Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 44, 4, pp. 767-782, Warsaw 2006 MODELING OF STRAIN LOCALIZATION IN QUASI-BRITTLE MATERIALS WITH A COUPLED ELASTO-PLASTIC-DAMAGE MODEL Jerzy Bobiński Jacek Tejchman Faculty of Civil and Environmental Engineering, Gdansk University of Technology e-mail: bobin@pg.gda.pl; tejchmk@pg.gda.pl The paper presents results of numerical simulations of strain localization in quasi-brittle materials (like concrete) under plane strain conditions. Tomodel thematerial behaviour, an isotropic elasto-plastic-damagemo- del combining elasto-plasticity and scalar damage was used. An elasto- plastic constitutive law using a Drucker-Prager yield surface (in com- pression) andRankine yield surface (in tension)was defined. Amodified failure criterion by Rankine for the equivalent strain using an exponen- tial evolution law was assumed within damage mechanics. To obtain mesh-independent results of strain localization, themodel was enhanced by non-local terms in the softening regime. A four-point bending test of a concrete beamwith a single notch was numerically simulated using the finite element method. FE-results were compared with laboratory experiments. Key words: damagemechanics, elasto-plasticity, non-local theory, strain localization 1. Introduction Analysis of concrete elements is complex due to occurrence of strain locali- zation which is a fundamental phenomenon under both quasi-static and dy- namic conditions (Bazant, 1984, 2003; Wittmann et al., 1992; van Vliet and van Mier, 1996; Chen et al., 2001). It can occur in the form of cracks (if cohesive properties are dominant) or shear zones (if frictional properties pre- vail). The determination of the width and spacing of strain localization is crucial to evaluate thematerial strength at peak and in the post-peak regime. Concrete behaviour canbemodeledwithin continuummechanicsmodels using e.g.: non-linear elasticity (Palaniswamy andShah, 1974), fracture (Bazant and 768 J. Bobiński, J. Tejchman Cedolin, 1979; Hilleborg, 1985), endochronic theory (Bazant and Bhat, 1976; Bazant and Shieh, 1978), micro-plane theory (Bazant and Ozbolt, 1990; Jirasek, 1999), plasticity (Willam andWarnke, 1975; Pietruszczak et al., 1988; Menetrey and Willam, 1995; Bobinski and Tejchman, 2004), damage theory (Dragon and Mroz, 1979; Peerlings et al., 1998; Chen, 1999; Bobinski and Tejchman, 2005) and the coupled plastic-damage approach (Lemaitre, 1985; de Borst et al., 1999; Ibrahimbegovic et al., 2003, Salari et al., 2004), as well as discrete models using the lattice approach (Herrmann et al., 1989; Vervu- urt et al., 1994; Schlangen andGarboczi, 1997; Kozicki andTejchman, 2006a) and DEM (Sakaguchi and Mühlhaus, 1997; D’Addetta et al., 2002; Donze et al., 1999). To properly describe the strain localization within continuum mechanics, the models should be enhanced by the characteristic length of a micro-structure (de Borst et al., 1992; Chen et al., 2001). There are several approaches within continuum mechanics to include the characteristic length and to preserve well-posedness of the underlying incremental boundary value problem (de Borst et al., 1992) in quasi-brittle materials as: second-gradient (Chen et al., 2001; Peerlings et al., 1998; Pamin and de Borst, 1998; Pamin, 2004), non-local (Pijaudier-Cabot andBazant, 1987; Chen, 1999; Akkermann, 2000; Bobinski and Tejchman, 2004; Jirasek, 2004), and viscous ones (Sluys, 1992; Sluys and de Borst, 1994). Owing to them, objective and properly co- nvergent numerical solutions for localized deformation (mesh-insensitive load- displacementdiagramandmesh-insensitive deformationpattern) are achieved. Otherwise, FE-results are completely controlled by the size and orientation of the mesh, and thus produce unreliable results, i.e. strain localization beco- mes narrower uponmesh refinement (element size becomes the characteristic length) and computed force-displacement curves change considerably depen- ding on the width of the calculated localization. In addition, a premature divergence of incremental FE-calculations is often met. The aim of the present paper is to show the capability of an isotropic elasto-plastic-damage continuummodel to describe the strain localization and stiffness degradation in a concrete element during monotonous and cyclic four-point bending. Themodel is enhanced by the internal length of a