Microplane model for concrete with relaxed kinematic constraint
Introduction
In recent years, a significant effort in modeling of concrete-like materials for general stress–strain histories has been expended. Presently the available models for concrete can roughly be classified into two categories: (1) macroscopic models, in which the material behavior is considered to be an average response of a rather complex microstructural stress transfer mechanism and (2) the microscopic models, where the micromechanics of deformations is described by stress–strain relations on the microlevel. No doubt, from the physical point of view, more promising are microscopic models. However, they are computationally extremely demanding. Therefore, in practical applications we have to use macroscopic models.
At the macroscale, the model has to correctly describe microstructural phenomena such as cohesion, friction and aggregate interlock. These phenomena make the consumption of energy, which is released as a consequence of cracking, possible. They are microstructural (volume) and not point material properties. Consequently, the macroscopic models have to be related to the characteristic volume in which these phenomena take place i.e. they have to be nonlocal (Pijaudier-Cabot and Bažant, 1987, Bažant, 1991, de Borst, 1991; Ožbolt and Bažant, 1996). From the microscopic point of view (micro models), the nonlocality has no meaning. However, the same as the macroscopic model, itself, the nonlocality is an important tool (concept, model, etc.) which makes representation of the microscopic phenomena on the macroscale realistic. The problems of nonlocality and regularization procedures are beyond the scope of the present paper and are not discussed here in detail.
The evolution of damage (cracking) is closely related to the structure geometry and its size. Due to the fact that the material testing is performed on specimens of a finite size (structures), from experimental results it is difficult to filter out the macroscopic material properties. Typical example is the uniaxial compressive test. As soon as the peak load is reached (approximately uniaxial compressive strength), a number of vertical or inclined cracks arise. The specimen (structure) fails after smaller cracks result into one (or more) diagonal-shear-splitting crack(s), whose propagation is controlled by the specimen size and boundary conditions. In the critical (the weakest) cross section (crack plane), the material ruptures i.e. discontinuous change of the strain field takes place (continuum → discontinuum). Perpendicular to the discontinuity plane as well as in the three-dimensional space around it, the stresses and strains relax (unload) approximately to zero (Fig. 1).
Traditionally, the macroscopic models are formulated by total or incremental formulation between the σij and εij components of the stress and strain tensor, using the theory of tensorial invariants (Chen and Chen, 1975, Willam and Warnke, 1974, Gerstle et al., 1980, Gerstle, 1981, Ortiz, 1985). In the framework of the theory, there are various possible approaches for modeling of concrete, such as plasticity, plastic-fracturing theory, continuum damage mechanics, endocronic theory and their various combinations. Due to the complexity of the concrete, presently exists no model based on the stress and strain tensor and their invariants which is capable to realistically predict the behavior of concrete, not only for the three-dimensional monotonic loading, but also for the general three-dimensional cyclic loading. For instance, the invariant type of the models have difficulties with correct modeling of concrete expansion for triaxial compressive load, which in some applications governs the failure mechanism and is a consequence of cracking (discontinuity). Such models are based on the continuum mechanics and are generally not capable to simulate complex stress–strain states, which involve cracking, using only a few available invariants. Moreover, based on the plasticity type of the flow rules, which is most commonly in these models used, it is difficult to model complex three-dimensional cyclic response of concrete. These, as well some other drawbacks of the constitutive laws based on the theory of tensorial invariants is the main motivation for the use of the microplane theory as an alternative approach for macroscopic modeling of concrete.
Section snippets
Microplane material model
In the microplane model, the material is characterized by a relation between the stress and strain components on planes of various orientations. These planes may be imagined to represent the damage planes or weak planes in the microstructure, such as contact layers between aggregate pieces in concrete (Fig. 2). In the model, the tensorial invariance restrictions need not be directly enforced. They are automatically satisfied by superimposing in a suitable manner, the responses from all the
Basic assumptions and macro–micro relationships
To keep the conceptual simplicity of the improved model, most of the assumptions originally introduced by Bažant and Prat (1988) and Ožbolt and Bažant (1992) are kept the same. However, instead of working with the microplane strain calculated from the total strain tensor (continuous strain field), for dominant tensile load the microplane stress is calculated from the effective microplane strain. The main assumptions are as follows:
(I) Each microplane resists normal and shear strain components (ε
Verification of the model
To verify the material model for concrete, one has to compare experimental results with the model response. Due to the fact that each test specimen is a structure which possibly exhibits size effect on the peak load and post-peak response, the calibration and verification of the macroscopic material models for quasi-brittle materials is difficult. On the macroscale, cracking phenomena are not a point but a volume property. Consequently, when the model response is compared with the test data,
Conclusions
The present microplane model is the macroscopic three-dimensional material model for concrete. The model is aimed to be used for modeling of damage and fracture phenomena in concrete and reinforced concrete structures loaded under general three-dimensional state of stresses and strains.
The simplest and computationally most efficient form of the model formulation is based on the kinematic constraint approach which, however, in combination with realistic stress–strain relationships for concrete
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