Mesh-free Galerkin simulations of dynamic shear band propagation and failure mode transition
Introduction
In 1980s, Kalthoff and Winkler published their experimental work on a pre-notched plate subjected to high speed impact (Kalthoff, 1987; Kalthoff and Winkler, 1987). In the experiment (here after referred to as KW problem), a thin metal plate with two pre-notches is impacted by a cylindrical projectile with a flat end. The projectile contacts with the plate in the region between the two notches (see Fig. 1) with speeds from 10 to 100 m/s. The impact initiates a compression wave in the plate, which first creates a predominantly mode II dynamic loading at the notch tips, and then raises temperature around the notch tip regions. The thermo-mechanical behavior of the plate is very complicated and unexpected. The shear dominated loading leads to three different outcomes depending the magnitude of the impact speed: (1) when the impact velocity is low, there is no fracture in the specimen, but massive plastic deformation at notch tip; (2) when the impact velocity exceeds a certain limit, say VTF (subscript “TF” denotes “tensile fracture”), a locally mode I crack initiates from the notch tip and propagates in a direction that forms roughly a 70° angle with the notch line direction (Fig. 1) (in the single notch plate experiment, the cleavage crack path forms 30° angle with the notch line (Zhou et al., 1996b)); (3) when the impact velocity exceeds another (higher) velocity limit, say VSB (subscript “SB” denotes “shear band”), the final failure mode changes: a shear band initiates at the notch tip and propagates through the specimen, which is a purely ductile failure phenomenon.
The significance of the KW problem is that it reveals that under high strain rate shear loading, material failure seems to be dominated by a ductile failure, and the failure mode tends to change from brittle to ductile as the impact velocity increases. The change of failure mode with increase of impact speed is usually referred to as dynamic failure mode transition. The failure mode transition demonstrated in KW experiment contradicts the traditional belief that material failure mode should change from ductile to brittle as the strain rate increases.
After Kalthoff and Winkler's pioneering work, similar experiments have been conducted by several research groups with various materials: mild steel (C300 Maraging steel by Mason et al. (1994); Zhou et al. (1996b), Ti–6Al–4V alloy by Zhou et al., 1996b, Zhou et al., 1998, and polymer (polycarbonate) by Ravi-Chandar (1995)). The same failure mode transition has been observed. Zhou et al., 1996a, Zhou et al., 1996b, Zhou et al., 1998 re-designed the experiment, to refine and improve experimental conditions. In Zhou–Rosakis–Ravichandran experiment (here after referred to as ZRR problem), a single notch plate is used as the target. The main advantage of a single notch specimen is that it eliminates the interference as well as interaction of the diffracted waves between the notches, which occurs in KW's double notch experimental setting, therefore a longer and “cleaner” shear dominated loading can be achieved at the notch tip.
Nonetheless, the main characteristics of the single notch experiment are similar to the double notch experiment, except for an unusual mode switching phenomenon in the intermediate impact velocity range. In the intermediate range impact velocity (20<V<29.6 m/s), at first a shear band initiates from the notch tip, and is arrested in the middle of the plate; then suddenly a cleavage type crack initiates from the arrested shear band tip. This differs from the behavior observed by Kalthoff that a crack directly initiates from the notch tip. This is a failure mode transition from ductile failure (shear band propagation) to brittle failure (cleavage fracture).1 At higher impact velocities (V>29.6 m/s), they confirmed Kalthoff's result, i.e. a shear band propagates and penetrates through the specimen without the cleavage fracture. It indicates the dominance of the ductile failure at high strain rate (see Fig. 2).
Reported simulations of the KW and the ZRR experiments have not yet been entirely satisfactory. The challenges are to reproduce the observed shear band propagation speeds and to predict the failure mode transition. The first numerical simulation was reported by Needleman and Tvergaard (1995). To explain the brittle-to-ductile mode transition as impact velocity increases, they noted that high strain rate loading enhances thermal softening, which in turn suppresses the build-up of the maximum hoop stress; thus the system is more prone to ductile failure rather than brittle failure, i.e., the dynamic shear band formation prevails over cleavage crack formation. In their simulation, Zhou et al. (1996a) introduced a viscous fluid-type of constitutive equation for the damage material inside shear band, to mimic its drastic reduction of shear force carrying capability. However, neither computation was able to accurately predict the failure mode transition. In both finite element computations (Needleman and Tvergaard, 1995; Zhou et al., 1996a), the results were mesh dependent. The work reported in Zhou et al. (1996a) is the only successful simulation of the time history of dynamic shear band growth. However, in that work, the shear band was forced to propagate along the finite element edges, a straight line parallel with the pre-notch, in contrast to the curved shear band observed in the real experiment.
