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A stabilized finite element approach to model contact conditions in fractured subsurface media
Abstract
We propose a stabilized approach based on Nitsche's method for enforcing contact constraints over crack surfaces. The proposed method addresses the shortcomings of conventional penalty and augmented Lagrange multiplier approaches by combining their attractive features. Similar to an augmented Lagrange multiplier approach, the proposed method has a consistent variational basis resulting in stronger enforcement of the non-interpenetration constraint. At the same time, the proposed method is purely displacement-based and alleviates the stability challenges common to mixed methods. The method also retains the computational efficiency of penalty approaches by eliminating the outer augmentation loop necessary for augmented Lagrangian approaches and resulting in smaller system matrices.
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Dynamic crack growth is analysed numerically for a plane strain block with an initial central crack subject to tensile loading. The continuum is characterized by a material constitutive law that relates stress and strain, and by a relation between the tractions and displacement jumps across a specified set of cohesive surfaces. The material constitutive relation is that of an isotropic hyperelastic solid. The cohesive surface constitutive relation allows for the creation of new free surface and dimensional considerations introduce a characteristic length into the formulation. Full transient analyses are carried out. Crack branching emerges as a natural outcome of the initial-boundary value problem solution, without any ad hoc assumption regarding branching criteria. Coarse mesh calculations are used to explore various qualitative features such as the effect of impact velocity on crack branching, and the effect of an inhomogeneity in strength, as in crack growth along or up to an interface. The effect of cohesive surface orientation on crack path is also explored, and for a range of orientations zigzag crack growth precedes crack branching. Finer mesh calculations are carried out where crack growth is confined to the initial crack plane. The crack accelerates and then grows at a constant speed that. for high impact velocities, can exceed the Rayleigh wave speed. This is due to the finite strength of the cohesive surfaces. A fine mesh calculation is also carried out where the path of crack growth is not constrained. The crack speed reaches about 45% of the Rayleigh wave speed, then the crack speed begins to oscillate and crack branching at an angle of about 29 from the initial crack plane occurs. The numerical results are at least qualitatively in accord with a wide variety of experimental observations on fast crack growth in brittle solids.
In this work, we propose a novel weighting for the interfacial consistency terms arising in a Nitsche variational form. We demonstrate through numerical analysis and extensive numerical evidence that the choice of the weighting parameter has a great bearing on the stability of the method. Consequently, we propose a weighting that results in an estimate for the stabilization parameter such that the method remains well behaved in varied settings; ranging from the configuration of embedded interfaces resulting in arbitrarily small elements to such cases where a large contrast in material properties exists. An important consequence of this weighting is that the bulk as well as the interfacial fields remain well behaved in the presence of (a) elements with arbitrarily small volume fractions, (b) large material heterogeneities and (c) both large heterogeneities as well as arbitrarily small elements. We then highlight the accuracy and efficiency of the proposed formulation through numerical examples, focusing particular attention on interfacial quantities of interest.
Modeling hydraulic fracturing in the presence of a natural fracture network is a challenging task, owing to the complex interactions between fluid, rock matrix, and rock interfaces, as well as the interactions between propagating fractures and existing natural interfaces. Understanding these complex interactions through numerical modeling is critical to the design of optimum stimulation strategies. In this paper, we present an explicitly integrated, fully coupled discrete‐finite element approach for the simulation of hydraulic fracturing in arbitrary fracture networks. The individual physical processes involved in hydraulic fracturing are identified and addressed as separate modules: a finite element approach for geomechanics in the rock matrix, a finite volume approach for resolving hydrodynamics, a geomechanical joint model for interfacial resolution, and an adaptive remeshing module. The model is verified against the Khristianovich–Geertsma–DeKlerk closed‐form solution for the propagation of a single hydraulic fracture and validated against laboratory testing results on the interaction between a propagating hydraulic fracture and an existing fracture. Preliminary results of simulating hydraulic fracturing in a natural fracture system consisting of multiple fractures are also presented. Copyright © 2012 John Wiley & Sons, Ltd.
