# American Institute of Mathematical Sciences

December  2013, 6(6): 1473-1485. doi: 10.3934/dcdss.2013.6.1473

## On the Cosserat model for thin rods made of thermoelastic materials with voids

 1 Department of Mathematics, University "A.I. Cuza" of Iaşi, 700506 Iaşi, Romania 2 Faculty of Mechanical Engineering, Otto-von-Guericke-University, 39106 Magdeburg, Germany

Received  June 2012 Revised  September 2012 Published  April 2013

In this paper we employ a Cosserat model for rod-like bodies and study the governing equations of thin thermoelastic porous rods. We apply the counterpart of Korn's inequality in the three-dimensional elasticity theory to prove existence and uniqueness results concerning the solutions to boundary value problems for thermoelastic porous rods, both in the dynamical theory and in the equilibrium case.
Citation: Mircea Bîrsan, Holm Altenbach. On the Cosserat model for thin rods made of thermoelastic materials with voids. Discrete & Continuous Dynamical Systems - S, 2013, 6 (6) : 1473-1485. doi: 10.3934/dcdss.2013.6.1473
##### References:
 [1] H. Altenbach, K. Naumenko and P. A. Zhilin, A direct approach to the formulation of constitutive equations for rods and shells,, in, (2006), 87. Google Scholar [2] M. Bîrsan, Inequalities of Korn's type and existence results in the theory of Cosserat elastic shells,, J. Elasticity, 90 (2008), 227. doi: 10.1007/s10659-007-9140-2. Google Scholar [3] M. Bîrsan and H. Altenbach, A mathematical study of the linear theory for orthotropic elastic simple shells,, Math. Meth. Appl. Sci., 33 (2010), 1399. doi: 10.1002/mma.1253. Google Scholar [4] M. Bîrsan and H. Altenbach, Theory of thin thermoelastic rods made of porous materials,, Arch. Appl. Mech., 81 (2011), 1365. doi: 10.1007/s00419-010-0490-z. Google Scholar [5] M. Bîrsan and H. Altenbach, The Korn-type inequality in a Cosserat model for thin thermoelastic porous rods,, Meccanica, 47 (2011), 789. doi: 10.1007/s11012-011-9477-2. Google Scholar [6] M. Bîrsan and T. Bîrsan, An inequality of Cauchy-Schwarz type with application in the theory of elastic rods,, Libertas Mathematica, 31 (2011), 123. Google Scholar [7] H. Brezis, "Analyse Fonctionelle. Théorie et Applications,", (French) [Functional Analysis: Theory and Applications], (1983). Google Scholar [8] G. Capriz, "Continua with Microstructure,", Springer Tracts in Natural Philosophy, 35 (1989). doi: 10.1007/978-1-4612-3584-2. Google Scholar [9] G. Capriz and P. Podio-Guidugli, Materials with spherical structure,, Arch. Rational Mech. Anal., 75 (1981), 269. doi: 10.1007/BF00250786. Google Scholar [10] P. G. Ciarlet, "Mathematical Elasticity, Vol. I. Three-Dimensional Elasticity,", Studies in Mathematics and its Applications, 20 (1988). Google Scholar [11] P. G. Ciarlet, "Mathematical