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 and 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 "Shell Structures: Theory and Applications" (eds. W. Pietraszkiewicz and C. Szymczak), Taylor and Francis, London, (2006), 87-90.

[2]

M. Bîrsan, Inequalities of Korn's type and existence results in the theory of Cosserat elastic shells, J. Elasticity, 90 (2008), 227-239. doi: 10.1007/s10659-007-9140-2.

[3]

M. Bîrsan and H. Altenbach, A mathematical study of the linear theory for orthotropic elastic simple shells, Math. Meth. Appl. Sci., 33 (2010), 1399-1413. doi: 10.1002/mma.1253.

[4]

M. Bîrsan and H. Altenbach, Theory of thin thermoelastic rods made of porous materials, Arch. Appl. Mech., 81 (2011), 1365-1391. doi: 10.1007/s00419-010-0490-z.

[5]

M. Bîrsan and H. Altenbach, The Korn-type inequality in a Cosserat model for thin thermoelastic porous rods, Meccanica, 47 (2011), 789-794. doi: 10.1007/s11012-011-9477-2.

[6]

M. Bîrsan and T. Bîrsan, An inequality of Cauchy-Schwarz type with application in the theory of elastic rods, Libertas Mathematica, 31 (2011), 123-126.

[7]

H. Brezis, "Analyse Fonctionelle. Théorie et Applications," (French) [Functional Analysis: Theory and Applications], Collection Mathématiques Appliquées pour la Maîtrise, [Collection of Applied Mathematics for the Master's Degree], Masson, Paris, 1983.

[8]

G. Capriz, "Continua with Microstructure," Springer Tracts in Natural Philosophy, 35, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-3584-2.

[9]

G. Capriz and P. Podio-Guidugli, Materials with spherical structure, Arch. Rational Mech. Anal., 75 (1981), 269-279. doi: 10.1007/BF00250786.

[10]

P. G. Ciarlet, "Mathematical Elasticity, Vol. I. Three-Dimensional Elasticity," Studies in Mathematics and its Applications, 20, North-Holland Publishing Co., Amsterdam, 1988.

[11]

P. G. Ciarlet, "Mathematical Elasticity. Vol. III. Theory of Shells," Studies in Mathematics and its Applications, 29, North-Holland Publishing Co., Amsterdam, 2000.

[12]

P. G. Ciarlet, "An Introduction to Differential Geometry with Applications to Elasticity," Springer, Dordrecht, 2005.

[13]

E. Cosserat and F. Cosserat, "Théorie des Corps Déformables," (French) [Theory of deformable bodies], A. Herman et Fils, Paris, 1909.

[14]

S. C. Cowin and J. W. Nunziato, Linear elastic materials with voids, J. Elasticity, 13 (1983), 125-147. doi: 10.1007/BF00041230.

[15]

M. A. Goodman and S. C. Cowin, A continuum theory for granular materials, Arch. Rational Mech. Anal., 44 (1972), 249-266. doi: 10.1007/BF00284326.

[16]

A. E. Green and P. M. Naghdi, On thermal effects in the theory of rods, Int. J. Solids Struct., 15 (1979), 829-853. doi: 10.1016/0020-7683(79)90053-2.

[17]

L. P. Lebedev, M. J. Cloud and V. A. Eremeyev, "Tensor Analysis with Applications in Mechanics," World Scientific Publishing Co. Pte. Ltd., Hackensack, New Jersey, 2010. doi: 10.1142/9789814313995.

[18]

A. I. Lurie, "Theory of Elasticity," Foundations of Engineering Mechanics, Springer, Berlin, 2005. doi: 10.1007/978-3-540-26455-2.

[19]

P. Neff, On Korn's first inequality with non-constant coefficients, Proc. Roy. Soc. Edinb. A, 132 (2002), 221-243. doi: 10.1017/S0308210500001591.

[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), 363-392. doi: 10.1142/S0218202507001954.

[21]

J. W. Nunziato and S. C. Cowin, A nonlinear theory of elastic materials with voids, Arch. Rational Mech. Anal., 72 (1979), 175-201. doi: 10.1007/BF00249363.

[22]

G. Panasenko, "Multi-scale Modelling for Structures and Composites," Springer, Dordrecht, 2005.

[23]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[24]

M. B. Rubin, "Cosserat Theories: Shells, Rods, and Points," Solid Mechanics and Its Applications, 79, Springer, Netherlands, 2000. doi: 10.1007/978-94-015-9379-3.

[25]

J. G. Simmonds, A simple nonlinear thermodynamic theory of arbitrary elastic beams, J. Elasticity, 81 (2005), 51-62. doi: 10.1007/s10659-005-9003-7.

[26]

V. A. Svetlitsky, "Statics of Rods," Foundations of Engineering Mechanics, Springer-Verlag, Berlin, 2000.

[27]

D. Tiba and R. Vodák, A general asymptotic model for Lipschitzian curved rods, Adv. Math. Sci. Appl., 15 (2005), 137-198.

[28]

I. I. Vrabie, "$C_0$-Semigroups and Applications," North-Holland Mathematics Studies, 191, North-Holland Publishing Co., Amsterdam, 2003.

[29]

P. A. Zhilin, Nonlinear theory of thin rods, in "Advanced Problems in Mechanics" (eds. D.A. Indeitsev, E.A. Ivanova and A.M. Krivtsov), Vol. 2, Instit. Problems Mech. Eng. R.A.S. Publ., St. Petersburg, (2006), 227-249.

[30]

P. A. Zhilin, "Applied Mechanics: Theory of Thin Elastic Rods," (in Russian), Politekhn. Univ. Publ., St. Petersburg, 2007.

show all references

References:
[1]

H. Altenbach, K. Naumenko and P. A. Zhilin, A direct approach to the formulation of constitutive equations for rods and shells, in "Shell Structures: Theory and Applications" (eds. W. Pietraszkiewicz and C. Szymczak), Taylor and Francis, London, (2006), 87-90.

