European Congress on Computational Methods in Applied Sciences and Engineering ECCOMAS 2004 P. Neittaanmäki, T. Rossi, S. Korotov, E. Oñate, J. Périaux, and D. Knörzer (eds.) Jyväskylä, 24—28 July 2004
COMPUTATIONAL FRACTURE MECHANICS: A SURVEY OF THE FIELD Anthony R. Ingraffea, Paul A. Wawrzynek School of Civil and Environmental Engineering Hollister Hall, Cornell University, Ithaca, NY 14853, USA e-mail:
[email protected], web page: www.cfg.cornell.edu
Key words: computational mechanics, cracks, simulation, fracture, fatigue, survey. Abstract. This paper surveys examples of historical and state-of-the-art approaches of computational fracture mechanics. A global taxonomy of these approaches is first defined. The two main branches of this taxonomy are approaches based on geometrical representation and numerical representation of cracks. Approaches are briefly described and compared for their use in both the traditional role of computational fracture mechanics—calculation of crack front fields—and in the emerging role of prediction of material damage evolution and toughness. This paper is a much condensed version of a more thorough review found in Ingraffea [6].
1
Anthony R. Ingraffea, Paul A. Wawrzynek
1 INTRODUCTION The overall objective of fracture mechanics is the determination of the rate of change of the shape of an existing crack. Will it propagate under given loading and environmental conditions, and, if it does propagate, at what rate and into what configuration? The corresponding computational requirement has been to obtain the fields--displacement, strain, stress, and energy—from which the driving force for crack propagation might be extracted. One can think of this classical role of computational fracture mechanics (CFM) as quantifying the driving force provided by crack front fields for crack growth, while experimental fracture mechanics traditionally quantified the resisting force provided by the material containing the crack. However, the role of CFM has been expanding, especially in the last decade. Not only does it continue to encompass its classical responsibility to compute driving forces, but it is now also frequently employed to predict a material’s resistance to crack growth, and even the process of crack nucleation itself. Like so many other fields such as physics, biology and chemistry, fracture mechanics is riding the waves of ever-increasing computational power and ability to observe at smaller length and time scales. At the time of the publication of a previous overview on computational fracture mechanics [1], routine finite element analyses (FEA) of crack problems involving tens-of-thousands of degrees-of-freedom (DOF) required mainframe computing and minutes to complete. Today, similar analyses occur in seconds on laptops of traveling fracture mechanicians. It is now increasingly commonplace to apply supercomputers or large cluster computers to fracture problems involving millions of DOF. Simulation, the process of reproducing birth-to-death mimicry of complex physical processes, is becoming as commonplace as analysis, the traditional process of determining only a description of a current state. The purpose of this paper is to survey examples of historical and state-of-the-art techniques of computational fracture mechanics for both of its roles. It is intended for the student of computational fracture mechanics, trying to make sense of its vast literature and myriad methods. It is hoped that the paper will provide a basis for deciding which of these methods would be better suited to providing support for improved physics or mechanics, or at least improved efficiency, in solving the problem, whether research or practice oriented, the student is addressing. The paper will not cover methods for extracting driving forces for crack growth. Techniques for computing such classical driving forces as stress intensity factors, crack tip opening displacements and angles, energy release rates, and elastic and elasto-plastic crack front integrals are well described in such recent treatises as [2] and [3] for linear finite and boundary element methods (BEM), respectively, and [4] and [5] for nonlinear methods. Rather, this chapter will be organized around that aspect of CFM that fundamentally differentiates current approaches: their method of representation of cracking in a numerical model. This perspective will then become a roadmap to current implementations of these approaches.
2
Anthony R. Ingraffea, Paul A. Wawrzynek
In principle, any of the above-mentioned driving force extraction techniques can be used with any representational approach. Some of these approaches are currently limited to 2space, or to linear material behavior. The most significant of the representational approaches will be described here, and some current examples of application, pro/con comparisons of them as they are applied to both roles of fracture mechanics, and representative citations are presented. Of course, breadth and depth of coverage will be space-limited, and, admittedly, slanted towards the author’s point of view. This paper is a much condensed version of [6]. 2 TAXONOMY OF APPROACHES FOR REPRESENTATION OF CRACKING PROCESSES Figure 1 shows a taxonomy of approaches for representation of cracking processes. In this context, representation simply means the manner in which the presence of a crack, or a nucleation process, is accommodated in the numerical procedure. Currently available approaches fall naturally into two representational formats. In the first approach, the crack is a geometrical entity; the geometry model and the discretization model, if needed, are updated with crack growth. This is the left main branch of the taxonomy in Figure 1. In the other approach, the underlying geometry model does not contain the crack, and neither it nor the discretization model, if needed, changes during crack growth. Rather, the crack is represented either in the material constitutive model, or in a kinematic model as an intense localization of strain. This is the right main branch of the taxonomy in Figure 1. These fundamentally different representational schemes result in markedly different software, database, and data structure designs. 3 GEOMETRICAL REPRESENTATION APPROACHES 3.1 Constrained Shape Methods With the geometrical approach, crack growth can be constrained or arbitrary. If growth is restricted by the discretization method, or restricted to certain analytical shapes, such as flat and/or elliptical, growth is said to be constrained. Among the constrained shape methods are prescribed, analytical geometry, and known solution techniques. In the prescribed method, the crack surfaces are constrained to be coincident with existing faces of elements used in the finite element method (FEM). The element mesh is not changed during crack growth, but nodes common to two or more elements are decoupled as a crack passes through them. This is probably the first approach used to model cracking in the FEM [7, 8, 9]. Clearly, with this prescribed approach the mechanics of crack trajectory are constrained by the discretization. Crack spacing is controlled by element size, and trajectory by mesh topology. Moreover, in the era that this method was first used, there was no automatic meshing capability, and decoupling could increase bandwidth of the FEM coefficient matrix at a time when computer power was relatively low. This drawback set the
3
Anthony R. Ingraffea, Paul A. Wawrzynek
