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September  2014, 19(7): 2335-2352. doi: 10.3934/dcdsb.2014.19.2335

On the theory of viscoelasticity for materials with double porosity

1. 

Institute for Fundamental and Interdisciplinary Mathematics Research, Ilia State University, K. Cholokashvili Ave., 3/5, 0162, Tbilisi, Georgia

Received  April 2013 Revised  September 2013 Published  August 2014

In this paper the linear theory of viscoelasticity for Kelvin-Voigt materials with double porosity is presented and the basic partial differential equations are derived. The system of these equations is based on the equations of motion, conservation of fluid mass, the effective stress concept and Darcy's law for materials with double porosity. This theory is a straightforward generalization of the earlier proposed dynamical theory of elasticity for materials with double porosity. The fundamental solution of the system of equations of steady vibrations is constructed by elementary functions and its basic properties are established. Finally, the properties of plane harmonic waves are studied. The results obtained from this study can be summarized as follows: through a Kelvin-Voigt material with double porosity three longitudinal and two transverse plane harmonic attenuated waves propagate.
Citation: Merab Svanadze. On the theory of viscoelasticity for materials with double porosity. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2335-2352. doi: 10.3934/dcdsb.2014.19.2335
References:
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J. G. Berryman and H. F. Wang, The elastic coefficients of double-porosity models for fluid transport in jointed rock,, Journal of Geophysical Research, 100 (1995), 24611.  doi: 10.1029/95JB02161.  Google Scholar

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J. G. Berryman and H. F. Wang, Elastic wave propagation and attenuation in a double-porosity dual-permiability medium,, International Journal of Rock Mechanics and Mining Sciences, 37 (2000), 63.  doi: 10.1016/S1365-1609(99)00092-1.  Google Scholar

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D. E. Beskos and E. C. Aifantis, On the theory of consolidation with double porosity - II,, International Journal of Engineering Science, 24 (1986), 1697.  doi: 10.1016/0020-7225(86)90076-5.  Google Scholar

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M. A. Biot, General theory of three-dimensional consolidation,, Journal of Applied Physics, 12 (1941), 155.  doi: 10.1063/1.1712886.  Google Scholar

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M. A. Biot, Theory of propagation of elastic waves in a fluid-saturated porous solid, I. Low frequency range, II. Higher frequency range,, Journal of the Acoustical Society of America, 28 (1956), 179.  doi: 10.1121/1.1908241.  Google Scholar

[9]

M. A. Biot, Mechanics of deformation and acoustic propagation in porous media,, Journal of Applied Physics, 33 (1962), 1482.  doi: 10.1063/1.1728759.  Google Scholar

[10]

M. A. Biot, Theory of finite deformations of porous solids,, Indiana University Mathematics Journal, 21 (1972), 597.   Google Scholar

[11]

S. Chiriţă, C. Galeş and I. D. Ghiba, On spatial behavior of the harmonic vibrations in Kelvin-Voigt materials,, Journal of Elasticity, 93 (2008), 81.  doi: 10.1007/s10659-008-9167-z.  Google Scholar

[12]

R. M. Christensen, Theory of Viscoelasticity,, 2nd ed., (2010).   Google Scholar

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M. Ciarletta, F. Passarella and M. Svanadze, Plane waves and uniqueness theorems in the coupled linear theory of elasticity for solids with double porosity,, Journal of Elasticity, 114 (2014), 55.  doi: 10.1007/s10659-012-9426-x.  Google Scholar

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M. Ciarletta and A. Scalia, On some theorems in the linear theory of viscoelastic materials with voids,, Journal of Elasticity, 25 (1991), 149.  doi: 10.1007/BF00042463.  Google Scholar

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S. C. Cowin, The viscoelastic behavior of linear elastic materials with voids,, Journal of Elasticity, 15 (1985), 185.  doi: 10.1007/BF00041992.  Google Scholar

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S. C. Cowin, Bone poroelasticity,, Journal of Biomechanics, 32 (1999), 217.  doi: 10.1016/S0021-9290(98)00161-4.  Google Scholar

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S. De Cicco and M. Svanadze, Fundamental solution in the theory of viscoelastic mixtures,, Journal of Mechanics of Materials and Structures, 4 (2009), 139.   Google Scholar

