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Modelling and Simulation in Engineering
Volume 2011, Article ID 179467, 5 pages
http://dx.doi.org/10.1155/2011/179467
Research Article

Thermal Effect on Elastic Waves of Anisotropic Saturated Porous Solid

School of Civil Engineering and Architecture, Zhejiang University of Science and Technology, Hangzhou 310023, China

Received 6 March 2011; Accepted 25 October 2011

Academic Editor: Ahmed Rachid

Copyright © 2011 S. H. Guo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The motion equations of anisotropic media, coupled to the mass conservation and thermoequilibrium equations of fluid, are studied here based on the standard space of physical presentation for thermoelastic dynamics of anisotropic saturated porous solids. By introducing a new compressible thermo-elastic model, a set of uncoupled equations of elastic waves are deduced. The results show that the elastic waves and speeds of elastic waves are affected by both anisotropic subspaces of solids and thermal and compressive coupling coefficients between fluid and solid. Based on these laws, we discuss the propagation behaviour of elastic waves for various anisotropic solids.

1. Introduction

A general theory of three-dimensional propagation of elastic waves in a fluid-saturated porous solid was presented by Biot [13]. In this theory, the continuum consists of both solid constituent and fluid one, which are two entirely different materials, and the coupled, dilatational waves can be generated in both solid constituent and fluid one at a sound source. The thermal effects on the elastic waves were discussed by other works [4, 5]. But in fact, for most of geological materials, such as rock and soil, it is difficult to distinguish the solid constituent and the fluid one and also impossible to get elastic waves of both solid constituent and fluid one by common measuring methods. In this paper, a new theory of elastic waves in a fluid-saturated porous solid subjected to thermal effects is given, in which the idea of standard spaces [613] is used to deal with the motion equation, thermoequilibrium equation, and the mass conservation equation. By this method, the classical Newton’s equation of motion, thermoequilibrium equation, and the mass conservation equation under the geometric presentation can be transformed into the eigen ones under the physical presentation. The former is in the form of tensor, and the latter is in the form of scalar. By introducing a new compressible thermoelastic model, a set of uncoupled modal equations of elastic waves are obtained, each of which shows the existence of elastic subwaves; meanwhile, the propagation speed, propagation direction and space pattern of these subwaves can be completely determined by the modal equations.

2. Constitutive Equations of Anisotropic Porous Media

According to Terzaghi’s effective stress theory and the compressible relationship of porous elastic materials, the constitutive equations of fluid saturated linear porous solids with thermal effects are as follows:𝜎𝑖𝑗=𝑐𝑠𝑖𝑗𝑘𝑙𝜀𝑠𝑘𝑙+𝑎𝑖𝑗𝑝𝑏𝑖𝑗𝜃,𝑛=𝜈𝑝+𝜅𝜃𝑎𝑘𝑙𝜀𝑠𝑘𝑙,𝜂=𝜏𝜃+𝑏𝑘𝑙𝜀𝑠𝑘𝑙+𝜅𝑝,(1) where, 𝑝 is the pore pressure, 𝜃 is the temperature, 𝑐𝑠𝑖𝑗𝑘𝑙 elastic tensor of solid constituent of porous material, 𝜈 compressive coefficient of porous material, 𝜏 heat coefficient, 𝑎𝑖𝑗 coupling coefficient between solid constituent and fluid one, 𝑏𝑖𝑗 thermal coupling coefficient, 𝜅 coupling coefficient between porous deformation and heat, 𝑛 porosity, 𝜂 entropy.

Rewriting (1) in the Voigt’s notation, we have 𝝈=𝐜𝑠𝐒𝑠+𝐚𝑝𝐛𝜃,(2)𝑛=𝜈𝑝+𝜅𝜃𝐚𝑇𝐒𝑠,(3)𝜂=𝜏𝜃+𝐛𝑇𝐒𝑠+𝜅𝑝.(4) The skeleton elastic matrix 𝐜𝑠 can be spectrally decomposed as follows [69]:𝐜𝑠=𝚽𝚲𝚽𝑇,(5) where Λ=diag[𝜆1,𝜆2,𝜆3,𝜆4,𝜆5,𝜆6] are the matrixes of eigen elasticity [1013] of porous skeleton. Φ={𝝋1,𝝋2,𝝋𝟑,𝝋4,𝝋5,𝝋6} are the modal matrixes, which are both orthogonal and positive definite matrixes, and satisfy Φ𝑇Φ=𝐈.

