# Theory of space-time dissipative elasticity and scale effects

Nanoscale Systems: Mathematical Modeling, Theory and Applications (2013)

- Volume: 2, page 166-178
- ISSN: 2299-3290

## Access Full Article

top## Abstract

top## How to cite

topS.A. Lurie, and P.A. Belov. "Theory of space-time dissipative elasticity and scale effects." Nanoscale Systems: Mathematical Modeling, Theory and Applications 2 (2013): 166-178. <http://eudml.org/doc/267333>.

@article{S2013,

abstract = {In this article a model of irreversible dynamic thermoelasticity of an ideal continuua is constructed from an elasticity theory of asymmetrical, transversely isotropic in time direction, dissipative defectless 4D-continuum. In the model the fourth component of the 4D-displacement vector is locally irregular time R. The kinematic model comprises 3D-tensor of distortion, 3Dvector of velocity, 3D-gradient vector of local irregular time and entropy in unified tensor object which is an asymmetrical 4D-tensor of distortion of second rank. Consequently, the force model comprises 3D-tensor of stress, 3D-vector of impulses, 3D-vector of heat flow and temperature in unified tensor object which is an asymmetrical 4D-stress tensor of second rank. Hooke’s law equations have been formulated which connect components of asymmetrical 4D-tensors of stress and distortion. Physical interpretations have been given to the tensors’ components of thermomechanical properties of formulated continuum. Therefore, the article formulate an irreversible dynamic thermoelasticity covariant model of ideal (defectless) continua in which basic kinematic and force variables are components of unified tensor objects and theory is represented by 4D-vector equation. Sedov’s equation has been derived and resulted into Euler’s equations, space projections of which determine motion equations, and time projection determines heat conductivity equation as well as the whole spectrum of the space-time boundary value problems. The proposed theory allows one to describe the scale effects in the thermal processes and opens prospects for studying the scale effects of the coupled dynamic thermoelasticity and its nanoscience applications. A temperature-scale refinement can also broaden the range of applicability of the law of heat conduction in solids to allow for design of small-sized components, devices and nano-systems.},

author = {S.A. Lurie, P.A. Belov},

journal = {Nanoscale Systems: Mathematical Modeling, Theory and Applications},

keywords = {variation model; dissipation; space time elasticity; extended thermodynamic; relaxation time; scale effects; small-sized components, devices and nano-systems},

language = {eng},

pages = {166-178},

title = {Theory of space-time dissipative elasticity and scale effects},

url = {http://eudml.org/doc/267333},

volume = {2},

year = {2013},

}

TY - JOUR

AU - S.A. Lurie

AU - P.A. Belov

TI - Theory of space-time dissipative elasticity and scale effects

JO - Nanoscale Systems: Mathematical Modeling, Theory and Applications

PY - 2013

VL - 2

SP - 166

EP - 178

AB - In this article a model of irreversible dynamic thermoelasticity of an ideal continuua is constructed from an elasticity theory of asymmetrical, transversely isotropic in time direction, dissipative defectless 4D-continuum. In the model the fourth component of the 4D-displacement vector is locally irregular time R. The kinematic model comprises 3D-tensor of distortion, 3Dvector of velocity, 3D-gradient vector of local irregular time and entropy in unified tensor object which is an asymmetrical 4D-tensor of distortion of second rank. Consequently, the force model comprises 3D-tensor of stress, 3D-vector of impulses, 3D-vector of heat flow and temperature in unified tensor object which is an asymmetrical 4D-stress tensor of second rank. Hooke’s law equations have been formulated which connect components of asymmetrical 4D-tensors of stress and distortion. Physical interpretations have been given to the tensors’ components of thermomechanical properties of formulated continuum. Therefore, the article formulate an irreversible dynamic thermoelasticity covariant model of ideal (defectless) continua in which basic kinematic and force variables are components of unified tensor objects and theory is represented by 4D-vector equation. Sedov’s equation has been derived and resulted into Euler’s equations, space projections of which determine motion equations, and time projection determines heat conductivity equation as well as the whole spectrum of the space-time boundary value problems. The proposed theory allows one to describe the scale effects in the thermal processes and opens prospects for studying the scale effects of the coupled dynamic thermoelasticity and its nanoscience applications. A temperature-scale refinement can also broaden the range of applicability of the law of heat conduction in solids to allow for design of small-sized components, devices and nano-systems.