stage for the so-called “smeared crack” taxonomy branch to be described in Section 4.1. Despite these constraints on the mechanics of crack growth, modern computational power has allowed this approach, also known as node decoupling or splitting in the literature, to continue to be used today in a variety of forms. Today one can use many more and smaller elements, and automatically connect them in a manner that decreases trajectory constraint [10]. There are now automatic nodal bandwidth minimizers and iterative solution strategies that minimize the computational burden of adding nodes for the decoupling process. One modern variant of nodal decoupling is one in which interface elements are inserted [11, 12, 13, 14, 15, 16, 17] to open gradually previously coincident element faces through a non-linear cohesive cracking model. The crack is then constrained to follow a finite number of possible shapes determined by best-fitting the predicted shape to the existing mesh. The prescribed approach, also known as node splitting or node decoupling in the literature, has endured because of its simplicity. Standard FEM or BEM methods can be used. Initial structural configuration and boundary conditions are limited only by available geometry and mesh modeling capability and by the FEM/BEM software. As will be shown in Section 3.2, below, the intrinsic shortcomings of the prescribed method associated with fixed element topology can be overcome, at the expense of local remeshing and, perhaps, field mapping. If a crack is constrained to propagate in a mathematically simple shape, analytical geometry and known-solution approaches are available. An example of the former is the finite element alternating method (FEAM) [18, 19, 20]. In this method, the structure containing the crack(s) can be arbitrary, and the finite element method is used to calculate all field quantities in the uncracked structure. In addition to a finite element solution for the field quantities, a weight function solution for stress intensity factors or other driving forces is required. Analytical weight functions have been developed for a number of analytical crack geometries, e.g. flat ellipse. The influences of the weight functions on the finite element solution and vice versa are then computed to satisfactory error tolerance using an iterative, self-consistency technique. Like the prescribed method, the alternating method does not require any remeshing to accommodate or propagate a crack, and it can be used with any standard finite (or boundary) element code without modification. However, additional programming is needed to extract the needed fields from the finite element solution in the region containing a crack, to perform the weight function solution using these fields, and to iterate between the finite element and weight function solutions to satisfactory error tolerance. If the structure containing the crack is sufficiently simple, weight functions might exist that would obviate the need for field solution using the finite or boundary element methods. In this case, known solution methods can be employed. The simplest and probably most often used constrained geometry approach is based on known analytical and/or numerical solutions. The key characteristics of this approach are:
4
Anthony R. Ingraffea, Paul A. Wawrzynek
• It is available for only a small number of relatively simple combinations of structural geometries and boundary conditions; • Crack shapes are restricted to straight lines in 2D, and flat, analytical surfaces in 3D; and • Crack growth in most cases is restricted to LEFM assumptions. Under these restrictions, crack driving forces can be stored in look-up tables, or the weight function approach can be used to quickly generate them. They can then be processed quickly to predict stability and velocity in such codes as NASGRO [21], AFGROW [22] and FASTRAN [23]. The known solution method is computationally simple, and can certainly often be used for preliminary, approximate analysis of combinations of geometry and boundary conditions for which previous solutions do not exist. At the other extreme of computational intensity are the arbitrary geometrical approaches described next. 3.2 Arbitrary Shape Approaches If simulated growth is arbitrary, the crack takes the shape predicted by the physics and mechanics of growth, constrained mainly by the assumptions therein. Either there are no restrictions on shape imposed by discretization, or the discretization occurs at a much lower length scale than the structure in question. The latter case is actually an extreme version of prescribed representation described in Section 3.1, above. The discretization does not change during propagation, but elements are so small relative to crack length that trajectory appears to be unrestricted. There are currently three approaches in this category of simulations: those that do not require spatial discretization with continuous elements of volume, such as meshfree methods; those that require discretization but modify the mesh to conform to evolutionary geometry of the crack, such as adaptive FEM and BEM methods; and those that discretize with distinct rather than continuous elements of volume, such as lattice, particle, and atomistic methods. The latter approaches achieve arbitrariness of geometry in an approximate sense by using elements whose effective length scale is orders-of-magnitude smaller than that of the continuum they represent. One approach to solving the problem of the discretization interfering with desired crack growth is to remove meshing from the simulation process. The 1990’s saw an explosion of developments in so-called meshless, meshfree or element-free approaches and their application to, among many fields, computational fracture mechanics. The relationships among the meshless methods are described in [24], and their various applications in fracture mechanics in [25]. Unlike the representational methods discussed so far, meshfree methods rely on a field solver different from standard FEM. For example, the element-free Galerkin method (EFGM) [26] is a meshfree method in which the approximating function is moving least squares type
5
Anthony R. Ingraffea, Paul A. Wawrzynek
(MLS). In MLS, the approximation is a linear combination of basis functions and is fit to data by a weighted quadratic function. The MLS approximation is constructed entirely in terms of a set of interior nodes and a description of the boundaries of the model. The value of the desired functions, say displacements, at any point is obtained by solving a set of linear equations; the size of the system is determined by the number of nodes which influence the approximation at the point. MLS nodes do not have the usual FEM connectivity. In fact, nodal arrangement can be arbitrary, although arrangement does influence local accuracy. For example, the arrangement of nodes in the vicinity of a crack tip is known to have a strong influence on accuracy of stress intensity factor computation. The method is well-suited to the modeling of cracks in 2D [27] and 3D [28] because arbitrary surfaces across which the displacement function is discontinuous can readily be incorporated in the model. The presence of the crack, a surface of discontinuity, only affects how nodes influence the displacement at a given point. A visibility criterion can be used to select these points: any node which is not visible from a point is omitted from the evaluation of the displacement at that point. A drawback of the EFG is that straightforward implementation of the visibility criterion results in some interior discontinuities around the crack tip. It has been proven that the resulting approximation is still convergent, but smoothing by diffraction or transparency methods is preferable [24]. Another drawback of the EFG method is difficulty in enforcing essential boundary conditions. A workaround for this difficulty is to couple EFG with standard FEM, and use the latter in regions needing essential boundary conditions [29]. Adaptive FEM/BEM approaches permit completely arbitrary geometry of both structure and cracks by updating the mesh to conform to the evolving crack geometry. These approaches adhere strictly to the notion that they must contain an accurate geometry database, as required for practical industrial-strength problems. They rely on current state-of-the-art FE and BE formulations for the field solution, and on the advanced technology for automatic surface and volume mesh generation and for mapping state information from old to new mesh segments. A key component in the development of adaptive FEM/BEM approaches was the introduction of topological databases [30, 31] to organize and to control the interplay among the solid geometry and mesh models of the cracked structure, and the BEM or FEM solution technique. Simulation of crack growth is more complicated than many other applications of computational mechanics because the geometry (always) and topology (sometimes) of the structure evolve during the simulation. For this reason, a geometric description of the body that is independent of any numerical discretization should be maintained and updated as part of the simulation process. In adaptive FEM/BEM, the geometry database is a vital part of the representational database and contains an explicit description of the solid model, including the cracks.
