# Three-dimensional finite element modeling of ductile crack initiation and propagation

- H. R. Javani
^{1, 2}, - R. H. J. Peerlings
^{2}Email author and - M. G. D. Geers
^{2}

**3**:19

**DOI: **10.1186/s40323-016-0071-y

© The Author(s) 2016

**Received: **9 February 2016

**Accepted: **9 May 2016

**Published: **7 June 2016

## Abstract

A crack initiation and propagation algorithm driven by non-local ductile damage is proposed in a three-dimensional finite strain framework. The evolution of plastic strain and stress triaxiality govern a non-local ductile damage field via constitutive equations. When the damage reaches a critical threshold, a discontinuity in the form of a crack surface is inserted into the three-dimensional continuum. The location and direction of the introduced discontinuity directly result from the damage field. Crack growth is also determined by the evolution of damage at the crack tip and the crack surface is adaptively extended in the computed direction. Frequent remeshing is used to computationally track the initiation and propagation of cracks, as well as to simultaneously maintain a good quality of the finite elements undergoing large deformations. This damage driven remeshing strategy towards fracture allows one to simulate arbitrary crack paths in three-dimensional evolving geometries. It has a significant potential for a wide range of industrial applications. Numerical examples are solved to demonstrate the ability of the proposed framework.

### Keywords

Finite element method Ductile fracture Remeshing Nonlocal damage## Background

Controlling crack initiation and propagation is one of the important aspects in maintaining the integrity of an engineering structure. In some other cases, however, cracks are introduced on purpose. Examples can be found in forming processes such as cutting or blanking. Computational models are indispensable for the predictive analysis of the mechanics of ductile fracture. Algorithms for dealing with two-dimensional (2D) crack propagation problems are by now well established. However, at present, three-dimensional (3D) problems cannot be analyzed routinely, particularly if they are accompanied by large (plastic) strains. This is due to the complex topology and geometry changes, accompanied with localized deformation and material degradation. At the same time, full 3D modelling of cracks provides a more realistic prediction tool for studying true 3D structures, as well as local features like crack tunneling, e.g. [1].

There is an extensive literature on modelling cracks in general. They can either be modelled in a continuous way, by degrading and/or deleting elements, or by introducing a true discontinuity. A discontinuity can be implicitly modelled by element or nodal enrichment [2–14]. However, most of these methods are applicable for small displacements and cannot be directly applied for large deformations. In a second category of discontinuous approaches remeshing is used to explicitly model the discontinuity, i.e. by alignment of the mesh with the crack and nodal decoupling perpendicular to the crack [15–23].

Here we concentrate on the second category and extend a continuum damage mechanics approach to 3D crack initiation and propagation. Along these lines, Mediavilla et al. [22] suggested a continuous-discontinuous methodology for modeling cracks in 2D problems, in which the crack geometry is incorporated in the mesh by frequent remeshing. This algorithm is attractive especially when dealing with ductile failure, where large local deformations occur and remeshing is necessary even for the continuous part of the problem. Incorporating the additional geometrical changes due to crack growth then requires only a limited intervention in the algorithms used.

In this study, we develop an extension of Mediavilla et al.’s remeshing strategy to 3D problems in which damage growth and 3D crack propagation occur in a large deformation setting. Remeshing is used to deal with geometrical changes due to large deformations as well as crack growth [22]. Crack initiation and crack growth are governed by a continuum damage model which is intrinsically coupled to the underlying elasto-plastic constitutive model. The damage formulation is nonlocal (of the implicit gradient type) to ensure proper localization properties [24]. Once the damage reaches a critical level somewhere in the geometry, a discrete crack is introduced in the geometrical description of the body. This crack is extended when the damage field at its front becomes critical, whereby the orientation is governed by the direction of maximum nonlocal damage driving variable. As a result, no additional fracture criterion is required to control the crack growth. The crack surface is constructed by computing the propagation direction and distance for each node on the crack-front. By splitting the nodes along the crack surface, discontinuities are allowed along the element faces. Robustness of the simulations is ensured by temporarily applying the element internal forces as external forces on the crack nodes and gradually reducing them to zero.

