Numerical simulation of crack curving and branching in brittle materials under dynamic loads using the extended non-ordinary state-based peridynamics
Introduction
Crack curving and branching is one of the typical characteristics in dynamic brittle fracture problems (Bowden et al., 1967, Ramulu and Kobayashi, 1985, Ravi-Chandar and Knauss, 1984a, Ravi-Chandar and Knauss, 1984b). Rocks are the brittle and heterogeneous materials. Particularly, when rock is subjected to dynamic tensile loads, instantaneous energy release results in stress redistribution, which triggers dramatic deterioration of the mechanical properties of rocks and rapid propagation of cracks (Yang et al., 2012). Therefore, studies of crack curving and branching in brittle materials is helpful to understand the dynamic properties of rocks under dynamic loads.
In past decades, numerous physical experiments and theoretical models were conducted to investigate the mechanism of crack curving and branching in brittle materials under dynamic loads (Cox et al., 2005, Bouchbinder et al., 2010, Bouchbinder et al., 2014, Fineberg and Bouchbinder, 2015, Bobaru and Zhang, 2015). Based on the elastodynamic theoretical results, a crack branching criterion was proposed by Yoffe (1951). Meanwhile, Yoffe (1951) found that for a crack of constant length translating with a constant velocity in an infinite medium, the maximum circumferential stress shifts from the symmetry line to lines that make an angle of 60° with the direction of propagation of the crack when the crack speed exceeds the 0.73 fraction of the Rayleigh wave speed. Crack propagation for semi-infinite cracks was studied by Eshelby (1969) and Freund (1972) in an infinite 2D linear elastic medium. Dally (1979) employed high-speed photographic systems with photoelastic methods to obtain a sequence of isochromatic-fringe patterns, which represent the state of stress associated with the rapid propagating crack. In the dynamic fracture experiments carried by Ravi-Chandar and Knauss, 1984a, Ravi-Chandar and Knauss, 1984b, Fineberg et al., 1991 and Cramer et al. (2000), the ‘mirror–mist–hackle’ phenomenon was observed in that the dynamic crack growth successively experiences the smooth, rough and zigzag stage. Ramulu and Kobayashi (1985) found that the increase in fracture surface roughness is tightly connected to macroscopic and microscopic crack path instability, which was also observed in experiments prior to branching. The microscopic crack path instability, which eventually leads to crack curving and branching, is triggered by the dynamic evolution process zone (Ravi-Chandar and Knauss, 1984a). In this process zone, nucleation, growth and coalescence of microcracks and microvoids occur (Ramulu and Kobayashi, 1983, Ravi-Chandar and Knauss, 1984a, Ravi-Chandar and Knauss, 1984b, Bobaru and Zhang, 2015). Similar mechanisms were also observed in glassy polymers like PMMA (Ravi-Chandar and Yang, 1997) and polystyrene (Hull, 1994, Hull, 1999). Furthermore, the mechanism of microvoids and microcracks-induced crack path instabilities, which trigger the asymmetrical phenomenon during crack propagation, were deeply studied by researchers. For example, Fineberg et al. (1991) suggested that the break-up of a simple crack to multiple cracks occurred at a critical velocity, at which a single crack became unstable. This instability, which became known as the microbranching instability, was later researched to obtain the characteristics of many brittle amorphous materials (Gross et al., 1993, Sharon et al., 1995, Boudet et al., 1995, Sharon and Fineberg, 1996, Sharon and Fineberg, 1998, Sharon and Fineberg, 1999, Fineberg and Marder, 1999, Cramer et al., 2000, Livne et al., 2005). Fineberg et al. (1991) applied the method of the resistance change of a thin aluminum layer evaporated on a specimen surface to measure crack speed and found the fluctuation of the crack propagation speed at the crack tips. The fluctuation of crack propagation speed in dynamic brittle fracture were considered to be caused by the occurrence of microbranching (Sharon and Fineberg, 1996).
Recent studies of understanding crack instability were carried in explaining the oscillatory instability (Livne et al., 2007), which occurs at high crack propagation speeds. It was found that tensile cracks, once reaching about 0.9cs (cs is the shear wave velocities of brittle materials), are unstable due to shear perturbations. Upon losing stability, the oscillatory wavelength is linked to the scale of the nonlinear elastic fields surrounding the crack tip (Bouchbinder, 2009, Goldman et al., 2012). Goldman et al. (2015) also found that once a simple tensile crack is subjected to shear perturbations, cracks undergo the microbranching instability above a finite velocity-dependent threshold.
Numerical approaches prove necessary to study crack curving and branching in dynamic brittle fracture problems. There are two main ways to research the dynamic brittle fracture problems involving atomistic modeling (classical and quantum) and continuum mechanics modeling (Cox et al., 2005, Bobaru and Zhang, 2015).
