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Cosserat Media

  • Chapter
Generalized Continua from the Theory to Engineering Applications

Part of the book series: CISM International Centre for Mechanical Sciences ((CISM,volume 541))

Abstract

In this chapter we discuss the three-dimensional Cosserat-type theories of continua. Originally this approach belongs to Cosserat brothers, Eugèn and François, who introduced an elastic media with kinematically independent translational and rotational degrees of freedom in their centurial book “Théorie des corps déformables” (Cosserat and Cosserat, 1909). Within the framework of the Cosserat media each point of the media can be represented as an infinitesimal rigid body. This means that in this media there exist stresses and couple stresses as responses for translational and rotational degrees of freedom. These characteristic features of the Cosserat continuum model give a possibility to describe more complex media, for example, micro-inhomogeneous materials, polycrystalline and cellular solids, foams, lattices, masonries, particle assemblies, magnetic rheological fluids, liquid crystals, etc. The aim of this chapter consists of a brief presentation of basics of Cosserat continuum mechanics including kinematics, dynamics and constitutive modeling.

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Bibliography

  • E. L. Aero, A. N. Bulygin, and E. V. Kuvshinskii. Asymmetric hydromechanics. Journal of Applied Mathematics and Mechanics, 29(2):333–346, 1965.

    MATH  Google Scholar 

  • M. Agranovich. Elliptic boundary problems. In M. Agranovich, Y. Egorov, and M. Shubin, editors, Partial Differential Equations IX: Elliptic Boundary Problems. Encyclopaedia of Mathematical Sciences, volume 79, pages 1–144. Springer, Berlin, 1997.

    Google Scholar 

  • H. Altenbach, V. A. Eremeyev, L. P. Lebedev, and L. A. Rendón. Acceleration waves and ellipticity in thermoelastic micropolar media. Archive of Applied Mechanics, 80(3):217–227, 2010a.

    Google Scholar 

  • H. Altenbach, G. A. Maugin, and V. Erofeev, editors. Mechanics of Generalized Continua, volume 7 of Advanced Structured Materials. Springer, Berlin, 2011.

    Google Scholar 

  • J. Altenbach, H. Altenbach, and V. A. Eremeyev. On generalized Cosserattype theories of plates and shells. A short review and bibliography. Archive of Applied Mechanics, 80(1):73–92, 2010b.

    MATH  Google Scholar 

  • S. S. Antman. Nonlinear Problems of Elasticity. Springer Science Media, New York, 2nd edition, 2005.

    MATH  Google Scholar 

  • V. I. Arnold. Mathematical Methods of Classical Mechanics. Springer, New York, 2nd edition, 1989.

    Google Scholar 

  • O. A. Bauchau and L. Trainelli. The vectorial parameterization of rotation. Nonlinear Dynamics, 32(1):71–92, 2003.

    MathSciNet  MATH  Google Scholar 

  • A. Bertram. Elasticity and Plasticity of Large Deformations: An Introduction. Springer, Berlin, 3rd edition, 2012.

    Google Scholar 

  • A. Bertram and B. Svendsen. On material objectivity and reduced constitutive equations. Archive of Mechanics, 53(6):653–675, 2001.

    MathSciNet  MATH  Google Scholar 

  • A. Bertram and B. Svendsen. Reply to Rivlin’s Material symmetry revisited or Much Ado About Nothing. GAMM Mitteilungen, 27(1):88–93, 2004.

    MathSciNet  MATH  Google Scholar 

  • D. Besdo. Towards a Cosserat-theory describing motion of an originally rectangular structure of blocks. Archive of Applied Mechanics, 80(1): 25–45, 2010.

    MATH  Google Scholar 

  • J. Besson, G. Cailletaud, J.-L. Chaboche, S. Forest, and M. Blétry. Non-Linear Mechanics of Materials, volume 167 of Solid Mechanics and Its Applications. Springer, Dordrecht, 2010.

    Google Scholar 

  • D. Bigoni and W. J. Drugan. Analytical derivation of Cosserat moduli via homogenization of heterogeneous elastic materials. Transactions of ASME. Journal of Applied Mechanics, 74(4):741–753, 2007.

