JOURNAL OF THEORETICAL AND APPLIED MECHANICS 44, 3, pp. 485-503, Warsaw 2006 EFFECT OF CHARACTERISTIC LENGTH ON NONLOCAL PREDICTION OF DAMAGE AND FRACTURE IN CONCRETE Halina Egner Jacek J. Skrzypek Władysław Egner Institute of Applied Mechanics, Cracow University of Technology, Poland e-mail: halina.egner@pk.edu.pl The paper deals with a new nonlocal integral-typemodel for simulation of an anisotropic, localised damage and for prediction of combined failuremodes in a plane-notched concrete specimen. The nonlocal incremental-type model of the elastic-brittle-damagematerial is an extension of the relevant localmodel originated byMurakami andKamiya (1997), modified later to the incremen- tal form by Kuna-Ciskał and Skrzypek (2004). In order to avoid the mesh- dependence and ensure stability and convergence, two localisation limiters are examined: the concept of Nonlocal Averaging (NA) and the additional Cut- offAlgorithm (CA), applied to damage conjugate thermodynamic forces. The elastic-brittle damage constitutive equations are formulated in an incremen- tal and nonlocal fashion, by the use of a damage dissipation potential defined in the space of averaged regularised damage variables instead of the corre- sponding local ones. The Gauss distribution function is taken as the weight function for the definition of a nonlocal continuum. In order to assess how much the new nonlocal model is capable of describing localised strain and damage fields, an example of the plane double-notched specimen of Nooru- Mohammed (1992) is examined.Much emphasis is put to proper choice of the characteristic length of the nonlocal continuum.Convergence of themesh size is proved for both, the damage incubation period and fracture, when a single localisation limiter (NA) is active. Key words: nonlocal approach, anisotropic damage, characteristic length, mesh-dependence 1. Introduction The use of classical local constitutive models is insufficient for problems whe- re a strong strain softening effect occurs. In case of inelastic material beha- 486 H. Egner et al. viour, two dissipative processes are responsible for the strain softening: (vi- sco)plasticity and/or damage (cf. Hansen and Schreyer, 1994; Abu Al-Rub and Voyiadjis, 2003). In the case of Local Models (LM), the stress at a gi- ven point is assumed to be uniquely determined by the strain history at this point only. However, as the (visco)plasticity and damage frequently localize over narrow zones of a continuum, statistical homogeneity in a representative volume element is lost. Hence, the characteristic length scale has to be intro- duced into the nonlocal model (NL) in order to account for the influence of an internal state variable also at neighbouring points. From a computational point of view, in the case of the localised phenomena, ill-posedness of the bo- undary value problem andmesh sensitivity of finite element computations are met. In particular, incorporation of viscosity retains ellipticity of the problem, such that the well-posedness is preserved because the viscosity implicitly in- troduces length-scalemeasures that reduce the strain and damage localisation (cf.Wang et al., 1996; Dornowski and Perzyna, 2000; Glema et al., 2000). On the other hand, if (visco)plasticity is not accounted for, some computational localisation limiters should be used provided by the concept of nonlocal we- ighted averaging (cf. Bažant, 1984; Pijaudier-Cabot andBažant, 1987; Bažant andPijaudier-Cabot, 1988; Jirasek, 1998; Comi, 2001; Comi andPerego, 2004; Voyijadis and AbuAl-Rub, 2002). If f(x) is a local field in a volume V , the corresponding nonlocal field is defined as f(x)= 1 V ∫ V h(x,ξ)f(ξ) dξ (1.1) where h(x,ξ) is a monotonically decreasing weight function, defined in such a way that a uniform field is not altered by it V (ξ)= ∫ V h(r) dV r = |x−ξ| (1.2) As the weight function h(r), the Gauss distribution function h(r)= exp ( − r2 2l2 ) (1.3) or the bell-shape function h(r)=      ( 1− r2 R2 )2 0¬ r ¬ R 0 R ¬ r (1.4) Effect of characteristic length on nonlocal prediction... 