Mesh refitting approach: a simple method to model mixedmode crack propagation in nonlinear elastic solids
 Y. Sudhakar^{1}Email author and
 Wolfgang A. Wall^{1}
DOI: 10.1186/s403230170088x
© The Author(s) 2017
Received: 30 August 2016
Accepted: 5 June 2017
Published: 24 June 2017
Abstract
We devise a finite element methodology to trace quasistatic throughthickness crack paths in nonlinear elastic solids. The main feature of the proposed method is that it can be directly implemented into existing large scale finite element solvers with minimal effort. The mesh topology modifications that are essential in propagating a crack through the finite element mesh are accomplished by utilizing a combination of a mesh refitting procedure and a nodal releasing approach. The mesh refitting procedure consists of two steps: in the first step, the nodes are moved by solving the elastostatic equations without touching the connectivity between the elements; in the next step, if necessary, quadrilateral elements attached to crack tip nodes are split into triangular elements. This splitting of elements allows the straightforward modification of element connectivity locally, and is a key step to preserve the quality of the mesh throughout the simulation. All the geometry related operations required for crack propagation are addressed in detail with full emphasis on computer implementation. Solving several examples involving single and multiple cracks, and comparing them with experimental or other numerical approaches indicate that the proposed method captures crack paths accurately.
Keywords
Mesh refitting method Crack propagation in nonlinear materials Jintegral Nodal releasing technique Fracture of nonlinear solidsBackground
Objective and motivation

to update the existing large scale structural mechanics solver into a robust tool to handle single and multiple quasistatic cracks

to couple the crack propagation method with existing FSI approach to model FSFI.
Brief overview of relevant methods

Adaptive remeshing

Enriched partition of unity
Adaptive remeshing methods
These methods, as the name implies, adaptively refine the mesh in the crack tip vicinity where the solution dictates the dynamics of crack propagation, and coarsen the mesh away from the crack tip. They make use of special data structures [2, 3, 15], together with either a globally adaptive remeshing procedure [4–6] or a local mesh modification algorithm [7] to accommodate crack propagation at arbitrary directions within the computational domain. Each time when a crack extends, these methods introduce several new nodes into the mesh. As a result, they require mesh generation related algorithms to modify the mesh appropriately. Usually in a large scale FE code which is generalized to address multiscale and multiphysics problems, such fracturespecific and meshmodification routines are neither available nor easy to implement in a generalized way.
Enriched partition of unity methods (EPUM)
These methods represent the recent developments in computational fracture mechanics. They include extended finite element methods (XFEM) [16–19] and generalized finite element methods (GFEM) [20, 21], and these class of methods are originally developed with an objective to eliminate the adaptive remeshing and its associated complex and timeconsuming operations. The fundamental idea behind this method is to enrich the finite element solution space with additional problemspecific enrichment functions. The crack can propagate within the interior of an element, and hence it is possible to simulate crack propagation without modifying the underlying discretization. Though these methods are demonstrated to be powerful, the following points hamper their easier implementation into an existing structural mechanics solver. The numerical integration of singular enrichment functions is still an active area of research in EPUM [22–28]. Moreover, these methods require the implementation of complicated geometrymesh intersections for which robustness is always an issue, especially in 3D. Also, the number of degrees of freedom attached to a few nodes changes each time when the crack advances. Most importantly, except a few (e.g. [29–31]), all studies are focused on crack propagation through linear elastic materials mainly because the formulation of enrichment functions in nonlinear regime is still an active area of research.
Owing to the aforementioned implementation issues, neither EPUM nor adaptive remeshing methods are ideal for developing fluidstructurefracture interaction methods. This is because FSFI methods have to handle the combination of challenges from two sources: those arising from crack propagation besides another big challenge from FSI. The aim of this work is to devise a simple crack propagation approach that is suitable specifically for developing FSFI.
The method developed in this work, as will be explained later, shares a similarity with arbitrary Lagrangian Eulerian (ALE) based methods that it involves a meshdeformation step. Therefore, ALE based methods for fracture mechanics are briefly recalled. The use of ALE in computational fracture mechanics is not widespread. Only few studies employed ALE to address crack propagation problems, and a brief account of majority of such works is provided below.
Existing ALE based crack propagation methods
To summarize, to our knowledge neither a complex trajectory of a single crack nor simple propagation of multiple cracks within a material is modeled until now using ALE based methods. This is predominantly due to the fact that the mesh modification method used in ALE, in its classical sense, cannot handle the mesh topology changes that are introduced by advancing a crack through the FE mesh. Continuous remeshing is mandatory to eliminate the associated mesh tangling problems. In this work, this is avoided by using an additional step in the mesh refitting procedure that allows to modify the element connectivity locally to preserve the quality of mesh, as will be explained in the later section.
