. Plasticity

Author

Morro, Angelo, Giorgi, Claudio, Bellomo, Nicola, Series Editor, Tezduyar, Tayfun E., Series Editor, Aoki, Kazuo, Editorial Board Member, Bazilevs, Yuri, Editorial Board Member, Chaplain, Mark, Editorial Board Member, Degond, Pierre, Editorial Board Member, Deutsch, Andreas, Editorial Board Member, Gibelli, Livio, Editorial Board Member, Herrero, Miguel Ángel, Editorial Board Member, Hughes, Thomas J.R., Editorial Board Member, Koumoutsakos, Petros, Editorial Board Member, Prosperetti, Andrea, Editorial Board Member, Rajagopal, K.R., Editorial Board Member, Takizawa, Kenji, Editorial Board Member, Tao, Youshan, Editorial Board Member, van Brummelen, Harald, Editorial Board Member, Morro, Angelo, and Giorgi, Claudio

Published
2023
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11. Micropolar Media

Author

Morro, Angelo, Giorgi, Claudio, Bellomo, Nicola, Series Editor, Tezduyar, Tayfun E., Series Editor, Aoki, Kazuo, Editorial Board Member, Bazilevs, Yuri, Editorial Board Member, Chaplain, Mark, Editorial Board Member, Degond, Pierre, Editorial Board Member, Deutsch, Andreas, Editorial Board Member, Gibelli, Livio, Editorial Board Member, Herrero, Miguel Ángel, Editorial Board Member, Hughes, Thomas J.R., Editorial Board Member, Koumoutsakos, Petros, Editorial Board Member, Prosperetti, Andrea, Editorial Board Member, Rajagopal, K.R., Editorial Board Member, Takizawa, Kenji, Editorial Board Member, Tao, Youshan, Editorial Board Member, van Brummelen, Harald, Editorial Board Member, Morro, Angelo, and Giorgi, Claudio

Published
2023
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12. Rate-Type Models

Author

Morro, Angelo, Giorgi, Claudio, Bellomo, Nicola, Series Editor, Tezduyar, Tayfun E., Series Editor, Aoki, Kazuo, Editorial Board Member, Bazilevs, Yuri, Editorial Board Member, Chaplain, Mark, Editorial Board Member, Degond, Pierre, Editorial Board Member, Deutsch, Andreas, Editorial Board Member, Gibelli, Livio, Editorial Board Member, Herrero, Miguel Ángel, Editorial Board Member, Hughes, Thomas J.R., Editorial Board Member, Koumoutsakos, Petros, Editorial Board Member, Prosperetti, Andrea, Editorial Board Member, Rajagopal, K.R., Editorial Board Member, Takizawa, Kenji, Editorial Board Member, Tao, Youshan, Editorial Board Member, van Brummelen, Harald, Editorial Board Member, Morro, Angelo, and Giorgi, Claudio

Published
2023
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13. Aging and Higher-Order Grade Materials

Author

Morro, Angelo, Giorgi, Claudio, Bellomo, Nicola, Series Editor, Tezduyar, Tayfun E., Series Editor, Aoki, Kazuo, Editorial Board Member, Bazilevs, Yuri, Editorial Board Member, Chaplain, Mark, Editorial Board Member, Degond, Pierre, Editorial Board Member, Deutsch, Andreas, Editorial Board Member, Gibelli, Livio, Editorial Board Member, Herrero, Miguel Ángel, Editorial Board Member, Hughes, Thomas J.R., Editorial Board Member, Koumoutsakos, Petros, Editorial Board Member, Prosperetti, Andrea, Editorial Board Member, Rajagopal, K.R., Editorial Board Member, Takizawa, Kenji, Editorial Board Member, Tao, Youshan, Editorial Board Member, van Brummelen, Harald, Editorial Board Member, Morro, Angelo, and Giorgi, Claudio

Published
2023
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14. Porous Materials

Author

Morro, Angelo, Giorgi, Claudio, Bellomo, Nicola, Series Editor, Tezduyar, Tayfun E., Series Editor, Aoki, Kazuo, Editorial Board Member, Bazilevs, Yuri, Editorial Board Member, Chaplain, Mark, Editorial Board Member, Degond, Pierre, Editorial Board Member, Deutsch, Andreas, Editorial Board Member, Gibelli, Livio, Editorial Board Member, Herrero, Miguel Ángel, Editorial Board Member, Hughes, Thomas J.R., Editorial Board Member, Koumoutsakos, Petros, Editorial Board Member, Prosperetti, Andrea, Editorial Board Member, Rajagopal, K.R., Editorial Board Member, Takizawa, Kenji, Editorial Board Member, Tao, Youshan, Editorial Board Member, van Brummelen, Harald, Editorial Board Member, Morro, Angelo, and Giorgi, Claudio

