Solutions for Biot's Poroelastic Theory in Key Engineering Fields. Theory and Applications PDF
Preview Solutions for Biot's Poroelastic Theory in Key Engineering Fields. Theory and Applications
SOLUTIONS FOR BIOT’S POROELASTIC THEORY IN KEY ENGINEERING FIELDS SOLUTIONS FOR BIOT’S POROELASTIC THEORY IN KEY ENGINEERING FIELDS Theory and Applications YUANQIANG CAI HONGLEI SUN Elsevier Radarweg29,POBox211,1000AEAmsterdam,Netherlands TheBoulevard,LangfordLane,Kidlington,OxfordOX51GB,UnitedKingdom 50HampshireStreet,5thFloor,Cambridge,MA02139,UnitedStates ©2017ElsevierInc.Allrightsreserved. Nopartofthispublicationmaybereproducedortransmittedinanyformorbyanymeans,electronic ormechanical,includingphotocopying,recording,oranyinformationstorageandretrievalsystem, withoutpermissioninwritingfromthepublisher.Detailsonhowtoseekpermission,furtherinformation aboutthePublisher’spermissionspoliciesandourarrangementswithorganizationssuchasthe CopyrightClearanceCenterandtheCopyrightLicensingAgency,canbefoundatourwebsite: www.elsevier.com/permissions. ThisbookandtheindividualcontributionscontainedinitareprotectedundercopyrightbythePublisher (otherthanasmaybenotedherein). 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LibraryofCongressCataloging-in-PublicationData AcatalogrecordforthisbookisavailablefromtheLibraryofCongress BritishLibraryCataloguing-in-PublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary ISBN:978-0-12-812649-3 ForinformationonallElsevierpublicationsvisitourwebsite athttps://www.elsevier.com/books-and-journals Publisher:GlynJones AcquisitionEditor:SimonTian EditorialProjectManager:KatieChan ProductionProjectManager:AnushaSambamoorthy CoverDesigner:VictoriaPearson TypesetbySPiGlobal,India Introduction Allstructuresmadebyhumanbeingshavetobeplacedonorinthesoil.Atthevery beginning,these soil foundations areonly subjectedto static loads,which areloads that buildupgraduallyovertimeorwithnegligibledynamiceffects,alsoknownasmonotonic loads.Inthe1930s,duetotherapiddevelopmentofmachinemanufacturingandtrans- portation industries, the dynamic interaction between structural foundations and the underlying soil behavior under the action of cyclic loads started to receive considerable attentioninanumberofengineeringfields.Cyclicloadsareloadswhichexhibitadegree of regularity both in their magnitude and frequency. Stress reversals, rate effects and dynamic effects are the important factors that distinguish cyclic loads from static loads. Practicallyspeaking,norealcyclicloadsexistinnature;however,manykindsofloads canbesimplifiedintocyclicloadsfortheconvenienceofstudy,analysisanddesign.For example,theoperationofareciprocatingorarotarymachinetypicallyproducesacyclic load.Thepassingofalongtraincanbeconsideredacyclicload.Evencarsrunningona roadonthesamelinecanbesimplifiedasacyclicload.Cyclicloadsactingonthestruc- turesandsoilcanproduceelasticwavesinthegroundwhichwillactonthesurrounding foundations and soil. These actions can cause environmental and safety problems. As a result, it is very important to take a deep look into this area to advance the knowledge regarding the theory of vibrations, the principles of wave propagation, and