Microplane Model M5 with Kinematic and Static Constraints for Concrete Fracture and Anelasticity. I: Theory

Transcription

1 Microplane Model M5 with Kinematic and Static Constraints or Concrete Fracture and Anelasticity. I: Theory Zdeněk P. Bažant, F.ASCE, 1 and Ferhun C. Caner 2 Abstract: Presented is a new microplane model or concrete, labeled M5, which improves the representation o tensile cohesive racture by eliminating spurious excessive lateral strains and stress locking or ar postpeak tensile strains. To achieve improvement, a kinematically constrained microplane system simulating hardening nonlinear behavior (nearly identical to previous Model M4 stripped o tensile sotening) is coupled in series with a statically constrained microplane system simulating solely the cohesive tensile racture. This coupling is made possible by developing a new iterative algorithm and by proving the conditions o its convergence. The special aspect o this algorithm (contrasting with the classical return mapping algorithm or hardening plasticity) is that the cohesive sotening stiness matrix (which is not positive deinite) is used as the predictor and the hardening stiness matrix as the corrector. The sotening cohesive stiness or racturing is related to the racture energy o concrete and the eective crack spacing. The postpeak sotening slopes on the microplanes can be adjusted according to the element size in the sense o the crack band model. Finally, an incremental thermodynamic potential or the coupling o statically and kinematically constrained microplane systems is ormulated. The data itting and experimental calibration or tensile strain sotening are relegated to a subsequent paper in this issue, while all the nonlinear triaxial response in compression remains the same as or Model M4. DOI: /(ASCE) (2005)131:1(31) CE Database subject headings: Concrete; Fracture; Inelastic action; Damage; Sotening; Finite element method; Numerical models. Introduction, Background, and Objective Except or the growth o spherical voids and collapse o spherical pores, which is not typical o concrete, almost all o the inelastic deormations in concrete microstructure, such as slip, riction, tensile microcrack opening, axial splitting, and lateral spreading in compression, occur on well deined planes taking any spatial orientation. Although these deormations occur at dierent points in the microstructure, the microplane models concentrate all such deormations, occurring in a small representative volume o the microheterogeneous material, into one point o the macroscopic smoothing continuum. Thus the constitutive properties characterizing these oriented inelastic phenomena (as well as the spherical voids and pores, i any) can be described by means o stress and strain vectors acting on a plane o arbitrary spatial orientation, called the microplane (Bažant 1984). The microplanes may be imagined as the tangent planes o an elemental sphere surrounding every continuum point [Fig. 1]. 1 McCormick School Proessor and W.P. Murphy Proessor o Civil Engineering and Materials Science, Northwestern Univ., 2145 Sheridan Rd., Tech A135, Evanston, IL z-bazant@northwestern.edu 2 Ramón y Cajal Fellow, ETSECCPB-ETCG, Univ. Politecnica de Catalunya, Jordi Girona 1-3, Ed.D2 D.305, Barcelona 08034, Spain. erhun.caner@upc.es; ormerly, Visiting Scholar, Northwestern Univ., 2145 Sheridan Rd., Evanston, IL Note. Associate Editor: Franz-Jose Ulm. Discussion open until June 1, Separate discussions must be submitted or individual papers. To extend the closing date by one month, a written request must be iled with the ASCE Managing Editor. The manuscript or this paper was submitted or review and possible publication on January 27, 2003; approved on February 13, This paper is part o the Journal o Engineering Mechanics, Vol. 131, No. 1, January 1, ASCE, ISSN / 2005/ /$ The strain and stress vectors on the microplanes, e and, must be assumed to be constrained in some way to the strain and stress tensors o the macrocontinuum, e ij and ij (the indices reer to components in Cartesian coordinates x i,i=1,2,3). The constraint is said to be kinematic (or static) i the strain (or stress) vector on each microplane is the projection o the continuum strain (or stress) tensor, i.e., e j =n i e ij and j =n i ij (where n i is the unit normal vector o the microplane). The stresses (or strains) in a kinematically (or statically) constrained microplane model cannot be the stress tensor projections but are related to the e (or ) only by a weak variational constraint, represented by the principle o virtual work (or complementary virtual work). The condition o tensorial invariance is automatically satisied by considering planes o all orientations. Compared to the classical tensorial constitutive models based on tensorial invariants, the microplane concept has potent advantages: (1) As realized already by Taylor (1938), a constitutive law in terms o vectors rather than tensors is clearer, conceptually simpler, and easier to ormulate. (2) The vector representation can directly characterize the aorementioned oriented deormations as well as their localization into one preerred orientation (note or example that, by contrast, a relationship between the hydrostatic pressure and the second deviatoric stress invariant cannot describe rictional slip on a plane o a speciic orientation). (3) The so-called vertex eect, an essential characteristic o concrete (Caner et al. 2002) which is generally missed by the classical tensorial models, is exhibited automatically. (4) Apparent deviations rom normality in the sense o tensorial plastic models are automatic, since a microplane model or plasticity is equivalent to a large number o simultaneously active yield suraces, or each o which the normality rule can be satisied. (5) The constraint o the microplanes automatically provides all the cross eects such as the shear dilatancy and pressure sensitivity. (6) Combinations JOURNAL OF ENGINEERING MECHANICS ASCE / JANUARY 2005 / 31

2 1959; Rice 1968; Hillerborg et al. 1976; Petersson 1981; Hillerborg 1985; RILEM 1985; Bažant and Planas 1998) or crack band theory (Bažant and Oh 1983a;b). Need or a Mixed Static-Kinematic Constraint Fig. 1. Let: Coupling o kinematically and statically constrained microplane systems or hardening and sotening responses. Right: components o strain or stress vectors on microplane. o loading and unloading on dierent microplanes provide a complex path dependence and automatically reproduce the Bauschinger eect and hysteresis under cyclic loading. (7) The dependence o the current yield limits on the strain components (rather than scalar hardening-sotening parameters) is easy to take into account. (8) In cyclic loading, atigue is automatically simulated by accumulation o residual stresses on the microplanes ater each load cycle. (9) Finally, anisotropy, while not requisite or concrete, can be captured easily, simply by making the constitutive properties o a microplane dependent on its orientation (in more detail, see Caner and Bažant 2000). These advantages outweigh the burden o a greater amount o computations, a burden that has been waning rom year to year with the relentless advance in computer power. Since its inception (Bažant and Oh 1983a,b, 1985), the microplane constitutive model or concrete has developed into a powerul and robust computational tool or three-dimensional inite element analysis o concrete structures. Initially ormulated or concrete as an extension and modiication o a groundbreaking idea o Taylor (1938), the evolution o the model has advanced through several progressively improved versions, which were labeled as