    MathSciNet  Google Scholar 

  • J. Chróścielewski, J. Makowski, and W. Pietraszkiewicz. Statics and Dynamics of Multyfolded Shells. Nonlinear Theory and Finite Elelement Method (in Polish). Wydawnictwo IPPT PAN, Warszawa, 2004.

    Google Scholar 

  • I. Cielecka, M. Wo´zniak, and C. Wo´zniak. Elastodynamic behaviour of honeycomb cellular media. Journal of Elasticity, 60(1):1–17, 2000.

    MATH  Google Scholar 

  • J. D. Clayton. Nonlinear Mechanics of Crystals, volume 177 of Solid Mechanics and Its Applications. Springer, Dordrecht, 2011.

    Google Scholar 

  • E. Cosserat and F. Cosserat. Théorie des corps déformables. Herman et Fils, Paris, 1909.

    Google Scholar 

  • E. Cosserat and F. Cosserat. Sur la théorie de l’elasticité. Ann. Toulouse, 10:1–116, 1896.

    MathSciNet  Google Scholar 

  • S. Diebels. A micropolar theory of porous media: constitutive modelling. Transport in Porous Media, 34(1–3):193–208, 1999.

    Google Scholar 

  • J. Dyszlewicz. Micropolar Theory of Elasticity. Springer, Berlin, 2004.

    MATH  Google Scholar 

  • W. Ehlers, E. Ramm, S. Diebels, and G. D. A. d’Addetta. From particle ensembles to Cosserat continua: Homogenization of contact forces towards stresses and couple stresses. International Journal of Solids and Structures, 40(24):6681–6702, 2003.

    MathSciNet  MATH  Google Scholar 

  • V. V. Eliseev. Mechanics of Elastic Bodies (in Russian). Politekhn. Univ. Publ., St. Petersburg, 1996.

    Google Scholar 

  • V. A. Eremeyev. Acceleration waves in micropolar elastic media. Doklady Physics, 50(4):204–206, 2005.

    Google Scholar 

  • V. A. Eremeyev and W. Pietraszkiewicz. Local symmetry group in the general theory of elastic shells. Journal of Elasticity, 85(2):125–152, 2006.

    MathSciNet  MATH  Google Scholar 

  • V. A. Eremeyev and W. Pietraszkiewicz. Material symmetry group of the non-linear polar-elastic continuum. International Journal of Solids and Structures, doi: 10.1016/j.ijsolstr.2012.04.007, 2012.

    Google Scholar 

  • V. A. Eremeyev and L. M. Zubov. On the stability of elastic bodies with couple stresses. Mechanics of Solids, 29(3):172–181, 1994.

    Google Scholar 

  • V. A. Eremeyev and L. M. Zubov. On constitutive inequalities in nonlinear theory of elastic shells. ZAMM, 87(2):94–101, 2007.

    MathSciNet  MATH  Google Scholar 

  • V. A. Eremeyev and L. M. Zubov. Principles of Viscoelastic Micropolar Fluid Mechanics (in Russian). SSC of RASci Publishers, Rostov on Don, 2009.

    Google Scholar 

  • V. A. Eremeyev, L. P. Lebedev, and H. Altenbach. Foundations of Microplar Mechanics. Springer, Berlin, 2012.

    Google Scholar 

  • J. L. Ericksen and C. Truesdell. Exact theory of stress and strain in rods and shells. Archive for Rational Mechanics and Analysis, 1(1):295–323, 1958.

    MathSciNet  MATH  Google Scholar 

  • A. C. Eringen. Microcontinuum Field Theory, volume II. Fluent Media. Springer, New York, 2001.

    Google Scholar 

  • A. C. Eringen. A unified continuum theory for electrodynamics of polymeric liquid crystals. International Journal of Engineering Science, 38(9–10): 959–987, 2000.

    MathSciNet  MATH  Google Scholar 

  • A. C. Eringen. Nonlocal continuum field theories. Springer, New York, 2002.

    MATH  Google Scholar 

  • A. C. Eringen. Theory of micropolar fluids. Journal of Mathematics and Mechanics, 16(1):1–18, 1966.

    MathSciNet  Google Scholar 

  • A. C. Eringen. A unified continuum theory of liquid crystals. ARI – An International Journal for Physical and Engineering Sciences, 73–84(2): 369–374, 1997a.