Structure of the paper
The remainder of the paper is structured as follows. The next section briefs the governing equations and the boundary conditions for the problem. Then the complete crack propagation algorithm together with all the details necessary for computer implementation are presented. Finally several numerical examples to demonstrate the accuracy of the proposed method are described.
Governing equations
The mesh refitting approach
The complete numerical methodology, together with the computer implementation aspects, of the present approach are presented in this section. It is assumed that the fracture behavior of the material is completely characterized by the Jintegral.
The focus of the present work is to simulate throughthickness mixedmode quasistatic crack propagation within a structure. The current work can be considered as an extension of Tabiei and Wu [40] which describes the implementation of a crack module in DYNA3D FE package, and shares similarities with the method of Miehe and Gürses [3]. Both works address crack propagation through linear elastic materials only. Moreover the complex geometry related operations, like deciding the new crack tip nodes, are not addressed in depth. These details are crucial for implementation of the method. The approach proposed in [3] was called an radaptive method, but in order to avoid confusion with complex radaptive mesh redistribution methods [41, 42], the present method is labelled as mesh refitting approach. The following section presents the complete implementation details of the present method. Though we simulate throughthickness cracks, for simplicity we explain the geometryrelated operations in 2D.
Solve the governing equations
The first step is to solve the structural dynamic equations by freezing the location of the crack. The strong form given in Eq. (1) is multiplied by appropriate test functions (\(\delta \mathbf d ^s\)) and are integrated over the structural domain to obtain the weak form which is stated as,
Perform computational crack propagation procedure
The displacement solution obtained from the previous step is used to perform crack propagation procedure by computing vector \(\mathbf J \)integral. It involves seven discrete steps, each of which are detailed below.
Step 1: Construct local coordinate system at crack tip
To compute fracture mechanics quantities from the FE solution, and to decompose these quantities into their corresponding modes in a mixedmode problem, it is essential to construct a local coordinate system (\(\underline{\xi },\underline{\eta }\)) at the crack tip (x \(_c\)) as shown in Fig. 2. The base vectors (\(\mathbf e _1,\mathbf e _2\)) associated with (\(\underline{\xi },\underline{\eta }\)) can be easily constructed because \(\mathbf e _1\) is the symmetry line of the crack; \(\mathbf e _2\) can be obtained by computing the normal to \(\mathbf e _1\) in a right hand coordinate system.
Step 2: Compute vector \(\varvec{J}\)integral
In order to construct q, using nodal connectivity information, all the elements that are located on \(n\)layers around the crack tip (see Fig. 3b with \(n=4\)) are considered. From this, the elements connected to the crack tip are deleted. Then, all the nodes that are on the outer boundary of this element set are located, and among these nodes, the one which has the shortest distance (\(r_{\mathrm {min}}\)) from the crack tip is chosen. Then the support function is initialized to take a value of unity at the inner layer of nodes, and drops smoothly to zero when the distance of a node from crack tip is more than or equal to \(r_{\mathrm {min}}\). The distribution of q within the integration domain is given in Fig. 3b.
Note In this work, we assume that the crack surfaces are tractionfree. However, in FSFI applications, fluid loads are acting on the crack faces, and as a result an additional term appear in the computation of Jintegral. This is explained further in [1].
Step 3: Check crack propagation criterion
The crack propagation criterion determines whether the existing crack propagates through the structure under the current stress state. Crack propagation occurs when the driving force reaches or exceeds the material resistance. The Jintegral provides a measure of driving force, and its critical value (\(J_c\)) is assumed to be a material property, which quantifies the material’s resistance to crack propagation.
Step 4: Obtain the direction of crack propagation
After confirming that the crack propagates, the next logical step is to determine along which direction it is going to advance through the material, which is provided by the crack kinking criterion.
Step 5: Find new crack tip nodes
Having computed the crack propagation direction from \(\varvec{J}\), the next essential step is to determine the new crack tip nodes i.e, nodes in the FE mesh through which the crack must be propagated. A geometrybased method is used in this work to identify the new tip nodes.
The first step is to identify all the elements that are connected to the current tip node (Fig. 4). Among the edges of these elements, the edge that is intersected by the propagation vector is found. This intersecting edge is drawn with a thick continuous line in Fig. 4. Then the angles formed by the line joining the current tip node to the edge nodes, and the crack propagation direction are calculated (\(\phi _1\) and \(\phi _2\) in figure).
