# PVDF Based Pressure Sensor for the Characterisation of the Mechanical Loading during High Explosive Hydro Forming of Metal Plates

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{®}—(Numerics Gmbh) based on computational fluid dynamic (CFD) and—RADIOSSv2017

^{®}—(ALTAIR Hyperworks) that uses a finite element method in an explicit scheme. The third objective is to give analytical interpretations, based on the shock polar technique, of experimental observations carried out on large and thick metal plate. For confidential reasons, the explosive charge and pool details are not evoked. Thus, all time, pressure, and distance results implying the explosive are dimensionless.

## 2. Materials and Methods

#### 2.1. PVDF Sensor Description and Implementation

- R#1 M$\Omega $
- C # 1500 pF
- $R\ast C$ # 1.5 ms

#### 2.2. HEHF Experiments

#### 2.3. Analytical Method

^{®}numerical file was coded; it calculates the value of the incident pressure knowing the value of the measured pressure peak. Indeed, state 3 results from the interaction of a shock coming from aluminium shield and transmitted into steel. According to the planar shock interaction, pressure induced by the shock will be equal in steel and aluminium.

## 3. Results

#### 3.1. Explosive Charge Directly above the Sensor

#### 3.2. Angle of Incidence Effects

## 4. Discussion

^{®}and RADIOSS

^{®}. Shock physics explicit Eulerian/Lagrangian dynamics (SPEED

^{®}) is developed by Numerics Gmbh, and has capability to perform 2D and 3D multi-material Eulerian simulations, 2D Lagrangian models and 3D ideal gas calculations. It is based on an explicit solver for non-linear problems and is proficient for shocks and explosion analysis. In addition, an adaptive mesh, and mesh activation method are implemented and are revealed to be very efficient for saving computational time. RADIOSS

^{®}is an ALTAIR

^{®}code based on a finite element method using an explicit solver, mainly for crashes and multi-physics analysis. It has multiple possibilities such as Eulerian, Lagrangian, Arbitrary Langrangian Eulerian (ALE), Coupled Eulerian-Lagrangian (CEL) and Smooth Particles Hydrodynamics (SPH) simulations in 2D and 3D.

^{®}shock EoS based on Mie-Grüneisen and shock EoS. SPEED

^{®}Johnson-Cook constitutive law (a combination of shock EoS and visco-elasto-plastic and fracture Johnson-Cook models) is used for aluminium (AA2024-T351) and steel (Steel 1006) [24,25]. The material parameters for shock EoS are mentioned in Table 1—for aluminium, steel and water—and in Table 4 for PVDF. The explosive is modelled using the Jones-Wilkins-Lee (JWL) equation of state [26]. The JWL equation of state corresponds to the equation of state of detonation products. It is an usual formulation for dealing with detonation by numerical modelling, implemented in numerous software, under this formula:

^{®}, the EoS is defined by a linear relation used to calculate the Hugoniot pressure with the shock Equation (3), as explained in SPEED

^{®}reference guide:

^{®}, the hydrodynamic pressure P of an element can be calculated using a polynomial expression as explained in its reference guide:

^{®}software. An explosive cord is placed in the centre of a 90 charge radii square computational domain of water and a numerical gauge is put at 40 charge radii from the centre of the explosive (Figure 12). All mesh elements are squares. The mesh sensitivity of the model is studied and the element size was fixed to 2 mm for the coarser model, to 0.05 mm for the finest one. The pressure history during wave propagation from the detonating cord is monitored (Figure 13) by implementing numerical sensors placed in the mesh.

^{®}software, with quad elements of size of 100 μm.

^{®}and RADIOSS

^{®}: an explosive is set into a 210 charge radii square domain of water with numerical gauges spaced every 5 charge radii (cf. Figure 16). The goal is to compute the peak of overpressure and to compare it with measurements with the analytical method presented in Section 2.3 to get the incident pressure in a range of 40 to 100 charge radii (cf. Figure 5 and Figure 17).

^{®}and SPEED

^{®}with the measurement is calculated at each distance with respect to measurement values in Table 5. The relative difference is mainly under 10% up to 90 radii except for one shot at 40 radii. The computed pressure tends to be higher than the measured pressure under 60 radii and under the measured pressure from 70 radii away from the explosive. However, this statement has to be taken carefully since not many shots were performed.

