The diffuse Nitsche method: Dirichlet constraints on phase-field boundaries

We explore diffuse formulations of Nitsche's method for consistently imposing Dirichlet boundary conditions on phase-field approximations of sharp domains. Leveraging the properties of the phase-field gradient, we derive the variational formulation of the diffuse Nitsche method by transferring all integrals associated with the Dirichlet boundary from a geometrically sharp surface format in the standard Nitsche method to a geometrically diffuse volumetric format. We also derive conditions for the stability of the discrete system and formulate a diffuse local eigenvalue problem, from which the stabilization parameter can be estimated automatically in each element. We advertise metastable phase-field solutions of the Allen-Cahn problem for transferring complex imaging data into diffuse geometric models. In particular, we discuss the use of mixed meshes, that is, an adaptively refined mesh for the phase-field in the diffuse boundary region and a uniform mesh for the representation of the physics-based solution fields. We illustrate accuracy and convergence properties of the diffuse Nitsche method and demonstrate its advantages over diffuse penalty-type methods. In the context of imaging based analysis, we show that the diffuse Nitsche method achieves the same accuracy as the standard Nitsche method with sharp surfaces, if the inherent length scales, i.e., the interface width of the phase-field, the voxel spacing and the mesh size, are properly related. We demonstrate the flexibility of the new method by analyzing stresses in a human vertebral body

Computational fracture modelling by an adaptive cracking particle method

Conventional element-based methods for crack modelling suffer from remeshing and mesh distortion, while the cracking particle method is meshless and requires only nodal data to discretise the problem domain so these issues are addressed. This method uses a set of crack segments to model crack paths, and crack discontinuities are obtained using the visibility criterion. It has simple implementation and is suitable for complex crack problems, but suffers from spurious cracking results and requires a large number of particles to maintain good accuracy. In this thesis, a modified cracking particle method has been developed for modelling fracture problems in 2D and 3D. To improve crack description quality, the orientations of crack segments are modified to record angular changes of crack paths, e.g. in 2D, bilinear segments replacing straight segments in the original method and in 3D, nonplanar triangular facets instead of planar circular segments, so continuous crack paths are obtained. An adaptivity approach is introduced to optimise the particle distribution, which is refined to capture high stress gradients around the crack tip and is coarsened when the crack propagates away to improve the efficiency. Based on the modified method, a multi-cracked particle method is proposed for problems with branched cracks or multiple cracks, where crack discontinuities at crack intersections are modelled by multi-split particles rather than complex enrichment functions. Different crack propagation criteria are discussed and a configurational-force-driven cracking particle method has been developed, where the crack propagating angle is directly given by the configuration force, and no decomposition of displacement and stress fields for mixed-mode fracture is required. The modified method has been applied to thermo-elastic crack problems, where the adaptivity approach is employed to capture the temperature gradients around the crack tip without using enrichment functions. Several numerical examples are used to validate the proposed methodology

A hp-adaptive discontinuous Galerkin finite element method for accurate configurational force brittle crack propagation

Engineers require accurate determination of the configurational force at the crack tip for fracture fatigue analysis and accurate crack propagation. However, obtain- ing highly accurate crack tip configuration force values is challenging with numer- ical methods requiring knowledge of the stress field around the crack tip a priori. In this thesis, the symmetric interior penalty discontinuous Galerkin finite element method is combined with a residual based a posteriori error estimator which drives a hp-adaptive mesh refinement scheme to determine accurate solutions of the stress field about about the crack. This facilitates the development of a novel method to calculate the crack tip configurational force that is accurate, requires no a priori knowledge of the stress field about the crack tip with, its error bound by an error estimator which is calculated a posteriori. Benchmark values of the crack tip con- figurational force are presented for problems containing multiple mixed mode cracks in both isotropic and anisotropic materials. Additionally, the hp-adaptivity is com- bined with a mathematical analysis of the stress field at the crack tip to critique the convergence and limitations of other methods in the literature to calculate the crack tip configurational force. Two methods for staggered quasi-static crack prop- agation are also presented. An rp-adaptive method which is simple to implement and computationally inexpensive, element edges aligned with the crack propagation path with the exploitation of the discontinuous Galerkin edge sti↵ness terms exist- ing along element interfaces to propagate a crack. The second method is denoted the hpr-adaptive method which combines the accurate computation of the crack tip configuration force with r-adaptivity to produce a computationally expensive but accurate method to propagate multiple cracks simultaneously. Further, for indeter- minate systems, an average boundary condition that restrains rigid body motion and rotation is introduced to make the system determinate

