A finite element strategy coupling a gradient-enhanced damage model and cohesive cracks for quasi-brittle materials

A new combined strategy to describe failure of quasi-brittle materials is presented thus allowing the complete description of the process, from initiation of damage to crack propagation. For the early stages of the process, and in order to overcome the well-known problems characterising local descriptions of damage (e.g. mesh-dependence), a gradient-enhanced model based on smoothed displacements is employed. In order to deal with material separation, this continuous description is coupled to a cohesive crack when damage parameter exceeds a critical value. Some difficulties may arise when dealing with the transition from regularised damage models to evolving cracks: crack initiation, crack-path direction, energetic equivalence... In this work, a discrete cohesive crack is introduced when the damage parameter exceeds a critical value. On the one hand, and to determine the crack-path direction, the medial axis of the already damaged profile is computed. That is, a geometric tool widely used in the computer graphics field is used here to track the crack surface. Since this technique is exclusively based on the shape of the regularised damage profile, no mesh sensitivity is observed when determining the crack direction. On the other hand, and to define the cohesive law, an energy balance is imposed thus ensuring that the fracture energy not yet dissipated in the damage zone is transferred to the crack

A new continuous-discontinuous damage model: cohesive cracks via an accurate energy-transfer process

A new continuous-discontinuous strategy to describe failure of quasi-brittle materials is presented. For the early stages of the failure process, a gradient-enhanced model based on smoothed displacements is employed. As soon as the damage parameter exceeds a critical value Dcrit<1, a cohesive crack is introduced. A new criterion to estimate the energy not yet dissipated by the bulk when switching models-from continuous to continuous-discontinuous-is proposed. Then, this energy is transferred to the cohesive crack thus ensuring that the continuous and the continuous-discontinuous strategies are energetically equivalent. Compared to other existing techniques, this new strategy accounts for the different unloading branches of damage models and thus, a more accurate estimation of the energy that has to be transferred is obtained. The performance of this technique is illustrated with one- and two-dimensional examples.Peer ReviewedPostprint (author’s final draft

A continuous-discontinous model for softening and cracking based on non-local displacements

A continuous-discontinuous model to simulate numerically an entire failure process is presented. Crack inception and its propagation is modelled by means of a gradient non-local model based on non-local displacements. To simulate properly the final stages of the process, a discrete crack approach (X-FEM) is used, where both local and non-local displacements are modelled as discontinuous fields. In this paper, this new combined approach is studied in detail and one- and two-dimensional examples are carried out to validate it.Postprint (published version

A geometric bridge between regularised damage and energetically equivalent cracks

In order to achieve a better characterisation of a whole failure process, models which combine damage and fracture mechanics have recently been proposed. Here, a new combined methodology is presented: in order to describe damage inception and its diffuse propagation, a gradient-enhanced continuum model based on smoothed displacements is used, which is coupled to a discontinuous one to describe the final stages of the process. Special emphasis should be placed on the difficulties concerning the transition between continuous damage growth and fracture. On the one hand, and in order to conserve the energy dissipation through the change of models, an appropriate cohesive law must be defined. In this paper, the proposed technique to define this law is explained. On the other hand, the direction of the crack path should be determined. Here, a new strategy is proposed: the discontinuity is propagated following the direction dictated by the medial axis of the damaged domain. That is, a geometric tool, widely used in the computer graphics field, is used here to locate cracks.Peer ReviewedPostprint (published version

Continuous-discontinuous modelling for quasi-brittle failure: propagating cracks in a regularised bulk

