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An overview of computational methods to model fracture in brittle and quasi-brittle materials is given. The overview focuses on continuum models for fracture. First, numerical difficulties related to modelling fracture for quasi-brittle materials will be discussed. Different techniques to eliminate or circumvent those difficulties will be described subsequently. In that context, regularization techniques such as nonlocal models, gradient enhanced models, viscous models, cohesive zone models, and smeared crack models will be discussed. The main focus of this paper will be on computational methods for discrete fracture (discrete cracks). Element erosion technques, inter-element separation methods, the embedded finite element method (EFEM), the extended finite element method (XFEM), meshfree methods (MMs), boundary elements (BEMs), isogeometric analysis, and the variational approach to fracture will be reviewed elucidating advantages and drawbacks of each approach. As tracking the crack path is of major concern in computational methods that preserve crack path continuity, one section will discuss different crack tracking techniques. Finally, cracking criteria will be reviewed before the paper ends with future research perspectives.

The prediction of material failure is of major importance in engineering and materials Science. In the past two decades, a huge effort was made to develop novel, efficient, and accurate computational methods for fracture, and an enormous progress was made. This paper will give an overview on those advances. The paper focuses on continuum models for brittle and quasi-brittle fracture. It will not discuss issues related to ductile fracture with plastic deformation prior to localization; see for example, the excellent work of the group of Prof. Li [

There are three failure modes see Figure

Different failure modes.

The equivalence principe of continuum damage mechanics.

The use of strain-softening material models leads to an ill-posed boundary value problem (BVP) [

The difficulties that emerge with the material instabilities due to the ill-posedness of the BVP can be avoided by regularization techniques such as higher order continuum models, gradient based models, and polar theories (the most well-known theory is probably the Cosserat continuum), nonlocal models, viscous models or cohesive zone models. These regularization techniques introduce a characteristic length into the discretization. In other words, the local character of the deformation is lost, and the fracture is “smeared” over a certain domain involving several elements. Therefore, a very fine mesh has to be used in order to resolve the crack. As the crack introduces a jump in the displacement field, methods that smear the crack over a certain region are not capable of representing the correct crack kinematics. Moreover, the different length scales (of the structure and the characteristic length) may significantly increase the computational cost. A lot of effort has been devoted to develop methods that are capable of capturing the crack and take advantage of regularization techniques that can be coupled to these strong discontinuity approaches. Modeling fracture with strong-discontinuity approaches require two key ingredients:

a method that is capable of capturing the crack kinematics and

a cracking criterion that determines the orientation and the length of the crack.

Gradient-enhanced models, or briefly called gradient models, are typically described by differential equations that contain higher order spatial derivatives. The coefficients multiplying the terms of different order have different physical dimensions. It is possible to deduce the characteristic length from their ratio. Gradient models can be incorporated into plasticity format, [

Strongly non-local models are models of the integral type, see Figure

Principle of nonlocal constitutive models with the typical bell-shaped domain of influence.

The introduction of a viscosity can also restore the well-posedness of the BVP or initial BVP (IBVP). It can be regarded as introducing higher order time derivatives, similar to the gradient models. Considering the dimensions of the viscosity

The introduction of a viscosity can sometimes be physically motivated. The strain rate effect and the corresponding dynamic strength, increase, for example, can be captured by viscous damage models [

Cohesive zone models (CZMs) also restore the well-posedness of the (I)BVP. In contrast to the models described before, CZMs can be combined with computational methods that maintain the local character of the crack. In cohesive cracks, a traction-separation model is applied across the crack surface that links the cohesive traction transmitted by the discontinuity surface to the displacement jump, characterized by the separation vector. CZMs go back to the 60’s and were originally developed from Dugdale [

The principal idea of cohesive cracks is shown in Figure

Principle of cohesive crack models.