487 are frequently used. In (1.3) l is the internal length of the nonlocal continuum, whereas R is called the interaction length that is related, but not equal, to the internal length l. If the orientation ofmaterial fibres is considered, amore complicated averaging operator might be used, where not only the distance between the points x and ξ, but also the orientation of principal axes at these points are accounted for (cf. Bažant, 1994). In the present paper, the nonlocal measures (NA) of the force conjugates {Y,B} are defined instead of the local ones {Y,B} previously used (LA). The use of another Cut-off Algorithm (CA), originated byMurakami and Liu (1995) Ŷ = kY, where k = 1 if Yeq ¬ Yu or k = Yu/Yeq if Yeq > Yu is also tested from the point of view of convergence. The nonlocal measure Y of the strain energy release rate Y was earlier used by Pijaudier-Cabot and Bažant (1987) and by Comi and Perego (2004) for a simple isotropic elastic-damage model. By contrast, amore extendedcase is considered in thepresentpaper,where the anisotropic damagemeasure D is used and an additional scalar parameter β stands for the damage hardening. The other possibility is to directly avera- ge the damage as suggested by Bažant and Pijaudier-Cabot (1988) or strain (cf. Bažant and Liu, 1988). The nonlocal variables {Y,B}, with the cut-off algorithm {Ŷ, B̂} if necessary, affect the nonlocal definition of the damage dissipation potential F(Y,B) instead of the traditional one F(Y,B), when the local approach is used (cf. Murakami and Kamiya, 1997). The developed model is capable of capturing the damage anisotropy and deactivation (incu- bation period) as well as the failuremechanism (fracture). The essential point is how to properly choose the internal length parameter of the nonlocal con- tinuum. This length may be assessed by experimental comparison of energy in a specimen where damage is constrained to remain diffuse, and another one where damage localizes to yield a single crack (cf. Mazars and Pijaudier- Cabot, 1989, 1996). It may also be established from the maximum size of the aggregate in concrete da, such that l ≈ 3da holds (cf. also Saouridis and Mazars, 1992). Some particular suggestions can be found in the comparative study on different models for concrete: local (Ottosen) or nonlocal (nonlocal damage and gradient plasticity). The internal length values are thus set to l ≈ 5mm for the gradient plasticity model, and to l ≈ 8mm for the non- local damage model. On the other hand, Comi and Perego (2004) used for the nonlocal concrete model the value l = 1.1mm. In what follows, a proper characteristic length is numerically assessed from simulation tests on damage and fracture prediction in the double-notched specimen of Nooru-Mohammed (1992). Different values of characteristic lengths for concrete are numerically 488 H. Egner et al. tested in the present paper, ranging from 0mm (local) to 20mm, to finally assess the value l = 7.5mm as the ”optimal” one, to preserve characteristic damage incubation and ultimate localised failure prediction without violating the stability andmesh convergence. 2. Total form of the local elastic brittle damage constitutive model When a total stress-strain formulation is used, the general thermodynamically based theory of local constitutive and evolution equations of an elastic-brittle damagedmaterial is the key for further extension (cf.Murakami andKamiya, 1997). The Helmholtz free energy is defined in a local fashion as a function of the elastic strain tensor εe, the second rank damage tensor D, and the scalar damage hardening variable β. The Helmholtz free energy, decomposed into the elastic ψe(εe,D) and damage ψd(β) terms, is postulated as a state potential ρψ(εe,D,β)= ρψe(εe,D)+ρψd(β) (2.1) FollowingMurakami andKamiya assumptions, both terms of free energy (2.1) are represented as ρψe(εe,D)= 1 2 λ(trεe)2+µtr(εe ·εe)+η1(trε e)2 trD+ +η2 tr(ε e ·εe)trD+η3 trε e tr(εe ·D)+η4 tr(ε ∗e ·ε∗e ·D) (2.2) ρψd(β)= 1 2 Kdβ 2 where λ and µ are Lamè constants for the undamagedmaterial, whereas η1, η2, η3, η4 and Kd are the damage material constants. In order to properly describe the unilateral damage response under tension or compression, themodified elastic strain tensor ε∗e is defined in theprincipal strain co-ordinate system ε∗eI = 〈ε e I〉+ ζ〈−ε e I〉= k(ε e I)ε e I ζ ∈ 〈0,1〉 k(εeI)= kI = H(ε e I)+ ζH(−ε e I) I =1,2,3 (2.3) where 〈·〉 denotes the Macauley bracket, H(·) is the Heaviside step function, εeI(I = 1,2,3) are the principal values of ε e, and ζ is an additional material constant responsible for the unilateral damage response effect under tension or Effect of characteristic length on nonlocal prediction... 