Uniaxial compression of plates of brittle materials containing pre-existing planar cracks oriented at certain angles with respect to the direction of overall compression has revealed that the relative frictional sliding of the faces of the pre-existing cracks may produce, at their tips, tension cracks which deviate at sharp angles from the sliding plane. These tension cracks then continue to grow in a stable manner with increasing axial compression, curving toward an orientation parallel to the direction of axial compression. Within the framework of linear fracture mechanics, the out-of-plane extension of a pre-existing straight crack, induced by overall far-field compression, is analyzed, and various parameters which characterize the growth process are quantified. It is shown analytically that, for a wide range of pre-existing crack orientations, the out-of-plane crack extension initiates at an angle close to 70o from the direction of the pre-existing crack; the exact value of this angle, of course, depends on the friction factor and the orientation of the pre-existing crack. It is found that the growth process is stable initially, but the rate of increase of the length of the extended portion with respect to the increasing axial compression dramatically increases after a certain extension length is attained, and in fact, this length becomes unbounded if a small lateral tension also exists. -Authors
Ket qualitative features of solutions exhibiting strong discontinuities in rate-independent inelastic solids are identified and exploited in the design of a new class of finite element approximations. The analysis shows that the softening law must be re-interpreted in a distributional sense for the continuum solutions to make mathematical sense and provides a precise physical interpretation to the softening modulus. These results are verified by numerical simulations employing a regularized discontinuous finite element method which circumvent the strong mesh-dependence exhibited by conventional methods, without resorting to viscosity or introducing additional ad-hoc parameters. The analysis is extended to a new class of anisotropic rate-independent damage models for brittle materials.
A scalable algorithm for modeling dynamic fracture and fragmentation of solids in three dimensions is presented. The method is based on a combination of a discontinuous Galerkin (DG) formulation of the continuum problem and cohesive zone models (CZM) of fracture. Prior to fracture, the flux and stabilization terms arising from the DG formulation at interelement boundaries are enforced via interface elements, much like in the conventional intrinsic cohesive element approach, albeit in a way that guarantees consistency and stability. Upon the onset of fracture, the traction–separation law (TSL) governing the fracture process becomes operative without the need to insert a new cohesive element. Upon crack closure, the reinstatement of the DG terms guarantee the proper description of compressive waves across closed crack surfaces.The main advantage of the method is that it avoids the need to propagate topological changes in the mesh as cracks and fragments develop, which enables the indistinctive treatment of crack propagation across processor boundaries and, thus, the scalability in parallel computations. Another advantage of the method is that it preserves consistency and stability in the uncracked interfaces, thus avoiding issues with wave propagation typical of intrinsic cohesive element approaches.A simple problem of wave propagation in a bar leading to spall at its center is used to show that the method does not affect wave characteristics and, as a consequence, properly captures the spall process. We also demonstrate the ability of the method to capture intricate patterns of radial and conical cracks arising in the impact of ceramic plates, which propagate in the mesh impassive to the presence of processor boundaries.
An improvement of a new technique for modelling cracks in the finite element framework is presented. A standard displacement-based approximation is enriched near a crack by incorporating both discontinuous fields and the near tip asymptotic fields through a partition of unity method. A methodology that constructs the enriched approximation from the interaction of the crack geometry with the mesh is developed. This technique allows the entire crack to be represented independently of the mesh, and so remeshing is not necessary to model crack growth. Numerical experiments are provided to demonstrate the utility and robustness of the proposed technique.
This is the second edition of the valuable reference source for numerical simulations of contact mechanics suitable for many fields like civil engineering, car design, aeronautics, metal forming, or biomechanics. Boundary value problems involving contact are of great importance in industrial applications in engineering such as bearings, metal forming processes, rubber seals, drilling problems, crash analysis of cars, rolling contact between car tires and the road, cooling of electronic devices... Other applications are related to biomechanical engineering design where human joints, implants or teeth are of consideration. Due to this variety, contact problems are today combined either with large elastic or inelastic deformations including time dependent responses. Thermal coupling also might have to be considered. Even stability behaviour has to be linked to contact. The topic of computational contact is described in depth providing an up-to-date treatment of different formulations, algorithms and discretisation techniques for contact problems which are established in the geometrically linear and nonlinear range. This book provides the necessary continuum mechanics background which includes the derivation of the contact constraints. Constitutive equations stemming from tribology which are valid at the contact interface are discussed in detail. Discretization schemes for small and finite deformations are discussed in depth. Solid and beam contact is considered as well as contact of unstable systems and thermomechanical contact. The algorithmic side covers a broad range of solution methods. Additionally adaptive discretisation techniques for contact analysis are presented as a modern tool for engineering design simulations. The present text book is written for graduate, Masters and PhD students, but also for engineers in industry, who have to simulate contact problems in practical application and wish to understand the theoretical and algorithmic background of contact treatment in modern finite element systems. For this second edition, illustrative simplified examples and new discretisation schemes as well as adaptive procedures for coupled problems are added. © Springer-Verlag Berlin Heidelberg 2006. All rights are reserved.
We investigate a finite element method for frictional sliding along embedded interfaces within a weighted Nitsche framework. For such problems, the proposed Nitsche stabilized approach combines the attractive features of two traditionally used approaches: viz. penalty methods and augmented Lagrange multiplier methods. In contrast to an augmented Lagrange multiplier method, the proposed approach is primal; this allows us to eliminate an outer augmentation loop as well as additional degrees of freedom. At the same time, in contrast to the penalty method, the proposed method is variationally consistent; this results in a stronger enforcement of the non-interpenetrability constraint. The method parameter arising in the proposed stabilized formulation is defined analytically, for lower order elements, through numerical analysis. This provides the proposed approach with greater robustness over both traditional penalty and augmented Lagrangian frameworks. Through this analytical estimate, we also demonstrate that the proposed choice of weights, in the weighted Nitsche framework, is indeed the optimal one. We validate the proposed approach through several benchmark numerical experiments.