A new strategy to describe failure of quasi-brittle materials -concrete, for example- is presented. Traditionally, numerical simulation of quasi-brittle failure has been tackled from two different points of view: damage mechanics and fracture mechanics. The former, which belongs to the family of continuous models, describes fracture as a process of strain localisation and damage growth. The latter, which falls in the family of discontinuous models, explicitly introduces displacement discontinuities. Recently, some new approaches that merge these two classical theories have been devised. Although these combined approaches allow a better characterisation of the whole failure process, there are still some issues that need to be addressed, specially regarding the model switching -from the continuous to the continuous-discontinuous strategy. The goal of this thesis is to present a new contribution in this direction. Our main concern is to properly account for the three main difficulties that emerge when dealing with combined strategies: (1) the pathological mesh-dependence exhibited by local softening models needs to be corrected; (2) the crack-path location has to be determined and (3) the switching from the continuous to the continuous-discontinuous strategy should be done in such a way that the two approaches are energetically equivalent. First, we extend the applicability to a two- and three-dimensional setting of an alternative approach to regularise strain-softening -where non-locality is introduced at the level of displacements rather than some internal variable. To this end, we propose new combined boundary conditions for the regularisation equation (for the smoothed displacement field). As illustrated with different two- and three-dimensional examples, these boundary conditions allow to obtain physical realistic results for the first stages of the failure process. Second, we present a new combined formulation that allows the propagation of cracks through a regularised bulk. To define the crack-path, instead of the classical mechanical criteria, we propose to use a geometrical criterion. More specifically, given a regularised damage field D(x), the discontinuity propagates following the direction dictated by the medial axis of the isoline (or isosurface in 3D) D(x) = D*. That is, a geometric tool widely used for image analysis, computer vision applications or mesh generation purposes is used here to locate cracks. We illustrate the capabilities of this new approach by carrying out different two- and three-dimensional numerical tests. Last, we propose a new criterion to estimate the energy not yet dissipated by the bulk when switching models, so it can be transferred to the cohesive crack. This ensures that the continuous and the continuous-discontinuous strategies are energetically equivalent. Compared to other existing techniques, we present a strategy that accounts for the different unloading branches of damage models thus better estimating the energy that has to be transferred. We illustrate the performance of this technique with one- and two-dimensional examples.En aquesta tesi, presentem una nova estratègia per tal de descriure el procés de fallida de materials quasi-fràgils, com ara el formigó. Típicament la simulació numèrica d'aquest procés s'ha dut a terme mitjançant models de dany o models de fractura. Els primers |models continus| descriuen la fractura com un procés de localització de deformacions on el dany creix i es propaga. Els models de fractura, en canvi, són models discontinus que introdueixen de manera explícita discontinuïtats en el camp de desplaçaments. Recentment s'han proposat estratègies que combinen aquestes dues teories clàssiques. Tot i que aquestes formulacions alternatives permeten simular millor el procés de fallida, encara queden alguns aspectes per aclarir, especialment pel que fa al canvi de models |de l’estratègia contínua a la discontínua. En aquesta tesi es presenta una nova estratègia contínua-discontínua. El nostre principal objectiu és proposar nous mètodes per tal de resoldre tres de les dificultats que presenten aquests models combinats: (1) solucionar la dependència patològica de la malla d'elements finits que presenten els models locals amb reblaniment; (2) determinar la trajectòria de la fissura i (3) assegurar-se que el canvi de models del continu al discontinu| es fa de manera que les dues estratègies siguin energèticament equivalents. En primer lloc, ampliem l’ús |per tal de poder simular problemes dos i tres dimensionals d'una estratègia alternativa que regularitza el reblaniment de les lleis de tensió-deformació. Aquí la no-localitat s'introdueix a nivell del camp de desplaçaments i no a través d'una variable interna com succeeix en les formulacions estàndards. Per aquest motiu, proposem noves condicions de contorn combinades per l’equació de regularització (pel camp de desplaçaments suavitzat). Tal com s'observa en diferents exemples dos i tres dimensionals, aquestes condicions permeten simular de manera físicament realista les primeres etapes del procés de fallida. En segon lloc, presentem una nova formulació combinada on les fissures es propaguen a través del medi regularitzat. Per tal de definir la trajectòria d'aquestes fissures, utilitzem un criteri geomètric, a diferència dels criteris mecànics clàssics. En particular, sigui D(x) un camp regularitzat de dany, les discontinuats es propaguen seguint la direcció marcada per l'eix mitjà de la isolínia (o isosuperfície mitjana en 3D) D(x) = D_. _Es a dir, utilitzem aquí aquesta eina geomètrica, molt emprada en d'altres aplicacions com ara l’anàlisi d'imatges, la visió artificial o la generació de malles| per tal de propagar les fissures. En aquest cas, donem també exemples dos i tres dimensionals. Finalment, proposem un nou criteri per tal d'estimar l'energia que l'estructura encara no ha dissipat en el moment en que canviem de model, per tal que pugui ser transferida a la fissura cohesiva. D'aquesta manera, s'assegura que l’estratègia contínua i la contínua-discontínua siguin energèticament equivalents. En comparació amb d'altres tècniques, aquesta estratègia té en compte les diferents branques de descàrrega dels models de dany i permet estimar de manera més precisa l'energia que cal transmetre. Per tal de mostrar aquest balanç energètic es duen a terme diferents exemples en una i dues dimensions

Discontinuous failure in a gradient-enhanced continuous damage model: a regularised displacement framework

To simulate numerically a failure process, a new kind of model which combines the two traditional approaches (damage and fracture mechanics) has been proposed in the literature. The basic idea of these hybrid strategies is to employ regularised continuous models to describe the first stages of failure and discontinuous models to deal with the possible development of cracks. Here, a new combined approach is presented. In order to describe damage inception and its diffuse propagation, an implicit gradient-enhanced continuum model based on smoothed displacements is used, where two different displacement fields coexist: (a) the standard displacements u u u and (b) the gradient-enriched displacement field u u u, which is the solution of a partial differential equation with u u u as the source term. Once the damage parameter exceeds a critical value, the continuous model is coupled to a discontinuous one. The eXtended Finite Element Method (X-FEM) is used to describe the growing cracks, whose direction of propagation is prescribed by the steepest descent direction of the damage profile and whose cohesive law is defined according to an energy balance. Therefore, the energy not yet dissipated by the continuous bulk is transmitted to the cohesive interface thus ensuring that the energy dissipated by the structure remains constant through the transition