Elasticity. Vol. III. Theory of Shells,", Studies in Mathematics and its Applications, 29 (2000). Google Scholar [12] P. G. Ciarlet, "An Introduction to Differential Geometry with Applications to Elasticity,", Springer, (2005). Google Scholar [13] E. Cosserat and F. Cosserat, "Théorie des Corps Déformables,", (French) [Theory of deformable bodies], (1909). Google Scholar [14] S. C. Cowin and J. W. Nunziato, Linear elastic materials with voids,, J. Elasticity, 13 (1983), 125. doi: 10.1007/BF00041230. Google Scholar [15] M. A. Goodman and S. C. Cowin, A continuum theory for granular materials,, Arch. Rational Mech. Anal., 44 (1972), 249. doi: 10.1007/BF00284326. Google Scholar [16] A. E. Green and P. M. Naghdi, On thermal effects in the theory of rods,, Int. J. Solids Struct., 15 (1979), 829. doi: 10.1016/0020-7683(79)90053-2. Google Scholar [17] L. P. Lebedev, M. J. Cloud and V. A. Eremeyev, "Tensor Analysis with Applications in Mechanics,", World Scientific Publishing Co. Pte. Ltd., (2010). doi: 10.1142/9789814313995. Google Scholar [18] A. I. Lurie, "Theory of Elasticity,", Foundations of Engineering Mechanics, (2005). doi: 10.1007/978-3-540-26455-2. Google Scholar [19] P. Neff, On Korn's first inequality with non-constant coefficients,, Proc. Roy. Soc. Edinb. A, 132 (2002), 221. doi: 10.1017/S0308210500001591. Google Scholar [20] P. Neff, A geometrically exact planar Cosserat shell-model with microstructure: Existence of minimizers for zero Cosserat couple modulus,, Math. Models Meth. Appl. Sci., 17 (2007), 363. doi: 10.1142/S0218202507001954. Google Scholar [21] J. W. Nunziato and S. C. Cowin, A nonlinear theory of elastic materials with voids,, Arch. Rational Mech. Anal., 72 (1979), 175. doi: 10.1007/BF00249363. Google Scholar [22] G. Panasenko, "Multi-scale Modelling for Structures and Composites,", Springer, (2005). Google Scholar [23] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar [24] M. B. Rubin, "Cosserat Theories: Shells, Rods, and Points,", Solid Mechanics and Its Applications, 79 (2000). doi: 10.1007/978-94-015-9379-3. Google Scholar [25] J. G. Simmonds, A simple nonlinear thermodynamic theory of arbitrary elastic beams,, J. Elasticity, 81 (2005), 51. doi: 10.1007/s10659-005-9003-7. Google Scholar [26] V. A. Svetlitsky, "Statics of Rods,", Foundations of Engineering Mechanics, (2000). Google Scholar [27] D. Tiba and R. Vodák, A general asymptotic model for Lipschitzian curved rods,, Adv. Math. Sci. Appl., 15 (2005), 137. Google Scholar [28] I. I. Vrabie, "$C_0$-Semigroups and Applications,", North-Holland Mathematics Studies, 191 (2003). Google Scholar [29] P. A. Zhilin, Nonlinear theory of thin rods,, in, (2006), 227. Google Scholar [30] P. A. Zhilin, "Applied Mechanics: Theory of Thin Elastic Rods,", (in Russian), (2007). Google Scholar

