The cohesive-frictional crack model applied to the bi-material interface between a dam and the foundation rock Silvio Valente 1, Andrea Alberto 1 and Fabrizio Barpi 1 1 Department of Structural, Geotechnical and Building Engineering, Politecnico di Torino, Italy E-mail: silvio.valente@polito.it, andrea.alberto@polito.it, fabrizio.barpi@polito.it Keywords: Asymptotic field, Cohesive crack, Coulomb friction, Bi-material interface, Dam-foundation joint,fictitious crack SUMMARY. The most realistic method used today for numerical analyses of concrete fracture is the cohesive crack model. In this context, in case of large scale problems with friction, the incremental solution can lose uniqueness and consequently the Newton-Raphson procedure can fail. An effective way to prevent these difficulties is to enforce an asymptotic solution at the tip of the fictitious crack. In this direction Karihaloo and Xiao proposed an asymptotic field for a cohesive crack growing in Mixed Mode (Mode I and II) conditions. In the present paper the above mentioned solution is generalized to the case of a cohesive crack growing at bi-material interface. This enhancement allows us to obtain a more realistic simulation of the fracture process occurring at the dam-foundation joint.numerical results in the case of a gravity dam proposed as a benchmark by the International Commission for Large Dams are shown. 1 INTRODUCTION Cohesive crack models are an important means of describing localization and failure in engineering structures,with reference to quasi-brittle materials. When these models are adopted, the stresses acting on the non-linear fracture process zone are considered as decreasing functions of the displacement discontinuity. These functions are assumed to be material properties through the use of a pre-defined softening law. Although this standard formulation of the cohesive crack model is highly simplified, it is able to capture the essence of the fracture process in concrete specimens and structures [1]. De Borst et al. [2] have given an overview of the various ways of numerically implement the cohesive 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 any mesh bias. The XFEM enriches the standard local FE approximations with known information about the problem. Zi and Belytschko [6] enriched all cracked linear or quadratic triangular finite elements including the elements containing the crack tip by the sign function. Alfaiate et al. [7] embedded displacements 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 typical cusp-like shape of the process zone at the tip of a cohesive crack. However the cubic function does not 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 expansion at the cohesive crack tip, analogous to Williams expansions at a traction free crack tip. Coulomb friction on the cohesive crack faces is also considered. The main advantage of the above mentioned expansion, compared to the work of Zhang and Deng [10], is that the softening law can be expressed in a special polynomial form which can be calibrated on many commonly-used traction-separation law, e.g. rectangular, linear, bi-linear and exponential. Many studies on mixed-mode cohesive cracks can be found in the literature, for example Valente 1
[11] and Cocchetti et al [12], but there is doubt about the accuracy of the cohesion-sliding relation because it is difficult to isolate it from frictional forces between the rough cohesive crack faces in quasi-brittle materials such as concrete. The frictional cohesive cracks are different from the frictional 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 this paper with reference to the joint between a gravity dam and the foundation rock (see [13] and [14]). 