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The cohesive frictional crack model applied to

the analysis of the dam-foundation joint

F. Barpi a,∗, S. Valente a

aDipartimento di Ingegneria Strutturale e Geotecnica, Politecnico di Torino,Corso Duca degli Abruzzi 24, 10129 Torino (Italy)

Abstract

The mechanical behaviour of dam-foundation joints plays a key role in concretedam engineering since it is the weakest part of the structure and therefore the evo-lutionary crack process occurring along this joint determines the global load bearingcapacity. The reference volume involved in the above mentioned process is so largethat it cannot be tested in a laboratory: structural analysis has to be carried on bynumerical modelling. The use of the asymptotic expansions proposed by Karihaloo& Xiao (2008) at the tip of a crack with normal cohesion and Coulomb frictioncan overcome the numerical difficulties that appear in large scale problems whenthe Newton-Raphson procedure is applied to a set of equilibrium equations basedon ordinary shape functions (Standard Finite Element Method). In this way it ispossible to analyze problems with friction and crack propagation under the constantload induced by hydro-mechanical coupling. For each position of the fictitious cracktip, the condition K1 = K2 = 0 allows us to obtain the external load level and thetangential stress at the tip. If the joint tangential strength is larger than the valueobtained, the solution is acceptable, because the tensile strength is assumed negligi-ble and the condition K1 = 0 is sufficient to cause the crack growth. Otherwise, theload level obtained can be considered as an overestimation of the critical value anda special form of contact problem has to be solved along the fictitious process zone.For the boundary condition analyzed (ICOLD benchmark on gravity dam model),after an initial increasing phase, the water lag remains almost constant and themaximum value of load carrying capacity is achieved when the water lag reaches itsconstant value.

Key words: Cohesive crack, Concrete, Dam, Fluid driven fracture, Foundation,Frictional crack, Fracture, Hydro mechanical coupling, ICOLD, Joint, Water lag

1 Nomenclature

• ′: derivative with respect to z• a1n, a2n, b1n, b2n: real coefficients• An = a1n + i a2n, Bn = b1n + i b2n: complex coefficients• α1, α2, . . .: best fitting constants• c: joint cohesion• E: Young modulus• ν: Poisson’s ratio• δ: crack sliding displacement• δc: critical value of δ• ft: ultimate tensile strength• GII

F : conventional Mode II fracture energy• hiff : imminent failure flood water level (Fig. 3)• hovt = hiff − hc: over-topping water heigth• hc: dam crest height (Fig. 3)• i: imaginary unit, iteration number• K1: mode I stress intensity factor• K2: mode II stress intensity factor• λi: eigenvalues• κ: Kolosov constant• µ = E/(2 (1 + ν)): shear modulus• µf = −τxy

σy|θ=π: stress ratio used in the asymptotic expansion

• φ(z): analytic function• Φ: Coulomb friction angle in joint failure criterion (Fig. 4)• χ(z): analytic function• p: water pressure along the crack (Fig. 3)• tn: traction along the crack (Fig. 3)• r: polar coordinate• σx: stress along x direction• σy: stress along y direction• σc: critical value of σy (corresponding to w = 0)• τxy: tangential stress• θ: polar coordinate• u: displacement along x direction• v: displacement along y direction• w: crack opening displacement• wc: critical value of w• weff =

√w2 + δ2: effective joint opening

• weff,c: critical value of weff

• z = r ei θ: complex variable

∗ Corresponding author.Email addresses: fabrizio.barpi@polito.it, silvio.valente@polito.it (S.

Valente).

2 Introduction

The mechanical behaviour of dam-foundation joints plays a key role in concretedam engineering since it is the weakest part in the structure and thereforethe evolutionary crack process occurring along the joint determines the globalload bearing capacity. In the scientific literature two problems on load-bearingcapacity are discussed:

• the problem of sliding along a pre-existing compressed discontinuity (see,among others, Barton et al. (1985), Plesha (1987), Gens et al. (1990), Stup-kiewicz & Mroz (2001)),

• the problem of crack initiation and propagation along an undamaged inter-face (see Carol et al. (1997), Cervenka et al. (1998), Barpi & Valente (2008),Cocchetti et al. (2002))