M1, M2, M3, and M4 or concrete (as described in Bažant et al. 2000a) and M4R or rock (Bažant and Zi 2003). The evolution then ramiied to other complex materials such as sand, clay, rigid oam, shape-memory alloys, and iber composites (Brocca and Bažant 2000, 2001a,b; Brocca et al. 2001). The model was extended to inite strain (Bažant et al. 2000a) and to the rate eect or creep (Bažant et al. 2000c), and has been used in dynamic inite element analyses with up to several million inite elements (Bažant et al. 2000b). Recently, Model M4 (see Part II) introduced into Model M4 a racture energy based recalibration o sotening boundaries in the sense o the crack band theory. The latest version, Model M4 (Bažant et al. 2000a; Caner and Bažant 2000) or M4 describes satisactorily all the experimentally documented nonlinear triaxial behavior o concrete under compression and shear, and also simulates well the distributed tensile cracking that occurs or strains in the prepeak, peak, and early postpeak regimes. Problems have nevertheless arisen in predicting the ar postpeak tensile response in which the tensile stress across cracks is getting reduced to zero. The objective o this paper is to overcome these problems by ormulating a new microplane model, labeled M5. The new model enhances Models M4 and M4 with a statically constrained system o microcracks directly simulating the sotening stress-separation law o cohesive racture (Barenblatt Although the behavior or strains in the ar post-peak tail o the stress strain diagram is normally irrelevant or predicting the load capacities o structures, it is very important or dynamic response, especially or correctly assessing the energy absorption capability. For the ar postpeak tensile strains and or the ormation o complete ractures in which the crack bridging stresses are reduced to zero, Model M4, unortunately, does not perorm as well as the simple cohesive crack model or the simple classical smeared cracking models. It is plagued by two problems: (1) At ar postpeak tensile sotening, the lateral contraction in the directions parallel to the cracks is much larger than the Poisson eect in the material between the cracks and must be judged excessive (even though good test data are lacking, due to the diiculty o capturing strain at a random place where the crack band localizes); and (2) the model exhibits the so-called stress locking; in other words, a small but inite tensile stress across the crack band is retained even at extremely large tensile strains. The cause does not lie in the classical problem o spurious localization and mesh sensitivity, which can be readily avoided by either applying a nonlocal operator or by modiying the postpeak constitutive behavior according to the crack band theory (Bažant and Oh 1983a,b; also Bažant and Planas 1998; Jirásek and Bažant 2002). Rather, the cause lies in the necessity o a kinematic constraint or a sotening damage model. The kinematic constraint assumes that the strain vectors on the microplanes are the projections o the strain tensor. Because o this hypothesis, the crack opening produces large tensile strains not only in the microplanes nearly parallel to the cracks but also in those which are signiicantly inclined to the crack. These inclined microplanes produce large lateral contraction and, since they never soten to zero, they contribute a tensile stress across the cracks even i widely opened. The problem is aggravated by the necessity o a volumetric deviatoric split (proposed in Bažant and Prat 1988a,b), i.e., the split o each microplane normal strain into its volumetric and deviatoric components. Such a split is inevitable or a realistic representation o nonlinear triaxial behavior in compression and the transition between compression and tension. In particular, the split is necessary to capture the act that uniaxial compression and weakly conined compression lead to strain sotening with a large lateral expansion, whereas the hydrostatic compression (and compression at zero lateral strain) never leads to any strain sotening. This is a troublesome but inevitable dichotomy, which is ignored by most tensorial-type nonlinear triaxial constitutive models or concrete and rock. The split causes that, in ar postpeak sotening, most o the tensile strain concentrates into the deviatoric component which is tensile in the direction normal to the crack but equally large and compressive in directions parallel to the crack. Eliminating the split merely or tension does not help because it destroys the continuity o transition between the tensile and compressive responses, and restricting the magnitude o volumetric strain (by a volumetric boundary) only alleviates but does not avoid the problem The kinematic constraint, when introduced in Bažant (1984) and Bažant and Oh (1983b), represented a cardinal departure rom the classical Taylor models or metals and nonsotening (overconsolidated) soils (Batdor and Budianski 1949; Budianski 32 / JOURNAL OF ENGINEERING MECHANICS ASCE / JANUARY 2005

3 and Wu 1962; Lin and Ito 1965, 1966; Brown 1970; Hill and Rice 1972; Zienkiewicz and Pande 1977; Pande and Sharma 1982; Pande and Xiong 1982; Bronkhorst et al. 1992; Butler and Mc- Dowell 1998). In Taylor models, the constraint o the planes called here the microplanes is static, i.e., the stress vector on these planes is assumed to be the projection o the stress tensor. As learned rom the irst studies o sotening in racturing materials at Northwestern University during the early 1980s, the main reason or replacing at that time the static constraint with a kinematic constraint was to stabilize the system o sotening microplanes. I there is no sotening, as in plastic metals, the static constraint causes no instability because the strain corresponding to a given stress reduction is uniquely deined by the elastic unloading. But i there is sotening, the static constraint inevitably leads to instability because the microplane strains caused by a given stress reduction are not unique, corresponding to either the sotening branch or the unloading branch. The consequence is that all the microcracking strains (unlike plastic strains) suddenly localize, under static constraint, into a single microplane o one orientation only. This makes it impossible to simulate a system o microcracks o many orientations, developing simultaneously in the peak stress region. Later on, in the ar postpeak response, the cracking o course does localize into one dominant orientation, and that is why we will strive here to enhance Model M4 with a static constraint. There is also a secondary, physical, reason or the governing role o the kinematic constraint in the peak stress region. Unlike plastic yielding, happening at constant stress, the average local strain tensor o the material around and between the diuse microcracks is essentially the same as the strain tensor o the macroscopic smoothing continuum. The ormation o small microcracks causes the continuum stress to drop without aecting the strain tensor (the same observation underlies the classical Kachanov-type theory o continuum damage mechanics). Assuming all the behavior to adhere to a strict kinematic constraint is also arguable rom the viewpoint o stiness bounds. From the theory o composite materials (e.g., Christensen 1979), it is well known that the kinematic constraint, corresponding to Voigt s (1889) parallel coupling model, and the static constraint, corresponding to Reuss (1929) series coupling model, represent the upper and lower bounds on the material stiness, which are generally not close to each other. The real behavior is somewhere inbetween. Thereore, a microplane