Introduction Plasticity based analytical modeling

Introduction
Plasticity-based analytical modeling and finite order NCT-501 methods (FEM) may be used to predict the fragmentation pattern of warheads. However, the viability of the predictions relies on the material constitutive models describing the plastic flow stress and fracture. For an expanding thin wall casing, the tangential strain rates are typically in the range of 104–105/s and the quasi static established material model may not be viable. Main research issues are the dependency of fracture strain on triaxiality (that means on the proportion of invariant I1 to J2), the influence of the third invariant, i.e., strain rate, on ductility, element size and the connection to adiabatic shear bands at high strain rate, and whether statistical failure predicts the size distribution of fragments better than a homogeneous failure model [1–6].
Failure process of ductile materials is caused by the nucleation, growth and coalescence of voids to fracture. The fracture coalescence depends on pressure or triaxiality (that means on the proportion of invariant I1 to J2) [7]. In general, the larger the triaxiality is, the smaller the fracture strain at failure becomes. This is in agreement with theoretical models for void growth [8,9]. Recently, Bao and Wierzbicki [10,11] compared the different models to cover the influence of triaxiality. They concluded that none of the models were able to capture the fracture behavior in the entire range of triaxiality. The void growth was the dominated ductile failure mode at large triaxialities (say above 0.4), while the shear of voids dominates at low triaxialities. The main conclusion was that there was indeed a possible slope discontinuity in the fracture locus corresponding to the point of fracture transition [11]. A dependency of the third invariant has been forecasted.
Both yield strength and ultimate tensile strength usually increase with strain rate for steel materials. The ductility of quenched and tempered steel may increase with strain rate, while the ductility of the material which high strength is achieved by precipitation hardening process may decrease with strain rate. Body-centered cubic (bcc) materials can also behave different from face-centered cubic (fcc) materials. Thermal softening decreases strength and increases ductility. Thus the ductility of materials could increase with small strain rates but could decrease with higher strain rates due to thermal softening. Decreased ductility at higher strain rates may be explained by shear localization due to adiabatic heating [12]. Unstable adiabatic shear transfers the entire burden of strain to a finite number of these shear planes (adiabatic shear bands). Due to restriction on computational time, the element sizes are traditionally too coarse to resolve the shear bands by direct simulation.
Wilkins et al. [13] concluded many years ago that the order of the applied loads, i.e. hydrostatic pressure followed by shear or vice versa, should be important in failure modeling. To account for the order of the applied loads, the cumulative damage criterion has been applied [13]. Fracture occurs at a point of the material order NCT-501 where a weighted measure of the accumulated plastic strain reaches a critical value. The weighing function depends on the triaxiality and/or the third invariant I3. Finding an appropriate weighting function is still an active field of research [14,15]. In the Johnson–Cook (J–C) model [16], an uncoupled (passive) damage evolution formulation with no third invariant dependency is adopted, which entails that there is no coupling between the stress-strain behavior and the damage evolution until fracture occurs at the critical damage.
The split-Hopkinson bar (SHB), which may be used in compression, tension and torsion testing, is the most widespread method for material high strain rate characterization. However the strain rate never exceeds 103/s and is thus much lower than that achieved under explosive loadings. Many ductile materials display an increase in yield stress for strain rates above 103/s [17,18]. It is challenging to conduct material tests at the strain rates of larger than 103/s. The flyer plate impact testing produce uniform stress and strain rates but the testing is expensive. The Taylor testing is relatively inexpensive and data could be obtained from simple post-test measurements. However, the Taylor test produces non-uniform stress and strain fields and the results are not so easily interpreted for material modeling.