489 compression (cf. Krajcinovic, 1996). For ζ =1 themodified strain tensor ε∗e is identical to εe, so that the unilateral damage (crack opening/closure) effect is not accounted for. In contrast, for ζ =0, the strain tensor ε∗e is modified in such a way that the negative principal strain components are replaced by zeros, whereas the positive ones remain unchanged. When the general coordinate system is used, the modified strain tensor ε∗e is expressed as follows (cf. Hayakawa andMurakami, 1997 ) ε∗eij = 3 ∑ I=1 ε∗eI QIiQIj = 3 ∑ I=1 k(εeI)ε e IQIiQIj = Bijklε e kl (2.4) where Bijkl = ∑3 I=1 k(εI)QIiQIjQIkQIl is a fourth rank tensor built of the direction cosines between the principal strain axes and the current spatial system. The following local, total stress–strain relations of the anisotropic elasticity coupled with damage are furnished from (2.2)1 according to the conventional procedure of the thermodynamic formalism σ= ∂(ρψ) ∂εe = [λ(trεe)+2η1(trε e)(trD)+η3 tr(ε e ·D)]I+ (2.5) +2[µ+η2(trD)]ε e+η3 tr(ε e)D+η4(ε ∗e ·D+D ·ε∗e) : ∂ε∗e ∂εe When the vector-matrix notation is used, the above reduces to {σ}= [Λs(D)]{εe} (2.6) where [Λs(D)] stands for the locally defined secant stiffness matrix. The thermodynamic force conjugates of the internal state variables {D,β} are defined in a local form as follows Y=−ρ ∂ψe ∂D =−[η1(trε e)2+η2 tr(ε e ·εe)]I−η3(trε e)εe−η4ε ∗e ·ε∗e (2.7) B = ρ ∂ψd ∂β = Kdβ Thedamagedissipationpotential, defined in the space of local force conjugates {Y,−B}, is assumed in the form (cf. Murakami andKamiya, 1997) F(Y,B)= Yeq− (B0+B)=0 Yeq = √ 1 2 Y :L :Y (2.8) 490 H. Egner et al. where B0 and B stand for the initial damage threshold and the subsequent damage force conjugate, respectively. The fourth-rank tensor L is definedhere in a simplified way Lijkl = 1 2 (δikδjl+ δilδjk) (2.9) However, in a more general case, it may also be assumed as a tensor function of damage L(D), linear in D (cf. Hayakawa and Murakami, 1997; Bielski et al., 2006). Finally, the local damage evolution equations are established from the damage potential by the normality rule Ḋ= λ̇d ∂F ∂Y β̇ = λ̇d (2.10) where the consistency condition is used to eliminate λ̇d λ̇d = α∂F ∂Y : Ẏ ∂B ∂β = α L :Y 2KdYeq : Ẏ (2.11) 3. Nonlocal formulation of the damage dissipation potential and evolution equations Local state equations (2.1) through (2.6), combined with the local evolution law for Ḋ and β̇, were applied by Kuna-Ciskał and Skrzypek (2004). This approach, however, is not capable of predicting damage evolution in the case of localised damage and strain fields because of a spuriousmesh effect. Inwhat follows, the Nonlocal Approach (NA) to damage dissipation and evolution is proposed. In general, to avoid the singularity of Y at the crack tip when the mesh size tends to zero, the Cut-off Algorithm (CA) may optionally be used in the neighbourhood of the crack tip, according to the scheme (cf. Skrzypek et al., 2005a, 2005b) Ŷ= kY k =      1 if Yeq ¬ Yu Yu Yeq if Yeq > Yu (3.1) where the cut-off factor k is defined as follows k = Yu Yeq = B0+B Yeq (3.2) Effect of characteristic length on nonlocal prediction... 491 and B0 stands for the initial damage threshold. The new variable Ŷ is next subjected to the nonlocal treatment (NA) Ŷ, according to the following for- mula (cf. e.g. Pijaudier-Cabot and Bažant, 1987; Comi and Perego, 2004) Ŷ= ∫ Ωd Ŷ(ξ)ϕ(x,ξ) dΩd ∫ Ωd ϕ(x,ξ) dΩd ϕ(x,ξ)= exp [ − (d(x,ξ) d∗ )2] (3.3) where ϕ(x,ξ) is the weight function and d∗ stands for the internal length of the nonlocal continuum. The damage dissipation potential in the space of nonlocal force conjugates {Ŷ,−B} is assumed inananalogous formas in the local space {Y,−B},where only the isotropic hardening is accounted for (2.8) F(Ŷ,B)= Ŷ eq− (B0+B)= 0 Ŷ eq = √ 1 2 Ŷ :L : Ŷ (3.4) In equation (3.4), B stands for the nonlocal damage force conjugate of the nonlocal damage hardening variable β. The nonlocal evolution equations for β̇ and Ḋ are finally established from the normality rule instead of (2.10) Ḋ= λ̇d ∂F ∂Ŷ β̇ = λ̇d (3.5) where the consistency condition is used to calculate λ̇d Ḟ =0= ∂F ∂Ŷ : . Ŷ+ ∂F ∂B Ḃ λ̇d = α L : . Ŷ 2KdŶ eq : . Ŷ (3.6) A factor α = 1 or α = 0 is used for the active or passive damage growth, respectively. 