In 3D, compared with the 2D case considered by Mediavilla et al. [22], the remeshing strategy which we follow presents a number of important additional computational challenges. (1) A reliable tetrahedral finite element is required to enable robust automatic remeshing of complex geometries. We adopt a bubble-enhanced mixed finite element formulation of the continuum model [25, 26]. (2) An accurate transfer operator is required to map history data from one mesh to the next. Here special precautions need to be taken to ensure consistency between the transferred fields [27, 28]. (3) Algorithms are needed to manipulate the 3D geometrical description of the problem upon initiation of a crack, as well as for every increment of crack growth. This is the main topic of the present paper.

The algorithm developed here is based on a geometrical description by a surface mesh, which is adapted according to the computed nonlocal damage field. To initiate a crack, elements with damage values higher than a critical limit are first identified. They form a cloud which is either completely inside the body or in contact with a surface. For internal clouds we use an averaging technique to compute the center of the cloud. This point is taken as the center of the emerging crack surface. Using the damage distribution, a plane is defined for inserting a discontinuity. For clouds which are in contact with an external boundary, a crack-front is constructed and this front is connected to the external surface by a discontinuity surface. When crack propagation is predicted by the damage evolution ahead of a crack, that part of the surface mesh which represents the crack faces is extended. For this the strategy followed in 2D by Mediavilla et al. [22] is applied in planes perpendicular to the crack front. Care needs to be taken to ensure the consistency of the crack front and to respect the outer surface of the body. At all stages of the simulation, the damage field is also used in order to refine the discretisation in critical regions of the geometry. We illustrate the methodology by showing two numerical examples, one illustrating crack initiation inside a body (i.e. a rectangular bar under tension) and one at the surface (of a double notched specimen).

This paper is structured as follows. In the next section, the continuum damage model, element technology, remeshing and transfer are briefly reviewed. We then first present the 3D crack propagation algorithm, since elements of it are used in the crack initiation algorithm, which is subsequently discussed for internal as well as surface cracks. After presenting two numerical examples, we conclude by highlighting the newly added features of the algorithm.

## Continuum model and finite element discretisation

### Continuum nonlocal damage model

*z*is a local damage driving variable, the evolution of which depends on the effective plastic strain \(\varepsilon _{p}\) and the (effective) stress triaxiality \(\hat{\tau }_{h}/\hat{\tau }_{eq}\);

*A*and

*B*are material constants as introduced by Goijaerts et al. [30]. Equation (9 10) uses the local damage driving variable

*z*together with a Neumann boundary condition (with normal \(\vec {n}\)) to compute a nonlocal damage driving variable \(\bar{z}\), which controls the damage evolution. The use of this nonlocal quantity is necessary to regularise the localisation of deformation and damage, which would otherwise become pathological [31]. The final link to the damage evolution is made via a history variable, \(\kappa \), and the evolution law (11).

### Finite element form of the equations

### Remeshing

Our strategy to deal with 3D crack growth, as well as the large deformations which we wish to model, necessitates frequent remeshing on a global level. After a predefined number of increments, the surface mesh of the 3D body is extracted from the model. If crack growth is detected, the surface mesh must be modified to incorporate a new crack segment, see the next section. Otherwise, the existing surface mesh is used as input to the 3D tetrahedral mesh generator TetGen [32], together with an indicator field for the desired element size. The remeshing aims to produce smaller elements in areas where the damage evolves significantly and larger elements in undamaged regions or regions where the damage growth has stopped. The damage rate is used as a pointwise indicator to set the element size. Elements with the largest damage rate have the smallest volume and vice versa. For the intermediate damage rates, the volume of the elements is interpolated between the maximum and minimum values, proportional to their damage rate.

*a*and

*b*in Fig. 2). Vertex

*a*and the two adjacent faces vanish from the topology. Vertex

*b*is moved to a new position

*c*which is the midpoint between

*a*and

*b*. After collapsing an edge, we measure the dihedral angle between the neighboring newly produced faces and if overlapping occurs, the edge collapse is canceled and the original surface is recovered. This process is repeated until the desired coarsened surface is produced.

