# Using an Internal State Variable Model Framework to Investigate the Influence of Microstructure and Mechanical Properties on Ballistic Performance of Steel Alloys

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## Abstract

**:**

## 1. Introduction

#### 1.1. Ballistic Impact Modeling and Experiments

#### 1.2. Internal State Variable-Based Constitutive Model for Ductile Materials

## 2. Materials and Methods

#### 2.1. Part I: Ballistic Impact of Rolled Homogeneous Armor Steel Plates by Spherical Projectiles

#### 2.1.1. Finite Element Simulation Framework

#### 2.1.2. Constitutive Model

**A**), rate functions are denoted by a dot accent ($\dot{A}$), and frame-indifferent second-rank tensors are denoted by an overbar and dot accent ($\dot{\overline{\mathit{A}}}$).

**σ**, the Lamé elastic constants, λ and μ, and the elastic spin tensor,

**W**. Here, the plastic spin is assumed to be zero. In Equations (2) and (3)

_{e}**D**,

**D**, and

_{e}**D**are the total, elastic, and inelastic rates of deformation tensors, respectively. The void volume fraction, $\varphi $, is used to represent damage in Equation (3). Void volume fraction increases compliance (Equation (1)) and inelastic flow rate (Equation (3)). The inelastic flow rate,

_{in}**D**, is a function of the isotropic hardening, R, kinematic hardening tensor,

_{in}**α**, the deviatoric Cauchy stress tensor,

**σ′**, void volume fraction, $\varphi $, and a collection of yield-related parameters, Y(T), V(T), and f(T) described in.

**α**and R) are used to represent the effects of geometrically necessary and statistically stored dislocation densities, respectively, on the material’s plastic response. The objective kinematic hardening rate, $\dot{\overline{\mathsf{\alpha}}}$, is given through [76] as:

_{s}(T) and r

_{d}(T) represent the temperature-dependent static and dynamic recovery for kinematic hardening [52], GS

_{0}and GS represent the reference and initial grain size, respectively, and Z is a dimensionless grain-size sensitivity parameter. Similarly, ref. [76] expressed the isotropic hardening rate, $\dot{R}$, as

_{s}(T) and R

_{d}(T) account for the static and dynamic recoveries for isotropic hardening [52].

_{1}is the first stress invariant, and J

_{2}is the second deviatoric stress invariant. Equation (8) introduces rate sensitivity through the yield terms V(T) and Y(T), and m is the Cocks–Ashby damage coefficient [75].

_{IC}is the fracture toughness, and J

_{3}is the third deviatoric stress invariant. Here, invariant ratios containing I

_{1}correspond to stress triaxiality, while terms containing J

_{3}capture shear effects. The parameters a, b, and c are used to calibrate void nucleation stress-state sensitivity and, C

_{ηT}is used to capture void nucleation temperature dependence consistent with the findings in [84]. Chandler et al. [85] introduced the parameters H

_{B}and m

_{h}to represent the interfacial hydrogen concentration in atomic parts per million (APPM) and a given material’s fracture sensitivity due to the presence of hydrogen, respectively. Chandler et al. [85] related the lattice hydrogen concentration in stressed regions (H

_{σ}) to the trapped hydrogen concentration at the interfaces of grain and particle boundaries (H

_{B}) using a theoretical approach developed in [86] and [87], i.e.,

_{B}is the binding energy of hydrogen at trapping sites, R is the universal gas constant, and ${H}_{\sigma}$ is the hydrogen concentration in stressed regions. The hydrogen concentration in a stressed region (H

_{σ}) is a function of the lattice hydrogen in unstressed regions (H

_{L}) and the hydrostatic pressure (I

_{1}):

_{0}is the average initial diameter of voids in the material. Stress-invariant ratios are used in conjunction with coefficients A

_{void}, B

_{void}, and n to capture stress-state sensitivity. The coefficient C

_{Tν}controls the void growth ISV’s temperature dependence.

_{2}is a void nucleation and growth-sensitivity coefficient.

#### 2.2. Part II: Parameter Sensitivity Study

#### 2.2.1. Second Phase Particle Number Density and Size

_{0}) are treated as independent design variables and average particle area fraction (f) is calculated as the product of particle number density and cross-section area:

^{2}–10

^{3}particles/mm

^{2}. The nominal upper and lower bounds for η were selected as 250 and 4000 particles/mm

^{2}, respectively, for the parameter sensitivity study.

