The cohesive-frictional crack model applied to the bi-material interfacebetween a dam and the foundation rock

Silvio Valente1, Andrea Alberto1 and Fabrizio Barpi11Department of Structural, Geotechnical and Building Engineering, Politecnico di Torino, ItalyE-mail: silvio.valente@polito.it, andrea.alberto@polito.it, fabrizio.barpi@polito.it

Keywords: Asymptotic field, Cohesive crack, Coulomb friction, Bi-material interface, Dam-foundationjoint,Fictitious crack

SUMMARY. The most realistic method used today for numerical analyses of concrete fracture is thecohesive crack model. In this context, in case of large scale problems with friction, the incrementalsolution can lose uniqueness and consequently the Newton-Raphson procedure can fail. An effec-tive way to prevent these difficulties is to enforce an asymptotic solution at the tip of the fictitiouscrack. In this direction Karihaloo and Xiao proposed an asymptotic field for a cohesive crack grow-ing in Mixed Mode (Mode I and II) conditions. In the present paper the above mentioned solutionis generalized to the case of a cohesive crack growing at bi-material interface. This enhancement al-lows us to obtain a more realistic simulation of the fracture process occurring at the dam-foundationjoint.Numerical results in the case of a gravity dam proposed as a benchmark by the InternationalCommission for Large Dams are shown.

1 INTRODUCTIONCohesive crack models are an important means of describing localization and failure in engi-

neering structures,with reference to quasi-brittle materials. When these models are adopted, thestresses acting on the non-linear fracture process zone are considered as decreasing functions of thedisplacement discontinuity. These functions are assumed to be material properties through the useof a pre-defined softening law. Although this standard formulation of the cohesive crack model ishighly simplified, it is able to capture the essence of the fracture process in concrete specimens andstructures [1].

De Borst et al. [2] have given an overview of the various ways of numerically implement thecohesive zone method. They concluded that the extended/generalized finite element method (XFEM)([3] , [4] , [5]) provides a proper representation of the discrete character of the method avoiding anymesh bias. The XFEM enriches the standard local FE approximations with known information aboutthe problem. Zi and Belytschko [6] enriched all cracked linear or quadratic triangular finite elementsincluding the elements containing the crack tip by the sign function. Alfaiate et al. [7] embeddeddisplacements jumps which do not need to be homogeneous within each FE. Mariani and Perego [8]introduced in a standard FE model a displacement discontinuity, in order to reproduce the typicalcusp-like shape of the process zone at the tip of a cohesive crack. However the cubic function doesnot represent the true angular distribution of the displacement adjacent to the tip.

In order to overcome this problem, Karihaloo and Xiao [9] obtained an asymptotic expansionat the cohesive crack tip, analogous to Williams expansions at a traction free crack tip. Coulombfriction on the cohesive crack faces is also considered. The main advantage of the above mentionedexpansion, compared to the work of Zhang and Deng [10], is that the softening law can be expressedin a special polynomial form which can be calibrated on many commonly-used traction-separationlaw, e.g. rectangular, linear, bi-linear and exponential.

Many studies on mixed-mode cohesive cracks can be found in the literature, for example Valente

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[11] and Cocchetti et al [12], but there is doubt about the accuracy of the cohesion-sliding relationbecause it is difficult to isolate it from frictional forces between the rough cohesive crack facesin quasi-brittle materials such as concrete. The frictional cohesive cracks are different from thefrictional contact of crack faces because the friction operates when the crack faces are open.

A new asymptotic expansion, which can be applied at a bi-material interface,is presented in thispaper with reference to the joint between a gravity dam and the foundation rock (see [13] and [14]).

2 THE MODELThe adopted mathematical formulation closely follows that used by Karihaloo and Xiao [9].

Figure 1: A cohesive crack at a bi-material inter-face (for simplicity the traction component τxy

is not drawn).Figure 2: Cohesive law assumed according to[15]

Muskhelishvili showed that, for plane problems, the stress and displacements in the Cartesiancoordinate system (see Figure 1) can be expressed in terms of two analytic functions, φ(z) and χ(z),of the complex variable z = reiθ.