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G. Del Piero and L. Deseri, On the concepts of state and free energy in linear viscoelasticity,, Archive for Rational Mechanics and Analysis, 138 (1997), 1.  doi: 10.1007/s002050050035.  Google Scholar

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M. Di Paola and M. Zingales, Exact mechanical models for fractional viscoelastic material,, Journal of Rheology, 56 (2012), 983.   Google Scholar

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M. Di Paola and M. Zingales, A discrete mechanical model of fractional hereditary materials,, Meccanica, 48 (2013), 1573.  doi: 10.1007/s11012-012-9685-4.  Google Scholar

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M. Fabrizio, C. Giorgi and A. Morro, Free energies and dissipation properties for systems with memory,, Archive for Rational Mechanics and Analysis, 125 (1994), 341.   Google Scholar

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M. Fabrizio and B. Lazzari, On the existence and the asymptotic stability of solutions for linearly viscoelastic solids,, Archive for Rational Mechanics and Analysis, 116 (1991), 139.  doi: 10.1007/BF00375589.  Google Scholar

[26]

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

[27]

D. Graffi and M. Fabrizio, Sulla nozione di stato per materiali viscoelastici di tipo "rate",, Atti Acc. Lincei, 83 (1989), 201.   Google Scholar

[28]

L. Hörmander, The Analysis of Linear Partial Differential Operators I: Differential Operators with Constant Coefficients,, Grundlehren der mathematischen Wissenschaften, (1998).  doi: 10.1007/978-3-642-96750-4.  Google Scholar

[29]

D. Ieşan, On the theory of viscoelastic mixtures,, Journal of Thermal Stresses, 27 (2004), 1125.  doi: 10.1080/01495730490498575.  Google Scholar

[30]

D. Ieşan, A theory of thermoviscoelastic composites modelled as interacting Cosserat continua,, Journal of Thermal Stresses, 30 (2007), 1269.   Google Scholar

[31]

D. Ieşan, On a theory of thermoviscoelastic materials with voids,, Journal of Elasticity, 104 (2011), 369.  doi: 10.1007/s10659-010-9300-7.  Google Scholar

[32]

D. Ieşan and L. Nappa, On the theory of viscoelastic mixtures and stability,, Mathematics and Mechanics of Solids, 13 (2008), 55.  doi: 10.1177/1081286506072351.  Google Scholar

[33]

R. Lakes, Viscoelastic Materials,, Cambridge University Press, (2009).  doi: 10.1017/CBO9780511626722.  Google Scholar

[34]

M. Y. Khaled, D. E. Beskos and E. C. Aifantis, On the theory of consolidation with double porosity - III,, International Journal for Numerical and Analytical Methods in Geomechanics, 8 (1984), 101.   Google Scholar

[35]

N. Khalili, Coupling effects in double porosity media with deformable matrix,, Geophysical Research Letters, 30 (2003).  doi: 10.1029/2003GL018544.  Google Scholar

[36]

N. Khalili and S. Valliappan, Unified theory of flow and deformation in double porous media,, European Journal of Mechanics - A/Solids, 15 (1996), 321.   Google Scholar

[37]

F. Martínez and R. Quintanilla, Existence, uniqueness and asymptotic behaviour of solutions to the equations of viscoelasticity with voids,, International Journal of Solids and Structures, 35 (1998), 3347.  doi: 10.1016/S0020-7683(98)00018-3.  Google Scholar

[38]

F. Passarella, V. Tibullo and V. Zampoli, On microstretch thermoviscoelastic composite materials,, Europ. J. Mechanics, 37 (2013), 294.  doi: 10.1016/j.euromechsol.2012.07.002.  Google Scholar

[39]

S. R. Pride and J. G. Berryman, Linear dynamics of double-porosity dual-permeability materials - I,, Physical Review E, 68 (2003).  doi: 10.1103/PhysRevE.68.036603.  Google Scholar

[40]

R. Quintanilla, Existence and exponential decay in the linear theory of viscoelastic mixtures,, European Journal of Mechanics A/Solids, 24 (2005), 311.  doi: 10.1016/j.euromechsol.2004.11.008.  Google Scholar

