Abstract
The traditional rheological method is supplemented by a new element—rigid contact, which serves to take into account different resistance of a material to tension and compression. A rigid contact describes mechanical properties of an ideal granular material involving rigid particles for an uniaxial stress state. Combining it with elastic, plastic, and viscous elements, one can construct rheological models of different complexity.
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References
Badiche, X., Forest, S., Guibert, T., Bienvenu, Y., Bartout, J.D., Ienny, P., Croset, M., Bernet, H.: Mechanical properties and non-homogeneous deformation of open-cell nickel foams: application of the mechanics of cellular solids and of porous materials. Mater. Sci. Eng. A Struct. Mater. Prop. Microstruct. Process. 289 (1–2), 276–288 (2000)
Banhart, J., Baumeister, J.: Deformation characteristics of metal foams. J. Mater. Sci. 33 (6), 1431–1440 (1998)
Biot, M.A.: Theory of finite deformations of porous solids. Indiana Univ. Math. J. 21, 597–735 (1972)
Biot, M.A.: Nonlinear and semilinear rheology of porous solids. J. Geophys. Res. 78 (23), 4924–4937 (1973)
de Boer, R.: Theory of Porous Media: Highlights in the Historical Development and Current State. Springer, New York (1999)
Borja, R.I.: Multiscale modeling of pore collapse instability in high-porosity solids. In: Soares, C.A.M., Martins, J.A.C., Rodrigues, H.C. (eds.) Computational Mechanics: Solids, Structures and Coupled Problems, pp. 165–172. Springer, Netherlands (2006)
Breuer, S., Onat, E.T.: On uniqueness in linear viscoelasticity. Q. Appl. Math. 19 (4), 355–359 (1962)
Bulavskii, V.A.: Metody Relaksaczii dlya Sistem Neravenstv (Relaxation Methods for Systems of Inequalities). Izd. Novosib. Univ., Novosibirsk (1981)
Bykovtsev, G.I., Ivlev, D.D.: Teoriya Plastichnosti (Plasticity Theory). Dal’nauka, Vladivostok (1998)
Carcione, J.M.: Viscoelastic effective rheologies for modeling wave propagation in porous media. Geophys. Prospect. 46, 249–270 (1998)
Carroll, M.M., Holt, A.C.: Static and dynamic pore-collapse relations for ductile porous materials. J. Appl. Phys. 43, 1626–1636 (1972)
Christensen, R.M.: Theory of Viscoelasticity, 2nd edn. Dover Publications Inc., Mineola (2010)
Cole, K.S., Cole, R.H.: Dispersion and absorption in dielectrics—I. Alternating current characteristics. J. Chem. Phys. 9, 341–351 (1941)
Day, W.A.: The Thermodynamics of Simple Materials with Fading Memory, Springer Tracts in Natural Philosophy, vol. 22. Springer, New York (1972)
Ehlers, W. (ed.): Theoretical and Numerical Methods in Continuum Mechanics of Porous Materials, Solid Mechanics and Its Applications. IUTAM Symposium. Kluwer Academic Publishers, Netherlands (2001)
Gibson, L.J.: Properties and applications of metal foams. In: Kelly, A., Zweben, C. (eds.) Comprehensive Composite Materials, Metal Matrix Composites. vol. 3, pp. 821–842. Pergamon Press, Oxford (2000)
Gibson, L.J., Ashby, M.F.: Cellular Solids: Structure and Properties. Cambridge Solid State Science Series. Cambridge University Press, Cambridge (1997)
Gnoevoi, A.V., Klimov, D.M., Chesnokov, V.M.: Osnovy Teorii Techenii Bingamovskikh Sred (Foundations of the Theory of Flows of Bingham Media). Fizmatlit, Moscow (2004)
Green, R.J.: A plasticity theory for porous solids. Int. J. Mech. Sci. 14, 215–224 (1972)
Nesterenko, V.F.: Dynamics of Heterogeneous Materials. Springer, New York (2001)
Rabotnov, Y.N.: Elements of Hereditary Solid Mechanics. Mir Publishers, Moscow (1980)
Reiner, M.: Deformation, Strain and Flow: an Elementary Introduction to Rheology. H. K. Lewis, London (1960)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Sadovskaya, O.V., Sadovskii, V.M.: Rheological models of uniaxial deformation of porous media. Vestnik Krasnoyarsk. Univ.: Fiz.-Mat. Nauki 9, 202–206 (2006)
Sadovskaya, O.V., Sadovskii, V.M.: Models of rheologically compicated media with different resistances to tension and compression. In: Mathematical Models and Methods of Continuum Mechanics: Collected Papers, pp. 224–238. IAPU DVO RAN, Vladivostok (2007)
Sadovskii, V.M.: Numerical modeling in problems of the dynamics of granular media. In: Proceedings of the Mathematical Centre N. I. Lobachevskii, Izd. Kazansk. Mat. Obshhestva, Kazan, vol. 15, pp. 183–198 (2002)
Sadovskii, V.M.: To the theory of elastic-plastic waves propagation in granular materials. Doklady Phys. 47 (10), 747–749 (2002)
Sadovskii, V.M.: Rheological models of hetero-modular and granular media. Dal’nevost. Mat. Zh. 4 (2), 252–263 (2003)
Sevostianov, I., Kachanov, M.: On the yield condition for anisotropic porous materials. Mater. Sci. Eng. A 313 (1–2), 1–15 (2001)
Vinogradov, G.V., Malkin, A.Y.: Rheology of Polymers. Springer, New York (1980)
Zintchenko, V.A., Sadovskaya, O.V., Sadovskii, V.M.: A numerical algorithm and a computer system for the analysis of rheological schemes. Numer. Methods Program. Adv. Computing 7 (2), 125–132 (2006)
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Sadovskaya, O., Sadovskii, V. (2012). Rheological Schemes. In: Mathematical Modeling in Mechanics of Granular Materials. Advanced Structured Materials, vol 21. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29053-4_2
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DOI: https://doi.org/10.1007/978-3-642-29053-4_2
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