^{®}and in SPEED

^{®}models. The smaller element size obtained with SPEED

^{®}could be achieved by using the mesh activation feature, saving a lot of CPU time. Both software give similar trends, and are close to the estimated incident pressure from experimental data, except for the longest distances (Figure 17). An explanation could be that the pressure of elements cannot reach ${P}_{CJ}$ (Chapman-Jouguet pressure) in the explosive during the detonation stage and could imply not only a reduction in pressure at the source but also a dissipation of energy due to the phenomena of numerical dissipation (Figure 13) for the longest distances. The method for estimating the incident pressure could also contribute to discrepancies since it is based on the 1D approximation.

## 5. Conclusions

- -
- The measurement method and signal interpretation is described by a physical and analytical description. It pointed out the relevance of the measured peaks of over-pressure. The pressure amplitudes given by the sensor are close to the ones deduced by the analytical approach and computed with numerical modelling. The analysis presented allowed calculating loading pressure exerted on the metal plate. Considerations on incident and reflected shocks were efficient to take into account the multi-layer PVDF gauge and its intrusiveness.
- -
- Measurements of the peaks of over-pressure versus distance correlate with two different numerical models performed with two different commercial codes based on different numerical methods.
- -
- The formation of a Mach stem at about between 30${}^{\circ}$ and 45${}^{\circ}$ propagating on the metal anvil has been evidenced in underwater environment and results in an additional mechanical loading on the fluid/metal plate interaction that has to be considered in the HEHF process.
- -
- Its implementation does not disturb the forming process of large and thick metal plates and allows an in-situ dynamic pressure monitoring during the HEHF process.
- -
- In the case of rigid plates, the sensor can be reusable several times, for about 30 shots before it broke.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

^{®}. At last, authors want to thanks Thomas Hartmann from Numerics Gmbh for his support.

## Conflicts of Interest

## Abbreviations

PVDF | Polyvinyliden Fluorid |

EoS | Equation of State |

BNC | Bayonet Neill–Concelman |

HEHF | High Explosive HydroForming |

UNDEX | Underwater Explosion |

3DMF | 3D Metal Forming |

ALE | Arbitrary Lagrangian Eulerian |

CEL | Coupling Eurler Lagrange |

SPH | Smooth Particle Hydrodynamic |

CFD | Computational Fluid Dynamics |

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**Figure 1.**Bauer $S25$ PVDF bare shock gauge. Leads are on each side of the tape, sensing area is located at their facing portions.

**Figure 2.**Calibration of electrical charges per unit of area produced by PVDF versus loading pressure under mono-dimensional shock pressure (inspired from [14]).

**Figure 3.**Shielded PVDF sensor stuck with tape on the steel anvil. Detonating cord positioned above with plastic posts and tape. Scheme (

**a**) and picture (

**b**).

**Figure 4.**x-t (distance-time) diagram (

**a**) and P-u (pressure-particle velocity) diagram (

**b**) describing the interaction of an assumed planar shock with constitutive layers of the sensor. Shocks and release waves are in red. The primary shock comes from water, goes through aluminium shield, PVDF and aluminium before being reflected. The successive states reached by the PVDF are represented through the P-u diagram.

**Figure 5.**P-u diagram showing the successive states of water, aluminium, and steel under a 200 MPa incident shock load. The shock comes from the water, is transmitted into the aluminium and then into the steel.

**Figure 6.**Flowchart of Matlab

^{®}numerical file used to calculate the incident pressure from transmitted pressure to steel using 1D shock theory.

**Figure 7.**Measured peak pressures with sensor on steel anvil versus distance with one detonating cord (dimensionless units).

**Figure 8.**Measured pressure with sensor onto steel anvil using thin (30 μm) and thick (300 μm) double sided adhesive tape, from one detonating cord (dimensionless unit).

**Figure 9.**Signals obtained with a PVDF gauge (dimensionless units) on aluminium anvil and on steel anvil using explosive at 100 charge radii.

**Figure 10.**Variants of the setup: pressure sensor on steel anvil (

**a**), pressure sensor on aluminium anvil attached with steel hooks (

**b**), pressure sensor on aluminium anvil clamped over steel anvil (

**c**).

**Figure 12.**Configuration of the mesh sensitivity study simulation study performed with SPEED

^{®}. An explosive is set in the centre of the mesh domain with a numerical gauge at 40 radii away from it. The element size varies from $0.05$ mm to 2 mm, and the domain is a 90 radii square.

**Figure 13.**Mesh convergence study using 2D SPEED

^{®}simulation (dimensionless unit), the explosive is set at 40 charge radii from the numerical gauge.

**Figure 14.**SPEED

^{®}modelling of detonating cord underwater explosion above PVDF gauge placed on steel anvil.