Coupled deformation, fluid flow and fracture propagation in porous media

Polygonal faults are non-tectonic fault systems which are layer-bound (at some vertical scale) and are widely developed in fine-grained sedimentary basins. Although several qualitative mechanisms have been hypothesised to explain the formation of these faults, there is a weak general consensus that they are formed by the coupled deformation and fluid expulsion of the host sediments (consolidation). This thesis presents a numerical framework that can be extended to investigate the role consolidation plays in the development and evolution of these faults. The method is also applicable to reservoir engineering and CO2 storage. An understanding of the coupled mechanical response and fluid flow is critical in determining compaction and subsidence in oil reservoirs and fault-seal integrity during CO2 disposal and storage. The technique uses a fracture mapping approach (FM) and the extended finite element method (XFEM) to modify the single phase FEM consolidation formulation. A key feature of FM-XFEM is its ability to include discontinuities into a model independently of the computational mesh. The fracture mapping approach is used to simulate the flow interaction between the matrix and existing fractures via a transfer function. Since fractures are represented using level set data, the need for complex meshing to describe fractures is not required. The XFEM component of the method simulates the influence of the pore fluid on the mechanical behaviour of the fractured medium. In XFEM, enrichment functions are added to the standard finite element approximation to ensure an accurate approximation of discontinuous fields within the simulation domain. FM-XFEM produces results comparative to the discrete fracture method on relatively coarse meshes. FM-XFEM has also been extended to model the propagation of existing fractures using a mixed-mode criterion applicable to geological media. Stress concentrations at the tips of existing fractures show good agreement with an analytical solution found in literature

Multiscale formulation for material failure accounting for cohesive cracks at the macro and micro scales

This contribution presents a two-scale formulation devised to simulate failure in materials with het- erogeneous micro-structure. The mechanical model accounts for the activation of cohesive cracks in the micro-scale domain. The evolution/propagation of cohesive micro-cracks can induce material instability at the macro-scale level. Then, a cohesive crack is activated in the macro-scale model which considers, in a homogenized sense, the constitutive response of the intricate failure mode taking place in the smaller length scale.The two-scale model is based on the concept of Representative Volume Element (RVE). It is designed following an axiomatic variational structure. Two hypotheses are introduced in order to build the foundations of the entire two-scale theory, namely: (i) a mechanism for transferring kinematical information from macro- to-micro scale along with the concept of “Kinematical Admissibility”, relating both primal descriptions, and (ii) a Multiscale Variational Principle of internal virtual power equivalence between the involved scales of analysis. The homogenization formulae for the generalized stresses, as well as the equilibrium equations at the micro-scale, are consequences of the variational statement of the problem.The present multiscale technique is a generalization of a previous model proposed by the authors and could be viewed as an application of a general framework recently proposed by the authors. The main novelty in this article lies on the fact that failure modes in the micro-structure now involve a set of multiple cohesive cracks, connected or disconnected, with arbitrary orientation, conforming a complex tortuous failure path. Tortuosity is a topic of decisive importance in the modelling of material degradation due to crack propagation. Following the present multiscale modelling approach, the tortuosity effect is introduced in order to satisfy the “Kinematical Admissibility” concept, when the macro-scale kinematics is transferred into the micro-scale domain. There- fore, it has a direct consequence in the homogenized mechanical response, in the sense that the proposed scale transition method (including the tortuosity effect) retrieves the correct post-critical response.Coupled (macro-micro) numerical examples are presented showing the potentialities of the model to sim- ulate complex and realistic fracture problems in heterogeneous materials. In order to validate the multiscale technique in a rigorous manner, comparisons with the so-called DNS (Direct Numerical Solution) approach are also presented

A statistical approach for fracture property realization and macroscopic failure analysis of brittle materials

Lacking the energy dissipative mechanics such as plastic deformation to rebalance localized stresses, similar to their ductile counterparts, brittle material fracture mechanics is associated with catastrophic failure of purely brittle and quasi-brittle materials at immeasurable and measurable deformation scales respectively. This failure, in the form macroscale sharp cracks, is highly dependent on the composition of the material microstructure. Further, the complexity of this relationship and the resulting crack patterns is exacerbated under highly dynamic loading conditions. A robust brittle material model must account for the multiscale inhomogeneity as well as the probabilistic distribution of the constituents which cause material heterogeneity and influence the complex mechanisms of dynamic fracture responses of the material. Continuum-based homogenization is carried out via finite element-based micromechanical analysis of a material neighbor which gives is geometrically described as a sampling windows (i.e., statistical volume elements). These volume elements are well-defined such that they are representative of the material while propagating material randomness from the inherent microscale defects. Homogenization yields spatially defined elastic and fracture related effective properties, utilized to statistically characterize the material in terms of these properties. This spatial characterization is made possible by performing homogenization at prescribed spatial locations which collectively comprise a non-uniform spatial grid which allows the mapping of each effective material properties to an associated spatial location. Through stochastic decomposition of the derived empirical covariance of the sampled effective material property, the Karhunen-Loeve method is used to generate realizations of a continuous and spatially-correlated random field approximation that preserve the statistics of the material from which it is derived. Aspects of modeling both isotropic and anisotropic brittle materials, from a statistical viewpoint, are investigated to determine how each influences the macroscale fracture response of these materials under highly dynamic conditions. The effects of modeling a material both explicitly by representations of discrete multiscale constituents and/or implicitly by continuum representation of material properties is studies to determine how each model influences the resulting material fracture response. For the implicit material representations, both a statistical white noise (i.e., Weibull-based spatially-uncorrelated) and colored noise (i.e., Karhunen-Loeve spatially-correlated model) random fields are employed herein