Fundamental solutions in the theory of elasticity for triple porosity materials. (English) Zbl 1344.74017
Summary: This paper concerns with the full coupled linear theory of elasticity for triple porosity materials. The system of the governing equations based on the equations of motion, conservation of fluid mass, the constitutive equations and Darcy’s law for materials with triple porosity. The cross-coupled terms are included in the equations of conservation of mass for the fluids of the three levels of porosity and in the Darcy’s law for materials with triple porosity. The system of general governing equations of motion is expressed in terms of the displacement vector field and the pressures in the three pore systems. Five spatial cases of the dynamical equations are considered: equations of steady vibrations, equations in Laplace transform space, equations of quasi-static, equations of equilibrium and equations of steady vibrations for rigid body with triple porosity. The fundamental solutions of the systems of these PDEs are constructed explicitly by means of elementary functions and finally, the basic properties of the fundamental solutions are established.
MSC:
| 74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |
| 76S05 | Flows in porous media; filtration; seepage |
| 35E05 | Fundamental solutions to PDEs and systems of PDEs with constant coefficients |
| 35J47 | Second-order elliptic systems |
| 35Q35 | PDEs in connection with fluid mechanics |
References:
| [1] | Abdassah D, Ershaghi I (1986) Triple-porosity systems for representing naturally fractured Reservoirs. SPE Form Eval. (April) 113-127. SPE-13409-PA |
| [2] | Al-Ahmadi HA, Wattenbarger RA (2011) Triple-porosity models: One further step towards capturing fractured reservoirs heterogeneity. In: Paper SPE 149054-MS Presented at SPE/DGS Saudi Arabia Section Technical Symposium and Exhibition. Al-Khobar, Saudi Arabia. doi:10.2118/149054-MS · Zbl 0493.76094 |
| [3] | Bai M, Elsworth D, Roegiers JC (1993) Multiporosity/multipermeability approach to the simulation of naturally fractured reservoirs. Water Resour Res 29:1621-1633 · doi:10.1029/92WR02746 |
| [4] | Bai M, Roegiers JC (1997) Triple-porosity analysis of solute transport. J. Cantam. Hydrol. 28:189-211 |
| [5] | Barenblatt GI, Zheltov IP, Kochina IN (1960) Basic concept in the theory of seepage of homogeneous liquids in fissured rocks (strata). J. Appl. Math. Mech. 24:1286-1303 · Zbl 0104.21702 · doi:10.1016/0021-8928(60)90107-6 |
| [6] | Beskos DE, Aifantis EC (1986) On the theory of consolidation with double porosity -II. Int J Eng Sci 24:1697-1716 · Zbl 0601.73109 · doi:10.1016/0020-7225(86)90076-5 |
| [7] | Biot MA (1941) General theory of three-dimensional consolidation. J Appl Phys 12:155-164 · JFM 67.0837.01 · doi:10.1063/1.1712886 |
| [8] | Bitsadze L, Tsagareli I (2015) The solution of the Dirichlet BVP in the fully coupled theory of elasticity for spherical layer with double porosity. Meccanica. doi:10.1007/s11012-015-0312-z · Zbl 1332.74015 · doi:10.1007/s11012-015-0312-z |
| [9] | Burchuladze TV, Gegelia TG (1985) The development of the potential methods in the elasticity theory. Metsniereba, Tbilisi (Russian) · Zbl 0594.73026 |
| [10] | Ciarletta M, Passarella F, Svanadze M (2014) Plane waves and uniqueness theorems in the coupled linear theory of elasticity for solids with double porosity. J Elast 114:55-68 · Zbl 1451.74074 · doi:10.1007/s10659-012-9426-x |
| [11] | Cowin SC (ed) (2008) Bone mechanics handbook. Informa Healthcare USA Inc., New York |
| [12] | Cowin SC (1999) Bone poroelasticity. J Biomech 32:217-238 · doi:10.1016/S0021-9290(98)00161-4 |
| [13] | Cowin SC, Gailani G, Benalla M (2009) Hierarchical poroelasticity: movement of interstitial fluid between levels in bones. Philos Trans R Soc A 367:3401-3444 · Zbl 1185.74057 · doi:10.1098/rsta.2009.0099 |
| [14] | Gegelia T, Jentsch L (1994) Potential methods in continuum mechanics. Georgian Math J 1:599-640 · Zbl 0815.35120 · doi:10.1007/BF02254683 |
| [15] | Gentile M, Straughan B (2013) Acceleration waves in nonlinear double porosity elasticity. Int J Eng Sci 73:10-16 · doi:10.1016/j.ijengsci.2013.07.006 |
| [16] | Hamed E, Lee Y, Jasiuk I (2010) Multiscale modeling of elastic properties of cortical bone. Acta Mech 213:131-154 · Zbl 1320.74078 · doi:10.1007/s00707-010-0326-5 |
| [17] | Hörmander L (2005) The analysis of linear partial differential operators II: differential operators with constant coefficients. Springer, Berlin · Zbl 1062.35004 |