4. Incremental formulation of the nonlocal model and the failure criterion In the case of a nonlocal continuum, new nonlocal variables Y and B or, al- ternatively, Ŷ and B are used in the evolution equations only, instead of the local ones Y and B. Hence, the internal damage variables are also defined in 492 H. Egner et al. a nonlocal sense, e.g. D and β. Other state variables, σ and εe are not sub- jected to the nonlocal averaging, however the locally-defined siffness matrix Λs(D) in (2.6) has to be replacedby the new, nonlocally prescribed secantma- trix Λ s (D) that accounts for damage nonlocality. In the paper byKuna-Ciskał and Skrzypek (2004), the incremental constitutive equations were derived in a local sense, to enable introduction of the general failure criterion based on Drucker’s material stability postulate, and to ensure convergence and nume- rical stability.When the nonlocal approach is used, the new nonlocal effective tangent stiffness matrix Λ t (εe,D) has to be defined instead of the local one Λt(εe,D). To this end, the local secant stiffness matrix Λs(D) in (2.6) has to bemodified in a nonlocal sense, Λ s (D), accounting for damage nonlocality (3.5). Then, the new nonlocal effective tangent stiffness matrix Λ t (εe,D) is established to yield the incremental constitutive equation as follows dσ=Λ s : dεe+εe : ∂Λ s ∂D : dD (4.1) Finally, applying (3.5) to obtain nonlocal damage increments dD on dεe, the following incremental state equation is derived (cf. Kuna-Ciskał and Skrzypek 2004) dσ= [ Λ s +αεe : ∂Λ s ∂D : ∂D ∂εe ] : dεe =Λ t (εe,D) : dεe or (4.2) {dσ}= [Λ t (εe,D)]{dεe} The square bracket in (4.2) represents the newnonlocal effective tangent stiff- ness Λ t (εe,D) that follows damage nonlocality D (3.5). In order to introduce the general failure criterion, Drucker’s material stability postulate is adopted dσijdεij > 0 (4.3) Substituting for dσij formula (4.2) into stability criterion (4.3), we obtain ∂2ψ ∂εij∂εkl dεijdεkl = Hijkldεijdεkl > 0 (4.4) The nonlocally defined quadratic form (∂2ψ/∂εij∂εkl)dεijdεkl must be posi- tive definite for arbitrary values of the components dεij. Hence, eventually, condition (4.4) requires that thenonlocalHessianmatrix Hbepositivedefinite (cf. Chen and Han, 1995). Effect of characteristic length on nonlocal prediction... 493 The element tangent stiffness matrix is used for the quasi-Newton algo- rithm for the first iteration step of solving non-linear equation (4.2) as long as failure criterion (4.4) holds. The stiffness of the element in the FE mesh that have come to failure is next reduced to zero. As a consequence, the failed element is completely released fromstress andanappropriate stress redistribu- tionoccurs in theneighbouring elements to ensure the global equilibrium.Note that the above failure criterion, (4.4), assumes a brittle failuremechanism.Ho- wever, when a broader class of materials is considered, a post-peak softening regime can also be admittedwhichwould result in strain localisation such that a smooth drop in stiffness of elements that come to failure is met (cf. Bielski et al., 2006). 5. Numerical simulation of nonlocal damage and fracture in concrete under plane stress conditions 5.1. Computational algorithm for nonlocal description of damage and fracture in an elastic-damage material The iteration of the global equilibrium of a system is performed by ABAQUSwith the use of Newton-Raphsonmethod.All variables are updated by the end of an increment, after the convergence is achieved. The physical relations are integrated at a point level (Gauss point of an element) bymeans of the user-supplied subroutine UMAT, starting from the known equilibrium state and for the elastic strain increment given in each iteration. The output information – stresses and all other state variables – is updated by the end of the integration increment and so are both stiffness matrices, Λ s (D) (secant) and Λ t (εe,D) (tangent), accounting for damage nonlocality. The integration is performedwith explicit forwardEuler’s scheme.The de- rivatives (stiffness) are knownat the startingpoint andkept constant along the increment. Such an approach may successfully be used only for a sufficiently small incremental step. The particular form of the stiffness matrix depends on the state variables as well as on the kind of deformation process taking place through a strain increment. Namely, it depends on whether the process is active or passive. ”Active” (loading) denotes a process which implies evolution of the limit sur- face (damage evolution); ”passive” (unloading) stands for changes inside the limit surface (no damage evolution); ”neutral” denotes a process tangent to the limit surface (no damage evolution). 494 H. Egner et al. The integral-type nonlocal definition of variables is furnished by means of another subroutine URDFIL, which gives an access to the file with results during the analysis. The subroutine is called up at the end of any increment in which new information is written to the results file. The local variables from all integration points are written to an array and then subjected to the nonlocal treatment (NA) according to formulae (3.1) through (3.3). The array is placed in theCOMMONblock, hence the nonlocal variables from the end of the present increment are accessible in all user routines for the next increment. When applying the algorithm (CA)+(NA) (see Section 3), onemore nume- rical operation is neccessary. After ”cutting-off” CAand”weighted averaging” NA, the value of Ŷ eq has to be additionally shifted to meet the critical sur- face at a particular integration point in which it was maximal before cutting; otherwise the damage evolution process would be blocked. 5.2. Material data, geometry and loading A double-edge notched plane-stress specimen follows an experiment car- ried out by Nooru-Mohammed experiment (1992) and Nooru-Mohammed et al. (1993). The experiment enabled analysis of various combinations of shear and tension under controlled displacement. Themodel was investigated by di Prisco et al. (2000) to simulate fracturebymeans of three approaches: the local model, the gradient plasticitymodel and the nonlocal damagemodel. Inwhat follows, in order to prevailMode I crack growth under tension, the shear com- ponent was excluded (δt =0). Thematerial data for a high strength concrete that describe the basic Murakami-Kamiya model are taken after Murakami andKamiya (1997) (see Fig.1). Fig. 1. Double-edge notched specimen configuration (cf di Prisco et al., 2000) and material data (cf Murakami and Kamiya, 1997) Effect of characteristic length on nonlocal prediction... 495 5.3. Simulation of fracture in a double-notched specimen Assuming a uniform normal displacement δn applied at the top of the specimen shown in Fig.1 (δt =0), a complete process of damage growth and fracture is simulated until the ultimate failure of the specimen is predicted. A combined non-symmetric tension/shear failure mode is developed due to the non-symmetric boundary conditions used (Fig.1). Two zones of failed elements where the ellipticity is lost (checking nonlocal Hessian matrix H (4.4)) are spreading inwards in opposite directions from the notches as long as the ultimate fracturemechanism is not achieved. The releasing of consecutive failed elements from stresses results in stress redistribution in neighbouring (non-failed) elements. The distribution of stresses along the MN line (Fig.1) becomesmore andmore non-uniform, finally yielding strong stress localisation in front of two failed zones that come into touch when the ultimate fracture is met. Fig. 2. FEMmeshes for convergence tests 496 H. Egner et al. To check the influence of characteristic length of nonlocal continuum (4.2) on the damage evolution and fracture processes, the following values of d∗ = 2.5, 5.0, 7.5, 10.0, 20.0mm are tested. The range of values is taken on the basis of di Prisco et al. (2000), where for the nonlocal damage model d∗ ≈ 8mm is adopted. The finite element