A simple FEM–DEM technique for fracture prediction in materials and structures
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Abstract
This paper presents a new computational technique for predicting the onset and evolution of fracture in a continuum in a simple manner combining the finite element method (FEM) and the discrete element method (DEM). Onset of cracking at a point is governed by a simple damage model. Once a crack is detected at an element side in the FE mesh, discrete elements are generated at the nodes sharing the side and a simple DEM mechanism is considered to follow the evolution of the crack. The combination of the DEM with simple 3noded linear triangular elements correctly captures the onset of fracture and its evolution, as shown in several examples of application in two and three dimensions.
Keywords
Discrete elements Finite elements Fracture mechanics FEM–DEM technique1 Introduction
The development of a fracture within a continuous medium is a topic of much interest in the strength analysis of brittle and ductile materials. One of the most recent methodologies to simulate the fracture process is the discrete element method (DEM) [8, 13, 23, 24, 25, 26, 27, 28, 29]. However, the inherent difficulty for calibrating the material parameters in the DEM, as well as the need for a large number of discrete elements for solving practical problems, questions its effectiveness for largescale fracture mechanics analysis, even though the qualitative results of the DEM for predicting fracture patterns are pretty good.
Much research has been invested in recent years in the development of the finite element method (FEM) for modeling the onset and propagation of cracks in frictional materials [4, 5, 6, 9, 11, 14, 15, 16, 18, 19, 20]. However, FEM procedures for crack prediction use sophisticated element formulations and often require to remesh in the vicinity of the possible cracks paths [7, 11, 16].
The paper describes the basis of the simple FEM–DEM procedure proposed. The method extends a welldefined crack opening methodology termed element elimination technique (EET) [12, 17, 30] that creates discrete elements at the crack lips. Onset of cracking at the midpoint of the element sides is governed by a standard single parameter damage model. This is followed by the removal of the side and the generation of a discrete element at the nodes sharing the side. Some important aspects inherent to the formulation here presented guarantee the good results obtained like a smoothed stress field, mass conservation, and the use of a simple algorithm to ensure the postfracture contact. The FEM–DEM approach proposed is applied to a collection of benchmark problems in two (2D) and three (3D) dimensions which evidence the good performance of this numerical technique.
2 Analogy between dem and fem
The main feature of the DEM versus the FEM is its ability to generate a fracture in any direction by selectively breaking the bonds between the individual discrete elements. A time explicit integration scheme and the adequate definition of the DEM material parameters at the contact interface between the discrete elements are the key ingredients of most DEM procedures.
Despite the many advantages of the DEM, the material parameters used at the contact interface between discrete elements are not able to represent properly all the properties of a continuous domain. Additionally, and perhaps most importantly, the simple law that defines the crack appearance at the contact interface of discrete elements is not comparable to the sophisticated failure criteria used in fracture mechanics. As a consequence despite recent progress in this field [23], it is difficult to define the stress state on a continuum via the cohesive bonds of discrete elements.
In view of above facts the question arises: Is there any way to define conditions at the particle interface in the DEM so that they yield the same displacement field as in the FEM?. Figure 1 shows that the answer is yes for 3noded linear triangles. The stiffness matrix of a linear triangular element can be defined using Green’s theorem in terms of integrals along the element sides [10, 21, 22]. The integration over each element side ij yields the stiffness that each cohesive link must have in the DEM.
In this way, the stiffness required by a cohesive link in the DEM to represent a continuum via the FEM can be defined. However, if both approaches are identical, what is the advantage of using discrete elements?. Obviously, the finite element formulation is more complete and flexible. The displacement field is defined over the entire domain. Even more, the stress field in the FEM is more accurate and easier to obtain. It is also possible to prove convergence and stability for the numerical solution. However, there are distinct features in the DEM that make it a powerful numerical technique for modeling multifracture situations in materials and structures.
3 From fem to dem
The DEM is a very powerful tool when it is used for analysis of granular materials. Its main advantage when applied to a continuous domain is its capability for predicting random cracking paths, which is useful for reproducing correctly the fracture behavior of materials such as soils, rocks, ceramics, and concrete, among others [23, 24, 25, 26, 27, 28]. Thus, the rationale of the FEM–DEM approach proposed in this work is to apply the DEM methodology for modeling the onset and evolution of a crack to the standard FEM formulation. The direct application of the FEM (or the DEM) using the stiffness matrix described in Figure 1 holds as long as no cohesive link is removed. The problem arises when there is a need to remove (or break) a cohesive link, coinciding with an element side. The stiffness matrix of a finite element is obtained as the balance of internal forces of the element. Hence by eliminating the stiffness contributed by a link, the forces between the two nodes involved are unbalanced which affects the entire finite element mesh, or all neighboring particles in the DEM. The right way to eliminate the cohesive bond is by calculating the stiffness loss associated to the removal area. In other words, the initial stiffness of the element is reduced by eliminating the area between the two nodes sharing the broken side and the centroid of the element, as shown in Figure 2.