| [18] | Ieşan D (2015) Method of potentials in elastostatics of solids with double porosity. Int J Eng Sci 88:118-127 · Zbl 1423.74096 · doi:10.1016/j.ijengsci.2014.04.011 |
| [19] | Ieşan D, Quintanilla R (2014) On a theory of thermoelastic materials with a double porosity structure. J Therm Stress 37:1017-1036 · doi:10.1080/01495739.2014.914776 |
| [20] | Ji B, Gao H (2006) Elastic properties of nanocomposite structure of bone. Compos Sci Technol 66:1212-1218 · doi:10.1016/j.compscitech.2005.10.017 |
| [21] | Khaled MY, Beskos DE, Aifantis EC (1984) On the theory of consolidation with double porosity -III. Int J Numer Anal Methods Geomech 8:101-123 · Zbl 0586.73116 · doi:10.1002/nag.1610080202 |
| [22] | Khalili N, Selvadurai APS (2003) A fully coupled constitutive model for thermo-hydro-mechanical analysis in elastic media with double porosity. Geophys Res Lett 30:2268 |
| [23] | Khalili N, Valliappan S (1996) Unified theory of flow and deformation in double porous media. Eur J Mech A Solids 15:321-336 · Zbl 0942.74013 |
| [24] | Kupradze VD (1965) Potential methods in the theory of elasticity. Israel Program Sci. Transl, Jerusalem · Zbl 0188.56901 |
| [25] | Kupradze VD, Gegelia TG, Basheleishvili MO, Burchuladze TV (1979) Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity. North-Holland, Amsterdam · Zbl 0406.73001 |
| [26] | Liu CQ (1981) Exact solution for the compressible flow equations through a medium with triple-porosity. Appl Math Mech 2:457-462 · Zbl 0606.76108 · doi:10.1007/BF01875921 |
| [27] | Liu JC, Bodvarsson GS, Wu YS (2003) Analysis of pressure behaviour in fractured lithophysical reservoirs. J Contam Hydrol 62-63:189-211 · doi:10.1016/S0169-7722(02)00169-9 |
| [28] | Moutsopoulos KN, Konstantinidis AA, Meladiotis I, Tzimopoulos ChD, Aifantis EC (2001) Hydraulic behavior and contaminant transport in multiple porosity media. Trans. Porous Media 42:265-292 · doi:10.1023/A:1006745924508 |
| [29] | Rohan E, Naili S, Cimrman R, Lemaire T (2012) Multiscale modeling of a fluid saturated medium with double porosity: relevance to the compact bone. J Mech Phys Solids 60:857-881 · doi:10.1016/j.jmps.2012.01.013 |
| [30] | Scarpetta E, Svanadze M, Zampoli V (2014) Fundamental solutions in the theory of thermoelasticity for solids with double porosity. J Therm Stresses 37:727-748 · doi:10.1080/01495739.2014.885337 |
| [31] | Straughan B (2008) Stability and wave motion in porous media. Springer, New York · Zbl 1149.76002 |
| [32] | Straughan B (2013) Stability and uniqueness in double porosity elasticity. Int J Eng Sci 65:1-8 · Zbl 1423.74286 · doi:10.1016/j.ijengsci.2013.01.001 |
| [33] | Straughan B (2015) Convection with local thermal non-equilibrium and microfluidic effects. Springer, Switzerland, Heidelberg · Zbl 1325.76005 · doi:10.1007/978-3-319-13530-4 |
| [34] | Svanadze M (1996) The fundamental solution of the oscillation equations of the thermoelasticity theory of mixture of two elastic solids. J Therm Stresses 19:633-648 · doi:10.1080/01495739608946199 |
| [35] | Svanadze M (2005) Fundamental solution in the theory of consolidation with double porosity. J Mech Behav Mater 16:123-130 · doi:10.1515/JMBM.2005.16.1-2.123 |
| [36] | Svanadze M (2014) Uniqueness theorems in the theory of thermoelasticity for solids with double porosity. Meccanica 49:2099-2108 · Zbl 1299.74047 · doi:10.1007/s11012-014-9876-2 |
| [37] | Svanadze M, De Cicco S (2013) Fundamental solutions in the full coupled linear theory of elasticity for solid with double porosity. Arch Mech 65:367-390 · Zbl 1446.74116 |
| [38] | Svanadze M, Scalia A (2013) Mathematical problems in the coupled linear theory of bone poroelasticity. Comput Math Appl 66:1554-1566 · Zbl 1381.74155 · doi:10.1016/j.camwa.2013.01.046 |
| [39] | Tsagareli I, Bitsadze L (2015) Explicit solution of one boundary value problem in the full coupled theory of elasticity for solids with double porosity. Acta Mech 226:1409-1418 · Zbl 1329.74035 · doi:10.1007/s00707-014-1260-8 |
| [40] | Wilson RK, Aifantis EC (1982) On the theory of consolidation with double porosity -I. Int J Eng Sci 20:1009-1035 · Zbl 0493.76094 · doi:10.1016/0020-7225(82)90036-2 |
| [41] | Wu YS, Liu HH, Bodavarsson GS (2004) A triple-continuum approach for modelling flow and transport processes in fractured rock. J Contam Hydrol 73:145-179 · doi:10.1016/j.jconhyd.2004.01.002 |
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