After getting the required intersecting edge, the next step is to check whether the crack propagates along the diagonal. This is realized by the condition, \(\phi _1\le \) diagtol (\(=\)0.25 radians in all the simulations). In this case, the diagonal node corresponding to \(\phi _1\) is marked as new tip node. Since the crack propagates through a diagonal of the element, this element must be split along this diagonal to accommodate crack propagation through the mesh, as explained in the next step.
If \(\phi _1>\) diagtol, then the nondiagonal node of the intersecting edge will be the next new tip node. In either cases, all the new tip nodes (if there are multiple cracks, each crack tip will have its own new tip node) are stored in \(\mathcal {R}_{ale}\). Moreover, the distance between the intersection point and the new tip nodes, marked as \(\varvec{\delta }_{ale}\) in the figure, are computed and will be used in the next step.
Step 6: Mesh refitting procedure
 1.
Nodal repositioning
 2.
Splitting quadrilateral (Quad) elements into triangular (Tri) elements
where \(\varvec{\sigma }^m\) is the fictitious Cauchy stress tensor, \(\mathbf d ^m\) denotes the displacements at each node within the mesh, \(\Omega ^s\) represents the whole structural domain, and \(\partial \Omega _{ale}\) denotes the boundary for ALE computations: \(\partial \Omega _{ale}=\Gamma ^s_D\cup \Gamma ^s_N\cup \Gamma ^s_c\). Displacements at the new crack tip nodes are set to be \(\varvec{\delta }_{ale}\) that is computed in the previous step.
The above equations are solved to obtain the mesh displacement \(\mathbf d ^m\), which is used to move each node in the mesh to its new location. After this mesh movement operation, the new crack tip node is moved to the intersection point along the intersection edge (see Fig. 4).
In the next step of the mesh refitting procedure, the Quad elements that are marked to be split are cut into two Tri elements. This happens when the crack propagates very close to the diagonal of a Quad element (Fig. 5a). This process does not involve introducing new nodes into the mesh. By comparing Figs. 4 and 5a, the effect of the mesh refitting procedure is clear: the new tip nodes are first moved to the desired location using the nodal repositioning step, and then the Quad element is appropriately split into Tri elements to enable crack propagation along the diagonal. In short, the combination of nodal repositioning and element splitting ensure that after the mesh modifications, the crack propagates along an existing edge in the new mesh.
One of the main reasons for the failure of ALE based methods in handling large deformation or topology change is that such methods maintain their nodal connectivity during the entire simulation. The element splitting operations used in the present work alleviates this problem by enabling us to modify the connectivity between the elements locally. This is an essential step without which the nodal repositioning method cannot handle the change in mesh topology that is inherent to crack propagation problems, without resorting to complicated and timeconsuming remeshing procedures.
Step 7: Nodal releasing technique
The two previous steps have enabled us to identify new tip nodes, and to move these nodes to match the computed propagation angle. However, the material separation is not yet included within the FE procedure. In order to achieve this, and to form physical crack surfaces, the nodal releasing technique is used.
In order to represent the material separation, the element connectivity at the current tip node must be modified; a duplicate node is created at the same location where the current tip resides. Few elements are released from the current tip node, and are assigned with a new duplicate node. This, in turn, generates new crack surfaces. In order not to destroy the FE mesh during this process, a consistent way of determining which elements get duplicate nodes is used.
In this procedure, two angles are defined: one is \(\phi _p\) already defined in Fig. 4, and the other is the angle formed by the negative normal at crack tip to the propagation vector (\(\phi _{nn}\) in Fig. 5a). Then, for each element, the angle (\(\phi _g\)) formed by the line connecting current tip to the centroid of the element and the normal is computed. The element is released and gets the duplicate node, if \(\phi _g\notin ~[\phi _p,\phi _{nn}]\). The elements that retain the current tip node are shaded in Fig. 5a. After nodal releasing and modifying element connectivity, the mesh close to the crack tip is plotted in Fig. 5b. At this point, the material separation is introduced and all the crack propagation operation are completed.
Numerical examples
Several examples of varying complexity are solved to demonstrate the effectiveness of the proposed method. These examples exhibit single and mixedmode behavior, involving mono and multimaterials. In order to closely examine the accuracy of the method, crack paths obtained from the present method are compared with experiments or results obtained from other methods in literature.
The first two examples consider stationary cracks, and the quantities calculated are compared with XFEM studies [30, 31]. These examples consider highly nonlinear effects evident from the crack tip blunting observed in the results. All the other examples involve complex crack propagation through the structure.