2 THE MODEL The adopted mathematical formulation closely follows that used by Karihaloo and Xiao [9]. Figure 1: A cohesive crack at a bi-material interface (for simplicity the traction component τ xy is not drawn). Figure 2: [15] Cohesive law assumed according to Muskhelishvili showed that, for plane problems, the stress and displacements in the Cartesian coordinate system (see Figure 1) can be expressed in terms of two analytic functions, φ(z) and χ(z), of the complex variable z = re iθ. σ x + σ y = 2[φ (z) + φ (z)] (1) σ y σ x + 2iτ xy = 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 conjugate. In Eq. (3), µ = E/[2(1 + ν)] is the shear modulus; the Kolosov constant 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) and χ(z) can be chosen as series of complex eigenvalue Goursat functions (Sih and Liebowitz [16]) φ 1 (z) = A n z λn = A n r λn e iλnθ, χ 1 (z) = B n z λn+1 = B n r λn+1 e i(λn+1)θ (4) 2
φ 2 (z) = G n z λn = G n r λn e iλnθ, χ 2 (z) = H n z λn+1 = H n r λn+1 e i(λn+1)θ (5) where the complex coefficients A n = a 1n + ia 2n and B n = b 1n + ib 2n are related to material 1 (see Figure 1 when (0 θ π)) and the complex coefficients G n = g 1n + ig 2n and H n = h 1n + ih 2n are related to material 2 (when -π θ 0). The eingenvalues, λ n and coefficients a 1n, a 2n, b 1n, b 2n, g 1n, g 2n,h 1n and h 2n are real. 2.1 Boundary conditions at the bi-material interface The following conditions need to be satisfied (θ = ±0, two materials are bonded): u ϑ=0 + = u ϑ=0, v ϑ=0 + = v ϑ=0, σ y ϑ=0 + = σ y ϑ=0, τ xy ϑ=0 + = τ xy ϑ=0 (6) Eq.(6) gives the following constraints on the coefficients: ( µ2 (k 1 λ n ) g 1n = + λ ) ( n + 1 µ2 ( λ n 1) a 1n + + λ n + 1 (k 2 + 1)µ 1 k 2 + 1 (k 2 + 1)µ 1 k 2 + 1 ( µ2 (k 1 + λ n ) g 2n = + 1 λ ) ( n µ2 (λ n + 1) a 2n + λ n + 1 (k 2 + 1)µ 1 k 2 + 1 (k 2 + 1)µ 1 k 2 + 1 ( µ2 (k 1 λ n ) h 1n = + k ) ( 2 λ n µ2 (λ n + 1) a 1n + + k 2 λ n (k 2 + 1)µ 1 k 2 + 1 (k 2 + 1)µ 1 k 2 + 1 ) b 1n (7) ) b 2n (8) ) b 1n (9) ( (1 λn )µ 2 (k 1 + λ n ) h 2n = + (k ) ( 2 + λ n )(λ n 1) (1 λn )µ 2 a 2n + + (k ) 2 + λ n ) b 2n (k 2 + 1)(λ n + 1)µ 1 (k 2 + 1)(λ n + 1) (k 2 + 1)µ 1 (k 2 + 1) (10) Since the coefficients related to material 2 can be expressed as a function of coefficients related to material 1, afterward only the second group of coefficients will be mentioned. 2.2 Boundary conditions along the cohesive zone σ y ϑ=π = σ y ϑ= π 0, τ xy θ=π = τ xy θ= π = µ f σ y ϑ=π 0 (11) The asymptotic solution of Eq. (11) is composed of two parts: (a) Integer eigenvalues: λ n = n + 1, n = 0, 1, 2,... b 2n = n n + 2 a 2n µ f (a 1n + b 1n ) (12) σ y ϑ=±π = τ xy ϑ=±π = r n (n + 2)(n + 1)(a 1n + b 1n ) cos(nπ) (13) µ f exactly as in the mono-material case presented in [9]; COD = w = CSD = δ = 0 (14) 3
(b) Fractional eigenvalues: λ n = n + 3 2, n = 0, 1, 2,... b 2n = (µ 1k 2 λ n + µ 1 + µ 2 k 1 + µ 2 λ n ) (µ 2 λ n + µ 2 + µ 1 k 2 λ n + µ 1 k 2 ) a 2n (15) b 1n = ( µ 1k 2 λ n + µ 1 + µ 2 k 1 µ 2 λ n ) (µ 2 λ n + µ 2 + µ 1 k 2 λ n + µ 1 k 2 ) a 1n (16) w = δ = r 2n+3 2 r 2n+3 2 a 2n = a 1n µ f [ ] (17) 2(µ 1 + µ 2 k 1 ) a 1n sin 2n + 3 π µ 1 µ 2 2 (18) [ ] 2(k 1k 2 1) a 2n sin 2n + 3 π µ 1 k 2 µ 2 2 (19) ˆσ y = σ y ϑ=±π = τ xy ϑ=±π = e n r 2n+1 2 (20) σ 0 µ f σ 0 e n = 1 [ ( ] 2n + 3 µ2 (k 1 1) µ 1 (k 2 1) )a 2n sin 2n + 3 π (21) σ 0 2 µ 2 + µ 1 k 2 2 e n coefficients vanish in the homogeneous case.this is the main difference which characterizes the case of crack between dissimilar materials. Further details are presented in [17] and [18]. 2.3 Cohesive law In 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 process zone (shortened FPZ) and the ratio σy ϑ=±π τ xy ϑ=±π remains constant while both components are decreasing functions of the displacement discontinuity (proportional softening). This behavior follows a pre-defined cohesive law shown in Figure 2. 3 NUMERICAL RESULTS Figure 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, w eff,c = 2.56mm, µ f σ u = 0.95 MP a, µ f = 45 ). The failure criterion applied at the fictitious crack tip is shown in Figure 5. 