On the use of quarter-point tetrahedral finite elements in linear elastic fracture mechanics

Nejati, Morteza; Paluszny, Adriana; Zimmerman, Robert W.

doi:10.1016/j.engfracmech.2015.06.055

Cited by 60 publications

(29 citation statements)

References 60 publications

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“…Triangle and tetrahedral elements around the crack tip are taken to be quarter point elements (Banks‐Sills & Sherman, ), which capture the singularity ensuing at the fracture tip by shifting the midside node one quarter toward the fracture tip. These have been found to significantly improve the measured stress intensity factor values in unstructured tetrahedral meshes (Nejati et al, ). In addition, the created mesh is refined near the fracture tip to improve sampling of the displacement field.…”

Section: Methodsmentioning

confidence: 99%

Quantification of Fracture Interaction Using Stress Intensity Factor Variation Maps

2017

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Accurate and flexible models of fracture interaction are sought after in the fields of mechanics and geology. Stress intensity factors (SIFs) quantify the energy concentrated at the fracture tips and are perturbed from their isolated values when two fractures are close to one another. Using a three‐dimensional finite element fracture mechanics code to simulate static fractures in tension and compression, interaction effects are examined. SIF perturbations are characterized by introducing three interaction measures: the circumferential and maximum SIF perturbation provide the “magnitude” of the effect of interaction, and the amplification to shielding ratio quantifies the balance between increased and decreased SIFs along the tip. These measures are used to demonstrate the change in interaction with fracture separation and to find the separation at which interaction becomes negligible. Interaction maps are constructed by plotting the values of the interaction measures for a static fracture as a second fracture is moved around it. These maps are presented for several common fracture orientations in tension. They explore interaction by highlighting regions in which growth is more likely to occur and where fractures will grow into nonplanar geometries. Interaction maps can be applied to fracture networks with multiple discontinuities to analyze the effect of geometric variations on fracture interaction.

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Section: Methodsmentioning

confidence: 99%

Quantification of Fracture Interaction Using Stress Intensity Factor Variation Maps

2017

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“…Accounting for the fluid pressure applied on fracture faces in energy-based extraction methods like the J-integral, 55 the interaction integral, 56 the contour integral method, 57 or the cut-off function method 58,59 is error prone. In this work, we use a DCM 60,61 to extract SIFs at the fracture front. As shown below, this method does not need to explicitly account for the boundary conditions on fracture faces and its implementation is quite straightforward.…”

Section: Appendix A: Displacement Correlation Methods For Sif Extractionmentioning

confidence: 99%

“…For a fracture front vertex j, the displacement jump across the fracture faces, Δu (r a ) = u (r a , = )−u (r a , = − ), is computed at a distance r a behind the fracture front in the local fracture front coordinate system, as shown in Figure A1. The SIFs at fracture front vertex j are then approximated using the following relations 61…”

Section: Appendix A: Displacement Correlation Methods For Sif Extractionmentioning

confidence: 99%

Coupled hydromechanical‐fracture simulations of nonplanar three‐dimensional hydraulic fracture propagation

Gupta

Duarte

2017

Num Anal Meth Geomechanics

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SummaryThis paper presents an algorithm and a fully coupled hydromechanical-fracture formulation for the simulation of three-dimensional nonplanar hydraulic fracture propagation. The propagation algorithm automatically estimates the magnitude of time steps such that a regularized form of Irwin's criterion is satisfied along the predicted 3-D fracture front at every fracture propagation step.A generalized finite element method is used for the discretization of elasticity equations governing the deformation of the rock, and a finite element method is adopted for the solution of the fluid flow equation on the basis of Poiseuille's cubic law. Adaptive mesh refinement is used for discretization error control, leading to significantly fewer degrees of freedom than available nonadaptive methods. An efficient computational scheme to handle nonlinear time-dependent problems with adaptive mesh refinement is presented. Explicit fracture surface representations are used to avoid mapping of 3-D solutions between generalized finite element method meshes. Examples demonstrating the accuracy, robustness, and computational efficiency of the proposed formulation, regularized Irwin's criterion, and propagation algorithm are presented.

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“…A virtual integration technique avoids the need of a structured mesh by using an artificial 'virtual' mesh to integrate around the tips, using a 'virtual' cylinder (Cervenka and Saouma 1997;Paluszny and Zimmerman 2011) or a 'virtual' disk (Nejati et al 2015b). When aided by isoparametric quadratic elements (Daimon and Okada 2014) and with the quarter-point optimisation (Nejati et al 2015a), this technique allows swift and accurate computation of stress intensity factors on unstructured tetrahedral meshes. The integration yields high quality solutions when the integration domain radius approximates the mesh size around the fracture (Nejati et al 2015b).…”

Section: Stress Intensity Factors and Growth Criteriamentioning

confidence: 99%

“…Side-and corner quarter-point tetrahedral elements are placed along the crack front. These are a type of quadratic elements in which the centre node is shifted towards the tip; they are placed in order to better capture the stress singularity at the fracture tip (Nejati et al 2015a). Displacements are computed at the nodes, material properties are defined at Gaussian integration points, and stresses and strains are computed at the integration point locations.…”

Section: Discretisationmentioning

confidence: 99%

Modelling of primary fragmentation in block caving mines using a finite-element based fracture mechanics approach

Paluszny

Zimmerman

2017

Geomech. Geophys. Geo-energ. Geo-resour.

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The growth of fractures around an undercut of a block cave is simulated. A finite element based approach is used, in which fractures are represented as non-planar 3D surfaces that grow in response to boundary stresses and interaction. A new mesh is recreated at each step to compute the displacement field. Stress intensity factors are computed around fracture tips using a technique that computes the interaction integral over a virtual disk. Fracture geometry is updated using Paris and Schöllmann propagation laws, and a geometric fracture pattern ensues from the simulation. The growth of fractures is examined in the lower 20 m of a mine at 812 m depth. The growth of 30, 60 and 90 fractures is examined. Realistic extraction schedules for over 100 draw points control the rate of mass extraction. The effect of rock bridges as overburden stress shields is investigated. Bridges are modelled by constraining the vertical displacement of the top boundary. This case is compared to a Neumann-type overburden stress boundary condition in which the overburden is felt throughout the top of the cave. In both cases, fractures grow to form a dome shape above and around the cave during extraction. For the case of a fixed top boundary, fracture growth is observed away from the cave, while in the direct overburden stress case, fractures tend to grow close to the cave. Over-arching fractures concentric to the undercut continue to grow as the cave progresses.

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On the use of quarter-point tetrahedral finite elements in linear elastic fracture mechanics

Cited by 60 publications

References 60 publications

Quantification of Fracture Interaction Using Stress Intensity Factor Variation Maps

Quantification of Fracture Interaction Using Stress Intensity Factor Variation Maps

Coupled hydromechanical‐fracture simulations of nonplanar three‐dimensional hydraulic fracture propagation

Modelling of primary fragmentation in block caving mines using a finite-element based fracture mechanics approach

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