^{1/2}/f

^{1/3}relationship, thus increasing particle size for a given volume fraction increases void nucleation rates. In steels, secondary-phase particle diameters have been observed to range from the order of 10

^{−4}to 10

^{−2}mm (Table 2). The values for particle size, d, correspondingly range from 10

^{−4}to 10

^{−2}mm in the parameter sensitivity study.

#### 2.2.2. Grain Size

_{0}, GS) and the grain-size sensitivity exponent (Z). The grain size sensitivity exponent, Z, has been calibrated for a variety of materials, including steel [112]. Grain sizes in steel have been shown to vary from 1 μm in diameter for high-strength steels (and even less than 1 μm in specially designed ultra-fine-grain materials) to greater than a 100 μm size for mild steels [98,105,113]. The bounds for the grain size term, GS, were selected as 1 μm and 100 μm, accordingly. The reference grain size, GS

_{0}, was designated a control variable with a nominal value of 10 μm.

#### 2.2.3. Initial Void Volume Fraction

^{−2}to 10

^{−1}for cast steels [116,117,118]. Conversely, wrought steels may exhibit very initial porosities on the order of 10

^{−4}(cf. [82]). Horstemeyer and Ramaswamy [72] noted that the experimental quantification of void volume fractions lower than 10

^{−4}is difficult, but numerically demonstrated the significant effects of a microporosity of 10

^{−6}on void growth rates in aluminum and 304 L stainless steel alloys. For metal alloys, minimal strain is required for failure beyond aggregate void volume fractions of 10

^{−1}. Therefore, the upper and lower bounds for initial porosity (ϕ

_{0}) in the parameter sensitivity study were selected as 10

^{−2}and 10

^{−6}, respectively.

#### 2.2.4. Lattice Hydrogen Concentration

^{−5}and 10

^{−4}atomic parts per million (APPM) in 1518 spheroidized steel were studied and showed a significant increase in nucleation rate relative to unhydrogenated materials. Lee and Gangloff [129] observed lattice hydrogen concentrations as high as 10

^{−2}APPM in hydrogen embrittled steels. In the present study, the upper and lower bounds for lattice hydrogen concentration, H

_{L}, were selected as 10

^{−3}and 10

^{−5}APPM, respectively. Values for the binding energy, W

_{B}= 56 kJ·mol

^{−1}, gas constant, R = 8.31 J·mol

^{−1}·K

^{−1}, molar hydrogen volume, V = 2.0 cm

^{3}·mol

^{−1}, hydrogen sensitivity, m

_{h}= 3.0, and coefficients used in Equations (10) and (11) were obtained from [85].

#### 2.2.5. Material Hardness

_{03}, C

_{09}, and C

_{15}were varied to achieve the desired mechanical hardness levels and the values of the constants are shown in Table 3.

#### 2.2.6. Design of Experiments

_{0}), particle diameter (d), grain diameter (GS), initial void volume fraction (ϕ

_{0}), lattice hydrogen concentration (H

_{L}), and Brinell hardness. Due to the nonlinear relationship between hardness and ballistic merit observed by [63], each parameter was assigned five possible levels, the bounds of which are discussed in previous sections. A finite element simulation of the impact event was simulated for each unique parameter set and, in each instance, the predicted residual velocity was assessed. The corresponding orthogonal array is the L

_{25}(5

^{6}), or L

_{25}, array which allows up to six independent parameters with five levels. A full factorial set of calculations would require 5

^{6}= 15,625 separate calculations; the L

_{25}array, however, requires only 25. Similar DOE-based computational approaches for studying void growth and nucleation using an ISV constitutive model can be found in [79].