6
Anthony R. Ingraffea, Paul A. Wawrzynek
Several topological data structures have been proposed for manifold objects. A B-rep modeler is capable of representing both solid and thin shell structures. However, modeling thin shells imposes some constraints on the choice of the topological data structure. Specifically, non-manifold topologies are often created; that is, the topologies cannot exist on a two-manifold representation. Other features that introduce non-manifold conditions include internal surfaces, such as bi-material interfaces, and some crack configurations. A nonmanifold data structure, such as the radial-edge [32], is needed to model these types of structures. This data structure is able to represent model topologies consisting of branching or intersecting cracks in addition to bi-material interfaces and shell structures – a necessary requirement for a general purpose fracture simulator. Simpler, edge-based data structures are also available for operations in two-dimensions [33]. Example uses of adaptive FEM/BEM approaches are: • • • •
2D FEM- and BEM-based simulations for static and dynamic LEFM and non-LEFM problems [33, 34, 35, 36, 37]; 3D BEM-based simulations of quasi-static and fatigue crack growth in geometrically complex solids [3, 32, 38, 39, 40, 41]; 3D FEM-based simulations of crack growth in shells under non-LEFM and large displacement conditions [42, 43, 44]; 3D FEM-based simulations of arbitrary crack growth in geometrically complex solids [45, 46].
There is a world-wide community of researchers in automatic mesh generation, and there is continuous progress in reliability and efficiency of meshers to support adaptive remeshing approaches [47, 48, 49, 50]. In choosing between so-called meshfree and adaptive remeshing approaches, one has to decide where to invest software complexity, in the numerical solution procedure itself, or in the meshing process. Boundary element methods are space-encapsulating; finite element methods are spacefilling. Discrete element methods (DEM) are generally neither. DEM’s have been described in [51] Cundall and Hart (1992) as those in which discrete, contacting bodies are allowed finite rotations and displacements, including complete detachment, and among which new contacts are recognized automatically during simulation. That is, DEM’s represent discontinua in which initial voids are the result of the choice of the shape and size of the discrete element being used, or represent emerging discontinua in which surfaces of weakness are initially represented by non-linear springs connecting surfaces of the discrete elements. By their nature, DEM’s introduce an additional, initial length scale, that of the discrete element, usually much shorter than the structure being modeled. The kinematics in the DEM are usually governed by an explicit implementation of Newton’s second law: each discrete element is allowed to rotate and displace in a time step as a result of numerical integration of this law. Mechanical forces on the elements result from rigid or spring-like contacts with
7
Anthony R. Ingraffea, Paul A. Wawrzynek
adjacent elements. In the latter case, needed for crack growth simulation, element debonding follows from a cohesive constitutive model. In contrast to the cohesive disbonding models used in BE and FE methods, wherein the relationship is between generalized traction and generalized relative displacement, for the DEM the relationship is between generalized force and generalized relative displacement. DEM approaches are usually intended for the nontraditional role of computational fracture mechanics: elucidation of fundamental cracking mechanisms and prediction of toughness. An overview of the DEM is given in [52]. Among the discrete element approaches are lattice, particle, and atomistic representations. Lattice models are those in which the discrete elements are truss elements or beam elements. These are given geometrical and constitutive properties which allow the lattice to approximate continuum elastic or plastic behavior in the large, and can be given “breaking” rules based on maximum force or energy dissipation to mimic the fracturing process on a mesoscale [53, 54, 55, 56]. Particle models are those in which the discrete elements can be disks, spheres, polygons, or polyhedra [57, 58, 59, 60, 61]. Like the lattice models, the particles themselves are given constitutive properties which allow their bonded assemblage to approximate continuum elastic or plastic behavior in the large. The bonds at the points of contact among the particles are given the limiting strength or energy dissipation rules which enable decohesion and crack growth. These are controllable through appropriate selection of inter-particle bond strengths and toughnesses. Lattice and particle approaches both suffer from a similar drawback. In order to use “element” shapes that are simple, like a beam or a sphere, and that are sufficiently small to allow damage to occur on the desired length-scale, one has to contrive artificial local elastic and plastic constitutive properties to produce observed global properties. For each topology or packing arrangement and size distribution of discrete elements, a different set of artificial local properties must first be determined by a calibration process. There are constraints on topology/arrangement/size that must be observed. Further, one has to transform actual micromechanical damage processes, such as grain boundary decohesion and slip or polycrystal plasticity, into equivalent actions on the end of a beam, or at a point contact between particles. There is no free lunch. Atomistic models are those in which the discrete elements are individual atoms. One can imagine that such a model is a limiting length-scale case of a particle model. The atomistic approach is the currently the most fundamental geometrical approach to crack propagation. An example is the Modified Embedded Atom Method (MEAM), a semi-empirical numerical tool used to study the deformation of various classes of materials at the atomic size scale [62, 63, 64, 65]. The MEAM is an effective device for studying crack growth and fracture since its framework assures the proper modeling of free surfaces and the anisotropic elastic response of materials, components critical to crack propagation under linear-elastic conditions. Although they do not suffer from the drawbacks of “artificial” constitutive properties intrinsic to lattice
8
Anthony R. Ingraffea, Paul A. Wawrzynek
and particle approaches, atomistic approaches have their own set of problems. Obviously, one requires many atoms to fill even a very small volume. The current atomistic simulation record is about a billion atoms [62, 63], enough to fill about a cubic micron. A number of modifications to straightforward atomistic modeling are being developed to address this length-scale problem. Among these is the quasicontinuum approach [66, 67, 68] in which a conventional continuum mechanics formulation, such as FEM, can have its length-scale of applicability extended through enrichment via atomistically derived constitutive information. This length limitation is one of a number of factors that currently limit atomistic approaches to simulation of fundamentally important cracking problems. Another is time-scale. The natural frequency of thermal vibration of atoms is on the order of femtoseconds. To perform simulations at reasonable temperatures, time steps of that order are necessary. Consequently, atomistic approaches to simulating cracking processes that occur within even milliseconds are currently prohibitive. Accurate many-body potentials are not yet available for many elements of practical interest. Finally, since real cracking processes are always subject to crack front chemistry effects, atomistic approaches need to be extended to include multiple-species interactions. Finally, the separation into two main branches of crack representation shown in Figure 1 can appear to break down in the case of the atomistic approach. This approach is simultaneously geometrical and constitutive. Clearly, the only way to represent breakage of atomic bonds is to separate atoms geometrically. However, the limit of this separation process is embedded within the interatomic potential, and in itself is the constitutive model for material behavior at this length scale. The discrete approaches were invented specifically to overcome the perceived difficulty in starting with a continuum approach, like the FEM, and arriving at some form of discontinuum, like fragmentation. At first, it might appear that so-called continuum approaches and the discrete approaches are radically different when applied to cracking problems. However, the differences arise only from the method of representation of cracking, and not from the numerical methods themselves. The fundamental mechanics of fracture in these alternative approaches are not limited by the mathematical nature of the solution methodology, FEM, lattice, or particle, but on the size and shape of the “elements” used. The former is effectively limited by available software and computer power, the latter by the taste of the analyst. 4 NON-GEOMETRICAL REPRESENTATION APPROACHES There are two classes of non-geometrical representations available within the framework of the FEM, constitutive and kinematic, Figure 1. In the former, the material stiffness is appropriately degraded locally to mimic the displacement discontinuity created by a crack, while the underlying geometry and mesh models are left unchanged. In the latter the effect of a crack on the surrounding strain and/or displacement fields is embedded in the local
