# American Institute of Mathematical Sciences

September  2014, 19(7): 2169-2187. doi: 10.3934/dcdsb.2014.19.2169

## Strain gradient theory of porous solids with initial stresses and initial heat flux

 1 Department of Mathematics, "Al.I. Cuza" University, and Octav Mayer Institute of Mathematics (Romanian Academy), 700508, Iaşi, Romania

Received  March 2013 Revised  May 2013 Published  August 2014

In this paper we present a strain gradient theory of thermoelastic porous solids with initial stresses and initial heat flux. First, we establish the equations governing the infinitesimal deformations superposed on large deformations. Then, we derive a linear theory of prestressed porous bodies with initial heat flux. The theory is capable to describe the deformation of chiral materials. A reciprocity relation and a uniqueness result with no definiteness assumption on the elastic constitutive coefficients are presented.
Citation: Dorin Ieşan. Strain gradient theory of porous solids with initial stresses and initial heat flux. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 2169-2187. doi: 10.3934/dcdsb.2014.19.2169
##### References:
 [1] E. C. Aifantis, Exploring the applicability of gradient elasticity to certain micro/ nano reliability problems, Microsystem Technology, 15 (2009), 109-115. doi: 10.1007/s00542-008-0699-8. [2] G. Amendola, M. Fabrizio and J. M. Golden, Thermodynamics of a non-simple heat conductor with memory, Quart. Appl. Math., 69 (2011), 787-806. doi: 10.1090/S0033-569X-2011-01228-5. [3] G. Amendola, M. Fabrizio and J. M. Golden, Second gradient viscoelastic fluids: Dissipation principle and free energies, Meccanica, 47 (2012), 1859-1868. doi: 10.1007/s11012-012-9559-9. [4] H. Askes and E. C. Aifantis, Gradient elasticity in statics and dynamics: An overview of formulations, length scale identification procedures, finite element implementations and new results, Int. J. Solids Struct., 48 (2011), 1962-1990. doi: 10.1016/j.ijsolstr.2011.03.006. [5] O. Brulin and S. Hjalmars, Linear grade consistent micropolar theory, Int. J. Eng. Sci., 19 (1981), 1731-1738. doi: 10.1016/0020-7225(81)90163-4. [6] L. Brun, Methodes energetiques dans les systemes evolutifs lineaires, J. Mecanique, 8 (1969), 125-166. [7] D. E. Carlson, Linear Thermoelasticity, in Handbuch der Physik, vol. VIa/2, (ed. C. Truesdell), Springer-Verlag, Berlin-Heidelberg-New York, 1972. [8] S. Chirita, Uniqueness and continuous dependence results for the incremental thermoelasticity, J. Thermal Stresses, 5 (1982), 331-346. doi: 10.1080/01495738208942154. [9] O. Coussy, Mechanics and Physics of Porous Solids, John Wiley and Sons, Chichester, 2010. doi: 10.1002/9780470710388. [10] S. C. Cowin and J. W. Nunziato, Linear elastic materials with voids, J. Elasticity, 13 (1983), 125-147. doi: 10.1007/BF00041230. [11] T. Dillard, S.Forest and P. Ienny, Micromorphic continuum modelling of the deformation and fracture behaviour of nickel foams, Eur.J. Mech.-A/Solids, 25 (2006), 526-549. doi: 10.1016/j.euromechsol.2005.11.006. [12] A. C. Eringen and E. S. Suhubi, Nonlinear theory of simple microelastic solids, Int. J. Eng. Sci., 2 (1964), 189-203. doi: 10.1016/0020-7225(64)90004-7. [13] A. C. Eringen, Microcontinuum Field Theories. I: Foundations and Solid, Springer- Verlag, New York, Berlin, Heidelberg, 1999. doi: 10.1007/978-1-4612-0555-5. [14] M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity, SIAM Studies in Applied Mathematics 12, Philadelphia, PA, USA, 1992. doi: 10.1137/1.9781611970807. [15] S. Forest, J. M. Cardona and R. Sievert, Thermoelasticity of second- grade media, in Continuum Thermomechanics, The Art and Science of Modelling Material Behaviour, Paul Germain's Anniversary