The latter problem is discussed below in the framework of the cohesive crackmodels, introduced by Barenblatt (1959) and Dugdale (1960) for elastoplas-tic materials, and by Hillerborg et al. (1976) for quasi-brittle materials. Inthis model, the fracture process zone (due to degradation mechanisms suchas plastic micro-voiding or micro-cracking) in front of the actual crack tip islumped into a discrete line (two-dimensional) or plane (three-dimensional) andis represented by a nonlinear traction-separation law across this line or plane.When the tangential components of the tractions are present the solution canlose uniqueness. Therefore numerical difficulties occur if the Newton-Raphsonprocedure is applied to a set of equilibrium equations based on ordinary shapefunctions (Standard Finite Element Method). In order to overcome these dif-ficulties Strouboulis et al. (2001) proposed an approximation which employsknowledge about the character of the solution (Generalized Finite ElementMethod). In this direction we take advantage from the work of Karihaloo& Xiao (2008) on the asymptotic fields at the tip of a cohesive crack. Inthis model frictional forces operate when the crack faces are open. Therefore,these forces are different from those operating in a contact problem. In thiscontext Karihaloo & Xiao (2008) obtained asymptotic expansions at a cohe-sive crack tip analogous to the Williams (1957) expansions at a traction-freecrack tip for any traction-separation law that can be expressed in a specialpolynomial form.

3 Polynomial cohesive law for quasi-brittle materials

In order to obtain the separable asymptotic field at a cohesive crack tip (interms of r and θ functions, see Fig. 1) in quasi-brittle materials, Karihaloo &Xiao (2007) reformulate the softening law into the following polynomial form:

σy

ft= 1 +

5∑

i=1

αi

(

w

wc

) 2i3

−(

1 +5∑

i=1

αi

)(

w

wc

)4

(1)

where σy and ft are the stress normal to the cohesive crack faces and theuniaxial tensile strength, respectively; w and wc are the opening displacementof the cohesive crack faces and the critical displacement at the real crack tip;αi are fitting parameters. Equation 1 can represent a wide variety of softeninglaws. For example, Karihaloo & Xiao (2007) showed that the experimentalresults of Cornelissen et al. (1986) for normal concrete can be fitted very wellby Eq. 1 with: α1 = −0.872, α2 = −16.729, α3 = 67.818, α4 = −110.462,α5 = 83.158 (see Fig. 2). The above mentioned shape coefficients are used inthe present work.

4 Asymptotic fields at the tip of a crack with normal cohesion and

Coulomb friction

The mathematical formulation follows closely that used by Karihaloo & Xiao(2008), so only a brief description will be given here. Muskhelishvili (1953)showed that, for plane problems, stresses and displacements in the Cartesiancoordinate system (see e.g. Fig. 1) can be expressed in terms of two analyticfunctions φ(z) and χ(z) of the complex variable z = reiθ

σx + σy = 2[φ′(z) + φ′(z)] (2)

σy − σx + 2iτxy = 2[zφ′′(z) + χ′′(z)] (3)

2µ(u+ iv) = κφ(z)− zφ′(z)− χ′(z) (4)

where a prime denotes differentiation with respect to z and an overbar complexconjugate. In Eq. 4, µ = E/[2(1 + ν)] is the shear modulus; the Kolosovconstant is κ = 3−4ν for plane strain and κ = (3−ν)/(1+ν) for plane stress;E and ν are Young’s modulus and Poisson’s ratio, respectively.

For a general mixed mode I+II problem, the two analytic functions φ(z) andχ(z) can be chosen as series of complex eigenvalue Goursat functions (Sih &Liebowitz (1968))

φ(z) =∑

n=0

Anzλn =

∑

n=0

Anrλneiλnθ (5)

χ(z) =∑

n=0

Bnzλn+1 =

∑

n=0

Bnrλn+1ei(λn+1)θ (6)

where the complex coefficients are An = a1n + ia2n and Bn = b1n + ib2n . Theeigenvalues λn and coefficients a1n, a2n, b1n and b2n are real.

Substituting complex functions 5 and 6 into 2, 3 and 4, the complete seriesexpansions of the displacements and stresses near the tip of the crack can bewritten:

2µu =∑

n=0

rλn

{

κ[a1n cosλnθ − a2n sinλnθ] + λn[−a1n cos(λn − 2)θ+

a2n sin(λn − 2)θ] + (λn + 1)[−b1n cosλnθ + b2n sinλnθ]}

(7)

2µ v =∑

n=0

rλn

{

κ[a1n sinλnθ + a2n cosλnθ] + λn[a1n sin(λn − 2)θ+

a2n cos(λn − 2)θ] + (λn + 1)[b1n sinλnθ + b2n cosλnθ]}

(8)