micro- structure in the softening regime bymeans of non-local theory.TheFE-results are compared to corresponding laboratory experiments byHordijk (1991). The paper is a continuation ofFE-investigations of the strain localization in concre- te elements performed with the elasto-plastic model with non-local softening (Bobiński and Tejchman, 2004, 2006) and the non-local damage model with non-local softening (Bobiński and Tejchman 2005). These investigations have shown that the non-local theory allows us to obtain fully objective numerical solutions for boundary value problems including the strain localization. Modeling of strain localization in quasi-brittle materials... 769 2. Constitutive continuum model 2.1. Isotropic elasto-plastic model Anelasto-plasticmodelwith the isotropic hardeningand softening consists of two yield conditions. In the compression regime, the linear Drucker-Prager criterion is defined as (Abaqus, 1998; Bobinski and Tejchman, 2004, 2006) f1 = q+ptanϕ− ( 1− 1 3 tanϕ ) σc(κ1) (2.1) wherein q is the vonMises equivalent stress, p –mean stress, ϕ – internal fric- tion angle, σc – uniaxial compression yield stress, κ1 – hardening (softening) parameter equal to plastic strain in uniaxial compression ε p 11. The invariants p and q are p = 1 3 σkk and q = √ 3 2 sijsji (2.2) where σij – stress tensor and sij – deviatoric stress tensor. The flowpotential function is taken as g1 = q+ptanψ (2.3) where ψ denotes the dilatancy angle. The flow rule is assumed as dε p ij = dκ1 1− 1 3 tanψ ∂g1 ∂σij (2.4) In the tensile regime, the Rankine criterion is assumed (Bobinski and Tejch- man, 2006) with the yield function f2 =max{σ1,σ2,σ3}−σt(κ2) (2.5) where σ1, σ2 and σ3 denote the principal stresses, σt – tensile yield stress and κ2 – hardening (softening) parameter (equal to the maximum principal plastic strain ε p 1). The associated flow rule is assumed. Tomodel the concrete softening in tension, the exponential curve by Hordijk (1991) is chosen σt(κ2)= ft[(1+A1κ 3 2)exp(−A2κ2)−A3κ2] (2.6) where ft stands for the tensile strength of the concrete. The constants A1, A2 and A3 are A1 = c1 κu A2 = c2 κu A3 = 1 κu (1+ c31)exp(−c2) (2.7) where κu denotes the ultimate value of the softening parameter, c1 = 3 and c2 =6.93. The constitutive isotropic elasto-plastic model for concrete requires two elastic parameters: modulus of elasticity E and Poisson’s ratio ν, one com- presion plastic function σc = f(κ1), one tensile plastic function σt = f(κ2), internal friction angle ϕ and dilatancy angle ψ. 770 J. Bobiński, J. Tejchman 2.2. Isotropic damage model Themodel describes degradation of amaterial due tomicro-cracking with the aid of a single scalar damage parameter D growing from zero (undamaged state) to one (completely damaged state) (Bobinski andTejchman, 2005). The stress-strain relationship is represented by the following relationship σij =(1−D)Ceijklεkl (2.8) where Ceijkl is the linear elasticmaterial stiffnessmatrixand εkl – strain tensor. The damage parameter D acts as a stiffness reduction factor (Poisson’s ratio is not affected by damage and it remains constant). The growth of the damage variable is controlled by the damage threshold parameter κ, which is defined as the maximum of the equivalent strain measure ε̃ reached during the load history up to time t κ =max τ¬t ε̃(τ) (2.9) The loading function of damage is equal to f(ε̃,κ)= ε̃−max{κ,κ0} (2.10) where κ0 is the initial value of κ when damage starts. If the loading func- tion f is negative, damage does not develop. During monotonic loading, the parameter κ grows (it coincides with ε̃) and during unloading and reloading it remains constant. To describe the equivalent strain measure ε̃, a definition corresponding to the Rankine failure criterion (Jirasek, 2004) is adopted ε̃ = 1 E max{σeffi } (2.11) where E is the modulus of elasticity and σ eff i are the principal values of the effective stress σ eff ij σ eff ij = C e ijklε e kl (2.12) To describe the evolution of the damage parameter D, an exponential softe- ning law proposed by Peerlings et al. (1999) is assumed D =1− κ κ0 ( 1−α+αe−β(κ−κ0) ) (2.13) where α and β are material parameters. The constitutive isotropic damagemodel for concrete requires the following parameters: modulus of elasticity E, Poisson’s ratio ν, ft, κ0, α and β. Modeling of strain localization in quasi-brittle materials... 