size must be lower than the charac- teristic length to make the nonlocal approach active, so the rectangular mesh 2.5mm×2.5mm is adoptedhere (Fig.2a). Othermeshes, shown inFig.2b and Fig.2c are defined for the convergence test. Theeffect of increasingvalue of d∗ on Y eq is shown inFig.3 andFig.4.The characteristic length defines the area over which Yeq is averaged, according to (3.3). The bigger d∗ – the larger area, the more balanced and lower values of the averaged variable around the present integration point (Fig.3), and the less advanced damage and fracture process at a chosen level of load (Fig.4). Fig. 3. Effect of the characteristic length on Yeq localization – precritical stage The increasing value of d∗ clearlymakes the fracture progress slower, as shown in Fig.4. Depending on the value of d∗, at the same post-critical loading step (54), a different advance of fracture is met. In other words, an increase of d∗ results in an increase of both critical displacements shown in Fig.5. The value of the ”incubation” displacement (Fig.5) denotes the displace- ment of the edge DC (Fig.1) at the instant of macro-crack appearance, while the ”fracture” displacement means the displacement of the same edge at the Effect of characteristic length on nonlocal prediction... 497 Fig. 4. Effect of the characteristic length on Yeq localization – postcritical stage Fig. 5. Effect of the characteristic length on critical displacements: incubation and failure overall failure of the structural element. It is observed that, for d∗ ¬ 7.5mm, the second critical displacement at fracture becomes (almost) non-sensitive to the characteristic length size. 5.4. Effect of the characteristic length on the equivalent crack shape Simulations of the crack growth by the nonlocal approach developed in the present paper depend on the characteristic length of a nonlocal continuum (cf. Nooru-Mohammed et al., 1993; Skrzypek et al., 2005a, 2005b). When the (NA) algorithm is used, the increasingwidth of the crack is observedwhen the 498 H. Egner et al. characteristic length grows. At the same time, the direction ofmacro-cracking changes: the lower d∗ the more the crack shape tends to a straight form (see Fig.6). Fig. 6. Effect of d∗ onmesh deformation (crack pattern) at failure Fig. 7. A test on the mesh shape dependence (d∗ =7.5mm): (a) regular rectangular mesh, (b) irregular triangularmesh Letus consider themesh-dependenceof thenumerical simulations ofdama- ge and fracture. The local approach is element shape-dependent. For different finite element shapes, different crack patterns are usually obtained. Namely, when the local approach is used, the crack may propagate from the current failed element to the neighbouring one, the edge of which is shared with the corresponding edge of the current failed element. In other words, the element shape restricts possible directions of crack propagation (cfMurakami andLiu, Effect of characteristic length on nonlocal prediction... 499 1995;Kuna-Ciskał, 1999). By contrast, the nonlocal approachpresented in this paperallows obtaining the crackpattern independentof the element shape (see Fig.7). Eventually, changing the size of an element a convergence test was per- formed. Results shown in Fig.8 proved capability of the developed nonlocal model to properly describe both the pre-critical damage evolution during in- cubation period as well as the post–critical crack growth period. It is shown that the use of an additional localisation limiter, the Cut-offAlgorithm (CA), is not necessary tomeet a convergence. The use of a single localization limiter, by the Nonlocal Averaging (NA) only, is capable of convergent prediction of both critical displacements – at the crack initiation and the fracture. Fig. 8. Convergence tests: (a) no localization limiters (local approach), (b) one localization limiter (NA), (c) two localization limiters (NA+CA) 500 H. Egner et al. 6. Conclusions: capability of localization limiters • The modified local Murakami-Kamiya model of an elastic-damage ma- terial is capable of qualitative simulating secondary crack growth, but a severe mesh dependence is observed when the local approach is used. • Asufficient nonlocal