A new plasticity and failure model for ballistic application

Abstract

Ballistic phenomena give rise to a plethora of failure modes that compete. Johnson–Cook (JC) plasticity and failure models have been extremely successful because, while being conceptually simple, they capture the essence of the operative mechanics and they provide reasonably good predictions for ballistic limits. Nevertheless, the Johnson–Cook models, due to their isotropic flow and failure surface, cannot reproduce certain failure modes commonly seen in quasistatic tests: cup and cone failure, slanted failure in tensile specimens, and slanted failure in compression specimens. This work shows that by introducing the third invariant (or Lode angle) in both the JC plasticity and damage models, cup and cone, and slanted failure modes arise naturally. After the model is presented it is exercised with a material taken from the literature to predict successfully Taylor anvil and ballistic penetration failure patterns.

Introduction

When simulating ballistic or explosive events, hydrocode users are often expected to decide which plasticity and failure material models are best suited for their particular problem. Actually this is not a trivial task. Because of the large number of models available in the literature it can be a rather confusing endeavor to determine the most appropriate model. Available models range from the very simple one parameter models to very complex models with many material constants.

Simple one parameter plasticity models like Von Mises or Tresca (see circle and hexagone in Fig. 1), are very successful in predicting the yield point for ductile materials but are unable to reproduce patterns observed in some experiments; e.g., localization of plastic deformation in a slanted plane in tensile or compression tests, or differences in the strain-hardening between tensile and shear tests of a material. For homogeneous isotropic materials the yield stress or flow surface should depend only on the three invariants of the stress tensor: I₁, I₂ and I₃. If pressure independent plasticity is assumed, the usual approach is to work with the two invariants of the deviatoric stress tensor J₂ and J₃. For ductile metals the assumption of pressure independent flow-stress has been demonstrated to work very well as a first approximation, although Bridgman [1] did observe a change in the strain-hardening slope of steel under different pressures. Pressure is important in brittle materials, for example ceramics or rocks, and models like Drucker–Prager [2] incorporate the first invariant $I_1=-3P$, where P is the pressure.

Johnson–Cook (JC) is probably the most popular plasticity and failure model used in ballistics. A comprehensive summary of the model is presented in Ref. [3]. The original plasticity model can be found in [4], while the original failure model is in [5]. The plasticity model is a “J₂ model”, i.e. the flow surface does not depend on pressure or the third invariant. The failure envelope of the JC model does depend on pressure and J₂ through the triaxiality parameter ($\text{the}\text{ratio}{\sigma }^{\ast }=I_1/3/\sqrt{3J_2}$) but does not depend on the third invariant. The model is reasonably good at predicting phenomena like the ballistic limit but fails, due to its J₂-only nature, to reproduce failure patterns like, for example, slanted failure of a smooth tensile or a compression specimen. Also JC is unable to capture, among others, the fact that many materials clearly have a different response (equivalent stress vs. equivalent plastic strain) in torsion than in tension. The usual way to treat this fact when calibrating the JC material models is to average the torsion and tensile behavior, see [4]. Wilkins et al. [6] actually addressed this problem by blending the tensile and torsion tests in the flow surface he proposed. Their flow surface and failure envelope are functions of J₂ and J₃. Unfortunately, many tests and computations are required to calibrate Wilkins’ model, so it is not extensively used in the ballistics community.

There have been many attempts at reproducing failure patterns and ballistic failure as seen in tests. A review of the models of interest for ballistic applications is presented in Chocron and Anderson [7]. A simple single parameter model like Cockcroft-Latham [8] can actually perform fairly well as shown by Børvik et al. [9], [10] when reproducing the ballistic limit. But it is not thought that it could reproduce a rich set of patterns like slanted, or cup and cone failures. Tuler-Butcher [11] is also a very simple failure model that captures the time dependency in spall failure but, again, probably cannot reproduce other failure modes. More sophisticated models like Gurson [12] or the Gurson–Tvergaard–Needleman (GTN) model [13] aim at capturing better the micromechanics, but are not suited for ballistics because of lack of simple physics like shear failure. This aspect is corrected in the recently published models by Xue [14] (known as the Xue-GTN model) and Nahshon and Hutchinson [15]. Other interesting models have been published by Børvik et al. [16], Bao and Wierzbicki [17], Wierzbicki et al. [18], and Bai and Wierzbicki [19]. All these models have the physics to be able to reproduce failure patterns seen in tensile and compression tests and in ballistics phenomena but, as far as the authors know, such a complete check has yet to be performed on these models.

The objective of this work is to present a new plasticity and failure model that consists of a combination and slight modification of the JC model and Xue-Wierzbicki model [18]. In creating the model the following constraints were applied:

• Because many materials are already characterized as JC and constants are readily available, the model should, as much as possible, take advantage of the known constants and add a minimum of additional constants over the JC model.

• To keep the model simple and reduce the number of constants, damage was not directly coupled with the plasticity model.¹ Although this might not be completely realistic, it does avoid the use of ad-hoc constants or damage functions. A simple linear evolution of damage was adopted.

• The model should capture features like slanted failure in tensile or compression tests, cup and cone, etc. It also should capture failures typically seen in ballistic phenomena like shear, spall, etc.

• The model should capture properly the strain to failure vs. triaxiality function as well as the difference between the hardening functions in tension and shear.

• The model should capture the experimental observation that effective strain at failure can be different for pure shear and tensile tests.

The reader can find below a detailed description of the postulated model as well as the results obtained when applying the model. The results will show that the model can capture a slanted failure pattern for the compression tests as well as the cup and cone pattern on the cylindrical tensile specimen. The model is also validated with Taylor anvil tests published in the literature.