CZMs can be classified into initially rigid models (sometimes also called extrinsic models), Figure

Typical one-dimensional cohesive models (a) The model by Dugdale1 for fracture across grain boundaries in metal, (b) a modified Dugdale model, (c) linear decaying cohesive model, (d) exponential decaying model, (e) model with positive initial stiffness for brittle fracture, and (f) non-admissible cohesive model.

Initially rigid models cause numerical difficulties when elastic unloading occurs at an early stage such that the stiffness for the unloading case tends to infinity. It is believed that initially rigid models are better suited particularly in the context of dynamic fracture. A good overview and a discussion on initially elastic and initially rigid models are given, for example, in Papoulia et al. [

An important material parameter of the CZM is the fracture energy

The main idea of smeared crack models is to spread the energy release along the width of the localization band usually within a single element so that it be objective. This is achieved by calibrating the width of the band such that the dissipated energy is the correct one. This introduces a characteristic length into the discretization that depends on the size of the elements.

A milestone in smeared crack models is the

Adjustment of the stress-strain curve in the crack band model, (a) master curve and decomposition of the strains into prelocalizaton and post-localization parts, (b) horizontal scaling for an element larger than the physical process zone, and (c) snap back for too large elements reproduced from Bažant and Jirásek [

The smeared crack models can be further classified into rotating and fixed crack models. In the

Principle of (a) rotating crack models, (b) fixed crack models.

The orientation of the crack is typically chosen to be perpendicular to the direction of the principal tensile strain or principal tensile stress. Since the direction of the principal tensile strain changes during loading, fixed crack models result in too stiff system responses. Fixing the crack orientation leads to stress locking, meaning stresses are transmitted even over wide open cracks mainly caused due to shear stresses generated by a rotation of the principal strain axes after the crack initiation. Therefore,

Some models combine the rotating and fixed smeared crack approach [

Smeared crack models in finite element analysis often exhibit a so-called mesh alignment sensitivity or mesh orientation bias, see Figure

Mesh orientation bias for smeared crack models.

The easiest way to deal with discrete fracture is the element erosion algorithm [

For brittle fracture, Pandolfi and Ortiz [

Standard finite elements have difficulties to capture the crack kinematics since they use continuous trial functions that are not particularly well adapted for solutions with discontinuous displacement fields. One of the first models capable of modelling cracks within the FEM are the so-called

Many interesting problems have been studied by interelement-separation methods [

Special crack tip elements for linear fracture problems were developed in the early 70’s [

In 1987, Ortiz et al. [

Element with (a) one weak discontinuity, (b) two weak discontinuities, and (c) one strong discontinuity.

The first version of the EFEM is often called statical optimal symmetric (SOS) since traction continuity is fulfilled, but it is not possible to capture the correct crack kinematics. The correct relative rigid body motion of the element is not guaranteed and it has been shown by Jirásek [

The approximation of the displacement field in the EFEM is given by [

(a) Piecewise constant crack opening in embedded elements and (b) linear crack opening for XFEM.

Oliver et al. [

Methods that model the crack as continuous surface and therefore contain a “physical” crack tip are expected to be more accurate. However, the requirement to represent the crack surface is a drawback of those methods, in particular for complex crack patterns (e.g., in three-dimensions or for branching cracks). EFEM in principle does not require crack path continuity [

A very flexible method that can handle linear and nonlinear crack openings is the extended finite element method (XFEM) developed by Belytschko et al. [

Principle of XFEM for strong discontinuities in 1D.

The basic idea of XFEM is to decompose the displacement field into a continuous part

For elements completely cut by the crack, the jump function is often chosen to cause a discontinuous displacement field

Representation of the crack surface by level sets

In LEFM, the crack tip enrichment is choosen according to the analytical solution:

(a) Topological enrichment versus (b) and (c) geometrical enrichment.

