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

Presented is a new microplane model for concrete, labeled M5, which improves the representation of tensile cohesive fracture by eliminating spurious excessive lateral strains and stress locking for far postpeak tensile strains. To achieve improvement, a kinematically constrained microplane system simulating hardening nonlinear behavior (nearly identical to previous Model M4 stripped of tensile softening) is coupled in series with a statically constrained microplane system simulating solely the cohesive tensile fracture. This coupling is made possible by developing a new iterative algorithm and by proving the conditions of its convergence. The special aspect of this algorithm (contrasting with the classical return mapping algorithm for hardening plasticity) is that the cohesive softening stiffness matrix (which is not positive definite) is used as the predictor and the hardening stiffness matrix as the corrector. The softening cohesive stiffness for fracturing is related to the fracture energy of concrete and the effective crack spacing. The postpeak softening slopes on the microplanes can be adjusted according to the element size in the sense of the crack band model. Finally, an incremental thermodynamic potential for the coupling of statically and kinematically constrained microplane systems is formulated. The data fitting and experimental calibration for tensile strain softening are relegated to a subsequent paper in this issue, while all the nonlinear triaxial response in compression remains the same as for Model M4. DOI: 10.1061/(ASCE)0733-9399(2005)131:1(31) CE Database subject headings: Concrete; Fracture; Inelastic action; Damage; Softening; Finite element method; Numerical models. Introduction, Background, and Objective Except for the growth of spherical voids and collapse of spherical pores, which is not typical of concrete, almost all of the inelastic deformations in concrete microstructure, such as slip, friction, tensile microcrack opening, axial splitting, and lateral spreading in compression, occur on well defined planes taking any spatial orientation. Although these deformations occur at different points in the microstructure, the microplane models concentrate all such deformations, occurring in a small representative volume of the microheterogeneous material, into one point of the macroscopic smoothing continuum. Thus the constitutive properties characterizing these oriented inelastic phenomena (as well as the spherical voids and pores, if any) can be described by means of stress and strain vectors acting on a plane of arbitrary spatial orientation, called the microplane (Bažant 1984). The microplanes may be imagined as the tangent planes of an elemental sphere surrounding every continuum point [Fig. 1]. McCormick School Professor and W.P. Murphy Professor of Civil Engineering and Materials Science, Northwestern Univ., 2145 Sheridan Rd., Tech A135, Evanston, IL 60208. E-mail: [email protected] Ramón y Cajal Fellow, ETSECCPB-ETCG, Univ. Politecnica de Catalunya, Jordi Girona 1-3, Ed.D2 D.305, Barcelona 08034, Spain. E-mail: [email protected]; formerly, Visiting Scholar, Northwestern Univ., 2145 Sheridan Rd., Evanston, IL 60208. Note. Associate Editor: Franz-Josef Ulm. Discussion open until June 1, 2005. Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on January 27, 2003; approved on February 13, 2004. This paper is part of the Journal of Engineering Mechanics, Vol. 131, No. 1, January 1, 2005. ©ASCE, ISSN 0733-9399/ 2005/1-31–40/$25.00. JOUR The strain and stress vectors on the microplanes, e and s, must be assumed to be constrained in some way to the strain and stress tensors of the macrocontinuum, eij and sij (the indices refer to components in Cartesian coordinates xi , i=1, 2, 3). The constraint is said to be kinematic (or static) if the strain (or stress) vector on each microplane is the projection of the continuum strain (or stress) tensor, i.e., sed j =nieij and ssd j =nisij (where ni is the unit normal vector of 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 s) only by a weak variational constraint, represented by the principle of virtual work (or complementary virtual work). The condition of tensorial invariance is automatically satisfied by considering planes of 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 of vectors rather than tensors is clearer, conceptually simpler, and easier to formulate. (2) The vector representation can directly characterize the aforementioned oriented deformations as well as their localization into one preferred orientation (note for example that, by contrast, a relationship between the hydrostatic pressure and the second deviatoric stress invariant cannot describe frictional slip on a plane of a specific orientation). (3) The so-called vertex effect, an essential characteristic of concrete (Caner et al. 2002) which is generally missed by the classical tensorial models, is exhibited automatically. (4) Apparent deviations from normality in the sense of tensorial plastic models are automatic, since a microplane model for plasticity is equivalent to a large number of simultaneously active yield surfaces, for each of which the normality rule can be satisfied. (5) The constraint of the microplanes automatically provides all the cross effects such as the shear dilatancy and pressure sensitivity. (6) Combinations NAL OF ENGINEERING MECHANICS © ASCE / JANUARY 2005 / 31 of loading and unloading on different microplanes provide a complex path dependence and automatically reproduce the Bauschinger effect and hysteresis under cyclic loading. (7) The dependence of the current yield limits on the strain components (rather than scalar hardening-softening parameters) is easy to take into account. (8) In cyclic loading, fatigue is automatically simulated by accumulation of residual stresses on the microplanes after each load cycle. (9) Finally, anisotropy, while not requisite for concrete, can be captured easily, simply by making the constitutive properties of a microplane dependent on its orientation (in more detail, see Caner and Bažant 2000). These advantages outweigh the burden of a greater amount of computations, a burden that has been waning from year to year with the relentless advance in computer power. Since its inception (Bažant and Oh 1983a,b, 1985), the microplane constitutive model for concrete has developed into a powerful and robust computational tool for three-dimensional finite element analysis of concrete structures. Initially formulated for concrete as an extension and modification of a groundbreaking idea of Taylor (1938), the evolution of the model has advanced through several progressively improved versions, which were labeled as M1, M2, M3, and M4 for concrete (as described in Bažant et al. 2000a) and M4R for rock (Bažant and Zi 2003). The evolution then ramified to other complex materials such as sand, clay, rigid foam, shape-memory alloys, and fiber composites (Brocca and Bažant 2000, 2001a,b; Brocca et al. 2001). The model was extended to finite strain (Bažant et al. 2000a) and to the rate effect or creep (Bažant et al. 2000c), and has been used in dynamic finite element analyses with up to several million finite elements (Bažant et al. 