771 2.3. Coupled elasto-plastic damage model This model combines elasto-plasticity with scalar damage enhanced by non-locality and enables one to simulate the stiffness degradation during cyclic loading due to cracks and plastic strains. It follows the second-gradientmodel formulated byPamin anddeBorst (1999). It assumes that the total strains εij are equal to strains in an undamaged skeleton ε eff ij (effective strains). The plastic flow can occur only in the undamaged specimen, so the elasto-plastic model is defined in terms of effective stresses (Eq. (2.11)). As a consequence, the damage degradation does not affect plasticity. Equation (2.8) is modified as σij =(1−D)σeffij =(1−D)C e ijklε e kl (2.14) First, elasto-plastic calculations in the effective stress space are performed. After that the damage parameter D is calculated. The equivalent strain me- asure ε̃ (Eq. (2.11)) can be defined in terms of the total strain εij or in terms of the elastic strain εeij. In the first case, the equivalent strain measu- re ε̃ (Eq. (2.11)) is obtained by replacing the elastic strain εeij by the total strain εij only in Eq. (2.12). Finally, the stresses are obtained according to Eq. (2.14). 3. Non-local theory Todescribe the strain localization, to preservewell-posedness of the boundary value problem and to obtainmesh-independent FE-results, a non-local theory was used as a regularization technique (Pjaudier-Cabot and Bazant, 1987; Bobinski andTejchman, 2004). Usually, it is sufficient to treat non-locally only the variable controlling the material softening (Brinkgreve, 1994) (whereas stresses and strains remain local). It is assumed in elasto-plasticity that the softening parameter κ was non-local (Bobinski and Tejchman, 2004) κ(xk)= 1 A ∫ ω(r)κ(xk+r) dV with A = ∫ ω(r) dV (3.1) where xk are the coordinates of the considered (actual) point, r – distance measured from the point xk to other material points, ω – weighting function and A – weighted volume. As a weighting function ω, the Gauss distribution function for 2D problems is used ω(r)= 1 l √ π exp [ − (r l )2] (3.2) 772 J. Bobiński, J. Tejchman where l denotes the characteristic (internal) length connected to the micro- structure of the material. The averaging in Eq. (3.1) is restricted to a small representative area around eachmaterial point. The influence of points at the distance of r = 3l is only about 0.1% (Fig.1). The softening rates dκi are assumed according to themodified formula (Brinkgreve, 1994) (independently for both yield surfaces) dκi(xk)= (1−m)dκi(xk)+ m A ∫ [ω(r)dκi(xk +r)] dV (3.3) where m is the additional material parameter which should be greater than 1 to obtain mesh-independent results (Bobinski and Tejchman, 2004). Eq. (3.3) can be rewritten as (Brinkgreve, 1994) dκi(xk)= dκi(xk)+m (1 A ∫ [ω(r)dκi(xk +r)] dV −dκi(xk) ) (3.4) Since the rates of the hardening parameter are not known at the beginning of each iteration, extra sub-iterations are required to solve Eq. (3.4). To simplify the calculations, thenon-local rates are replacedby their approximations dκesti calculated on the basis of the known total strain rates (Brinkgreve, 1994) dκi(xk)≈ dκi(xk)+m (1 A ∫ [ω(r)dκesti (xk +r)] dV −dκesti (xk) ) (3.5) Fig. 1. Region of the influence of the characteristic length l and weighting function w In thedamagemechanicsmodel, the equivalent strainmeasure ε̃ is replaced in Eq. (2.11) by its non-local definition ε ε(xk)= 1 A ∫ ω(r)ε̃(xk+r) dV (3.6) In the coupled elasto-plastic-damage model, the non-locality can be intro- duced in elasto-plasticity or in damage.All three non-localmodelswere imple- mented in the Abaqus Standard program (Abaqus, 1999) with the aid of the Modeling of strain localization in quasi-brittle materials... 773 subroutine UMAT (user constitutive law definition) and UEL (user element definition) (Bobinski and Tejchman, 2004, 2005, 2006). The FE-simulations were performedunder plane strain conditions. A geometric nonlinearity (large displacements) was taken into account. A quadrilateral elements composed of fourdiagonally crossed triangleswere used to avoid volumetric locking (Groen, 1997). 