Abstract
The constitutive relationships for a granular material involving absolutely rigid and elastic particles as well as the constitutive equations for a heteromodular elastic material are generalized to the case of a spatial stress-strain state under small strains. Versions of constraints on admissible stress tensors for an isotropic material, which are defined with the help of the Coulomb–Mohr and von Mises–Schleicher cones, are considered. Dual cones of admissible strain tensors are constructed. The projection operators, which are used further in algorithms for numerical realization of spatial models, are presented.
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References
Drucker, D.C., Prager, W.: Soil mechanics and plastic analysis or limit design. Q. Appl. Math. 10(2), 157–165 (1952)
Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Academic Press, San Diego (2000)
Haar, A, von Kármán, T.: Zur Theorie der Spannungszustände in plastischen und sandartigen Medien. Nachrichten von der Königlichen Gesellschaft der Wissenschaften, pp. 204–218 (1909)
Hencky, H.: Zur Theorie plastischer Deformationen und der hierdurch im Material hervorgerufenen Nachspannungen. ZAMM 4, 323–335 (1924)
Koiter, W.T.: General theorems for elastic-plastic solids. In: Sneddon, I.N., Hill, R. (eds.) Progress in Solid Mechanics, pp. 165–221. North-Holland, Amsterdam (1960)
Mróz, Z., Szymanski, C.:Non-associated flow rules in description of plastic flow of granular materials. In: Olszak W. (ed.) Limit Analysis and Rheological Approach in Soil Mechanics, CISM Courses and Lectures No 217, pp. 23–41. Springer, New York (1979)
Myasnikov, V.P., Oleynikov, A.I.: Equations of the elasticity theory and the yield condition for granular linearly dilatational media. Fiz.-Tekhn. Probl. Razrab. Pol. Iskop. 6, 14–19 (1984)
Myasnikov, V.P., Oleynikov, A.I.: Deformation model of ideally granular medium. Dokl. Akad. Nauk SSSR 316(3):565–568 (1991)
Myasnikov, V.P., Oleynikov, A.I.: Basic general relationships of the model of an isotropic elastic heteroresistant medium. Dokl. Akad. Nauk SSSR 322(1), 57–60 (1992)
Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I.: Integrals and Series, vol. 2–3. Gordon and Breach Science Publishers, Amsterdam (1990)
Reynolds, O.: Papers of Mechanical and Physical Subjects, vol. II. Cambridge University Press, Cambridge (1901)
Sadovskii, V.M.: Problems of the dynamics of granular media. Mat. Modelirovanie 13(5), 62–74 (2001)
Sokolovskii, V.V.: Statics of Granular Media. Pergamon Press, Oxford (1965)
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Sadovskaya, O., Sadovskii, V. (2012). Spatial Constitutive Relationships. 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_4
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DOI: https://doi.org/10.1007/978-3-642-29053-4_4
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