2000b). Recently, Model M4f (see Part II) introduced into Model M4 a fracture energy based recalibration of softening boundaries in the sense of the crack band theory. The latest version, Model M4 (Bažant et al. 2000a; Caner and Bažant 2000) or M4f describes satisfactorily all the experimentally documented nonlinear triaxial behavior of concrete under compression and shear, and also simulates well the distributed tensile cracking that occurs for strains in the prepeak, peak, and early postpeak regimes. Problems have nevertheless arisen in predicting the far postpeak tensile response in which the tensile stress across cracks is getting reduced to zero. The objective of this paper is to overcome these problems by formulating a new microplane model, labeled M5. The new model enhances Models M4 and M4f with a statically constrained system of microcracks directly simulating the Fig. 1. Left: Coupling of kinematically and statically constrained microplane systems for hardening and softening responses. Right: components of strain or stress vectors on microplane. softening stress-separation law of cohesive fracture (Barenblatt 32 / JOURNAL OF ENGINEERING MECHANICS © ASCE / JANUARY 2005 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 for a Mixed Static-Kinematic Constraint Although the behavior for strains in the far post-peak tail of the stress–strain diagram is normally irrelevant for predicting the load capacities of structures, it is very important for dynamic response, especially for correctly assessing the energy absorption capability. For the far postpeak tensile strains and for the formation of complete fractures in which the crack bridging stresses are reduced to zero, Model M4, unfortunately, does not perform as well as the simple cohesive crack model or the simple classical smeared cracking models. It is plagued by two problems: (1) At far postpeak tensile softening, the lateral contraction in the directions parallel to the cracks is much larger than the Poisson effect in the material between the cracks and must be judged excessive (even though good test data are lacking, due to the difficulty of 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 finite tensile stress across the crack band is retained even at extremely large tensile strains. The cause does not lie in the classical problem of spurious localization and mesh sensitivity, which can be readily avoided by either applying a nonlocal operator or by modifying 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 of a kinematic constraint for a softening damage model. The kinematic constraint assumes that the strain vectors on the microplanes are the projections of the strain tensor. Because of 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 significantly inclined to the crack. These inclined microplanes produce large lateral contraction and, since they never soften to zero, they contribute a tensile stress across the cracks even if widely opened. The problem is aggravated by the necessity of a volumetric– deviatoric split (proposed in Bažant and Prat 1988a,b), i.e., the split of each microplane normal strain into its volumetric and deviatoric components. Such a split is inevitable for a realistic representation of nonlinear triaxial behavior in compression and the transition between compression and tension. In particular, the split is necessary to capture the fact that uniaxial compression and weakly confined compression lead to strain softening with a large lateral expansion, whereas the hydrostatic compression (and compression at zero lateral strain) never leads to any strain softening. This is a troublesome but inevitable dichotomy, which is ignored by most tensorial-type nonlinear triaxial constitutive models for concrete and rock. The split causes that, in far postpeak softening, most of 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 for tension does not help because it destroys the continuity of transition between the tensile and compressive responses, and restricting the magnitude of 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 from the classical Taylor models for metals and nonsoftening (overconsolidated) soils (Batdorf and Budianski 1949; Budianski 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 McDowell 1998). In Taylor models, the constraint of the planes called here the microplanes is static, i.e., the stress vector on these planes is assumed to be the projection of the stress tensor. As learned from the first studies of softening in fracturing materials at Northwestern University during the early 1980s, the main reason for replacing at that time the static constraint with a kinematic constraint was to stabilize the system of softening microplanes. If there is no softening, as in plastic metals, the static constraint causes no instability because the strain corresponding to a given stress reduction is uniquely defined by the elastic unloading. But if there is softening, the static constraint inevitably leads to instability because the microplane strains caused by a given stress reduction are not unique, corresponding to either the softening 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 of one orientation only. This makes it impossible to simulate a system of microcracks of many orientations, developing simultaneously in the peak stress region. Later on, in the far postpeak response, the cracking of 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 for the governing role of the kinematic constraint in the peak stress region. Unlike plastic yielding, happening at constant stress, the average local strain tensor of the material around and between the diffuse microcracks is essentially the same as the strain tensor of the macroscopic smoothing continuum. The formation of small microcracks causes the continuum stress to drop without affecting the strain tensor (the same observation underlies the classical Kachanov-type theory of continuum damage mechanics). Assuming all the behavior to adhere to a strict kinematic constraint is also arguable from the viewpoint of stiffness bounds. From the theory of 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 stiffness, which are generally not close to each other. The real behavior is somewhere inbetween. Therefore, a microplane model incorporating a mixture of the kinematic and static constraints should be physically more realistic. There are further reasons for seeking a way to formulate a hybrid constraint. Just like the strains due to plastic slip on a long slip line, the strains due to the opening of continuous cracks (unlike the strains due to small diffuse microcracks) are additive to the elastic strains of the material inbetween and both happen at the same stress. Consequently, the interaction of 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 of the strains due to long cracks of different orientations. This means that the microplane system for the interaction of the opening of 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 diffuse microcracking in the peak stress region. A mixture of both constraints was attempted at Northwestern already in the early 1980s but seemed unworkable. It took the JOUR form of a series coupling of two microplane systems [Fig. 1], in which the strain tensor eij was decomposed as