4. FE-simulations The problem of a notched beam under four-point bending was experimental- ly investigated by Hordijk (1991) and numerically simulated both by Pamin (1994) with a second-gradient elasto-plasticmodel andby Simone et al. (2002) with a second-gradient damage model. The geometry of the specimen is gi- ven in Fig.2. The beam has a small notch (5× 10mm2) at the mid-span. The thickness of the beam is 50mm. The deformation is induced by impo- sing a vertical displacement of ∆v =0.25mm at two nodes at the top in the central part of the beam. The FE-calculations are carried out under plane strain conditions. Themodulus of elasticity is assumed to be E =40GPa and Poisson’s ratio ν = 0.2. FE-analyses are carried out with 3 different meshes: coarse (2152 triangular elements),medium (2332 triangular elements) andfine (4508 triangular elements) mesh (Fig.3). Fig. 2. Geometry and boundary conditions of the notched beam under bending [mm] Figure 4 shows load-displacement curves for different meshes with the elasto-plastic model using non-local softening. The plastic tensile curve is as- sumed according to Eqs. (2.6) and (2.7) with ft =2.3MPa and κu =7 ·10−3. Theplasticmaterial parameters in a compression regime has not any influence on the FE-results. The characteristic length is taken as l = 10mm and the parameter m =2.The calculated curves inFig.4 are similar for all FE-meshes and are in a good agreement with the experimental outcome (Hordijk, 1991). In the case of the coupled elasto-plastic-damage model, the calculations have been carried out only with the hardening elasto-plastic model combined 774 J. Bobiński, J. Tejchman Fig. 3. FE-meshes used in numerical simulations: coarse (a), medium (b) and fine mesh (c) Fig. 4. Calculated load-displacement curves for a beam under four-point bending for different meshes using the elasto-plastic model with non-local softening compared to the experiment (Hordijk, 1991) with damage, which has a better physical motivation (on the basis of the FE- results byPamin and deBorst (1999)). The crucial point is to obtain the start of damage and yielding almost at the same time. In the first step, for the sake of simplicity, the vonMises yield criterion with the yield stress σy =4.5MPa and a linear hardening parameter (with the modulus hp = E/2) are defined in the compression regime. In the damage regime, the following parameters are assumed: κ0 =4.6 ·10−5, α =0.92, β =200 and l =5mm. Modeling of strain localization in quasi-brittle materials... 775 Figure 5 shows load-displacement diagrams for the damagemodel and the coupled elasto-plastic-damagemodelwith the equivalent strainmeasure as the total strain and as the elastic strain. The damage model well agrees with the experimental curve, while the curve calculated with the coupled model with the total strain ε̃(εij) lies under, and the curve computed with the coupled model with the elastic strain ε̃(εeij) lies above the experimental curve. Fig. 5. Load-displacement curves for the damagemodel and coupled elasto-plastic-damagemodels using the total and elastic strains (κ0 =4.6 ·10−5, α =0.92, β =200 and l =5mm) Fig. 6. Simulated load-displacement curves for the coupled elasto-plastic-damage model with non-local softening using the total strain (κ0 =6 ·10−5, β =150, l =5mm) and the elastic strain (κ0 =5 ·10−5, β =400, l =5mm) compared to the experiment by Hordijk (1991) To obtain a better agreement with experiments, newmaterial parameters have been calibrated for the coupled elasto-plastic-damage model. The follo- wing new parameters κ0 and β have been chosen (l = 5mm): κ0 = 6 ·10−5 and β = 150 for the model with the equivalent strain measure equal to the total strain, and κ0 =5 ·10−5 and β =400 for themodel with the equivalent strainmeasure equal to the elastic strain.The improved load-displacement cu- 776 J. Bobiński, J. Tejchman rves are presented in Fig.6. An insignificant effect of the mesh discretization on the load-displacement curves is demonstrated in Fig.7. Fig. 7. Calculated load-displacement curves for different meshes (Fig.3) with the coupled elasto- plastic-damagemodel with non-local softening using the elastic (κ0 =6 ·10−5, β =150, l =5mm) (a) and the total strain (κ0 =5 ·10−5, β =400, l =5mm) (b) compared to the experiment by Hordijk (1991) The calculated contours of the non-local parameter κ above the notch are shown in Fig.8 for three models. The obtained results do not depend on the mesh size. The width of the localized zone is approximately equal to 25mm (2.5× l) within elasto-plasticity, 31mm (6.2× l) in damage mechanics, and 31mm (6.2× l) within elasto-plasticity combined with damage. A comparison with the experimental