numerical methods in finding appropriate solutions for problems of practical interest. In the past few decades, many studies have been carried out on soil-structure inter- actionsundercyclicloads.Mostofthemhavetreatedthesoilasanelasticorviscoelastic medium.However,thereisundergroundwaterinwhatisconsideredsoilmedium,such that the soil is actually a two-phase medium. Biot [1] pioneered the development of an elastodynamic theory for a fluid-filled elastic porous medium. Since its publication, Biot’stheoryhashadwideapplicationsinthegeotechnicalprofessionsforanalyzingwave propagationcharacteristicsundercyclicloads.Theaimofthisbookistoprovideatutorial and a state-of-the-art compilation of the advances in the applications of Biot’s theory. vii CHAPTER 1 Basic Equations and Governing Equations This chapterintroduces the basicequations, the governingequations and theboundary conditions of Biot’s theory and its transformed forms, which are essential for solving engineering problems in a fully saturated poroelastic medium. Weusetwokindsofcoordinatesystemsinthisbook,aCartesiancoordinatesystem andacylindricalcoordinatesystem.Therelationshipsbetweenthesecoordinatesystems are given in Table 1.1.1, in which angle θ (0(cid:1)θ(cid:1)2π) is measured from the positive direction of the x-axis to that of the y-axis. In these coordinates, we use (u , u , u ), x y z (ur,uθ,uz)todenotethedisplacementsatapointinthesolidinCartesianandcylindrical coordinatesystems,respectively.Thefootnotesdemonstratealongwhichdirectionthedis- placementoccurs.Wecanalsousetheminmatrixformas{u}¼[ux,uy,uz]Tor[ur,uθ,uz]T, respectively, where the superscript T stands for transpose. We write the stresses and strains in the Cartesian and cylindrical systems, respectively, as 2 3 1 1 ε γ γ 2 3 6 xx 2 xy 2 zx7 σ τ τ 6 7 4ττxxyx στxyyy στzyzx5 and 666612γxy εyy 12γyz7777 zx yz zz 4 5 1 1 γ γ ε 2 zx 2 yz zz 2 3 1 1 ε γ γ 2 3 6 rr 2 rθ 2 zr7 σ τ τ 6 7 4ττrrθr στθrθθ στθzzr 5 and 666612γrθ εθθ 12γθz7777 zr θz zz 4 5 1 1 γ γ ε 2 zr 2 θz zz where σ is the stress tensor, τ is the shear stress, γ is the shear strain, and ε is the ij ij ij ij strain tensor. SolutionsforBiot’sPoroelasticTheoryinKeyEngineeringFields ©2017ElsevierInc. 1 http://dx.doi.org/10.1016/B978-0-12-812649-3.00001-0 Allrightsreserved. 2 SolutionsforBiot'sPoroelasticTheoryinKeyEngineeringFields Table1.1.1 DirectioncosinesbetweencoordinateaxesinCartesianandcylindricalcoordinates x y z r cosθ sinθ 0 θ (cid:3)sinθ cosθ 0 z 0 0 1 The transformation rule between the stresses and strains in different coordinates is σp0q0¼lp0ilq0jσij (1.1.1) εp0q0¼lp0ilq0jεij (1.1.2) whereσijdenotesthestressesinCartesiancoordinates,σp0q0standsforthestressesinanew Cartesiancoordinatesystemafterrotation,andlp0iandlq0jrepresentthedirectioncosines between two coordinate axes. Combining Eq. (1.1.1) with Table 1.1.1, the relationships between the stresses in cylindrical coordinates and Cartesian coordinates can be easily derived: σ ¼σ cos2θ+σ sin2θ+2τ sinθcosθ r x y xy σα¼(cid:2)σxsin2θ(cid:3)+σycos2θ(cid:3)2τx(cid:2)ysinθcosθ (cid:3) τrα¼ σy(cid:3)σx sinθcosθ+τxy cos2θ(cid:3)sin2θ (1.1.3) τ ¼τ cosθ+τ sinθ zr zx yz ταz¼(cid:3)τzxsinθ+τyzcosθ σ ¼σ z z 1.1. BASIC EQUATIONS Thebasicequationsforalinearelasticmediumaregeometricequations(strain-displacement relations),equationsofmotionandconstitutiveequations(stress-strainrelations). 