model incorporating a mixture o the kinematic and static constraints should be physically more realistic. There are urther reasons or seeking a way to ormulate a hybrid constraint. Just like the strains due to plastic slip on a long slip line, the strains due to the opening o continuous cracks (unlike the strains due to small diuse microcracks) are additive to the elastic strains o the material inbetween and both happen at the same stress. Consequently, the interaction o the elastic strains with the strains due to large and continuous (or almost continuous) cracks should be described by a series coupling model, and so should the interaction o the strains due to long cracks o dierent orientations. This means that the microplane system or the interaction o the opening o wide and long cracks should have a static rather than kinematic constraint, and that this system should be coupled in series with the microplane system that simulates nonlinear triaxial behavior and diuse microcracking in the peak stress region. A mixture o both constraints was attempted at Northwestern already in the early 1980s but seemed unworkable. It took the orm o a series coupling o two microplane systems [Fig. 1], in which the strain tensor ij was decomposed as ij = e ij + ij where e ij, ij strain tensors or the kinematically and statically constrained (nonsotening and sotening) microplane systems, both subjected to the same stress tensor ij. Such a coupling, however, caused the microplane strains and stresses o both systems to become tied by many implicit nonlinear equations. This appeared to pose at that time ormidable computational diiculties. To circumvent the problem, an iterative algorithm akin to that used in plastic inite element analysis was tried but was ound to diverge because o sotening. It was or this reason that the pursuit o series coupling model with hybrid constraints was at that time abandoned. Nevertheless, Carol and Bažant (1995) later demonstrated advantages o a series coupling o racturing microplane system to an elastic microplane system. Challenge: How to Achieve Convergence o Sotening or Static Constraint? The iterations in each loading step o plastic inite element analysis commonly employ the return mapping algorithm (Hughes 1984; Simo and Hughes 1998), in which the elastic deormation is used as the predictor and a subsequent return to the current yield surace as the corrector. A special case is the radial return algorithm or the J 2 low theory o plasticity (Wilkins 1964; Krieg and Key 1976; Krieg and Krieg 1977; Simo and Taylor 1985). In these algorithms, the corrector has in act an ininite negative stiness since the drop o stress point to the yield surace in the stress space is carried out at constant strain. A direct application o this kind o algorithm to the series coupling o two microplane systems [Fig. 1] is not a easible approach because a return at constant strain would require solving a system o many nonlinear equations. To emulate at least the spirit o this algorithm or the series coupling model [Fig. 1], it would seem natural to use the kinematically constrained microplane system with a positive incremental stiness [Fig. 1, let] as the predictor, and the racturing statically constrained microplane system [Fig. 1, right] as the corrector. Such a corrector has a inite negative stiness, and this stiness is typically smaller in magnitude than the stiness o the predictor. Unortunately, though, the convergence o this analog o the return mapping algorithm breaks down (see Algorithm A in Part II). To achieve convergence, a new kind o iterative algorithm will be introduced. When the statically constrained microplane system [Fig. 1, let] is active (i.e., when cracks are opening), its negative (or negative deinite) incremental stiness will be employed as the predictor, and the positive (or positive deinite) stiness o the kinematically constrained microplane system [Fig. 1, right] will be employed as the corrector. Such an iterative algorithm converges, and does so in a geometric progression, as proven in a later section (Algorithm B or B ). Adopting the sotening inelastic part o response as the predictor is the opposite o the algorithms established in computational plasticity, and the reason that this works is that the negative stiness o the inelastic corrector is not too large in magnitude. An acceleration giving quadratic convergence can urther be obtained by using the Newton-Raphson method to subdivide the strain between the two microplane systems. 1 JOURNAL OF ENGINEERING MECHANICS ASCE / JANUARY 2005 / 33

4 Review o Basic Relations or Microplane Systems The microplane system with strain tensor ij [Fig. 1] is constrained statically and describes only the strain sotening due to the opening o cohesive cracks o various possible orientations. The microplane system with strain tensor e ij = ij ij is constrained kinematically and describes the elastic deormations and all the remaining inelastic deormations, which exclude cohesive crack sotening (but include deviatoric sotening needed to simulate the sotening in uniaxial compression). According to these constraints e N = N ij e ij e M = M ij ij e L = L ij e ij N = N ij ij 2 with In numerical calculations ij = 3 u ij d 2 u ij = N N ij + L L ij + M M ij ij = 3 2 N m u ij d 6 w u ij =1 A quadrature ormula with N m =37 microplanes per hemisphere (Stroud 1971) has been used in the present calculations. Bažant and Oh (1986) studied the accuracy o various ormulas in dierent situations and developed a new, more eicient (albeit less accurate), ormula with 21 integration points. [see Fig. 1 where the circular points placed on an icosahedron deine the directions o microplane normals.] 7 8 M = M ij ij 3 Microplane Constitutive Relations o Model M5 L = L ij ij in which N ij =n i n j ; M ij = m i n j +m j n i /2; L ij = l i n j +l j n i /2; and subscripts M and N label two suitably chosen coordinate directions given by orthogonal unit coordinate vectors o components m i and l i, lying in the microplane [Fig. 1]. Microplane strain e N is urther split into its volumetric and deviatoric components e D = e N e V e V = 1 3 e kk 4 For strains e N, e V, e D, e M, and e L, the microplane constitutive law or the kinematically constrained system gives the corresponding stresses s N, s V, s D, s M, and s L (which are not the same as the stresses N, V, D, M, and L in the statically constrained part, except by chance). According to the principle o virtual work (Bažant et al. 2000a) with ij = s V ij + s ij D s D ij = 3 s ij d 2 s ij = s D N ij ij 3 + s L L ij + s M M ij where surace o a unit hemisphere (in detail, see Bažant et al. 2000a). This integral is evaluated numerically using an optimal Gaussian quadrature ormula or a spherical surace (Bažant and Oh 1985, 1986) which represents a weighted sum over the stresses on N m microplanes with discretely distributed orientations =1,2, N m ; N m ij 6 w s ij + s V ij =1 N where w quadrature weights m =1 =1. For the statically constrained microplane system, no volumetric deviatoric split is introduced. According to the principle o complementary virtual work 5 6 The inelastic constitutive laws on the microplanes are deined by the so-called stress strain boundaries (or strain-dependent sotening yield limits, introduced in Bažant et al. 1996a,b). For e ij, these laws are nearly the same as in M4, with minor dierences to be described later. Within the boundaries, the incremental response is elastic (like in M4) and described as ṡ V =E V ė V,ṡ D =E D ė D,ṡ M =E T ṡ M,ṡ L =E T ṡ L, where the superior dots denote the time rates and E V,E D,E T microplane elastic moduli, the relationship