Thermal Effect on Elastic Waves of Anisotropic Saturated Porous Solid

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 [1–3]. 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 [6–13] 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:𝜎_𝑖𝑗𝑛==𝑐𝜈𝑝^{𝑠𝑖𝑗𝑘𝑙}𝜀+^𝑠𝑘𝑙𝜅+𝜃𝑎−_𝑖𝑗𝑎𝑝−_𝑘𝑙𝑏_𝑖𝑗𝜃,𝜂=𝜏𝜃+𝑏_𝑘𝑙𝜀^𝑠𝑘𝑙,𝜀^𝑠𝑘𝑙+𝜅𝑝,(1) where, 𝑝 is the pore pressure, 𝜃 is the temperature, 𝑐^{𝑠𝑖𝑗𝑘𝑙} elastic tensor of solid constituent of porous material, 𝜈 compressive coefficient of porous material, 𝜏 heat coefficient, 𝑎_𝑖𝑗 coupling coefficient between solid constituent and fluid one, 𝑏_𝑖𝑗 thermal coupling coefficient, 𝜅 coupling coefficient between porous deformation and heat, 𝑛 porosity, 𝜂 entropy.

Rewriting (1) in the Voigt’s notation, we have 𝑛𝝈==𝜈𝐜⋅^𝑠𝑝⋅𝐒+^𝑠𝜅+𝜃𝐚−⋅𝑝𝐚_𝑇−𝐛𝜃,(2)𝜂=𝜏𝜃+𝐛_𝑇⋅𝐒^𝑠,(3)⋅𝐒^𝑠+𝜅𝑝.(4) The skeleton elastic matrix 𝐜^𝑠 can be spectrally decomposed as follows [6–9]:𝐜^𝑠=𝚽𝚲𝚽^𝑇,(5) where Λ=diag[𝜆₁,𝜆₂,𝜆₃,𝜆₄,𝜆₅,𝜆₆] are the matrixes of eigen elasticity [10–13] of porous skeleton. Φ={𝝋₁,𝝋₂,𝝋_𝟑,𝝋₄,𝝋₅,𝝋₆} are the modal matrixes, which are both orthogonal and positive definite matrixes, and satisfy Φ^𝑇Φ=𝐈.

Projecting the elastic physical qualities of the geometric presentation, such as the stress vector 𝝈 and strain vector 𝐒^𝑠, into the standard spaces of the physical presentation, we get𝐒𝝈^∗=𝚽^𝑇⋅𝝈,_∗=𝚽^𝑇⋅𝐒^𝑠.(6) Rewriting (6) in the form of scalar, we have𝑆𝜎^∗𝑖=𝝋^∗𝑖𝑇⋅𝝈,𝑖=1−𝑚,(7)_∗𝑖=𝝋^∗𝑖𝑇⋅𝐒^𝑠,𝑖=1−𝑚,(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 have𝚽^𝑇𝝈𝑛==𝚽𝜈^𝑇⋅𝐜𝑝^𝑠𝚽+𝐒𝜅^∗⋅+𝜃𝚽−^𝑇𝐚𝐚_𝑇𝑝−𝚽^𝑇𝐛𝜃,𝜂=𝜏⋅𝜃+𝐛_𝑇𝚽𝐒^∗,𝚽𝐒^∗+𝜅⋅𝑝.(9) Using (5), (6), we get𝑛𝝈^∗==𝜈𝑝𝚲+𝐒^∗𝜅+𝜃𝐚−^∗𝑝𝐚−_∗𝑇𝐛^∗𝜃,𝜂=𝜏𝜃+𝐛_∗𝑇⋅𝐒^∗,⋅𝐒^∗+𝜅𝑝.(10) Rewriting the above equations in the form of scalar, we have𝑛=𝜎𝜈^∗𝑖𝑝=+𝜆𝜅_𝑖𝜃𝑆−^∗𝑖+𝑎_∗𝑘𝑎^∗𝑖𝑝−𝑏^∗𝑖𝜃,𝑖=1−𝑚,𝜂=𝜏𝜃+𝑏_∗𝑘𝑆^∗𝑘,𝑘=1−𝑚sumto𝑘,𝑆^∗𝑘+𝜅𝑝,𝑘=1−𝑚sumto𝑘.(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: 𝜕𝑛𝛾_𝑤𝜕𝑡+^⇀∇⋅𝛾𝑉_𝑤⇀=0,(12) where, 𝑛 is porosity, 𝛾_𝑤 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=𝑑𝑠_𝑒+𝑑𝑠_𝑖=0,(14) where 𝑠 is total entropy of system. By using the entropy density, the above can be written as follows𝜌𝜂̇=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 [6–8]:Δ^∗𝑖𝜎^∗𝑖=𝜌_𝑠∇_𝑡𝑡𝑆^∗𝑖,𝑖=1−𝑚,(16) where, Δ^∗𝑖={𝜑^∗𝑖}^𝑇[Δ]{𝜑^∗𝑖} is the stress operator, in which[Δ]=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝜕¹¹000𝜕₃₁𝜕₂₁0𝜕₂₂0𝜕₃₂0𝜕₂₁00𝜕0₂₃𝜕₃₃𝜕₃₂𝜕₃₁0𝜕₂₃𝜕₂₂+𝜕₃₃𝜕₂₁𝜕₃₁𝜕₁₃0𝜕₁₃𝜕₁₂𝜕₁₁+𝜕₃₃𝜕₃₂𝜕₁₂𝜕₁₂0𝜕₁₃𝜕₂₃𝜕₂₂+𝜕₁₁⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,(17) where 𝜕_𝑖𝑗=𝜕_𝑗𝑖=𝜕²/𝜕𝑥_𝑖𝜕𝑥_𝑗.

