On the theory of viscoelasticity for materials with double porosity

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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

Abstract
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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.

Keywords: Kelvin-Voigt material., double porosity, Viscoelasticity.

Mathematics Subject Classification: Primary: 74D05, 74F10; Secondary: 74A60, 74J0.

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:

[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.
[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.
[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.
[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.*
[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.
[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.
[7]	M. A. Biot, General theory of three-dimensional consolidation,, Journal of Applied Physics, 12 (1941), 155. doi: 10.1063/1.1712886.
[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.
[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.
[10]	M. A. Biot, Theory of finite deformations of porous solids,, Indiana University Mathematics Journal, 21 (1972), 597.
[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.
[12]	R. M. Christensen, Theory of Viscoelasticity,, 2nd ed., (2010).
[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.*
[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.
[15]	S. C. Cowin, The viscoelastic behavior of linear elastic materials with voids,, Journal of Elasticity, 15 (1985), 185. doi: 10.1007/BF00041992.
[16]	S. C. Cowin, Bone poroelasticity,, Journal of Biomechanics, 32 (1999), 217. doi: 10.1016/S0021-9290(98)00161-4.
[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.*
[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.
[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.
[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.*
[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.*
[22]	M. Di Paola and M. Zingales, Exact mechanical models for fractional viscoelastic material,, Journal of Rheology, 56 (2012), 983.
[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.
[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.*
[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.*
[26]	M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity,, SIAM, (1992). doi: 10.1137/1.9781611970807.
[27]	D. Graffi and M. Fabrizio, Sulla nozione di stato per materiali viscoelastici di tipo "rate",, Atti Acc. Lincei, 83 (1989), 201.
[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.
[29]	D. Ieşan, On the theory of viscoelastic mixtures,, Journal of Thermal Stresses, 27 (2004), 1125. doi: 10.1080/01495730490498575.
[30]	D. Ieşan, A theory of thermoviscoelastic composites modelled as interacting Cosserat continua,, Journal of Thermal Stresses, 30 (2007), 1269.
[31]	D. Ieşan, On a theory of thermoviscoelastic materials with voids,, Journal of Elasticity, 104* (2011), 369. doi: 10.1007/s10659-010-9300-7.*
[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.
[33]	R. Lakes, Viscoelastic Materials,, Cambridge University Press, (2009). doi: 10.1017/CBO9780511626722.
[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.
[35]	N. Khalili, Coupling effects in double porosity media with deformable matrix,, Geophysical Research Letters, 30 (2003). doi: 10.1029/2003GL018544.
[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.
[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.
[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.
[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.
[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.
[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.
[42]	A. Scalia, Shock waves in viscoelastic materials with voids,, Wave Motion, 19 (1994), 125. doi: 10.1016/0165-2125(94)90061-2.
[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.
[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.
[45]	M. M. Svanadze, Potential method in the linear theories of viscoelasticity and thermoviscoelasticity for Kelvin-Voigt materials,, Technische Mechanik, 32 (2012), 554.
[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.*
[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.
[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.
[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.*
[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.
[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.*
[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.
[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.
[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.
[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.
[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.

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.
[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.
[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.
[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.*
[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.
[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.
[7]	M. A. Biot, General theory of three-dimensional consolidation,, Journal of Applied Physics, 12 (1941), 155. doi: 10.1063/1.1712886.
[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.
[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.
[10]	M. A. Biot, Theory of finite deformations of porous solids,, Indiana University Mathematics Journal, 21 (1972), 597.
[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.
[12]	R. M. Christensen, Theory of Viscoelasticity,, 2nd ed., (2010).
[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.*
[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.
[15]	S. C. Cowin, The viscoelastic behavior of linear elastic materials with voids,, Journal of Elasticity, 15 (1985), 185. doi: 10.1007/BF00041992.
[16]	S. C. Cowin, Bone poroelasticity,, Journal of Biomechanics, 32 (1999), 217. doi: 10.1016/S0021-9290(98)00161-4.
[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.*
[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.
[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.
[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.*
[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.*
[22]	M. Di Paola and M. Zingales, Exact mechanical models for fractional viscoelastic material,, Journal of Rheology, 56 (2012), 983.
[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.
[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.*
[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.*
[26]	M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity,, SIAM, (1992). doi: 10.1137/1.9781611970807.
[27]	D. Graffi and M. Fabrizio, Sulla nozione di stato per materiali viscoelastici di tipo "rate",, Atti Acc. Lincei, 83 (1989), 201.
[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.
[29]	D. Ieşan, On the theory of viscoelastic mixtures,, Journal of Thermal Stresses, 27 (2004), 1125. doi: 10.1080/01495730490498575.
[30]	D. Ieşan, A theory of thermoviscoelastic composites modelled as interacting Cosserat continua,, Journal of Thermal Stresses, 30 (2007), 1269.
[31]	D. Ieşan, On a theory of thermoviscoelastic materials with voids,, Journal of Elasticity, 104* (2011), 369. doi: 10.1007/s10659-010-9300-7.*
[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.
[33]	R. Lakes, Viscoelastic Materials,, Cambridge University Press, (2009). doi: 10.1017/CBO9780511626722.
[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.
[35]	N. Khalili, Coupling effects in double porosity media with deformable matrix,, Geophysical Research Letters, 30 (2003). doi: 10.1029/2003GL018544.
[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.
[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.
[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.
[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.
[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.
[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.
[42]	A. Scalia, Shock waves in viscoelastic materials with voids,, Wave Motion, 19 (1994), 125. doi: 10.1016/0165-2125(94)90061-2.
[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.
[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.
[45]	M. M. Svanadze, Potential method in the linear theories of viscoelasticity and thermoviscoelasticity for Kelvin-Voigt materials,, Technische Mechanik, 32 (2012), 554.
[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.*
[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.
[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.
[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.*
[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.
[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.*
[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.
[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.
[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.
[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.
[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.

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