    Google Scholar 

  • A. C. Eringen. A unified continuum theory of electrodynamics of liquid crystals. International Journal of Engineering Science, 35(12–13):1137–1157, 1997b.

    MathSciNet  MATH  Google Scholar 

  • A. C. Eringen. Microcontinuum Field Theory, volume I. Foundations and Solids. Springer, New York, 1999.

    MATH  Google Scholar 

  • A. C. Eringen and C. B. Kafadar. Polar field theories. In A. C. Eringen, editor, Continuum Physics, volume IV, pages 1–75. Academic Press, New York, 1976.

    Google Scholar 

  • V. I. Erofeev. Wave Processes in Solids with Microstructure. World Scientific, Singapore, 2003.

    Google Scholar 

  • R. P. Feynman, R. B. Leighton, and M. Sands. The Feynman Lectures on Physics, volume 1. Addison-Wesley, Reading, Massachusetts, 6th edition, 1977.

    Google Scholar 

  • G. Fichera. Existence theorems in elasticity. In S. Flügge, editor, Handbuch der Physik, volume VIa/2, pages 347–389. Springer, Berlin, 1972.

    Google Scholar 

  • S. Forest. Mechanics of generalized continua: construction by homogenizaton. Journal de Physique IV France, 8(PR4):Pr4–39–Pr4–48, 1998.

    Google Scholar 

  • S. Forest and Kh. T. Duy. Generalized continua and non-homogeneous boundary conditions in homogenisation methods. ZAMM, 91(2):90–109, 2011.

    MathSciNet  MATH  Google Scholar 

  • S. Forest and K. Sab. Cosserat overal modelling of heterogeneous materials. Mechanics Research Communications, 25(4):449–454, 1998.

    MathSciNet  MATH  Google Scholar 

  • S. Forest and R. Sievert. Nonlinear microstrain theories. International Journal of Solids and Structures, 43(24):7224–7245, 2006.

    MathSciNet  MATH  Google Scholar 

  • R. D. Gauthier and W. E. Jahsman. Quest for micropolar elastic-constants. Transactions of the ASME. Journal of Applied Mechanics, 42(2):369–374, 1975.

    Google Scholar 

  • R. D. Gauthier and W. E. Jahsman. Quest for micropolar elastic-constants. 2. Archives of Mechanics, 33(5):717–737, 1981.

    MATH  Google Scholar 

  • D. H. Hodges. Nonlinear Composite Beam Theory, volume 213 of Progress in Astronautics and Aeronautics. American Institute of Aeronautics and Astronautics, Inc., Reston, 2006.

    Google Scholar 

  • L. Hörmander. Linear Partial Differential Equations, volume 116 of A Series of Comprehensive Studies in Mathematics. Springer, Berlin, 4th edition, 1976.

    Google Scholar 

  • R. A. Horn and Ch. R. Johnson. Matrix Analysis. Cambridge University Press, Cambridge, 1985.

    MATH  Google Scholar 

  • D. Ieşan. Existence theorems in the theory of micropolar elasticity. International Journal of Engineering Science, 8:777–791, 1970.

    MathSciNet  MATH  Google Scholar 

  • D. Ieşan. Existence theorems in micropolar elastostatics. International Journal of Engineering Science, 9:59–78, 1971.

    MATH  Google Scholar 

  • E. A. Ivanova, A. M. Krivtsov, N. F. Morozov, and A. D. Firsova. Description of crystal packing of particles with torque interaction. Mechanics of Solids, 38(4):76–88, 2003.

    Google Scholar 

  • M. Jabareen and M. Rubin. An improved 3-D brick Cosserat point element for irregular shaped elements. Computational Mechanics, 40(6): 979–1004, 2007.

    MathSciNet  MATH  Google Scholar 

  • M. Jabareen and M. Rubin. Modified torsion coefficients for a 3-D brick Cosserat point element. Computational Mechanics, 41(4):517–525, 2008.

    MathSciNet  MATH  Google Scholar 

  • J. Jeong and P. Neff. Existence, uniqueness and stability in linear Cosserat elasticity for weakest curvature conditions. Mathematics and Mechanics of Solids, 15(1):78–95, 2010.

    MathSciNet  MATH  Google Scholar 

  • C. B. Kafadar and A. C. Eringen. Micropolar media – I. The classical theory. International Journal of Engineering Science, 9(3):271–305, 1971.