[41]

E. Rohan, S. Naili, R. Cimrman and T. Lemaire, Multiscale modeling of a fluid saturated medium with double porosity: Relevance to the compact bone,, Journal of the Mechanics and Physics of Solids, 60 (2012), 857.  doi: 10.1016/j.jmps.2012.01.013.  Google Scholar

[42]

A. Scalia, Shock waves in viscoelastic materials with voids,, Wave Motion, 19 (1994), 125.  doi: 10.1016/0165-2125(94)90061-2.  Google Scholar

[43]

K. Sharma and P. Kumar, Propagation of plane waves and fundamental solution in thermoviscoelastic medium with voids,, Journal of Thermal Stresses, 36 (2013), 94.  doi: 10.1080/01495739.2012.720545.  Google Scholar

[44]

B. Straughan, Stability and uniqueness in double porosity elasticity,, International Journal of Engineering Science, 65 (2013), 1.  doi: 10.1016/j.ijengsci.2013.01.001.  Google Scholar

[45]

M. M. Svanadze, Potential method in the linear theories of viscoelasticity and thermoviscoelasticity for Kelvin-Voigt materials,, Technische Mechanik, 32 (2012), 554.   Google Scholar

[46]

M. M. Svanadze, Potential method in the linear theory of viscoelastic materials with voids,, Journal of Elasticity, 114 (2014), 101.  doi: 10.1007/s10659-013-9429-2.  Google Scholar

[47]

M. M. Svanadze, On the solutions of equations of the linear thermoviscoelasticity theory for Kelvin-Voigt materials with voids,, Journal of Thermal Stresses, 37 (2014), 253.  doi: 10.1080/01495739.2013.839851.  Google Scholar

[48]

M. Svanadze, Fundamental solution in the theory of consolidation with double porosity,, Journal of the Mechanical Behavior of Materials, 16 (2005), 123.  doi: 10.1515/JMBM.2005.16.1-2.123.  Google Scholar

[49]

M. Svanadze, Plane waves and boundary value problems in the theory of elasticity for solids with double porosity,, Acta Applicandae Mathematicae, 122 (2012), 461.  doi: 10.1007/s10440-012-9756-5.  Google Scholar

[50]

M. Svanadze, The boundary value problems of the full coupled theory of poroelasticity for materials with double porosity,, Proceedings in Applied Mathematics and Mechanics, 12 (2012), 279.  doi: 10.1002/pamm.201210130.  Google Scholar

[51]

M. Svanadze, Fundamental solution in the linear theory of consolidation for elastic solids with double porosity,, Journal of Mathematical Sciciences, 195 (2013), 258.  doi: 10.1007/s10958-013-1578-0.  Google Scholar

[52]

M. Svanadze and S. De Cicco, Fundamental solutions in the full coupled linear theory of elasticity for solid with double porosity,, Archives of Mechanics, 65 (2013), 367.   Google Scholar

[53]

M. Svanadze and G. Iovane, Fundamental solution in the linear theory of thermoviscoelastic mixtures,, European Journal of Applied Mathematics, 18 (2007), 323.  doi: 10.1017/S0956792507006961.  Google Scholar

[54]

M. Svanadze and A. Scalia, Mathematical problems in the coupled linear theory of bone poroelasticity,, Computers and Mathematics with Applications, 66 (2013), 1554.  doi: 10.1016/j.camwa.2013.01.046.  Google Scholar

[55]

R. K. Wilson and E. C. Aifantis, On the theory of consolidation with double porosity - I,, International Journal of Engineering Science, 20 (1982), 1009.   Google Scholar

[56]

Y. Zhao and M. Chen, Fully coupled dual-porosity model for anisotropic formations,, International Journal of Rock Mechanics and Mining Sciences, 43 (2006), 1128.  doi: 10.1016/j.ijrmms.2006.03.001.  Google Scholar

show all references

References:
[1]

G. Amendola, M. Fabrizio and J. M. Golden, Thermodynamics of Materials with Memory: Theory and Applications,, Springer, (2012).  doi: 10.1007/978-1-4614-1692-0.  Google Scholar

[2]