o which to the Young s modulus E and Poisson ratio is best taken as E V =E/ 1 2, E D =E/ 1+ ] and E T =E D (as justiied in Bažant and Prat 1988a; Carol et al. 1991; Carol and Bažant 1997). Although, upon reaching a boundary curve, the microplane stress strain curve has a sharp sudden change o slope, the resulting macrocontinuum stress strain curve is quite smooth because the transitions to the boundary curve happen on dierent microplanes at dierent times. Cohesive Fracture in Tension and Shear We will assume that a cohesive crack can develop in the direction o any microplane i the tensile strength is exhausted. The relative normal and shear displacements across the crack, N, L, and M, are modeled by normal and shear strains on the microplane N = N /h L = L /h M = M /h where h eective width o the band o inite elements representing the racture (coinciding with the actual size o the elements i the racture propagates along the mesh line o a square grid). I, however, there is a system o parallel cracks (stabilized or instance by reinorcement or adjacent compressed zone), then h should represent the typical spacing o the parallel cracks. For pure mode (opening) racture, a cohesive crack is normally characterized by a sotening stress-separation unction N = N.We will consider, more generally, a mixed opening-shear cohesive racture (mixed Modes I, II, and III), or which the vector o cohesive stress is in general a unction o the displacement vector. We will adopt here a recent simpliication o Camacho 9 34 / JOURNAL OF ENGINEERING MECHANICS ASCE / JANUARY 2005

5 and Ortiz (1996), which assumes the existence o a potential or the cohesive stresses, (ree energy density), where = h, 10 = 2 N L + 2 M Factor, very small or concrete, expresses the act that concrete much preers to racture in tension than in shear (according to Bažant et al. 1988, the ratio o Mode III to Mode I racture energies o concrete appears to be about 30, and so 1/30). Based on the existence o potential, N =h 1 / N, L =h 1 / L, and M =h 1 / M. This yields where N = N /, L = 2 L /, M = 2 M / 11 = 2 N + 2 L + 2 M / 2 12 The last equation may be veriied by substituting Eq. (11). Note that in this model, based on a potential, the cohesive stress vector is always parallel to the stress displacement vector (or to the racture strain vector) because N / = N /, etc. It must be admitted that this property is not realistic in the case o pure shear on the macrocontinuum level, because it implies zero dilatancy. For the shear on a microplane, however, this is not a problem since dilatancy is automatically created by the interaction o microplanes o dierent orientations, simply as a result o the act that the strength is lower in tension than in compression (Bažant and Gambarova 1984; Bažant et al. 2000a). During the initial elastic and inelastic stress increase, the cracks remain closed and so the strains on the statically constrained microplane system remain zero. When the microplane normal stress N reaches the strength limit t in tension, a coplanar cohesive crack is assumed to orm on that microplane. N is imagined to represent the cohesive (crack-bridging) stress transmitted between the aces o this crack, which is determined by the sotening law N = w where w is the separation between the crack aces (crack opening); w can be related to the racturing part o the total strain by N =w/s where s is the eective crack spacing (between parallel cracks o about the same orientation as the microplane). The sotening stress-separation relation is assumed to be bilinear. Furthermore, microplane racturing compliances on every microplane o the statically constrained system and a racturing compliance tensor on the macro scale must be deined in order to implement an eicient algorithm or the additive split o the strains. To this end, it is convenient to write the split o the increment o total strain as i = e i i + i =0 using the convenience o Voigt notation (where i=1,2,...6; e1,... e6 correspond to e11,...2 e31 ). Fixing i at a given load step, and using i =K in e n and i =C in n where K in is the elastic 6 6 stiness matrix and C in is the racturing compliance matrix, the lin- earized problem becomes i e 0 n +C in e n 0 in which Fig. 2. Bilinear sotening relation or normal microplane stresses and strains, and relation or crack unloading and reloading G ij = i e j = i e j ij = i n n e j ij = C in K nj ij 13 Here, C in = i / n can be calculated using Eq. (7) as C in = 2 3 N N N + L L L n L i d which simpliies to n N i + M M M n M i 14 C in = 3 C N N i N n + C M M i M n + C L L i L n d 15 2 where C N = / is normal compliance and C M =C L = 2 / shear racturing compliances deined on each microplane o the statically constrained microplane system, and C in corresponding compliance tensor. Fracturing Properties o Statically Constrained Microplanes The sotening response in pure tension or tension combined with shear is expressed by means o the racturing strain tensor ij and is described by the statically constrained microplane system. No elastic response is included in ij. Since N = N /s where N crack opening, the microplane sotening law is deined as a scaled sotening law or the cohesive crack model. The bilinear sotening law (Fig. 2), well justiied or concrete racture (Petersson 1981; Hillerborg 1985; RILEM 1985; Guinea et al. 1994; Bažant and Planas 1998), is adopted: N = t max 1 1 N, max 1 N N1 N1 N2 N1,0 16 where 0 1 and the points 0, t, N1, t, and N2,0 in the diagram o normal stress versus racturing strain on the microplane deine the bilinear cohesive law (Fig. 2). The oregoing sotening law is deined only or N 0 because negative crack opening can never occur. As soon as the axial response reaches t on one microplane, the crack on that microplane begins to open while all the other microplanes normally begin unloading. Thus the racture process JOURNAL OF ENGINEERING MECHANICS ASCE / JANUARY 2005 / 35

6 normally localizes into one single microplane. As the crack or that microplane continues to open, the normal stress and crack opening displacement sweep the area under the prescribed sotening curve. The areas under the entire sotening curve = w and under the initial tangent o this curve represent the Mode I racture energies G F = 0 N dw = s 0 N2 N d N 17 G = t N1 2 1 (the integral or G F was justiied by Rice 1968); G racture energy that normally governs the maximum loads o notched specimens and structures with large stress-ree cracks (Bažant and Planas 1998; Bažant et al., private communication 2002) because the cohesive stress at the notch tip at maximum load is normally not yet reduced to a value less than t. I both G F and G have been measured or the given concrete (Bažant et al., private communication 2002), it suices to choose a suitable value or and then the sotening law is ully deined. Analysis o notched racture test data gives roughly 1/4 (Bažant and Planas 1998); however, van Mier s uniaxial tensile test data are optimally itted with As a compromise, all the present data itting has been done with =0.3, even though it does not yield optimum its. I only G F has been measured, one must also choose the ratio G /G F (a suitable choice is 0.4; Bažant and Planas 1998; Bažant and Becq-Giraudon 2002), or the value N1. Instead o G F, one can speciy N2 (a suitable value may be roughly 0.03). Using the tensile strength data or concrete specimens o dierent sizes, one can determine G or the type o concrete used rom the size eect law. The values o and N2 can be assumed to be ixed at certain values or all concretes (e.g., =0.44 and N2 =0.03), and one can thus obtain the approximation N1 1.34G 1 /s t where actor 1.34 is obtained