The role of heterogeneity in two long-range systems is explored with a focus on the interplay of this heterogeneity with the component system interactions. The first will be the heterogeneous Ising model with long-range interactions. Earthquake fault systems under long-range stress transfer with varying types of heterogeneity will be the second system of interest. First I will review the use of the intervention method to determine the time and place of nucleation and extend its use as an indicator for spinodal nucleation. The heterogeneous Ising model with fixed magnetic sites will then be reformulated as a dilute random field Ising model. This reformulation will allow for the application of spinodal nucleation theory to the heterogeneous Ising model by correcting the spinodal field and the critical exponent sigma describing the critical behavior of clusters in spinodal nucleation theory. The applicability of this correction is shown by simulations that determine the cluster scaling of the nucleating droplets near the spinodal. Having obtained a reasonable definition of the saddle point object describing the nucleation droplet, the density profile of the nucleating droplet is measured and deviations from homogeneous spinodal nucleation are found due to the excess amount of sparseness in the nucleating droplet due to the heterogeneity. Earthquake fault systems are then introduced and a connection is shown of two earthquake models. Heterogeneity is introduced in the form of asperities with the intent of modeling the effect of hard rocks on earthquake statistics. The asperities are observed to be a crucial element in explaining the behavior of aftershocks resulting in Omori's law. A second form of heterogeneity is introduced by coupling the Olami-Feder-Christensen model to an invasion percolation model for the purpose of modeling an earthquake fault system undergoing hydraulic fracturing. The ergodicty and event size statistics are explored in this extended model. The robustness of the event size statistics results are explored by allowing for the dissipation parameter in the Olami-Feder-Christensen model to vary.

Motivated by the structure of the normal surface stress of the extended Hertzian approach [1] given due to terms of the form r^2n*(a^2-r^2)^(1/2) (n=0, 2, 4, 6…) it seems attractive to evaluate the complete elastic field also for shear loadings of this form. The reason for this lays in the demand for analytical tools for the description of mixed loading conditions as they appear for example in scratch experiments. [1] N. Schwarzer, "Elastic Surface Deformation due to Indenters with Arbitrary symmetry of revolution", J. Phys. D: Appl. Phys., 37 (2004) 2761-2772

In this paper it is shown that a completely analytical theory based on the extended Hertzian approach together with additional considerations taking into account the geometrical conditions of a curved surface provide an appropriate model for the theoretical “simulation” of a variety of asperity contact problems. This model yields relatively fast and easy to use tools for the analysing of contact problems arising in connection with rough surfaces. In this study the results are shown on the example of a 3µm-DLC-coating on a steel substrate with asperities of about 100µm in diameter and 15µm height. It is found, that – under a general average pressure of 1GPa – the ideal asperity tip contact situation would lead to severe damage due to plastic flow within the steel substrate. On the other hand a rather conforming contact situation appears to be completely non critical.