A general framework for softening regularisation based on gradient elasticity

A general non-local approach to regularise strain-softening continua is presented. The key idea is to introduce the gradient-type enrichment at the level of displacements (rather than some internal variable), so the model is formulated with two distinct displacement fields. In fact, gradient models based on two displacement fields are usual in non-local elasticity, where the goal is to avoid the shortcomings of classical (local) elasticity (i.e. strain singularities in statics, non-dispersive behaviour in dynamics). We show that such a gradient elasticity backbone model can be combined with any standard nonlinear constitutive driver to render a regularised model for softening inelasticity. To illustrate the generality of the approach, two prototype models (isotropic damage and von Mises plasticity) are discussed. The numerical examples show that the regularised models exhibit all of the desired features: mesh insensitivity, imperfection size insensitivity and description of size effects.Postprint (published version

Impact of model complexity and multi-scale data integration on the estimation of hydrogeological parameters in a dual-porosity aquifer

This study investigates the impact of model complexity and multi-scale prior hydrogeological data on the interpretation of pumping test data in a dual-porosity aquifer (the Chalk aquifer in England, UK). In order to characterize the hydrogeological properties, different approaches ranging from a traditional analytical solution (Theis approach) to more sophisticated numerical models with automatically calibrated input parameters are applied. Comparisons of results from the different approaches show that neither traditional analytical solutions nor a numerical model assuming a homogenous and isotropic aquifer can adequately explain the observed drawdowns. A better reproduction of the observed drawdowns in all seven monitoring locations is instead achieved when medium and local-scale prior information about the vertical hydraulic conductivity (K) distribution is used to constrain the model calibration process. In particular, the integration of medium-scale vertical K variations based on flowmeter measurements lead to an improvement in the goodness-of-fit of the simulated drawdowns of about 30%. Further improvements (up to 70%) were observed when a simple upscaling approach was used to integrate small-scale K data to constrain the automatic calibration process of the numerical model. Although the analysis focuses on a specific case study, these results provide insights about the representativeness of the estimates of hydrogeological properties based on different interpretations of pumping test data, and promote the integration of multi-scale data for the characterization of heterogeneous aquifers in complex hydrogeological settings

A continuous‐discontinuous model for crack branching

A new continuous‐discontinuous model for fracture that accounts for crack branching in a natural manner is presented. It combines a gradient‐enhanced damage model based on nonlocal displacements to describe diffuse cracks and the extended finite element method (X‐FEM) for sharp cracks. Its most distinct feature is a global crack tracking strategy based on the geometrical notion of medial axis: the sharp crack propagates following the direction dictated by the medial axis of a damage isoline. This means that, if the damage field branches, the medial axis automatically detects this bifurcation, and a branching sharp crack is thus easily obtained. In contrast to other existing models, no special crack‐tip criteria are required to trigger branching. Complex crack patterns may also be described with this approach, since the X‐FEM enrichment of the displacement field can be recursively applied by adding one extra term at each branching event. The proposed approach is also equipped with a crack‐fluid pressure, a relevant feature in applications such as hydraulic fracturing or leakage‐related events. The capabilities of the model to handle propagation and branching of cracks are illustrated by means of different two‐dimensional numerical examples

An experimental study of the influence of stress history on fault slip during injection of supercritical CO2

The injection of super-critical CO2 into a depleted reservoir will alter the pore pressure of the basin, which if sufficiently perturbed could result in fault slip. Therefore, knowledge of the acceptable pressure limits is required in order to maintain fault stability. A two-part laboratory study was conducted on fully saturated kaolinite fault gouge to investigate this issue. Previously, we showed that fault slip occurred once pore-pressure within the gouge was sufficient to overcome the normal stress acting on the fault. For kaolinite, this behaviour occurred at a pressure similar to the yield stress. The current study shows that following a slow-reduction in the maximum principal stress, as would be expected through changes in effective stress, the reactivation pressure shows a stress memory. Consequently, the pressure necessary to initiate fault slip is similar to that required at the maximum stress encountered. Therefore, fault slip is at least partially controlled by the previous maximum stress and not the current stress state. During the slow reduction in normal stress, the flow characteristics of the fault remain unchanged until pore-pressure exceeds shear stress and does not increase significantly until it exceeds normal stress. This results in fault slip, which slows the rate of flow increase as shear is an effective self-sealing mechanism. These observations lead to the conclusion that stress history is a vital parameter when considering fault stability