**P**]:

**P**], {R}, and {A} are described in [134] as:

**P**] values of +1, +0.5, 0, −0.5, and −1 represent levels 1, 2, 3, 4, and 5, respectively. The values of each column of [

**P**] are selected such that the inner product between any two columns is zero, satisfying the orthogonality condition. Orthogonality of [

**P**] ensures that each value of {A} uniquely describes test result sensitivity to one test variable (columns of [

**P**]). Table 4 maps each parameter’s respective values to the levels used to populate [

**P**].

**P**], was multiplied by both sides of Equation (15). Since [

**P**] is a non-square matrix of dimensions m × n = 25 × 6, where m (25) > n (6), the inverse of [

**P**] is formulated as a left inverse matrix, that is,

#### 2.3. Part III: Modeling the Microstructurally Driven Transition of Penetration Modes for Increasing Material Hardness

_{0}, particle size, d, particle volume fraction, f, and grain size, GS. Sources suggest that lower quench rates and higher tempering temperatures used to achieve softer materials lead to a coarsening of second-phase particles and precipitates accompanied by a reduction in undissolved carbides [133,136,137,138,139]. The rapid quench rates and lower tempering temperatures necessary for achieving and sustaining high volume fraction martensitic grain (high-hardness) steels could plausibly result in finely dispersed, relatively high number density distributions of second-phase particles. Grain size is correlated to thermomechanical processing temperature and time due to recrystallization (cf. [140]). Grain size and hardness are inversely correlated for a given steel alloy, consistent with the findings of [105]. The trends of increasing particle number density and decreasing particle and grain size with increasing hardness were assumed for this study. Particle volume fraction was calculated as a function of size and number density using Equation (14). Specific values for each microstructure property were selected to qualitatively predict the perforation mode and ballistic performance trends discussed in [63]. The ranges of microstructure property values used in this study are bounded by experimentally observed properties for various steel alloys contained in Table 2.

^{2}) is based on the steel literature findings contained in Table 2. Second-phase particle diameters in steel can vary widely from less than 5 nm [139] to approximately 10 μm [101], which were selected as the particle diameter bounds for this study. These bounds exclude high volume fraction second-phase grains, which can exceed 10 µm. Future studies would benefit from the quantitative microstructural characterization of RHA steel alloys over the 250–550 BHN range to develop physically representative ISV model calibrations for non-idealized materials.

_{p}is the perforation velocity of the current material system and is normalized by a nominal perforation velocity (taken to be V

_{p}at 250 BHN), V

_{p}

_{0}.

## 3. Results and Discussion

#### 3.1. Part I: Validation of Internal State Variable Finite Element Framework

#### 3.2. Part II: Parameter Sensitivity Study

**P**] to solve Equation (18) and determine the sensitivity of residual velocity to parameters (1)–(6). The results of the parameter sensitivity array {A} were normalized by the maximum value of {A} and are shown in Figure 6.

**P**]. Under these assumptions, the normalized sensitivity of residual velocity to material hardness (A

_{6}) was determined to be 0.78. Thus, material hardness is a strong secondary influence on the ballistic performance of metal targets. Future studies would benefit from considering the correlation between mechanical strength and ductility.

_{pore}= 0.01) in test 8 (see Table 5) could have played a dominant role in the test’s high residual velocity (518.94 m/s) because of the otherwise low lattice hydrogen concentration (10

^{−5}APPM) and moderate target and projectile hardness (350 BHN). However, high residual projectile velocities only occurred at initial porosities under 10

^{−2}in the presence of either significant lattice hydrogen concentrations (10

^{−3}and 5·10

^{−4}APPM in tests 7 and 13, respectively) or low material hardness (250 BHN in tests 7, 13, 18, and 19). Table 5 shows no conclusive correlation between particle number density, particle diameter, and grain diameter for tests resulting in the highest and lowest projectile residual velocities.

#### 3.3. Part III: Modeling the Microstructurally Driven Transition of Penetration Modes for Increasing Target Material Hardness

**ε**) and grey hues denote regions of material experiencing

_{p}**ε**> 0.5.

_{p}_{0}= 200 particles/mm

^{2}, K

_{IC}= 2800 MPa·mm

^{1/2}) impacted at a ballistic limit velocity of 850 m/s, perforation tends to occur due to rupture resulting from large bending deformation, consistent with [63]. Figure 7b shows that perforation of the 400 BHN plate (grain diameter = 7.5 μm, d = 3.5 μm, f = 0.0011, η

_{0}= 200 particles/mm

^{2}, K

_{IC}= 2625 MPa·mm

^{1/2}) due to impact at 1000 m/s ballistic limit velocity occurs via combined bending and shear petaling perforation modes. Figure 8 shows that for these theoretical materials, the targets sustain large plastic deformation that facilitates impact energy dissipation resulting in a domain of increasing Ballistic Merit (BM = 1.0625 at 300 BHN and BM = 1.25 at 400 BHN—Figure 8). These materials can sustain relatively large plastic strain fields due to the low initial particle density distributions and high fracture toughness, which contribute to low initial void nucleation rates and delayed onset of strain localization and fracture.

