# Modeling a projectile penetrating a steel plate

#### Background

A steel projectile impacting and penetrating a steel plate is a dynamic problem involving materials subjected to high strains, large strain rates and, possibly, high temperature changes. A numerical simulation analyzing this type of problem will require the use of a constitutive model that is able to capture these phenomena. There are various models that can be used to describe these effects. However, one that is popular in the fracture mechanics literature and deals with ductile materials, as in the case of steel, is the plasticity and damage model developed by Gordon R. Johnson and William H. Cook: the Johnson-Cook model.

In 1983, Johnson and Cook presented their plasticity model, mainly based on the relationship shown in Equation 1 for the von Mises flow stress:[1]

Equation 1: Johnson-Cook plasticity equation for von Mises flow stress.

Some of the terms in Equation 1 are defined in the table below:

 $\bar \epsilon^p$ effective plastic strain $\dot{\epsilon^*}$ non-dimensional ratio of effective plastic strain rate to reference strain rate $T^*$ homologous temperature: quotient of the difference between the current and room temperatures to the difference between the melting and current temperatures

The parameters A, B, n, C and m are material constants that can be obtained from physical experiments or from computational methods; this discussion focuses on using numerical methods common to the Integrated Computational Materials Engineering (ICME) community for obtaining these parameters.

Each term enclosed in the square brackets represents a particular effect from a material subjected to relatively high plastic strain. The first of these is related to strain hardening, which strengthens the material as it deforms. The second corresponds to strain-rate effects, while the third one is related to thermal softening, which deals with the weakening of a material at high temperatures.

Johnson and Cook expanded their original model in 1985 to allow accumulation of damage and fracture on the material.[2] The relationship that they developed for their damage model is presented in Equation 2:

Equation 2: Johnson-Cook damage equation for failure strain.

Where σ* is the ratio of the pressure to the effective stress and D1 to D5 are material damage constants. This cumulative-damage fracture model is very similar to the plasticity flow stress in the sense that it combines three terms in a multiplicative manner. The three terms in this case are enclosed in square brackets in the equation above and include the effects of stress triaxiality, strain rate, and local heating.

#### Approach

Figure 1: Multiscale modeling of a projectile penetrating a steel plate.

Most of the parameters in the plasticity and damage model can be experimentally obtained from tension, torsion and Hopkinson bar tests. However, testing requires having access to the material of interest as well as the equipment needed to perform the experiment. In the case of not having these at hand, computational methods offer an attractive alternative for coming up with these parameters. It is the intent of the student that the different effects described by the two equations above can be obtained using an ICME approach. This will require analyzing the problem from a downscaling perspective and extracting information at different length scales. The information that should be extracted pertains with strain hardening, strain-rate effects, thermal softening, stress triaxiality effects and elastic modulus.

At the electron scale, density functional theory can be used to extract information about the elasticity, which is one of the parameters needed for the material model at the macroscale continuum. This information is further passed to the nanoscale, along with details on the surface adsorption and cohesive energies.

At the nanoscale level, atomistic calculations can provide information about high rate mechanisms that can be fed back to the macroscale as it is highly relevant in a penetration problem. Also at this level, molecular dynamics calculation using the cohesive energy obtained at the electron scale can be used for cohesive fracture at the microscale. Additionally, this scale can provide aspects about the dislocation mobility, which is required for higher length scales.

At the microscale, information about temperature effects on voids should be fed back to the macroscale continuum. This scale can also provide information about the hardening rules for crystal plasticity to the mesoscale and used in material models at the continuum level. Other relevant parameters at this scale for properly solving a penetration problem are the void-crack nucleation and dislocation interaction.

At the mesocale, pore coalescence interactions with particles, cracks or other pores should dominate the focus of the analysis. Although lower length scales should have provided all of the information for building a material model at the macroscale continuum using Johnson-Cook parameters, the interaction of voids at this scale can provide information about damage evolution based on some fracture energy that can be used to describe how a material may fail.

All of the upscaled information collected at these length scales can be used for defining a material model for steel using Johnson-Cook parameters at the continuum level. This material model could then be used in a finite element analysis code for simulating a projectile penetrating a plate.

The proposed ICME approach is more flexible when compared to traditional experiments for obtaining material constants in the sense that it allows one to easily track how a material is affected as a result of changes in its microstructure. This allows one to engineer materials that meet specific design requirements at a faster pace. Moreover, the possibilities for applying this method to a design optimization process are endless.

#### References

1. G.R. Johnson and W.H. Cook. "A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures." 7th International Symposium on Ballistics, 1983.
2. G.R. Johnson and W.H. Cook. "Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures." Engineering Fracture Mechanics, Vol. 21, No. 1, 1985.