XFEM has been mainly applied to problems involving crack growth of a single crack or a few cracks. There are only a few contributions dealing with fracture problems that involve the growth of numerous (several hundreds of) cracks [

XFEM is a method that ensures crack path continuity. Enforcing crack path continuity is especially challenging in three dimensions. The crack path can be represented explicitly, usually with piecewise straight/planar crack segments [

Level sets are particularly attractive when a crack tip enrichment is used as geometry quantities can be calculated through the level sets. For example, the distance of a material point to the crack tip is

Curved crack and its representation with level set functions when the level set is discretized with linear triangles.

Numerical integration in XFEM requires special attention, in particular, around the crack tip when employing non-polynomial enrichment functions. Laborde et al. [

Another major concern in XFEM are the so-called blending elements, Figure

XFEM discretization with enriched domain

Applying XFEM to linear problems in elastostatics, the final discrete equations for a single crack are obtained by substituting the test and trial functions into the weak form of the equilibrium equation yielding

Although the XFEM approximation is capable of representing crack geometries that are independent of element boundaries, it relies on the interaction between the mesh and the crack geometry to determine the sets of enriched nodes [

(a) and (b) Crack length that approach the local element size cannot be accurately represented by the standard XFEM approximation. Dots denote single enriched nodes and squares denote double (in our case, the node will contain the enrichment of two-crack tips) enriched nodes; (c) the dashed line shows the effective crack length; (d) even if no crack tip enrichment is used, in order to close the crack within a single element, no nodes have to be enriched with a step function.

effective crack length

Admissible crack representation.

More details on XFEM and its applications can be found in the excellent review papers by Karihaloo and Xiao [

An alternative to the standard XFEM for strong discontinuities was proposed by A. Hansbo and P. Hansbo [

It avoids the “mixed” terms

It leads to a better conditioned system matrix.

Standard mass lumping schemes such as the row-sum technique can be used. Note that, efficient mass lumping schemes have meanwhile been proposed for XFEM [

Based on the full interpolation basis of the overlapping elements, the Hansbo-XFEM can straightforwardly integrate enhanced techniques that are more difficult to integrate in an enriched XFEM formulation [

The implementation of the Hansbo XFEM is simpler. It has already been implemented in the commercial software package ABAQUS. In ABAQUS, it has to be used in combination with cohesive zone models. The ABAQUS implementation of the Hansbo-XFEM can handle numerous cracks (crack propagation) in statics. In contrast, XFEM with tip enrichment has also been incorporated in ABAQUS. However, a crack propagation algorithm for the tip-enriched XFEM is yet not available in ABAQUS. A plugin-in (Morfeo) that can handle three-dimensional crack growth in LEFM was recently developed by the company Cenaero.

It is difficult to apply the Hansbo XFEM to other problems besides cracks.

It is also difficult to incorporate a tip enrichment for the Hansbo-XFEM. A crack tip element has been developed for problems in 2D [

The principle of the phantom node method in which the hatched area is integrated to build the discrete momentum equation; the solid circles represent real nodes and the empty ones phantom nodes.

Consider a body that is cracked as shown in Figure

Remmers et al. [

Thanks to the absence of a mesh, meshfree methods (MMs) offer another alternative to model fracture. They are particularly well suited for dynamic fracture and large deformations. Moreover, adaptive h-refinement can be easily implemented in a meshfree context [

Most meshfree shape functions depend on a weighting or kernel function, which is denoted by

Due to the absence of a mesh, fracture in meshfree methods can occur naturally [

The visibility method is the first approach that introduces a discrete crack into the meshfree discretization. In the visibility method, the crack boundary is considered to be opaque. Thus, the displacement discontinuity is modeled by excluding the particles on the opposite side of the crack in the approximation of the displacement field, see Figure

Principle of the visibility, diffraction, and transparency methods with corresponding shape functions, from Belytschko et al. [

(a) Undesired introduced discontinuities by the visibility method, (b) Difficulties with the visibility method for concave boundaries and kinks.

