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 & 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.  doi: 10.1007/s00542-008-0699-8.  Google Scholar

[2]

G. Amendola, M. Fabrizio and J. M. Golden, Thermodynamics of a non-simple heat conductor with memory,, Quart. Appl. Math., 69 (2011), 787.  doi: 10.1090/S0033-569X-2011-01228-5.  Google Scholar

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G. Amendola, M. Fabrizio and J. M. Golden, Second gradient viscoelastic fluids: Dissipation principle and free energies,, Meccanica, 47 (2012), 1859.  doi: 10.1007/s11012-012-9559-9.  Google Scholar

[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.  doi: 10.1016/j.ijsolstr.2011.03.006.  Google Scholar

[5]

O. Brulin and S. Hjalmars, Linear grade consistent micropolar theory,, Int. J. Eng. Sci., 19 (1981), 1731.  doi: 10.1016/0020-7225(81)90163-4.  Google Scholar

[6]

L. Brun, Methodes energetiques dans les systemes evolutifs lineaires,, J. Mecanique, 8 (1969), 125.   Google Scholar

[7]

D. E. Carlson, Linear Thermoelasticity,, in Handbuch der Physik, (1972).   Google Scholar

[8]

S. Chirita, Uniqueness and continuous dependence results for the incremental thermoelasticity,, J. Thermal Stresses, 5 (1982), 331.  doi: 10.1080/01495738208942154.  Google Scholar

[9]

O. Coussy, Mechanics and Physics of Porous Solids,, John Wiley and Sons, (2010).  doi: 10.1002/9780470710388.  Google Scholar

[10]

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

[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.  doi: 10.1016/j.euromechsol.2005.11.006.  Google Scholar

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A. C. Eringen and E. S. Suhubi, Nonlinear theory of simple microelastic solids,, Int. J. Eng. Sci., 2 (1964), 189.  doi: 10.1016/0020-7225(64)90004-7.  Google Scholar

[13]

A. C. Eringen, Microcontinuum Field Theories. I: Foundations and Solid,, Springer- Verlag, (1999).  doi: 10.1007/978-1-4612-0555-5.  Google Scholar

[14]

M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity,, SIAM Studies in Applied Mathematics 12, (1992).  doi: 10.1137/1.9781611970807.  Google Scholar

[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, (2000), 163.   Google Scholar

[16]

P. Giovine, Linear wave motions in continua with nano-pores, in Wave Processes in Classical and New Solids,, (ed. P. Giovine), (2012), 62.   Google Scholar

[17]

A. E. Green, Thermoelastic stresses in initially stressed bodies,, Proc. Roy. Soc. London, 266 (1962), 1.  doi: 10.1098/rspa.1962.0043.  Google Scholar

[18]

A. E. Green and R. S. Rivlin, Multipolar continuum mechanics,, Arch. Rational Mech.Anal., 17 (1964), 113.   Google Scholar

[19]

S. Hjalmars, Non-linear micropolar theory, in Mechanics of Micropolar Media,, (eds. O. Brulin and R.K.T. Hsieh), (1982), 147.   Google Scholar

[20]

D. Iesan, Incremental equations in thermoelasticity,, J. Thermal Stresses, 3 (1980), 41.   Google Scholar

[21]

D. Iesan, Prestressed Bodies,, Pitman Research Notes in Mathematics Series 195, (1989).   Google Scholar

[22]

D. Iesan, Thermoelastic Models of Continua,, Kluwer Academic, (2004).  doi: 10.1007/978-1-4020-2310-1.  Google Scholar

[23]

R. J. Knops and E. W. Wilkes, Theory of elastic stability,, in Handbuch der Physik, (1973).   Google Scholar

[24]

R. J. Knops and L. E. Payne, Uniqueness Theorems in Linear Elasticity,, Springer Tracts in Natural Philosophy, (1971).   Google Scholar

[25]

R. J. Knops, Uniqueness and continuous data dependence in the elastic cylinders,, Int. J. Non-Linear Mech., 36 (2001), 489.  doi: 10.1016/S0020-7462(00)00078-0.  Google Scholar

[26]

F. Martinez, F. and R. Quintanilla, On the incremental problem in thermoelasticity of nonsimple materials,, Zeit. Angew.Math. Mech., 78 (1998), 703.   Google Scholar

[27]

R. D. Mindlin, Microstructure in linear elasticity,, Arch.Rational Mech.Anal., 16 (1964), 51.   Google Scholar

[28]