_{0}= 2000 particles/mm

^{2}, K

_{IC}= 1800 MPa·mm

^{1/2}) impacted at a ballistic limit velocity of 950 m/s shows perforation via localized shear plugging (Figure 7c). The increase in second-phase particle number density and decrease in fracture toughness (Figure 3) contribute to the increased nucleation rates under plastic strain relative to softer target materials. Through Figure 6, we infer that the increase initial particle number density and decreasing grain size have a compounding contribution to the total damage accumulation rate (thus adversely affecting ballistic performance) through the void nucleation and void coalescence rates, respectively. Logically, increasing second-phase particle number density (2000 particles/mm

^{2}at 500 BHN, 200 particles/mm

^{2}at 400 and 300 BHN) increases the number of potential void nucleation sites, while decreasing the material grain size (2.5 μm at 500 BHN, 12.5 and 7.5 μm at 300 and 400 BHN, respectively) and decreasing the distance between defects (particles, subgrains, and high-angle grain boundaries) and thus stimulates more rapid onset of void interaction and instability. While the property distributions for these simulation materials are theoretical, each value selected is representative of documented microstructure characteristics in steels shown through Table 2.

**D**, and damage, $\varphi $, in Equation (3)) and ultimately produces localized shear fracture modes (plugging). The localized fracture dissipates less energy than the large plastic-ductility perforation modes, and the target shows perforation at lower velocities (950 m/s at 500 BHN) than softer targets (1000 m/s at 400 BHN). This tendency reflects non-monotonic ballistic performance with the material hardness trend documented in [63].

_{in}- Increasing lattice hydrogen concentration (from 10
^{−5}to 10^{−3}APPM) exponentially increases void nucleation rates (Equation (9)) in materials subjected to greater than zero stress triaxiality through [85]. As the material expands under tension, hydrogen is freer to migrate through the lattice to preferential trapping sites near grain or inclusion boundaries. The hydrogen reduces the local fracture toughness contributing to increasing void nucleation rates. Macroscopically, this corresponds to more localized fracture and perforation modes in high-tensile pressure regions, less energy absorption through global plastic strain accumulation, and reduced perforation velocities. - Perforation velocity is strongly sensitive to target material hardness. Increasing hardness (and corresponding yield and rate hardening characteristics) from 250 BHN to 550 BHN increases the mechanical work for a given strain, thus reducing greater energy from the impact event, and subsequently increasing the velocity required to perforate the material.
- Perforation velocity shows finite normalized sensitivity to initial void volume fraction (0.38), grain size (0.33), and particle number density (0.25) in descending order. These sensitivities are relatively smaller than the lattice hydrogen concentration (1.0) and hardness sensitivities (0.78) but are still significant. Increasing levels of initial porosity from 10
^{−6}to 10^{−2}increases material compliance (Equation (3)) and reduces the void nearest-neighbor distance, thus accelerating the onset of void coalescence and unstable fracture (consider the coupled effects of Equations (6) and (13)). Reducing the grain size exponentially increases void coalescence rate (GS in Equation (13)) through reduction in defect nearest-neighbor distances, contributing to the earlier onset of coalescence with strain once voids have nucleated. Increasing the second-phase particle number density from 250 to 4000 voids/mm^{2}increases the number of initial points of localized stress concentration and void nucleation in the microstructure. Physically, increasing the particle number density reduces the neighbor distance once voids have begun to nucleate, contributing to earlier material instability. - Perforation velocity shows negligible sensitivity to particle size. Mathematically, nucleation rate, $\dot{\eta}$ is correlated to particle diameter, d, through a square root relationship $\dot{\eta}$ ∝ d
^{1/2}, while the nucleation rate’s relationship to lattice hydrogen concentration and particle number density is exponential and linear, respectively. Physically, the fracture of larger particles would produce conditions conducive to void growth (Equation (12)), However, ballistic penetration is a high strain rate event that favors a large quantity of nucleation events leading to final rupture through coalescence rather than the growth of large voids. - Diminished ballistic performance trends in high hardness (>450 BHN) targets experimentally observed by [63] can be qualitatively predicted using an ISV based framework featuring decreasing grain size, particle size, particle volume fraction, and fracture toughness and increasing particle number density with increasing material hardness in accord with qualitative trends documented in the relevant literature. A single order of magnitude increase in particle number density (250 to 2000 particles/mm
^{2}) and corresponding reduction in grain size (5 to 2.5 μm) from 450 to 500 BHN hardness degrades the Ballistic Merit from 1.24 to 1.13, despite the increase in yield and flow strength of the material. The reduction in ballistic resistivity is caused by a transition from large ductility perforation modes to localized shear plugging (Figure 7). - Traditional constitutive modeling approaches can struggle to successfully predict changes in perforation behavior for alloys of similar element composition and varying processing history [56,63,64]. This work demonstrates that a deformation history- and microstructure property-dependent ISV constitutive model can predict changes in ballistic perforation modes and nonmonotonic trends in ballistic limit for increasing target strength.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