Published
2023
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15. Balance Equations

Author

Morro, Angelo, Giorgi, Claudio, Bellomo, Nicola, Series Editor, Tezduyar, Tayfun E., Series Editor, Aoki, Kazuo, Editorial Board Member, Bazilevs, Yuri, Editorial Board Member, Chaplain, Mark, Editorial Board Member, Degond, Pierre, Editorial Board Member, Deutsch, Andreas, Editorial Board Member, Gibelli, Livio, Editorial Board Member, Herrero, Miguel Ángel, Editorial Board Member, Hughes, Thomas J.R., Editorial Board Member, Koumoutsakos, Petros, Editorial Board Member, Prosperetti, Andrea, Editorial Board Member, Rajagopal, K.R., Editorial Board Member, Takizawa, Kenji, Editorial Board Member, Tao, Youshan, Editorial Board Member, van Brummelen, Harald, Editorial Board Member, Morro, Angelo, and Giorgi, Claudio

Published
2023
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16. Solids

Author

Morro, Angelo, Giorgi, Claudio, Bellomo, Nicola, Series Editor, Tezduyar, Tayfun E., Series Editor, Aoki, Kazuo, Editorial Board Member, Bazilevs, Yuri, Editorial Board Member, Chaplain, Mark, Editorial Board Member, Degond, Pierre, Editorial Board Member, Deutsch, Andreas, Editorial Board Member, Gibelli, Livio, Editorial Board Member, Herrero, Miguel Ángel, Editorial Board Member, Hughes, Thomas J.R., Editorial Board Member, Koumoutsakos, Petros, Editorial Board Member, Prosperetti, Andrea, Editorial Board Member, Rajagopal, K.R., Editorial Board Member, Takizawa, Kenji, Editorial Board Member, Tao, Youshan, Editorial Board Member, van Brummelen, Harald, Editorial Board Member, Morro, Angelo, and Giorgi, Claudio

Published
2023
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17. Generalities on Constitutive Models

Author

Morro, Angelo, Giorgi, Claudio, Bellomo, Nicola, Series Editor, Tezduyar, Tayfun E., Series Editor, Aoki, Kazuo, Editorial Board Member, Bazilevs, Yuri, Editorial Board Member, Chaplain, Mark, Editorial Board Member, Degond, Pierre, Editorial Board Member, Deutsch, Andreas, Editorial Board Member, Gibelli, Livio, Editorial Board Member, Herrero, Miguel Ángel, Editorial Board Member, Hughes, Thomas J.R., Editorial Board Member, Koumoutsakos, Petros, Editorial Board Member, Prosperetti, Andrea, Editorial Board Member, Rajagopal, K.R., Editorial Board Member, Takizawa, Kenji, Editorial Board Member, Tao, Youshan, Editorial Board Member, van Brummelen, Harald, Editorial Board Member, Morro, Angelo, and Giorgi, Claudio

Published
2023
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18. Kinematics

Author

Morro, Angelo, Giorgi, Claudio, Bellomo, Nicola, Series Editor, Tezduyar, Tayfun E., Series Editor, Aoki, Kazuo, Editorial Board Member, Bazilevs, Yuri, Editorial Board Member, Chaplain, Mark, Editorial Board Member, Degond, Pierre, Editorial Board Member, Deutsch, Andreas, Editorial Board Member, Gibelli, Livio, Editorial Board Member, Herrero, Miguel Ángel, Editorial Board Member, Hughes, Thomas J.R., Editorial Board Member, Koumoutsakos, Petros, Editorial Board Member, Prosperetti, Andrea, Editorial Board Member, Rajagopal, K.R., Editorial Board Member, Takizawa, Kenji, Editorial Board Member, Tao, Youshan, Editorial Board Member, van Brummelen, Harald, Editorial Board Member, Morro, Angelo, and Giorgi, Claudio

Published
2023
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32. Solids

Author

Morro, Angelo, primary and Giorgi, Claudio, additional

Published
2023
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36. Resilience of Cultural Heritage in Extreme Weather Conditions: The Case of the UNESCO Villa Romana del Casale Archaeological Site’s Response to the Apollo Medicane in October 2021

Author

Prestileo, Fernanda, Mascitelli, Alessandra, Meli, Guido, Petracca, Marco, Giorgi, Claudio, Melfi, Davide, Puca, Silvia, Dietrich, Stefano, Goos, Gerhard, Founding Editor, Hartmanis, Juris, Founding Editor, Bertino, Elisa, Editorial Board Member, Gao, Wen, Editorial Board Member, Steffen, Bernhard, Editorial Board Member, Yung, Moti, Editorial Board Member, Gervasi, Osvaldo, editor, Murgante, Beniamino, editor, Misra, Sanjay, editor, Rocha, Ana Maria A. C., editor, and Garau, Chiara, editor

Published
2022
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40. Strain-Rate and Stress-Rate Models of Nonlinear Viscoelastic Materials.

