Solutions for Biot's Poroelastic Theory in Key Engineering Fields. Theory and Applications

174 Pages·2017·9.294 MB·English

by  Yuanqiang Cai and Honglei Sun (Auth.)

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
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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