extension of themodifiedMurakami-Kamiyamodel of the elastic-brittle damage (NMMK) is achieved by the use of the nonlocal damage variable definition (NA). No additional application of the Cut-off Algorithm (CA) is necessary to meet a convergence (mesh independence). • The appropriate choice of the internal length of a nonlocal continuum depends on the characteristic dimension of the model. If too small. i.e. close to the local approach– the convergence is lost. If too large – the spe- cific crack topology is lost (smeared too much). In the example used for simulation of the crack growth in a double-notched concrete specimen, the notch dimension is 5mm×25mm, whereas the ”optimal” internal length is established to 7.5mm. • The topology of secondary cracks, obtained by simulation when the NMMKmodel is used, is similar to these obtained inbenchmark tests by di Prisco et al. The same double-notched sample geometry is incorpora- ted, but thematerial used in the reference test (concrete) slightly differs from the one used in the present simulation. Besides, simple loading is used, δt =0. • The mesh size convergence is proved for both critical displacements, at incubation and at fracture. Acknowledgements TheGrantPB807/T07/2003/25andMNiI/SGI2800/PK/079/2005supported by the State Committee for Scientific Research (KBN) in Poland are gratefully acknow- ledged. References 1. Abu Al.-Rub R.K., Voyiadjis G.Z., 2003, On the coupling of anisotropic damage and plasticity models for ductile materials, Int. J. Solids Struct., 40, 2611-2643 Effect of characteristic length on nonlocal prediction... 501 2. Bažant Z.P., 1984, Imbricate continuum and its variational derivation, J. Eng. Mechanics, ASCE, 110, 1593-1712 3. BažantZ.P., 1994,Nonlocal damage theorybasedonmicromechanicsof crack interaction, J. Eng. Mech., ASCE, 120, 593-617 4. Bažant Z.P., Liu F.-B., 1988,Nonlocal smeared crackingmodel for concrete fracture, J. Eng. Mechanics, ASCE, 114, 2493-2510 5. Bažant Z.P., Pijaudier-Cabot G., 1988, Nonlocal continuum damage, lo- calization, instability and convergence, J. Appl. Mech., ASME, 55, 287-293 6. Bielski J., Skrzypek J.J., Kuna-Ciskał H., 2006, Implementation of a model of coupled elastic-plastic unilateral damagematerial to finite element code, Int. J. Damage Mech., 15, 5-39 7. Chaboche J.L., 1993,Development of continuumdamagemechanics of elastic solids sustaining anisotropic and unilateral damage, Int. J. Damage Mech., 2, 311-329 8. Chaboche J.L., Lesne P.M., Maire J.F., 1995, Continuum damage me- chanics, anisotropy and damage deactivation for brittle materials like concrete and ceramic composites, Int. J. Damage Mech., 4, 5-21 9. Chen W.F., Han D.J., 1995, Plasticity for Structural Engineers, Springer- Verlag, NewYork 10. Comi C., 2001, A nonlocal model with tension and compression damage me- chanisms,Eur. J. Mech. A/Solids, 20, 1-22 11. Comi C., Perego U., 2004, Criteria for mesh refinement in nonlocal damage finite element analysis,Eur. J. Mech. A/Solids, 23, 615-632 12. DeBorstR., 2002, Fracture in quasi-brittlematerials: a review of continuum damage-based approaches,Eng. Fracture Mech., 69, 95-112 13. Di Prisco M., Ferrara L., Meftah F., Pamin J., de Borst R., Ma- zars J.,Reynouard J.M., 2000,Mixedmode fracture in plain and reinforced concrete: some results on benchmark tests, Int. J. Fracture, 103, 127-148 14. Dornowski W., Perzyna P., 2000, Localization phenomena in thermo- viscoplastic flow processes under cyclic dynamic loadings,Comp. Assisted Me- chanics in Eng. Sciences, 7, 117-160 15. Glema A., Łodygowski T., Perzyna P., 2000, Interaction of deformation waves and localization phenomena in inelastic solids,Comp.Meth. Appl. Mech. Eng., 183, 123-140 16. HansenN.R., SchreyerH.L., 1994,A thermodynamically consistent frame- work for theories of elastoplasticity coupledwith damage, Int. J. Solids Struct., 34, 359-389 502 H. Egner et al. 17. Hayakawa K., Murakami S., 1997, Thermodynamical modeling of elastic- plastic damage and experimental validation of damage potential, Int. J. Da- mage Mech., 6, 333-363 18. Jirasek M., 1998, Nonlocal models for damage and fracture: comparison of approaches, Int. J. Solids Structures, 35, 4133-4145 