J. A. D. Appleby, M. Fabrizio, B. Lazzari and D. W. Reynolds, On exponential asymptotic stability in linear viscoelasticity,, Mathematical Models and Methods in Applied Sciences, 16 (2006), 1677.  doi: 10.1142/S0218202506001674.  Google Scholar

[3]

G. I. Barenblatt, I. P. Zheltov and I. N. Kochina, Basic concept in the theory of seepage of homogeneous liquids in fissured rocks (strata),, Journal of Applied Mathematics and Mechanics, 24 (1960), 1286.  doi: 10.1016/0021-8928(60)90107-6.  Google Scholar

[4]

J. G. Berryman and H. F. Wang, The elastic coefficients of double-porosity models for fluid transport in jointed rock,, Journal of Geophysical Research, 100 (1995), 24611.  doi: 10.1029/95JB02161.  Google Scholar

[5]

J. G. Berryman and H. F. Wang, Elastic wave propagation and attenuation in a double-porosity dual-permiability medium,, International Journal of Rock Mechanics and Mining Sciences, 37 (2000), 63.  doi: 10.1016/S1365-1609(99)00092-1.  Google Scholar

[6]

D. E. Beskos and E. C. Aifantis, On the theory of consolidation with double porosity - II,, International Journal of Engineering Science, 24 (1986), 1697.  doi: 10.1016/0020-7225(86)90076-5.  Google Scholar

[7]

M. A. Biot, General theory of three-dimensional consolidation,, Journal of Applied Physics, 12 (1941), 155.  doi: 10.1063/1.1712886.  Google Scholar

[8]

M. A. Biot, Theory of propagation of elastic waves in a fluid-saturated porous solid, I. Low frequency range, II. Higher frequency range,, Journal of the Acoustical Society of America, 28 (1956), 179.  doi: 10.1121/1.1908241.  Google Scholar

[9]

M. A. Biot, Mechanics of deformation and acoustic propagation in porous media,, Journal of Applied Physics, 33 (1962), 1482.  doi: 10.1063/1.1728759.  Google Scholar

[10]

M. A. Biot, Theory of finite deformations of porous solids,, Indiana University Mathematics Journal, 21 (1972), 597.   Google Scholar

[11]

S. Chiriţă, C. Galeş and I. D. Ghiba, On spatial behavior of the harmonic vibrations in Kelvin-Voigt materials,, Journal of Elasticity, 93 (2008), 81.  doi: 10.1007/s10659-008-9167-z.  Google Scholar

[12]

R. M. Christensen, Theory of Viscoelasticity,, 2nd ed., (2010).   Google Scholar

[13]

M. Ciarletta, F. Passarella and M. Svanadze, Plane waves and uniqueness theorems in the coupled linear theory of elasticity for solids with double porosity,, Journal of Elasticity, 114 (2014), 55.  doi: 10.1007/s10659-012-9426-x.  Google Scholar

[14]

M. Ciarletta and A. Scalia, On some theorems in the linear theory of viscoelastic materials with voids,, Journal of Elasticity, 25 (1991), 149.  doi: 10.1007/BF00042463.  Google Scholar

[15]

S. C. Cowin, The viscoelastic behavior of linear elastic materials with voids,, Journal of Elasticity, 15 (1985), 185.  doi: 10.1007/BF00041992.  Google Scholar

[16]

S. C. Cowin, Bone poroelasticity,, Journal of Biomechanics, 32 (1999), 217.  doi: 10.1016/S0021-9290(98)00161-4.  Google Scholar

[17]

S. C. Cowin, G. Gailani and M. Benalla, Hierarchical poroelasticity: Movement of interstitial fluid between levels in bones,, Philosophical Transactions of the Royal Society A: mathematical, 367 (2009), 3401.  doi: 10.1098/rsta.2009.0099.  Google Scholar

[18]

S. De Cicco and L. Nappa, Singular surfaces in thermoviscoelastic materials with voids,, Journal of Elasticity, 73 (2003), 191.  doi: 10.1023/B:ELAS.0000029961.09749.2b.  Google Scholar

[19]

S. De Cicco and M. Svanadze, Fundamental solution in the theory of viscoelastic mixtures,, Journal of Mechanics of Materials and Structures, 4 (2009), 139.   Google Scholar