σx =∑

n=0

rλn−1{

2λn[a1n cos(λn−1)θ−a2n sin(λn−1)θ]−λn(λn−1)[a1n cos(λn−3)θ

− a2n sin(λn − 3)θ]− (λn + 1)λn[b1n cos(λn − 1)θ − b2n sin(λn − 1)θ]}

(9)

σy =∑

n=0

rλn−1{

2λn[a1n cos(λn−1)θ−a2n sin(λn−1)θ]+λn(λn−1)[a1n cos(λn−3)θ

− a2n sin(λn − 3)θ] + (λn + 1)λn[b1n cos(λn − 1)θ − b2n sin(λn − 1)θ]}

(10)

τxy =∑

n=0

rλn−1

{

λn(λn − 1)[a1n sin(λn − 3)θ + a2n cos(λn − 3)θ]+

(λn + 1)λn[b1n sin(λn − 1)θ + b2n cos(λn−1)θ]}

(11)

w = v

∣

∣

∣

∣

θ=π

− v

∣

∣

∣

∣

θ=−π

=∑

n=0

rλn

µ[(κ+ λn)a1n + (λn + 1)b1n] sinλnπ (12)

δ = u∣

∣

∣

∣

θ=π

− u∣

∣

∣

∣

θ=−π

=∑

n=0

rλn

µ[(λn − κ)a2n + (λn + 1)b2n] sinλnπ (13)

The imposition of continuity conditions on normal stress component of Eq. 10(σy|θ=π = σy|θ=−π) along the cohesive zone gives:

(a2n + b2n) sin(λn − 1)π = 0 (14)

The imposition of continuity conditions on tangential stress component ofEq. 11 (τxy|θ=π = τxy|θ=−π) along the cohesive zone gives:

[(λn − 1)a1n + (λn + 1)b1n] sin(λn − 1)π = 0 (15)

Equations 14 and 15 are satisfied for sin(λn − 1)π = 0 or for b2n = −a2n. Inother words the asymptotic solutions can be collected in two classes. The firstclass is characterized by integer eigenvalues:

λn = n+ 1, n = 0, 1, 2 . . . , w = 0, δ = 0 (16)

the second class is characterized by the remaining cases (non integer eigenval-ues):

b2n = −a2n, b1n = −λn − 1

λn + 1a1n, w 6= 0, δ 6= 0 (17)

The imposition of the Coulombian friction condition (τxy|θ=π = −µf σy|θ=π)along the cohesive zone, for the first class of solutions, gives:

λn = n+1, na2n + (n+2)b2n = −µf (n+2)(a1n + b1n) n = 0, 1, 2 . . . (18)

and, for the second class of solutions, gives:

(µfa1n − a2n) cos(λn − 1)π = 0 (19)

Since both factors in Eq. 19 may vanish independently of each other, it ap-pears that, for the crack with normal cohesion and Coulombian friction, theeigenvalues and asymptotic fields are not unique. Additional assumptions haveto be made to ensure uniqueness. Assuming that µfa1n−a2n 6= 0, Eq. 19 gives:

cos(λn − 1)π = 0, λn =2n+ 3

2, n = 0, 1, 2 . . . (20)

This assumption does not lead to any loss of generality. Now, it is possible tocomplete the expressions of the asymptotic fields.

In the case of integer eigenvalues, substituting Eq. 18 in 10 gives:

σy|θ=±π = −τxy|θ=±π

µf

=∑

n=0

(n+ 2)(n+ 1)rn(a1n + b1n) cos(nπ) (21)

In the case of non-integer eigenvalues, substituting Eqs. 17 and 20 in 12 and 13gives:

w =∑

n=0

r2n+3

2

µ

[(

κ+2n+ 3

2

)

a1n +2n+ 5

2b1n

]

sin2n+ 3

2π (22)

δ =∑

n=0

r2n+3

2

µ

[(

2n+ 3

2− κ

)

a2n +2n+ 5

2b2n

]

sin2n+ 3

2π (23)

In Eq. 20 n = −1 corresponds to the singular terms, which are excluded apriori (K1 = K2 = 0).

5 The iterative solution procedure

For each position of the fictitious crack tip (shortening FCT) the followingiterative procedure is applied:

w

δ

i+1

= f

σy

τxy

i

,

σy

τxy

i+1

= g

w

δ

i+1

i = 0, 1, 2 . . . (24)

Since the material outside the fracture process zone (shortening FPZ) is linear,it is possible to compute the external load multiplier (hovt) and the tangential

stress at the FCT (τxy,FCT ) by imposing that the stress field is not singular(stress intensity factorsK1 = K2 = 0). All these linear constraints are includedin the operator f .