or =0.44. Note that i the initial slope o the sotening curve is chosen too steep, convergence problems may result. The unloading and reloading o the cohesive crack are assumed to be linear, as sketched in Fig. 2. They are deined by the relation N = Nu 1+k u N / Nu 18 where N 0 or unloading; Nu maximum N attained so ar (which must be saved, or each microplane); Nu corresponding stress, and k u coeicient deining the degree o reversibility o crack opening. For k u =1 the opening is ully reversible and or k u =0 it is irreversible. In reality, the opening is partially reversible 0 k u 1, because o the microscopic debris deposited between the crack aces and because o irreversibility o the rictional slip in the racture process zone. For computations, k u =1 has been assumed. I k u 1 is chosen, or the sake o simplicity and also due to lack o good test data, the dierence between the unloading and reloading paths may be neglected (although it would not be diicult to program it). Note that, regardless o the value o k u, the unloading is governed by the kinematically constrained part. Constitutive Properties o Kinematically Constrained Microplanes For the kinematically constrained microplanes subjected to strain tensor ij, we try to ollow microplane Model M4 or concrete as Fig. 3. Stress strain boundaries o kinematically constrained microplane system, co-opted rom Model M4 with modiications ar as possible. What needs to be removed rom Model M4 (Fig. 3) is the sotening portions o the stress strain boundaries or tensile normal strain and tensile volumetric strain on the microplanes because the sotening in pure tension and in tension with shear is already represented by the sotening cohesive law or the statically constrained microplanes. In consequence, the rictional shear boundary used in M4 needs to be adjusted as well. It is ortunate that the necessary modiications have no eect on the nonlinear triaxial response in compression and shear, and so most o the eatures o Model M4, tediously calibrated in previous work, can be retained. Normal Boundary The microplane normal boundary (i.e, the boundary on s N ), which in M4 and M4 governs primarily the tensile sotening, might seem redundant since the tensile sotening is modeled by the stati- 36 / JOURNAL OF ENGINEERING MECHANICS ASCE / JANUARY 2005

7 cally constrained microplanes. Not quite, however, since this boundary is also important or the postpeak sotening in uniaxial compression. Thus, the normal boundary is kept the same as it is in Model M4 or M4 (Fig. 3), even though it now serves only to restrict excessively large lateral expansion under uniaxial compression. It has no eect on the tensile response in the current model, because the cohesive sotening law or the statically constrained microplanes is activated beore this normal boundary is approached. Once the sotening on the statically constrained microplanes becomes activated, the kinematically constrained microplane system unloads, due to series coupling. Volumetric Boundary In Models M4 and M4, the normal boundary that bounds the normal stress s N was alone insuicient to prevent volumetric stress s V rom growing too large because the sotening eect o the normal boundary was oset by increasing s V. Thereore, it was necessary to also impose a sotening tensile volumetric boundary (Bažant et al. 2000a). In the current model, the problem o growing tensile stress at large V never occurs because the volumetric and deviatoric stresses unload while the statically constrained microplanes provide sotening in tension. Thus, the series coupling in the present model makes the tensile volumetric boundary in tension superluous. Shear Boundary The shear boundary has dierent roles in tension and compression. In compression, it serves as a rictional yield surace in the plane o shear stress versus normal stress, with the cohesion gradually approaching zero as the racturing damage increases. In the tensile range, however, the rictional boundary cannot be closed, as in M4, because it would cause excessive prepeak plastic strain and shit the stress peak too ar. Rather, the shear response in tension must remain almost elastic until the peak, which is ensured by elevating the tensile portion o the rictional boundary as shown in Fig. 3. Thus, in tension, the rictional boundary is ormulated as a shear stress bound proportional to the tensile normal stress. The modiication o the rictional shear boundary o M4 (Fig. 3) is mathematically described as ollows: or s N 0 where s T b = s T 0 + c 22 V eq s N 19 Making the slope a unction o V eq or tensile s N makes it possible to distinguish important qualitative dierences between compression and tension. For compression, this unction causes the slope to be negative and thus the yield surace to be closed. For tension, this slope is positive, thus allowing a linear shear response until the tensile strength is reached on a statically constrained microplane, and in this way the statically constrained microplane system is made to control the tensile sotening. Incremental Thermodynamic Potential Two kinds o thermodynamic potentials are o interest: (1) the complete potential, which is a unction o the reversible parts o total strains and involves separate internal state variables or irreversible strains governing energy dissipation; and (2) the incremental potential associated with the tangential (inelastic) moduli tensor, considered as given and thus equivalent to an elastic moduli tensor. The latter, which does not separate the reversible (elastic) and irreversible (inelastic) deormations, cannot capture energy dissipation but can capture the entropy produced by deviations rom equilibrium, which is what is needed to detect biurcation and identiy the actual postbiurcation equilibrium path (as explained in Bažant 1988 and in Section 10.2 o Bažant and Cedolin 1991). Only the latter will be presented here. The ormer could be obtained i the latter is generalized by separation o reversible and irreversible deormations. This would require extending the analysis o Carol et al. (2001, 2004) rom a kinematically constrained to a hybrid microplane model, and is beyond the scope o this paper. Note that the microplane ormulation presented does not rest on the incremental potential but is derived rom the principles o virtual work and complementary virtual work. Because o mixing the kinematic and static constraints, an incremental potential representing the speciic ree energy, o either Helmholtz or Gibbs type, does not exist on the macrocontinuum level (the term speciic means per unit volume, and density per unit mass ). Nevertheless, a potential or microplane models with mixed constraints can be deined. From the theoretical and computational viewpoints, this is an attractive aspect, which will now be demonstrated. The total variation o the speciic Helmholtz ree energy o a continuum with a constitutive law described by the microplane Model M5 may be expressed as s T 0 = c 10 s 0 1+c 10 s 0 / k 1 k 2 E T 20 E T k 1 c 11 s 0 = 1+c 12 V /k 1 Here, c 22 = and c 12 =0.155, while all the other parameters have the same values as given in Caner and Bažant (2000). The shear boundary or compressive s N is as given in Bažant et al. (2000a,b,c): or s N 0 s b T = E Tk 1 k 2 c 10 s N + s 0 N E T k 1 k 2 + c 10 s N + s 0 21 N where X =max X,0 positive part o argument. I V is large, c 10 = ds b b T /ds N sn =0 initial slope o the boundary, and lim sn s T =E T k 1 k 2, which represents a horizontal asymptote. in which n e = n d n + d = n e n en d + n d 22 s = e en e e d + n d 23 s e = n / e n 24 JOURNAL OF ENGINEERING MECHANICS ASCE / JANUARY 2005 / 37