Author

Giorgi, Claudio and Morro, Angelo

Subjects
*DIFFERENTIAL forms, *VISCOELASTIC materials, *NONLINEAR equations, *ENTROPY, *EQUATIONS
Abstract

The paper is devoted to the modeling of nonlinear viscoelastic materials. The constitutive equations are considered in differential form via relations between strain, stress, and their derivatives in the Lagrangian description. The thermodynamic consistency is established by using the Clausius–Duhem inequality through a procedure that involves two uncommon features. Firstly, the entropy production is regarded as a positive-valued constitutive function per se. This view implies that the inequality is in fact an equation. Secondly, this statement of the second law is investigated by using an algebraic representation formula, thus arriving at quite general results for rate terms that are usually overlooked in thermodynamic analyses. Starting from strain-rate or stress-rate equations, the corresponding finite equations are derived. It then emerges that a greater generality of the constitutive equations of the classical models, such as those of Boltzmann and Maxwell, are obtained as special cases. [ABSTRACT FROM AUTHOR]

Published
2024
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42. Steady states of elastically-coupled extensible double-beam systems

Author

Dell'Oro, Filippo, Giorgi, Claudio, and Pata, Vittorino

Subjects
Mathematics - Analysis of PDEs
Abstract

Given $\beta\in\mathbb{R}$ and $\varrho,k>0$, we analyze an abstract version of the nonlinear stationary model in dimensionless form $$\begin{cases} u"" - \Big(\beta+ \varrho\int_0^1 |u'(s)|^2\,{\rm d} s\Big)u" +k(u-v) = 0 v"" - \Big(\beta+ \varrho\int_0^1 |v'(s)|^2\,{\rm d} s\Big)v" -k(u-v) = 0 \end{cases} $$ describing the equilibria of an elastically-coupled extensible double-beam system subject to evenly compressive axial loads. Necessary and sufficient conditions in order to have nontrivial solutions are established, and their explicit closed-form expressions are found. In particular, the solutions are shown to exhibit at most three nonvanishing Fourier modes. In spite of the symmetry of the system, nonsymmetric solutions appear, as well as solutions for which the elastic energy fails to be evenly distributed. Such a feature turns out to be of some relevance in the analysis of the longterm dynamics, for it may lead up to nonsymmetric energy exchanges between the two beams, mimicking the transition from vertical to torsional oscillations.

Published
2016

43. A model of viscoelasticity with time-dependent memory kernels

Author

Conti, Monica, Danese, Valeria, Giorgi, Claudio, and Pata, Vittorino

Subjects
Mathematics - Dynamical Systems
Abstract

We consider the model equation arising in the theory of viscoelasticity $$\partial_{tt} u-h_t(0)\Delta u -\int_{0}^\infty h_t'(s)\Delta u(t-s)d s+ f(u) = g.$$ Here, the main feature is that the memory kernel $h_t(\cdot)$ depends on time, allowing for instance to describe the dynamics of aging materials. From the mathematical viewpoint, this translates into the study of dynamical systems acting on time-dependent spaces, according to the newly established theory of Di Plinio et al. In this first work, we give a proper notion of solution, and we provide a global well-posedness result. The techniques naturally extend to the analysis of the longterm behavior of the associated process, and can be exported to cover the case of general systems with memory in presence of time-dependent kernels.

Published
2016

48. Long-Term Damped Dynamics of the Extensible Suspension Bridge

Author

Bochicchio, Ivana, Giorgi, Claudio, and Vuk, Elena

Subjects
Mathematical Physics
Abstract

This work is focused on the doubly nonlinear equation, whose solutions represent the bending motion of an extensible, elastic bridge suspended by continuously distributed cables which are flexible and elastic with stiffness k^2. When the ends are pinned, long-term dynamics is scrutinized for arbitrary values of axial load p and stiffness k^2. For a general external source f, we prove the existence of bounded absorbing sets.When f is timeindependent, the related semigroup of solutions is shown to possess the global attractor of optimal regularity and its characterization is given in terms of the steady states of the problem., Comment: 19 pages, 1 figure

Published
2011
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