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##### References:
 [1] H. Altenbach, K. Naumenko and P. A. Zhilin, A direct approach to the formulation of constitutive equations for rods and shells,, in, (2006), 87. Google Scholar [2] M. Bîrsan, Inequalities of Korn's type and existence results in the theory of Cosserat elastic shells,, J. Elasticity, 90 (2008), 227. doi: 10.1007/s10659-007-9140-2. Google Scholar [3] M. Bîrsan and H. Altenbach, A mathematical study of the linear theory for orthotropic elastic simple shells,, Math. Meth. Appl. Sci., 33 (2010), 1399. doi: 10.1002/mma.1253. Google Scholar [4] M. Bîrsan and H. Altenbach, Theory of thin thermoelastic rods made of porous materials,, Arch. Appl. Mech., 81 (2011), 1365. doi: 10.1007/s00419-010-0490-z. Google Scholar [5] M. Bîrsan and H. Altenbach, The Korn-type inequality in a Cosserat model for thin thermoelastic porous rods,, Meccanica, 47 (2011), 789. doi: 10.1007/s11012-011-9477-2. Google Scholar [6] M. Bîrsan and T. Bîrsan, An inequality of Cauchy-Schwarz type with application in the theory of elastic rods,, Libertas Mathematica, 31 (2011), 123. Google Scholar [7] H. Brezis, "Analyse Fonctionelle. Théorie et Applications,", (French) [Functional Analysis: Theory and Applications], (1983). Google Scholar [8] G. Capriz, "Continua with Microstructure,", Springer Tracts in Natural Philosophy, 35 (1989). doi: 10.1007/978-1-4612-3584-2. Google Scholar [9] G. Capriz and P. Podio-Guidugli, Materials with spherical structure,, Arch. Rational Mech. Anal., 75 (1981), 269. doi: 10.1007/BF00250786. Google Scholar [10] P. G. Ciarlet, "Mathematical Elasticity, Vol. I. Three-Dimensional Elasticity,", Studies in Mathematics and its Applications, 20 (1988). Google Scholar [11] P. G. Ciarlet, "Mathematical Elasticity. Vol. III. Theory of Shells,", Studies in Mathematics and its Applications, 29 (2000). Google Scholar [12] P. G. Ciarlet, "An Introduction to Differential Geometry with Applications to Elasticity,", Springer, (2005). Google Scholar [13] E. Cosserat and F. Cosserat, "Théorie des Corps Déformables,", (French) [Theory of deformable bodies], (1909). Google Scholar [14] S. C. Cowin and J. W. Nunziato, Linear elastic materials with voids,, J. Elasticity, 13 (1983), 125. doi: 10.1007/BF00041230. Google Scholar [15] M. A. Goodman and S. C. Cowin, A continuum theory for granular materials,, Arch. Rational Mech. Anal., 44 (1972), 249. doi: 10.1007/BF00284326. Google Scholar [16] A. E. Green and P. M. Naghdi, On thermal effects in the theory of rods,, Int. J. Solids Struct., 15 (1979), 829. doi: 10.1016/0020-7683(79)90053-2. Google Scholar [17] L. P. Lebedev, M. J. Cloud and V. A. Eremeyev, "Tensor Analysis with Applications in Mechanics,", World Scientific Publishing Co. Pte. Ltd., (2010). doi: 10.1142/9789814313995. Google Scholar [18] A. I. Lurie, "Theory of Elasticity,", Foundations of Engineering Mechanics, (2005). doi: 10.1007/978-3-540-26455-2. Google Scholar [19] P. Neff, On Korn's first inequality with non-constant coefficients,, Proc. Roy. Soc. Edinb. A, 132 (2002), 221. doi: 10.1017/S0308210500001591. Google Scholar [20] P. Neff, A geometrically exact planar Cosserat shell-model with microstructure: Existence of minimizers for zero Cosserat couple modulus,, Math. Models Meth. Appl. Sci., 17 (2007), 363. doi: 10.1142/S0218202507001954. Google Scholar [21] J. W. Nunziato and S. C. Cowin, A nonlinear theory of elastic materials with voids,, Arch. Rational Mech. Anal., 72 (1979), 175. doi: 10.1007/BF00249363. Google Scholar [22] G. Panasenko, "Multi-scale Modelling for Structures and Composites,", Springer, (2005). Google Scholar [23] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar [24] M. B. Rubin, "Cosserat Theories: Shells, Rods, and Points,", Solid Mechanics and Its Applications, 79 (2000). doi: 10.1007/978-94-015-9379-3. Google Scholar [25] J. G. Simmonds, A simple nonlinear thermodynamic theory of arbitrary elastic beams,, J. Elasticity, 81 (2005), 51. doi: 10.1007/s10659-005-9003-7. Google Scholar [26] V. A. Svetlitsky, "Statics of Rods,", Foundations of Engineering Mechanics, (2000). Google Scholar [27] D. Tiba and R. Vodák, A general asymptotic model for Lipschitzian curved rods,, Adv. Math. Sci. Appl., 15 (2005), 137. Google Scholar [28] I. I. Vrabie, "$C_0$-Semigroups and Applications,", North-Holland Mathematics Studies, 191 (2003). Google Scholar [29] P. A. Zhilin, Nonlinear theory of thin rods,, in, (2006), 227. Google Scholar [30] P. A. Zhilin, "Applied Mechanics: Theory of Thin Elastic Rods,", (in Russian), (2007). Google Scholar
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