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Material Modeling

Modeling of reinforced concrete (RC) structures is a problem having complex character. It involves both modeling of concrete and reinforcing steel and also the interaction of these materials. As it is widely known, concrete belongs among so called quasi-brittle materials. Its behaviour is strongly influenced by a stress state to which it is exposed. The behaviour of concrete varies from brittle to very ductile according to the lateral confinement which can be provided either by outside constraints or by transversal reinforcement, which is the common case. All types of transversal reinforcement such as different ties, stirrups or steel sections can be used to provide this lateral confinement. The change of concrete behaviour according to lateral confinement yields the need of triaxial modeling of concrete. Without respecting the triaxial behaviour of concrete one cannot describe the reality of RC structures. There is also one very important feature of concrete. General loading cases lead to the phenomenon called softening. Softening of concrete is characterized by a progressive loss of material integrity, which yields the descending load-deflection diagram.

Modeling of reinforcing steel must include elastoplastic behaviour of the material. Longitudinal reinforcement is usually placed near the surface of concrete and fixed laterally by transversal reinforcement. If the RC structure is subjected to the compressive load the longitudinal reinforcement can buckle. Thus, the model of longitudinal reinforcement must include also the possibility of buckling of steel.

What are the material models capable to describe triaxial nonlinear behaviour of concrete? Well, there is not much choices. Classical models are based mainly on the theory of plasticity. Theory of plasticity was firstly developed for modeling of metals, but it was also enlarged for modeling of concrete (see e.g. [1,2,3]). Theory of plasticity provides nonlinear description of concrete including loading, unloading and path dependence. The key point of these models is the definition of yield condition which is usually formulated in the stress space. This formulation can be very complicated and moreover multidimensional formulation can be hardly imagined and physically interpreted. Another type of models are based on continuum damage mechanics. These models are able to describe material stiffness degradation according to the certain damage parameter, which can be defined as a single scalar parameter or a tensor of higher orders. A popular damage model was proposed by Mazars in [4]. Some models combine the attitude of plasticity and damage mechanics (see [5]).

All these types of models suffer from some kind of insufficiency. One of the major errors can be caused by not respecting of anisotropy development within the material microstructure. Deficiency of these models is that they are usually derived in principal strain space. They do not respect the rotation of principal axes during loading process that can lead to big errors. Especially description of softening is very sensitive to it, because damage development changes the material from isotropic to highly anisotropic. The right way to solve this problem is to link damage with its orientation in the material and compute the material response directly for this concrete orientation. This is done so in concept of microplane model [6,7,8]). Final response is then given by combination of the responses from different orientations.

The classical approach to the constitutive modeling is based on a direct relationship between strain and stress tensors and their invariants. In contrary to it, constitutive relations of microplane model are formulated in terms of strain and stress components on planes of arbitrary spatial orientations, so called microplanes. This attitude excels in conceptual simplicity and allows straightforward modeling of anisotropy and other processes connected with planes with different orientations. The penalty to be paid is a great increase in computational effort. The relationship between micro and macro level is obtained by projecting strain tensor to the particular microplanes (so called kinematic constraint) or by projecting stress tensor (static constraint). Then constitutive relations between microstrains and corresponding microstresses are evaluated. The missing link (between microstresses and macrostress for kinematic constraint and between micro and macro strain for static constraint, respectively) is obtained by application of principle of virtual work. That kind of material model is capable to describe triaxial nonlinear behaviour of concrete including tensional and compressive softening, damage of the material, different types of loading, unloading or cyclic loading.

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Daniel Rypl