Mathematical Physics - Volume II - Numerical Methods

Chapter 6. Introduction to Computational Mechanics of Discontinua

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Modeling using computer simulations is more flexible than analytical modeling and has the advantage over experimental research in that the data are available at every stage of the virtual experiment. This flexibility extends to the ability to configure loads, initial and boundary conditions, and to tailor the custom-made models in accordance with the topological, geometric, and structural disorder of material. These, so called, virtual experiments are in main aspects very similar to the laboratory experiments. First, a sample is prepared from the subject material rendering a simulation object in virtual space ("numerical" or "virtual" material). The sample created in this way is then connected to the necessary "virtual" measuring instruments so that the parameters of the state can be recorded over a period of time. Since most measurements are subject to statistical variations, the more time is available to average the results, the more accurate the measurements become. However, the virtual measurement resolution is inversely proportional to the size of the averaging period and it is necessary to find a compromise between these conflicting requirements taking into account the nature of the physical phenomenon being simulated. Consequently, the most common errors made during virtual experiments are very similar to those that can contaminate the results of actual laboratory experiments: the sample is not adequately prepared; the measurement is too short; due to conceptual oversights, we do not measure what we intend to measure... All CMD methods offer some common advantages in damage and fracture analysis compared to corresponding conventional computational methods based on continuum mechanics (typically, FEM). Damage and its evolution are presented explicitly through broken bonds or contacts; it is not necessary to use any empirical relations to define damage or determine its effect on material behavior. Variety of structural flaws nucleate, extend and merge into different types of macro scopic damage without the need to use numerical "ingenuities" such as convenient mesh orientation, mesh reformulation or constant adoptive meshing. There is no need to develop constitutive laws or damage models in order to represent complex nonlinear responses of materials as they emerge naturally through the collective behavior of discrete units whose interaction is guided by relatively simple rules. A summary of the CMD models that will be considered herein is shown in Figure 6.2 in conjunction with the natural spatial scale on which they are most commonly used.

Figure 6.2: Spatial scales and corresponding traditional CMD models as tentatively classified herein for typical brittle materials with random microstructure (e.g., concrete).