Numerical simulation of crack curving and branching in brittle materials under dynamic loads using the extended non-ordinary state-based peridynamics

Highlights

• The extended NOSB-PD is applied to simulate and investigated dynamic brittle fracture.

• Numerical convergence of NOSB-PD is investigated.

• Effects of geometric and loading conditions on crack curving and branching are studied.

• Effects of non-overlapping length on crack branching and coalescence are analyzed.

Abstract

In this paper, the stress-based failure criteria are implemented into the extended non-ordinary state-based peridynamics (NOSB-PD). When the mean stresses between the interacting material points satisfy the stress-based failure criteria, the breakage of bonds between the interacting material points occurs. The phenomena of crack curving and branching in brittle materials subjected to dynamic loads are investigated using the proposed method. A benchmark example and the Kalthoff-Winkler experiment are firstly simulated to prove the ability and accuracy of the proposed numerical method. Then, one single crack curving and branching in brittle materials subjected to dynamic biaxial loads are also simulated. The effects of geometric and loading conditions on crack curving and branching are studied. The present numerical results are in good agreement with the previous experimental and numerical results. Finally, the phenomena of multiple crack propagation, branching and coalescence under biaxial dynamic loads are also investigated. The effects of non-overlapping length on crack propagation, branching and coalescence under biaxial dynamic loads are analyzed.

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 0.9c_s (c_s 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.