# Modeling rapidly growing cracks in planar materials with a view to micro structural effects

## Abstract

Dynamic fracture behavior in both fairly continuous materials and discontinuous cellular materials is analyzed using a hybrid particle model. It is illustrated that the model remarkably well captures the fracture behavior observed in experiments on fast growing cracks reported elsewhere. The material’s microstructure is described through the configuration and connectivity of the particles and the model’s sensitivity to a perturbation of the particle configuration is judged. In models describing a fairly homogeneous continuous material, the microstructure is represented by particles ordered in rectangular grids, while for models describing a discontinuous cellular material, the microstructure is represented by particles ordered in honeycomb grids having open cells. It is demonstrated that small random perturbations of the grid representing the microstructure results in scatter in the crack growth velocity. In materials with a continuous microstructure, the scatter in the global crack growth velocity is observable, but limited, and may explain the small scattering phenomenon observed in experiments on high-speed cracks in e.g. metals. A random perturbation of the initially ordered rectangular grid does however not change the average macroscopic crack growth velocity estimated from a set of models having different grid perturbations and imply that the microstructural discretization is of limited importance when predicting the global crack behavior in fairly continuous materials. On the other hand, it is shown that a similar perturbation of honeycomb grids, representing a material with a discontinuous cellular microstructure, result in a considerably larger scatter effect and there is also a clear shift towards higher crack growth velocities as the perturbation of the initially ordered grid become larger. Thus, capturing the discontinuous microstructure well is important when analyzing growing cracks in cellular or porous materials such as solid foams or wood.

### Keywords

Dynamic fracture Crack growth velocity Particle method Heterogeneous material## 1 Introduction

Recently the authors proposed a three-dimensional mechanical model to capture dynamic fractures in heterogeneous materials (Persson and Isaksson 2014). The focus in the previous study was on discontinuous fiber-based materials, but the model is equally well suited for more continuous materials, as will be further revealed here. Modeling dynamic fractures has evolved from the classical static fracture criterion by Griffith (1920) via quasi-static crack growth to full dynamics (cf. Nilsson 2001; Freund 1998). Some of the different methods of modeling dynamic cracks are analytical (Gehlen et al. 1987; Popelar and Gehlen 1987), while others are numerical, such as the finite element method (Ramulu and Kobayashi 1985), the extended finite element method (Belytschko et al. 2009) and so-called “mesh-free methods” (Mos et al. 1999). Lattice models (LM) pioneered by Hrennikoff (1941) are conceptually near to the FE models but with a different history. While FEM stems from discretizing a continuum, LMs share a history with molecular dynamics (MD) and has later been adapted to deal with effective material points rather than atoms cf. Ostoja-Starzewski et al. (1996). A third class of models is the morphology based models where the discretization is made to correspond to actual physical micro structures. In resent years particle methods have entered the scene, e.g. hybrid particle element methods (Fahrenthold and Horban 2001; Rabb and Fahrenthold 2010; Monteiro Azevedo and Lemos 2006) and peridynamics (Silling and Bobaru 2005; Silling and Askari 2005; Ha and Bobaru 2010) or discrete element methods (Persson and Isaksson 2013, 2014). The distinction between particle and hybrid particle element method is not absolute, however the hybrid methods often have the possibility to model new contacts and use interactions known from mechanics such as trusses and shells (Rabb and Fahrenthold 2010) whilst in the general case other potentials are common such as the Lenord-Jones cf. (Gao 1996), or longer reaching interactions cf. (Silling and Bobaru 2005, Wang et al. 2009). There is also a class of morphology based particle methods, such as the one used in this paper. All three classes of particle models are examples of LM. The traditional analytical solutions have limited usefulness for practical problems, but they provide a basis on which the computational (numerical) models stand and serves as benchmark problems. Finite element models are perhaps the most established methods to model stationary cracks and fractures. However, they are less suited for modeling growing cracks for mainly two reasons: (1) the material description needs to be continuous and, (2) re-meshing strategies are needed to capture the changed geometry when a crack grows. The first problem has been partly solved by the introduction of the extended finite element method, also known as the generalized finite element method, in which a crack can extend through split elements. The second problem has been met by introduction of “mesh-free” methods such as the boundary element method. However, a significant drawback of the boundary element method is the difficulty to include material nonlinearities and anisotropy, cf. Zang and Gudmundson (1988). Moreover, while analytical methods are computationally inexpensive they are of limited practical use when analyzing fracture in engineering structures of arbitrary geometry because of the complex interactions between the crack and other surfaces (e.g. boundaries). In this case, numerical models become necessary (cf. Nilsson 2001). Even though e.g. crack paths often can be fairly accurate estimated by a quasistatic method, the crack growth velocities needs to be determined by other means cf. (Ooi and Yang 2011). In the LM class any lattice-lattice interaction may be used, and for modeling linear elastic fracture a popular choice is spring network models (Ostoja-Starzewski et al. 1996). Some extensions to the simple springs are made to adjust the Poisons ratio by the introduction of torsion spring and beams cf. (Ostoja-Starzewski et al. 1996; Wang et al. 2009). Such models are commonly used to model fracture with implicit solutions cf. (Ostoja-Starzewski et al. 1996; Suiker et al. 2001; Marder and Liu 1993) or by solving embedded differential equations to compute favorable crack speeds cf. (Marder and Liu 1993). To model a growing crack with a physical dynamic behavior, it is possible to use a particle representation of the material with mechanical interaction laws and advance in time using Newton’s equation of motion rather than the quasi-static motion favored in e.g. traditional finite element models. The interaction laws might be based on e.g. piecewise continuum mechanics, like the hybrid particle element method, or non-local interaction laws such in peridynamics cf. (Silling and Bobaru 2005; Silling and Askari 2005). These methods are inherently dynamic in the sense that the crack velocity is a result of the material parameters, rather than a separate material parameter to be set (cf. Persson and Isaksson 2014). If the previously mentioned models are sorted by mathematical similarities new groupings would arise, with the major groupings; non linear versus linear radial interactions, radial versus radial and angular interactions eg. spring type versus beam type interactions, in the case of angular interactions rigid versus elastic versus freely rotating connections between elements, whether or not the discretization represent a physical entity or is an arbitrary discretization. Finally there is the option whether to advance in time explicit or implicit, which refers to advancing in time using Newtonian mechanics or some quasistatic means of time dependent evolution and an explicit integration does not exclude the possibility of an implicit integration scheme such as Rungekutta.

