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We present an open-source software framework called PERMIX for multiscale modeling and simulation of fracture in solids. The framework is an object oriented open-source effort written primarily in Fortran 2003 standard with Fortran/C++ interfaces to a number of other libraries such as LAMMPS, ABAQUS, LS-DYNA and GMSH. Fracture on the continuum level is modeled by the extended finite element method (XFEM). Using several novel or state of the art methods, the piece software handles semi-concurrent multiscale methods as well as concurrent multiscale methods for fracture, coupling two continuum domains or atomistic domains to continuum domains, respectively. The efficiency of our open-source software is shown through several simulations including a 3D crack modeling in clay nanocomposites, a semi-concurrent FE-FE coupling, a 3D Arlequin multiscale example and an MD-XFEM coupling for dynamic crack propagation.

This paper presents an adaptive strategy for phase-field simulations with transition to fracture. The phase-field equations are solved only in small subdomains around crack tips to determine propagation, while an extended finite element method (XFEM) discretization is used in the rest of the domain to represent sharp cracks, enabling to use a coarser discretization and therefore reducing the computational cost. Crack-tip subdomains move as cracks propagate in a fully automatic process. The same mesh is used during all the simulation, with an h -refined approximation in the elements in the crack-tip subdomains. Continuity of the displacement between the refined subdomains and the XFEM region is imposed in weak form via Nitsche’s method. The robustness of the strategy is shown for some numerical examples in 2D and 3D, including branching and coalescence tests.

This paper shows an inf-sup stability property for several well-known 2D and 3D Stokes elements on triangulations which are not fitted to a given smooth or polygonal domain. The property implies stability and optimal error estimates for a class of unfitted finite element methods for the Stokes and Stokes interface problems, such as Nitsche-XFEM or cutFEM. The error analysis is presented for the Stokes problem. All assumptions made in the paper are satisfied once the background mesh is shape-regular and fine enough.

The cohesive zone model (CZM) honors the softening effects and plastic zone at the fracture tip in a quasi-brittle rock, e.g., shale, which results in a more precise fracture geometry and pumping pressure compared to those from linear elastic fracture mechanics. Nevertheless, this model, namely the planar CZM, assumes a predefined surface on which the fractures propagate and therefore restricts the fracture propagation direction. Notably, this direction depends on the stress interactions between closely spaced fractures and can be acquired by integrating CZM as the segmental contact interaction model with a fully coupled pore pressure–displacement model based on extended finite element method (XFEM). This integrated model, called XFEM-based CZM, simulates the fracture initiation and propagation along an arbitrary, solution-dependent path. In this work, we modeled a single stage of 3D hydraulic fracturing initiating from three perforation clusters in a single-layer, quasi-brittle shale formation using planar CZM and XFEM-based CZM including slit flow and poroelasticity for fracture and matrix spaces, respectively, in Abaqus. We restricted the XFEM enrichment zones to the stimulation regions as enriching the whole domain leads to extremely high computational expenses and unrealistic fracture growths around sharp edges. Moreover, we validated our numerical technique by comparing the solution for a single fracture with KGD solution and demonstrated several precautionary measures in using XFEM in Abaqus for faster solution convergence, for instance the initial fracture length and mesh refinement. We demonstrated the significance of the injection rate and stress contrast in fracture aperture, injection pressure, and the propagation direction. Moreover, we showed the effect of the stress distribution on fracture propagation direction comparing the triple-cluster fracturing results from planar CZM with those from XFEM-based CZM. We found that the stress shadowing effect of hydraulic fractures on each other can cause these fractures to coalesce, grow parallel, or diverge depending on cluster spacing. We investigated the effect of this arbitrary propagation direction on not only the fractures’ length, aperture, and the required injection pressure, but also the fractures’ connection to the wellbore. This connection can be disrupted due to the near-wellbore fracture closure which may embed proppant grains on the fracture wall or screen out the fracture at early times. Our results verified that the near-wellbore fracture closure strongly depends on the following: (1) the implemented model, planar or XFEM-based CZM; and (2) fracture cluster spacing. Ultimately, we proposed the best fracturing scenario and cluster spacing to maintain the fractures connected to the wellbore.

