Abstract
This paper presents a new aspect in generalized continuum theory, namely micropolar media showing structural change. Initially, the necessary theoretical framework for a micropolar continuum is presented. To this end, the standard macroscopic equations for mass and linear and angular momentum are complemented by a recently proposed kinetic equation for the moment of inertia tensor containing a production term. An example for this term is studied: A continuous stream of matter through a crusher is considered. The matter is milled, and consequently, the total number of particles will change. This structural change is the reason for the production of microinertia. The matter is modeled as a Hookean as well as a linear viscous material. The equations are solved numerically based on a finite difference technique.
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References
Chen, K.: Microcontinuum balance equations revisited: the mesoscopic approach. J. Non-Equilib. Thermodyn. 32, 435–458 (2007)
Dłużewski, P.H.: Finite deformations of polar elastic media. Int. J. Solids Struct. 30(16), 2277–2285 (1993)
Eremeyev, V.A., Lebedev, L.P., Altenbach, H.: Foundations of Micropolar Mechanics. Springer, Heidelberg (2012)
Eringen, A.: Continuum Physics, vol. IV. Academic Press, New York (1976)
Eringen, A.: A unified continuum theory of electrodynamics of liquid crystals. Int. J. Eng. Sci. 35(12/13), 1137–1157 (1997)
Eringen, A.: Microcontinuum Field Theory I. Foundations and Solids. Springer, New York (1999)
Eringen, A.C., Kafadar, C.B.: Polar field theories. In: Continuum physics IV. Academic Press, London (1976)
Glane, S., Rickert, W., Müller, W.H., Vilchevskaya, E.: Micropolar media with structural transformations: Numerical treatment of a particle crusher. In: Proceedings of XLV International Summer School–Conference APM 2017, pp. 197–211. IPME RAS (2017)
Hamilton, E.: Elastic properties of marine sediments. J. Geophys. Res. 76(2), 579–604 (1971)
Ivanova, E., Vilchevskaya, E., Müller, W.H.: Time derivatives in material and spatial description—What are the differences and why do they concern us? In: Naumenko, K., Aßmus, M. (eds.) Advanced Methods of Mechanics for Materials and Structures, pp. 3–28. Springer, Berlin (2016)
Ivanova, E.A., Vilchevskaya, E.N.: Micropolar continuum in spatial description. Contin. Mech. Thermodyn. 28(6), 1759–1780 (2016)
Mindlin, R.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16(1), 51–78 (1964)
Müller, W.H., Vilchevskaya, E.N., Weiss, W.: A meso-mechanics approach to micropolar theory: a farewell to material description. Phys. Mesomech. 20(3), 13–24 (2017)
Oevel, W., Schröter, J.: Balance equation for micromorphic materials. J. Stat. Phys. 25(4), 645–662 (1981)
Truesdell, C., Toupin, R.A.: The Classical Field Theories. Springer, Heidelberg (1960)
Vilchevskaya, E.N., Müller, W.H.: Some remarks on recent developments in micropolar continuum theory. In: Proceedings of 5th International Conference on Topics Problems of Continuum Mechanics, Armenia, pp. 1–10. Journal of Physics: Conference Series, IOPScience (2018)
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Support of this work by a Grant from the Russian Foundation for Basic Research (16-01-00815) is gratefully acknowledged.
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Communicated by Andreas Öchsner.
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Fomicheva, M., Vilchevskaya, E.N., Müller, W.H. et al. Milling matter in a crusher: modeling based on extended micropolar theory. Continuum Mech. Thermodyn. 31, 1559–1570 (2019). https://doi.org/10.1007/s00161-019-00772-4
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DOI: https://doi.org/10.1007/s00161-019-00772-4