9
Anthony R. Ingraffea, Paul A. Wawrzynek
approximant within elements, again leaving the geometry and mesh models unaltered. The major objective in both approaches is to obviate the need for repetitive remeshing. 4.1 Constitutive Non-Geometrical Approaches Constitutive non-geometrical approaches are known as smeared crack representations. This name derives from the fact that the resulting “crack” is actually a distributed softened zone around a stiffness quadrature point within a finite element. In the smeared crack class of constitutive methods, the cracked material is idealized as a damaged continuum. The very substantial body of approaches known as damage mechanics is a super-set of smeared crack approaches [69, 70]. Rashid [71] apparently first suggested the smeared crack approach as a simplified means of representing cracks in reinforced concrete, albeit without regard to fracture mechanics. That is, his criterion for crack growth was exceeding a specified limiting value of tensile principal stress normal to the direction of crack growth. Bažant and Cedolin [72] noted the need for an energy basis for the smeared crack approach to make it objective with respect to meshing: increased mesh refinement would otherwise trigger crack growth at increasingly lower loads. They integrated the smeared crack representational model with classical LEFM. Introduction of an energy-based criterion necessarily and fundamentally introduced a difficulty with this approach that is still not completely resolved at this time. That difficulty is that element size has to be included in the energy-based criterion for crack growth. Clearly, the volume of material associated with a quadrature point relates to both element size and to stored strain energy. This intertwining of constitutive and meshing models complicates a crack growth simulation with the smeared crack approach. This approach makes even a problem of LEFM materially non-linear. Hence, there is a tradeoff in such a case between, for example, extra computations for remeshing using the geometrically adaptive method, and extra computations for repeated equation solving using the smeared crack approach. Rots and de Borst [73] and Willam, Pramano, and Sture [74] highlight other principal shortcomings of the classical smeared crack approach. Among these are: • Directional bias: crack pattern is dependent on element topology; • Spurious kinematic modes: artificial softening of elements may cause negative eigenvalues, loss of numerical stability; • Stress locking: results are always too stiff; residual strength too high. Many variants of the smeared crack approach have been offered in the literature, all seeking to eliminate or at least diminish these shortcomings. de Borst [75] presents comparative studies among these variants, while Weihe, Kroplin, and de Borst [76] present a comprehensive classification of smeared crack models. The crack band approach is the largest sub-class in which the crack is implicitly modeled by the assumption that its opening is distributed over a prescribed length [61]. The concept of a crack band model has been applied to various types of constitutive representations such as elasticity-based fixed [71] and rotating
10
Anthony R. Ingraffea, Paul A. Wawrzynek
crack models [77, 78], elasticity-based damage models [69], multi-directional fixed crack models [79], and plasticity-based crack models. Another sub-class of constitutive representation is the enhanced continuum approach. These models all have in common the introduction of an internal length scale that reflects the physical interaction between two points of a material. In this group there are non-local, gradient models, micro-polar and rate-dependent models [80, 75, 81]. Nevertheless, this approach has seen extensive and effective use in the concrete and geomechanics communities, and has found its way in one form or another, into many commercial FE programs. Moreover, the essence of this approach can be seen in some of its siblings, such as element extinction and computational cell, described next. An extreme version of the constitutive method is element extinction. In this FE approach, when a criterion for crack advance has been met in a finite element, the element is simply removed from the model. See, for example, Beissel, Johnson, and Popelar [82]. The resulting crack width and pattern is clearly mesh dependent, and energy balance rules must still be followed for mesh objectivity. A variant of the smeared crack/element extinction approach which has been used very effectively for the more modern role of computational fracture mechanics, relating resistance to microstructure, is the method of computational cells. This approach has been very successfully applied to the problem of ductile crack growth in steels, both in 2D [83, 84, 85] and in 3D [86, 87]. Constitutive approaches are one of the two major branches of the non-geometrical road to crack representation. With constitutive approaches, a crack is represented in an abstract manner within a finite element by appropriate modification of its material stiffness matrix in Equation 1:
[k ] = ∫∫∫vol [B]T [D ][B] J dVol
(1)
where [k] is an element stiffness matrix [D] is the material stiffness matrix [B] is the strain-displacement matrix J is the Jacobian of transformation Vol is the volume of the element
The other non-geometrical branch represents a crack within an element by appropriate modification of the strain-displacement matrix and/or the Jacobian in Equation 1, as described next.
11
Anthony R. Ingraffea, Paul A. Wawrzynek
4.2 Kinematic non-geometrical approaches The kinematic approach is a FE method in which crack representation is embedded in the local finite element approximants, and consequently appears as changes in the straindisplacement matrix, [B], or in the Jacobian, J, of equation 1. There were many attempts in the early history of the application of the FEM to fracture mechanics to enrich a finite element with knowledge of crack presence within its domain. Many researchers investigated special finite element formulations that incorporated singular basis functions in [B], mappinginduced singularities, or stress intensity factors as nodal variables (e.g., [88, 89, 90, 91, 92, 93, 94, 95, 96]). Perhaps the best known of these formulations is the so-called ¼-point, singular, isoparametric element [94, 95] which is found today in the element libraries of most commercial FE programs. There has been a resurgence in interest in kinematic approaches kindled by the work of Belytschko and Black [97], in what has become known as the Extended Finite Element Method (XFEM), and that of Strouboulis et al. [98] and Duarte et al. [99, 100], in what has become known as the Generalized Finite Element Method (GFEM). A comprehensive review of XFEM and GFEM methods for fracture mechanics is Karihaloo and Xiao [101]. The basic idea in both cases is to introduce an intra-element crack-like discontinuity in either the strain field (origins of this approach can be seen in [102, 103]), or in the displacement field (origins of this approach can be seen in [104]). Enrichment with higher order terms and direct calculation of stress intensity factors has been shown by Karihaloo and Xiao and Karihaloo [105]. Enrichment with non-LEFM terms has been described in Moës and Belytschko [106] for cohesive crack growth. One complication in the implementation of this approach involves the numerical quadrature for enriched elements. Completely arbitrary transversal of an element by a crack creates highly irregular sub-domains within the element within which quadrature must be performed. It has been found that, in many cases, and especially in 3D, standard, low-order Gauss quadrature is unacceptable. Consequently, a scheme for discretizing enriched elements into triangles or tetrahedra, within which much higher order quadrature schemes are employed, is a practical, but cumbersome, necessity. Example applications of the XFEM to 3D arbitrary crack growth under LEFM conditions are shown in [107]. These simulations also used a level-set extension to the XFEM [108]. The XFEM or GFEM kinematic approaches are developing quickly because of the promise of substantial reduction in, or even complete elimination of repeated applications of the discretization process. Standard FEM methods and programs can be used for the nonenriched domain. However, substantial additional programming and calculations are needed to:
12
Anthony R. Ingraffea, Paul A. Wawrzynek
• • • •
5
Identify those elements to be enriched; Perform the necessary coordinate transformations on the enriched interpolants; Discretize each enriched element into regular sub-cells for quadrature; Perform non-standard quadrature on each sub-cell.