σx + σy = 2[φ′(z) + φ′(z)] (1)

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

2µ(u + iv) = kφ(z)− zφ′(z)− χ′(z) (3)

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

For a general mixed mode I+II problem, the two analytic functions φ(z) and χ(z) can be chosenas series of complex eigenvalue Goursat functions (Sih and Liebowitz [16])

φ1(z) =∑n=0

Anzλn =∑n=0

Anrλneiλnθ, χ1(z) =∑n=0

Bnzλn+1 =∑n=0

Bnrλn+1ei(λn+1)θ (4)

2

φ2(z) =∑n=0

Gnzλn =∑n=0

Gnrλneiλnθ, χ2(z) =∑n=0

Hnzλn+1 =∑n=0

Hnrλn+1ei(λn+1)θ (5)

where the complex coefficients An = a1n + ia2n and Bn = b1n + ib2n are related to material1 (see Figure 1 when (0 ≤ θ ≤ π)) and the complex coefficients Gn = g1n + ig2n and Hn =h1n + ih2n are related to material 2 (when -π ≤ θ ≤ 0). The eingenvalues, λn and coefficientsa1n, a2n, b1n, b2n, g1n, g2n,h1n and h2n are real.

2.1 Boundary conditions at the bi-material interfaceThe following conditions need to be satisfied (θ = ±0 , two materials are bonded):

u|ϑ=0+ = u|ϑ=0− , v|ϑ=0+ = v|ϑ=0− , σy|ϑ=0+ = σy|ϑ=0− , τxy|ϑ=0+ = τxy|ϑ=0− (6)

Eq.(6) gives the following constraints on the coefficients:

g1n=(

µ2(k1 − λn)(k2 + 1)µ1

+λn + 1k2 + 1

)a1n +

(µ2(−λn − 1)(k2 + 1)µ1

+λn + 1k2 + 1

)b1n (7)

g2n=(

µ2(k1 + λn)(k2 + 1)µ1

+1− λn

k2 + 1

)a2n +

(µ2(λn + 1)(k2 + 1)µ1

− λn + 1k2 + 1

)b2n (8)

h1n=(−µ2(k1 − λn)

(k2 + 1)µ1+

k2 − λn

k2 + 1

)a1n +

(µ2(λn + 1)(k2 + 1)µ1

+k2 − λn

k2 + 1

)b1n (9)

h2n =(

(1− λn)µ2(k1 + λn)(k2 + 1)(λn + 1)µ1

+(k2 + λn)(λn − 1)(k2 + 1)(λn + 1)

)a2n +

((1− λn)µ2

(k2 + 1)µ1+

(k2 + λn)(k2 + 1)

)b2n

(10)Since the coefficients related to material 2 can be expressed as a function of coefficients related tomaterial 1, afterward only the second group of coefficients will be mentioned.

2.2 Boundary conditions along the cohesive zone

σy|ϑ=π = σy|ϑ=−π 6= 0, τxy|θ=π = τxy|θ=−π = −µfσy|ϑ=π 6= 0 (11)

The asymptotic solution of Eq. (11) is composed of two parts:(a) Integer eigenvalues: λn = n + 1, n = 0, 1, 2, ...

b2n = − n

n + 2a2n − µf (a1n + b1n) (12)

σy|ϑ=±π =τxy|ϑ=±π

−µf=

∑n=0

rn(n + 2)(n + 1)(a1n + b1n) cos(nπ) (13)

COD = w = CSD = δ = 0 (14)

exactly as in the mono-material case presented in [9];

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(b) Fractional eigenvalues: λn = n + 32 , n = 0, 1, 2, ...

b2n = − (µ1k2λn + µ1 + µ2k1 + µ2λn)(µ2λn + µ2 + µ1k2λn + µ1k2)

a2n (15)

b1n =(−µ1k2λn + µ1 + µ2k1 − µ2λn)(µ2λn + µ2 + µ1k2λn + µ1k2)

a1n (16)

a2n = −a1n

µf(17)

w =∑n=0

r2n+3

2

[2(µ1 + µ2k1)