[2]

M. Bîrsan, Inequalities of Korn's type and existence results in the theory of Cosserat elastic shells, J. Elasticity, 90 (2008), 227-239. doi: 10.1007/s10659-007-9140-2.

[3]

M. Bîrsan and H. Altenbach, A mathematical study of the linear theory for orthotropic elastic simple shells, Math. Meth. Appl. Sci., 33 (2010), 1399-1413. doi: 10.1002/mma.1253.

[4]

M. Bîrsan and H. Altenbach, Theory of thin thermoelastic rods made of porous materials, Arch. Appl. Mech., 81 (2011), 1365-1391. doi: 10.1007/s00419-010-0490-z.

[5]

M. Bîrsan and H. Altenbach, The Korn-type inequality in a Cosserat model for thin thermoelastic porous rods, Meccanica, 47 (2011), 789-794. doi: 10.1007/s11012-011-9477-2.

[6]

M. Bîrsan and T. Bîrsan, An inequality of Cauchy-Schwarz type with application in the theory of elastic rods, Libertas Mathematica, 31 (2011), 123-126.

[7]

H. Brezis, "Analyse Fonctionelle. Théorie et Applications," (French) [Functional Analysis: Theory and Applications], Collection Mathématiques Appliquées pour la Maîtrise, [Collection of Applied Mathematics for the Master's Degree], Masson, Paris, 1983.

[8]

G. Capriz, "Continua with Microstructure," Springer Tracts in Natural Philosophy, 35, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-3584-2.

[9]

G. Capriz and P. Podio-Guidugli, Materials with spherical structure, Arch. Rational Mech. Anal., 75 (1981), 269-279. doi: 10.1007/BF00250786.

[10]

P. G. Ciarlet, "Mathematical Elasticity, Vol. I. Three-Dimensional Elasticity," Studies in Mathematics and its Applications, 20, North-Holland Publishing Co., Amsterdam, 1988.

[11]

P. G. Ciarlet, "Mathematical Elasticity. Vol. III. Theory of Shells," Studies in Mathematics and its Applications, 29, North-Holland Publishing Co., Amsterdam, 2000.

[12]

P. G. Ciarlet, "An Introduction to Differential Geometry with Applications to Elasticity," Springer, Dordrecht, 2005.

[13]

E. Cosserat and F. Cosserat, "Théorie des Corps Déformables," (French) [Theory of deformable bodies], A. Herman et Fils, Paris, 1909.

[14]

S. C. Cowin and J. W. Nunziato, Linear elastic materials with voids, J. Elasticity, 13 (1983), 125-147. doi: 10.1007/BF00041230.

[15]

M. A. Goodman and S. C. Cowin, A continuum theory for granular materials, Arch. Rational Mech. Anal., 44 (1972), 249-266. doi: 10.1007/BF00284326.

[16]

A. E. Green and P. M. Naghdi, On thermal effects in the theory of rods, Int. J. Solids Struct., 15 (1979), 829-853. doi: 10.1016/0020-7683(79)90053-2.

[17]

L. P. Lebedev, M. J. Cloud and V. A. Eremeyev, "Tensor Analysis with Applications in Mechanics," World Scientific Publishing Co. Pte. Ltd., Hackensack, New Jersey, 2010. doi: 10.1142/9789814313995.

[18]

A. I. Lurie, "Theory of Elasticity," Foundations of Engineering Mechanics, Springer, Berlin, 2005. doi: 10.1007/978-3-540-26455-2.

[19]

P. Neff, On Korn's first inequality with non-constant coefficients, Proc. Roy. Soc. Edinb. A, 132 (2002), 221-243. doi: 10.1017/S0308210500001591.

[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), 363-392. doi: 10.1142/S0218202507001954.

[21]

J. W. Nunziato and S. C. Cowin, A nonlinear theory of elastic materials with voids, Arch. Rational Mech. Anal., 72 (1979), 175-201. doi: 10.1007/BF00249363.

[22]

G. Panasenko, "Multi-scale Modelling for Structures and Composites," Springer, Dordrecht, 2005.

[23]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[24]

M. B. Rubin, "Cosserat Theories: Shells, Rods, and Points," Solid Mechanics and Its Applications, 79, Springer, Netherlands, 2000. doi: 10.1007/978-94-015-9379-3.

[25]

J. G. Simmonds, A simple nonlinear thermodynamic theory of arbitrary elastic beams, J. Elasticity, 81 (2005), 51-62. doi: 10.1007/s10659-005-9003-7.

[26]

V. A. Svetlitsky, "Statics of Rods," Foundations of Engineering Mechanics, Springer-Verlag, Berlin, 2000.

[27]

D. Tiba and R. Vodák, A general asymptotic model for Lipschitzian curved rods, Adv. Math. Sci. Appl., 15 (2005), 137-198.

[28]

I. I. Vrabie, "$C_0$-Semigroups and Applications," North-Holland Mathematics Studies, 191, North-Holland Publishing Co., Amsterdam, 2003.

[29]

P. A. Zhilin, Nonlinear theory of thin rods, in "Advanced Problems in Mechanics" (eds. D.A. Indeitsev, E.A. Ivanova and A.M. Krivtsov), Vol. 2, Instit. Problems Mech. Eng. R.A.S. Publ., St. Petersburg, (2006), 227-249.

[30]

P. A. Zhilin, "Applied Mechanics: Theory of Thin Elastic Rods," (in Russian), Politekhn. Univ. Publ., St. Petersburg, 2007.

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