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Some Issues Related to the Modeling of Dynamic Shear Localization-assisted Failure

Engineering design of structures to withstand accidental events involving high strain rates and/or impact loading requires predictive modeling capabilities for reproducing numerically potential premature failure following adiabatic shear banding (ASB). The purpose of the present chapter is to review ASB-oriented modeling approaches available in the literature (while not pretending to be exhaustive) that provide a better understanding of ASB and its consequences in structural metals and alloys.

1.1. Introduction

ASB is a mechanism of plastic flow localization known to be triggered by a thermo-mechanical instability in the context of dynamic plasticity (see, for example, Woodward [WOO 90] and Bai and Dodd [BAI 92]). It may be particularly encountered in high-strength metals and alloys including, but not restricted to:

• – steels: martensitic steel (Zener and Hollomon [ZEN 44]); HY100 (Marchand and Duffy [MAR 88]); Maraging C300 (Zhou et al. [ZHO 96a]); 4340VAR (Minnaar and Zhou [MIN 98]); ARMOX500T (Roux et al. [ROU 15]), etc.;
• – titanium alloys: various titanium alloys (Mazeau et al. [MAZ 97]); Ti-6Al-4V (Liao and Duffy [LIA 98]); β-CEZ (Sukumar et al. [SUK 13]); UFG pure Ti (Wang et al. [WAN 14]), etc.;
• – aluminum alloys: AA25XX (Liang et al. [LIA 12]); AA50XX (Yan et al. [YAN 14]); AA60XX (Adesola et al. [ADE 13]); AA70XX (Mondal et al. [MON 11]), etc.

Shown as causing either a loss of the ballistic performance of a protection (armor) plate made of high-strength steel and alloys (see, for example, Backman and Goldsmith [BAC 78]) or an increase of the ballistic performance of a kinetic energy penetrator made of depleted uranium, due to the self-sharpening effect (see, for example, Magness and Farrand [MAG 90] and Gsponer [GSP 03]), ASB has been widely studied for defense applications, mostly from a metallurgical viewpoint with the aim to possibly reduce/increase material ASB sensitivity. In parallel, for a long time, a condition for ASB initiation has been considered as a failure criterion in the design of protection structures undergoing impact and other high-strain rate loadings. However, this approach generally leads to over-conservative design since the structure is still able to consume energy in the post-localization stage. ASB is also seen to control chip serration in high-speed machining of, for example, high-strength steel and titanium alloys (see, for example, Molinari et al. [MOL 13]), having mitigated effect in the sense that it reduces the cutting force magnitude while generating fluctuations of the cutting force and degrading the surface roughness. Numerically optimizing the cutting conditions implies accounting for ASB.

It has thus become indispensable to explicitly deal with this progressive, irreversible softening mechanism of localized deformation to the same extent as it has become necessary to account for damage-induced softening for related applications.

In this chapter, we present selected ASB-oriented modeling approaches available in the literature (while not pretending to be exhaustive) for guiding researchers and engineers who need to consider and address ASB and its consequences in structural metals and alloys. The inability of standard engineering thermo-viscoplasticity models (see, for example, the Johnson-Cook model) to reproduce ASB-assisted failure (see Batra and Stevens [BAT 98] or Longère et al. [LON 09]) has led to the development of enriched models, i.e. models embedding discontinuity either at the constitutive equations level (see Longère et al. [LON 03]) or at the FE kinematics level (see, for example, Areias and Belytschko [ARE 07]). There are two classes of approaches depending on the modeling scale: a first class in which the RVE/FE characteristic length is smaller than the bandwidth, and a second class in which the RVE/FE characteristic length is greater than the bandwidth. RVE stands for the representative volume element for a given material. In the sequel, the former approach is referred to as “small-scale postulate”, whereas the latter is referred to as “large-scale postulate” (see Longère et al. [LON 18a]). It must be noted that a similar distinction, but with a different nomenclature, can be found in Belytshko et al. [BEL 88].