From the atomistic modeling perspective, some researchers applied the Molecular Dynamics (MD) to simulate and study crack branching in dynamic brittle fracture problems (Abraham et al., 1994, Abraham et al., 1997, Nakano et al., 1995, Abraham, 2005). And quantum mechanical calculations are required to enable an accurate quantitative description of mode I crack propagation at low speeds of a single crack in single crystal silicon (Kermode et al., 2008). However, atomistic simulation sometimes produced puzzling results (Zhou et al., 1996, Bobaru and Zhang, 2015). The phenomenon of crack branching cannot be reproduced by 2D atomistic models (Procaccia and Zylberg, 2013) and the realistic crystalline potentials failed to reproduce the experimental results (Hauch et al., 1999, Bouchbinder et al., 2014). Furthermore, atomistic models in simulations of dynamic brittle fracture have some obvious limitations (Zhou et al., 1996, Bobaru and Zhang, 2015): the small size of the sample modeled (usually micrometer and sub-micrometer) and the short time scales (usually nanoseconds).
For the continuum mechanical modeling, numerous researchers have successfully simulated crack curving and branching in dynamic brittle fracture. The finite element method (FEM) and various modified versions of FEM have been performed to simulate the crack curving and branching with element-erosion (Yang et al., 2012, Yang et al., 2015). Alternatives to element-erosion techniques are cohesive zone FEM models (Camacho and Ortiz, 1996, Xu et al., 2008), which remove the need of pre-knowledge of the crack path. Since the cracks can only propagate along the element boundaries (Camacho and Ortiz, 1996, Ortiz and Pandolfi, 1999, Xu and Needleman, 1994), the actual crack paths during crack propagation process may not be computed correctly. The extended finite element theory (XFEM) (Moes et al., 1999, Belytschko and Black, 1999, Dolbow and Belytschko, 1999) was applied to simulate and investigate crack curving and branching without re-meshing elements (Belytschko et al., 2003, Song et al., 2008, Song and Belytschko, 2009, Timon et al., 2009, Meng and Wang, 2015). However, in dynamic brittle fracture, the input fracture energy needs to be modified in the numerical model to obtain the crack propagation speed similar to those obtained from experiments (Song et al., 2008, Bobaru and Zhang, 2015). Moreover, the additional branching criteria should be inputted in the numerical models. In addition, models based on the “Principle of Local Symmetry” (Movchan et al., 2005, Bouchbinder and Procaccia, 2007) and models based on nonlinear constitutive behavior near the crack tip (Gao, 1996, Buehler et al., 2003, Buehler and Gao, 2006) were applied to investigate the phenomenon of crack curving and branching in dynamic brittle fracture.
Several kinds of meshfree methods (Nguyen et al., 2008, Liu, 2010), all of which only involve nodes, are available to simulate dynamic fracture of brittle materials. The most popular meshfree method are the Material Point Method (MPM) (Sulsky et al., 1994; Pandolfi et al., 2013), Smoothed Particle Hydrodynamics (SPH) (Lucy, 1977, Chakraborty and Shaw, 2013, Raymond et al., 2014) and the Element Free Galerkin Method (EFGM) (Belytschko and Tabbara, 1996, Rabczuk et al., 2007, Zi et al., 2007). A novel cracking particle model (CPM) has been developed in meshfree methods by Rabczuk and Belytschko, 2004, Rabczuk and Belytscko, 2007) to simulate and study the crack branching in dynamic brittle fracture problems. Dynamic failure and fragmentation has also been studied using Reproducing Kernel Particle Method (RKPM) by Guan et al. (2011). However, for the dynamic brittle damage of the concrete block, not all the failure features observed in experiments are reproduced by the numerical simulation.
Peridynamics (PD) is a nonlocal reformulation based on integro-differential governing equations in the classical continuum mechanics that allows a natural treatment of discontinuities by employing the concept of nonlocal interactions within a finite region (Silling, 2000). The PD theory reformulates the equation of motion such that no spatial derivatives are required. The original version of PD (Silling, 2000, Silling and Askari, 2005) was named as the bond-based peridynamics (BB-PD). The state-based peridynamics (SB-PD) (Silling et al., 2007, Silling and Lehoucq, 2010) is a generalization of the BB-PD theory, which is restricted to a fixed Poisson's ratio. The SB-PD can be classified into two types: ordinary state-based peridynamics (OSB-PD) and non-ordinary state-based peridynamics (NOSB-PD). All bonds in NOSB-PD are able to carry stresses in all direction which is much more realistic for modeling a continuum (Warren et al., 2009). It is also possible to simulate general non-linear anisotropic materials using NOSB-PD. Crack curving and branching problems in brittle materials under dynamic loads were modeled and studied using the BB-PD by Ha and Bobaru, 2010, Ha and Bobaru, 2011. The numerical convergence results in terms of the crack propagation speed and the predicted crack paths were also provided by Ha and Bobaru (2010). Agwai et al. (2011) conducted a comparative study of dynamic brittle fracture using BB-PD, where the predicted results were closed to the previous numerical results. Dipasquale et al. (2014) studied 2D dynamic brittle fracture using BB-PD with adaptive grid refinement. Huang et al. (2015) modified the bond stiffness in BB-PD to successfully simulate the crack branching problems. Ren et al. (2016) proposed dual-horizon peridynamics (DH-PD) to model dynamic brittle fracture problems. In addition, Amani et al. (2016) proposed a NOSB-PD formulation for thermoplastic fracture to simulate Taylor bar impact and Kalthoff-Winkler tests. The results showed good agreements with the previous experimental data and numerical simulations.