After remeshing, data which are available on the Gauss points of the old mesh are transferred to the Gauss points of the new mesh. For this we first use global smoothing, i.e. continuously interpolated, piecewise linear fields are determined which fit the integration point data best in a least squares sense. These data are subsequently interpolated at the new nodal coordinates and finally, using the element shape functions, the new Gauss point data are retrieved. In order to ensure a robust transfer, only a minimum set of data is transferred and the remaining data are reconstructed using the constitutive equations. This operation, which is an indispensable ingredient of the remeshing algorithm, is explained in detail in Ref. [28]. After transfer and reconstruction, balancing iterations are done to restore global equilibrium in the finite element sense. Since these iterations are not representative for any physical deformation, the material behavior is assumed to be elastic in order to guarantee convergence. Finally, because of the elastic nature of the balancing iterations, it is checked if the stress state obtained by them is on or inside the yield surface; otherwise the yield surface is corrected to restore yield consistency.

## Crack propagation

In computational fracture mechanics, a critical stress intensity factor or J-integral value is typically used as a criterion for crack growth. In addition, a maximum hoop stress (MHS), minimum strain energy density or maximum energy release rate criterion is used to determine the crack growth direction. A different approach is employed in this study, where the evolution of the continuum damage variable, \(\omega _{p}\), governs the propagation of a crack. This has the advantage that crack initiation and propagation can be dealt with using the same (continuum) equations and no separate fracture criteria are necessary. Once the crack has been initiated, it follows the damage evolution ahead of it wherever the damage has become critical, i.e. \(\omega _{p}=1\). This concept has been successfully applied to 2D crack growth simulations in shear dominated problems like blanking [17, 22].

This section summarizes the required steps for extending these algorithms to 3D problems. In 2D, the crack-front is a point, whereas in 3D it is a curve. For each node lying on this curve, a growth direction is determined in a plane perpendicular to the front. By using the nonlocal damage driving variable field in this plane, a direction vector is computed for all nodes lying on the crack-front. Using all these vectors, the extended crack surface is constructed. We discuss the numerical treatment of crack initiation in the next section, since it employs concepts developed here for crack propagation.

### Crack propagation direction and distance

At each converged loading increment, the damage at a point lying on the crack-front is compared to the critical damage, \(\omega _{p}^{c}=0.99\), on the basis of which the crack is extended (or not). This value has been found to be sufficiently close to the theoretical value of \(\omega _p^c = 1\) to ensure that most of the energy dissipation due to damage growth has taken place and the stress level has dropped sufficiently for it to not to be affected significantly by the insertion of a new crack segment. For a detailed study of the influence of these numerical parameters, in 2D, we refer to Ref. [22].

Using a tetrahedral discretisation of the 3D geometry, crack-front points coincide with finite element nodes. The damage variable is extrapolated from Gauss points to these nodes using a global smoothing procedure, i.e. a continuous, piecewise linear field is determined which fits the integration point data best in a least squares sense. The crack is predicted to grow over a distance which depends on the damage field ahead of the considered crack node and its direction is evaluated differently at the crack-front compared to the crack-front corners. Both cases are therefore explained separately below, followed by the distance by which the crack is extended.

#### Propagation of a crack-front node

*o*in Fig. 4, is used as the normal to this reference plane. This normal is determined from the discretised crack-front as follows.

*o*, the vectors \(\vec {v}_{1}\) and \(\vec {v}_{2}\) are the vectors connecting the considered crack-front vertex to its neighboring vertices in the discretised geometry. The tangent vector is then computed as

*N*points in a semi-circle located in the reference plane. A comparison has shown that using the nonlocal damage driving variable instead of the damage variable as used by Mediavilla et al. [22] avoids abrupt changes in the crack growth direction due to small local (numerical) variations between adjacent nodes. Vectors \(\vec {d}_{1}\) and \(\vec {d}_{2}\) in Fig. 4c are obtained from the intersection of the reference plane with the tetrahedral crack face edges of the discretised geometry. These two vectors are used to compute the vector \(\vec {d}\) that sets the central direction of the considered semi circle via

*N*different angles ranging from \(-\frac{\pi }{2}\) to \(\frac{\pi }{2}\). For each \(r_{i}\) an angle \(\theta _{i}\) is defined as (Fig. 4c)

#### Propagation of a crack-front corner

*N*different angles \(\theta _{j}\), ranging from \(-\pi /2\) to \(\pi /2\) are selected. A piecewise linear curve is obtained for each of these planes by intersecting it with the outer surface. Along these curves the nonlocal damage driving variable \(\bar{z}\) is sampled at four different distances \(r_{i}\) measured along the piecewise linear curve. For each \(r_{i}\) (the same sampling distances are used as in the previous section but then relative to the crack-front corner) there exists a plane with normal vector \(\vec {n}_i\), where \(\bar{z}\) has its maximum on the intersection line of this plane with the external surface–cf. Eq. (21). Finally using Eq. (22), the average of these normals constitutes the growth direction for the crack-front corner.