8 n = n / and e n / e=n. Here = 2 /3 = 2 /3 sin d d (, being spherical coordinates); mass density; n or n contributions to the variation o the Helmholtz ree energy density rom microplane o orientation n, surace o a unit hemisphere; e n,s e strain and stress vectors on kinematically constrained microplane o orientation n, and, n strain and stress vectors on statically constrained microplane o orientation n (corresponding to spherical coordinates,, the latitude, and longitude); e microplane stress vector corresponding to e n in the kinematically constrained part, and microplane strain vector corresponding to n in the statically constrained part; e n =e n, n = n, =e+, where, macrocontinuum stress and strain tensors; and e, partial strain tensors o macrocontinuum corresponding to the kinematically and statically constrained microplane systems. Substituting the tensor projection e n =e n into the irst integral in Eq. (23), we can actor out e in ront o the integral and get an integral that is ully determined by the strain tensor variation e and thus, taken alone, represents a potential. As or the second integral, however, an analogous operation is impossible because / n, in contrast to Eq. (24). Indeed, substituting the tensor projection o n = n does not yield an integral ully determined by the variation s o the strain tensor that provides the kinematic constraint, and thus is not a total variation o a potential in terms o the macroscopic variables e and s. To overcome this problem, recall the Legendre transormation which is, or instance, used to obtain the Gibbs ree energy rom the Helmholtz ree energy. To this end, we rewrite Eq. (23) as ollows: or B e, s = e en e : e d n : d 25 : e : = s e: e nd : n d 26 in which n / =n and B e, = :de :d = e n : d 27 Now we denote e s = e e n d = n d B e, = e 28 All these expressions represent total variations, and so e,, and B e, represent thermodynamic potentials; e and B e, potentials as unctions o internal macroscopic variable e which is not directly measurable, even or homogeneous deormation. Only is a potential as a unction o a measurable continuum variable ; it represents speciic Gibbs ree energy associated with the statically constrained microplane system. The total potential (per unit volume) is B e, = 1 n d = e 29 It is a mixed potential (per unit volume) representing a superposition o speciic Helmholtz and Gibbs ree energies (Bažant 2002) or the isothermal material properties (or adiabatic material properties, this potential represents a mixture o speciic total energy and enthalpy). The validity o this mixed potential is veriied by the act that the conditions o stationarity o this potential with respect to arbitrary variations and e yield, respectively, the expressions or and in terms o microplane stresses and strains, i.e., Eqs. (5) and (7). Integration o this potential over the structure volume and addition o external work yields a mixed potential o the structure load system, which allows detecting biurcation and deciding stability o the postbiurcation equilibrium paths [Bažant 1988; Bažant and Cedolin 1991, Eq. ( )]. The energy dissipation inequality, o course, cannot be checked rom the present potential based on tangential stiness. However, rom the model coupling (visualized in Fig. 1 o Bažant and Caner 2005) it is physically clear that i the behavior on each microplane o a system, constrained either kinematically or statically, ensures non-negative energy dissipation, the dissipation inequality must be satisied. Closing At this point, the constitutive model has been theoretically ormulated. What remains are the computational aspects, particularly the computational algorithm and the calibration by test data. This is taken up in Part II o this study (Bažant and Caner 2005) which ollows. Reerences Barenblatt, G. I. (1959). The ormation o equilibrium cracks during brittle racture. General ideas and hypothesis, axially symmetric cracks. Prikl. Mat. Mekh., 23(3), Batdor, S. B., and Budianski, B. (1949). A mathematical theory o plasticity based on the concept o slip. Technical Note No. 1871, National Advisory Committee or Aeronautics, Washington, D.C. Bažant, Z. P. (1984). Microplane model or strain controlled inelastic behavior. Mechanics o engineering materials, C. S. Desai and R. H. Gallagher, eds., Chap. 3, Wiley, London, Bažant, Z. P. (1988). Stable states and paths o structures with plasticity or damage. J. Eng. Mech., 114(12), Bažant, Z. P. (2002). Thermodynamic potential or microplane model M5 with kinematic and static constraints. Progress Note, Dept. o Civil Engineering, Northwestern Univ., Evanston, Ill. Bažant, Z. P., Adley, M. D., Carol, I., Jirásek, M., Akers, S. A., Rohani, B., Cargile, J. D., and Caner, F. C. (2000b). Large-strain generalization o microplane model or concrete and application. J. Eng. Mech., 126(9), Bažant, Z. P., and Becq-Giraudon, E. (2002). Statistical prediction o racture parameters o concrete and implications or choice o testing standard. Cem. Concr. Res. 32(4), Bažant, Z. P., and Caner, F. C. (2005). Microplane model M5 with kinematic and static constraints or concrete racture and anelasticity. II: Computation. J. Eng. Mech., 131(1), Bažant, Z. P., Caner, F. C., Adley, M. D., and Akers, S. A. (2000c). Fracturing rate eect and creep in microplane model or dynamics. J. Eng. Mech., 126(9), Bažant, Z. P., Caner, F. C., Carol, I., Adley, M. D., and Akers, S. 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9 (2000a). Microplane model M4 or concrete: I. Formulation with work-conjugate deviatoric stress. J. Eng. Mech., 126(9), Bažant, Z. P., and Cedolin, L. (1991). Stability o structures: elastic, inelastic, racture and damage theories, Oxord University Press, New York; and 2nd Ed. (2003), Dover, New York. Bažant, Z. P., and Gambarova, P. (1984). Crack shear in concrete: Crack band microplane model. J. Struct. Eng., 110(9), Bažant, Z. P., and Oh, B.-H. (1983a). Crack band theory or racture o concrete. Mater. Struct., 16, Bažant, Z. P., and Oh, B.-H. (1983b). Microplane model or racture analysis o concrete structures. Proc., Symp. on the Interaction o Non-Nuclear Munitions with Structures, United States Air Force Academy, Colorado Springs, Colo., Bažant, Z. P., and Oh, B.-H. (1985). Microplane model or progressive racture o concrete and rock. J. Eng. Mech., 111(4), Bažant, Z. P., and Oh, B.-H. (1986). Eicient numerical integration on the surace o a sphere. Z. Angew. Math. Mech., 66(1), Bažant, Z. P., and Planas, J. (1998). Fracture and size eect in concrete and other quasibrittle materials, CRC, Boca Raton, Fla. Bažant, Z. P., and Prat, P. C. (1988a). Microplane model or brittle plastic material: I. Theory. J. Eng. Mech., 114(10), Bažant, Z. P., and Prat, P. C. (1988b). Microplane model or brittle plastic material: II. Veriication. J. Eng. Mech., 114(10), Bažant, Z. P., Șener, S., and Prat, P. C. (1988). Size eect tests o torsional ailure o plain and reinorced concrete beams. Mater. Struct., 21, Bažant, Z. P., Xiang, Y., Adley, M. D., Prat, P. C., and Akers, S. A. (1996a). Microplane model or concrete. II. Data delocalization and veriication. J. Eng. Mech., 122(3), Bažant, Z. P., Xiang, Y., and Prat, P. C. (1996b). Microplane model or concrete. I. Stress strain boundaries and inite strain. J. Eng. Mech., 122(3), Bažant, Z. P., and Zi, G. (2003). Microplane constitutive model or porous isotropic rock. Int. J. Numer. Analyt. Meth. Geomech., 27, Brocca, M., and Bažant, Z. P. (2000). Microplane constitutive model and metal plasticity. Appl. Mech. Rev., 53(10), Brocca, M., and Bažant, Z. P. (2001a). Microplane inite element analysis o tube-squash test o concrete with shear angles up to 