In this paper, the extended NOSB-PD proposed by Zhou et al. (2016) and Wang et al. (2016) was applied to simulate and investigate the crack curving and branching in brittle materials subjected to dynamic loads. In the original NOSB-PD, a critical stretch of the bond between two interacting material points is derived from fracture energy G0, which is used to determine the breakage of bonds. However, the stress-based failure criteria are commonly applied to investigate the failure of materials in actual engineering. Therefore, the maximum tensile stress criterion and the Mohr-Coulomb criterion are implemented into the extended NOSB-PD. The maximum tensile stress criterion is applied to determine the tensile failure of the bond between interacting material points. While the Mohr-Coulomb failure criterion is employed to define the shear failure of the bond between interacting material points. Firstly, benchmark examples of dynamic brittle fracture are simulated to prove the ability and accuracy of the extended NOSB-PD. In addition, numerical m− convergence of crack branching problems is performed. Then, one single crack propagation and branching in the rock-like samples under dynamic biaxial loads are investigated. The effects of crack inclination angle and loading conditions on the crack branching are studied and discussed. Finally, multiple crack branching and coalescence in the rock-like samples subjected to dynamic biaxial loads are also researched. The effect of non-overlapping length on crack branching and coalescence is also discussed.
This paper is structured as follows: the extended NOSB-PD method is described in section 2, a benchmark example of a classical dynamic fracture problem is simulated in section 3, the Kalthoff-Winkler experiment is modeled in section 4, crack curving and branching in rock-like samples containing a single pre-existing crack under dynamic biaxial loading is studied in section 5, multiple crack branching and coalescence in rock-like samples subjected to dynamic biaxial loads are investigated in section 6, and the conclusions are drawn in section 7.
Section snippets
Governing equation
Based on the works by Silling et al. (2007), NOSB-PD theory is briefly described as follows: Consider a continuum body which occupies a region in the reference configuration and is under the action of external and internal forces. The PD model describes the dynamics of a body in its reference configuration and in the current configuration. A schematic of the body is depicted in Fig. 1. The bond vector between a material point and its neighbor point , defined as , gets
A benchmark example
In this section, a classical experiment of dynamic fracture problem (Ravi-Chandar and Knauss, 1984a) is simulated by the extended NOSB-PD. The predicted crack growth paths are compared with the previous experimental and numerical results. Moreover, the crack propagation speeds are quantitatively compared with the previous experimental and numerical results.
The numerical model
Kalthoff (1988) found that two kinds of cracking modes occur, i.e. the brittle fracture and shear-zone propagation mode when a specimen of maraging steel is subjected to impacts from a collision block with various speeds.
In the experiment, a plate with two initial edge notches was impacted by a rigid block, as shown in Fig. 12. The dimensions and the mechanical parameters of the plate are listed in Table 3 and Table 4, respectively. Two different failure mode were observed under the various
The numerical model
In this section, five numerical samples under dynamic biaxial loads are conducted by the extended NOSB-PD. The configuration of the numerical samples is plotted in Fig. 16. The dimension of the numerical model is 254 × 254 mm2 and the length of the pre-existing crack is equal to 40 mm. The relationship between the horizontal and vertical loads is expressed as:where λ denotes the dynamic biaxial loading ratio of the vertical loads to the horizontal loads. The dynamic loads are suddenly
Multiple crack branching under dynamic biaxial loading
In rock engineering, multiple cracks commonly exist in the rock masses. In this section, the effect of non-overlapping length on crack propagation, branching and coalescence under dynamic biaxial loading is investigated using the proposed numerical method.
Conclusions
In the extended NOSB-PD, the maximum tensile stress criterion is applied to determine the tensile failure of the bond between interacting material points. While, the Mohr-Coulomb failure criterion is employed to define the shear failure of the bond between interacting material points. The extended NOSB-PD is applied to simulate crack curving and branching in dynamic brittle fracture problems. A benchmark example and the Kalthoff-Winkler experiment are simulated to prove the ability and accuracy
Acknowledgements
The present work is supported by National Natural Science Foundation of China (Grant Nos.51325903 and 51679017), Project 973 (Grant No. 2014CB046903), Graduate Scientific Research and Innovation foundation of Chongqing, China (Grant No. CYB16012) and Natural Science Foundation Project of CQ-CSTC (Grant Nos. cstc2015jcyjys30001, cstc2015jcyjys30006 and cstc2016jcyjys0005), which are gratefully acknowledgement.
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