Comi, Claudia

Mariani, Stefano

Perego, Umberto

COMI, CLAUDIA

MARIANI, STEFANO

PEREGO, UMBERTO

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COHESIVE CRACK PROPAGATION IN DAMAGING CONCRETESTRUCTURES DISCRETIZED BY EXTENDED FINITE ELEMENTSC. COMI1, S. MARIANI2, U. PEREGO3Dipartimento di Ingegneria Strutturale, Politecnico di Milano, Milano (ITALY)1claudia.comi@polimi.it, 2stefano.mariani@polimi.it, 3umberto.perego@polimi.itABSTRACTA coupled continuum damage-discrete crack propagation strategy to simulate strength degradation and fracture inconcrete structures is presented. At fixed crack length, mesh-independent results are obtained through an integral non-local regularization of the damage model. For increasing loading conditions, as soon as a critical damage threshold isattained in the process zone, cohesive cracks are introduced to allow for a discontinuous displacement field. Withinthe class of cohesive laws that feature the same dissipation properties of the continuum damage model, constitutiveparameters are calibrated through an energy equivalence. Results concerning the wedge splitting test are shown todemonstrate the efficiency of the proposed procedure.1 INTRODUCTIONThe response of concrete structures to increasing loadings is characterized in the early stage by the spreadingof micro-cracks in localized regions (like, e.g., ahead of a pre-existing notch); these micro-cracks eventuallycoalesce to create only a few macro-cracks.To simulate numerically this complex behavior, a continuum damage approach proves efficient in the firststage. But, for increasing levels of damage, the width of the band where damage still continues to grow tendsto become narrower and narrower. To accurately resolve the process zone, the characteristic dimension offinite elements needs to be a fraction of the bandwidth. If it is chosen small enough from the beginning of theanalysis, computational costs become excessive for real-life structures like, e.g., concrete dams. Otherwise,the mesh can be adaptively refined during the analysis so as to simulate the process zone response withthe required degree of accuracy; in this case the computational burden is mainly due to the continuousprojection of current results from the old meshes to the new ones.In this work an alternative approach is followed. From a simplified bifurcation analysis of a one-dimensionalelastic damage continuum subject to a uniform stress and strain distribution, an estimate of the process zonewidth as a function of damage is obtained [1]. At fixed discretization, as soon as the mesh is considered toocoarse to resolve the band on the basis of the above estimate, a cohesive crack is allowed for through thepartition of unity concept [2, 3, 4]. As shown in [4], under mixed mode loading conditions this methodologyallows to simulate crack paths completely independent of the mesh layout.Concerning the constitutive modeling of concrete, an integral non-local regularization is adopted to restorethe well-posedeness of the set of equations governing the problem [5, 6]. This allows to fix the fractureenergy, that is the energy to be dissipated in order to completely annihilate the interaction between the twofaces of the crack. The cohesive model can thus be calibrated by allowing the crack to dissipate only thefraction of the concrete toughness not already dissipated by the continuum prior to crack insertion.This procedure is numerically tested with an analysis of the failure mechanism in the wedge splitting test[7], where a pre-existing edge notch is subject to mode I loading. It is shown that, when the displacementdiscontinuity is plugged into the simulation, a realistic damage pattern around the propagating macro-crackis obtained.2 GOVERNING RELATIONSLet us consider a two-dimensional (2D) solid Ω, with boundary Γ = Γu ∪ Γt (Fig. 1). Tractions t¯ areprescribed on Γt, whereas displacements u¯ are assigned on Γu. A displacement discontinuity locus Γd can1WGtGuGdtnGd-Gd+ms[ ]us[ ]unFigure 1: 2D solid. Geometry and notation.nucleate and propagate inside Ω.Under the assuption of linearized kinematics, equilibrium and compatibility relations for Ω are:CTσ + b¯ =0 in Ω\Γd (1)Nσ =t¯ on Γt (2)Mσ = −t+ on Γ+d , Mσ =t− on Γ−d , t ≡ t− = −t+ (3)ε =Cu in Ω\Γd (4)u =u¯ on Γu (5)[u] =u∣∣Γ+d− u∣∣Γ−d(6)where: u is the displacement vector; [u] is the displacement discontinuity vector along Γd; ε and σ are thestrain and stress vectors; t are the tractions acting upon the boundary ofΩ (Γd included); C is the differentialcompatibility operator; N andM are matrices gathering the components of unit vectors n andm (see Fig.1); T stands for transpose; Γ+d and Γ−d denote the two sides of Γd, according to Fig. 1.An isotropic elastic damage constitutive law is adopted for Ω\Γd. The local format of this law reads:σ = (1−D)E : ε (7)f = 2µ2e : e− 9K2a ε2v + 3Kb lnn2(c1−D)εv − k lnn(c1−D)≤ 0 (8)D˙ ≥ 0, fD˙ = 0 (9)where: E is the matrix of elastic moduli of the virgin material; D is the isotropic damage variable; µ andK are the shear and bulk elastic moduli; f is the damage activation function; εv is the volumetric strain,whereas e is the deviator of ε; a, b, c, k and n are model parameters, to be adjusted in order to match the2concrete behavior in the softening regime. The continuum model (7)-(9) locally dissipates a specific energy:gf =12σ20E{1 +2ncnn−1G [n, ln c]}(10)Here: E is the Young’s modulus of the virgin material; σ0 is the peak strength under uniaxial tensileloadings; G[n, ln c] ≡ ∫∞ln czn−1 exp(−z) dz is the incomplete gamma function.To restore the well-posedeness of the problem (1)-(9) in the softening regime (when D˙ > 0) and to achievemesh-independent results an integral non-local regularization of the strain invariants is adopted [5, 6].As for Γd, the cohesive law t = t([u]) is formulated so as to feature a softening envelope similar to thatof the damage model for the bulk [8]. Model parameters are calibrated in order to get the same energydissipation in the following two cases: (a) a continuum analysis allowing for bulk damage only, withoutcrack growth; (b) an enhanced analysis with bulk damage coupled to crack propagation as soon as a criticaldamage threshold is attained in the process zone (for details see [8]). It turns out that, under mode I loadings:tn ([u]n) =c exp−b+√b2 + 4(13 − a)k2kE(εc +[u]nLe)2nE (εc + [u]nLe)+12E (1−Dc) ε2c[u]nLeexp[− [u]nLe] (11)where: tn and [u]n are, respectively, the opening traction and displacement discontinuity (see Fig. 1); Dc isthe mentioned critical damage threshold and εc the relevant deformation in Ω; Le is the width of the processzone corresponding to Dc [1].3 FINITE ELEMENT FORMULATIONWithin the frame of the extended finite element method, the assumed discretized displacement field is:uh (x) =∑i∈Iφi (x)u0i +∑j∈JH (x)φj (x)uEj (12)where: x is the position vector in the 2D space; the node set I covers the whole domain Ω, while the setJ gathers only those nodes whose support ωj is even partially cut by Γd; φi is the piecewise linear nodalshape function; u0i are the basic degrees of freedom (DOFs) and uEj are the additional, or extended ones;H (x) is the generalized Heaviside step-function, defined according to:H (x) ={+1 if (x− x∗)T m > 0−1 if (x− x∗)T m < 0 (13)x∗ being the closest point projection of x onto Γd.To avoid locking of the discrete formulation when a collapse mechanism is developed in the structure aftercomplete crack growth, all the nodes belonging to the tip element are included in the set J .Now, let e1, e2, e3 be the vertex nodes of a generic triangular element e, whose domain is Ωe. If all the three3nodes belong to set J , the in-plane displacement field uh (x) reads:uh (x) ={uhx (x)uhy (x)}=[Φe... HΦe]u0e· · ·uEe x ∈ Ωe (14)where u0e and uEe respectively gather all the basic and extended nodal DOFs relevant to element e and Φeis the standard matrix of interpolation functions, i.e.:Φe =[φe1 0 φe2 0 φe3 00 φe1 0 φe2 0 φe3](15)The strain field is thus given by:εh (x) =εhx (x)εhy (x)2εhxy (x) =[Be... HBe]u0e· · ·uEe x ∈ Ωe\Γde (16)Γde being the portion of Γd inside element e and:Be =[φe1,x 0 φe2,x 0 φe3,x 00 φe1,y 0 φe2,y 0 φe3,yφe1,y φe1,x φe2,y φe2,x φe3,y φe3,x](17)Here a comma means derivative.The discretized displacement discontinuity along Γde is:[u]h (x) = uh∣∣∣∣Γ+de− uh∣∣∣∣Γ−de=∑j∈J2φj (x)uEj =[0... 