Projecting the elastic physical qualities of the geometric presentation, such as the stress vector 𝝈 and strain vector 𝐒𝑠, into the standard spaces of the physical presentation, we get𝝈=𝚽𝑇𝐒𝝈,=𝚽𝑇𝐒𝑠.(6) Rewriting (6) in the form of scalar, we have𝜎𝑖=𝝋𝑖𝑇𝑆𝝈,𝑖=1𝑚,(7)𝑖=𝝋𝑖𝑇𝐒𝑠,𝑖=1𝑚,(8) where 𝑚(6) is the number of the elastic independent subspaces. Equations (7) and (8) show the elastic physical qualities under the physical presentation.

Substituting (6) into (2)–(4), respectively, and multiplying them with the transpose of modal matrix in the left, we have𝚽𝑇𝝈=𝚽𝑇𝐜𝑠𝚽𝐒+𝚽𝑇𝐚𝑝𝚽𝑇𝐛𝜃,𝑛=𝜈𝑝+𝜅𝜃𝐚𝑇𝚽𝐒,𝜂=𝜏𝜃+𝐛𝑇𝚽𝐒+𝜅𝑝.(9) Using (5), (6), we get𝝈=𝚲𝐒+𝐚𝑝𝐛𝜃,𝑛=𝜈𝑝+𝜅𝜃𝐚𝑇𝐒,𝜂=𝜏𝜃+𝐛𝑇𝐒+𝜅𝑝.(10) Rewriting the above equations in the form of scalar, we have𝜎𝑖=𝜆𝑖𝑆𝑖+𝑎𝑖𝑝𝑏𝑖𝜃,𝑖=1𝑚,𝑛=𝜈𝑝+𝜅𝜃𝑎𝑘𝑆𝑘,𝑘=1𝑚sumto𝑘,𝜂=𝜏𝜃+𝑏𝑘𝑆𝑘+𝜅𝑝,𝑘=1𝑚sumto𝑘.(11) Equations (11) are just the modal constitutive equations for anisotropic saturated linear porous elastic media with thermal effects.

3. Mass Conservation, Heat, and Motion Equations

The mass conservation equation of liquid in porous media is the following: 𝜕𝑛𝛾𝑤+𝜕𝑡𝛾𝑤𝑉=0,(12) where, 𝑛 is porosity, 𝛾𝑤 specific gravity of liquid, 𝑉 flow velocity of liquid, and Hamilton operator.

For uncompressible and stable flowing of liquid, (12) can also be written as follows:𝑡𝑛=0.(13) For thermoequilibrium state of an isolated system, the second law of thermodynamics is the following:𝑑𝑠=𝑑𝑠𝑒+𝑑𝑠𝑖=0,(14) where 𝑠 is total entropy of system. By using the entropy density, the above can be written as follows𝜌̇𝜂=0,or𝜌𝜂=𝐶,(15) where 𝐶 is an arbitrary constant, and we take it to be zero.

The eigen equation of motion of solids can be written as follows [68]:Δ𝑖𝜎𝑖=𝜌𝑠𝑡𝑡𝑆𝑖,𝑖=1𝑚,(16) where, Δ𝑖={𝜑𝑖}𝑇[Δ]{𝜑𝑖} is the stress operator, in which[Δ]=𝜕11000𝜕31𝜕210𝜕220𝜕320𝜕2100𝜕33𝜕32𝜕3100𝜕23𝜕23𝜕22+𝜕33𝜕21𝜕31𝜕130𝜕13𝜕12𝜕11+𝜕33𝜕32𝜕12𝜕120𝜕13𝜕23𝜕22+𝜕11,(17) where 𝜕𝑖𝑗=𝜕𝑗𝑖=𝜕2/𝜕𝑥𝑖𝜕𝑥𝑗.

