## THERMAL SCIENCE

International Scientific Journal

### ONE MATHEMATICAL MODEL OF THERMAL CONDUCTIVITY FOR MATERIALS WITH A GRANULAR STRUCTURE

**ABSTRACT**

The creation of new materials based on nanotechnology is an important direction of modern materials science development. Materials obtained by using nanotechnology can possess unique physicomechanical and thermophysical properties, al-lowing to use them effectively in structures exposed to high-intensity thermomechanical effects. An important step of the creation and usage of new materials is the construction of mathematical models to describe the behavior of these materials in a wide range of changes in external influences. One of the possible models for describing the process of thermal conductivity in structurally sensitive materials is proposed in this paper. The model is based on the laws of rational thermodynamics of irreversible processes and models of a continuous medium with internal state parameters. A qualitative study of the constructed model is carried out. A difference scheme is constructed in order to find the solution of the non-stationary heat conduction problem with allowance for the spatial non-locality effect. The analysis of the solutions is carried out.

**KEYWORDS**

PAPER SUBMITTED: 2019-01-18

PAPER REVISED: 2019-02-02

PAPER ACCEPTED: 2019-04-11

PUBLISHED ONLINE: 2019-09-22

**THERMAL SCIENCE** YEAR

**2019**, VOLUME

**23**, ISSUE

**Supplement 4**, PAGES [S1273 - S1280]

- Andrievskiy, R. A., Nanomaterials: The Concept and Modern Problems, G. Ros. chem. J. Society for them. D.I. Mendeleyev, XLVI (2002), 5, pp. 50-56
- Roduner, E., Size Matters: Why Nanomaterials are Different. J. Chem Soc Rev., 35 (2006), 7, pp. 83-92
- Andrievskii, R. A., Ragulya, A. V., Nanostructural Materials, Akademiya, Moscow, 2005
- Gusev, A. N., Nanomaterials, Nanostructures and Nanotechnologies, Fizmatlit, Moscow, 2005
- Gusev, A. N., Rempel, A. A., Nanocrystalline Materials, Fizmatlit, Moscow, Russia, 2001
- Rusanov, A. I., A Wonderful World of Nanostructures, Journal of General Chemistry, 72 (2002), 4, pp. 532-549
- Suzdalev, I. P., Nanotechnology Physics and Chemistry of Nanoclusters, Komkniga, Moscow, 2005
- Kunin, I. A., Elastic Media with Microstructure I, Springer, Berlin, Germany,1982
- Golovin, N. N., Kuvyrkin, G. N., Mathematical Models of Carbon-Carbon Composite Deformation, Mechanics of Solids, 51 (2016), 5, pp. 596-605
- Zarubin, V. S., et al., Mathematical Model of a Nonlocal Medium with Internal State Parameters, Journal of Engineering Physics and Thermophysics, 86 (2013), 4, pp. 768-773
- Kuvyrkin, G. N. , Savelieva, I. Yu., Thermomechanical Model of of Nonlocal Deformation of a Solid, Mechanics of Solids, 51 (2016), 3, pp. 256-262
- Savelieva, I. Yu., Influence of Medium Nonlocality on Distribution of Temperature and Stresses in Elastic Body under Pulsed Heating, Mechanics of Solids, 53 (2018), 3, pp. 277-283
- Zarubin, V. S ., et al, Mathematical Model of Thermostatic Coating with Thermoelectric Modules, Journal of Engineering Physics and Thermophysics, 88 (2015), 6, pp. 1328-1335
- Kuvyrkin, G. N. et al., Mathematical Model of the Heat Transfer Process Taking into Account the Consequences of Nonlocality in Structurally Sensitive Materials, Journal of Physics: Conference Series, 991 (2018), 1, 012050
- Eringen, A. C., Nonlocal Continuum Field Theories, New York-Berlin-Heidelberg, Springer-Verlag, Germany, 2002
- Zarubin, V. S., Kuvyrkin, G. N., Mathematical Models of Continuum Mechanics and Electrodynamics, Izdat. MGTU im. Baumana, Moscow, 2008
- Pisano, A. A., Fuschi, P., Closed Form Solution for a Nonlocal Elastic Bar in Tension, International Journal of Solids and Structures, 40 (2013), 1, pp. 13-23
- Polizzotto, C., Nonlocal Elasticity and Related Variational Principles, International Journal of Solids and Structures, 38 (2001), 42-43, pp. 7359-7380
- Kalitkin, N. N., Numerical Methods, Nauka, Moscow, 1978
- Galanin, M. P., Savenkov, E. B., Metody Chislennogo Analiza Matematicheskikh Modeley. (Numerical Analysis of Mathematical Models - in Russian), MGTU im. N. E. Baumana., Moscow, 2010
- Samarskiy, A. A., Teoriya Raznostnykh Skhem (Difference scheme theory - in Russian), Nauka, Moscow, 1977