Ein numerisches Homogenisierungsverfahren zur Bestimmung makroskopischer Materialkennwerte von Böden und Gesteinen basierend auf Oberflächeneigenschaften
Final Report Abstract
The object of this research project was to develop a numerical homogenization procedure for the identification of macroscopic material parameters based on the microscopic surface properties of the considered material, i.e., pre-existing micro-cracks and micro-shear bands. Materials of interest were geomaterials, such as soils and rocks, in which the macroscopic, or average, stresses are strongly influenced by the aforementioned micro-defects. Clearly, the possibly anisotropic distribution of those defects plays an important role. For instance, the ultimate load of jointed rocks is significantly influenced by the roughness and orientation of pre-existing slip surfaces. Obviously, a direct macroscopic, phenomenological description of such complex mechanical responses is very complicated and in many cases, it is even impossible. However, by using a micro-scale model most physically essential effects can be consistently accounted for. In the present project, the roughness of slip planes was modeled by using a traction-separation law. Subsequently, a homogenization technique was utilized for computing the resulting average or macroscopic material response. Conceptually, for each macroscopic material point, a boundary value problem associated with the microscale is solved (FE2 ). For that purpose, the Finite Element (FE) method was adopted. Since the micro-defects are characterized by a pronounced localization behavior (the deformation is restricted to very narrow bands), the application of the standard finite element method is not possible. In the present project, a numerical model based on strong discontinuities (deformation jumps) was adopted and further elaborated. In contrast to classical constitutive models, approaches based on strong discontinuities, allow to use traction-separation laws (instead of stress-strain-relationships). Hence, friction models capturing the essential physics at the micro-scale can directly be utilized. The developed numerical implementation relies on the Strong Discontinuity Approach (SDA) framework. Without going to much into detail, the displacement jump is included within the finite element formulation by means of the Enhanced Assumed Strain (EAS) method. Consequently, the displacement discontinuities are modeled in an incompatible fashion and can be eliminated at the element level. Hence, the resulting scheme is computationally very efficient. This is particularly important, since more than thousand simultaneously active slip planes have to be taken into account. Though the SDA is well established for the modeling of single (or few) cracks and shear bands, the modifications necessary for a large number of micro-defects interacting or crossing each other has not been given yet. For this reason, a novel, fully threedimensional Multiple Localization Surface Approach (MLSA) was derived. It allows to model multiple micro-defects in each finite element and it is numerically very efficient. Furthermore, it is variationally consistent, i.e., it can be recast into a form such that all state variables follow jointly from minimizing a certain energy functional. The resulting three–dimensional finite element formulation was applied to analyze the influence of pre-existing and evolving micro-defects such as micro-shear bands on the macroscopic material response. For that purpose, a numerical homogenization strategy has been adopted. The application of the novel three-dimensional finite element implementation is mainly restricted by two different constraints. First, the concept of homogenization can only be adopted, if the assumption of scaleseparation is valid. However, if a micro-shear band runs through the entire representative volume element, this assumption does not make sense anymore. Hence, for the modeling of the post-peak behavior, or softening in general, new concepts are required. At least, the area of application of the proposed scheme has to be quantified in the near future. Second, even before the onset of softening, the employed numerical homogenization method (FE2 ) is very time-consuming. Consequently, novel, more efficient methods are needed. For instance, the results obtained from the homogenization procedure could be used to calibrate a macroscopic material model. By doing so, performance of the multi-scale approach could be increased significantly.
Publications
- An Efficient Finite Element Formulation Based on the Strong Discontinuity Approach for Modelling Material Failure at Finite Strains. Proceedings of the VIII International Conference on Computational Plasticity , 2005, Barcelona, Spanien
L. Stankovic, J. Mosler
- Locally Embedded Strong Discontinuities at finite strains - Numerical Implementation, 76. Jahrestagung der Gesellschaft für Angewandte Mathematik und Mechanik (GAMM), 2005, Luxembourg, Luxembourg
J. Mosler, L. Stankovic
- Locally Embedded Strong Discontinuities at finite strains - Numerical Implementation. PAMM, Vol. 5, 347-348, 2005
J. Mosler, L. Stankovic
- Modeling strong discontinuities at finite strains - A novel numerical implementation. Computer Methods in Applied Mechanics and Engineering, Vol. 195, Issues 33-36, 4396-4419, 2006
J. Mosler
- Numerical Prediction of Macroscopic Material Failure, 77. Jahrestagung der Gesellschaft für Angewandte Mathematik und Mechanik (GAMM), 2006, Berlin, Deutschland
L. Stankovic, J. Mosler
- Numerical Prediction of Macroscopic Material Failure. PAMM, Vol. 6, 197-198, 2006
L. Stankovic , J. Mosler
- Prediction of macroscopic Material Failure Based on Microscopic Cohesive Laws. Proceedings of the III European Conference on Computational Mechanics, Solids, Structures and Coupled Problems in Engineering, 2006, Lisbon, Portugal
L. Stankovic, J. Mosler
- Variational cohesive fracture models and threedimensional crack tracking. Proceedings of the 7th World Congress on Computational Mechanics, 2006, Los Angeles, USA
J. Mosler, M. Ortiz, A. Pandolfi
- Variational cohesive fracture models and threedimensional crack tracking. Proceedings of the 8th Congress on SIMAI, 2006, Baia Samuele (Ragusa), Sizilien, Italien
A. Pandolfi, J. Mosler, M. Ortiz
- Numerical analysis of shear bands in solids by means of computational homogenization, 78. Jahrestagung der Gesellschaft für Angewandte Mathematik und Mechanik (GAMM), 2007, Zürich, Schweiz
L. Stankovic, J. Mosler
- Numerical analysis of shear bands in solids by means of computational homogenization. PAMM, Vol. 7, 4060021-4060022, 2007
L. Stankovic, J. Mosler
- On the numerical modeling of localized material failure at finite strains by means of variational mesh adaption and cohesive elements. Habilitation, Ruhr-Universität Bochum, Lehrstuhl für Technische Mechanik, 2007
J. Mosler
- A Variationally Consistent Approach for Crack Propagation Based on Configurational Forces. Proceedings of the IUTAM Symposium on Progress in the Theory and Numerics of Con?gurational Mechanics, 2008, Nürnberg, Deutschland
J. Mosler
- Describing multiple surface localized failure by means of strong discontinuities at finite strains. Dissertation, Ruhr-Universität Bochum, Lehrstuhl für Technische Mechanik, 2008
L. Stankovic