It should also be noted that the visibility criterion leads to discontinuities in shape functions near nonconvex boundaries such as kinks, crack edges, and holes, as shown in Figure

An efficient implementation of the visibility method in 2D is given, for example, in [

The diffraction method is an improvement of the visibility method. It eliminates the

The diffraction method.

The idea of the diffraction method is not only to treat the crack as opaque but also to evaluate the length of the ray

The transparency method was developed as an alternative to the diffraction method by Organ et al. [

An additional requirement is usually imposed for particles close to the crack. Since the angle between the crack and the ray from the node to the crack tip is small, a sharp gradient in the weight function across the line ahead of the crack is introduced. In order to reduce this effect, Organ et al. [

The “see-through” method was proposed by Terry [

In the continuous line method from Krysl and Belytschko [

Belytschko and Fleming [

Domain of influence near a wedge-shaped non-convex boundary. The boundary is enforced if

Enrichment in meshfree methods was used before the development of XFEM. A good overview paper on enriched meshfree methods in the context of LEFM is given by Fleming et al. [

An extended element-free Galerkin (XEFG) method based on vector level sets was first proposed by Ventura et al. [

While all previous meshfree approaches for fracture require crack path continuity, the cracking particles method [

To model cracks in the cracking particles method, the displacement is decomposed into continuous and discontinuous parts:

The crack is modelled by a set of discrete cracks as shown in Figure

Schematic on the right shows a crack model for the crack on the left.

The approximation of the displacement field in the cracking particles method is given by

The jump in the displacement across the crack depends only on the discontinuous part of the displacement field

The cracking particles method might suffer from spurious crack patterns [

(a) Spurious cracking and (b) improved crack pattern.

Crack prevention: nodes

Recently, the cracking particles method has been presented without enrichment to model fracture in continua [

In the Boundary Element Method (BEM), the weak form is formulated in boundary integral form. It reduces therefore the dimension of the problem, for example, a three-dimensional problem reduces to two dimensions. However, the BEM is only applicable to problems where Green’s functions can be computed restricting the application range of the method. Due to the dimension-reduction, the remeshing procedure is fairly simple as the crack is part of the boundary. Interesting applications of the BEM to fracture can be found, for example, in [

Isogeometric analysis (IGA) [

Variational methods in fracture mechanics are a relatively new development in the field. The underlying theory was laid down by Francfort and Marigo [

In classical fracture mechanics, crack propagation is assumed to occur under thermodynamic equilibrium. In particular for crack formation occurring under constant load, the external work done by the applied loading is equal to twice the energy in the bulk, so that the total energy of the system may be expressed as

Despite improvements made by subsequent authors to account for nonlinearity and inelasticity, the original form of the fracture criterion remained, that is,

In order to address the above mentioned weaknesses of classical fracture mechanics, Francfort and Marigo [

Numerical experiments on the proposed formulation were carried out by Bourdin et al. [

A phase field model for mode III dynamic fracture was devised by Karma et al. [

Thermodynamically consistent phase field models of fracture were developed by Miehe et al. [

An important addition of Miehe et al. [

One challenge with the use of phase field models for fracture is the computational expense associated with mesh size requirements, since the use of a mesh having a characteristic length that is not small enough compared to the crack regularization parameter yields erroneous results with regard to the energy. Based on numerical experiments utilizing the stationary phase field equation, Miehe et al. [

Methods that ensure crack path continuity such as XFEM require the representation of the crack surface and algorithms to track the crack path. While those tasks are relatively easy to implement in two dimensions, their implementation in 3D is challenging.

The topology of the crack surface is commonly represented either explicitly by piece-wise planar crack segments or implicitly by level set functions. Meshfree methods and the phantom node method usually use the former method while many XFEM-implementations are based on an implicit representation of the crack surface using level sets.

There are three major approaches to track the evolving crack surface:

local methods,

global methods,

level set method.