    MATH  Google Scholar 

  • V. Kouznetsova, M. G. D. Geers, and W. A. M. Brekelmans. 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, 2002.

    MATH  Google Scholar 

  • E. Kröner, editor. Mechanics of Generalized Continua. Proceedings of the IUTAM-Symposium on the Generalized Cosserat Continuum and the Continuum Theory of Dislocations with Applications, Freudenstadt and Stuttgart (Germany), 1967, Berlin, 1968. Springer.

    MATH  Google Scholar 

  • R. S. Lakes. Experimental methods for study of Cosserat elastic solids and other generalized continua. In H. Mühlhaus, editor, Continuum Models for Materials with Micro-Structure, pages 1–22. Wiley, N. Y., 1995.

    Google Scholar 

  • R. S. Lakes. Experimental microelasticity of two porous solids. International Journal of Solids and Structures, 22(1):55–63, 1986.

    Google Scholar 

  • R. S. Lakes. Experimental micro mechanics methods for conventional and negative Poisson’s ratio cellular solids as Cosserat continua. Transactions of the ASME. Journal of Engineering Materials and Technology, 113(1): 148–155, 1991.

    Google Scholar 

  • R. Larsson and S. Diebels. A second-order homogenization procedure for multi-scale analysis based on micropolar kinematics. International Journal for Numerical Methods in Engineering, 69(12):2485–2512, 2007.

    MathSciNet  MATH  Google Scholar 

  • R. Larsson and Y. Zhang. Homogenization of microsystem interconnects based on micropolar theory and discontinuous kinematics. Journal of the Mechanics and Physics of Solids, 55(4):819–841, 2007.

    MathSciNet  MATH  Google Scholar 

  • K. C. Le and H. Stumpf. Strain measures, integrability condition and frame indifference in the theory of oriented media. International Journal of Solids and Structures, 35(9–10):783–798, 1998.

    MathSciNet  MATH  Google Scholar 

  • L. P. Lebedev, M. J. Cloud, and V. A. Eremeyev. Tensor Analysis with Applications in Mechanics. World Scientific, New Jersey, 2010.

    MATH  Google Scholar 

  • L. Librescu and O. Song. Thin-Walled Composite Beams: Theory and Applications, volume 131 of Solid Mechanics and Its Applications. Springer, Dordrecht, 2006.

    MATH  Google Scholar 

  • J.-L. Lions and E. Magenes. Problèmes aux limites non homogènes et applications. Dunod, Paris, 1968. G. Lukaszewicz. Micropolar Fluids: Theory and Applications. Birkhäuser, Boston, 1999.

    MATH  Google Scholar 

  • A. I. Lurie. Theory of Elasticity. Springer, Berlin, 2005.

    Google Scholar 

  • A. I. Lurie. Nonlinear Theory of Elasticity. North-Holland, Amsterdam, 1990.

    MATH  Google Scholar 

  • A. I. Lurie. Analytical Mechanics. Springer, Berlin, 2001.

    Google Scholar 

  • B. Markert, editor. Advances in Extended and Multifield Theories for Continua, volume 59 of Lecture Notes in Applied and Computational Mechanics. Springer, Berlin, 2011.

    Google Scholar 

  • G. A. Maugin. Configurational Forces: Thermomechanics, Physics, Mathematics, and Numerics. CRC Press, Boca Raton, 2010.

    Google Scholar 

  • G. A. Maugin. Nonlinear Waves on Elastic Crystals. Oxford University Press, Oxford, 1999.

    Google Scholar 

  • G. A. Maugin and A. V. Metrikine, editors. Mechanics of Generalized Continua: One Hundred Years After the Cosserats. Springer, New York, 2010.

    MATH  Google Scholar 

  • T. Merlini. A variational formulation for finite elasticity with independent rotation and biot-axial fields. Computational Mechanics, 19(3):153–168, 1997.

    MATH  Google Scholar 

  • N. P. Migoun and P. P. Prokhorenko. Hydrodynamics and Heattransfer in Gradient Flows of Microstructured Fluids (in Russian). Nauka i Technika, Minsk, 1984.

    Google Scholar 

  • R. Mora and A. M. Waas. Measurement of the Cosserat constant of circularcell polycarbonate honeycomb. Philosophical Magazine A. Physics of Condensed Matter Structure Defects and Mechanical Properties, 80(7): 1699–1713, 2000.