Belytschko and Tabbara (1996) used the element-free Galerkin (EFG) method to simulate the KW's double notch experiment by assuming that it is an elasto-dynamic process. Recently, Klein et al. (2000) again used mesh-free methods to simulate the KW problem, with a cohesive micromechanics model––the internal virtual bond constitutive model (Gao and Klein, 1999). In both computations, a crack was found to initiate from the notch tip at about a 70° angle as observed in Kalthoff's experiment. Neither of the above computations was able to replicate the failure mode transition, nor the dynamic shear band propagation. Furthermore, most of the computations conducted so far are two-dimensional (2D).
The present work is aimed at a comprehensive simulation of the ZRR problem (single notch problem). The goal is to investigate the major aspects of the physical deformation and failure processes, which include simulating failure mode transition, dynamic structures of temperatures distribution, stress distribution, inelastic strain/strain rate distributions, and identifying shear band propagation mechanism.
The arrangement of the paper is as follows: Mesh-free methods, in particular the reproducing kernel particle method (RKPM) (Liu et al., 1995a, Liu et al., 1995b, Liu et al., 1997a) is reviewed in Section 2. An explicit mesh-free Galerkin formulation is also outlined in the same section. A thermo-elasto-viscoplastic constitutive model is adopted, which is the adiabatic version of the model used by Zhou et al., 1996a, Zhou et al., 1998. By doing so, the rate tangent modulus method of Peirce et al. (1984) is extended to include adiabatic heating. These are discussed in Section 3. Section 4 focuses on the new discoveries of the simulations, such as failure mode transition, temperature reflection, asymptotic strain rate/stress field, and micro-structure of adiabatic shear band.
Section snippets
An explicit mesh-free Galerkin formulation
Several mesh-free methods are currently used in computational mechanics. These include the smoothed particle hydrodynamics (Gingold and Monaghan, 1977), the diffuse element method (Nayroles et al., 1992), the EFG method (Belytschko et al., 1994), the RKPM (Liu et al., 1995b, Liu et al., 1996, Liu et al., 1997b), etc. The particular mesh-free method used in this simulation is the RKPM. A detailed account of the method is provided in Liu et al. (1997a) and Li and Liu (1999).
It has been shown
Constitutive modeling
It is a well-known fact that the classical rate-independent plasticity theory does not possess an intrinsic length scale, which leads numerical pathologies in simulation of strain localization, such as mesh size and mesh alignment sensitivities.
Several regularization mechanism have been introduced in constitutive modeling. They include: viscoplastic model (Needleman, 1988, Needleman, 1989); thermal dissipation model (LeMonds and Needleman, 1986a, LeMonds and Needleman, 1986b; Oliver, 1989;
Overview
The computations carried out in this work focus on the experiments conducted by Rosakis and his co-workers, i.e. the ZRR problem. The experiment involves the asymmetric loading of a pre-notched plate (single notch) by a cylindrical projectile as shown in Fig. 3. In this numerical study, two configurations have been used to simulate plate specimens of different sizes, which correspond to two different sets of experiments. The first configuration models the experiment conducted by Zhou et al.
Concluding remarks
Mesh-free Galerkin methods have been used to simulate the dynamic shear propagation in a pre-notched plate subjected to asymmetric impact loading. The numerical results are compared to experiments. A number of new computational findings are noteworthy.
First, failure mode transition has been replicated in numerical computations. For intermediate impact velocities (20<V<30 m/s), a shear band initiates first from the notch tip, and then arrests in the middle of the plate. Finally a cleavage crack
Acknowledgements
The authors would like to thank Mr. Dong Qian for helping in post-processing many numerical data. This work is supported by grants from the Army Research Office, National Science Foundations, and Tull Family Endowment. A.J. Rosakis would like to acknowledge the support of the Office of Naval Research (Dr., Y.D.S. Rajapakse, program monitor) through grant no. N00014-95-1-0453.
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