result by Hordijk (1991) for a beam subject to cycling loading has also been performed. Figures 9 and 10 show the obtained curves for the coupled elasto-plastic-damage model with total strains ε̃(εij) and elastic strains ε̃(ε e ij), respectively, using the improved set of thematerial parameters. The result with the coupled elasto-plastic damage model using total strains ε̃(εij) fits the experimental outcome (load reversals exhibit a proper gradual increase of the elastic stiffness degradation), whereas the results with the coupled elasto-plastic damage model using elastic strains ε̃(εeij) slightly overestimate the grade of the stiffness degradation. To obtain more accurate results in the second case, further improvement of thematerial parameters is needed again. The obtained FE-results with the non-local model are quantitatively in good agreement with corresponding numerical results by Pamin and de Borst (1999) obtained for the second-gradient elasto-plastic-damagemodel.Both the shapeof load-displacements curves and thewidthof localized zones are similar. Thus, the non-local model is as effective as the second-gradient model in the numerical description of strain localization. Modeling of strain localization in quasi-brittle materials... 777 Fig. 8. Calculated contours of the non-local parameter κ near the notch for a beam under four-point bending for different FE-meshes (a) coarse, (b) medium and (c) fine mesh and different models (A) elasto-plasticity, (B) damage, (C) elasto-plasticity-damage (with elastic strains) 778 J. Bobiński, J. Tejchman Fig. 9. Calculated load-displacement curves for the coupled elasto-plastic-damage model with non-local softening using the total strain (κ0 =6 ·10−5, β =150, l =5mm) during cycling beam loading Fig. 10. Calculated load-displacement curves for the coupled elasto-plastic-damage model with non-local softening using the elastic strain (κ0 =5 ·10−5, β =400, l =5mm) during cycling beam loading 5. Conclusions The FE-calculations show that the hardening isotropic elasto-plastic model with softening isotropic damage enhanced by the characteristic length of a micro-structure can properly reproduce the experimental load-displacement diagrams and strain localization in quasi-brittle materials during monotonic and cyclic bending. During cycling loading, the fully coupled elasto-plastic damage model with fewmodel parameters is able to reflect both the stiffness degradation and the irreversible strains. The FE-results with respect to the load-displacement curve and width of strain localization do not suffer from mesh sensitivity. The most realistic results are obtained with the equivalent strain measure assumed as the total strain. The calculated width of the loca- lized zone in the concrete element is larger for the damage model and elasto- plastic model combined with damage (6.2× l) than with the elasto-plastic model (2.5× l) for the same characteristic length. 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Willam K.J.,Warnke E.P., 1975, Constitutivemodel for the triaxial beha- viour of concrete, IABSE Seminar onConcrete Structures Subjected to Triaxial Stress, Bergamo, Italy, 1-31 50. Wittmann F.H., Mihashi H., Nomura N., 1992, Size effect on fracture energy using three-point bend tests,Materials and Structures, 25, 327-334 Modelowanie lokalizacji odkształceń w materiałach quasi-kruchych z zastosowaniem modelu sprężysto-plastycznego Streszczenie Wartykule przedstawionowyniki symulacji numerycznych lokalizacji odkształceń w materiałach quasi-kruchych (jak beton) w płaskim stanie odkształcenia. Do opi- sumateriału przyjęto izotropowymodel sprężysto-plastyczny-zniszczeniowyuwzględ- niający prawo sprężysto-plastyczne ze skalarną degradacją sztywności. W przypad- ku prawa sprężysto-plastycznego przyjęto kryterium plastyczności Druckera-Pragera w ściskaniu i kryteriumplastyczności Rankine’a w rozciąganiu. Degradację sprężystą opisano z wykorzystaniem definicji odkształcenia zastępczego według warunku Ran- kine’a i wykładniczymprawemewolucji.W celu otrzymaniawynikówniezależnych od siatki elementówskończonych,wobszarzeosłabieniaprzyjęto teorięnielokalną.Przed- stawiono wyniki symulacji numerycznych dla belki betonowej z nacięciem obciążonej dwoma siłami skupionymi. Wyniki numeryczne porównane z wynikami doświadczal- nymi. Manuscript received February 2, 2006; accepted for print July 6, 2006