µ1µ2a1n

]sin

2n + 32

π (18)

δ =∑n=0

r2n+3

2

[− 2(k1k2 − 1)

µ1k2 − µ2a2n

]sin

2n + 32

π (19)

σy =σy|ϑ=±π

σ0=

τxy|ϑ=±π

−µfσ0=

∑n=0

enr2n+1

2 (20)

en =1σ0

[2n + 3

2

(µ2(k1 − 1)− µ1(k2 − 1)

µ2 + µ1k2

)a2n

]sin

2n + 32

π (21)

en coefficients vanish in the homogeneous case.This is the main difference which characterizesthe case of crack between dissimilar materials. Further details are presented in [17] and [18].

2.3 Cohesive lawIn order to make Figure 1 easier, only the normal components of cohesive stresses are drawn.

Nevertheless Equations 13 and 20 show that both components are operating along the fracture pro-cess zone (shortened FPZ) and the ratio σy|ϑ=±π

τxy|ϑ=±πremains constant while both components are de-

creasing functions of the displacement discontinuity (proportional softening). This behavior followsa pre-defined cohesive law shown in Figure 2.

3 NUMERICAL RESULTSFigure 3 show a gravity dam model proposed as a benchmark by the Int. Commission On Large

Dams [13],[14] (dam height 80 m, base 60 m, weff,c = 2.56mm, −µfσu = 0.95 MPa, µf = −45). The failure criterion applied at the fictitious crack tip is shown in Figure 5.

3.1 The Water LagAccording to the experimental results of Reich et al. [19], it is assumed that the water penetrates

into the FPZ up to the conventional knee point of the softening law (w > weff,c × 2/9 = 2.56 ×2/9 = 0.569 mm). The fraction of FPZ not reached by the water is called water lag (see Figure4). At the points where the water penetrates, the pressure is the same as in the reservoir at the samedepth. The concrete and the rock are assumed to be impervious. The asymptotic expansion used isbased on the assumption τxy|ϑ=±π = −µf σy|ϑ=±π therefore it can be applied only in the regionnot reached by the water. The free parameters of the expansion are calibrated in this region. Forexample, when the distance of the FCT from upstream edge is 12 m Figures 7, 8 show that thetotal solution perfectly fits the asymptotic curve in terms of crack opening and sliding displacement.Figure 9 shows the contour lines of τxy on a deformed mesh. Figures 10 and 11 show the angulardistribution of stresses.

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3.2 The iterative solution procedureFor each position of the fictitious crack tip (shortening FCT) the following iterative procedure is

applied: [wδ

]i+1

= f

( [σy

τxy

]i ),

[σy

τxy

]i+1

= g

( [wδ

]i+1 )i = 0, 1, 2 . . . (22)

Since the material outside the fracture process zone (shortening FPZ) is linear, it is possibleto compute the overtopping water height (hovt) and the tangential stress at the FCT (τxy,FCT ) byimposing that the stress field is not singular (stress intensity factors K1 = K2 = 0). All these linearconstraints are included in the operator f .

Since w,δ,σy ,τxy are compatible with the asymptotic expansion presented in this paper, operatorg includes the constraints previously described .

At the first iteration (i = 0) w = δ = 0 is assumed along the FPZ. According to this approachhovt and τxy,FCT are not defined a priori but are obtained from the analysis related to a pre-definedposition of the FCT.

3.3 Comparison between the results based on two different asymptotic expansionsThe results obtained through the asymptotic expansion proposed by Karihaloo and Xiao [9] for a

crack between similar media are based on the mean values of the elastic properties shown in Figure6 (Young’s modulus 32500 MPa, Poisson’s ratio 0.125, see Barpi and Valente [20]). In order tofacilitate the comparison with the results based on the expansion proposed in this paper, the sameposition of the fictitious crack tip (12 m from upstream edge) and the same length of the water lag(6.48 m see Figure 4) is assumed in both cases. Table 1 shows the main results. Since an increment ofwater penetration always increases the stress level at the fictitious crack tip it is possible to concludethat the solution obtained through the bi-material model is more conservative. In fact this modelpredicts the same stress level (see Table 1) for a lower overtopping water height and for a largerCOD in comparison to the case of similar media. Figure 12 shows also that the stress componentσx parallel to the crack is the largest component in the FPZ. If this component achieves the rockstrength, the crack can suddenly branch downwards into the rock mass.

