JTAM 3 2012 - Българска академия на ... · presence of discontinuities in...

Journal of Theoretical and Applied Mechanics, Sofia, 2012, vol. 42, No. 3, pp. 75–100

COMBINED

FIELDS

ELASTIC WAVEFIELD EVALUATION IN

DISCONTINUOUS POROELASTIC MEDIA BY BEM:

SH-WAVES*

Petia S. Dineva

Institute of Mechanics, Bulgarian Academy of Sciences,

Acad. G. Bonchev St., Bl. 4, 1113 Sofia, Bulgaria,

e-mail:[email protected]

Frank Wuttke

Chair of Geomechanical Modeling, Bauhaus University

99421 Weimar, Germany,


George D. Manolis

Department of Civil Engineering, Aristotle University,

54124 Thessaloniki, Greece,


[Received 21 April 2012. Accepted 14 May 2012]

Abstract. This work examines the anti-plane strain elastodynamicproblem for poroelastic geological media containing discontinuities in theform of cavities and cracks. More specifically, we solve for: (i) a modeIII crack; (ii) a circular cylindrical cavity, both embedded in an infiniteporoelastic plane; and (iii) a mode III crack in a finite-sized poroelasticblock. The source of excitation in all cases are time-harmonic, horizon-tally polarized shear (SH) waves. These three cases depict a situationwhereby propagating elastic waves are diffracted and scattered by thepresence of discontinuities in poroelastic soil, and this necessitates thecomputation of stress concentration factors (SCF) and stress intensityfactors (SIF). Thus, the sensitivity of the aforementioned factors to vari-ations in the material parameters of the surrounding poroelastic contin-uum must be investigated. Bardet’s model is introduced by assumingsaturated soils as the computationally efficient viscoelastic isomorphism

*The first author (PSD) wishes to acknowledge financial support provided through theMC-IEF Grant No. PIEF-GA-2010-270889. Also, the other two authors (FW, GDM) ac-knowledge the NATO Collaborate Linkage Grant No. CLG-984136.

Corresponding author e-mail: [email protected]

76 Petia S. Dineva, Frank Wuttke, George D. Manolis

to Biot’s equations of dynamic poroelasticity, and stress fields are thenevaluated for an equivalent one-phase viscoelastic medium. The compu-tational tool employed is an efficient boundary element method (BEM)defined in terms of the non-hypersingular, traction-based formulation.Finally, the results obtained herein demonstrate a marked dependenceof the SIF and the SCF on the mechanical properties of the poroelasticcontinuum, while the advantages of the proposed method as compared toalternative analytical and/or numerical approaches are also discussed.Key words: SH-waves, cavities, cracks, poroelasticity, Bardet’s model,anti-plane strain, stress concentration factor, stress intensity factor, bound-ary element method.

1. Introduction

Elastic wave propagation phenomena in geological media involve a num-ber of important factors such as material inhomogeneity, the presence of a sec-ond liquid phase in the soil, the presence of discontinuities, the existence ofburied seismic sources, and finally the presence of a free-surface relief withinlayered soil deposits. Various numerical simulations and measurements haveclearly demonstrated the influence of all the above factors on the resultingscattered wave field, and identified the generation of localized stress concen-tration phenomena [1]. In here, we examine time harmonic, anti-plane strainwave motion in poroelastic geological media with discontinuities in the formof cylindrical cavities and mode III cracks. The purpose is to quantify the in-fluence of soil poroelasticity on the SCF that develops at the cavity perimeterand of the SIF at the crack tips. The soil is approximated as a single-phasematerial with mechanical properties computed on the basis of the viscoelastic-poroelastic similarity.

Dynamic stress concentrations around discontinuities comprise an im-portant class of problems in solid mechanics [2]. Diffraction of seismic wavesaround buried cavities and geological cracks has also been a research topic ofinterest to earthquake engineering [3]. Elastostatic analysis of these problems,as given in the classical treatises of Neuber [4], Muskhelishvili [5], Savin [6]and Timoshenko and Goodier [7] is valid only when mass inertia effects areunimportant. In recent years, research activity on dynamic stress concentra-tion has greatly accelerated, resulting in a plethora of papers starting withthe works of Pao and Mow [2], Achenbach [8], Miklowitz [9] and Zhang andGross [10] among others. Various specialized computational methods were alsodeveloped and are classified here as follows: (a) quasi-analytical methods suchas the wave function expansion [2], the matched asymptotic expansion [11],and the complex function method [12–14]; (b) numerical methods such as var-

Elastic Wavefield Evaluation . . . 77

ious boundary element techniques, see Refs. [15–20]. The benchmark problemof an embedded cavity in the infinite space engulfed by body waves has beensolved by Pao and Mow [2], Datta and Shah [11] and Kobayashi and Nishimura[15] among others. Another such benchmark, namely the cavity and/or tunnelburied in a half-space has been studied by various researchers, see Refs. [15,17, 21]; the case of multiple circular holes can be found in Providakis et al.[19]. Finally, a comprehensive study of dynamic fracture in elastic isotropichomogeneous media containing cracks can be found in Zhang and Gross [10].