RHA Steel (250 BHN) | Value |
---|---|

Density (tonne/mm^{3}) | 7.83 × 10^{−9} |

Elastic Modulus (MPa) | 205,000 |

Poisson’s Ratio | 0.29 |

Thermal Conductivity (W·m·K^{−1}) | 42.5 |

Specific Heat (J·kg^{−1}·K^{−1}) | 480 |

Thermal Expansion (K^{−1}) | 1.15 × 10^{−5} |

Inelastic Heat Fraction | 0.3336 |

Melt Temperature (K) | 1803.15 |

ISV Model Coefficients | - |

C_{01} (MPa) | 5 |

C_{02} (K) | 0 |

C_{03} (MPa) | 690 |

C_{04} (K) | 22 |

C_{05} (MPa^{−1}) | 0.3 |

C_{06} (K) | 0 |

C_{07} (MPa^{−1}) | 0.3 |

C_{08} (K) | 150 |

C_{09} (MPa) | 4416 |

C_{10} (K) | 2 |

C_{11} (s·MPa^{−1}) | 0 |

C_{12} (K) | 0 |

C_{13} (MPa^{−1}) | 0.07 |

C_{14} (K) | 121.5 |

C_{15} (MPa) | 700 |

C_{16} (K) | 0 |

C_{17} (s*MPa^{−1}) | 0 |

C_{18} (K) | 0 |

C_{19} | 0.006 |

C_{20} (K^{−1}) | 1100 |

C_{21} | 0 |

C_{a} | −0.3 |

C_{b} | 0 |

A_{void} | 0 |

B_{void} | 0 |

a | 32,000 |

b | 10,800 |

c | 36,000 |

η_{0} (#/mm^{2}) | 200 |

K_{IC} (MPa·mm^{½}) | 2751 |

d (mm) | 0.0035 |

f | 0.00065 |

NND (mm) | 0.16 |

d_{0} (mm) | 0.002 |

cd_{2} | 1.5 |

GS_{0} (mm) | 0.01 |

GS (mm) | 0.01 |

ζ | 1 |

Initial Porosity | 0.00065 |

C_{TN} (K) | 300 |

C_{TC} (K^{−1}) | 0.002 |

McClintock Growth, n | 0.3 |

R_{0} (mm) | 0.001 |

Cocks-Ashby Growth, m | 20 |

RHA Steel | Hardness (BHN) | ||||||
---|---|---|---|---|---|---|---|

250 | 300 | 350 | 400 | 450 | 500 | 550 | |

Density (tonne/mm^{3}) | 7.83 × 10^{−9} | 7.83 × 10^{−9} | 7.83 × 10^{−9} | 7.83 × 10^{−9} | 7.83 × 10^{−9} | 7.83 × 10^{−9} | 7.83 × 10^{−9} |

Elastic Modulus (MPa) | 205,000 | 205,000 | 205,000 | 205,000 | 205,000 | 205,000 | 205,000 |