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Figure 3: Gravity dam proposed as benchmarkby ICOLD [13]

Figure 4: Water lag vs. FCT position

Figure 5: Failure criterion applied at the ficti-tious crack tip

Figure 6: Material properties assumed ac-cording to the benchmark problem proposed byICOLD [13]

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Figure 7: Crack opening displacement vs. dis-tance r

Figure 8: Crack sliding displacement vs. dis-tance r

Figure 9: Contour lines of τxy on a deformed mesh

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Figure 10: Comparison between analytical(λ ≤ 3.5) and numerical results in the bi-material case (r=0.24 m)

Figure 11: Comparison between analytical(λ ≤ 3.5) and numerical results in the bi-material case (r=0.48 m)

Figure 12: Comparison between the mono-material case and the bi-material case

Mono-material Bi-materialmax τ 0.85 MPa 0.875 MPamax σx 2 MPa 2 MPaOvertopping water height 4.91 m 3.65 mCOD at 6.48m from FCT 0.545 mm 0.608 mm−µf = τxy/σy 45 45

Table 1 Comparison between the main results.

4 CONCLUSIONSRemark 1. With reference to a large scale problem with friction the classical cohesive crack modelfails to converge to an equilibrium state, during the Newton-Raphson iterations.Remark 2. Through the new asymptotic expansion proposed it is possible to stabilize the numericalprocess for a crack growing at a bi-material interface.Remark 3. In this way it is possible to determine the critical value of the overtopping water levelfor a gravity dam proposed as a benchmark by the International Commission On Large Dams.Remark 4. The value obtained with the new expansion is more conservative than the value obtainedby assuming the expansion related to a cohesive crack growing between two similar materials.

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[6] Zi G. and Belytschko T., ”New crack-tip elements for XFEM and applications to cohesivecracks”, International Journal of Numerical Methods in Engineering, 57:2221-2240 (2003).

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[11] Valente S., ”Influence of friction on cohesive crack propagation”, RILEM/ESIS Conf.on Frac-ture Processes in Concrete Rock and Ceramics, 695-704. E&FN Spon (1991).

[12] Cocchetti G., Maier G. and Shen X., ”Piecewise linear models for interfaces and mixed modecohesive cracks”, Journal of Engineering Mechanics (ASCE), 3:279-298 (2002).

[13] ICOLD, ”Imminent failure flood for a concrete gravity dam”, in 5th International BenchmarkWorkshop on Numerical Analysis of Dams, Denver,CO (1999).

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[15] Cornelissen H.A.W., Hordijk D.A. and Reinhardt H.W., ”Experimental determination of cracksoftening characteristics of normal and lightweight concrete”, Heron, 31:45-56 (1986).

[16] Sih G.C. and Liebowitz H., Mathematical theories of brittle fracture. In H. Liebowitz, editor,Fracture (vol. II), 67-190. Academic Press, New York (1968).

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[17] Alberto A. and Valente S., ”Asymptotic fields at the tip of a cohesive frictional crack growingat the bi-material interface between a dam and the foundation rock”, Engineering FractureMechanics, 108C:135-144 (2013).

[18] Alberto A., ”Cohesive-frictional crack model applied to bi-material interfaces”, PhD thesis,XXV Ciclo, Politecnico di Torino (Italy) (2013).

[19] Reich W., Bruhwiler E., Slowik V. and Saouma V.E., ”Experimental and computational aspectsof a water/fracture interaction”, Dam fracture and damage, pages 123–131, Balkema, TheNetherlands (1994).

[20] Barpi F. and Valente S., ”The cohesive frictional crack model applied to the analysis of thedam-foundation joint”, Engineering Fracture Mechanics, 77:2182-2191 (2010).

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