We propose a formulation for tracking general crack paths in elastodamaging materials without mesh adaptivity and broadening of the damage band. The idea is to treat in a unified way both the damaging process and the development of displacement discontinuities by means of the regularized finite element method. With respect to previous authors’ contributions, a novel damage evolution law and an original crack tracking framework are proposed. We face the issue of mesh objectivity through several two-dimensional tests, obtaining smooth crack paths and reliable structural results.

The increased risk for fracture with age is associated not only with reduced bone mass but also with impaired bone quality. At the microscale, bone quality is related to porosity, microstructural organization, accumulated microdamage and intrinsic material properties. However, the link between these characteristics and fracture behavior is still missing. Bone tissue has a complex structure and as age-related compositional and structural changes occur at all hierarchical length scales it is difficult to experimentally identify and discriminate the effect of each mechanism. The aim of this study was therefore to use computational models to analyze how microscale characteristics in terms of porosity, intrinsic toughness properties and microstructural organization affect the mechanical behavior of cortical bone. Tensile tests were simulated using realistic microstructural geometries based on microscopy images of human cortical bone. Crack propagation was modelled using the extended finite element method where cement lines surrounding osteons were modelled with an interface damage law to capture crack deflections along osteon boundaries. Both increased porosity and impaired material integrity resulted in straighter crack paths with cracks penetrating osteons, similar to what is seen experimentally for old cortical bone. However, only the latter predicted a more brittle failure behavior. Furthermore, the local porosity influenced the crack path more than the macroscopic porosity. In conclusion, age-related changes in cortical bone affect the crack path and the mechanical response. However, increased porosity alone was not driving damage in old bone, but instead impaired tissue integrity was required to capture brittle failure in aging bone.

This paper presents the practical application of fracture mechanics in investigating the possibility of crack propagation in a brake calliper bracket mounted in a vehicle bogie. The extended finite element method available in the Abaqus software was used. This method allows the modelling of material damage and its propagation independently of the finite element mesh. Damage can arise in any area of finite elements without changing the mesh. Numerical simulation of crack propagation was performed in order to analyse how crack changes as a result of the location change of damage initiation.

This paper presents the practical application of fracture mechanics in investigating the possibility of crack propagation in a brake calliper bracket mounted in a vehicle bogie. The extended finite element method available in the Abaqus software was used. This method allows the modelling of material damage and its propagation independently of the finite element mesh. Damage can arise in any area of finite elements without changing the mesh. Numerical simulation of crack propagation was performed in order to analyse how crack changes as a result of the location change of damage initiation.

A sharp-interface Eulerian formulation for modeling the propagation of localized pitting corrosion is presented. This formulation allows for an accurate representation of the corrosion front independent of the underlying finite element mesh and handles complex morphological transitions such as pit merging without requiring remeshing or mesh-moving procedures. First, the governing equations of the moving interface problem associated with pit growth are derived for the two-phase (metal-solution) system. Next, the implementation of the combined extended finite element and level set method and the procedure to enforce interface conditions are discussed. Finally, the method is validated by conducting several benchmark numerical studies, and comparing the results with published experimental data and existing numerical studies. Simulation studies indicate that: during diffusion-controlled corrosion an isolated pit grows as a semi-circular shape, whereas closely spaced pits merge and grow into an elongated elliptical shape; and only during activation-controlled corrosion the initial pit morphology is preserved.

Microcracking in trabecular bone is responsible both for the mechanical degradation and remodeling of the trabecular bone tissue. Recent results on trabecular bone mechanics have demonstrated that bone tissue microarchitecture, tissue elastic heterogeneity and tissue-level mechanical anisotropy all should be considered to obtain detailed information on the mechanical stress state. The present study investigated the influence of tissue microarchitecture, tissue heterogeneity in elasticity and material separation properties and tissue-level anisotropy on the microcrack formation process. Microscale bone models were executed with the extended finite element method. It was demonstrated that anisotropy and heterogeneity of the bone tissue contribute significantly to bone tissue toughness and the resistance of trabecular bone to microcrack formation. The compressive strain to microcrack initiation was computed to increase by a factor of four from an assumed homogeneous isotropic tissue to an assumed anisotropic heterogenous tissue.