Mathematical Modeling of the Coupled Processes in Nanoporous Bodies

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The methods of irreversible thermomechanics and functional analysis are used to formulate the mathematical model of thermoelastic solid body taking account of structural heterogeneity of the body material and geometric irregularity of its surface. The density and the chemical potential of skeleton among others are included into the state parameters space. The source of skeleton mass reconciles the actual and reference body states and may be associated with real surface forming method. The analysis of model problem solutions shows that the model is appropriate to describe coupled processes in porous and nanoporous bodies. It allows studying the size effects of strength, elastic moduli, etc. caused by near-surface non-homogeneity.

1. Abeyaratne R., Knowles J.K. (1991), Kinetic relations and the propagation of phase boundaries in solids, Archive for Rational Mechanics and Analysis, 114(2), 119–154.

2. Aifantis E.C. (2011b), On the gradient approach–relation to Eringen’s nonlocal theory, International Journal of Engineering Science, 49(12), 1367–1377.

3. Aifantis E.C. (2011a), Gradient nanomechanics: applications to deformation, fracture, and diffusion in nanopolycrystals, Metallurgical and Materials Transactions A, 42(10), 2985.

4. Bao Y., Wen T., Samia A.C.S., Khandhar A., Krishnan K.M. (2016), Magnetic nanoparticles: material engineering and emerging applications in lithography and biomedicine, Journal of Materials Science, 51(1), 513–553.

5. Berezovski A., Engelbrecht J., Maugin G.A. (2007), Front dynamics in inhomogeneous solids. Proc. Estonian Acad. Sci. Phys. Math., 56(2), 155–161.

6. Bhattacharya A., Calmidi V.V., Mahajan R.L. (2002), Thermo-physical properties of high porosity metal foams, International Journal of Heat and Mass Transfer, 45(5), 1017–1031.

7. Biot M.A. (1941), General theory of three dimensional consolidation, Journal of Applied Physics, 12, 155–164.

8. Bozhenko B., Nahirnyj T., Tchervinka K. (2016), To modeling admixtures influence on the size effects in a thin film, Mathematical Modeling and Computing, 3(1), 12–22.

9. Burak Y.I., Nagirnyi T. (1992), Mathematical modeling of local gradient processes in inertial thermomechanical systems, International applied mechanics 28(12), 775–793.

10. Charalambakis N. (2010), Homogenization techniques and micro-mechanics. A survey and perspectives, Applied Mechanics Reviews, 63(3), 030803.

11. Coussy O. (2004), Poromechanics, John Wiley & Sons.

12. Dönmez A., Bažant Z.P. (2017), Size effect on punching strength of reinforced concrete slabs with and without shear reinforcement, ACI Structural Journal, 114(4), 875.

13. Elliott J.A. (2011), Novel approaches to multiscale modelling in materials science, International Materials Reviews, 56(4), 207–225.

14. Eringen A.C. (2002), Nonlocal continuum field theories, Springer Science & Business Media.

15. Eringen A.C., Edelen D.G.B. (1972), On nonlocal elasticity, International Journal of Engineering Science, 10(3), 233–248.

16. Geers M.G., Kouznetsova V., Brekelmans W.M. (2002), Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme, International Journal for Numerical Methods in Engineering, 54(8), 1235–1260.

17. Geers M.G.D., De Borst R., Peerlings R.H.J., Brekelmans W.A.M. (2001), A critical comparison of nonlocal and gradient-enhanced softening continua, International Journal of Solids and Structures, 38(44), 7723–7746.

18. Hu H., Onyebueke L., Abatan A. (2010), Characterizing and modeling mechanical properties of nanocomposites-review and evaluation, Journal of Minerals and Materials Characterization and Engineering, 9(04), 275.

19. Kachanov M., Sevostianov I. (2018), Quantitative Characterization of Microstructures in the Context of Effective Properties, In Micromechanics of Materials, with Applications (pp. 89–126), Springer, Cham.

20. Kalamkarov A.L., Andrianov I.V., Danishevsâ V.V. (2009), Asymptotic homogenization of composite materials and structures, Applied Mechanics Reviews, 62(3), 030802.

21. Markov K.Z. (2000), Elementary micromechanics of heterogeneous media, In Heterogeneous Media (pp. 1–162), Birkhäuser, Boston, MA.

22. Maugin G.A. (1979), Nonlocal theories or gradient-type theories-a matter of convenience, Archiv of Mechanics, Archiwum Mechaniki Stosowanej, 31, 15–26.

23. Nahirnyj T., Tchervinka K. (2012), Thermodynamical models and methods of thermomechanics taking into account nearsurface and structural nonhomogeneity. Bases of nanomechanics I, Spolom, Lviv (In Ukrainian).

24. Nahirnyj T., Tchervinka K. (2013), Structural inhomogeneity and size effects in thermoelastic solids, J. Coupled Syst. Multiscale Dyn., 1, 216–223.

25. Nahirnyj T., Tchervinka K. (2014), Basics of mechanics of local non-homogeneous elastic bodies. Bases of nanomechanics II, Rastr-7, Lviv (In Ukrainian).

26. Nahirnyj T., Tchervinka K. (2015), Mathematical Modeling of Structural and Near-Surface Non-Homogeneities in Thermoelastic Thin Films, International Journal of Engineering Science, 91, 49–62.

27. Pindera M.J., Khatam H., Drago A.S., Bansal Y. (2009), Microme-chanics of spatially uniform heterogeneous media: a critical review and emerging approaches, Composites Part B: Engineering, 40(5), 349–378.

28. Polizzotto C. (2003), Unified thermodynamic framework for nonlocal / gradient continuum theories, European Journal of Mechanics-A / Solids, 22(5), 651–668.

29. Rabotnov Yu.N. (1980), Elements of Hereditary Solid Mechanics, Mir Publ. Moscow (in Russian).

30. Rafii-Tabar H., Ghavanloo E., Fazelzadeh S.A. (2016), Nonlocal continuum-based modeling of mechanical characteristics of nanoscopic structures, Physics Reports, 638, 1–97.

31. Rosakis P., Knowles J.K. (1997), Unstable kinetic relations and the dynamics of solid-solid phase transitions, Journal of the Mechanics and Physics of Solids, 45(11), 2055–2081.

32. Tappan B.C., Steiner S.A., Luther E.P. (2010), Nanoporous metal foams, Angewandte Chemie International Edition, 49(27), 4544–4565.

33. Vafai K. (2015), Handbook of porous media, Crc Press.

34. Wang Y.M., Ma E. (2009), Mechanical properties of bulk nanostructured metals, Bulk Nanostructured Materials, 423–453.

35. Woźniak C. (1987), A nonstandard method of modelling of thermo-elastic periodic composites, International Journal of Engineering Science, 25(5), 483-498.

36. Young R., Kinloch I.A., Gong L., Novoselov K.S. (2012), The mechanics of graphene nanocomposites: a review, Composites Science and Technology, 72(12), 1459–1476.

Acta Mechanica et Automatica

The Journal of Bialystok Technical University

Journal Information

CiteScore 2017: 1.07

SCImago Journal Rank (SJR) 2017: 0.361
Source Normalized Impact per Paper (SNIP) 2017: 0.917


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