# Toward a Better Understanding of Shock Imprinting with Polymer Molds Using a Combination of Numerical Analysis and Experimental Research

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Section

^{3}, respectively. The SEP explosive was detonated by the No. 6 electric detonator manufactured by Kayak Japan Co., Ltd. The area of action of the underwater shock wave generated by this explosion was overwhelmingly larger than the molding area of the sample; therefore, the pressure applied to the workpiece and mold was considered uniform. The metal sample used in the experiment was an Al alloy (A1100) manufactured by Niraco Co., Ltd. (Tokyo, Japan), and its thickness was 100 µm. The mold was a PC plate provided by Takiron Co., Ltd. (Osaka, Japan), and its thickness was 2.0 mm. The PC plate contained a blind hole on one side with a diameter of 1.0 mm and a depth of 0.35 mm. The mold and the Al workpiece were placed in a vacuum polyethylene bag with a thickness of 70 µm and evacuated to an internal pressure of 0.01 MPa, as shown in Figure 2b. They were placed on a steel anvil at a distance H from the explosive. A set of imprinting experiments were conducted with distance H in the range of 20 to 100 mm. After imprinting, the sample was studied with JCM-5700, a scanning electron microscope (SEM) manufactured by JEOL Ltd. (Tokyo, Japan).

## 3. Numerical Simulation

_{1}, R

_{2}, and ω are the empirical coefficients.

_{H}is the Hugoniot pressure, Γ is the Gruneisen gamma, ρ is the density, e is the internal energy, ${e}_{H}$ is the Hugoniot internal energy, Γ

_{ρ}is the Gruneisen gamma of the reference state, ${\rho}_{0}$ is the initial density, ${c}_{0}$ is the bulk velocity of the sound, μ is the compression, and s is the linear Hugoniot gradient coefficient.

_{y}) is defined as follows:

_{p}is the equivalent plastic strain; $\dot{{\epsilon}_{p}^{*}}=\dot{\epsilon}/\dot{{\epsilon}_{ref}}$ is the dimensionless strain rate; and T* is the dimensionless temperature. The dimensionless temperature is defined as $\left\{{T}^{*}=\left(T-{T}_{room}\right)\u2044\left({T}_{m}-{T}_{room}\right)\right\}$. Additionally, $\dot{\epsilon}$ is the strain rate, $\dot{{\epsilon}_{ref}}$ is the reference strain rate, T is the current temperature, T

_{m}is the melting temperature of the alloy, and ${T}_{room}$ is the room temperature.

_{1}to D

_{5}are the material parameters, ${\sigma}^{*}={\sigma}_{m}\u2044{\sigma}_{eq}$ is the stress triaxial ratio, σ

_{m}is the mean stress, and σ

_{eq}is the equivalent von Mises stress. Depending on the equivalent fracture strain, the damage parameters can be calculated as follows:

## 4. Results and Discussion

#### 4.1. Experimental Results

#### 4.2. Numerical Simulation Results

^{7}and 10

^{6}/s for H = 20 mm and other conditions, respectively. This indicates the strain-rate region, where the dislocation avalanche can occur in the work piece. In the LSI studies using Si nanomolds, the workpiece deformation is constrained to the edges of the mold at strain rates >10

^{5}/s, resulting in a high dislocation density and dislocation avalanches in the workpiece (large deformation) [5,11,24]. On the other hand, when a polymer mold was used, the mold deformed significantly under pressure conditions in the order of gigapascals (Figure 7) (H = 20 mm). Therefore, it became difficult for the transition to concentrate on the work piece at the edge of the mold, and the dislocation avalanche did not occur. Even in the sub-micrometer-order shock-imprinting phenomenon, polymer molds could not maintain their shape under high-pressure conditions. The imprinting mechanism using the polymer mold was essentially different from that using the rigid silicon mold.

^{7}/s is achieved during the imprinting, Equation (6) indicates that the yield strength would only increase by a factor of 1.26 for Al and 1.84 for PC. Therefore, the increase in strength of the PC mold due to the increase in the strain rate does not significantly affect the deformation process in shock imprinting (at least when the pressure is of the order of 1 GPa (H = 20 mm)).