Since w,δ,σy,τxy are compatible with the asymptotic solution, operator g in-cludes the constraints described by Karihaloo & Xiao (2008) and not repeatedhere.

At the first iteration (i = 0) w = δ = 0 is assumed along the FPZ. Accordingto this approach hovt and τxy,FCT are not defined a priori but are obtainedfrom the analysis related to a pre-defined position of the FCT. If τxy,FCT is lessthan or equal to the local critical value, the solution obtained can be accepted.On the contrary, if τxy,FCT exceeds the local critical value, the associated loadlevel can be seen as an overestimation of the real critical value which remainsunknown. Of course it is possible to reduce the load level but in that case K1

becomes negative, a contact problem arises along the FPZ and the dilatancycondition has to be imposed. This special form of contact problem is beyondthe scope of the present work.

In the well established literature on mechanical behaviour of concrete joints(see Cervenka et al. (1998)), softening depends only on weff =

√w2 + δ2.

In the asymptotic expansion used, softening depends only on w. Therefore,during the iterative procedure, wc changes as follows:

wi+1c =

√

w2eff,c − (δi)2 (25)

In this model, the sliding rate (δ) is independent from the opening rate (w).With the terminology commonly used in plasticity, we can say that the failurecriterion adopted is non-associative.

6 Numerical example

As an example of application, the benchmark problem proposed in 1999 bythe International Commission On Large Dams ICOLD (1999) was analyzed(dam height 80 m, base 60 m, see Fig. 3).

For simplicity, the same value of Young’s modulus (E = 32.5 GPa) and Pois-son’s ratio (ν = 0.125) was assumed. Figure 4 shows the Mohr envelope of peakand residual strength for the joint (cohesion c=0.7MPa, Φ = 30o). The stressσx is positive (tension) along the lower edge of the crack. Figure 4 shows itscontribution to the achievement of the critical condition. As the crack grows,the value of σx at the FCT (also called T-stress) reduces. For conservative

reasons, the tensile strength of the joint and the related fracture energy areassumed as negligible. In case of linear softening the ICOLD benchmark sug-gests the assumption of a critical value of the crack sliding displacement equalto δc = 1 mm in the case of w = 0. Since the shape of the softening law as-sumed in the present paper is based on the results of Cornelissen et al. (1986),the previous value was increased to δc=2.56 mm. This choice is motivated bykeeping constant the fracture energy GII

F in the case w = 0. Since the crack isopen, beyond this value no stress transfer occurs.

6.1 Water lag

The well established literature on water driven fracture (see Desroches et al.(1994)) assumes that the water penetrates into the crack but does not reachthe FCT. The fraction of FPZ not reached by the water is called water lag.According to the experimental results of Reich et al. (1994), it is assumed thatthe water penetrates into the FPZ up to the conventional knee point of thesoftening law (w > weff,c × 2/9 = 2.56 × 2/9 = 0.569 mm). At the pointswhere the water penetrates, the pressure is the same as in the reservoir at thesame depth. The concrete and the rock are assumed to be impervious. Theasymptotic expansion used is based on the assumption τxy|θ=π = −µfσy|θ=π

therefore it can be applied only in the region not reached by the water. Figure 5shows the evolution of the water lag as a function of the FCT position. Thefree parameters of the expansion are calibrated in this region. In the remainingpart of the FPZ ordinary shape functions are used. For example, when thedistance of the FCT from upstream edge is 15 m Figs. 6, 7 and 8 show thatthe total solution perfectly fits the asymptotic curve in terms of crack openingand sliding displacement and in terms of tangential cohesive stress (the normalcomponent is negligible as required by the benchmark).

6.2 Loading conditions

The dam is analyzed under self-weight, reservoir filling and imminent failureflood loading conditions. In the numerical analysis the role of external loadmultiplier was played by the water level above the dam crest also called over-topping water heigth (shortening hovt = hiff − hc, see Fig. 3). Under theconservative assumptions previously described related to the material proper-ties, the crack starts before the water level reaches the dam crest (hovt < 0).