The problems of computing stress concentration effects becomes muchmore difficult when discontinuities occur in two-phase media such as water-saturated soils. As a consequence, relatively few results have been obtained,such as those of Kattis et al. [22] and Wang et al. [23], who studied diffrac-tion of elastic waves around cavities in a poroelastic medium, and of Lianget al. [24–26] who presented results for a cavity in a poroelastic half-spaceunder incident pressure and shear waves based on Biot’s [27] theory. The be-haviour of cracked poroelastic media is recognized as an area of high interestin geomechanics although somewhat ignored from a research viewpoint andin applications such as energy recovery from geological formations, see Refs.[28–30].

In a series of papers, Bardet [31, 32] demonstrated the applicability ofthe viscoelastic behaviour equivalent to Biot’s [27] model of dynamic poroelas-ticity (i.e., viscoelastic isomorphism) in the development of computational tech-niques for a well defined range of material parameters and in the low frequencyresponse range. More specifically, viscoelastic isomorphism was used by Mo-rochnik and Bardet [33] in the solution of wave scattering problems involvingpressure waves by a spherical poroelastic inhomogeneity, and results obtainedby both Biot and Bardet models were nearly identical in regions away from theimmediate vicinity of the inhomogeneity. More recently, a series of solutionsconducted by the authors [34] using the Bardet model show nearly identicalresults with those obtained by the Biot’s model for some benchmark type ofboundary-value problems (BVP).

In the following, the advantages and disadvantages of viscoelastic iso-morphism of Bardet are briefly discussed, since this is the preferred constitutivelaw for a boundary element method (BEM) treatment of elastic wave propa-gation problems.

(1) The main advantages of viscoelastic isomorphism is a mechanicalsimplicity (a one-phase instead of two-phase material) and a mathematicalsimplicity in terms of the resulting wave equation and its fundamental solu-tion. These allow for approximate solutions of complex BVP on seismic wave


propagation in fluid-saturated strata.

(2) The limitations of the Bardet model are (a) it is valid in the lowfrequency range, which however is typical for seismic waves, (b) it cannot ac-count for the second (slow) longitudinal wave that dampens out very fast;(c) it is not possible to account for boundary conditions involving pore fluidpressure; and (d) the consolidation process due to fluid-skeleton interaction isexpressed by a time-dependent deformation process that cannot be modelledby viscoelasticity. More specifically, fracture in poroelastic media is governedby a time-dependent pore pressure diffusion at the crack-tips, coupled with acorresponding time-dependent deformation of the soil skeleton. This processyields a SIF that depends on the pore pressure boundary conditions along thecrack faces.

We conclude adding up these limitations, that modelling fracture be-haviour of poroelastic media with viscoelastic isomorphism hinges on the use ofthe equivalent material parameters, which capture the basic characteristics ofthe problem but cannot account for time-dependent interaction between solidand fluid.

The basic BEM formulation using the linear elastic, anisotropic con-stitutive law has been presented in recent publications by the authors [35,36] for anti-plane and in-plane strain problems. The novelty here is an ex-tension of this BEM formulation to water-saturated soil media described byBardet’s viscoelastic isomorphism model. Briefly, the paper is structured as fol-lows: Section 2 defines the BVP for elastic waves propagating under anti-planestrain conditions in a poroelastic region containing discontinuities in the formof cracks and cavities. Next, the computational BEM model is discussed inSection 3, while Section 4 presents validation examples plus numerical resultsfrom a series of parametric studies. Finally, the paper ends with interpretationof the obtained results and a list of conclusions and a blueprint for future workin this area .

2. Problem formulation

2.1 Governing equation of motion and boundary conditions

In general, a mechanical model appropriate for soil-structure-interaction(SSI) phenomena comprises a finite local geological inclusion with discontinu-ities, embedded in a stack of soil layers, which in turn rests on the homogeneoushalf-space [37]. In here, however, we examine a series of sub-problems whichcomprise the general case, so as to separately examine the influence of discon-tinuities on SH-waves propagating through a poroelastic continuum. In the


Cartesian coordinate system Oxyz we consider a finite geological region withexternal boundary S swept by time harmonic SH-waves with a frequency ω(rad/sec), as shown in Fig. 1. The material is homogeneous, isotropic and wa-ter saturated with its mechanical properties described by Bardet’s [31] model,which represents poroelasticity as a single-phase, Kelvin-Voigt viscoelastic ma-terial, as discussed in the next section. The material now possesses complex-valued Lame constants λ∗, µ∗ and ρ is the (real) density based on the corre-spondence principle of viscoelasticity [38].

Fig. 1. Finite geological region with boundaryS containing discontinuities, such as amode III crack with boundary Γc and a cylindrical cavity with boundary ΓH , under

incident SH waves

Frequency domain viscoelastic BEM for solving dynamic problems wasused by Manolis and Beskos [39] in connection with the differential operatorform of the constitutive equations, by Kobayashi [40] in connection with theintegral form of the constitutive equations, and by Abascal and Dominguez[41] and Beskos et al. [42] in connection with the simple hysteretic dampingmodel.