With local crack tracking algorithms, the alignment of the crack surfaces is enforced with respect to its neighborhood. Local crack tracking algorithms are usually characterized by recursively “cutting” elements (or background cells in meshfree methods) and are especially effective in three dimensions. Local methods have difficulties to ensure a continuous crack surface in 3D. Jäger et al. [

In global methods, a linear heat conduction problem is solved each load step for the mechanical problem that needs to fulfill the condition

Feist and Hofstetter [

The level set method was originally developed to track interfaces that propagate orthogonal to their surface. To update the interface, the Hamilton-Jacobi equation was solved with respect to the level set. It makes the method particularly attractive for problems in fluid mechanics. However, the original level set method is not well suited to track crack surfaces for three reasons as noted, for example, by Ventura et al. [

The zero isobar of the level set must be updated behind the crack front to account for the fact once a material point is cracked, it remains cracked.

The crack surface is an open surface that extends during the crack propagation. It requires the introduction of another level set function (orthogonal to the level set function describing the crack surface) in order to uniquely determine the position of a material point with respect to the crack surface. When the crack propagates, this level set function needs to be updated as well. Moreover, both level set functions must be periodically reinitialized to the signed-distance functions to preserve stability [

The level set functions are not updated with the speed of the interface in the normal direction, and hence the Hamilton-Jacobi equation cannot be used. Instead, the level set function propagates with the speed of the crack front.

A very efficient and elegant crack propagation and track cracking algorithm in the context of XFEM was recently presented by Fries and Baydoun [

The fracture criterion is needed to determine whether or not a crack will propagate or nucleate. Moreover, the fracture criterion should provide the orientation and the “length” of the crack as well as whether or not cracks branch or join. Methods that ensure crack path continuity need to distinguish between crack nucleation and crack propagation. Detecting branching cracks in dynamic problems is particularly difficult in those methods. The fracture criterion is often met at several quadrature points in front of the crack tip, and reliable criteria to branch cracks in practice are still missing. The best results are obtained when the crack branches are known in advance.

In most applications, the “length” of the crack is controlled. Usually, it is correlated to the underlying discretization. To the best knowledge of the author, all of the methods assume a straight/planar extension of an existing crack surface.

In the following section, different cracking criteria to obtain the orientation of a newly (nucleated or propagated) crack segment are reviewed.

Besides approaches based on configurational forces [

Maximum hoop stress criterion or maximum principal stress criterion.

Minimum strain energy density criterion, Sih [

Maximum energy release rate criterion, Wu [

The zero

The first two criteria predict the direction of the crack trajectory from the stress state prior to the crack extension. The last two criteria require stress analysis for virtually extended cracks in various directions to find the appropriate crack-growth direction. Duflot and Hung [

In the maximum hoop stress or maximum principal stress criterion, the maximum circumferential stress

In LEFM, the local direction of the crack growth is determined by the condition that the local shear stress is zero that leads to the condition:

The minimum strain energy density criterion is based on a critical strain-energy-density factor

In the maximum energy release rate criterion, the crack propagates in the direction defined by the angle

Crack propagation with the energy release rate criterion.

For a Rankine material, a crack is introduced when the principal tensile stress reaches the uniaxial tensile strength. The crack is initiated perpendicular to the direction of the principal tensile stress. Usually, some kind of smoothing technique is applied that either averages the crack normal or the stress tensor, [

Fracture is caused by a material instability. A classical definition of material stability is based on the so-called Legendre-Hadamard condition, which establishes that for any non zero vectors

Loss of hyperbolicity (or loss of ellipticity or loss of material stability, resp.) is determined by minimizing

One difficulty is that the analysis of the acoustic tensor for isotropic materials generally will yield two directions

Minimum eigenvalue of the acoustic tensor as a function of the two angles for one material point.