Figure 9 shows the evolution of (τ/c) at the FCT as a function of the FCTposition. Based on the foregoing discussion, we can conclude that the asso-ciated load level hovt shown in Fig. 10 is just an overestimation of the reallevel. This model behaviour is due to the low value of cohesion suggested by

the benchmark. For higher values of cohesion the solution shown in Fig. 11and 10 is completely acceptable. Figure 10 gives the maximum value of hovt

which is also the maximum load carrying capacity of the dam. Figure 11 showsthe evolution of the horizontal crest displacement as a function of the FCTposition and Fig. 12 the deformed mesh along the joint.

7 Conclusions

• The reference volume involved in the fracture process of a dam joint is solarge that it cannot be tested in a laboratory: a numerical model is needed.

• The use of the asymptotic expansions proposed by Karihaloo & Xiao (2008)at the tip of a crack with normal cohesion and Coulomb friction can over-come the numerical difficulties that appear in large scale problems whenthe Newton-Raphson procedure is applied to a set of equilibrium equationsbased on ordinary shape functions (Standard Finite Element Method). Theassumption of a different law of sliding, for which the asymptotic expansionis not available in the literature, can induce a very slow rate of convergence.

• In this way it is possible to analyze problems with friction and crack prop-agation under the constant load induced by hydro-mechanical coupling.

• In the analysis of the dam-foundation joint penetrated by the water, for eachposition of the FCT, the two conditions K1 = K2 = 0 allow us to obtain theexternal load level and the tangential stress at the FCT. If the joint strengthis larger than the value obtained, the solution is acceptable, because thetensile strength is assumed negligible (according to the benchmark) and thecondition K1 = 0 is sufficient to cause the crack growth. Otherwise the loadlevel obtained can be considered as an overestimation of the critical value.

• For the boundary condition analyzed, after an initial increasing phase, thewater lag remains almost constant.

• For the boundary condition analyzed, the maximum value of load carryingcapacity is achieved when the water lag reaches its constant value.

8 Acknowledgments

The financial support provided by the Italian Ministry of Education, Uni-versity and Scientific Research (MIUR) to the research project on “Structural

monitoring, diagnostic inverse analyses and safety assessments of existing con-

crete dams” (grant number 20077ESJAP 003) is gratefully acknowledged.

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Cornelissen, H., Hordijk, D. & Reinhardt, H. (1986). Experimental determinationof crack softening characteristics of normal and lightweight concrete, Heron31: 45–56.

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Fig. 1. Stresses near the crack tip.

0 0.2 0.4 0.6 0.8 1Non dimensional opening (-)

0

0.2

0.4

0.6

0.8

1

Non

dim

ensi

onal

str

ess

(-)

Fig. 2. Non dimensional cohesive stress (σ/σc) vs. non dimensional crack opening(w/wc).

���

���

������� �

� �

����������

Fig. 3. Gravity dam proposed as benchmark by ICOLD (1999).

Fig. 4. Failure criterion.

0 5 10 15 20Distance of FCT from the upstream edge (m)

0

1

2

3

4

5

6

7

Wat

er la

g (m

)

Fig. 5. Water lag vs. FCT position.

0 5 10 15Distance (r) from the FCT (m)

0

0.0005

0.001

0.0015

0.002

Cra

ck o

peni

ng d

ispl

acem

ent (

m)

Total solutionAsymptotic component

Fig. 6. Crack opening displacement vs. distance r.

0 5 10 15Distance (r) from the FCT (m)

0

0.0002

0.0004

0.0006

0.0008

Cra

ck s

lidin

g di

spla

cem

ent (

m)

Total solutionAsymptotic component

Fig. 7. Crack sliding displacement vs. distance r.

0 5 10 15Distance (r) from the FCT (m)

0

0.2

0.4

0.6

0.8

1

Non

dim

ensi

onal

tang

entia

l coh

esiv

e st

ress Total solution

Asymptotic component

Fig. 8. Nondimensional tangential cohesive stress (τ/c) vs. distance r.

0 5 10 15 20Distance of FCT from upstream edge (m)

1.2

1.3

1.4

1.5

Tan

gent

ial s

tres

s ra

tio a

t FC

T (

-)

Fig. 9. Tangential stress ratio τxy/c at FCT vs. FCT position.

0 5 10 15 20Distance of FCT from upstream edge (m)

-2

0

2

4

6

Ove

rtop

ping

hei

ght (

m)

Fig. 10. Overtopping height hovt vs. FCT position.

0 5 10 15 20Distance of FCT from upstream edge (m)

0

0.5

1

Hor

izon

tal c

rest

dis

plac

emen

t (cm

)

Fig. 11. Horizontal crest displacement vs. FCT position.

Fig. 12. Deformed mesh.