4. Modal Equation of Elastic Waves with Thermal Effects

Substituting (11) into (13), (15), and (16), we haveΔ𝑖𝜆𝑖𝑆𝑖+𝑎𝑖𝑝𝑏𝑖𝜃=𝜌𝑠𝑡𝑡𝑆𝑖,(18)𝜈𝑝+𝜅𝜃𝑎𝑘𝑆𝑘=0,(19)𝜏𝜃+𝑏𝑘𝑆𝑘+𝜅𝑝=0.(20) From (19), we can get the following equation based on the operator principle:𝑎𝑝=𝑘𝜈𝑆𝑘𝜅𝜈𝜃.(21) Substituting it into (18) and (20), respectively, we haveΔ𝑖𝜆𝑖+𝑎𝑖𝛿𝑖𝑘𝑎𝑘𝜈𝑆𝑖𝜅𝜈𝑎𝑖+𝑏𝑖𝜃=𝜌𝑠𝑡𝑡𝑆𝑖,(22)𝜅𝜅𝜈𝑏𝜏𝜃𝑖+𝜅𝜈𝑎𝑖𝑆𝑖=0.(23) Substituting (23) into (22) again, we getΔ𝑖𝜆𝑖+𝑎𝑖𝛿𝑖𝑘𝑎𝑘𝜈𝜅𝜈𝑎𝑖+𝑏𝑖×𝜅𝜅𝜈𝜏1𝑏𝑖+𝜅𝜈𝑎𝑖𝑆𝑖=𝜌𝑠𝑡𝑡𝑆𝑖.(24) Rewriting it in the standard form of elastic waves, we haveΔ𝑖𝑆𝑖=1𝑐2𝑖𝑡𝑡𝑆𝑖,𝑖=1𝑚,(25) where𝑐𝑖=𝜆𝑖+𝑎𝑖𝛿𝑖𝑘𝑎𝑘/𝜈(𝜅/𝜈)𝑎𝑖+𝑏𝑖2((𝜅𝜅/𝜈)𝜏)1𝜌𝑠.(26) It is just the speed of elastic waves in porous media when thermal effects are considered.

5. Application

5.1. Isotropic Media

For isotropic media, the material tensors in (1) are represented by the following matrices under the compact notation𝑐11𝑐12𝑐12𝑐00012𝑐11𝑐12𝑐00012𝑐12𝑐11000000𝑐660000𝑐66000000𝑐66,𝑎11000𝑎11000𝑎11,𝑏11000𝑏11000𝑏11,(27) where 𝑐66=(𝑐11𝑐12).

There are two independent eigenspaces in an isotropic solids [69]𝑊=𝑊1(1)𝝋1𝑊2(5)𝝋2,,𝝋6,(28) where𝝋1=33[]1,1,1,0,0,0𝑇,𝝋2=22[]0,1,1,0,0,0𝑇𝝋3=66[]2,1,1,0,0,0𝑇,𝝋𝑖=𝝃𝑖(𝑖=4,5,6)(29) and 𝜉𝑖 is a vector of order 6 in which the 𝑖th element is 1 and others are 0.

The eigenelasticity of isotropic solids are𝜆1=𝑐11+2𝑐12𝜆=3𝜆+2𝜇,2=𝑐11𝑐12=2𝜇,(30) where 𝜆 and 𝜇 are Lame’s constants.

The coupled coefficients and stress operators for isotropic solids are calculated as𝑎1=3𝑎11,𝑎2𝑏=0,1=3𝑏11,𝑏2Δ=0,1=132III,Δ2=122III,(31) where 2III is Laplace’s operator of three dimensions.

It will be seen as follows that the modal strains in isotropic case represent the dilation and shear deformation, respectively.

Using (8), the modal strain of order 1 of isotropic solids is𝑆1=𝝋1𝑇𝐒=33𝑆11+𝑆22+𝑆33.(32) Equation (32) represents the relative change of the volume of elastic solids. So the wave equation of order 1 shows the motion of pure longitudinal wave.