Résumé: L’objectif de cette thèse est d’interpréter la dégradation des aspérités des joints rocheux sous différentes conditions de chargement. Pour cela, la variation des aspérités durant les différentes étapes du cisaillement d’un joint rocheux est observée. Selon le concept appelé “tiny windows”, une nouvelle méthodologie de caractérisation des épontes des joints a été développée. La méthodologie est basée sur les coordonnées tridimensionnelles de la surface des joints et elles sont mesurées après chaque essai. Après la reconstruction du modèle géométrique de la surface du joint, les zones en contact sont identifiées à travers la comparaison des hauteurs des “tiny windows” superposées. Ainsi, la distribution des zones de la surface en contact, endommagées et sans contact ont été identifiées. La méthode d’analyse d’image a été utilisée pour vérifier les résultats de la méthodologie proposée. Les résultats indiquent que cette méthode est appropriée pour déterminer la taille et la distribution des surfaces du joint en contact et endommagées à différentes étapes du cisaillement. Un ensemble de 38 répliques ont été préparées en coulant du mortier sans retrait sur une surface de fracture obtenue à partir d’un bloc de granite. Différentes conditions de chargement, incluant des chargements statiques et cycliques ont été appliquées afin d’étudier la dégradation des aspérités à différentes étapes du procédé de cisaillement. Les propriétés géométriques des “tiny windows” en contact en phase pré-pic, pic, post-pic et résiduelle ont été analysées en fonction de leurs angles et de leurs auteurs. Il a été remarqué que les facettes des aspérités faisant face à la direction de cisaillement jouent un rôle majeur dans le cisaillement. Aussi, il a été observé que les aspérités présentent différentes contributions dans le cisaillement. Les aspérités les plus aigües (“tiny windows” les plus inclinées) sont abîmées et les aspérités les plus plates glissent les unes sur les autres. Les aspérités d’angles intermédiaires sont définies comme “angle seuil endommagé” et “angle seuil en contact”. En augmentant la charge normale, les angles seuils diminuent d’une part et, d’autre part, le nombre de zones endommagées et en contact augmentent. Pour un petit nombre de cycles (avec faible amplitude et fréquence), indépendamment de l’amplitude, une contraction apparaît ; par conséquent, la surface en contact et les paramètres de résistance au cisaillement augmentent légèrement. Pour un grand nombre de cycles, la dégradation est observée à l’échelle des aspérités de second ordre, d’où une baisse des paramètres de résistance au cisaillement. Il a été aussi observée que les “tiny windows” avec différentes inclinaisons contribuent au processus de cisaillement, en plus des “tiny windows” les plus inclinées (aspérités plus aigües). Les résultats de la méthode proposée montrent que la différenciation entre les zones en contact et celles endommagées s’avère utile pour une meilleure compréhension du mécanisme de cisaillement des joints rocheux. / Abstract: The objective of the current research is to interpret the asperity degradation of rock joints under different loading conditions. For this aim, the changes of asperities during different stages of shearing in the three-dimensional joint surface are tracked. According to a concept named ‘tiny window’, a new methodology for the characterization of the joint surfaces was developed. The methodology is based on the three-dimensional coordinates of the joints surface that are captured before and after each test. After the reconstruction of geometric models of joint surface, in-contact areas were identified according to the height comparison of the face to face tiny windows. Therefore, the distribution and size of just in-contact areas, in-contact damaged areas and not in-contact areas are identified. Image analysis method was used to verify the results of the proposed method. The results indicated that the proposed method is suitable for determining the size and distribution of the contact and damaged areas at any shearing stage. A total of 38 replicas were prepared by pouring non-shrinking cement mortar on a fresh joint surface of a split granite block. Various loading conditions include monotonic and cyclic loading were applied to study the asperities degradation at different stages of shearing. The geometric properties of the in-contact tiny windows in the pre-peak, peak, post-peak softening and residual shearing stages were investigated based on their angle and height. It was found that those asperities facing the shear direction have the primary role in shearing. It is remarkable that different part of these asperities has their own special cooperation in shearing. The steepest parts (steeper tiny windows) are wore and the flatter parts (flatter tiny windows) are slid. The borderlines between these tiny windows defined as “damaged threshold angle” and “in-contact threshold angle”. By increasing normal load, both the amounts of threshold angles are decreased and contact and damaged areas increased. During low numbers of cycles (with low amplitude and frequency), independent of the type of cycle, contraction occurs and consequently the contact area and the shear strength parameters slightly increased. During larger number of cycles, degradation occurred on the second order asperities, therefore the shear strength parameters slowly decreased. It was also observed that tiny windows with different heights participate in the shearing process, not just the highest ones. The results of the proposed method indicated that considering differences between just in-contact areas and damaged areas provide useful insights into understanding the shear mechanism of rock joints.

Motivated by the structure of the normal surface stress of the extended Hertzian approach [1] given due to terms of the form r^2n*(a^2-r^2)^(1/2) (n=0, 2, 4, 6…) it seems attractive to evaluate the complete elastic field also for shear loadings of this form. The reason for this lays in the demand for analytical tools for the description of mixed loading conditions as they appear for example in scratch experiments. [1] N. Schwarzer, "Elastic Surface Deformation due to Indenters with Arbitrary symmetry of revolution", J. Phys. D: Appl. Phys., 37 (2004) 2761-2772

In this paper it is shown that a completely analytical theory based on the extended Hertzian approach together with additional considerations taking into account the geometrical conditions of a curved surface provide an appropriate model for the theoretical “simulation” of a variety of asperity contact problems. This model yields relatively fast and easy to use tools for the analysing of contact problems arising in connection with rough surfaces. In this study the results are shown on the example of a 3µm-DLC-coating on a steel substrate with asperities of about 100µm in diameter and 15µm height. It is found, that – under a general average pressure of 1GPa – the ideal asperity tip contact situation would lead to severe damage due to plastic flow within the steel substrate. On the other hand a rather conforming contact situation appears to be completely non critical.