Poisson’s Ratio | 0.29 | 0.29 | 0.29 | 0.29 | 0.29 | 0.29 | 0.29 |

Conductivity (W·m·K^{−1}) | 42.5 | 42.5 | 42.5 | 42.5 | 42.5 | 42.5 | 42.5 |

Specific Heat (J·kg^{−1}·K^{−1}) | 480 | 480 | 480 | 480 | 480 | 480 | 480 |

Thermal Expansion (K^{−1}) | 1.15 × 10^{−5} | 1.15 × 10^{−5} | 1.15 × 10^{−5} | 1.15 × 10^{−5} | 1.15 × 10^{−5} | 1.15 × 10^{−5} | 1.15 × 10^{−5} |

Inelastic Heat Fraction | 0.3336 | 0.3336 | 0.3336 | 0.3336 | 0.3336 | 0.3336 | 0.3336 |

Melt Temperature (K) | 1803.15 | 1803.15 | 1803.15 | 1803.15 | 1803.15 | 1803.15 | 1803.15 |

ISV Model Coefficients | - | - | - | - | - | - | - |

C_{01} (MPa) | 5 | 5 | 5 | 5 | 5 | 5 | 5 |

C_{02} (K) | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

C_{03} (MPa) | 690 | 803 | 1003 | 1113 | 1203 | 1300 | 1400 |

C_{04} (K) | 22 | 22 | 22 | 22 | 22 | 22 | 22 |

C_{05} (MPa^{−1}) | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 |

C_{06} (K) | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

C_{07} (MPa^{−1}) | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 |

C_{08} (K) | 150 | 150 | 150 | 150 | 150 | 150 | 150 |

C_{09} (MPa) | 4416 | 4416 | 4416 | 4416 | 4416 | 5216 | 6016 |

C_{10} (K) | 2 | 2 | 2 | 2 | 2 | 2 | 2 |

C_{11} (s·MPa^{−1}) | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

C_{12} (K) | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

C_{13} (MPa^{−1}) | 0.07 | 0.07 | 0.07 | 0.07 | 0.07 | 0.07 | 0.07 |

C_{14} (K) | 121.5 | 121.5 | 121.5 | 121.5 | 121.5 | 121.5 | 121.5 |

C_{15} (MPa) | 700 | 700 | 700 | 700 | 700 | 1000 | 1300 |

C_{16} (K) | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

C_{17} (s·MPa^{−1}) | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

C_{18} (K) | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

C_{19} | 0.006 | 0.006 | 0.006 | 0.006 | 0.006 | 0.006 | 0.006 |

C_{20} (K^{−1}) | 1100 | 1100 | 1100 | 1100 | 1100 | 1100 | 1100 |

C_{21} | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

C_{a} | −0.3 | −0.3 | −0.3 | −0.3 | −0.3 | −0.3 | −0.3 |

C_{b} | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

A_{void} | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

B_{void} | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

a | 32,000 | 32,000 | 32,000 | 32,000 | 32,000 | 32,000 | 32,000 |

b | 10,800 | 10,800 | 10,800 | 10,800 | 10,800 | 10,800 | 10,800 |

c | 360 | 360 | 360 | 360 | 360 | 360 | 360 |

η_{0} (#/mm^{2}) | 200 | 200 | 200 | 200 | 250 | 2000 | 4000 |

K_{IC} (MPa·mm^{½}) | 2846 | 2800 | 2751 | 2625 | 2530 | 1802 | 1500 |

d (mm) | 0.007 | 0.00525 | 0.0035 | 0.002625 | 0.0015 | 0.000035 | 0.0000035 |

f | 0.00245 | 0.001378 | 0.00065 | 0.000517 | 0.000375 | 1.23 × 10^{−8} | 6.13 × 10^{−7} |

NND (mm) | 0.08 | 0.08 | 0.08 | 0.08 | 0.08 | 0.08 | 0.08 |

d_{0} (mm) | 0.002 | 0.002 | 0.002 | 0.002 | 0.002 | 0.002 | 0.002 |

cd_{2} | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 |

GS_{0} (mm) | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 |

GS (mm) | 0.015 | 0.0125 | 0.01 | 0.0075 | 0.005 | 0.0025 | 0.001 |

ζ | 1.3 | 1.3 | 1.3 | 1.3 | 1.3 | 1.3 | 1.3 |

Initial Porosity | 0.00065 | 0.00065 | 0.00065 | 0.00065 | 0.00065 | 0.00065 | 0.00065 |