CONCLUSIONS
Crack propagation is a process of evolutionary geometry driven by relatively high values and gradients in crack front fields and concomitant material damage. The integration of all three fundamental aspects of the problem—computing local field quantities, ascertaining resulting material damage, and evolving the crack—can be had with a rich variety of approaches. One can certainly pick from one or more of the branches of the CFM tree of Figure 1 and get this integration job done with a satisfying degree of physical fidelity and economy of computation. However, for a CFM student or practicing engineer seeking more delectable fruit, to understand local material damage in a more basic way, or to find a practical solution to cracking in a real structure with complex geometry and boundary conditions, for which branch should one reach? In the author’s opinion, the answer to this question depends on where one is more comfortable investing R&D resources: in computational geometry to support the evolution of the representational database, or in numerical solution technology to extend the solution procedure in this equation. As has been shown, hopefully, in this paper, one’s view of the computational fracture mechanics field can be geometry-centric or meshcentric. The geometry-centric view springs from two observations. First, in application of CFM to practical problems of industry, a geometry database always exists and is of paramount importance. It is that from which manufacturing drawings are made and on which inspection procedures depend. Second, in application of CFM to better understanding of material damage processes, it is geometrical features at smaller and smaller length scales that almost invariably control these processes. Meshes, then, are mathematical artifacts that are mapped, and re-mapped onto geometry, and evolve with it, whether the evolution is driven by redesign, shape optimization, or macro-, meso-, micro-, or nano-crack growth. Crack growth simulation should be governed by relevant physical processes, and the discretization process should not lead or get in the way, but rather should follow. Let the many experts in geometrical modeling and mesh generation do their parts, and, with the support of traditional field solution procedures, let the CFM student or practitioner focus on improving the physics, mechanics, or efficiency of the mathematical model. In the geometry-centric view, the representation of a crack is its geometry, a thing quite independent from the field solver. The representation remains invariant with a change in the field solver. The mesh-centric view springs from the notion that the initial discretized form is the
13
Anthony R. Ingraffea, Paul A. Wawrzynek
mathematical model, and anything that can be done to simplify, or even eliminate, its evolution, is a good thing. One can, quite effectively, promote the solution function to that of paramount importance, innovate on it with constitutive or kinematic elegance, and virtually eliminate the need for further support from the geometrical modeling and meshing communities. One must do additional programming and spend some computational cycles doing calculations not needed, or not needed as much, in the more traditional field evaluators. But the payback is reduced calculations for remeshing and interaction with the geometry database. In the mesh-centric view, the representation of a crack is an intimate part of the field solver, and the representation would have to change if the field solver is changed. Many of the existing approaches described in this chapter were born of computational or modeling necessity, rather than from the notion that they might enable better understanding of the fundamental physics of crack growth. Further, most still focus on a single length scale, even though the length scales of material damage and the crack itself are usually vastly different. However, continuous improvements in computational power and physical observations of crack growth processes at increasingly smaller length and time scales are driving better integrations of these problem areas. The near future will see effective combinations of some of these approaches and even newer ones, both geometrical and nongeometrical, at different length scales. Finally, this author offers one more prediction of the future. Market and technological forces much larger than CFM are at work to change the way that information technology—in the forms of computing, data and information storage and access, and worldwide digital connectivity— will serve society. These changes will include software allocation, calculation, and data storage over a grid (maybe even worldwide) of distributed servers, and the use of the web as a bus for such activity. It is quite possible that a crack growth simulation in the not too distant future can be done by concatenation of a series of web services, not unlike the search engines and product/service orders forms which we now use in our everyday lives. Many of the branches in the tree in Figure 1 might become self-contained modules, distributed worldwide, and available for near-real-time, unique assembly for a specific application. Focus will shift away from the geometry/mesh centric issues discussed at length herein, and towards standards for interchange of modules and data, and multiscale and multiphysics adaptive simulations. Acknowledgements The authors want to thank their colleagues in the Cornell Fracture Group, and acknowledge the support of the Cornell Theory Center, the National Science Foundation ITR EIA-0085969, RI EIA-9972853, and NASA NAG-1-02051 and NCC3-994, the Institute for Future Space Transport. REFERENCES [1] Computational Methods in Mechanics: Vol 2, Computational Methods in the Mechanics of Fracture, S.
14
Anthony R. Ingraffea, Paul A. Wawrzynek
Atluri (ed), Elsevier Science Publishers: New York, 1986. [2] A. R. Ingraffea, P. A. Wawrzynek. Finite element methods for linear elastic fracture mechanics. In Comprehensive Structural Integrity, R. de Borst, and H. Mang (eds), Elsevier Science Ltd.: Oxford, England, 1-88, 2003. [3] M. H. Aliabadi. Boundary Element Methods in Linear Elastic Fracture Mechanics. In Comprehensive Structural Integrity, R. de Borst, and H. Mang (eds), Elsevier Science Ltd.: Oxford, England, 89-125, 2003. [4] B. Karihaloo, Q. Xiao. Linear and Nonlinear Fracture Mechanics. In Comprehensive Structural Integrity, R. de Borst, and H. Mang (eds), Elsevier Science Ltd.: Oxford, England, &&, 2003. [5] W. Brocks, A. Cornec. Computational Aspects of Nonlinear Fracture Mechanics. In Comprehensive Structural Integrity, R. de Borst, and H. Mang (eds), Elsevier Science Ltd.: Oxford, England, 127-210, 2003. [6] A. R. Ingraffea. Computational Fracture Mechanics. Mechanics, John Wiley and Sons: New York, 2004.