70. Int. J. Numer. Methods Eng., 52, Brocca, M., and Bažant, Z. P. (2001b). Size eect in concrete columns: Finite-element analysis with microplane model. J. Struct. Eng., 127(12), Brocca, M., Bažant, Z. P., and Daniel, I. M. (2001). Microplane model or sti oams and inite element analysis o sandwich ailure by core indentation. Int. J. Solids Struct., 38, Bronkhorst, C. A., Kalindindi, S. R., and Anand, L. (1992). Polycrystalline plasticity and the evolution o crystallographic texture in cc metals. Philos. Trans. R. Soc. London, Ser. A, 341, Brown, G. M. (1970). A sel-consistent polycrystalline model or creep under combined stress states. J. Mech. Phys. Solids, 18, Budianski, B., and Wu, T. T. (1962). Theoretical prediction o plastic strains o polycrystals. Proc., 4th U.S. National Congress o Applied Mechanics, ASME, New York, Butler, G. C., and McDowell, D. L. (1998). Polycrystal constraint and grain subdivision. Int. J. Plast., 14(8), Camacho, G. T., and Ortiz, M. (1996). Computational modeling o impact damage in brittle materials. Int. J. Solids Struct., 33(20 22), Caner, F. C., and Bažant, Z. P. (2000). Microplane model M4 or concrete: II. Algorithm and Calibration. J. Eng. Mech., 126(9), Caner, F. C., Bažant, Z. P., and Červenka, J. (2002). Vertex eect in strain-sotening concrete at rotating principal axes. J. Eng. Mech., 128(1), Carol, I., and Bažant, Z. P. (1995). New developments in microplane/ multicrack models or concrete. Proc., Fracture Mechanics o Concrete Structures (FraMCoS 2), F. H. Wittman, ed., Aediicatio, Freiburg, Germany, Carol, I., and Bažant, Z. P. (1997). Damage and plasticity in microplane theory. Int. J. Solids Struct., 34(29), Carol, I., Bažant, Z. P., and Prat, P. C. (1991). Geometric damage tensor based on microplane model. J. Eng. Mech., 117(10), Carol, I., Jirásek, M., and Bažant, Z. P. (2001). New thermodynamically consistent approach to microplane theory. Part I. Free energy and consistent microplane stresses. Int. J. Solids Struct., 38(17), Carol, I., Jirásek, M., and Bažant, Z. P. (2004). A ramework or microplane models at large strain, with application to hyperelasticity Int. J. Solids Struct., 41, Christensen, R. M. (1979). Mechanics o composite materials, Wiley, New York. Guinea, G. V., Planas, J., and Elices, M. (1994). A general bilinear it or the sotening curve o concrete. Mater. Struct., 27, Hill, R., and Rice, J. R. (1972). Constitutive analysis o elastic-plastic crystal at arbitrary strain. J. Mech. Phys. Solids, 20, Hillerborg, A. (1985). The theoretical basis o method to determine the racture energy G o concrete. Mater. Struct., 18(106), Hillerborg, A., Modéer, M., and Petersson, P. E. (1976). Analysis o crack ormation and crack growth in concrete by means o racture mechanics and inite elements. Cem. Concr. Res., 6, Hughes, T. J. R. (1984). Numerical implementation o constitutive models: Rate-independent deviatoric plasticity. Theoretical ormulation or large-scale computations o nonlinear material behavior, S. Nemat-Nasser, R. J. Assaro, and G. A. Hegemier, eds., Martinus Nijho, Dordrecht, The Netherlands, Jirásek, M., and Bažant, Z. P. (2002). Inelastic analysis o structures, Wiley, London. Krieg, R. D., and Key, S. W. (1976). Implementation o time dependent plasticity theory into structural computer programs. Constitutive equations in viscoplasticity: Computational and engineering aspects, J. A. Stricklin and K. J. Saczalski, eds., Vol. AMD-20, ASME, New York, Krieg, R. D., and Krieg, D. B. (1977). Accuracies o numerical solution methods or the elastic-perectly plastic model. J. Pressure Vessel Technol., 99(4), Lin, T. H., and Ito, M. (1965). Theoretical plastic distortion o a polycrystalline aggregate under combined and reversed stresses. J. Mech. Phys. Solids, 13, Lin, T. H., and Ito, M. (1966). Theoretical plastic stress-strain relationship o a polycrystal. Int. J. Eng. Sci., 4, Pande, G. N., and Sharma, K. G. (1982). Multilaminate model o clays A numerical evaluation o the inluence o rotation o the principal stress axis. Rep., Dept. o Civil Engineering, Univ. College o Swansea, Swansea, U.K. Pande, G. N., and Xiong, W. (1982). An improved multi-laminate model o jointed rock masses. Proc., Int. Symp. on Numerical Models in Geomechanics, R. Dungar, G. N. Pande, and G. A. Studder, eds., (Zürich, Switzerland), Balkema, Rotterdam, The Netherlands, Petersson, P. E. (1981). Crack growth and development o racture zones in plain concrete and similar materials. Rep. No. TVBM 1006, Lund Institute o Technology, Lund, Sweden. Reuss, A. (1929). Berechnung der Fließgrenze von Mischkristallen au Grund der Plastizitätsbedingung ür Einkristalle. Z. Angew. Math. Mech., 9, Rice, J. R. (1968). Mathematical analysis in the mechanics o racture. Fracture An advanced treatise, H. Liebowitz, ed., Vol. 2, Academic, New York, Réunion Internationale des Laboratoires d Essais et de Recherche sur les Matériaux (RILEM) Recommendation. (1985). Determination o racture energy o mortar and concrete by means o three-point bend tests o notched beams. Mater. Struct., 18(106), Simo, J. C., and Hughes, T. J. R. (1998). Computational plasticity, Springer, New York. Simo, J. C., and Taylor, R. L. (1985). Consistent tangent operators or JOURNAL OF ENGINEERING MECHANICS ASCE / JANUARY 2005 / 39

10 rate independent elastoplasticity. Comput. Methods Appl. Mech. Eng., 48, Stroud, A. H. (1971). Approximate calculation o multiple integrals, Prentice-Hall, Englewood Clis, N.J. Taylor, G. I. (1938). Plastic strain in metals. J. Inst. Met., 62, Voigt, W. (1889). Über die Beziehung zwischen den beiden Elastizitätskonstanten isotroper Körper. Wied. Ann., 38, Wilkins, M. L. (1964). Calculation o elastic-plastic low. Methods o computational physics, Vol. 3, B. Alder et al., eds., Academic, New York. Zienkiewicz, O. C., and Pande, G. N. (1977). Time-dependent multilaminate model o rocks A numerical study o deormation and ailure o rock masses. Int. J. Numer. Analyt. Meth. Geomech., 1, / JOURNAL OF ENGINEERING MECHANICS ASCE / JANUARY 2005

11 Microplane Model M5 with Kinematic and Static Constraints or Concrete Fracture and Anelasticity. II: Computation Zdeněk P. Bažant, F.ASCE, 1 and Ferhun C. Caner 2 Abstract: Following the ormulation o the constitutive model in the preceding Part I in this issue, the present Part II addresses the problems o computational algorithm and convergence o iterations. Typical numerical responses are demonstrated and the parameters o the model are calibrated by test data rom the literature. DOI: /(ASCE) (2005)131:1(41) CE Database subject headings: Concrete; Fracture; Inelastic action; Damage; Sotening; Finite element method; Numerical models. Introduction Ater ormulating the microplane constitutive model in the preceding Part I (Bažant and Caner 2005), we are now ready to introduce a localization limiter preventing spurious mesh sensitivity, develop the numerical algorithm, and investigate the conditions o convergence o iterations. Then we will demonstrate typical numerical simulations, compare the model to the characteristic test data rom the literature, and calibrate the model parameters. Adjustment o Postpeak Sotening Based on Fracture Energy To prevent postpeak strain sotening rom causing spurious mesh sensitivity in inite element analysis, one may apply the crack band model (Bažant and Oh 1983), in which the energy dissipation per unit volume is adjusted according to the ratio o the element size h to the characteristic crack bandwidth l (a material length), so as to achieve mesh-indendent energy dissipation per unit area o the crack plane. Postpeak strain sotening is known to engender spurious mesh sensitivity in inite