19. Krajcinovic D., 1996,Damage Mechanics, Elsevier, Amsterdam 20. Kuna-Ciskał H., 1999, Local CDM based approach to fracture of elastic- brittle structures,Technische Mechanik, 19, 4, 351-361 21. Kuna-CiskałH., SkrzypekJ.J., 2004,CDMbasedmodellingof damageand fracturemechanisms in concrete under tension and compression,Eng. Fracture Mech., 71, 681-698 22. Mazars J., Pijaudier-Cabot G., 1989, Continuum damage theory – appli- cation to concrete, J. Eng. Mech., ASCE, 115, 345-365 23. Mazars J., Pijaudier-Cabot G., 1996, Fromdamage to fracturemechanics and conversely: a combinedapproach, Int. J. Solids Structures,33, 20/22,3327- 3342 24. Murakami S., Kamiya K., 1997, Constitutive and damage evolution equ- ations of elastic-brittlematerials based on irreversible thermodynamics, Int. J. Solids Struct., 39, 4, 473-486 25. Murakami S., Liu Y., 1995, Mesh-dependence in local approach to creep fracture, Int. J. Damage Mech., 4, 230-250 26. Nooru-Mohammed M.B., 1992,Mixedmode fracture of concrete: an experi- mental approach, PhDThesis, Delft Univ. of Technology, p.151 27. Nooru-Mohammed M.B., Schlangen E., van Mier J.G.M., 1993, Expe- rimental and numerical study on the behaviour of concrete subjected to biaxial tension and shear,Adv. Cement Based Materials, 1, 22-37 28. Pijaudier-Cabot G., Bažant Z.P., 1987, Nonlocal damage theory, J. Eng. Mechanics, ASCE, 113, 1512-1533 29. SaouridisC.,MazarsJ., 1992,Predictionof failureandsizeeffect in concrete via a bi-scale damage approach,Eng. Comp. J., 9, 329-344 30. Skrzypek J., Kuna-CiskałH., 2003,Anisotropic elastic-brittle-damage and fracture models based on irreversible thermodynamics, In: Anisotropic Beha- viour of Damaged Materials, J. Skrzypek and A. Ganczarski (Edit.), 143-184, Berlin, Springer 31. Skrzypek J., Kuna-Ciskał H., Egner W., 2005a, Non-local description of anisotropic damage and fracture in elastic-brittle structures under plane stress conditions, Int. Conf. Fracture ICF 11, Torino,March 20-25, CD Effect of characteristic length on nonlocal prediction... 503 32. Skrzypek J., Kuna-Ciskał H., Egner W., 2005b, Non-local description of damage and fracture in double-notched concrete specimen under plane-stress conditions, Proc. III Symp. Mechaniki Zniszczenia Materiałów i Konstrukcji, Augustów, 381-384 33. Voyiadjis G.Z., Abu Al.-Rub R.K., 2002, Length scales in gradient pla- sticity, Proc. IUTAM Symp. on Multiscale Modeling and Characterisation of Elastic-Inelastic Behavior of Engineering Materials, S. Ahzi andM. Cherkaoui (Edit.), Marocco, Kluwer Acad. Publ. 34. Wang W.M., Sluys L.J., de Borst R., 1996, Interaction betweenmaterial length scale and imperfection size for localization phenomena in viscoplastic media, Eur. J. Mechanics A/Solids, 15, 3, 447-464 Wpływ długości charakterystycznej na prognozowanie rozwoju uszkodzeń i pękania w betonie przy zastosowaniu podejścia nielokalnego Streszczenie Wpracy opisany został nowy, nielokalnymodel typu całkowegodo symulacji roz- woju anizotropowych uszkodzeń w betonie. Przedstawionymodel nielokalny jest roz- winięciem modelu lokalnego zaproponowanego w pracy Murakami i Kamiya (1997), a zmodyfikowanego do formy przyrostowej w pracy Kuna-Ciskał i Skrzypek (2004). W celu uniknięcia zależności rozwiązania numerycznego od siatki MES oraz zapew- nienia stabilności i zbieżności zastosowanow obecnej pracy dwa sposoby ograniczania lokalizacji: nielokalne uśrednianie (NA) oraz algorytm obcinania (CA), oba zastoso- wane do sił termodynamicznych sprzężonych ze zmiennymi stanu uszkodzenia. Rów- nania konstytutywne materiału sprężysto-kruchego zapisane zostały w formie przy- rostowej z zastosowaniem zmiennych nielokalnych przy użyciu potencjału dyssypacji zdefiniowanegow przestrzeni uśrednionych zmiennych stanu uszkodzenia. Jako funk- cjęwagowąprzyjęto funkcjęGaussa. Przypomocy opisanegomodelu przeprowadzono numaryczną analizę rozwoju uszkodzeń i pękania w płaskim elemencie betonowym z podwójnym karbem, badanym eksperymentalnie przez Nooru-Mohammeda (1992). Przedyskutowano problem odpowiedniego doboru długości charakterystycznej konti- nuum nielokalnego oraz jej wpływu na rozwiązanie numeryczne. Manuscript received December 28, 2005; accepted for print April 20, 2006