To appear in Encyclopedia of Computational
[7] R. Clough. The Stress Distribution of Norfork Dam, Structures and Materials Research, Department of Civil Engineering, Series 100, Issue 19, U. of California, Berkeley, Ca., 1962. [8] F. Sims, J. Rhodes, R. Clough. Cracking in Norfolk Dam. J Amer Conc Inst, 61, 265-285, 1964. [9] D. Ngo, A. Scordelis. Finite Element Analysis of Reinforced Concrete Beams, J Amer Conc Inst, 64, 152163, 1967. [10]
C. Radin. The pinwheel tilings of the plane, Annals of Math, 139, 661-702, 1994.
[11]
G. Camacho, M. Ortiz. Computational modeling of impact damage in brittle materials, Int J Solids Structures, 33, 2899-2938, 1996.
[12]
M. Ortiz, A. Pandolfi A. Finite-Deformation irreversible cohesive elements for three-dimensional crack propagation analysis, Int J Numer Meth Engrg, 44, 1267-1282, 1999.
[13]
X-P. Xu, A. Needleman. Numerical Simulations of Fast Crack Growth in Brittle Solids, J Mech Phys Solids, 42, 1397-1434, 1994.
[14]
J. Hanson, A. R. Ingraffea. Using numerical simulations to determine the accuracy of the size-effect and two-parameter data reduction methods for fracture toughness tests of concrete, Engrg Fract Mech, 70, 1015-1027, 2002.
[15]
E. van der Giessen, V. Tvergaard. Effect of random variations in microstructure on the development of final creep failure in polycrystalline aggregates, Modeling Simul Mater Sci Engrg, 2, 721-738, 1994.
[16]
E. van der Giessen, P. Onck, M. W. D. van der Burg. Some effects of random microstructural variations on creep rupture, Engrg Fract Mech, 57, 205-226, 1997.
[17]
E. Iesulauro, A. R. Ingraffea, S. Arwade, P. Wawrzynek. Simulation of Grain Boundary Decohesion and Crack Initiation in Aluminum Microstructure Models, In Fatigue and Fracture Mechanics: 33rd Volume, ASTM STP 1417, Reuter W G and Piascik R S, (eds), American Society for Testing and Materials, West Conshohocken, PA, 715-728, 2002.
[18]
T. Nishioko, S. N. Atluri. Analytical solution for embedded elliptical cracks, and finite element alternating method for elliptical surface cracks, subjected to arbitrary loadings, Engrg Fracture Mech, 17, 247-268, 1983.
15
Anthony R. Ingraffea, Paul A. Wawrzynek
[19]
L.Wang, F. W. Brust, S. N. Atluri. The elastic-plastic finite element alternating method (EPFEAM) and the prediction of fracture under WFD conditions in aircraft structures Part I: EPFEAM theory. Computational Mechanics, 19, 356-369, 1997.
[20]
D. S. Pipkins, S. N. Atluri. Applications of the three dimensional alternating method. Finite Elements in Analysis and Design, 23, 133-152,1996.
[21]
NASGRO 4.0, Southwest Research Institute, San Antonio, Texas, USA. HTTP://vault.swri.org/Nasgro, 2002.
[22]
J. A. Harter. AFGROW Users Guide and Technical, Air Force Research Laboratory Report AFRL-VAWP-TR-1999-3016, 1999.
[23]
J. C. Newman, Jr. FASTRAN-II--A Fatigue Crack Growth Structural Analysis Program NASA Report TM-104159. NASA Langley Research Center, Hampton, VA, 1992.
[24]
T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, P. Krysl. Meshless methods: An overview and recent developments. Computer Meth Appld Mech Engrg 1996; 139: 3-47.
[25]
H. Askes, J. Pamin, R. de Borst. Dispersion analysis and element-free Galerkin solutions of second-and fourth order gradient enhanced damage models, Int J Num Meth Engrg, 49, 811-832, 2000.
[26]
T. Belytschko T, Y. Y. Lu, L. Gu. Element-free galerkin methods. Int J Num Meth Engrg, 37, 229-256, 1994.
[27]
T. Belytschko, L. Gu, and Y.Y. Lu. Fracture and crack growth by element-free Galerkin methods, Model Simul Mater Sci Engrg, 2, 519-534, 1994.
[28]
P. Krysl, T. Belytschko. Element free Galerkin method for dynamic propagation of arbitrary 3-D cracks, Int J Num Meth Engrg, 44, 767-800, 1999.
[29]
T. Belytschko, D. Organ, Y. Krongauz. Coupled finite element-element-free Galerkin method, Computational Mechanics, 17, 186-195, 1995.
[30]
P. Wawrzynek, A. R. Ingraffea. An edge - based data structure for two-dimensional finite element análisis, Engrg with Computers, 3, 13-20, 1987.
[31]
L. Martha, P. Wawrzynek, A. R. Ingraffea. Arbitrary crack propagation using solid modeling, Engrg with Computers, 9, 63-82, 1993.
[32]
B. Carter, P. Wawrzynek, A. R. Ingraffea. Automated 3D crack growth simulation, Int J Num Meth Engrg, 47, 229-253, 2000.
[33]
P. Wawrzynek, A. R. Ingraffea. Interactive finite element analysis of fracture processes: an integrated approach, Theor Appld Fract Mech, 8, 137-150, 1987.
[34]
A. R. Ingraffea, G. Blandford, J. Liggett. Automatic modelling of mixed - mode fatigue and quasi - static crack propagation using the boundary element method. In ASTM STP 791: Proc. of the 14th National Symposium on Fracture Mechanics. American Society for Testing and Materials: Philadelphia, USA, I 407 - I 426, 1983.
16
Anthony R. Ingraffea, Paul A. Wawrzynek
[35]
T. Bittencourt, P. Wawrzynek, J. Sousa, A. R. Ingraffea. Quasi-automatic simulation of crack propagation for 2D LEFM problems, Engrg Fract Mech, 55, 321-334, 1996.
[36]
M. James, D. Swenson. A software framework for two dimensional mixed mode-I/II elastic-plastic fracture. In Mixed-Mode Crack Behavior – ASTM STP 1359, Miller K J, McDowell D L (eds.), American Society for Testing and Materials: Philadelphia, USA, 111-126, 1999.
[37]
D. Swenson, A. R. Ingraffea. Modelling mixed-mode dynamic crack propagation using finite elements: theory and applications, Computational Mech, 3, 187-192, 1988.
[38]
R. Perucchio, A. R. Ingraffea. An integrated boundary element analysis system with interactive computer graphics for three dimensional linear - elastic fracture mechanics, J Comp Structures, 20, 157 – 171, 1985.
[40]
Y. Mi, M. Aliabadi. Three-dimensional crack growth simulation using BEM, Computers and Structures, 52, 871-878, 1994.
[41]
L. Spievak, D. Lewicki, P. Wawrzynek, A. R. Ingraffea. Simulating fatigue crack growth in spiral bevel gears, Engrg Fract Mech, 68, 53-76, 2001.