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Imprint experiment: an Al foil was compressed into a DVD mold by an underwater shock wave derived from an explosive, yielding a deeper imprinting shape than the DVD mold.

**Figure 2.**Schematic illustration of the shock-imprinting and pressure measurement experiments using a PC mold. (

**a**) The layout of an explosive and sample in water. (

**b**) Details of the placement of the PC mold and Al workpiece. (

**c**) Details of the PVDF gauge placement.

**Figure 3.**Numerical-simulation model using 2D Lagrangian and Eulerian solvers. (

**a**) Dimensions and shapes of the PC mold, Al workpiece, and water. (

**b**) Setting of the initial conditions. (

**c**) Underwater shock-wave model consisting of the explosive and water.

**Figure 4.**Final shape of an Al workpiece under varying pressure conditions. (

**a**) Surface of the Al workpiece at pressure-loading side. (

**b**) Cross-section of the workpiece.

**Figure 5.**Comparison of the pressure histories received from experimental and numerical simulations. (

**a**) Comparison of the peak pressure values and shapes at different values of H. (

**b**) Comparison of the pressure rise process for H = 20 mm.

**Table 1.**Jones–Wilkins–Lee (JWL) equation of state (EOS) parameters for the SEP explosive [16].

Parameter | Value |
---|---|

Reference density (g/cm^{3}) | 1.31 |

A (kPa) | 3.65 × 10^{8} |

B (kPa) | 2.31 × 10^{6} |

R_{1} | 4.3 |

R_{2} | 1.0 |

Ω | 0.28 |

C-J energy/unit volume e (kJ/m^{3}) | 3.761 × 10^{6} |

C-J detonation velocity (m/s) | 6.97 × 10^{3} |

C-J Pressure P_{CJ} (kPa) | 1.59 × 10^{7} |

A1100 | PC | Water | |
---|---|---|---|

Reference density (kg/m^{3}) | 2.707 | 1.197 | 1.00 |

Gruneisen gamma | 1.970 | 0.61 | 0.28 |

${c}_{0}$ (m/s) | 5386 | 1933 | 1483 |

s | 1.339 | 2.6050 | 1.75 |

Reference temperature (K) | 293 | 300 | - |

Specific heat (J/kg·K) | 884 | - | - |

A1100 | PC | |
---|---|---|

A (MPa) | 140 | 75.8 |

B (MPa) | 157 | 68.9 |

C | 0.016 | 0.052 |

n | 0.167 | 1 |

m | 1.7 | 1.85 |

D_{1} | 0.071 | - |

D_{2} | 1.248 | - |

D_{3} | −1.142 | - |

D_{4} | 0.0097 | - |

D_{5} | 0.0 | - |

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**MDPI and ACS Style**

Hasegawa, K.; Tanaka, S.; Bataev, I.; Inao, D.; Nishi, M.; Kubota, A.; Hokamoto, K.
Toward a Better Understanding of Shock Imprinting with Polymer Molds Using a Combination of Numerical Analysis and Experimental Research. *Materials* **2022**, *15*, 1727.
https://doi.org/10.3390/ma15051727

**AMA Style**

Hasegawa K, Tanaka S, Bataev I, Inao D, Nishi M, Kubota A, Hokamoto K.
Toward a Better Understanding of Shock Imprinting with Polymer Molds Using a Combination of Numerical Analysis and Experimental Research. *Materials*. 2022; 15(5):1727.
https://doi.org/10.3390/ma15051727

**Chicago/Turabian Style**

Hasegawa, Kouki, Shigeru Tanaka, Ivan Bataev, Daisuke Inao, Matatoshi Nishi, Akihisa Kubota, and Kazuyuki Hokamoto.
2022. "Toward a Better Understanding of Shock Imprinting with Polymer Molds Using a Combination of Numerical Analysis and Experimental Research" *Materials* 15, no. 5: 1727.
https://doi.org/10.3390/ma15051727