C_{TN} (K) | 300 | 300 | 300 | 300 | 300 | 300 | 300 |

C_{TC} (K^{−1}) | 0.002 | 0.002 | 0.002 | 0.002 | 0.002 | 0.002 | 0.002 |

McClintock Growth, n | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 |

D_{0} (mm) | 0.001 | 0.001 | 0.001 | 0.001 | 0.001 | 0.001 | 0.001 |

Cocks-Ashby Growth, m | 20 | 20 | 20 | 20 | 20 | 20 | 20 |

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**Figure 1.**Abaqus Explicit FEA mesh and boundary conditions for ballistic impact of a semi-infinite square target by spherical (

**a**) and cylindrical projectiles (

**b**).

**Figure 3.**Microstructure properties at increasing hardness levels of RHA steel alloy. Grain diameter (

**a**) particle diameter (

**b**), and particle volume fraction (

**c**) decrease with hardness while particle number density (

**d**) increases, following qualitative trends noted in the steel literature.

**Figure 4.**Comparison experimental (generated by ERDC) finite element model predicted normalized residual velocities for 12.7 mm RHA steel spheres impacting 9.53 mm and 12.7 mm thick RHA steel plates. The model shows strong agreement with experimental V50 perforation velocities and residual velocity slopes.

**Figure 5.**Residual velocity results of 25 impact simulations used to populate array {R} in the parametric sensitivity study. Each resultant value corresponds to a row of DOE test variable conditions presented in orthogonal array [

**P**].

**Figure 6.**Comparison of residual velocity sensitivity to six parameters for an RHA steel cylinder impacting a 6.35 mm thick RHA steel target. Lattice hydrogen concentration dominates material response and impactor–target material hardness as a highly influential secondary factor. Microstructure properties of initial void volume fraction, grain size, and particle number density are tertiary contributors to projectile residual velocity.

**Figure 7.**Plastic equivalent strain contours for impact perforation of varying Brinell hardness (BHN) RHA steel plates of (

**a**) 300 BHN at 850 m/s, (

**b**) 400 BHN at 1000 m/s, and (

**c**) 500 BHN at 950 m/s. Plates with BHN values of 300 and 400 exhibit damage evolution and perforation due to large plastic flow and combined tensile and shear fracture. The perforation mode for these materials resembles shear petaling. At 500 Brinell hardness, the damage evolution transitions to a plugging mode are dominated by localized shear concentrations. The low and intermediate hardness plates in (

**a**,

**b**) exhibit larger regions of significant plastic deformation relative to the high-hardness material exhibiting localized fracture (

**c**).

**Figure 8.**Comparison of Ballistic Merit for vacuum-induction-melted and electroslag-remelted 4340 steel alloys obtained from [63] and rolled homogeneous armor steel (RHA) at varying hardness. A threshold exists beyond 400 BHN where the penetration mode transitions from fracture due to plastic flow to localized shear plugging mechanisms. A reduction in ballistic merit is observed to correspond to the target material’s propensity to fail by localized damage evolution at high hardness. Relatively high impurity materials (those that are vacuum induction melted, VIM) exhibit less-favorable ballistic resistance properties than low-impurity materials (those that are electroslag remelted, ESR) due to the higher prevalence of stress-concentrating microstructural heterogeneities.

**Table 1.**Geometric properties of quarter-symmetry projectiles and targets used in Abaqus-Explicit finite element analysis simulations. Here, the quarter-symmetry values for target width are listed.

Case | Object | Shape | Thickness (mm) | Width (mm) | Diameter (mm) | # Elements |
---|---|---|---|---|---|---|

1 | Projectile | Sphere | - | - | 12.70 | 11,424 |

Target | Square | 9.53 | 127.0 | - | 206,400 | |

2 | Projectile | Sphere | - | - | 12.70 | 10,136 |

Target | Square | 12.70 | 170.0 | - | 238,680 | |

3 | Projectile | Cylinder | - | 12.70 | 6.35 | 3840 |

Target | Square | 6.35 | 63.50 | - | 97,344 |

**Table 2.**Second-phase particle property distributions in high-strength steels from literature sources.

Study | Material | Number Density (#/mm ^{2}) | Diameter (μm) | Vol. Fraction |
---|---|---|---|---|