2Φe]u0e· · ·uEe x ∈ Γde (18)During the analysis, at fixed crack length the damage Dh at Gauss points is continuously monitored; inall the elements where Dh exceeds the critical threshold, a crack segment should be introduced. However,according to the physics of the problem explained in the introduction, only one crack is allowed to propagateinside each damage band.To propagate Γd, the damage pattern in the process zone is recovered and best-fitted with a complete fourth-order polynomial. Crack growth is then chosen to be perpendicular to the direction of maximum curvatureof the interpolating polynomial at the crack tip.4 NUMERICAL SIMULATION OF THE WEDGE-SPLITTING TESTThe wedge splitting test configuration is shown in Fig. 2 (specimen thickness is 97 mm). Experimentally,a wedge is pushed from the left into the side notch so that two opposite forces (P ) act at the mouth ofthe initial crack and cause a mode I propagation. The ligament, where the process zone developes, is thussubject to bending.To simulate damage evolution in the bulk, the following values of model parameters have been adopted:E = 25200 MPa; ν = 0.2; a = 0.2; b = 0.1265 MPa; c = 10000; k = 6.838 · 10−7 MPa2; n = 12.The internal length scale for the non-local regularization has been assumed lc = 17.5 mm. These material430P200853070160PB=200Figure 2: Wedge splitting test. Geometry and loading conditions.Figure 3: Wedge splitting test. Load vs CMOD response.parameters correspond to a concrete tensile strength σM = 3.3 MPa and to a mode I fracture toughnessGf ≈ 100 J/m2.In the analysis, crack propagation is simulated under CMOD (crack mouth opening displacement) control.The numerical P vs CMOD response is compared in Fig. 3 to the available experimental data [7]. In viewof the rather rough calibration of the model, the softening branch can be considered to reproduce the actualresponse with an encouraging level of accuracy. The blue arrow in the figure points towards the onset ofcrack growth: the initial post peak strength reduction is thus caused by bulk damage only.In Fig. 4 level sets of damage D are reported in the deformed configuration, with displacements amplified100 times to appraise crack growth. It can be seen that, after an initial transient phase, damage evolvesmaintaining an almost constant bandwidth in the process zone ahead of the current crack tip.ACKNOWLEDGMENTSThis work has been carried out with the financial support of MIUR-PRIN 2002087915-2002 contract.REFERENCES[1] C. Comi and U. Perego. Criteria for mesh refinement in nonlocal damage finite element analyses.European Journal of Mechanics A/Solids, in press, 2004.510.90.80.70.60.50.40.30.20.10D(a) CMOD=0.1 mm10.90.80.70.60.50.40.30.20.10D(b) CMOD=0.2 mm10.90.80.70.60.50.40.30.20.10D(c) CMOD=0.3 mm10.90.80.70.60.50.40.30.20.10D(d) CMOD=0.4 mm10.90.80.70.60.50.40.30.20.10D(e) CMOD=0.5 mmFigure 4: Wedge splitting test. Simulated damage evolution and crack growth.[2] N. Moe¨s, J. Dolbow, and T. Belytschko. A finite element method for crack growth without remeshing.International Journal for Numerical Methods in Engineering, 46:131–150, 1999.[3] G.N. Wells and L.J. Sluys. A new method for modelling cohesive cracks using finite elements. Inter-national Journal for Numerical Methods in Engineering, 50:2667–2682, 2001.[4] S. Mariani and U. Perego. Extended finite element method for quasi-brittle fracture. InternationalJournal for Numerical Methods in Engineering, 58:103–126, 2003.[5] M. Jirasek. Nonlocal models for damage and fracture: Comparison of approaches. InternationalJournal of Solids and Structures, 35:4133–4145, 1998.[6] C. Comi. A non-local model with tension and compression damage mechanisms. European Journal ofMechanics A/Solids, 20:1–22, 2001.[7] E. Denarie´, V.E. Saouma, A. Iocco, and D. Varelas. Concrete fracture process zone characterizationwith fiber optics. ASCE Journal of Engineering Mechanics, 127:494–502, 2001.[8] C. Comi, S. Mariani, and U. Perego. From localized damage to discrete cohesive crack propagation innonlocal continua. In H.A. Mang, F.G. Rammerstorfer, and J. Eberhardsteiner, editors, Proceedings ofthe Fifth World Congress on Computational Mechanics (WCCM V). Vienna University of Technology,2002.6

Cohesive crack propagation in damaging concrete structures discretized by extended finite elements

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Cohesive crack propagation in damaging concrete structures discretized by extended finite elements

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