Minimum eigenvalue of the acoustic tensor

In [

Since the normal

Belytschko et al. [

The orientation of the crack can be obtained by global energy minimization. For different orientations of the crack, the global energy is computed, and the crack is propagated in the direction where the global energy has its minimum. This global fracture criterion can be expressed as:

Such criteria were used in the context of XFEM by Meschke and Dummerstorf [

As mentioned above, the easiest way is to control the “length” of the crack. In the context of XFEM, Belytschko et al. [

Numerous computational methods for fracture have been developed in the past two decades. Advances in partition of unity methods have provided effective and reliable tools to analyze fracture for numerous applications. Improvements were made particularly concerning the accuracy (through enrichment) and the efficiency (no remeshing). Some of those methods are already available in commercial CAE software packages and can be used for commercial applications as pointed out previously. Most of the partition of unity methods for fracture are well suited for static fracture when a moderate number of cracks occur. Many applications (of partition of unity methods) focus on crack propagation, mainly for propagation of single crack surfaces without branching cracks and crack coalescence. While the majority of the computational methods discussed above are capable of handling complex phenomena such as branching cracks, reliable criteria to branch a crack in practical FE-simulations are still missing. Those fracture criteria can be categorized in local criteria and global criteria. Local criteria are based on the stress state in the vicinity of the crack tip while global criteria are based on energy minimization. Both criteria should in principle predict complex fracture pattern naturally. For example, it is known that the ellipticity indicator

While—as stated above—the majority of the applications of computational methods for fracture focus on crack propagation problems, far less studies are concerned with crack nucleation. Methods as XFEM seem not well suited for dynamic fracture and fragmentation involving the nucleation, growth, and coalescence of an enormous number of cracks. An alternative pathway to dynamic fracture offer methods such as the distinct element method (DEM) [

The exact prediction of complex fracture patterns is nearly impossible for many applications due to its stochastic nature. The onset of crack nucleation and in fact also the direction in crack propagation depends on numerous factors such as the micro-structure of the material (imperfections, voids, microcracks, etc.) or loading conditions, that are not known exactly. The author has made the experience that results are particularly sensitive with respect to boundary conditions for example. Most of the novel computational methods for fracture are based on deterministic approaches. There are far less contributions on statistical computational methods for fracture [

deterministic methods in a stochastic setting and

full stochastic methods.

The first approach seems simpler as no modifications of the existing computational methods are needed. Sampling methods belong to the first category while variance reduction methods for example belong to the second category. One major challenge will be the design of

The key objective of computational methods is their application to “real-world” problems, for example, in order to support the design of new products. The choice of the method depends on the application. Most effort in the past was devoted to develop new methods. However, there has been minimal effort to assess those methods. In view of the growing number of computational methods for fracture, it would be of high practical relevance to provide a framework to quantitatively assess the quality of existing computational methods with respect to their accuracy, reliability, and robustness for a specific application. For example, computational methods (and constitutive models) have a number of uncertain input parameters such as the dilation parameter in meshfree methods. It is of utmost importance to quantitatively determine the sensitivity of the uncertain input parameters with respect to a quantity of interest (the output). It will also help to quantify the predictive capabilities of the computational methods.

One major application of computational methods for fracture is their use in the design of new materials. Obtaining a fundamental understanding of how materials fail was/is a main research direction in materials science. In failure, the response of a structure is driven by fine scale features (nano- or micro-structure of the material). For an accurate prediction of material failure, it is important to account for fine scale features in the process zone. Therefore, an important future research direction is the development of multiscale methods for fracture. While numerous multiscale methods (see e.g., [

Multiscale methods can be categorized into hierarchical, semiconcurrent, and concurrent methods [

The basic idea of semiconcurrent multiscale methods is illustrated in Figure

Schematic of a (a) hierarchical, (b) semiconcurrent, and (c) concurrent multiscale methods [

Numerous concurrent multiscale methods [

Future challenges will lie in the development of efficient methods to transfer length scales when fracture occurs, to adaptively choose the discretization and the model based on error estimation, and to bridge disparate time scale. To overcome the high computational cost will be another challenge, in particular, when these methods will be applied to “real-world” applications, for example, in computational materials design.