[20]

G. Del Piero and L. Deseri, On the concepts of state and free energy in linear viscoelasticity,, Archive for Rational Mechanics and Analysis, 138 (1997), 1.  doi: 10.1007/s002050050035.  Google Scholar

[21]

L. Deseri, M. Fabrizio and M. Golden, The concept of a minimal state in viscoelasticity: New free energies and applications to PDEs,, Archive for Rational Mechanics and Analysis, 181 (2006), 43.  doi: 10.1007/s00205-005-0406-1.  Google Scholar

[22]

M. Di Paola and M. Zingales, Exact mechanical models for fractional viscoelastic material,, Journal of Rheology, 56 (2012), 983.   Google Scholar

[23]

M. Di Paola and M. Zingales, A discrete mechanical model of fractional hereditary materials,, Meccanica, 48 (2013), 1573.  doi: 10.1007/s11012-012-9685-4.  Google Scholar

[24]

M. Fabrizio, C. Giorgi and A. Morro, Free energies and dissipation properties for systems with memory,, Archive for Rational Mechanics and Analysis, 125 (1994), 341.   Google Scholar

[25]

M. Fabrizio and B. Lazzari, On the existence and the asymptotic stability of solutions for linearly viscoelastic solids,, Archive for Rational Mechanics and Analysis, 116 (1991), 139.  doi: 10.1007/BF00375589.  Google Scholar

[26]

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

[27]

D. Graffi and M. Fabrizio, Sulla nozione di stato per materiali viscoelastici di tipo "rate",, Atti Acc. Lincei, 83 (1989), 201.   Google Scholar

[28]

L. Hörmander, The Analysis of Linear Partial Differential Operators I: Differential Operators with Constant Coefficients,, Grundlehren der mathematischen Wissenschaften, (1998).  doi: 10.1007/978-3-642-96750-4.  Google Scholar

[29]

D. Ieşan, On the theory of viscoelastic mixtures,, Journal of Thermal Stresses, 27 (2004), 1125.  doi: 10.1080/01495730490498575.  Google Scholar

[30]

D. Ieşan, A theory of thermoviscoelastic composites modelled as interacting Cosserat continua,, Journal of Thermal Stresses, 30 (2007), 1269.   Google Scholar

[31]

D. Ieşan, On a theory of thermoviscoelastic materials with voids,, Journal of Elasticity, 104 (2011), 369.  doi: 10.1007/s10659-010-9300-7.  Google Scholar

[32]

D. Ieşan and L. Nappa, On the theory of viscoelastic mixtures and stability,, Mathematics and Mechanics of Solids, 13 (2008), 55.  doi: 10.1177/1081286506072351.  Google Scholar

[33]

R. Lakes, Viscoelastic Materials,, Cambridge University Press, (2009).  doi: 10.1017/CBO9780511626722.  Google Scholar

[34]

M. Y. Khaled, D. E. Beskos and E. C. Aifantis, On the theory of consolidation with double porosity - III,, International Journal for Numerical and Analytical Methods in Geomechanics, 8 (1984), 101.   Google Scholar

[35]

N. Khalili, Coupling effects in double porosity media with deformable matrix,, Geophysical Research Letters, 30 (2003).  doi: 10.1029/2003GL018544.  Google Scholar

[36]

N. Khalili and S. Valliappan, Unified theory of flow and deformation in double porous media,, European Journal of Mechanics - A/Solids, 15 (1996), 321.   Google Scholar

[37]

F. Martínez and R. Quintanilla, Existence, uniqueness and asymptotic behaviour of solutions to the equations of viscoelasticity with voids,, International Journal of Solids and Structures, 35 (1998), 3347.  doi: 10.1016/S0020-7683(98)00018-3.  Google Scholar

[38]

F. Passarella, V. Tibullo and V. Zampoli, On microstretch thermoviscoelastic composite materials,, Europ. J. Mechanics, 37 (2013), 294.  doi: 10.1016/j.euromechsol.2012.07.002.  Google Scholar

[39]

S. R. Pride and J. G. Berryman, Linear dynamics of double-porosity dual-permeability materials - I,, Physical Review E, 68 (2003).  doi: 10.1103/PhysRevE.68.036603.  Google Scholar