Crack tip blunting
When the material deforms, the crack surfaces move apart, and the initially sharp crack will blunt significantly due to the material nonlinearity. The deformed configuration of the structure is shown in Fig. 6b. The vertical displacement of crack surface nodes are plotted against their horizontal position in the reference state in Fig. 6c; for comparison, XFEM simulation results are taken from [30]. It is directly evident that the results obtained from our simulations are matching well with the reported results. Moreover, the simulations using coarse and fine mesh yield identical values, which shows that the reported results are converged with mesh density. The coarse and fine mesh contain 1200 and 4800 uniform Cartesian elements respectively. It is to be mentioned that the XFEM study [30] was focused on incompressible materials, and the present results closely resembles the incompressible condition by taking \(\nu =0.49\).
Jintegral computation
Jintegral is a crucial parameter in our work because both the crack propagation and crack kinking criterion are entirely based on this single parameter. In order to study the accuracy of Jintegral evaluation, we consider an edge crack specimen under simple extension. A 2 mm \(\times \) 2 mm plate with \(\mu ^s=0.4425\,\hbox {MPa}\) and the equivalent Poisson ratio in the linear regime \(\nu =0.49\) is taken, and the crack occupies halfwidth of the specimen. The strain energy function and boundary conditions are same as that of the previous example. These details are taken from an XFEM study [31], which reports the value of Jintegral for large stretch ratios (\(\lambda \)). \(\lambda \) is defined as the ratio of deformed length to the original length. We intentionally chose [31] as the reference for our validation because comparing Jintegral values at high \(\lambda \) could be challenging.
It can be seen from Fig. 7a that the Jintegral values computed from our method are in excellent agreement with the XFEM results [31] even at large \(\lambda \). The coarse and fine mesh indicated in Fig. 7a contain 728 and 1600 uniform Cartesian elements respectively. At \(\lambda =2.5\), the difference between Jintegral values computed using the coarse and fine mesh is only 1.3%. At the same \(\lambda \), the difference between the value reported in [31] and the present simulation using the fine mesh is as low as 3.4%. This quantitative comparison shows that the Jintegrals computed in our work are very accurate even at large \(\lambda \).
The variation of J with respect to the number of layers chosen around the crack tip as the integration domain is given in Fig. 7b. It can be seen that after initial changes, the value of J is stable after 5 layers, after which only minute variations exist in J. In all the simulations presented in this work, 5 layers of elements around the crack tip are chosen to compute the Jintegral.
Having simulated a stationary crack, the remaining examples consider complex crack propagation through the structure. The comparison between the present results and the results obtained from the literature demonstrates the accuracy of the method. For all the following examples, the strain energy function is given by Eq. 4.
Single edge cracked plate under mixedmode loading
In this example, as shown in Fig. 8a, the plate which is fixed at the bottom edge, is subjected to shear stress \(\tau =1\) on the top. The initial edge crack length is half of the plate width. The Lame parameters are set such that \(E=30\) MPa and \(\nu =0.25\). The computational domain is discretized with 2736 elements with the whole area of crack propagation discretized with a fine mesh.
In this simulation, upon loading the crack propagates along a slightly curved path until it reaches the other end of the plate. In order to provide a detailed comparison, a zoomed view of crack path is plotted in Fig. 8c, and the result obtained from the present simulation is compared with two other studies: one utilizes a meshless method [53], and the other study is based on adaptive FEM [6]. It can be seen that the predicted crack tip trajectory matches very well with the results obtained from the other two studies.
Crack in a drilled plate
As reported in [7], the crack path follows different trajectories based on the choice of a and b, which are described as follows.
Simulation1
The location of the initial notch is given by \(a=5\) and \(b=1.5\) mm. The crack is initially attracted towards the bottom hole, propagates near this hole, and got deflected away to end in the middle hole as shown in Fig. 9b. This is in accordance with the experimental results of [7], and other numerical studies [3, 6, 54]. Comparison with the experimental results show that the present simulation produces very good results; even the crack deflection near the bottom hole is predicted well in the simulation as can be directly seen from Fig. 9b. This is one of the very challenging validation test cases, owing to the complex crack tip trajectory involved. The developed methodology can be said to be accurate as it produces results that are matching very well with the experimental values even for this complex configuration.
Simulation2
In this example, for which \(a=6\) and \(b=1\) mm, the crack is attracted towards the middle hole, and directly ends in it (Fig. 9c). There are no crack deflections observed, and for this example as well, the results match excellently with the experiment (Fig. 9c).