4. Modal Equation of Elastic Waves with Thermal Effects

Substituting (11) into (13), (15), and (16), we haveΔ^∗𝑖𝜆_𝑖𝑆𝜈𝑝^∗𝑖++𝜅𝑎^∗𝑖𝜃𝑝−−𝑎𝑏^∗𝑖_∗𝑘𝜃=𝜌_𝑠∇_𝑡𝑡𝑆^∗𝑖,(18)𝜏𝜃+𝑏_∗𝑘𝑆^∗𝑘=0,(19)𝑆^∗𝑘+𝜅𝑝=0.(20) From (19), we can get the following equation based on the operator principle:𝑝=𝑎^∗𝑘𝜈𝑆^∗𝑘−𝜅𝜈𝜃.(21) Substituting it into (18) and (20), respectively, we haveΔ^∗𝑖𝜆^𝑖+𝑎^∗𝑖𝛿_𝑖𝑘𝑎^∗𝑘𝜈𝑆^∗𝑖−𝜅𝜈𝑎^∗𝑖𝜅𝜅+𝑏^∗𝑖𝜃=𝜌^𝑠∇_𝑡𝑡𝑆^∗𝑖,(22)𝜈−𝜏𝜃−𝑏^∗𝑖+𝜅𝜈𝑎^∗𝑖𝑆^∗𝑖=0.(23) Substituting (23) into (22) again, we getΔ^∗𝑖𝜆^𝑖+𝑎^∗𝑖𝛿_𝑖𝑘𝑎^∗𝑘𝜈−𝜅𝜈𝑎^∗𝑖×𝜅𝜅+𝑏^∗𝑖𝜈−𝜏⁻¹𝑏^∗𝑖+𝜅𝜈𝑎^∗𝑖𝑆^∗𝑖=𝜌_𝑠∇_𝑡𝑡𝑆^∗𝑖.(24) Rewriting it in the standard form of elastic waves, we haveΔ^∗𝑖𝑆^∗𝑖=1𝑐^2𝑖∇_𝑡𝑡𝑆^∗𝑖,𝑖=1−𝑚,(25) where𝑐_𝑖=⎷𝜆^𝑖+𝑎^∗𝑖𝛿_𝑖𝑘𝑎^∗𝑘/𝜈−(𝜅/𝜈)𝑎^∗𝑖+𝑏^∗𝑖²((𝜅𝜅/𝜈)−𝜏)⁻¹𝜌_𝑠.(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 notation⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑐𝑐₁₁𝑐₁₂𝑐₁₂000₁₂𝑐𝑐₁₁𝑐₁₂000₁₂0𝑐0₁₂𝑐0₁₁𝑐0₆₆0000𝑐0₆₆000000𝑐0₆₆⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,⎡⎢⎢⎢⎢⎣𝑎₁₁0𝑎0₁₁000𝑎0₁₁⎤⎥⎥⎥⎥⎦,⎡⎢⎢⎢⎢⎣𝑏0₁₁𝑏0₁₁000𝑏0₁₁⎤⎥⎥⎥⎥⎦,(27) where 𝑐₆₆=(𝑐₁₁−𝑐₁₂).

There are two independent eigenspaces in an isotropic solids [6–9]𝑊=𝑊₁₍₁₎𝝋₁⊕𝑊₂₍₅₎𝝋₂,…,𝝋₆,(28) where𝝋₁=√33[1,1,1,0,0,0]^𝑇,𝝋₂=√22[0,1,−1,0,0,0]^𝑇𝝋₃=√66[2,−1,−1,0,0,0]^𝑇,𝝋_𝑖=𝝃_𝑖(𝑖=4,5,6)(29) and 𝜉_𝑖 is a vector of order 6 in which the 𝑖th element is 1 and others are 0.

The eigenelasticity of isotropic solids are𝜆₁=𝜆𝑐₁₁+2𝑐₁₂=3𝜆+2𝜇,₂=𝑐₁₁−𝑐₁₂=2𝜇,(30) where 𝜆 and 𝜇 are Lame’s constants.