We extend the weighted Nitsche’s method proposed in the first part of this study to include multiple intersecting embedded interfaces. These intersections arise either inside a computational domain – where two internal interfaces intersect; or on the boundary of the computational domain – where an internal interface intersects with the external boundary. We propose a variational treatment of both the interfacial kinematics and the external Dirichlet constraints within Nitsche’s framework. We modify the numerical analysis to account for these intersections and provide an explicit expression for the weights and the method parameters that arise in the Nitsche variational form in the presence of junctions. Finally, we demonstrate the performance of the method for both perfectly-tied interfaces and perfectly-plastic sliding interfaces through several benchmark examples.
Failure analyses of rock masses are frequently encountered in major civil engineering construction and mining, and in particular for projects related to slope excavation. The proposed procedure determines the seismic factor of safety against sliding along a joint plane.
In this paper, the shear behaviour of rock joints are numerically simulated using the discrete element code PFC2D. In PFC, the intact rock is represented by an assembly of separate particles bonded together where the damage process is represented by the breakage of these bonds. Traditionally, joints have been modelled in PFC by removing the bonds between particles. This approach however is not able to reproduce the sliding behaviour of joints and also results in an unrealistic increase of shear strength and dilation angle due to the inherent micro-scale roughness of the joint surface. Modelling of joints in PFC was improved by the emergence of the smooth joint model. In this model, slip surfaces are applied to contacts between particles lying on the opposite sides of a joint plane. Results from the current study show that this method suffers from particle interlocking which takes place at shear displacements greater than the minimum diameter of the particles. To overcome this problem, a new shear box genesis approach is proposed. The ability of the new method in reproducing the shear behaviour of rock joints is investigated by undertaking direct shear tests on saw-tooth triangular joints with base angles of 15°, 25° and 35° and the standard joint roughness coefficient profiles. A good agreement is found between the results of the numerical models and the Patton, Ladanyi and Archambault and Barton and Choubey models. The proposed model also has the ability to track the damage evolution during the shearing process in the form of tensile and shear fracturing of rock asperities.
This monograph aims to give a modern, comprehensive presentation of numerical methods for contact and impact problems in nonlinear solid and structural mechanics. The applicability of the algorithms discussed herein is broad, ranging from quasistatic frictionless and frictional contact applications, to fully transient problems involving inelastic impact, to problems in which a certain degree of frictional model sophistication (as manifested for example by thermal softening and/or rate and state dependence) is necessary to predict global structural response. It is the sincere hope of the author that this material will be of interest to graduate students, faculty researchers, engineering professionals, and scientists in a number of fields whose work requires quantification of contact phenomena as a part of their model development. Since the field of computational contact mechanics is rapidly evolving and remains immature from a research perspective, the focus of this presentation is on the underlying theory behind contact formulations, and on the details of the implementation of this theory in a finite element setting.
A new technique for the finite element modeling of crack growth with frictional contact on the crack faces is presented. The eXtended Finite Element Method (X-FEM) is used to discretize the equations, allowing for the modeling of cracks whose geometry are independent of the finite element mesh. This method greatly facilitates the simulation of a growing crack, as no remeshing of the domain is required. The conditions which describe frictional contact are formulated as a non-smooth constitutive law on the interface formed by the crack faces, and the iterative scheme implemented in the LATIN method [Nonlinear Computational Structural Mechanics, Springer, New York, 1998] is applied to resolve the nonlinear boundary value problem. The essential features of the iterative strategy and the X-FEM are reviewed, and the modifications necessary to integrate the constitutive law on the interface are presented. Several benchmark problems are solved to illustrate the robustness of the method and to examine convergence. The method is then applied to simulate crack growth when there is frictional contact on the crack faces, and the results are compared to both analytical and experimental results.
The breakage and shear behaviour of intermittent rock joints have been investigated in a series of direct shear tests with a new shear device, specifically designed for this purpose. The tests have been performed on specimens of rock-like material or hard rock, respectively, incorporating idealized non-persistent joints, made up of a number of short cracks in an en-échelon arrangement along the central shear axis.The shear behaviour of such a joint constellation has been found to be composed of different phases. The first phase of shearing is that of the actual rupture, initiated by the formation of wing cracks, starting from the existing cracks and growing into the material bridges, and concluded by the generation of additional new fractures connecting the initial cracks in the zone between the wing cracks. The second phase of shearing is characterized by friction processes and volume increase in the then continuous shear zone. Finally, the third phase of shearing, reached after large shear displacements, is determined by sliding processes inside the strongly fractured shear zone.In a large number of shear tests the geometrical parameters of the discontinuous joints as well as the loading conditions have been found to influence the activated shear resistance in each phase of shearing to a noticeably different extent. The orientation of the initial cracks and the normal stress, however, have been identified as the most influential parameters. Depending on the test conditions, an initially discontinuous rock joint can activate the largest shear resistance not just before rupture but in one of the two subsequent phases of shearing as well.The mechanisms which govern the different shear phases could be identified as (1) tensile rupturing, (2) rolling and sliding friction of dilatant joint zones and (3) sliding within the joint filling composed of brecciated material.