#### Directional smoothing

*k*on the crack-front is combined with that of the adjacent nodes using the following smoothing operation:

#### Growth distance

*k*at which the critical damage value \(\omega _{p}^{c}\) is exceeded, the crack is assumed to grow in the computed direction over a distance \(L_{k}\) until the damage drops below \(\omega _{p}=0.97~\omega _{p}^{c}\). To obtain a smoother crack surface for more stable (re)meshing and computation, we furthermore set a minimum and maximum growth distance as follows: \(L_{min}~=~0.1~\Delta a \); \(L_{max}~=~\Delta a\). This implies that for a point \(p_{k}^{o}\) on the old crack-front, the corresponding position on the new crack-front \(p_{k}^{n}\) is obtained as follows:

### Construction of the new crack surface

The propagation direction and distance have now been computed for all nodes on the crack-front. Next step is to construct a new segment of the crack surface, along which the crack will be opened. First, the intersection of the new crack segment with the outer surface is determined. This procedure, which is schematically shown in Fig. 6, ensures that the crack surface remains properly connected to the outer surface.

In order to modify the surface, the computed crack extension direction plane for the crack-front corner is intersected with the triangular outer surface elements. Starting from the old crack-front corner, surface elements are split along the direction plane until the predicted growth distance has been reached. Triangle edges which are cut by the direction plane are split by adding a node and the triangle is divided into two triangles, see Fig. 6. If the intersection point is within a certain distance (namely a tolerance which here is 0.1 times the element edge) of an existing edge or node, the node or edge is mapped onto the crack direction plane. This avoids the creation of excessively small surface elements. If the crack extension direction exactly passes through a node or an already available edge, then no modification is made. This process is repeated until the predicted growth distance is reached. If the new crack-front corner is inside a triangle, then this triangle is divided into three triangles and the node is stored as the new crack-front corner.

### Meshing of the new geometry

The constructed crack surface based on the crack propagation distances and directions is now used to discretise the geometry. The geometrical description consists of the outer surface of the volume, possibly including parts of the already existing crack surface, and an inner surface which defines the new crack growth segment. The new crack surface is treated as an internal boundary by the mesher, so that tetrahedral elements are generated on both sides of the surface without intersecting it.

In order to properly model the opening of the crack surface, a topological data structure is needed. This data structure is built using the connectivity of the elements and geometry of the discretised domain. Using this data structure, the elements connected to each node and their position with respect to the crack surface are identified. Details of this data structure are given in the Appendix.

### Crack opening

The mechanical insertion of the new crack surface is done by splitting the nodes generated on the new crack surface by the volumetric mesher. This implies that for each node, a corresponding node with the same coordinates is generated. The nodal connectivity of elements located on both sides of the crack is preserved, whereby the new node is used for the connectivity of the elements for one of the sides.

The two newly created surfaces are temporarily tied together by creating a dependency between their displacement degrees of freedom. While the crack is still closed, data from the last converged state is consistently transferred to the new mesh. Elastic equilibrium iterations are done in order to recover global consistency. During this iterative process the closed crack is treated as a new surface for Eqs. (15.3) and (16). However, the degrees of freedom for the pressure and the nonlocal damage driving variable are not tied. This improves stability of the simulation in the sense that the residual forces related to these two equations become zero in the elastic equilibrium iterations and artificial damage growth is prevented. This artificial damage growth, which is observed if all degrees of freedom are tied, may be caused by the sudden change in the boundary conditions for the nonlocal averaging Eq. (9 10).

Since the new crack surface is kept closed during the elastic equilibrium iterations using displacement tyings acting on the crack faces, a reaction force appears on these nodes. To mechanically open the crack, these reaction forces are first applied as external forces when the tyings are removed, and they are subsequently gradually released in a number of sub-steps, see Fig. 9 for an illustration (in which \(\vec {f}_{A}\) and \(\vec {f}_{B}\) represent these forces for one particular couple of nodes) and Ref. [22] for a more detailed description.