Non-zero field quantities for the anti-plane case with respect to theplane y = 0 are a displacement component uy(x, z) and two shear stresscomponents σyx(x, z), σyz(x, z). The corresponding traction is defined aspy = σyxnx + σyznz, where (nx, nz) is the outward pointing normal vectorat all surfaces, namely the external boundary of the finite domain, the crackline and the cavity perimeter. The common multiplier exp(iωt) is suppressedthrough-out the analysis since time-harmonic response is assumed. As a con-sequence, the governing equation of motion in the frequency domain simplifies


to:

(2.1) σiy,i + µ∗k2SHuy = 0, i = x, z

In the above, kSH =ω

cSHis the SH wave number with phase velocity

cSH , subscript commas denote spatial derivatives, vector quantities are denotedthrough the use of indices (i, j = x, z) and the summation convention overrepeated indices is implied.

The boundary conditions prescribed along the outer surface S is eithera displacement uy(x) = uy(x) for x ∈ SU or a traction py(x) = py(x) forx ∈ SP , where S = SU ∪ SP and SU ∩ SP = ∅. The region may contain cracksand/or cavities (holes), in which case traction-free conditions are satisfied alongthe crack lines Γc = Γ+c ∪Γ−c and hole perimeter ΓH . Sommerfeld’s radiationcondition must also be satisfied if an infinite region is under consideration topreclude waves moving into the core region from infinity.

2.2. Bardet’s viscoelastic isomorphism

In poroelasticity, we define a representative volume V for the solid-fluid continuum, which comprises an elastic, isotropic solid skeleton (matrix)with porosity n = VP /V , where VP is the pore volume. This volume may befilled with a fluid (water) or may be empty (vacuum). We consider here fullysaturated soil, which is a two-phase material since all pores are filled with fluid.Unsaturated soils are three-phased materials since air is also present in thepores, but this is beyond the scope of this work. The “dry rock” approximationis the case of an air-filled solid skeleton, while the “solid grain” characteristicsare the properties of the skeleton material. The following terminology is nowused for the components of the two-phase system: (a) dry rock (or soil), (b)solid grain and (c) fluid. The elastic bulk modulus and density respectively areKdry, ρdry = (1 − n)ρg; Kg, ρg; Kf , ρf , while the solid-fluid system density isρsat = ρdry + nρf = (1 − n)ρg + nρf . Next, the shear strength of the porousmaterial is provided by the solid skeleton and is not affected by the fluid, sincefluids sustain dilatational deformations only. Thus, both dry and saturatedsoils are described by the same shear modulus, namely µ = µdry = µsat =3(1 − 2ν)

2(1 + ν)Kdry, and ν is Poisson’s ratio for the dry skeleton.

The equation governing time-harmonic wave propagation in fluid-sa-turated soils considered as two-phase materials was derived by Biot [27]. Acharacteristic equation for the wave numbers can be obtained [31, 43] by sub-stituting the plane wave expansions for the displacements into Biot’s wave


equation without body forces. Three solutions to Biot’s wave equation havebeen identified, corresponding to (a) a shear wave S transmitted through thesolid skeleton, (b) a fast dilatational wave P1 and (c) a slow dilatational waveP2. The corresponding wave velocities are complex and frequency dependent,hence they correspond to dissipative and dispersive waves. Finally, the solidand fluid dilatations are in phase for the first arriving P1 wave, and in re-verse phase for the slower P2 wave, which damps out quickly. As mentionedpreviously, Bardet [31] proposed a single-phase viscoelastic material represen-tation for saturated soils. At first, the viscoelstic Lame constants are complex-valued with the same real part as in their elastic counterparts. The same holdstrue for the governing viscoelastic wave equation, with wave numbers whichare complex-valued, frequency dependent functions that satisfy causality con-ditions. More specifically, for a Kelvin-Voigt model, the wave numbers are

k2P =

ω2

c2P

(1 − iωξP ); k2

S =ω2

c2S

(1 − iωξS), where c

P, c

Sare (approximately) the

real part of the P- and S-wave velocities, while ξP ; ξS are the correspondingattenuation coefficients representing a small amount of hysteretic damping. Inthe low frequency range, i.e., ωξ ≪ 1, the wave numbers reduce to:

(2.2) kP ≈ ω (1 + 0.5ωξP )

cP

; kS ≈ ω (1 + 0.5ωξS)

cS

,

Bardet [31] introduced a poroelastic-viscoelastic isomorphism by equat-ing the wave numbers in Biot’s poroelastic model with the above viscoelasticones. This process gives the following equivalence relations between poroelasticand viscoelastic materials:

(2.3) cP

=

√

P + 2Q + R

ρsat

; cS

=

√µ

ρsat

;

(2.4) ξP =ρsat

b

(Q + R

P + 2Q + R· nρf

ρsat

)2

; ξS =ρsat

b

(nρf

ρsat

)2

.