The coupled coefficients and stress operators for isotropic solids are calculated as𝑎𝑏^∗1=√3𝑎₁₁,𝑎^∗2=0,_∗1Δ=√3𝑏₁₁,𝑏^∗2=0,_∗1=13∇^2III,Δ^∗2=12∇^2III,(31) where ∇²_III 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 is𝑆^∗1=𝝋^1∗𝑇⋅𝐒=√33𝑆¹¹+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 is𝑆^∗2𝝋^∗2=𝐒−𝑆^∗1𝝋^∗1.(33) Using the orthogonality condition of eigenvectors, we have||𝑆^∗2||=𝐒−𝑆^∗1𝝋^∗1^𝑇⋅𝐒−𝑆^∗1𝝋^∗1^1/2=13𝑆¹−𝑆₂²+𝑆₂−𝑆₃²+𝑆₃−𝑆₁²^1/2.(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=⎷𝐾+𝑎²¹¹/𝜈−(𝜅/𝜈)𝑎₁₁+𝑏₁₁²((𝜅𝜅/𝜈)−𝜏)⁻¹𝜌_𝑠𝑐₂,=𝐺𝜌_𝑠,(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:⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑐𝑐₁₁𝑐₁₂𝑐₁₃000₁₂𝑐𝑐₁₁𝑐₁₃000₁₃0𝑐0₁₃𝑐0₁₃𝑐0₄₄0000𝑐0₄₄000000𝑐0₆₆⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,⎡⎢⎢⎢⎢⎣𝑎₁₁0𝑎0₁₁000𝑎0₃₃⎤⎥⎥⎥⎥⎦,⎡⎢⎢⎢⎢⎣𝑏0₁₁𝑏0₁₁000𝑏0₃₃⎤⎥⎥⎥⎥⎦,(36) where 𝑐₆₆=(1/2)(𝑐₁₁−𝑐₁₂).

There are four independent eigenspaces in an transversely isotropic solids [6–9] as follows:𝑊=𝑊₁₍₁₎𝝋₁⊕𝑊₂₍₁₎𝝋₂⊕𝑊₃₍₂₎𝝋₃,𝝋₆⊕𝑊₄₍₂₎𝝋₄,𝝋₅,(37) where𝝋_𝟏,𝟐=𝑐¹³𝜆_1,2−𝑐₁₁−𝑐₁₂²+2𝑐₂₁₃×1,1,𝜆^1,2−𝑐₁₁−𝑐₁₂𝑐₃,0,0,0^𝑇𝝋_𝟑=√22[1,−1,0,0,0,0]^𝑇,𝝋_𝐢=𝝃_𝑖,𝑖=4,5,6.(38) The eigenelasticities of transversely isotropic solids are𝜆_1,2=𝑐¹¹+𝑐₁₂+𝑐₃₃2±𝑐¹¹+𝑐₁₂+𝑐₃₃2²𝜆₃+2𝑐₂₁₃,=𝑐₁₁−𝑐₁₂,𝜆₄=𝑐₄₄.(39) The coupled coefficients and stress operators for transversely isotropic solids are calculated as 𝑎^∗𝑖=𝑔_𝑖𝑎2𝑎₁₁+𝑑_𝑖𝑎₃₃,𝑖=1,2,_∗3𝑏=0,𝑎^∗4=0,_∗𝑖=𝑔_𝑖𝑏2𝑏₁₁+𝑑_𝑖𝑏₃₃,𝑖=1,2,_∗3Δ=0,𝑏^∗4=0,_∗1,2=𝑔^21,2∇^∗_III,Δ^∗3=23∇^2II,Δ^∗4=12∇^2III+2𝜕₁₂,(40) where, 𝑔_𝑖=𝑐₁₃/(𝜆_𝑖−𝑐₁₁−𝑐₁₂)²+2𝑐₂₁₃,𝑑_𝑖=𝜆_𝑖−𝑐₁₁−𝑐₁₂/𝑐₃.

It will be seen as follows that the modal strains in transversely isotropic case represent the quasi-dilation and quasi-shear deformation, respectively:𝑆_∗1,2=𝑐¹³𝜆_1,2−𝑐₁₁−𝑐₁₂²+2𝑐₂₁₃×𝑆₁₁+𝑆₂₂+𝜆^1,2−𝑐₁₁−𝑐₁₂𝑆_3∗𝑐₁₃𝑆³³,=12𝑆¹¹𝑆_4∗+𝑆₂₂²+𝑆²¹²,=12𝑆²³²+𝑆²³¹.(41) The speeds of transversely isotropic solids are the following:𝑐_1,2=⎷𝜆^1,2+𝑎^∗21,2/𝜈−(𝜅/𝜈)𝑎^∗1,2+𝑏^∗1,2²((𝜅𝜅/𝜈)−𝜏)⁻¹𝜌_𝑠𝑐₃,=𝑐₁₁−𝑐₁₂𝜌_𝑠𝑐₄,=𝑐₄₄𝜌_𝑠.(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.