    Google Scholar 

  • A. I. Murdoch. On objectivity and material symmetry for simple elastic solids. Journal of Elasticity, 60(3):233–242, 2000.

    MathSciNet  MATH  Google Scholar 

  • A. I. Murdoch. Objectivity in classical continuum physics: a rationale for discarding the ‘principle of invariance under superposed rigid body motions’ in favour of purely objective considerations. Continuum Mechanics and Thermodynamics, 15(3):309–320, 2003.

    MathSciNet  MATH  Google Scholar 

  • A. I. Murdoch. On criticism of the nature of objectivity in classical continuum physics. Continuum Mechanics and Thermodynamics, 17(2):135–148, 2005.

    MathSciNet  MATH  Google Scholar 

  • W. Muschik and L. Restuccia. Changing the observer and moving materials in continuum physics: objectivity and frame-indifference. Technische Mechanik, 22(2):152–160, 2002.

    Google Scholar 

  • B. Nadler and M.B. Rubin. A new 3-D finite element for nonlinear elasticity using the theory of a Cosserat point. International Journal of Solids and Structures, 40(17):4585–4614, 2003.

    MATH  Google Scholar 

  • M. N. L. Narasimhan. Principles of Continuum Mechanics. Wiley, New York, 1993.

    MATH  Google Scholar 

  • P. Neff and K. Chelmi´nski. Well-posedness of dynamic Cosserat plasticity. Applied Mathematics and Optimization, 56(1):19–35, 2007.

    MathSciNet  MATH  Google Scholar 

  • P. Neff and S. Forest. A geometrically exact micromorphic model for elastic metallic foams accounting for affine microstructure. Modelling, existence of minimizers, identification of moduli and computational results. Journal of Elasticity, 87(2–3):239–276, 2007.

    MathSciNet  MATH  Google Scholar 

  • P. Neff and J. Jeong. A new paradigm: the linear isotropic Cosserat model with conformally invariant curvature energy. ZAMM, 89(2):107–122, 2009.

    MathSciNet  MATH  Google Scholar 

  • P. Neff, K. Chelmi´nski, W. Müller, and Ch. Wieners. A numerical solution method for an infinitesimal elasto-plastic Cosserat model. Mathematical Models & Methods in Applied Sciences, 17(8):1211–1239, 2007.

    MathSciNet  MATH  Google Scholar 

  • P. Neff, J. Jeong, and A. Fischle. Stable identification of linear isotropic Cosserat parameters: bounded stiffness in bending and torsion implies conformal invariance of curvature. Acta Mechanica, 211(3–4):237–249, 2010.

    MATH  Google Scholar 

  • E. Nikitin and L. M. Zubov. Conservation laws and conjugate solutions in the elasticity of simple materials and materials with couple stress. Journal of Elasticity, 51(1):1–22, 1998.

    MathSciNet  MATH  Google Scholar 

  • L. Nirenberg. Topics in Nonlinear Functional Analysis. American Mathematical Society, New York, 2001.

    MATH  Google Scholar 

  • W. Noll. A mathematical theory of the mechanical behavior of continuous media. Archive for Rational Mechanics and Analysis, 2(1):197–226, 1958.

    MATH  Google Scholar 

  • W. Nowacki, editor. Theory of Micropolar Elasticity, volume 25 of CISM Courses and Lectures. Springer, Wien, 1970.

    Google Scholar 

  • W. Nowacki. Theory of Asymmetric Elasticity. Pergamon-Press, Oxford et al., 1986.

    MATH  Google Scholar 

  • W. Nowacki and W. Olszak, editors. Micropolar Elasticity, volume 151 of CISM Courses and Lectures. Springer, Wien, 1974.

    Google Scholar 

  • R. W. Ogden. Non-linear Elastic Deformations. Ellis Horwood, Chichester, 1984.

    Google Scholar 

  • H. C. Park and R. S. Lakes. Cosserat micromechanics of human bone: strain redistribution by a hydration-sensitive constituent. Journal of Biomechanics, 19(5):385–397, 1986.

    Google Scholar 

  • E. Pasternak and H.-B. Mühlhaus. Generalised homogenisation procedures for granular materials. Journal of Engineering Mathematics, 52(1):199–229, 2005.