We now have the various poroelastic material parameters related asfollows:

P =3 (1 − ν)

1 + νKdry +

Q2

R; Q =

n (1 − n − Kdry/Kg)

1 − n − Kdry/Kg + nKg/Kf

Kg;


R =n2Kg

1 − n − Kdry/Kg + nKg/Kf ,

(2.5)

N =3

2

1 − 2ν

1 + νKdry; Kdry =

2

3

µ (1 + ν)

1 − 2ν;

λsat = λdry +Q2

R=

3ν

1 + νKdry +

Q2

R.

Furthermore, b = n2gρf/k is the viscous dissipation coefficient, g is the

acceleration of gravity and k is the soil permeability with values in the range10−10 − 10−2 m/ sec.

Morochnik & Bardet [33] obtained the above approximate expressionsfor frequency values satisfying the condition (ω ρsat/b) << 1. This conditionis easily fulfilled for the frequency range considered important in earthquakeengineering since permeability values for most soils is small (e.g., for sandk = 10−6−10−4m/sec). The proposed model can also account for soil stiffening,in as much as the pore pressure induced by seismic loads helps in resisting thecompressive loads. This can be ascertained by inspecting Eq. (2.3) for the Pand S wave velocities, and Eq. (2.5) that shows increased λsat values comparedto λdry. The equivalent model also predicts changes in the damping mechanismfor a poroelastic material. This is clearly evident from Eq. (2.4) and (2.5),where phase velocities and attenuation coefficients for the propagating wavesdepend on porosity, bulk modulus of the dry skeleton, solid grain and fluid,and on soil hydraulic conductivity.

Finally, all numerical simulations presented in section 4 are for sand-stone, with Kg = 36.000 MPa; ρg = 2.650 kg/ m3; Kf = 2.000 MPa; ρf = 1.000kg/m3; see Lin et al. [43], who used the original 3D Biot model applied to thehomogeneous half-plane. The soil dry bulk modulus is calculated using an an-alytical expression derived from experimental data, with Kcr = 200 MPa beingthe bulk modulus at critical porosity ncr = 0.36:

(2.6) Kdry = Kcr + (1 − n/ncr) (Kg − Kcr) .

An earlier validation study of Bardet’ model [34] showed the wave fieldthat develops in the homogeneous poroelastic half-plane is nearly identical withthe solution obtained by using the Biot model. We demonstrate in Fig. 2 thedependence of the sandstone shear wave phase velocity of Eq. (2.3), for both


Fig. 2. SH-wave velocity variation with porosity for dry and saturated soil ascomputed by Bardet’s model

dry and saturated conditions, on the porosity n, where we observe the rapiddrop of cSH as the porosity increases.

2.3. Fundamental solution of the governing equation

The key role played by the fundamental solution in a BEM formulationis to reduce a given BVP to a system of boundary integral equations throughuse of a reciprocal theorem [44]. The displacement fundamental solution U∗

y

for the SH-wave problem satisfies the following partial differential equation:

(2.7) σ∗

iy,i + µ∗k2SHU∗

y = −µ∗δ(ζ − ζ0); i = x, z

In the above, the viscoelastic stress is defined as σ∗

iy = µ∗U∗

y,i andδ(ζ − ζ0) is the Dirac (generalized) delta function for a field and source pointconfiguration. This fundamental solution for the general anisotropic mediumwas derived in Manolis et al. [35] using the Radon transform. More specifically,after application of this transformation to Eq. (2.7), an ordinary differentialequation is obtained and the fundamental solution U∗

y is derived in a closedform, followed by the inverse Radon transform. Problems involving crackedanisotropic media have been successfully solved in the past by the authors [35,36], who also obtained results by Radon-type fundamental solutions.


Fig. 3. Scattering of SH- waves by a mode III crack of length 2c in the Oxz plane

3. BEM computational model

We use the non-hypersingular, traction-based BEM in the presence ofcracks with the frequency-dependent fundamental solution obtained in closedform by the Radon transform, as previously mentioned. This type of computa-tional method was first proposed for elastic, isotropic and homogeneous crackedsolids by Zhang and Gross [10] based on the J-integral for time-harmonic elas-todynamics. The reason it becomes necessary to switch to a non-hypersingulartraction BEM is because the conventional displacement-based BEM degener-ates in the presence of crack lines.

A BEM formulation for a bounded domain that is equivalent to theBVP defined by Eq. (2.1) plus the relevant boundary conditions is obtainedas follows: First, the boundary of a given problem is represented as Γ =S ∪ ΓH ∪ Γc, where S is the external boundary of the continuous mediumcontaining a cavity and/or a crack. Note, that a crack line is specified asΓc = Γ+c ∪ Γ−c, where Γ+c and Γ−crepresent the upper and lower lines of thecrack, see Fig. 3. The non-hypersingular traction BEM is defined as:

(3.1)

1

2p0

y(ζ) = µ∗ni(ζ)

∫

S

[(σ∗

η y(ζ, ζ0, ω)u0y,η(ζ0)

−ρω2U∗

y (ζ, ζ0, ω)u0y(ζ0)