[42]
C-S Chen, P. Wawrzynek, A. R. Ingraffea. Methodology for fatigue crack growth and residual strength prediction with applications to aircraft fuselages, Comp Mech, 19, 527-532, 1997.
[43]
C-S Chen, P. Wawrzynek, A. R. Ingraffea. Residual strength prediction of airplane fuselages using CTOA criterion, AIAA Journal, 40, 566-575, 2002.
[44]
C-S Chen, P. Wawrzynek, A. R. Ingraffea. Prediction of residual strength and curvilinear crack growth in aircraft fuselajes, AIAA Journal, 40, 1644-1652, 2002.
[45]
J. Ĉervenka, V. Saouma. Numerical evaluation of 3D SIF for arbitrary finite element meshes, Engrg Fract Mech, 57, 541-563, 1997.
[46]
A. Ural, A. R. Ingraffea, P.Wawrzynek, D. Lewicki. Three-dimensional, parallel, finite element simulation of fatigue crack growth in a spiral bevel gear. To appear in Engrg Fract Mech 2004.
[47]
P. Wawrzynek, A. R. Ingraffea. An interactive approach to local remeshing around a propagating crack, Finite Elem Analys Design, 5, 87 – 96, 1989.
[48]
D. Potyondy, P. Wawrzynek, A. R. Ingraffea. An algorithm to generate quadrilateral or triangular element surface meshes in arbitrary domains with applications to crack propagation, Int J Num Meth Engrg, 38, 2677-2701, 1995.
[49]
D. Nave, N. Chrisochoides, P. Chew. Guaranteed-quality parallel delaunay refinement for restricted polyhedral domains. To appear in Computational Geometry: Theory and Applications, 2004.
[50]
J. Cavalcante Neto, P. Wawrzynek, M. Carvalho, L. Martha, A. R. Ingraffea. An algorithm for threedimensional mesh generation for arbitrary regions with cracks, Engrg with Comp, 17, 75-91, 2001.
[51]
P. Cundall, R. Hart. Numerical modeling of discontinua, J Engrg Comp, 9, 101-113, 1992.
[52]
N. Bicanic. Fragmentation and discrete element methods. In Comprehensive Structural Integrity, de Borst
17
Anthony R. Ingraffea, Paul A. Wawrzynek
R, and Mang H A (eds), Elsevier Science Ltd.: Oxford, England, 427-458, 2003. [53]
B. Chiaia, A. Vervuurt, J. van Mier. Lattice model evaluation of progressive failure in disordered particle composites, Engrg Fract Mech, 57, 310-318, 1997.
[54]
van Mier J. Fracture processes of Concrete. CRC Press LLC: Boca Raton, FL, 1997.
[55]
E. Schlangen, E. Garboczi. Fracture simulations of concrete using lattice models: computational aspects, Engrg Fract Mech, 57, 319-332, 1997.
[56]
B. Karihaloo, P. Shao, Q. Xiao. Lattice modeling of the failure of particle composites, Engrg Fract Mech, 70, 2385-2406, 2003.
[57]
Disorder and fracture. J. Charmet, S. Roux, E. Guyon, (eds.), Plenum Press: New York, NY, 1989.
[58]
Statistical models for the fracture of disordered media. H. Herrmann, S. Roux S (eds.), North-Holland: New York, NY, 1990.
[59]
D. Potyondy, P. Cundall, C. Lee. Modeling rock using bonded assembles of circular particles. Proc Second North Amer Rock Mech Symp, Aubertin M. et al (eds), Balkema: Rotterdam, 1996.
[60]
D. Potyondy, P. Cundall. A bonded-particle model for rock. Int J Rock Mech Min Sci & Geomech Abstr, To appear, 2004.
[61]
Z. Bažant, and J. Planas. Fracture and Size Effect in Concrete and Other Quasibrittle Materials. CRC Press: Boca Raton, FL, 1997.
[62]
M. Baskes. Modified embedded atom potentials for cubic materials and impurities. Phys Rev, B 46, 27272742, 1992.
[63]
F. F. Abraham, R. Walkup, H. Gao, M. Duchaineau, T. D. de la Rubia. Simulating materials failure by using up to one billion atoms and the world’s fastest computer: brittle fracture, Proc Natl Acad Sci USA, 99, 5777-5782, 2002.
[64]
F. F. Abraham, R. Walkup, H. Gao, M. Duchaineau, T. D. de la Rubia. Simulating materials failure by using up to one billion atoms and the world’s fastest computer: work hardenimg, Proc Natl Acad Sci USA, 99, 5783-5787, 2002.
[65]
K. Gall, M. Horstemeyer, M. Baskes, M. Van Schlifgaarde. Atomistic simulations on the tensile debonding on an Al-Si interface, J Mech Physics Solids, 48, 2183-2185, 1999.
[66]
R. Miller, M. Ortiz, R. Phillips, V. Shenoy, E. Tadmor. Quasicontinuum models of fracture and plasticity, Engrg Fract Mech, 61, 427-444, 1998.
[67]
Shenoy V B, Miller R, Tadmor E B, Phillips R, Ortiz M. Quasicontinuum models of interfacial structure and deformation. Physical Review Letters 1998; 80; 742-745.
[68]
M. Ortiz, R. Phillips. Nanomechanics of defects in solids, Adv Appld Mech, 36, 1-79, 1996.
[69]
Krajcinovic D. Continuous damage mechanics revisited: basic concepts and definitions, J Appl Mech, 52, 829–834, 1985.
18
Anthony R. Ingraffea, Paul A. Wawrzynek
[70]
R. de Borst. Damage, material instabilities and failure. Encyclopedia Comp. Mech., E Stein, R de Borst, T Hughes (eds), John Wiley & Sons, Ltd., in press, 2004.
[71]
Y. R. Rashid. Analysis of prestressed concrete reactor vessels. Nuclear Engrg Design, 7, 334-344, 1968.
[72]
Z. Bažant, L. Cedolin. Blunt crack band propagation in finite element analysis, J Struct Div, ASCE, 105, 297-315, 1979.
[73]
J. Rots, R. de Borst. Analysis of mixed-mode fracture in concrete. J Engrg Mech ASCE, 113, 1739-1758, 1987.
[74]
K. Willam, E. Pramono, S. Sture. Fundamental issues of smeared crack models. SEM/RILEM Int. Conf. Fracture Concrete Rock. Shah S P, Swartz S (eds), 1987.
[75]
R. de Borst. Some recent developments in computational modeling of concrete fracture, Int J Fracture, 86, 5-36, 1997.
[76]
S. Weihe, B. Kroplin, R. de Borst. Classification of smeared crack models based on material and structural properties, Int J Solids Structures, 35, 1289-1308, 1998.