## Crack initiation

### Internal crack initiation

### Surface crack initiation

In some cases, a cloud of interconnected damaged elements contains nodes lying on the exterior surface of the geometry. If this is the case, a crack should nucleate from the exterior surface and propagate into the geometry with a proper propagation direction. For this purpose, the triangulated surface is modified to embed the new crack surface.

### Crack opening

A crack surface has been defined at this stage for the cracked topology. This surface should be opened to recover equilibrium first. The applied methodology is explained here for cracks located inside the body.

A similar technique is applied to open cracks in contact with the boundary of the geometry. The difference here is that the new crack front residing on the boundary is also opened.

## Examples

The developed algorithm has been employed for studying crack initiation and propagation in two examples. These examples have been selected in order to assess the performance of the methods developed above in dealing with crack initiation and propagation.

Shear modulus | 80.19 GPa |

Bulk modulus | 164.21 GPa |

Initial flow stress \(\tau _{\mathrm {y}0}\) | 0.45 GPa |

Residual flow stress \(\tau _{\mathrm {y}\infty }\) | 0.715 GPa |

Linear hardening coefficient | 0.129 GPa |

Saturation exponent \(\alpha \) | 16.93 |

Damage initiation threshold \(\kappa _\mathrm {i}\) | 0 |

Critical value of history parameter \(\kappa _\mathrm {c}\) | 0.4 |

Intrinsic length \(\ell \) | 1 mm |

Damage parameter | 3.9 |

Damage parameter | 0.63 |

The described constitutive law is implemented using a locking free mixed formulation of the tetrahedral element [25, 26], while a constant damage variable \(\omega _\mathrm {p}\) is used per element. In both examples, a vertical displacement is applied on the top surface of the model while the bottom surface is fixed. Frequent remeshing is used to maintain the quality of the elements and the damage rate \(\dot{\omega }_\mathrm {p}\) is employed as a point wise indicator for element size. Hence, the mesh is more refined in regions with a rapid evolution of damage.

### Crack initiation in a rectangular bar

As the necking progresses in the middle section of the specimen, a cloud of connected elements reveal a damage value higher than the critical level \(\omega _{p}^{c}\), as shown in Fig. 16a. The internal crack initiation algorithm is used to introduce a crack plane internally, see Fig. 16b. The geometry is therefore remeshed and by releasing the crack surface forces, a first crack appears inside the geometry.

### Surface crack initiation and propagation in a double notch specimen

*N*and the crack increment length \(\Delta a\) are 50 and 0.3 mm respectively.

## Conclusion

A large deformation 3D methodology has been developed to simulate the initiation and propagation of a crack in a ductile material, based on an underlying ductile damage mechanics formulation and a remeshing strategy.

An approach is presented to initiate a crack in 3D bodies undergoing large plastic deformations. Cracks start either internally or at the surface of the geometry, whereby a procedure is proposed for each case. In contrast to a traditional fracture mechanics approach, the size and direction of crack initiation and propagation are solely governed by the underlying damage model, and no extra criterion is therefore required.

Once a crack has been nucleated, it may propagate according to the damage field ahead of the crack tip. For each of the nodes on the current crack-front, a propagation direction and distance is computed. Depending upon the location of the node on the crack-front (corner or mid nodes), a slightly different method is used to identify the propagation vector. These propagation vectors, together with the old crack-front, are assembled to construct a new crack surface segment. The geometry is then discretised and refined in critical locations based on the damage rate.

The performance of the proposed method is shown by two examples where both cases for initiation/propagation of a crack (at the surface or internally) are demonstrated. Our results show that the method is promising in studying phenomena like internal fracture and other relevant applications. The characteristics of the proposed algorithm renders it promising for modelling 3D cracks in applications where remeshing is unavoidable. It presents two essential advantages over a conventional fracture mechanics approach: first, it uses only a single criterion (damage model) for both crack initiation and propagation (distance and direction) and, second, the mechanical strength of the structure has been already degraded by the damage, making it more convenient to introduce a crack.

## Declarations

### Author's contributions

HRJ carried out most of the study, including development of the methodology and its implementation, and drafted the manuscript. RHJP and MGDG conceived the study, participated in its design and coordination and critically reviewed the manuscript. All authors read and approved the final manuscript.

### Acknowledgements

This research was carried out under project number MC2.05205c in the framework of the Research Program of the Materials innovation institute M2i (http://www.m2i.nl).

### Competing interests

The authors declare that they have no competing interests.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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