**Figure 15.**Dimensionless measured pressure with pressure sensor on steel anvil with an explosive at 40 charge radii and SPEED

^{®}recored signal in PVDF with equivalent simulation, with 0.1 mm element size.

**Figure 16.**Configuration of the underwater explosion simulation of a detonating cord. The explosive is set in the centre of the mesh domain with numerical gauges from 40 to 100 radii away from it with a 5 radii step. The element size is equal to $0.05$ mm with SPEED

^{®}and $0.25$ mm with RADIOSS

^{®}, and the domain is a 210 radii square.

**Figure 17.**Comparison of incident pressure versus distance between experiments and simulation with RADIOSS

^{®}(0.25 mm square elements) and SPEED

^{®}(0.05 mm square elements) using dimensionless units. Experimental pressure is converted to incident pressure using 1D shock theory as explained in Section 2.3.

Material | Density (kg/m^{3}) | Sound Velocity (m/s) | s |
---|---|---|---|

Steel | 7896 | 4569 | 1.490 |

Aluminium | 2785 | 5328 | 1.338 |

Water | 998 | 1483 | 1.921 |

**Table 2.**Dimensionless peaks of overpressure measured by the sensor at various distances. The highest peak pressure corresponds to interaction between sensor aluminium shield and steel and the first peak corresponds to the interaction between sensor aluminium shield and water. The relative difference is given with respect to the highest peak.

Distance | Highest Peak Pressure Al/Steel | First Peak Pressure Al/Water | Pressure in Al by Analytical Approach | Rel. Diff. |
---|---|---|---|---|

40 | 1.210 | 0.791 | 0.845 | 6.4% |

50 | 0.886 | 0.644 | 0.622 | 3.5% |

60 | 0.791 | 0.548 | 0.544 | 1.1% |

70 | 0.767 | 0.512 | 0.538 | 4.8% |

80 | 0.690 | 0.477 | 0.484 | 1.4% |

90 | 0.578 | 0.387 | 0.406 | 4.7% |

100 | 0.581 | 0.387 | 0.408 | 5.1% |

**Table 3.**Dimensionless measured peak pressure with various angles of incidence (see Figure 11) at 50 charge radii above steel anvil.

Angle | 0° | 15° | 30° | 45° | 60° |

Pressure | 0.886 | 0.724 | 0.544 | 0.640 | 0.476 |

**Table 4.**Material parameters of PVDF required to model underwater explosion based on shock Equation of State [19].

Material | Density (kg/m^{3}) | Sound Vel (m/s) | s |
---|---|---|---|

PVDF | 1767 | 2579 | 1.586 |

**Table 5.**Relative difference (%) between the peak of overpressure from simulation with SPEED

^{®}and RADIOSS

^{®}simulation with incident pressure calculated with method presented in Section 2.3 and measurement (Figure 7). Since 2 shots were performed at 40 and 100 radii, both are presented with a specific label namely “(1)” and “(2)”. The percentage is computed with respect to measured values.

Distance | 40 (1) | 40 (2) | 50 | 60 | 70 | 80 | 90 | 100 (1) | 100 (2) |

SPEED^{®} | −1.2 | −15.5 | −6.2 | −1.8 | 8.2 | 9.2 | 8.4 | 13 | 18.1 |

RADIOSS^{®} | −1 | −15.2 | −10.8 | −3.8 | 6.7 | 9.1 | 9.2 | 18.3 | 23.1 |

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**MDPI and ACS Style**

Tartière, J.; Arrigoni, M.; Nême, A.; Groeneveld, H.; Van Der Veen, S.
PVDF Based Pressure Sensor for the Characterisation of the Mechanical Loading during High Explosive Hydro Forming of Metal Plates. *Sensors* **2021**, *21*, 4429.
https://doi.org/10.3390/s21134429

**AMA Style**

Tartière J, Arrigoni M, Nême A, Groeneveld H, Van Der Veen S.
PVDF Based Pressure Sensor for the Characterisation of the Mechanical Loading during High Explosive Hydro Forming of Metal Plates. *Sensors*. 2021; 21(13):4429.
https://doi.org/10.3390/s21134429

**Chicago/Turabian Style**

Tartière, Jérémie, Michel Arrigoni, Alain Nême, Hugo Groeneveld, and Sjoerd Van Der Veen.
2021. "PVDF Based Pressure Sensor for the Characterisation of the Mechanical Loading during High Explosive Hydro Forming of Metal Plates" *Sensors* 21, no. 13: 4429.
https://doi.org/10.3390/s21134429