Lets explore the last choice first: Depending on the available computational resources, advancing in time implicitly without inertia may be the only realistic possibility and it is thus a popular choice cf. (Ostoja-Starzewski et al. 1996; Suiker et al. 2001; Marder and Liu 1993; Gerstle et al. 2007; Bolander and Saito 1998). Including inertia and accelerate nodes explicitly comes with the added benefit of true dynamic and is made possible by modern computers even for large systems cf. (Zhang and Chen 2014). The availability of computational power is of the utmost importance when choosing complexity of interactions and the computers of today favors better descriptions, e.g. beams over springs since reading from memory takes more time than computing. There is also the added benefit of the possibility of solution stability with a larger time step. The benefits of including angular interactions and its effect on poissons ratio is extensively examined by cf. Ostoja-Starzewski et al. (1996). Each part of the theoretical assumptions used in the model in this paper is thus not novel by them self, but combined in the currently most favorable combination for computational purposes.

The strategy used in this study is based on a mechanical particle interaction model and utilize known physical interactions and explicit solver/integration in time. The model’s ability to capture real material behavior is first examined by comparing numerical results to experimental results found in literature. Then investigations are made on the microstructure’s particle discretization: the role of the discrete particles’ initial positions and the impact of any perturbations on the macroscopic global behavior.

## 2 The model

## 3 Comparisons to experiments

Two different geometries are used for numerical comparisons to experiments found in literature. In both examples it is assumed that linear elasticity prevails and that the material can be considered isotropic.