R. D. Mindlin and N. N. Eshel, On first strain gradient theories in linear elasticity,, Int. J. Solids Struct., 4 (1968), 109.  doi: 10.1016/0020-7683(68)90036-X.  Google Scholar

[29]

C. B. Navarro and R. Quintanilla, On existence and uniqueness in incremental thermoelasticity,, Zeit. Angew. Math. Mech., 35 (1984), 206.  doi: 10.1007/BF00947933.  Google Scholar

[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.  doi: 10.1007/s10659-007-9106-4.  Google Scholar

[31]

W. Nowacki, Theory of Asymmetric Elasticity,, Polish Scientific Publishers, (1986).   Google Scholar

[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.  Google Scholar

[33]

A. Ochsner, G. E. Murch and M. J. S. Lemos, Cellular and Porous Materials,, Wiley-VCH, (2008).   Google Scholar

[34]

S. A. Papanicolopulos, Chirality in isotropic linear gradient elasticity,, Int. J. Solids Struct., 48 (2011), 745.  doi: 10.1016/j.ijsolstr.2010.11.007.  Google Scholar

[35]

C. Rymarz, On the model of non-simple medium with rotational degrees of freedom,, Bull. Acad. Polon. Sci., 16 (1968), 271.   Google Scholar

[36]

G. Sciarra, F. Dell'Isola and O. Coussy, Second gradient poromechanics,, Int. J. Solids Struct., 44 (2007), 6607.  doi: 10.1016/j.ijsolstr.2007.03.003.  Google Scholar

[37]

R. A. Toupin, Elastic materials with couple stresses,, Arch.Rational Mech.Anal., 11 (1962), 385.  doi: 10.1007/BF00253945.  Google Scholar

[38]

R. A. Toupin, Theories of elasticity with couple-stress,, Arch. Rational Mech.Anal., 17 (1964), 85.   Google Scholar

[39]

J. R. Vinson and R. L. Sierakowski, The Behaviour of Structures Composed of Composite Materials,, Second edition, (2002).   Google Scholar

show all references

References:
[1]

E. C. Aifantis, Exploring the applicability of gradient elasticity to certain micro/ nano reliability problems,, Microsystem Technology, 15 (2009), 109.  doi: 10.1007/s00542-008-0699-8.  Google Scholar

[2]

G. Amendola, M. Fabrizio and J. M. Golden, Thermodynamics of a non-simple heat conductor with memory,, Quart. Appl. Math., 69 (2011), 787.  doi: 10.1090/S0033-569X-2011-01228-5.  Google Scholar

[3]

G. Amendola, M. Fabrizio and J. M. Golden, Second gradient viscoelastic fluids: Dissipation principle and free energies,, Meccanica, 47 (2012), 1859.  doi: 10.1007/s11012-012-9559-9.  Google Scholar

[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.  doi: 10.1016/j.ijsolstr.2011.03.006.  Google Scholar

[5]

O. Brulin and S. Hjalmars, Linear grade consistent micropolar theory,, Int. J. Eng. Sci., 19 (1981), 1731.  doi: 10.1016/0020-7225(81)90163-4.  Google Scholar

[6]

L. Brun, Methodes energetiques dans les systemes evolutifs lineaires,, J. Mecanique, 8 (1969), 125.   Google Scholar

[7]

D. E. Carlson, Linear Thermoelasticity,, in Handbuch der Physik, (1972).   Google Scholar

[8]

S. Chirita, Uniqueness and continuous dependence results for the incremental thermoelasticity,, J. Thermal Stresses, 5 (1982), 331.  doi: 10.1080/01495738208942154.  Google Scholar

[9]

O. Coussy, Mechanics and Physics of Porous Solids,, John Wiley and Sons, (2010).  doi: 10.1002/9780470710388.  Google Scholar

[10]

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

[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.  doi: 10.1016/j.euromechsol.2005.11.006.  Google Scholar

[12]

A. C. Eringen and E. S. Suhubi, Nonlinear theory of simple microelastic solids,, Int. J. Eng. Sci., 2 (1964), 189.  doi: 10.1016/0020-7225(64)90004-7.  Google Scholar

[13]

A. C. Eringen, Microcontinuum Field Theories. I: Foundations and Solid,, Springer- Verlag, (1999).  doi: 10.1007/978-1-4612-0555-5.  Google Scholar