Volume, (eds. G.A. Maugin, R. Drouot and F. Sidoroff.), Kluwer Academic Publishers, (2000), 163-176. [16] P. Giovine, Linear wave motions in continua with nano-pores, in Wave Processes in Classical and New Solids, (ed. P. Giovine), Publisher: InTech, (2012), 62-86. [17] A. E. Green, Thermoelastic stresses in initially stressed bodies, Proc. Roy. Soc. London, Ser. A, 266 (1962), 1-19. doi: 10.1098/rspa.1962.0043. [18] A. E. Green and R. S. Rivlin, Multipolar continuum mechanics, Arch. Rational Mech.Anal., 17 (1964), 113-147. [19] S. Hjalmars, Non-linear micropolar theory, in Mechanics of Micropolar Media, (eds. O. Brulin and R.K.T. Hsieh), World Scientific, Singapore, (1982), 147-189, [20] D. Iesan, Incremental equations in thermoelasticity, J. Thermal Stresses, 3 (1980), 41-56. [21] D. Iesan, Prestressed Bodies, Pitman Research Notes in Mathematics Series 195, Longman Scientific and Technical, Longman House, Harlow, Essex, UK and John Wiley & Sons, Inc., New York, 1989. [22] D. Iesan, Thermoelastic Models of Continua, Kluwer Academic, Dordrecht, 2004. doi: 10.1007/978-1-4020-2310-1. [23] R. J. Knops and E. W. Wilkes, Theory of elastic stability, in Handbuch der Physik, (ed. C. Truesdell), Springer-Verlag, Berlin Heidelberg-New York, 1973. [24] R. J. Knops and L. E. Payne, Uniqueness Theorems in Linear Elasticity, Springer Tracts in Natural Philosophy, vol. 19, Berlin-Heidelberg-New York, 1971. [25] R. J. Knops, Uniqueness and continuous data dependence in the elastic cylinders, Int. J. Non-Linear Mech., 36 (2001), 489-499. doi: 10.1016/S0020-7462(00)00078-0. [26] F. Martinez, F. and R. Quintanilla, On the incremental problem in thermoelasticity of nonsimple materials, Zeit. Angew.Math. Mech., 78 (1998), 703-710. [27] R. D. Mindlin, Microstructure in linear elasticity, Arch.Rational Mech.Anal., 16 (1964), 51-78. [28] R. D. Mindlin and N. N. Eshel, On first strain gradient theories in linear elasticity, Int. J. Solids Struct., 4 (1968), 109-124. doi: 10.1016/0020-7683(68)90036-X. [29] C. B. Navarro and R. Quintanilla, On existence and uniqueness in incremental thermoelasticity, Zeit. Angew. Math. Mech., 35 (1984), 206-215. doi: 10.1007/BF00947933. [30] P. Neff and S. Forest, A geometrically exact micromorphic model for elastic metallic foams accounting for affine microstructure. Modelling existence and minimizers, identification of moduli and computational results, J. Elasticity, 87 (2007), 239-276. doi: 10.1007/s10659-007-9106-4. [31] W. Nowacki, Theory of Asymmetric Elasticity, Polish Scientific Publishers, Warszawa and Pergamon Press, Oxford, New York, Paris, Frankfurt, 1986. [32] J. W. Nunziato and S. C. Cowin, A nonlinear theory of elastic materials with voids,, Arch.Rational Mech.Anal., 72 (): 175.  doi: 10.1007/BF00249363. [33] A. Ochsner, G. E. Murch and M. J. S. Lemos, Cellular and Porous Materials, Wiley-VCH, Weinheim, 2008. [34] S. A. Papanicolopulos, Chirality in isotropic linear gradient elasticity, Int. J. Solids Struct., 48 (2011), 745-752. doi: 10.1016/j.ijsolstr.2010.11.007. [35] C. Rymarz, On the model of non-simple medium with rotational degrees of freedom, Bull. Acad. Polon. Sci.,S. Sci. Techn., 16 (1968), 271-277. [36] G. Sciarra, F. Dell'Isola and O. Coussy, Second gradient poromechanics, Int. J. Solids Struct., 44 (2007), 6607-6629. doi: 10.1016/j.ijsolstr.2007.03.003. [37] R. A. Toupin, Elastic materials with couple stresses, Arch.Rational Mech.Anal., 11 (1962), 385-414. doi: 10.1007/BF00253945. [38] R. A. Toupin, Theories of elasticity with couple-stress, Arch. Rational Mech.Anal., 17 (1964), 85-112. [39] J. R. Vinson and R. L. Sierakowski, The Behaviour of Structures Composed of Composite Materials, Second edition, Kluwer Acad. Publ., Dordrecht, 2002.