[77]
R. Cope, P. Rao, L. Clark, P. Norris. Modelling of reinforced concrete behavior for finite element analysis of bridge slabs. In Numerical Methods for Nonlinear Problems I, Taylor et al. (eds), 457-470, 1980.
[78]
M. Jirásek, T. Timmerman. Rotating crack model with transition to scalar damage. J Enrg Mech ASCE, 124, 277-284, 1998.
[79]
R. de Borst, P. Nauta. Non-orthogonal cracks in a smeared finite element model. Engrg Computations, 2, 35-46, 1985.
[80]
I. Carol, Z. Bažant, P. Prat. Microplane-type constitutive models for distributed damage and localized cracking in concrete structures. In Int Conf Fracture Mechanics Concrete Structures, FRAMCOS 1, Bažant Z P (edit.) 299-304, 1992.
[81]
M. Jirásek. Nonlocal models for damage and fracture: comparison of approaches, Int J Solids Structures, 35, 4133-4145, 1998.
[82]
S. Beissel, G. Johnson, C. Popelar. An element-failure algorithm for dynamic crack propagation in general directions, Engrg Fracture Mech, 61, 407-426, 1998.
[83]
L. Xia, C. F. Shih. Ductile crack growth-I. A numerical study using computational cells with microstructurally-based length scales, J Mech Phys Solids, 43, 223-259, 1995.
[84]
L. Xia, C. F. Shih. Ductile crack growth-II. Void nucleation and geometry effects on macroscopic fracture behavior, J Mech Phys Solids, 43, 19533-1981, 1995.
[85]
L. Xia, C. F. Shih. Ductile crack growth-III. Statistical aspects of cleavage fracture after tearing, J Mech Phys Solids, 44, 603-639, 1996.
[86]
C. Ruggieri, T. Panontin, R. Dodds. Numerical modeling of ductile crack growth in 3-D using computational cell elements, Int J Fracture, 82, 67-95, 1996.
19
Anthony R. Ingraffea, Paul A. Wawrzynek
[87]
A. Gullerud, R. Dodds. Simulation of ductile crack growth using computational cells: numerical aspects, Engrg Fracture Mech, 66, 65-92, 2000.
[88]
E. Byskov. The calculation of stress intensity factors using the finite element method with cracked elements, Int J Fracture, 6, 159-167, 1970.
[89]
D. Tracey. Finite elements for determination of crack tip elastic stress intensity factors, Engrg Fracture Mech, 3, 255-256, 1971.
[90]
P. Tong , T. H. Pian, S. Lasry. A hybrid element approach to crack problems in plane elasticity, Int J Num Meth Engrg, 7, 1031-1036, 1973.
[91]
S. Papaioannou, P. Hilton, R. Lucas. A finite element method for calculating stress intensity factors and its application to composites, Engrg Fracture Mech, 6, 807-823, 1974.
[92]
S. Atluri, A. Kobayashi, M. Nakagaki. An assumed displacement hybrid finite element model for linear elastic fracture mechanics, Int J Fracture, 11, 257-271, 1975.
[93]
S. Benzley. Representation of singularities with isoparametric finite elements, Int J Num Meth Engrg, 8, 537-545, 1974.
[94]
R. Henshell, K. Shaw. Crack tip elements are unnecessary, J Num Meth Engrg, 9, 495-509, 1975.
[95]
R. Barsoum. On the use of isoparametric finite elements in lnear fracture mechanics, Int J Num Meth Engrg, 10, 25-37, 1976.
[96]
P. Lynn, A. R. Ingraffea. Transition element to be used with quarter - point crack tip elements, Int J Num Meth Engrg, 12, 1031-1036, 1978.
[97]
T. Belytchko, T. Black. Elastic crack growth in finite elements with minimal remeshing, J Num Meth Engrg, 45, 601-620, 1999.
[98]
T. Strouboulis, K. Copps, I. Babuska. The design and analysis of the generalized finite element method, Comput Meth Appl Mech Engrg, 181, 43-69, 2000.
[99]
C. Duarte, O. Hamzeh, T. Liszka, W. Tworzydlo. A generalized finite element method for the simulation of three dimensional dynamic crack propagation, Comp Meth Appld Mech Engrg, 190, 2227-2262, 2001.
[100]
C. Duarte, I. Babuska, J. Oden. Generalizad finite element methods for three dimensional structural mechanics problems, Comp Struct, 77, 215-232, 2000.
[101]
B. Karihaloo, Q. Xiao. Modelling of stationary and growing cracks in FE framework without remeshing: a state-of-the-art review, Computers and Structures, 81, 119-129, 2003.
[102]
T. Belytschko, J. Fish, B. Engelmann. A finite element with embedded localization zones, Computer Methods in Applied Mechanics and Engineering, 70, 59-89, 1988.
[103]
L. Sluys, A. Berends. Discontinuous failure analysis for mode-I and mode-II localization problems, Int J Solids Structures, 35, 4257-4274, 1998.
20
Anthony R. Ingraffea, Paul A. Wawrzynek
[104]
J. Simo, J. Oliver. A new approach to the analysis and simulation of strong discontinuities. In Fracture and Damage in Quasibrittle Structures, Bazant Z P et al. (eds), E&F N Spon, 25-39, 1994.
[105]
Q. Xiao, B. Karihaloo. Direct evaluation of accurate coefficients of the crack tip asympototic field. To appear in Fatigue Fract Engrg Mater Struct., 2004.
[106]
N. Moës, T. Belytschko. Extended finite element method for cohesive crack growth. Engrg Fract Mech, 69, 813-833, 2002.
[107]
A. Gravouil, N. Moës, T. Belytschko. Non-planar 3D crack growth by the extended finite element and level sets—part II: level set update, Int J Num Meth Engrg, 53, 2569-2586, 2002.
[108]
M. Stolarska, D. Chopp, N. Moës, T. Belytschko. Modeling crack growth by level sets in the extended finite element method, Int J Num Meth Engrg, 51, 943-960, 2001.
GEOMETRICAL REPRESENTATION
Constrained Shape Methods
Prescribed Methods Analytical Geometry Methods Known Solution Methods
NON-GEOMETRICAL REPRESENTATION
Arbitrary Shape Methods
Meshfree Methods Adaptive FEM/BEM Methods
Constitutive Methods
Kinematic Methods
Smeared Crack Methods
Enriched Element Methods
Element Extinction Methods
XFEM/GFEM
Lattice Methods Computational Cell Methods
Particle Methods Atomistic Methods
Figure 1. A taxonomy of methods of representation of cracking in a computational model.
21