4 Failure due to accumulated damage
In order to eliminate properly a cohesive bond, it is necessary to define a failure criterion. Many references can be found on this subject [4, 5, 6, 9, 11, 14, 15, 16, 18, 19, 20]. However, it is important to note that cohesive bonds are assumed to be placed at the element sides and not at the integration point within the element. Recalling that the stress field is discontinuous between elements, a smoothing procedure is needed to evaluate the stresses at the element edges and, subsequently, the failure criteria chosen at the edges. The smoothing procedure selected is the key point to have an accuracy stress field. In our work, we have followed the superconvergent patch recovery (SPR) method proposed by Zienkiewicz, and Zhu [32] which overcomes the need to add stabilization terms to the stress field as in alternative procedures [4, 5, 6]. The failure criterion chosen is based on the standard single parameter damage model, typically used for predicting the onset of fracture in concrete and ceramic materials [6, 14, 15, 18, 19, 20]. The damage model is summarized below.
4.1 Computation of the remaining stiffness for an element
A key issue in this approach derives from analyzing in detail Eq. (7). As it can be seen in Fig. 3, when an element has two fully damaged edges according to Eq. (7), the damaged stiffness is onethird of the original one. However, the fact is that a crack has already appeared within the element and, therefore, when two sides of an element are fully damaged, the whole element can be considered to be as fully damaged as well.
4.2 Damage evolution model
The damage model presented above is extremely simple in comparison to more sophisticated constitutive models for concrete and other frictional materials [4, 14, 15, 18, 19, 20].
The experimental characterization of the model is also simple, and the following material parameters are only required: Young modulus, tension, and compression limit strengths, and specific fracture energy per unit area obtained from uniaxial tests.
4.3 Definition of the characteristic length
Note that when a side is damaged, it affects all the elements that share the side, as shown in Fig. 3.
5 Generation of discrete particles
When a cohesive bond is fully removed (i.e., the side stiffness is neglected), two discrete elements (or particles) are created at the disconnected nodes. In our work, we have used circular disks (for 2D problems) and spheres (for 3D problems) for representing the discrete elements. The mass of each new discrete element corresponds to the nodal mass in the FEM and its radius will be the maximum one that guarantees the contact between the adjacent discrete elements without creating any overlappings between them. Indeed, this is not the only algorithm that can be used for generating discrete elements [13] but it has been proved to be a very effective procedure, as the main idea is to avoid that the new discrete elements created generate spurious contact forces.
Once a discrete element is created, the forces at the contact interfaces are used to define the interaction of the element with the adjacent ones. These forces are due only to the contact interaction in the normal and tangential directions. At the contact point, the minimum radius of the particles in contact are used to evaluate the contact forces [23].
In our work, we have used a local constitutive model for the normal and tangential forces at the contact interfaces between discrete elements as proposed in [23].
The extension of Eqs. (12) and (13) to the 3D case can be seen in [23, 31].
Some interesting facts are derived from this approach. Since the number of discrete elements generated in an analysis is only a fraction of the number of nodes in the mesh, the searching algorithm for evaluating the contact interactions between discrete elements does not consume much computational resources, as in the case of using discrete elements only. Additionally, the generated particles undergo relatively small displacements (due to the time increments used in the explicit integration scheme chosen here) so the list of possible contact points does not require a constant updating.
6 Examples
Four examples are presented to demonstrate the good behavior of the FEM–DEM approach described in the previous sections. The first example is the 2D study of a normalized tensile test in a concrete specimen. The second one is the 2D analysis of a mixedmode fracture benchmark in a concrete beam. Next, an indirect tensile test widely used in concrete and rock mechanics is analyzed in 2D and 3D using the FEM–DEM technique proposed. Finally, we present an example of compressive failure of a concrete specimen.
6.1 Normalized tensile test
The first example corresponds to the fracture analysis of a flat concrete specimen under tensile stress. The geometry is defined according to the norm D638 of the American Section of the International Association for Testing Materials (ASTM) [1] as shown in Fig. 6 where the three meshes of 3noded triangular elements used and the boundary conditions can be seen. A constant displacement field is imposed in the entire shadow area.
The study has been performed using the 2D FEM–DEM technique previously described. In order to localize the fracture, only one band of elements is allowed to break at the failure stress level corresponding to the tested material, using the linear damage model presented. The results obtained are analyzed by plotting the horizontal displacement of points PA and PB shown in Fig. 6.
The Young modulus, the Poisson ratio, and the density are, respectively, \(E_0=30\) GPa, \(\nu = 0.2\), and \(\gamma = 1.0 \times 10^3 \, \mathrm{N/m}^3\). The tensile strength is \(f_t = 10\) KPa. Two specific fracture energies per unit area have been considered \(G_{f_1}=0.0 \, \mathrm{J/m}^2\) and \(G_{f_2}=7.5 \times 10^{3} \, \mathrm{J/m}^2\).
The specimen deforms by applying a constant velocity displacement of \(0.5 \times 10^6 \, \mathrm{m/s}\). at the right tip of the specimen. Figure 7 shows the relationship between the imposed displacement and the load level for the brittle fracture case (\(G_{f_1}=0.0 \, \mathrm{J/m}^3\)). The behavior is exactly the same for the three meshes used and in agreement with the expected result. Figure 8 shows the damaged geometry. Note that where fracture appears, discrete elements are created at the crack lips as explained in the previous sections.