Also from (8), the modal strain of order 2 of isotropic solids is𝑆2𝝋2=𝐒𝑆1𝝋1.(33) Using the orthogonality condition of eigenvectors, we have||𝑆2||=𝐒𝑆1𝝋1𝑇𝐒𝑆1𝝋11/2=13𝑆1𝑆22+𝑆2𝑆32+𝑆3𝑆121/2.(34) Equation (34) represents the pure shear strain on the elastic solids. So the wave equation of order 2 shows the motion of pure transverse wave as follows:𝑐1=𝑎𝐾+211/𝜈(𝜅/𝜈)𝑎11+𝑏112((𝜅𝜅/𝜈)𝜏)1𝜌𝑠,𝑐2=𝐺𝜌𝑠,(35) where 𝐾 is the bulk modulus and 𝐺 shear one of solid. It is seen that the speed of pure transverse wave is not subject to the thermal effects.

5.2. Transversely Isotropic Media

For isotropic media, the material tensors in (1) are represented by the following matrices under the compact notation:𝑐11𝑐12𝑐13𝑐00012𝑐11𝑐13𝑐00013𝑐13𝑐13000000𝑐440000𝑐44000000𝑐66,𝑎11000𝑎11000𝑎33,𝑏11000𝑏11000𝑏33,(36) where 𝑐66=(1/2)(𝑐11𝑐12).

There are four independent eigenspaces in an transversely isotropic solids [69] as follows:𝑊=𝑊1(1)𝝋1𝑊2(1)𝝋2𝑊3(2)𝝋3,𝝋6𝑊4(2)𝝋4,𝝋5,(37) where𝝋𝟏,𝟐=𝑐13𝜆1,2𝑐11𝑐122+2𝑐213×𝜆1,1,1,2𝑐11𝑐12𝑐3,0,0,0𝑇𝝋𝟑=22[]1,1,0,0,0,0𝑇,𝝋𝐢=𝝃𝑖,𝑖=4,5,6.(38) The eigenelasticities of transversely isotropic solids are𝜆1,2=𝑐11+𝑐12+𝑐332±𝑐11+𝑐12+𝑐3322+2𝑐213,𝜆3=𝑐11𝑐12,𝜆4=𝑐44.(39) The coupled coefficients and stress operators for transversely isotropic solids are calculated as 𝑎𝑖=𝑔𝑖2𝑎11+𝑑𝑖𝑎33𝑎,𝑖=1,2,3=0,𝑎4𝑏=0,𝑖=𝑔𝑖2𝑏11+𝑑𝑖𝑏33𝑏,𝑖=1,2,3=0,𝑏4Δ=0,1,2=𝑔21,2III,Δ3=232II,Δ4=122III+2𝜕12,(40) where, 𝑔𝑖=𝑐13/(𝜆𝑖𝑐11𝑐12)2+2𝑐213,𝑑𝑖=𝜆𝑖𝑐11𝑐12/𝑐3.

It will be seen as follows that the modal strains in transversely isotropic case represent the quasi-dilation and quasi-shear deformation, respectively:𝑆1,2=𝑐13𝜆1,2𝑐11𝑐122+2𝑐213×𝑆11+𝑆22+𝜆1,2𝑐11𝑐12𝑐13𝑆33,𝑆3=12𝑆11+𝑆222+𝑆212,𝑆4=12𝑆232+𝑆231.(41) The speeds of transversely isotropic solids are the following:𝑐1,2=𝜆1,2+𝑎21,2(/𝜈𝜅/𝜈)𝑎1,2+𝑏1,22((𝜅𝜅/𝜈)𝜏)1𝜌𝑠,𝑐3=𝑐11𝑐12𝜌𝑠,𝑐4=𝑐44𝜌𝑠.(42) It is seen that the speeds of quasi-transverse waves are not subject to the thermal effects.

6. Conclusion

In this paper, a new elastic wave model for fluid-saturated porous solid with thermal effects under single-phase assumption is presented, in which the mechanical equations of motion coupled to mass conservation equation and thermoequilibrium equation are considered. A set of uncoupled elastic wave equations for anisotropic porous solid with thermal effects are deduced. The results show that the thermal and compressive coupling coefficients between solid and liquid constituents of system has and shear deformation of solids have not effects on the elastic wave equations and propagation speeds of elastic waves for anisotropic porous solids.

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