    MathSciNet  MATH  Google Scholar 

  • W. Pietraszkiewicz and V. A. Eremeyev. On natural strain measures of the non-linear micropolar continuum. International Journal of Solids and Structures, 46(3–4):774–787, 2009a.

    MathSciNet  MATH  Google Scholar 

  • W. Pietraszkiewicz and V. A. Eremeyev. On vectorially parameterized natural strain measures of the non-linear Cosserat continuum. International Journal of Solids and Structures, 46(11–12):2477–2480, 2009b.

    MathSciNet  MATH  Google Scholar 

  • S. Ramezani, R. Naghdabadi, and S. Sohrabpour. Constitutive equations for micropolar hyper-elastic materials. International Journal of Solids and Structures, 46(14–15):2765–2773, 2009.

    MATH  Google Scholar 

  • E. Reissner. On kinematics and statics of finite-strain force and moment stress elasticity. Studies in Applied Mathematics, 52:93–101, 1973.

    Google Scholar 

  • E. Reissner. Note on the equations of finite-strain force and moment stress elasticity. Studies in Applied Mathematics, 54:1–8, 1975.

    MathSciNet  MATH  Google Scholar 

  • R. S. Rivlin. Material symmetry and constitutive equations. Ingenieur-Archiv, 49(5-6):325–336, 1980.

    MATH  Google Scholar 

  • R. S. Rivlin. Material symmetry revisited. GAMM Mitteilungen, 25(1/2): 109–126, 2002.

    MathSciNet  MATH  Google Scholar 

  • R. S. Rivlin. Frame indifference and relative frame indifference. Mathematics and Mechanics of Solids, 10(2):145–154, 2005.

    MathSciNet  MATH  Google Scholar 

  • R. S. Rivlin. Some thoughts on frame indifference. Mathematics and Mechanics of Solids, 11(2):113–122, 2006.

    MathSciNet  MATH  Google Scholar 

  • R. E. Rosensweig. Magnetic fluids. Annual Review of Fluid Mechanics, 19: 437–461, 1987.

    Google Scholar 

  • M. B. Rubin. Cosserat Theories: Shells, Rods and Points. Kluwer, Dordrecht, 2000.

    MATH  Google Scholar 

  • M. B. Rubin. On the theory of a Cosserat point and its application to the numerical solution of continuum problems. Transactions of ASME. Journal of Applied Mechanics, 52(2):368–372, 1985.

    MATH  Google Scholar 

  • J. G. Simmonds. A simple nonlinear thermodynamic theory of arbitrary elastic beams. Journal of Elasticity, 81(1):51–62, 2005.

    MathSciNet  MATH  Google Scholar 

  • A. J. M. Spencer. Isotropic integrity bases for vectors and second-order tensors. Part II. Archive for Rational Mechanics and Analysis, 18(1): 51–82, 1965.

    MathSciNet  MATH  Google Scholar 

  • A. J. M. Spencer. Theory of invariants. In A. C. Eringen, editor, Continuum Physics, volume 1, pages 239–353. Academic Press, New-York, 1971.

    Google Scholar 

  • R. Stojanovi´c. Mechanics of Polar Continua: Theory and Applications, volume 2 of CISM Courses and Lectures. Springer, Wien, 1969a.

    Google Scholar 

  • R. Stojanovi´c. Recent Developments in the Theory of Polar Continua, volume 27 of CISM Courses and Lectures No. 27. Springer, Wien, 1969b.

    Google Scholar 

  • R. Stojanovi´c. Nonlinear micropolar elasticity. In W. Nowacki and W. Olszak, editors, Micropolar Elasticity, volume 151, pages 73–103. Springer, Wien, 1974.

    Google Scholar 

  • A. S. J. Suiker and R. de Borst. Enhanced continua and discrete lattices for modelling granular assemblies. Philosophical Transactions of the Royal Society A , 363(1836):2543–2580, 2005.

    MATH  Google Scholar 

  • B. Svendsen and A. Bertram. On frame-indifference and form-invariance in constitutive theory. Acta Mechanica, 132(1–4):195–207, 1999.

    MathSciNet  Google Scholar 

  • V. A. Svetlitsky. Statics of Rods. Springer, Berlin, 2000.