Since the fractured elements have a different size for each mesh, the displacement of point PB in the elastic region becomes smaller as the element size is reduced. However, once the crack initiates, the displacement of point PB is ruled by the elastic energy stored in the specimen. Beyond the limit load value, the displacement of the point follows the continuous branch of the theoretical static problem. As the problem has been solved in a dynamic fashion, the change in slope progresses gradually and has small fluctuations around the theoretical result. The crack pattern for this case is very similar to that shown in Fig. 8.
6.2 Fourpoint bending beam
The next example is the failure test of a double notch concrete beam analyzed under plane stress conditions. This is a good example of mixmode fracture. The beam is supported at two points and deforms in a bending mode by applying an imposed displacement at the two points depicted in Fig. 10 where the beam dimensions are also shown.
Figure 12 shows the crack path for the three meshes analyzed which coincide with the numerical results of Cervera et al. [7]. Figure 13 shows the relationship between the vertical reaction at a force support and the imposed displacement at any of the two points depicted in Fig. 10. The graphs are in good agreement with the results obtained in [7].
6.3 Indirect tensile test
The material properties are \(E_0=21\) GPa, \(\nu = 0.2\), \(\gamma = 7.8 \times 10^3\) N/m\(^3\), \(f_t =10\) KPa, and \(G_f =1\times 10^{3}\) J/m\(^2\). Using Eq. (14) this corresponds to a failure load of \(P=314.16\) N.
Three meshes of 890, 1989, and 7956 linear triangular elements each were used for the analysis, as shown in Fig. 14. The sample is deformed by imposing a constant velocity displacement at the top of the sample.
Figure 15 depicts the damaged geometry, as well as the crack and the discrete elements generated at a certain instant of the analysis. The cracking pattern is similar for the three meshes and in agreement with the expected result. Figure 16 shows the evolution of the vertical load versus the horizontal displacement at the center of the specimen up to the failure load. The numerical values for the tensile strength obtained using Eq. (14) for each three meshes (coarse to fine) were 10587, 10506, and 10481 Pa, respectively, which yield a maximum of \(5\,\%\) error versus the expected value of \(f_t = 10\) KPa.
The same example with identical geometry and mechanical properties was solved in 3D using an extension of the FEM–DEM technique presented in this work [31]. Three meshes were used with 9338, 31455, and 61623 4noded linear tetrahedra. Results of the crack pattern obtained for each of these meshes are depicted in Fig. 17. The numerical results for the load–displacement curve are presented in Fig. 18. The numerical values obtained for the tensile strength were (coarse to fine mesh) 10 693, 10 351, and 10 235 Pa which yielded a range of 62\(\,\%\) error versus the expected value of \(f_t = 10\) KPa.
6.4 Example of compressive failure
The usefulness of the FEM–DEM formulation is verified in the analysis of the compressive failure of a prismatic concrete specimen. The problem is analyzed in 2D. Figure 19 depicts the geometry of the specimen, the material properties, and the two meshes of 3noded triangles used for the analysis.
The material properties are \(E_0 = 30\) GPa, \(\nu = 0.2\), \(\gamma = 7.8 \times 103\) N/m\(^3\), \(f_t = 2.0\) MPa, \(G_t = 100\) J/m\(^2\), and \(f_c/f_t = 10\). This corresponds to a maximum uniaxial compression stress of 20.0 MPa.
Figure 20 shows the stress–strain curve up to failure. The failure compressive stress is around \(f_c =18.8\) MPa which agrees well with the expected value, taking into account the difficulty in modeling the correct boundary conditions.
Figure 21 shows the final failure pattern, displacements, and damage for the structured mesh. The fracture on the sample coincides with the theoretical case (Fig. 21a) [2] due to the symmetry of the mesh.
Figure 22 shows the final failure pattern, displacements, and damage for the unstructured mesh applying symmetry conditions. The fractures on the sample have a good agreement with the theoretical case [2] showing the bands of vertical cracks at \(45^{\circ }\).
7 Conclusions

The failure criterion is considered at the midpoint of the element sides using a smooth stress field which does not need any additional considerations such as stabilization, or complex mixed finite element formulations.

Damaging the element sides implies that the two elements sharing the side reduce its stiffness simultaneously.

There is no mass loss by eliminating the associated finite elements. This ensures the conservation of the domain mass during the fracturing process.

The implementation of the FEM–DEM technique presented is quite simple and has yielded promising results, both qualitatively and quantitatively, for predicting the onset and propagation of fracture in concrete samples under tension, compression and mixed failure modes.
Notes
Acknowledgments
The authors acknowledge the suggestions of Profs. M. Cervera and M. Chiumenti in the development of this research. This work was partially supported by the SAFECON project of the European Research Council. The results presented in this work have been obtained using the DEMPACK code of CIMNE (http://www.cimne.com/dempack) where the FEM–DEM procedure described has been implemented.
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