element analysis (Bažant 1976), with the energy dissipation per unit area o ailure surace depending on the chosen inite element size. There are two expedient ways to avoid it: (1) on the constitutive level, by applying the crack band model (Bažant and Oh 1983), in which the energy dissipation per unit volume is adjusted according to the ratio o the element size h to 1 McCormick School Proessor and W. P. Murphy Proessor o Civil Engineering and Materials Science, Northwestern Univ., 2145 Sheridan Rd., Tech A135, Evanston, IL z-bazant@northwestern.edu 2 Ramón y Cajal Fellow, ETSECCPB-ETCG, Univ. Politecnica de Catalunya, Jordi Girona 1-3, Ed.D2 D.305, Barclona 08034, Spain erhun.caner@upc.es; ormerly, Visiting Scholar, Northwestern Univ., 2145 Sheridan Rd., Evanston, IL Note. Associate Editor: Franz-Jose Ulm. Discussion open until June 1, Separate discussions must be submitted or individual papers. To extend the closing date by one month, a written request must be iled with the ASCE Managing Editor. The manuscript or this paper was submitted or review and possible publication on January 27, 2003; approved on February 13, This paper is part o the Journal o Engineering Mechanics, Vol. 131, No. 1, January 1, ASCE, ISSN / 2005/ /$ the characteristic crack bandwidth l (a material length); and (2) on the inite element level, by using a composite inite element in which a sotening inite element is coupled in three dimensions with unloading inite elements without adjusting the microplane constitutive law (Bažant et al. 2002, 2001b). The ormer, named Model M4 and developed by Bažant et al. (2002), is simpler and has been adopted or the present numerical studies. It may be briely described as ollows. Width l is the characteristic size o the representative volume (or characteristic length) o material or which the microplane constitutive model has been calibrated by tests (as described or M4 in Caner and Bažant 2000). The area W under the postpeak sotening portion o each microplane stress strain curve, multiplied by l, represents a contribution to the racture energy o the material, which must be a constant in order to avoid spurious mesh sensitivity (Bažant and Oh 1983). I element size h=l is chosen, the energy dissipation will be correct without any adjustment o microplane constitutive laws. However, an adjustment is needed when h l because the strain sotening tends to localize into a band o a single element width. Energy equivalence requires the adjustment to be such that W * h=w l or W * =W l/h, where W * is the area under the postpeak sotening curve o each microplane stress strain curve (Fig. 1). This may be achieved by a horizontal rescaling o all the sotening boundaries, tensile as well as compressive, in the ratio r =l/h; see Fig. 1 where Point 2 represents the point at which the sotening begins. Let Point P be any point on the sotening Boundary Curve 23, and let U be a point at the same level (same stress) on the Unloading Line 10. The rescaling must be such that Point P moves to Point P or which UP =rup. This represents a horizontal ainity transormation with respect to Axis 10, which transorms sotening Boundary Curve 23 to a new sotening Boundary Curve 2 3. When the element is too large, ratio r could be so small that the transormed sotening curve exhibits a snapback, i.e., the slope at some points changes rom negative to positive. Since static inite element analysis cannot cope with snapback instability, elements that large cannot be allowed. For mathematical implementation it is somewhat inconvenient that the axis o ainity is inclined. Since the inclination is very steep compared to the slope o the sotening boundary, it makes little dierence i the inclined ainity Axis 24 is replaced by the vertical ainity Axis 45 such that the area be equal to the JOURNAL OF ENGINEERING MECHANICS ASCE / JANUARY 2005 / 41

12 Fig. 1. (Top) Localized crack band; Adjustment o sotening branches o microplane constitutive laws according to crack band model (proposed in Bažant et al. 2001); (bottom let) accurate method; (bottom right) simpliied method area Such rescaling is simpler and has been adopted or calculations. Rescaling by this kind o ainity transormation is applied to tensile sotening boundary o the statically constrained microplane system, which emulates the cohesive crack model or tension. It is also applied to each sotening boundary in the kinematically constrained microplane system, i.e., to the sotening positive and negative deviatoric boundaries, the tensile volumetric boundary, and the tensile normal boundary the boundaries that control sotening o the material in compression. For a crack band running along the mesh lines o a rectangular or square mesh, h represents the width o the elements orming the crack band. For a crack band running through an irregular mesh, h cannot be deined exactly. However, it seems reasonable to simply assume that h=4 the ratio o the element area to the element perimeter, or h=6 the ratio o the element volume to its surace, in the case o two-or three-dimensional inite elements, respectively. Fig. 2. Optimum its o tensile sotening test data o Petersson (1981), with corresponding lateral strain (data at bottom has compressive strength c =27 MPa and other one c =41 MPa); parameter values used in Model M5 are k 1 = , k 2 =160, k 3 =10, k 4 =150 with =0.3, N1 =0.005, and N2 = , just like its predecessors, despite the new algorithm which represents a radical improvement in the series o microplane models or concrete. Convergence Criteria or Series Coupling Model with Sotening Since the ormulation o a convergent iterative algorithm has been the crucial step making the present series coupling model possible, a deeper analysis o the covergence problem is in order. It will be instructive to consider irst an algorithm which comes to mind more naturally (and has in act been tried long ago, albeit with disappointing results). Then we will try another algorithm and show why, and when, it works. Algorithm A. Predictor o Positive Stiness Consider a series coupling o two elements subjected to stress (Fig. 1). As the applied load is reduced, one element, with strain e, unloads with a positive tangent stiness K, and the other, with Numerical Comparisons with Test Data Although the present Model M5 has been veriied by comparisons with all the test data used in veriying Model M4 (Caner and Bažant 2000), comparisons o the tests with compressive or compressive-shear loading need not be reproduced here because they are identical or almost identical. The only signiicant change, and the advantage o this model, is ound in tensile racturing tests reaching into very large tensile strains. Two typical data sets have been selected or itting those o Petersson (1981) or uniaxial tension, and o Bresler and Pister (1958) or combined shear and uniaxial compression (produced by torsional axial loading). The latter data represent the compression shear envelope. Its points have been obtained by running simulations o response or dierent ratios o axial normal stress and shear stress, and collecting the peak points. Optimum itting o both these data has succeeded to produce a satisactory match; see Figs. 2 and 3. In Fig. 4, the eect o the load step size on the uniaxial tension response is shown. It is demonstrated in this igure that the model response is accurate with strain increments on the order o 1 Fig. 3. Optimum it o Bresler and Pister s (1958) test data or combined shear and uniaxial compression c =68 MPa ; parameter values used in M5 are k 1 =0.0001, k 2 =350, k 3 =10, k 4 =150 with =0.3, N1 =0.005, and N2 = / JOURNAL OF ENGINEERING MECHANICS ASCE / JANUARY 2005