To study the damage process of microscale to macroscale in coarse-grained granite specimen under uniaxial compressive stress, we have observed micro-damage localization and propagation by using a newly developed experimental system that allows us to observe the damaging process continuously.The results showed that pre-existing microcracks lead to macroscopic shear fracture through the damage development process. The mechanism of micro-damage initiation in a granite specimen under uniaxial compressive stress may be considered for two cases. One is that two grains such as quartz and feldspar contact each other in the same direction as the axial stress, and the other is that a biotite grain inclined to the axial stress direction is surrounded by feldspar grains. The homogenization theory was applied to verify numerically the micromechanics of stress-induced damage in the mineral contacts. Local stress distribution in the periodic-micro structure was also calculated by the homogenization theory. It is shown that this analysis, which takes into account the initial state of the specimen, is well adapted to the behavior of two grains for which microcracking is the fundamental mechanism of damage.
A Lagrangian finite element method of fracture and fragmentation in brittle materials is developed. A cohesive-law fracture model is used to propagate multiple cracks along arbitrary paths. In axisymmetric calculations, radial cracking is accounted for through a continuum damage model. An explicit contact/friction algorithm is used to treat the multi-body dynamics which inevitably ensues after fragmentation. Rate-dependent plasticity, heat conduction and thermal coupling are also accounted for in calculations. The properties and predictive ability of the model are exhibited in two case studies: spall tests and dynamic crack propagation in a double cantilever beam specimen. As an example of application of the theory, we simulate the experiments of Field (1988) involving the impact of alumina plates by steel pellets at different velocities. The calculated conical, lateral and radial fracture histories are found to be in good agreement with experiment.
Contact problem suffers from a numerical instability similar to that encountered in incompressible elasticity, in which the normal contact pressure exhibits spurious oscillation. This oscillation does not go away with mesh refinement, and in some cases it even gets worse as the mesh is refined. Using a Lagrange multipliers formulation we trace this problem to non-satisfaction of the LBB condition associated with equal-order interpolation of slip and normal component of traction. In this paper, we employ a stabilized finite element formulation based on the polynomial pressure projection (PPP) technique, which was used successfully for Stokes equation and for coupled solid-deformation–fluid-diffusion using low-order mixed finite elements. For the frictional contact problem the polynomial pressure projection approach is applied to the normal contact pressure in the framework of the extended finite element method. We use low-order linear triangular elements (tetrahedral elements for 3D) for both slip and normal pressure degrees of freedom, and show the efficacy of the stabilized formulation on a variety of plane strain, plane stress, and three-dimensional problems.
In this paper we present a finite element algorithm for the static solution of two-dimensional frictionless contact problems involving bodies undergoing arbitrarily large motions and deformations. A mixed penalty formulation is employed in approximating the resulting variational inequalities. The algorithm is applied to quadratic elements along with a rational scheme for determining the contacting regions. Several numerical simulations illustrate the applicability and accuracy of the proposed solution procedure.
The purpose of this paper is to establish a methodology to determine the equivalent elastic properties of fractured rock masses by explicit representations of stochastic fracture systems, and to investigate the conditions for the application of the equivalent continuum approach for representing mechanical behavior of the fractured rock masses. A series of numerical simulations of mechanical deformation of fractured rock masses at different scales were conducted with a large number of realizations of discrete fracture networks (DFN), based on realistic fracture system information and using the two-dimensional distinct element program, UDEC. General theory of anisotropic elasticity was used for describing the macroscopic mechanical behavior of fractured rock masses as equivalent elastic continua. Verification of the methodology for determining the elastic compliance tensor was conducted against closed-form solutions for regularly fractured rock mass, leading to very good agreements. The main advantage of the developed methodology using the distinct element method is that it can consider complex fracture system geometry and various constitutive relations of fractures and rock matrix, and their interactions. Two criteria for the applicability of equivalent continuum approach were adopted from the investigations: (i) the existence of a properly defined REV (representative elementary volume) and (ii) the existence of an elastic compliance tensor. For the problems with in situ conditions studied in this paper, the results show that a REV can be defined and the elastic properties of the fractured rock mass can be represented approximately by the elastic compliance tensor through numerical simulations.