)δiλ − σ∗

λ y(ζ, ζ0, ω)u0y,i(ζ0)

]

nλ(ζ0)dS

− µ∗ ni(ζ)

∫

S

U∗

y,i(ζ, ζ0, ω)p0y(ζ0)dS, ζ ∈ S,

for the outer boundary only. Also, the BEM equations in the presence ofdiscontinuities are:


(3.2)

py(ζ) = µ∗ni(ζ)

∫

S

[

(σ∗

η y(ζ, ζ0, ω)ucy,η(ζ0) − ρω2U∗

y (ζ, ζ0, ω)ucy(ζ0))δiλ−

− σ∗

λ y(ζ, ζ0, ω)ucy,i(ζ0)

]

nλ(ζ0)dS−

− µ∗ni(ζ)

∫

S

U∗

y,i(ζ, ζ0, ω)pcy(ζ0)dS+

+ µ∗ni(ζ)

∫

Γ+c

[

(σ∗

η y(ζ, ζ0, ω)∆ucy,η(ζ0) − ρω2U∗

y (ζ, ζ0, ω)∆ucy(ζ0))δiλ−

− σ∗

λ y(ζ, ζ0, ω)∆ucy,i(ξ0)

]

nλ(ζ0)dΓ+c+

+ µ∗ni(ζ)

∫

ΓH

[

(σ∗

η y(ζ, ζ0, ω)ucy,η(ζ0) − ρω2U∗

y (ζ, ζ0, ω)ucy(ζ0))δiλ−

− σ∗

λ y(ζ, ζ0, ω)ucy,i(ζ0)

]

nλ(ζ0)dΓH−

− µ∗ni(ζ)

∫

ΓH

U∗

y,i(ζ, ζ0, ω)pcy(ζ0)dΓH ,

ζ ∈ Γ = S ∪ Γ+c ∪ ΓH ,

involving boundaries S, ΓH and one side of the crack line, namely Γ+c. In theabove, we have that the free traction term in the boundary integral equationsis:

(3.3)

py(ζ) = −1

2pc

y(ζ), ζ ∈ S; py(ζ) = −1

2p0

y(ζ), ζ ∈ ΓH ;

py(ζ) = −1

2p0

y(ζ), ζ ∈ Γ+c.


Next, ∆ucy = uc

y

∣∣Γ+c−uc

y

∣∣Γ−c denotes the crack opening displacement

(COD), ni are all outward unit normal vectors at the surfaces, ζ(x, z) andζ0(x, z) are the observation and the source points, respectively. Furthermore,U∗

y (ζ, ζ0, ω) is the fundamental displacement solution of Eq. (2.1). u0y, p0

y

denote displacements and tractions in terms of prescribed boundary conditionsthat encompass all external loads across the boundary S of the region free ofholes or cracks. In the presence of discontinuities, uc

y, pcy are the fields induced

by the loads pcy = −p0

y acting on the cavity perimeter ΓH and/or the cracklines Γc, along with zero boundary conditions on the external boundary, i.e.,uc

y(ζ) = 0, ζ ∈ Su and pcy(ζ) = 0, ζ ∈ SP . This breakdown is a consequence of

the principle of superposition that is valid for linear problems and addition ofthe BVP components yields the correct boundary conditions along all surfacesof the problem.

The total wave field uy, py is a sum of the incident wave field fordisplacement and traction uin

y , piny and the scattered wave field usc

y , pscy , i.e.

uy = uiny + usc

y , py = piny + psc

y in the case when we consider inhomogeneitiessuch as cracks and/or cavities in an infinite domain. The incident wave dis-placement satisfies the governing equation (2.1), while the scattered wave dis-placement satisfies the wave equation (2.1), the boundary condition along theinhomogeneity’s perimeter py = 0 and the Sommerfeld radiation condition atinfinity.

The integro-differential system of Eqs. (3.1)–(3.2) is defined with re-spect to the unknown field quantities u0

y, p0y, uc

y, pcy and ∆uc

y for the generalBVP, which involves a finite external region with boundary S that encompassesa mode III crack and/or a cylindrical cavity. In the case of problem (i) for acrack in the infinite plane, the BIE (3.1)–(3.2) are reduced since there are nointegrals along the external boundary S and along the cavity perimeter ΓH .The only unknowns here are the CODs. Next, when we solve problem (ii) fora circular cylindrical cavity embedded in the infinite poroelastic plane, thenintegrals along S and Γ+c are absent and the only unknowns are displacementsalong the cavity perimeter. Finally, for case (iii) the BIE along ΓH are notused. An algebraic system of equations is obtained following discretization ofall surfaces with quadratic (i.e., three-node) boundary elements (BE) and spe-cial quarter-point boundary elements (QP–BE) that capture the asymptoticbehaviour of displacements and stresses near the crack tips by use of specialinterpolation functions. This system is in terms of the aforementioned un-knowns and is completely equivalent to Eq. (3.1)–(3.3). It is then solved usingnodal collocation for a discrete spectrum of frequency values. The followingprinciples hold true in the approximation of the displacement uy and traction


py across a BE: (a) Holder continuity conditions are satisfied at least at thecollocation points uy ∈ C1,α(S); p∈y C0,α(S); (b) near the crack-tip, the asymp-totic expressions for the displacement and for the traction is well known fromlinear fracture mechanics, i.e. uy ≈ O(