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##### References:
 [1] E. C. Aifantis, Exploring the applicability of gradient elasticity to certain micro/ nano reliability problems, Microsystem Technology, 15 (2009), 109-115. doi: 10.1007/s00542-008-0699-8. [2] G. Amendola, M. Fabrizio and J. M. Golden, Thermodynamics of a non-simple heat conductor with memory, Quart. Appl. Math., 69 (2011), 787-806. doi: 10.1090/S0033-569X-2011-01228-5. [3] G. Amendola, M. Fabrizio and J. M. Golden, Second gradient viscoelastic fluids: Dissipation principle and free energies, Meccanica, 47 (2012), 1859-1868. doi: 10.1007/s11012-012-9559-9. [4] H. Askes and E. C. Aifantis, Gradient elasticity in statics and dynamics: An overview of formulations, length scale identification procedures, finite element implementations and new results, Int. J. Solids Struct., 48 (2011), 1962-1990. doi: 10.1016/j.ijsolstr.2011.03.006. [5] O. Brulin and S. Hjalmars, Linear grade consistent micropolar theory, Int. J. Eng. Sci., 19 (1981), 1731-1738. doi: 10.1016/0020-7225(81)90163-4. [6] L. Brun, Methodes energetiques dans les systemes evolutifs lineaires, J. Mecanique, 8 (1969), 125-166. [7] D. E. Carlson, Linear Thermoelasticity, in Handbuch der Physik, vol. VIa/2, (ed. C. Truesdell), Springer-Verlag, Berlin-Heidelberg-New York, 1972. [8] S. Chirita, Uniqueness and continuous dependence results for the incremental thermoelasticity, J. Thermal Stresses, 5 (1982), 331-346. doi: 10.1080/01495738208942154. [9] O. Coussy, Mechanics and Physics of Porous Solids, John Wiley and Sons, Chichester, 2010. doi: 10.1002/9780470710388. [10] S. C. Cowin and J. W. Nunziato, Linear elastic materials with voids, J. Elasticity, 13 (1983), 125-147. doi: 10.1007/BF00041230. [11] T. Dillard, S.Forest and P. Ienny, Micromorphic continuum modelling of the deformation and fracture behaviour of nickel foams, Eur.J. Mech.-A/Solids, 25 (2006), 526-549. doi: 10.1016/j.euromechsol.2005.11.006. [12] A. C. Eringen and E. S. Suhubi, Nonlinear theory of simple microelastic solids, Int. J. Eng. Sci., 2 (1964), 189-203. doi: 10.1016/0020-7225(64)90004-7. [13] A. C. Eringen, Microcontinuum Field Theories. I: Foundations and Solid, Springer- Verlag, New York, Berlin, Heidelberg, 1999. doi: 10.1007/978-1-4612-0555-5. [14] M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity, SIAM Studies in Applied Mathematics 12, Philadelphia, PA, USA, 1992. doi: 10.1137/1.9781611970807. [15] S. Forest, J. M. Cardona and R. Sievert, Thermoelasticity of second- grade media, in Continuum Thermomechanics, The Art and Science of Modelling Material Behaviour, Paul Germain's Anniversary Volume, (eds. G.A. Maugin, R. Drouot and F. Sidoroff.), Kluwer Academic Publishers, (2000), 163-176. [16] P. Giovine, Linear wave motions in continua with nano-pores, in Wave Processes in Classical and New Solids, (ed. P. Giovine), Publisher: InTech, (2012), 62-86. [17] A. E. Green, Thermoelastic stresses in initially stressed bodies, Proc. Roy. Soc. London, Ser. A, 266 (1962), 1-19. doi: 10.1098/rspa.1962.0043. [18] A. E. Green and R. S. Rivlin, Multipolar continuum mechanics, Arch. Rational Mech.Anal., 17 (1964), 113-147. [19] S. Hjalmars, Non-linear micropolar