[40]

R. Quintanilla, Existence and exponential decay in the linear theory of viscoelastic mixtures,, European Journal of Mechanics A/Solids, 24 (2005), 311.  doi: 10.1016/j.euromechsol.2004.11.008.  Google Scholar

[41]

E. Rohan, S. Naili, R. Cimrman and T. Lemaire, Multiscale modeling of a fluid saturated medium with double porosity: Relevance to the compact bone,, Journal of the Mechanics and Physics of Solids, 60 (2012), 857.  doi: 10.1016/j.jmps.2012.01.013.  Google Scholar

[42]

A. Scalia, Shock waves in viscoelastic materials with voids,, Wave Motion, 19 (1994), 125.  doi: 10.1016/0165-2125(94)90061-2.  Google Scholar

[43]

K. Sharma and P. Kumar, Propagation of plane waves and fundamental solution in thermoviscoelastic medium with voids,, Journal of Thermal Stresses, 36 (2013), 94.  doi: 10.1080/01495739.2012.720545.  Google Scholar

[44]

B. Straughan, Stability and uniqueness in double porosity elasticity,, International Journal of Engineering Science, 65 (2013), 1.  doi: 10.1016/j.ijengsci.2013.01.001.  Google Scholar

[45]

M. M. Svanadze, Potential method in the linear theories of viscoelasticity and thermoviscoelasticity for Kelvin-Voigt materials,, Technische Mechanik, 32 (2012), 554.   Google Scholar

[46]

M. M. Svanadze, Potential method in the linear theory of viscoelastic materials with voids,, Journal of Elasticity, 114 (2014), 101.  doi: 10.1007/s10659-013-9429-2.  Google Scholar

[47]

M. M. Svanadze, On the solutions of equations of the linear thermoviscoelasticity theory for Kelvin-Voigt materials with voids,, Journal of Thermal Stresses, 37 (2014), 253.  doi: 10.1080/01495739.2013.839851.  Google Scholar

[48]

M. Svanadze, Fundamental solution in the theory of consolidation with double porosity,, Journal of the Mechanical Behavior of Materials, 16 (2005), 123.  doi: 10.1515/JMBM.2005.16.1-2.123.  Google Scholar

[49]

M. Svanadze, Plane waves and boundary value problems in the theory of elasticity for solids with double porosity,, Acta Applicandae Mathematicae, 122 (2012), 461.  doi: 10.1007/s10440-012-9756-5.  Google Scholar

[50]

M. Svanadze, The boundary value problems of the full coupled theory of poroelasticity for materials with double porosity,, Proceedings in Applied Mathematics and Mechanics, 12 (2012), 279.  doi: 10.1002/pamm.201210130.  Google Scholar

[51]

M. Svanadze, Fundamental solution in the linear theory of consolidation for elastic solids with double porosity,, Journal of Mathematical Sciciences, 195 (2013), 258.  doi: 10.1007/s10958-013-1578-0.  Google Scholar

[52]

M. Svanadze and S. De Cicco, Fundamental solutions in the full coupled linear theory of elasticity for solid with double porosity,, Archives of Mechanics, 65 (2013), 367.   Google Scholar

[53]

M. Svanadze and G. Iovane, Fundamental solution in the linear theory of thermoviscoelastic mixtures,, European Journal of Applied Mathematics, 18 (2007), 323.  doi: 10.1017/S0956792507006961.  Google Scholar

[54]

M. Svanadze and A. Scalia, Mathematical problems in the coupled linear theory of bone poroelasticity,, Computers and Mathematics with Applications, 66 (2013), 1554.  doi: 10.1016/j.camwa.2013.01.046.  Google Scholar

[55]

R. K. Wilson and E. C. Aifantis, On the theory of consolidation with double porosity - I,, International Journal of Engineering Science, 20 (1982), 1009.   Google Scholar

[56]

Y. Zhao and M. Chen, Fully coupled dual-porosity model for anisotropic formations,, International Journal of Rock Mechanics and Mining Sciences, 43 (2006), 1128.  doi: 10.1016/j.ijrmms.2006.03.001.  Google Scholar

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