√

r) and py ≈ O(1/√

r), where r is thedistance from the crack-tip; (c) points at which the boundary is not smooth,and correspondingly the normal vector does not exist, or the Holder conditionsfails, are defined as “irregular” points. The crack-tips in a cracked body andthe sharp corners in a body containing crack or hole are particular examplesof irregular points. These irregular points should not be used as collocationpoints since the traction boundary integral equations exist only at the pointswhere the interpolation functions are (Holder) continuously differentiable andthe boundary is smooth. This problem is circumvented by using the shiftednode concept [45], where odd-numbered nodes are offset from by a small dis-tance so that nodal collocation can proceed unimpeded; they still remain atthese locations for geometry interpolation purposes.

It should be mentioned that the COD plays a key role in dynamicfracture analysis. More specifically, all other field quantities at any arbitraryinternal point can be easily evaluated from the relevant integral representationformulas of elastodynamics if the COD’s and the displacements along the cavityperimeter have been computed from the BEM system of Eq. (3.1)–(3.2). Thus,both near- and far- field solutions can be determined, with the former yieldinginformation on the dynamic stress concentration distribution and the lattershowing the influence of discontinuities at the outer boundaries, which in thecase of seismic wave propagation correspond to the motions observed along thefree surface of geological media [1].

The singular integrals in Eq. (3.1)–(3.2) converge in the Cauchy prin-cipal value sense, if the appropriate smoothness requirements in the approxi-mation of the displacement and stress fields are fulfilled. The disadvantage ofthe standard quadratic approximation built within the ordinary BE is that itcannot impose smoothness at irregular points such as crack-tips, corner pointsand generally odd-numbered nodes of the mesh, which delineate the two endsof a given element. This problem is overcome by the use of the ’shifted point’method, as mentioned previously.

Finally, the most essential quantities that characterize the stress con-centration phenomena are the SIF for cracks and the SCF for cavities. Theformer are obtained directly from the nodal values of the traction ahead ofthe crack-tip as SIFIII = lim

︸︷︷︸

x1→c

py

√

2π(x1 ∓ c), where c is the half-length of

the crack (see Fig. 3) and the crack is orientated along the Ox axis and the


Fig. 4. Scattering of SH- waves by a circular cylindrical cavity of radius r = c in theOxz plane

center at the origin O. They are subsequently normalized by the maximumamplitude value for the traction resulting from the incident wave. The SCFare defined as the ratio of the stress along the circumference of a cavity tothe maximal amplitude of the incident stress field at the same point (see Fig.4). More specifically, the normalized dynamic SCF |σθγ/τ0| is computed byusing the formula σθγ = −σ1 sin (θ − γ)+σ2 cos (θ − γ), where σ1 = σxy +σin

xy,σ2 = σzy + σin

zy and τ0 = ω√

µρ is the amplitude of the maximum shear stressof the incident SH-wave for the isotropic homogeneous case. Also, γ is theangle of the observation point along the cavity perimeter with respect to thehorizontal direction and θ is the incident angle of the incoming wave, see Fig. 4.

4. Numerical examples

In this section we distinguish two parts, one focusing on validation-typestudies, while another collects the results of parametric studies for both cracksand cavities in the poroelastic continuum, which is modeled as a single-phase,viscoelastic material.

4.1. SH-wave scattering by a finite line crack in the infinite

poroelastic plane

This problem is depicted in Fig. 3, where a mode III crack of half-lengthc is located in the Oxz infinite plane and swept by an SH-wave propagatingat an incident angle θ with respect to the horizontal axis. Values for the shear


modulus and density of this elastic plane are as follows:

(4.1) µ = 2.71 × 1010 N/m2; ρ = 7.550 kg/m3.

Fig. 5. Normalized dynamic SIF validation for a single mode III crack versus normal-ized frequency Ω for a normally incident SH-wave propagating in the homogeneous,

isotropic plane

At first, we have the results of a validation study for the linear elastic,homogeneous isotropic background material. More specifically, Fig. 5 plotsthe SIF at the crack tip versus normalized frequency Ω = (ωc/cS), wherecS (m/sec) is the elastic wave speed. The results are parametric in terms ofthe number of the number of BE used for modelling the crack surface. Thesecomprise both conventional three-node quadratic BE on the regular surfaces,plus the special quarter point element (QP–BE) for modelling asymptotic be-haviour in the displacement field as ui ≈ O (

√r) near the crack tip. Figure

5 demonstrates the high accuracy achieved by the use of only seven BE, (fiveregular plus two special), which is further improved as the number of BE in-creases to fifteen. Concurrently, the results shown in Fig. 5 are obtained byDaros [46], who used a non-hypersingular, traction-based BEM, along withthose of Meguid and Wang [47] who used a singular integral equation method.We conclude that the accuracy of our proposed traction-based BEM is satis-factory.