[14]

M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity,, SIAM Studies in Applied Mathematics 12, (1992).  doi: 10.1137/1.9781611970807.  Google Scholar

[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, (2000), 163.   Google Scholar

[16]

P. Giovine, Linear wave motions in continua with nano-pores, in Wave Processes in Classical and New Solids,, (ed. P. Giovine), (2012), 62.   Google Scholar

[17]

A. E. Green, Thermoelastic stresses in initially stressed bodies,, Proc. Roy. Soc. London, 266 (1962), 1.  doi: 10.1098/rspa.1962.0043.  Google Scholar

[18]

A. E. Green and R. S. Rivlin, Multipolar continuum mechanics,, Arch. Rational Mech.Anal., 17 (1964), 113.   Google Scholar

[19]

S. Hjalmars, Non-linear micropolar theory, in Mechanics of Micropolar Media,, (eds. O. Brulin and R.K.T. Hsieh), (1982), 147.   Google Scholar

[20]

D. Iesan, Incremental equations in thermoelasticity,, J. Thermal Stresses, 3 (1980), 41.   Google Scholar

[21]

D. Iesan, Prestressed Bodies,, Pitman Research Notes in Mathematics Series 195, (1989).   Google Scholar

[22]

D. Iesan, Thermoelastic Models of Continua,, Kluwer Academic, (2004).  doi: 10.1007/978-1-4020-2310-1.  Google Scholar

[23]

R. J. Knops and E. W. Wilkes, Theory of elastic stability,, in Handbuch der Physik, (1973).   Google Scholar

[24]

R. J. Knops and L. E. Payne, Uniqueness Theorems in Linear Elasticity,, Springer Tracts in Natural Philosophy, (1971).   Google Scholar

[25]

R. J. Knops, Uniqueness and continuous data dependence in the elastic cylinders,, Int. J. Non-Linear Mech., 36 (2001), 489.  doi: 10.1016/S0020-7462(00)00078-0.  Google Scholar

[26]

F. Martinez, F. and R. Quintanilla, On the incremental problem in thermoelasticity of nonsimple materials,, Zeit. Angew.Math. Mech., 78 (1998), 703.   Google Scholar

[27]

R. D. Mindlin, Microstructure in linear elasticity,, Arch.Rational Mech.Anal., 16 (1964), 51.   Google Scholar

[28]

R. D. Mindlin and N. N. Eshel, On first strain gradient theories in linear elasticity,, Int. J. Solids Struct., 4 (1968), 109.  doi: 10.1016/0020-7683(68)90036-X.  Google Scholar

[29]

C. B. Navarro and R. Quintanilla, On existence and uniqueness in incremental thermoelasticity,, Zeit. Angew. Math. Mech., 35 (1984), 206.  doi: 10.1007/BF00947933.  Google Scholar

[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.  doi: 10.1007/s10659-007-9106-4.  Google Scholar

[31]

W. Nowacki, Theory of Asymmetric Elasticity,, Polish Scientific Publishers, (1986).   Google Scholar

[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.  Google Scholar

[33]

A. Ochsner, G. E. Murch and M. J. S. Lemos, Cellular and Porous Materials,, Wiley-VCH, (2008).   Google Scholar

[34]

S. A. Papanicolopulos, Chirality in isotropic linear gradient elasticity,, Int. J. Solids Struct., 48 (2011), 745.  doi: 10.1016/j.ijsolstr.2010.11.007.  Google Scholar

[35]

C. Rymarz, On the model of non-simple medium with rotational degrees of freedom,, Bull. Acad. Polon. Sci., 16 (1968), 271.   Google Scholar

[36]

G. Sciarra, F. Dell'Isola and O. Coussy, Second gradient poromechanics,, Int. J. Solids Struct., 44 (2007), 6607.  doi: 10.1016/j.ijsolstr.2007.03.003.  Google Scholar

[37]

R. A. Toupin, Elastic materials with couple stresses,, Arch.Rational Mech.Anal., 11 (1962), 385.  doi: 10.1007/BF00253945.  Google Scholar

[38]

R. A. Toupin, Theories of elasticity with couple-stress,, Arch. Rational Mech.Anal., 17 (1964), 85.   Google Scholar

[39]

J. R. Vinson and R. L. Sierakowski, The Behaviour of Structures Composed of Composite Materials,, Second edition, (2002).   Google Scholar

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