### 3.1 Geometries

### 3.2 Results

### 3.3 Discussion of the experimental comparisons

In the two compared examples, i.e the results in Figs. 6 and 7, it is shown that the model captures the dynamic fractures remarkably well. In the first example, the strip, the cracks advances as the external boundaries are held stationary when the crack growth has initiated, while in the second example, the impulse loaded beam, the crack propagates when the external boundaries moves, which demonstrates the diversity of the model. In Fig. 6 there is a considerable difference between the experimental data and the analytical model. There are a few plausible explanations for this, but energy dissipation is most likely the main reason. In the region of the experimental data the numerical model agrees well with the experiments, and outside this region the behavior agrees phenomenological with the analytical model, however with a shift in the maximum crack propagation speed. Nilsson (1972) speculate that plasticity processes in the crack tip region might have an important role in the crack propagation in this case. This might agree with our results since fairly large amounts of energy dissipation takes place in the model. The first data point in Fig. 6 is special in the sense that it does not represent a continuous growth but a series of interrupted cracks. It should thus not to be seen as having a lower crack speed than the limiting \(0.28c_2\) established by Marder and Liu (1993).

The modeling of the beam impacted by a drop hammer, i.e. the ballistic experiment of Zehnder and Rosakis (1990), also agree well with the experiments. However, the simulation shows some oscillations in the crack speed that the experimental resolution is to course to capture. At the instances when the crack grows, it grows at a constant speed of \(0.5c_2\). Between the times of growth are moments of crack arrest, and together this results in an average crack propagation of \(0.3c_2\) on the global scale which corresponds well to the reported experimental one, Fig. 7. Both this crack growth speed of \(0.5c_2\), and the slowing effect of the oscillations agrees with the work with double cantilever beam by Kanninen (1974). With a shear wave speed \(c_2\) of roughly \(3100\) m/s, the \(3 \mu \)s of the first diversion from the initial crack speed corresponds to the time a shockwave, originating at the first crack growth event, need to reach the closest free edge, reflect and meet the moving crack. This interference phenomenon seems to have a great influence and somewhat slows down the crack growth velocity.

## 4 Material discretization

### 4.1 Cellular microstructure

### 4.2 Results and discussion

The numerical crack propagation speed versus fracture surface energy is displayed in Fig. 9. The analytic solution for the undamped continuum of Nilsson (2001) is included as a reference. There is a phenomenological similarity between the analytic solution and the numerical results in that there is a significant increase in crack surface energy once certain crack speed is reached. Although there is a considerable difference in when this shift occurs, once the shift is reached the increase in fracture surface energy is similar.

## 5 Perturbation of the particles’ initial positions

### 5.1 Irregular grid configuration

### 5.2 Results and discussion

The computed crack growth velocity versus fracture surface energy for the perturbed grids are displayed in Figs. 12 and 13. The result in Fig. 12 demonstrates that even a small perturbation of initially ordered regular grids (continuous microstructure) may change the crack growth behavior, and with \(\tilde{r}_{\varDelta }/l_0=0.04\) the scatter in the results closely match the scattering in the experimental data. With \(\tilde{r}_{\varDelta }/l_0=0.16\) the scatter in the numerical model greatly exceeds those of the experiments in Nilsson (1974). However, even with the highest level of perturbation, there is no change in the shape of the curve and there is no categorical shift in the relation between crack speed and crack surface energy.

## 6 Conclusions

In the examples in Sect. 3.1 it has been shown that the model captures dynamic fracture both driven by elastic energy with stationary boundaries and cracks driven by energy supplied by moving boundaries, which demonstrates the diversity of the model. The comparison to the impact experiment is of particular interest since the numerical model have a higher resolution of the crack growth behavior and is on a length scale that is below the detection limit of most instruments, and which is essential to the crack growth. The model has been used to illustrate difference in crack growth behavior in materials having fairly continuous and discontinuous (cellular) microstructures. The influence of perturbations in a discontinuous microstructure is significantly larger than the influence of perturbations in continuous microstructures. A perturbed regular grid, i.e. a fairly continuous microstructure, might be interpreted as a grained material microstructure, and may explain the scatter in the experimental data observed in Nilsson (1974). It is interesting to note that although all fracture in this study has been of a truly brittle manner, it captures the fracture behavior of more ductile materials such as a steel plate. This support the hypothesis that heat and sound (vibrations) are important parts of the energy dissipation in dynamic fracture and may be of a greater importance than the energy lost due to plastic deformation. Finally, capturing a discontinuous microstructure well is very important when analyzing growing cracks in cellular materials such as solid foams or wood, and the accuracy of the predictions on solid foam fracture, which for most foams have a ductile component will be investigated in a forthcoming study.

## Notes

### Acknowledgments

The Swedish Research Council is acknowledged for funding this study through Grant No. 2010-4348.

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