Next, Fig. 6 plots the same SIF, but for a poroelastic material obeying


(a)

(b)

Fig. 6. Normalized SIF versus normalized frequency kc for a single mode III crackin an infinite poroelastic plane under an incident SH-wave: (a) small and (b) large

porosity n values for both dry and saturated soil

the Bardet viscoelastic isomorphism model. We show two sets of results inthis figure, one for porosity n < 0.25 and another for n > 0.30, while Poisson’sratio is fixed at ν = 0.10. The value of the normalized SIF increases in theformer plot of Fig. 6(a) with increasing dimensionless frequency to values thatexceed the elastic SIF by about 10%, and then starts to taper off. Porosityvalues of around n = 0.25 serve as an upper bound for this type of behaviour,


while low ones of n = 0.10 give results that run roughly parallel and slightlyabove those of the elastic SIF. Also, the difference between dry and saturatedsoil at the same porosity value is rather small. Similar behaviour as before isobserved in Fig. 6(b) up to a dimensionless frequency value of Ω = 0.40, pastwhich the SIF drops sharply, irrespective of what the porosity value is.

4.2. SH-wave scattering by a circular cavity in the infinite

poroelastic plane

Figure 4 depicts the problem, where a circular cylindrical cavity ofradius r = c is embedded in the infinite elastic Oxz plane and swept by a har-monic SH-wave. The boundary of the cavity is traction-free and the incomingwave scatters on the perimeter, producing a non-zero displacement field. TheBVP is solved by the BEM for the scattered wave field, whereupon the incidentwave produces a traction field along the cavity’s perimeter, with its negativevalue (i.e., opposite direction) imposed as a load. This way, the sum of incidentand scattered traction fields along the perimeter is zero, which is the correctboundary condition, with the BEM yielding the scattered displacement field.

Fig. 7. Dynamic SCF validation at a point A on the perimeter of the circular cavityversus normalized frequency Ω for an incoming SH-wave with incidence angle θ = 0

Figure 7 plots the dynamic SCF on the cavity perimeter as a functionof the incoming wave frequency (the non-dimensional value Ω is used), alongwith the classical results of Pao and Mow [2]. The SCF is normalized bythe magnitude of the stress value τ0 carried by the SH-wave. This validation


(a)

(b)

Fig. 8. Dynamic SCF at a point A on the perimeter of the circular cavity ver-sus normalized frequency Ω for an incoming SH-wave with incidence angle θ = 0:

(a) small and (b) large porosity n values for both dry and saturated soil

study assumes the surrounding material to be linearly elastic. The excellentconvergence of the BEM as a function of a small number of BE is clearly shownin Fig. 7, where just ten BE along the perimeter produces acceptable results,while thirty BE gives a nearly perfect match. Next, Fig. 8 is the same plot,but for a surrounding material that is poroelastic and represented by Bardet’sviscoelastic isomorphism. We show two sets of results as in the previous sub-


section, one for porosity n < 0.20 and another for 0.24 < n < 0.30, whilePoisson’s ratio is fixed at ν = 0.10. The poroelastic SCF drops off smoothly(maximum drop in the neighborhood of 15–20%) in the former case of Fig.8(a), with increasing frequency as compared to the elastic values. This drop ismore pronounced with higher porosity values, and the difference between dryand saturated soil for the same porosity remains quite small. We observe inFig. 8(b) that at higher porosities, the poroelastic SCF drops very rapidly tovalues that are about 80% less compared to the elastic SCF and remain lowat around SCF = 1.2. This drop is more rapid as the porosity increases ton = 0.30.

4.3. Cracked finite poroelastic region subjected to time-har-

monic tractions

A finite elastic region contains a centrally placed, mode III crack and issubjected to a uniform time-harmonic traction along its horizontal boundaries,as shown in Fig. 9. A validation study for this example is performed usingtruncation, i.e., by increasing the size of the material surrounding the crackuntil the outer boundary has practically no influence on the SIF that developsat the crack tip. We converge to a finite square region of size 2a, and giventhe mode III crack of half-length c, we require that a > 10c. In this case, theouter boundary no longer influences the elastic wave field, insofar that waves

Fig. 9. Finite rectangular poroelastic region with a centered crack with half-length cunder uniform, time-harmonic traction py = σ, where σ is the load stress amplitude


Fig. 10. Normalized dynamic SIF versus normalized frequency Ω for a mode III centredcrack in a rectangular poroelastic region as a function of porosity n

scattered by the crack dampen out before reaching the perimeter of the BEMmesh. The results are now the same as those recovered in sub-section 4.1 fora crack in infinite plane, see Fig. 5. The boundary element mesh consists of7 BE along the crack and 20 BE along the external boundary in the BEMsimulations.