1.1.1 Geometric Equations The geometric equations in Cartesian coordinates are given as: @u @u @u ε ¼ x, γ ¼ y + z x @x yz @z @y @u @u @u ε ¼ y, γ ¼ z + x (1.1.4) y @y zx @x @z @u @u @u ε ¼ z, γ ¼ x + y z @z xy @y @x In cylindrical coordinates, they become BasicEquationsandGoverningEquations 3 ε ¼@ur, γ ¼@uθ +@uz r @r θz @z r@θ εθ¼1r@@uθθ +urr, γzr¼@@urz +@@uzr (1.1.5) ε ¼@uz, γ ¼1@ur +@uθ(cid:3)uθ z @z rα r @θ @r r The tensor form of the geometric equations in Cartesian coordinates can be written as (cid:2) (cid:3) 1 ε ¼ u +u (1.1.6) ij i,j j,i 2 with 2ε ¼γ when i6¼j. ij ij The matrix form of the equations is given as ε¼ETðrÞu (1.1.7) in which u¼[u , u , u ]T and E(r) is an operator matrix defined by x y z 2 3 @ @ @ 6 0 0 0 7 6@x @z @y7 6 7 6 @ @ @ 7 EðrÞ¼66 0 0 0 77 (1.1.8) 6 @y @z @x7 6 7 4 @ @ @ 5 0 0 0 @z @y @x 1.1.2 Equations of Motion Whenthegoverningequationsofadynamicproblemaresetattheequilibriumposition, thebodyforceisomittedintheequations.InCartesiancoordinates,wehavetheequa- tions of motion as @σ @τ @τ @2u x + yx + zx¼ρ x @x @y @z @t2 @σ @τ @τ @2u y + xy + zy¼ρ y (1.1.9) @y @x @z @t2 @σ @τ @τ @2u z + xz + yz¼ρ z @z @x @y @t2 where ρ is the density of the material. Using Eq. (1.1.3), in cylindrical coordinates, the equations of motion become 4 SolutionsforBiot'sPoroelasticTheoryinKeyEngineeringFields @σr +1@τrθ +@τzr +σr(cid:3)σθ¼ρ@2ur @r r @θ @z r @t2 @τrθ +1@σθ +@τθz +2τrθ¼ρ@2uθ (1.1.10) @r r @θ @z r @t2 @τzr +1@τθz +@σz +τzr¼ρ@2uz @r r @θ @z r @t2 The tensor form of the equations of motion in Cartesian coordinates can be written as σ ¼ρu€, ði¼x,y,z or r,θ,zÞ (1.1.11) ij,j i where the dots above the symbols denote partial differentiation with respect to time t. 1.1.3 Constitutive Equations The tensor form of Hooke’s law in Cartesian coordinates can be written as σ ¼c ε , ði,j,k,l¼x,y,z or r,θ,zÞ (1.1.12) ij ijkl kl where c are components of a fourth-rank tensor including 81 components. Since the ijkl stressvectorsaresymmetric,theexchangeoftheindicesiandjdoesnotaltertheresult. Notingthatthestrainvectorsaresymmetricaswell,thesameprocesscanbedonetothe indices k and l; and then we have the relationships c ¼c and c ¼c ijkl jikl ijkl ijlk In addition, since we are considering the adiabatic process, we still have the following relationship: c ¼c ijkl klij Thus, among the 81 components of c , the maximum number of independent ones is ijkl 21.Forahomogeneousmedium,thenumberofindependentcomponentsgoesdownto 2,whicharetheLameconstantλandμ,orYoung’smodulusEandPoisson’sratioν.And the tensor form of Hooke’s law can be simplified as σ ¼2με +λδ ε (1.1.13) ij ij ij kk (cid:4) 1 wheni¼j where δ ¼ ij 0 wheni6¼j BasicEquationsandGoverningEquations 5 Expanding Eq. (1.1.13), we have σ ¼λe+2με x x σ ¼λe+2με y y σ ¼λe+2με z z (1.1.14) τ ¼μγ ¼2με xy xy xy τ ¼μγ ¼2με yz yz yz τ ¼μγ ¼2με zx zx zx νE where λ and μ are Lame’s constant with the relationship λ¼ . ð1+νÞð1(cid:3)2νÞ e¼ε +ε +ε , which is called the volume