[101] | 4340 Steel | 800–4000 | 4.5–9.7 | 0.060 |

[102] | 0.17-0.44 C Steel | 430 | 14.0 | 0.066 |

[103] | HY 180 Steel | 2600–6000 | 0.20–0.32 | 0.00019–0.00021 |

[82] | RHA Steel | 170 | 7.0 | 0.00065 |

**Table 3.**Mechanical properties and model coefficients for steel alloys of varying Brinell hardness (BHN).

Brinell Hardness (BHN) | ||||||
---|---|---|---|---|---|---|

Description | 250 | 350 | 450 | 500 | 550 | |

Yield (MPa) | - | 700 | 1075 | 1250 | 1400 | 1500 |

UTS (MPa) | - | 870 | 1150 | 1392 | 1600 | 1740 |

C03 (MPa) | Model constant affecting yield | 690 | 1003 | 1203 | 1300 | 1403 |

C09 (MPa) | Kinematic hardening modulus | 4416 | 4416 | 4416 | 5216 | 6016 |

C15 (MPa) | Isotropic hardening modulus | 700 | 700 | 700 | 1000 | 1300 |

Parameter Levels | |||||
---|---|---|---|---|---|

Parameter | +1 | +0.5 | 0 | −0.5 | −1 |

Particle No. Density (#/mm^{2}) | 250 | 500 | 1000 | 2000 | 4000 |

Particle Diameter (μm) | 0.1 | 0.5 | 1.0 | 5.0 | 10.0 |

Grain Diameter (μm) | 1.0 | 5.0 | 10.0 | 50.0 | 100.0 |

Initial Porosity | 10^{−6} | 10^{−5} | 10^{−4} | 10^{−3} | 10^{−2} |

Lattice Hydrogen (APPM) | 10^{−5} | 10^{−4} | 2.5 × 10^{−4} | 5 × 10^{−4} | 10^{−3} |

Brinell Hardness | 250 | 350 | 450 | 500 | 550 |

**Table 5.**Select test levels and resultant residual velocity values from parameter sensitivity study.

Test | Particle No. Density (μm) | Particle Diameter (μm) | Grain Diameter (μm) | Initial Porosity | Lattice Hydrogen (APPM) | Brinell Hardness (BHN) | Residual Velocity (m/s) |
---|---|---|---|---|---|---|---|

7 | 500 | 0.5 | 10 | 10^{−3} | 10^{−3} | 250 | 589.60 |

8 | 500 | 1.0 | 50 | 10^{−2} | 10^{−5} | 350 | 518.94 |

13 | 1000 | 1.0 | 100 | 10^{−5} | 5 × 10^{−4} | 250 | 559.44 |

19 | 2000 | 5.0 | 5 | 10^{−2} | 2.5 × 10^{−4} | 250 | 521.16 |

25 | 4000 | 10.0 | 50 | 10^{−4} | 10^{−4} | 250 | 560.12 |

9 | 500 | 5.0 | 100 | 10^{−6} | 10^{−4} | 450 | 0 |

10 | 500 | 10.0 | 1 | 10^{−5} | 2.5 × 10^{−4} | 500 | 0 |

15 | 1000 | 10.0 | 5 | 10^{−3} | 10^{−5} | 450 | 0 |

18 | 2000 | 1.0 | 1 | 10^{−3} | 10^{−4} | 550 | 0 |

24 | 4000 | 5.0 | 10 | 10^{−6} | 10^{−5} | 550 | 0 |

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**MDPI and ACS Style**

Peterson, L.; Horstemeyer, M.; Lacy, T.; Moser, R.
Using an Internal State Variable Model Framework to Investigate the Influence of Microstructure and Mechanical Properties on Ballistic Performance of Steel Alloys. *Metals* **2023**, *13*, 1285.
https://doi.org/10.3390/met13071285

**AMA Style**

Peterson L, Horstemeyer M, Lacy T, Moser R.
Using an Internal State Variable Model Framework to Investigate the Influence of Microstructure and Mechanical Properties on Ballistic Performance of Steel Alloys. *Metals*. 2023; 13(7):1285.
https://doi.org/10.3390/met13071285

**Chicago/Turabian Style**

Peterson, Luke, Mark Horstemeyer, Thomas Lacy, and Robert Moser.
2023. "Using an Internal State Variable Model Framework to Investigate the Influence of Microstructure and Mechanical Properties on Ballistic Performance of Steel Alloys" *Metals* 13, no. 7: 1285.
https://doi.org/10.3390/met13071285