Section 1.2 deals with preliminary considerations and the introduction of basic concepts. Sections 1.3 and 1.4 present some models based on the “small-scale postulate” and “large-scale postulate”, respectively. The summary and conclusions are given in Section 1.5.

1.2. Preliminary/fundamental considerations

The present work focuses on metals and alloys, even though most of the considerations and concepts presented in the following apply to a wider class of solid materials, including, for example, polymers (below glass transition). In addition, the numerical approach considered for the resolution of the initial boundary value problems involving structural materials susceptible to ASB is here restricted to the finite-element method, which is the most widely used method for engineering applications. Thus, there is a connection between the volume element and the integration point.

1.2.1. Localization and discontinuity

First, definitions of basic concepts are introduced:

Discontinuity

It should be recalled that according to the discontinuity theory (see Figure 1.1):

• – a “strong” discontinuity involves a discontinuity of the displacement/ velocity field;
• – a “weak” discontinuity involves a discontinuity of the gradient of the displacement/velocity field, i.e. of the strain/strain rate field.

For example, a crack generates a strong discontinuity, whereas strain localization produces a weak discontinuity. An ASB exhibiting a width with distinct boundaries is thus associated with strain localization involving a weak discontinuity, i.e. a discontinuity of the gradient of displacement/velocity field.

Strain localization

• – the “physical” strain localization, as observed experimentally, results from a thermo-mechanical instability due to, for example, thermal softening, damage, microstructural changes or their combination;
• – the “numerical” strain localization, as observed in numerical simulations, is characterized by the formation of a band whose width covers only a single (standard) finite element and results from the loss of uniqueness of the solution of the initial boundary value problem (IBVP) in the softening regime, having, as a result, mesh size and orientation dependence of the numerical results.

Ideally, the numerical strain localization would/should numerically reproduce the physical strain localization. However, it is rarely, actually scarcely ever, the case.

Figure 1.1. Displacement and strain fields in the absence of discontinuity and in the presence of weak and strong discontinuities. Body Ω with a discontinuity; w represents the discontinuity width and u and ∇u are the displacement/velocity and displacement/velocity gradient, respectively.

Source: Longère [LON 18a]

Based on the well-known experimental results obtained by Marchand and Duffy [MAR 88] on dynamic torsion loading of a thin-walled cylinder made of high-strength steel (see also Roux et al. [ROU 15] for impact loading), the following scenario is well established nowadays. During the shear loading of a viscoplastic material, we can distinguish three stages: a first stage of homogeneous deformation, a second stage of weakly heterogeneous deformation and a third stage of strongly heterogeneous deformation. It is during the third stage that ASB occurs and further develops, sometimes leading (see, for example, Longère and Dragon [LON 15] for titanium alloys) to void growth-induced damage and ultimately to fracture in the band wake. Thus, two characteristic lengths are involved: a large one W for weakly heterogeneous deformation and a small one w for strongly heterogeneous deformation (see Figure 1.2).

Figure 1.2. Zones of weakly and strongly heterogeneous deformations according to Marchand and Duffy’s terminology [MAR 88].

Source: Longère [LON 18a]

The non-local modeling framework has been specifically developed to respond to the need for incorporation of material length-scale measures in the constitutive description involving deformation process strongly affected by the presence of material or geometrical imperfections, distribution of defects and strain localization phenomena. A systematic design of the non-local gradient-enhanced continuum model for solving high-velocity impact-related problems has been attempted by Voyiadjis and co-workers. A thermo-viscoplastic and thermo-viscodamage model in this context was introduced by Abu Al-Rub and Voyiadjis [ABU 06a, ABU 06b] and further applied by Voyiadjis et al. [VOY 13]. Second-order gradients in the hardening variables and in the damage variable are introduced, and the coupling between these variables and their gradients are accounted for. The proposed theory leads to numerous...