strain or the matrix dilation. x y z The corresponding matrix form of the equations is given as σ¼Cε (1.1.15) whereσandεarevectorsofstressandengineeringstrain,respectively.InCartesiancoor- dinates, they become h i T σ¼ σ ,σ ,σ ,γ ,γ ,γ x y z xy yz zx h i T ε¼ ε ,ε ,ε ,γ ,γ ,γ x y z xy yz zx And C should be a nonsingular and reversible matrix, which can be written as 2 3 λ+2μ λ λ 0 0 0 6 7 6 λ λ+2μ λ 0 0 07 6 7 6 λ λ λ+2μ 0 0 07 C¼66 77 (1.1.16) 6 0 0 0 μ 0 07 6 7 4 0 0 0 0 μ 05 0 0 0 0 0 μ 1.2. GOVERNING EQUATIONS OF A FULLY SATURATED POROELASTIC MEDIUM Thissectionintroducesthegoverningequationsforafullysaturatedporoelasticmedium. 1.2.1 Governing Equations in Cartesian Coordinates Asfor modeling thedynamic responsesof a saturated porous medium, Biot [3] was the first to give the theory that presents three kinds of coupling (viscous, inertial, and mechanical) between the porous solid skeleton and pore fluid, and demonstrated that 6 SolutionsforBiot'sPoroelasticTheoryinKeyEngineeringFields two kinds of longitudinal waves (P1 and P2 waves) and one kind of rational wave (the Swave)existinthesaturatedporousmedium.TheexistenceoftheP2wavealwaysdis- tinguishesthetwo-phasemediumfromthesingle-phaseone.ThebasicvariablesofBiot’s theory are always the solid skeleton displacement (u) and the average displacement of pore fluid relative to the solid skeleton (w). Consideringtheconceptof effectivestressof thesaturated mixture,therelationship between effective stress, total stress and pore pressure can be expressed as σ0 ¼σ +αδ p (1.1.17) ij ij ij whereσ 0istheeffectivestresstensor,σ isthetotalstresstensor,δ istheKroneckerdelta, ij ij ij and α is the Biot constant that depends on the geometry of material voids. For the most part, in soil mechanics problems, α(cid:4)1 can be assumed. The relationship between total stress and effective stress becomes σ0 ¼σ +δ p (1.1.18) ij ij ij whichcorrespondstotheclassicaleffectivestressdefinitionbyTerzaghi.Thusthetensor formoftheequationsofmotionforafullysaturatedporoelasticmediumbecomes(omit- ting the body force): σ ¼ρu€+ρ w€ (1.1.19) ij,j i f i where the dots above the symbols denote partial differentiation with respect to time t; thus u€ is the acceleration of the solid part, w is the fluid displacement relative to i i thesolidpart,andw€ isthefluidaccelerationrelativetothesolidpart.Forfullysaturated i porous media (no air trapped inside), density is equal to ρ¼nρ +ð1(cid:3)nÞρ, where n is f s the porosity, and ρ and ρ are the soil particle and water densities, respectively. s f For the pore fluid, the equation of momentum balance can be expressed as p ¼(cid:3)bw_ (cid:3)ρ u€(cid:3)mw€ (1.1.20) ,i i f i i wheretheparameterb¼ρg/k ,wherek istheDarcypermeabilityofthesoilmedium f D D and g is the gravity; p is the pore water pressure. According to the classical effective stress definition by Terzaghi in Eq. (1.1.17), the constitutive Eq. (1.1.13) becomes σ ¼2με +λδ ε (cid:3)αδ p (1.1.21) ij ij ij kk ij The final equation is the mass conservation of the fluid flow, which is expressed by p_¼(cid:3)αMe_+Mς_ (1.1.22) where ς¼(cid:3)w (1.1.23) i,i
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