Next, Fig. 10 plots the normalized SIF versus dimensionless frequencyΩ for the poroelastic region of finite size 20x40 (m), parametric in terms ofporosity values n. We first observe that there is a definite pattern of peaks inthe SIF frequency response, caused by the finite size of the dry ‘box’, namelya sharp one at Ω = 0.2 and a lesser one at Ω = 0.6. These indicate verystrong resonance effects, and the former SIF overshoots its static value forthe infinite region forty-fold. The latter one is milder and the overshoot isroughly five-fold. The presence of a water phase in the saturated soil acts as adamping mechanism and causes a frequency shift to lower values for the SIFpeaks (i.e., to less than Ω = 0.1 depending on the porosity for the first peakand to Ω = 0.55 for the second peak). It also causes a substantial drop inthese peak SIF values, to about thirty-fold in the former case and to a just adoubling in the latter case.


5. Interpretation of results

In this work, we focus on elastic wave propagation in a poroelastic con-tinuum with discontinuities under anti-plane strain conditions. The objectivewas to study the development of the SIF at the tip of a crack and of the dy-namic SCF in the perimeter of a cavity, when the surrounding medium is watersaturated soil, and the excitation are incoming SH- axes. Based on the resultpresented herein, the following points can now be summarized:

(1) For small values of porosity around one-tenth, there are minor differencesin the SIF values at the crack-tips between dry and saturated soils. Thisdifference, however, also increases and starts to depend on the frequencyof excitation as porosity increases.

(2) Up to porosity values of one-quarter, the SIF values increase, both withrespect to increasing frequency and porosity values.

(3) For high porosity values of over one-third, there exists a ’turning’ fre-quency past which the SIF starts to drop after having reached is max-imum value. This ’turning’ frequency starts to shift to the left in thespectrum (i.e., decreases) with increasing porosity.

(4) In a finite-sized region, resonance effects are observed and very high SIFvalues result, with the presence of the fluid phase acting as a dampingmechanism that ameliorates this effect.

(5) For both dry and saturated soils with low porosities, the SCFs are lowerthan those observed for a purely elastic material, and this reductionis proportional with the frequency (at least in the frequency intervalconsidered herein).

(6) Also, the SCF computed for both dry and saturated soils with porositieshigher than one-quarter are lower than the corresponding elastic SCF inthe low (dimensionless) frequency range up to Ω = 0.5. Past that value,they start to exhibit an oscillatory behaviour with increasing frequency.

At this point, we stress that insofar as the Bardet model for one-phasematerials is concerned, it cannot take into account the time-dependent porepressure diffusion process at the crack-tip, nor the pore pressure boundary con-ditions along crack faces. Thus, there are limitations when the aforementionedmodel is applied to fracture in poroelastic media, but it is still useful for engi-neering practice. More specifically, the viscoelastic-poroelastic similarity is use-ful for extracting approximate solutions that can be used as benchmark casesto calibrate the accuracy of more advance (in terms of constitutive laws) com-putational methods applied to wave propagation problems in fluid-saturatedsoils. In sum, the dynamic mode III crack SIF is higher that the correspond-


ing static one for a wide range of frequencies of excitation, which means theinertial properties of the material should be taken into consideration. Mostimportant, however, is the presence of the fluid phase in the soil that altersthe SIF values dramatically and its influence cannot be easily surmised, unlessthe appropriate computations are performed. The same comments hold truefor the SCF that develops in the perimeter of buried cavities.

6. Conclusions

The novelty in the present work stems from the incorporation of theviscoelastic isomorphism concept in a mesh-reducing numerical method suchas the BEM as a way of examining the dynamic response of fractured and/ordiscontinuous poroelastic media. This offers the following computational ad-vantages:

(a) The modelling of a two-phase material by a single-phase viscoelastic one.This simplifies the governing equation and allows for recovery of its funda-mental solution, which is the key ingredient in a BEM formulation. Thisopens the gateway to study fracture in poroelastic media given the com-plexity of BVP in fracture mechanics. At the same time, the equivalentmodel can account for changes in the stiffness and damping mechanismin the porous soil, since Bardet’s equivalence formulas for the phase ve-locities show explicit dependence on the porosity, the bulk modulus ofthe dry skeleton, of the solid grain and of the fluid components, and onthe hydraulic conductivity.

(b) The use of the BEM also carries computational advantages in compari-son with other analytical or numerical methods, such as (i) a drop in thedimensionality of the problem with a corresponding reduction in the sizeof the resulting algebraic system following surface discretization; (ii) so-lution at prescribed internal points in the domain is selectively performedbased on the boundary values without recourse to domain discretization(e.g., finite elements or finite differences); (iii) high accuracy in the re-sults due to the semi-analytical character of the method insofar as it isbased on the fundamental solution of the underlying wave equation; (iv)the high level of accuracy is maintained because numerical quadrature isdirectly applied to the boundary integral equations, which are an exactsolution statement; (v) the satisfactory solution of problems exhibitingstress gradient fields is possible.

In sum, the present BEM development using viscoelastic isomorphismgives approximate solutions for complex BVP involving wave propagation in


fluid-saturated media. High accuracy in the BEM solution is maintained fol-lowing this viscoelastic approximation, along with a very efficient use of CPUtime and memory because of the mesh-reduction character of the BEM.

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