Four point beam with two notches
Geometric parameters defining notch location for Bittencourt’s drilled plate problem shown in Fig. 9a
Simulation  a  b 

1  5.0  1.5 
2  6.0  1.0 
The crack paths through the FE mesh is given in Fig. 10b. To demonstrate the accuracy of the simulation, the crack paths obtained from the present method are compared with the results reported using a meshless method that incorporates crack tip singular fields as enrichments [56]; results presented for the finest meshless node distribution is used for the comparison. The comparison of crack paths is plotted in Fig. 10c. It can be seen that for both crack tips, the tip trajectory obtained from the present simulations matches very well with the reported value.
Crack deflection due to inclusion
Crack growth in the presence of an inclusion is studied in this example. Geometry, loading, and boundary conditions are given in Fig. 11a; they are taken from [57]. The configuration consists of a rectangular plate which contains an offcentre circular inclusion. The Lame parameters of the plate are set such that \(E_{plate}\) \(=\) 20 MPa and \(\nu \) \(=\) 0.3. The objective of this study is to check whether the method is capable of accurately predicting the influence of this inclusion on crack propagation, which is already reported in [52, 57].
The inclusion is characterized by the ratio of Young’s modulus of the plate to that of the inclusion (\(r=E_{plate}/E_{incl.}\)). Two values are considered; \(r=10\) which means that the Young’s modulus of the inclusion is 10 times lower than that of the plate which is referred to as “soft” inclusion, and \(r=0.1\) that is referred to as “hard” inclusion. The Poisson ratio is assumed to be the same as that of the plate. The whole structural domain is discretized with 3213 elements.
The effect of the inclusion on the crack tip trajectory is shown in Fig. 11b. For soft inclusion, the crack is attracted towards the side of the inclusion; however, the crack does not end in it. In case of a hard inclusion, the crack deflects away from it. These observations are consistent with the already reported results [52, 57].
Nonlinear elastic plate with a hole
The above examples considered crack propagation with little material nonlinearity. This is evident from the fact that the crack remains sharp even after several propagation steps. The following example considers crack propagation involving high material nonlinearity under large deformation. A small offcentre hole is introduced in the geometric configuration considered for the first example, and this simulation allows the crack to propagate through the material. The Lame parameters are set to yield \(E=10\) GPa, \(\nu =0.3\); critical Jintegral, \(J_c=50\) kJm\(^{2}\). The top surface is subjected to a displacement of 0.5 mm. The geometric configuration of this example is presented in Fig. 12a.
As with the linear elastic examples, the loading (or the corresponding Dirichlet boundary condition here) is increased very smoothly from the zero initial value so that the influence of inertia is neglected. When the material starts deforming, as expected, the crack starts to blunt, and the Jintegral value starts to increase. When J reaches \(J_c\), then the crack starts to propagate; the deformed configuration of the structure at which the crack starts propagating is depicted in Fig. 12b. Due to the presence of the hole, the crack slightly deflects upwards, as can be seen from Fig. 12b, c. From all these plots, one can infer that the crack tip is always blunt owing to the material nonlinearity, and the present method is able to model fracture behavior in such scenarios.
Conclusion
A finite element methodology to model mixedmode crack propagation through nonlinear elastic materials is proposed in this work. The striking feature of this method is that it facilitates, with minimal implementation efforts, to update an existing large scale structural mechanics solver into a robust tool to handle single and multiple cracks. The method involves two steps: in the first step, the governing equations of the structure are solved using nonlinear FEM by freezing the crack in the structure; in the next step, the solution obtained from the FEM is used to propagate the crack based on the maximum energy release rate criterion. Advancing the crack through a FE mesh requires a continual change in topology of the mesh, which is achieved in this work by utilizing a mesh refitting approach. This method, as the name suggests, refits the mesh at each instant of crack advancement in such a way that the crack propagates through an existing edge in the modified mesh. The mesh deformation strategies (for example used in ALE based methods) usually result in mesh tangling issues when attempting to handle topology changes in the mesh. This problem is circumvented in this work by splitting the quadrilateral elements into triangular elements in the crack tip neighborhood, which allows the possibility of local mesh connectivity to be modified. This step is crucial to preserve the quality of the mesh throughout the simulation, without which the mesh movement methods will fail. Examples involving single and multimaterials with one or multiple cracks are reported. The obtained results are compared with experimental and other available computational methods. The comparison demonstrated that the present method accurately predicted the fracture behavior of all the examples considered.
Declarations
Author's contributions
YS developed the idea, conducted numerical experiments and wrote draft. WAW finetuned the research idea, suggested numerical experiments and revised the paper. Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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