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September 8th, 2015

The microplane model (Bazant 1984c) trades simplicity of concept for increased numerical work leftto the computer. This model represents a generalization of the basic idea of G. l. Taylor (1938), who proposed that the constitutive behavior of polycrystalline metals may be characterized by relations between the stress and strain vectors acting on planes of all possible orientations within the material, and that the macroscopic strain or stress tensor may then be obtained as a summation (or resultant) of all these vectors under the assumption of a static or kinematic micro-macro constraint.

Taylor’s idea was soon recognized as the most realistic way to describe the plasticity of metals, but the lack of computers prevented practical application in the early times. Batdorf and Budianski (1949) were first to describe hardening plasticity of polycrystalline metals by a model of this type, and many other researchers subsequently refined or modified this approach to metals (Kroner 1961; Budianski and Wu 1962; Lin and Ito 1965, 1966; Hill 1965, 1966; Rice 1970). Taylor’s idea was also developed for the hardening inelastic response of soils and rocks (Zienkiewicz and Pande 1977; Pande and Sharma 1981, 1982; Pande and Xiong 1982).

In all the aforementioned approaches, it was assumed that the stress acting on various planes in the material, called the slip planes, was the projection of the macroscopic stress tensor. This is a static constraint. As shown later, the static constraint prevents such models from being generalized to postpeak strain softening behavior or damage. In an effort to model concrete, it was realized that the extension to damage requires replacing the static constraint by a kinematic constraint, in which the strain vector on any inclined plane in the material is the projection of the macroscopic strain tensor (Bazant 1984c). The kinematic constraint makes it possible to avoid spurious localization among orientations in which all the strain softening localizes preferentially into a plane of only one orientation.

In all applications to metals, the formulations based on Taylor’s idea were called the slip theory of plasticity, and in applications to rock, the multi-laminate model. These terms, however, became unsuitable for the description of damage in quasibrittle materials. For example, the salient inelastic behavior of concrete docs not physically represent plastic slip (except under extremely high confining stresses), but microcracking. For this reason, the neutral term “microplane model”, applicable to any physical type of inelastic behavior, was coined (Bazant 1984c) (although a nondescriptive term such as “Taylor-Batdorf – Budianski model", possibly with the names of further key contributors, could also be used). The term “microplane” reflects the basic feature that the material properties arc characterized by relations between the stress and strain components independently for planes of various orientation within the microstructure of the material. This term also avoids confusion with the type of micro-macro constraint, which has always been static in the slip theory of plasticity but must be kinematic for strain-softening of concrete. Also, as introduced for the tnicroplane model (Bazant 1984c), the tensorially invariant macroscopic constitutive relations are obtained from the responses on the microplanes of all orientations in a more general manner than in the slip theory of plasticity—by means of a variational principle (or the principle of virtual work).

The microplanc model of concrete was developed in detail first for the tensile fracturing (Bazant and Oh 1983b, 1985; Bazant and Gambarova 1984), and later for nonlinear triaxial behavior in compression with shear (Bazant and Prat 1988b). The reason that these new models used the kinematic rather than static constraint for the microplanes was to avoid spurious instability of the constitutive model due to strain softening (which always occurs for the static constraint). Because the tangent! til material stiffness matrix loses positive definiteness (due to postpeak strain softening as well as lack or normality), the nonlocal approach, which prevents spurious excessive localization of damage in structures and spurious mesh sensitivity, was combined with the microplane model (Bazant and Ozbolt 1990, 1992; Ozbolt and Bazant 1991, 1992). An explicit formulation and an efficient numerical algorithm for the tnicroplane model of

Bazant and Prat (1988b) was recently presented by Carol, Prat and Bazant (1992). It was also shown that the microplane model with a kinematic constraint can be cast in the form of continuum damage mechanics in which the damage, understood as a reduction of the stress-resisting cross section area fraction in the material, represents a fourth-order tensor independent of the microplane material characteristics (Carol, Bazant and Prat 1991; Carol and Bazant 1997).

Although the microplane model of Bazant and Prat (1988b) was initially thought to perform well for postpeak softening damage in both compression and tension, Jirasek (1993) found that, in postpeak uniaxial tension, large positive lateral strains develop at large tensile strains. Me showed that this was caused by localization of tensile strain softening into the volumetric strain while the deviatoric strains on the strain softening microplanes exhibited unloading. It was recognized that this localization of tensile softening damage into one of the two normal strain components in tension (that is the volumetric one), was an inevitable consequence of separating the normal strains into the volumetric and deviatoric parts. However, this separation was previously shown necessary (Bazant and Prat 1988b) for correct modeling of triaxial behavior in compression as well as for achieving the correct elastic Poisson ratio. The problem was overcome by introducing a new concept—the stress-strain boundaries (Ba?.anl 1993c; Bazant, Jirasek et al. (1994); Bazant, Xiang and Prat 1996), which will be described in detail. This concept allows an explicit algorithm and is computationally efficient.

The basic philosophy of microplane model blends well with the philosophy of finite elements. Finite elements represent a discretization with respect to space (or distance), while the microplane model represents a discretization with respect to orientations. In both, the principle of virtual works is used, as will be seen, in analogous ways—to establish the equilibrium relations and stiffness for the postulated kinematic constraint, which is given by the shape (or interpolation) functions for finite elements or by the kinematic constraint between orientations. This analogous structure is suitable for explicit programs (Fig. 14.1.1).

In another sense, the microplane model can be regarded as complementary to the nonlocal concept. Whereas the nonlocal concept handles interactions at distance, the microplane model handles interactions between orientations (Fig. 14.1.2). The nonlocality prevents spurious localization in space, whereas the kinematic constraint of the microplane model prevents localizations between orientations, as will be pointed out.