theory, in Mechanics of Micropolar Media, (eds. O. Brulin and R.K.T. Hsieh), World Scientific, Singapore, (1982), 147-189, [20] D. Iesan, Incremental equations in thermoelasticity, J. Thermal Stresses, 3 (1980), 41-56. [21] D. Iesan, Prestressed Bodies, Pitman Research Notes in Mathematics Series 195, Longman Scientific and Technical, Longman House, Harlow, Essex, UK and John Wiley & Sons, Inc., New York, 1989. [22] D. Iesan, Thermoelastic Models of Continua, Kluwer Academic, Dordrecht, 2004. doi: 10.1007/978-1-4020-2310-1. [23] R. J. Knops and E. W. Wilkes, Theory of elastic stability, in Handbuch der Physik, (ed. C. Truesdell), Springer-Verlag, Berlin Heidelberg-New York, 1973. [24] R. J. Knops and L. E. Payne, Uniqueness Theorems in Linear Elasticity, Springer Tracts in Natural Philosophy, vol. 19, Berlin-Heidelberg-New York, 1971. [25] R. J. Knops, Uniqueness and continuous data dependence in the elastic cylinders, Int. J. Non-Linear Mech., 36 (2001), 489-499. doi: 10.1016/S0020-7462(00)00078-0. [26] F. Martinez, F. and R. Quintanilla, On the incremental problem in thermoelasticity of nonsimple materials, Zeit. Angew.Math. Mech., 78 (1998), 703-710. [27] R. D. Mindlin, Microstructure in linear elasticity, Arch.Rational Mech.Anal., 16 (1964), 51-78. [28] R. D. Mindlin and N. N. Eshel, On first strain gradient theories in linear elasticity, Int. J. Solids Struct., 4 (1968), 109-124. doi: 10.1016/0020-7683(68)90036-X. [29] C. B. Navarro and R. Quintanilla, On existence and uniqueness in incremental thermoelasticity, Zeit. Angew. Math. Mech., 35 (1984), 206-215. doi: 10.1007/BF00947933. [30] P. Neff and S. Forest, A geometrically exact micromorphic model for elastic metallic foams accounting for affine microstructure. Modelling existence and minimizers, identification of moduli and computational results, J. Elasticity, 87 (2007), 239-276. doi: 10.1007/s10659-007-9106-4. [31] W. Nowacki, Theory of Asymmetric Elasticity, Polish Scientific Publishers, Warszawa and Pergamon Press, Oxford, New York, Paris, Frankfurt, 1986. [32] J. W. Nunziato and S. C. Cowin, A nonlinear theory of elastic materials with voids,, Arch.Rational Mech.Anal., 72 (): 175.  doi: 10.1007/BF00249363. [33] A. Ochsner, G. E. Murch and M. J. S. Lemos, Cellular and Porous Materials, Wiley-VCH, Weinheim, 2008. [34] S. A. Papanicolopulos, Chirality in isotropic linear gradient elasticity, Int. J. Solids Struct., 48 (2011), 745-752. doi: 10.1016/j.ijsolstr.2010.11.007. [35] C. Rymarz, On the model of non-simple medium with rotational degrees of freedom, Bull. Acad. Polon. Sci.,S. Sci. Techn., 16 (1968), 271-277. [36] G. Sciarra, F. Dell'Isola and O. Coussy, Second gradient poromechanics, Int. J. Solids Struct., 44 (2007), 6607-6629. doi: 10.1016/j.ijsolstr.2007.03.003. [37] R. A. Toupin, Elastic materials with couple stresses, Arch.Rational Mech.Anal., 11 (1962), 385-414. doi: 10.1007/BF00253945. [38] R. A. Toupin, Theories of elasticity with couple-stress, Arch. Rational Mech.Anal., 17 (1964), 85-112. [39] J. R. Vinson and R. L. Sierakowski, The Behaviour of Structures Composed of Composite Materials, Second edition, Kluwer Acad. Publ., Dordrecht, 2002.
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