Numerical Homogenization of the Rigidity Tensor in Hooke's Law Using the Node-Based Finite Element Method |
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Authors: | Wouter Zijl Max A. N. Hendriks C. Marcel P. ''t Hart |
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Affiliation: | (1) Netherlands Institute of Applied Geoscience TNO, P.O. Box 80015, NL-3508 TA Utrecht, The Netherlands;(2) TNO Building and Construction Research, P.O. Box 49, NL-2600 AA Delft, The Netherlands |
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Abstract: | Combining a geological model with a geomechanical model, it generally turns out that the geomechanical model is built from units that are at least a 100 times larger in volume than the units of the geological model. To counter this mismatch in scales, the geological data model's heterogeneous fine-scale Young's moduli and Poisson's ratios have to be upscaled to one equivalent homogeneous coarse-scale rigidity. This coarse-scale rigidity relates the volume-averaged displacement, strain, stress, and energy to each other, in such a way that the equilibrium equation, Hooke's law, and the energy equation preserve their fine-scale form on the coarse scale. Under the simplifying assumption of spatial periodicity of the heterogeneous fine-scale rigidity, homogenization theory can be applied. However, even then the spatial variability is generally so complex that exact solutions cannot be found. Therefore, numerical approximation methods have to be applied. Here the node-based finite element method for the displacement as primary variable has been used. Three numerical examples showing the upper bound character of this finite element method are presented. |
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Keywords: | periodic media Poisson's ratio upscaling Young's modulus |
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