3.1 The Water Lag According 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 > w eff,c 2/9 = 2.56 2/9 = 0.569 mm). The fraction of FPZ not reached by the water is called water lag (see Figure 4). At the points where the water penetrates, the pressure is the same as in the reservoir at the same depth. The concrete and the rock are assumed to be impervious. The asymptotic expansion used is based on the assumption τ xy ϑ=±π = µ f σ y ϑ=±π therefore it can be applied only in the region not reached by the water. The free parameters of the expansion are calibrated in this region. For example, when the distance of the FCT from upstream edge is 12 m Figures 7, 8 show that the total solution perfectly fits the asymptotic curve in terms of crack opening and sliding displacement. Figure 9 shows the contour lines of τ xy on a deformed mesh. Figures 10 and 11 show the angular distribution of stresses. 4
3.2 The iterative solution procedure For each position of the fictitious crack tip (shortening FCT) the following iterative procedure is applied: [ ] i+1 w = f δ ( [ σy τ xy ] i ), [ σy τ xy ] i+1 = g ( [ ] i+1 ) w δ i = 0, 1, 2... (22) Since the material outside the fracture process zone (shortening FPZ) is linear, it is possible to compute the overtopping water height (h ovt ) and the tangential stress at the FCT (τ xy,f CT ) by imposing that the stress field is not singular (stress intensity factors K 1 = K 2 = 0). All these linear constraints are included in the operator f. Since w,δ,σ y,τ xy are compatible with the asymptotic expansion presented in this paper, operator g includes the constraints previously described. At the first iteration (i = 0) w = δ = 0 is assumed along the FPZ. According to this approach h ovt and τ xy,f CT are not defined a priori but are obtained from the analysis related to a pre-defined position of the FCT. 3.3 Comparison between the results based on two different asymptotic expansions The 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 Figure 6 (Young s modulus 32500 MPa, Poisson s ratio 0.125, see Barpi and Valente [20]). In order to facilitate the comparison with the results based on the expansion proposed in this paper, the same position 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 of water penetration always increases the stress level at the fictitious crack tip it is possible to conclude that the solution obtained through the bi-material model is more conservative. In fact this model predicts the same stress level (see Table 1) for a lower overtopping water height and for a larger COD in comparison to the case of similar media. Figure 12 shows also that the stress component σ x parallel to the crack is the largest component in the FPZ. If this component achieves the rock strength, the crack can suddenly branch downwards into the rock mass. 5
Figure 3: Gravity dam proposed as benchmark by ICOLD [13] Figure 4: Water lag vs. FCT position Figure 6: Material properties assumed according to the benchmark problem proposed by ICOLD [13] Figure 5: Failure criterion applied at the fictitious crack tip 6
Figure 7: Crack opening displacement vs. distance r Figure 8: Crack sliding displacement vs. distance r Figure 9: Contour lines of τxy on a deformed mesh 7
Figure 10: Comparison between analytical (λ 3.5) and numerical results in the bimaterial case (r=0.24 m) Figure 11: Comparison between analytical (λ 3.5) and numerical results in the bimaterial case (r=0.48 m) Mono-material Bi-material max τ 0.85 MPa 0.875 MPa max σ x 2 MPa 2 MPa Overtopping water height 4.91 m 3.65 m COD at 6.48m from FCT 0.545 mm 0.608 mm µ f = τ xy /σ y 45 45 Table 1 Comparison between the main results. Figure 12: Comparison between the monomaterial case and the bi-material case 4 CONCLUSIONS Remark 1. With reference to a large scale problem with friction the classical cohesive crack model fails 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 numerical process 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 level for 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 obtained by assuming the expansion related to a cohesive crack growing between two similar materials. 8
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