    MATH  Google Scholar 

  • C. Truesdell. Die Entwicklung des Drallsatzes. ZAMM, 44(4/5):149–158, 1964.

    MathSciNet  MATH  Google Scholar 

  • C. Truesdell. A First Course in Rational Continuum Mechanics. Academic Press, New York, 1977.

    MATH  Google Scholar 

  • C. Truesdell. Rational Thermodynamics. Springer, New York, 2nd edition, 1984.

    MATH  Google Scholar 

  • C. Truesdell and W. Noll. The nonlinear field theories of mechanics. In S. Flügge, editor, Handbuch der Physik, volume III/3, pages 1–602. Springer, Berlin, 1965.

    Google Scholar 

  • C. Truesdell and R. Toupin. The classical field theories. In S. Flügge, editor, Handbuch der Physik, volume III/1, pages 226–793. Springer, Berlin, 1960.

    Google Scholar 

  • O. van der Sluis, P. H. J. Vosbeek, P. J. G. Schreurs, and H. E. H. Meijer. Homogenization of heterogeneous polymers. International Journal of Solids and Structures, 36(21):3193–3214, 1999.

    MathSciNet  MATH  Google Scholar 

  • A. O. Varga. Stress-Strain Behavior of Elastic Materials: Selected Problems of Large Deformations. Interscience, New York, 1996.

    Google Scholar 

  • C. C. Wang and C. Truesdell. Introduction to Rational Elasticity. Noordhoof Int. Publishing, Leyden, 1973.

    MATH  Google Scholar 

  • H. Xiao. On symmetries and anisotropies of classical and micropolar linear elasticities: a new method based upon a complex vector basis and some systematic results. Journal of Elasticity, 49(2):129–162, 1998.

    MATH  Google Scholar 

  • J. F. C. Yang and R. S. Lakes. Experimental study of micropolar and couple stress elasticity in compact bone in bending. Journal of Biomechanics, 15(2):91–98, 1982.

    Google Scholar 

  • V. A. Yeremeyev and L. M. Zubov. The theory of elastic and viscoelastic micropolar liquids. Journal of Applied Mathematics and Mechanics, 63 (5):755–767, 1999.

    MathSciNet  Google Scholar 

  • Q. S. Zheng. Theory of representations for tensor functions – a unified invariant approach to constitutive equations. Applied Mechanics Reviews, 47(11):545–587, 1994.

    Google Scholar 

  • Q. S. Zheng and A. J. M. Spencer. On the canonical representations for Kronecker powers of orthogonal tensors with application to material symmetry problems. International Journal of Engineering Science, 31(4): 617–635, 1993.

    MathSciNet  MATH  Google Scholar 

  • L. M. Zubov. Nonlinear Theory of Dislocations and Disclinations in Elastic Bodies. Springer, Berlin, 1997.

    MATH  Google Scholar 

  • L. M. Zubov and V. A. Eremeev. Equations for a viscoelastic micropolar fluid. Doklady Physics, 41(12):598–601, 1996.

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Lehrstuhl Technische Mechanik, Institut für Mechanik, Fakultät für Maschinenbau, Otto-von-Guericke-Universität Magdeburg, Magdeburg, Germany

    Holm Altenbach & Victor A. Eremeyev

  2. South Scientific Center of RASci and South Federal University, Rostov on Don, Russian Federation

    Victor A. Eremeyev

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  1. Holm Altenbach
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  2. Victor A. Eremeyev
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  1. Otto Von Guericke University, Magdeburg, Germany

    Holm Altenbach

  2. South Federal University and South Scientific Center of Rasci, Rostov on Don, Russian Federation

    Victor A. Eremeyev

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© 2013 CISM, Udine

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Altenbach, H., Eremeyev, V.A. (2013). Cosserat Media. In: Altenbach, H., Eremeyev, V.A. (eds) Generalized Continua from the Theory to Engineering Applications. CISM International Centre for Mechanical Sciences, vol 541. Springer, Vienna. https://doi.org/10.1007/978-3-7091-1371-4_2

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  • DOI: https://doi.org/10.1007/978-3-7091-1371-4_2

  • Publisher Name: Springer, Vienna

  • Print ISBN: 978-3-7091-1370-7

  • Online ISBN: 978-3-7091-1371-4

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