Keywords: spatial and temporal multi-scale models, damage and fracture, multi-grid solver, heterogeneous materials, concrete.

The damage behaviour of heterogeneous materials, such as concrete, is often described by continuum damage theories on the macro-scale, using homogenized material parameters. These models have the advantage of being applied in simulations of large-scale structures, but the experimental determination of necessary material parameters, especially for the description of material damage effects, is demanding and often a direct identification of these parameters is not possible. Furthermore these models are not capable of describing all the physical effects applying to the continuum theories, for example, the decohesion effects between grain and matrix material. At the meso-scale the heterogeneous material structure is described explicitly, representing the aggregates and the matrix material, and thereby allowing a detailed description of the separate constituents. This paper describes various aspects of simulating the damage effects of concrete materials in heterogeneous multi-scale models, integrating aggregate-matrix models on the meso-scale into continuum damage models for the macro-scale. Beginning with the generation of grain-matrix models we will discuss numerical discretization techniques. Different iterative schemes following the nonlinear response paths, including softening branches and appropriate solver strategies such as multi-grid solvers will be presented. The material description on the meso-scale will be discussed, indicating the advantages of applied simulation techniques. Coupling between models on different spatial scales, for example meso and macro-scales, resulting into heterogeneous multi-scale models is proposed via constraint conditions. Thereby it becomes possible to simulate large-scale constructional components and to obtain detailed information on local micro-damage effects at the same time. An example of a multi-scale simulation is shown in Figure 1. In this example a priori information about the zone where the localised damage is expected can be used. This information can, for example, be obtained by a linear analysis in advance. The stress concentration in the corner region can be used as an indicator to model the sub-region in the vicinity of the re-entering corner with a meso-scale model, while the rest of the structure is kept as a macro-scale model.

The load is applied as displacement u at the lower right hand edge and is increased incrementally. The constitutive law for the matrix material is coupling a micro-plane model with an isotropic damage model for the softening regime. Figure 2 shows the propagation of the damage zone, starting from the reentering corner, through the model for different load situations. The corresponding load-displacement curve is given in Figure 3.

Similar strategies as applied for spatial multi-scale problems can be adopted for simulation of temporal multi-scale problems, as found for example in concrete creeping problems. The paper will give an outlook into adequate methods.

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