LA - eng

KW - variation model; dissipation; space time elasticity; extended thermodynamic; relaxation time; scale effects; small-sized components, devices and nano-systems

UR - http://eudml.org/doc/267333

ER -

## References

top- [1] S.A. Lurie and P.A. Belov Mathematical Models of the Mechanics of the Continuum and Physical Fields. Moscow: The Publishing House of the Russian Academy of Sciences Data Center. 2000. 151 p.
- [2] S.A. Lurie, I.F. Obraztsov, P.A. Belov, and Yu.G. Yanovskii. On Some Classes of Models of Thin Structures. Izv. Vyssh. Uchebn. Zaved. Estestv. Nauki. 3 110–118 (2000).
- [3] P. Belov and S. Laurie, A continuum model of microheterogeneous media. J. Appl. Math. Mech. 73(5), 599–608 (2009). Zbl1272.74024
- [4] S.A. Lurie, P.A.Belov, and N.P. Tuchkova. Gradient theory of media with conserved dislocations: application to microstructured materials, Particular models: generalized Cosserats media model with surface effects, porous media, media with free forming (media with “twinning”), generalized pseudo-continuum, In: One hundred years after the Cosserats. Series: Advances in Mechanics and Mathematics, 21 110–119 (2010).
- [5] S.A. Lurie and P.A. Belov Cohesion field: Barenblatt’s hypothesis as formal corollary of theory of continuous media with conserved dislocations Int. J. Fract. 50(1–2) 181–194 (2008) [WoS] Zbl1143.74012
- [6] S.A. Lurie, D.B. Volkov-Bogorodsky, V.I. Zubov, and N.P. Tuchkova. Advanced theoretical and numerical multiscale modeling of cohesion/adhesion interactions in continuum mechanics and its applications for filled nanocomposites Int. J. Comp. Mater. Sci. 45(3) 709–714 (2009).
- [7] S.A. Lurie, P. A. Belov, and Yu.G. Yanovskii. On Simulation of Heat Transmission in Dynamically Deformed Media. Mekh. Komp. Mater. Konstr. 6(3) 436–444 (2000) [J. Comp. Mech. Design (Engl. Transl.)].
- [8] S.A. Lurie and P.A. Belov. A Variational Model for Nonholonomic Media Mekh. Komp. Mater. Konstr. 7(2) 266–276 (2001) [J. Comp. Mech. Design (Engl. Transl.)].
- [9] P.A. Belov, A.G. Gorshkov, and S.A. Lurie. Variational Model of Nonholonomic 4D-Media. Izv. Akad. Nauk. Mekh. Tverd. Tela 6 29–46 (2006) [Mech. Solids (Engl. Transl.) 41(6), 22–35 (2006)].
- [10] P.A. Belov, S.A. Lurie. Ideal Nonsymmetric 4D-Medium as a Model of Invertible Dynamic Thermoelasticity. Mechanics of Solids, 47(5), 580–590 (2012). [Crossref][WoS]
- [11] J. C. Maxwell. Philos. Trans. Roy. Soc. London 157, 49 (1867).
- [12] C. Cattaneo, C R. Hebd. Seances Acad. Sci. 247, 431 (1958).
- [13] Z.-Y. Guo. Motion and transfer of thermal mass – thermal mass and thermon gas. J. Eng. Term 27, 631–634 (2006).
- [14] A. Sellitto, F.X. Alvarez. Non-Fourier heat removal from hot nanosystems through graphene layer. J. Nanoscale Systems: Mathematical Modeling, Theory and Applications 1, 38–47 (2012). Zbl1273.80004
- [15] F.X. Àlvarez, V.A. Cimmelli, D. Jou, A. Sellitto. Mesoscopic description of boundary effects in nanoscale heat transport. J. Nanoscale Systems: Mathematical Modeling, Theory and Applications 1, 112–142 (2012). Zbl1273.74003
- [16] A. Gusev and S. Lurie. Wave-relaxation duality of heat propagation in Fermi-Pasta-Ulam chains. Modern Physics Letters B. 26(22), 1250145 (10 pages) (2012) [Crossref][WoS]