# Coupled Modeling Approach for Laser Shock Peening of AA2198-T3: From Plasma and Shock Wave Simulation to Residual Stress Prediction

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Parameters

#### 2.2. Simulation Scheme

#### 2.2.1. Global LSP Model

#### 2.2.2. Finite Element Model

#### 2.3. Model Analysis

#### 2.3.1. Sensitivity Study

#### 2.3.2. Residual Stress Prediction

#### 2.3.3. Parameter Identification

## 3. Results and Discussion

#### 3.1. Pressure Profiles

#### 3.2. Residual Stresses

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

A | yield strength |

${A}_{\mathrm{p}}$ | coefficient of laser absorption by the plasma |

B | strengthening coefficient |

$\Gamma $ | mass flow |

C | rate dependency of the material |

D | shock wave velocity |

E | internal energy |

${E}_{\mathrm{Y}}$ | Young’s modulus |

${E}_{\Gamma}$ | energy exchange through mass flows |

${E}_{\mathrm{pk}}$ | kinetic plasma energy |

${E}_{\mathrm{pt}}$ | total plasma energy |

F | laser focus size |

I | laser intensity |

${I}_{\mathrm{max}}$ | peak of laser intensity |

L | plasma thickness |

P | pressure |

${P}_{\mathrm{max}}$ | maximum plasma pressure |

Q | specific phase change energy |

S | empiric parameters, which connect shock and particle velocities |

T | laser pulse FWHM |

${T}_{\mathrm{L}}$ | total pulse duration |

U | particles velocity |

${U}_{\mathrm{pm}}$ | expansion velocity of plasma in metal direction |

${U}_{\mathrm{pw}}$ | expansion velocity of plasma in water direction |

${U}_{\mathrm{s}}$ | velocity of sound |

${W}_{\mathrm{p}}$ | work done by plasma |

$\gamma $ | specific heat ratio |

${\epsilon}_{P}$ | plastic strain |

${\dot{\epsilon}}_{P}$ | plastic strain rate |

${\dot{\epsilon}}_{P,0}$ | reference plastic strain rate |

$\nu $ | Poisson’s ratio |

n | strain hardening exponent |

$\rho $ | mass density |

${\rho}_{\mathrm{pmax}}$ | maximum plasma density |

${\sigma}_{\mathrm{Y}}$ | yield stress |

${\sigma}_{xx}^{M}$ | measured residual stress in xx-direction |

${\sigma}_{yy}^{M}$ | measured residual stress in yy-direction |

${\sigma}_{xx}^{Sim}$ | calculated residual stress in xx-direction |

${\sigma}_{yy}^{Sim}$ | calculated residual stress in yy-direction |

t | time |

FE | finite element |

FWHM | full-width-at-half-maximum |

ICF | inertial confinement fusion |

LSP | laser shock peening |

RS | residual stress |

Subscript ’m’ | metal region |

Subscript ’0’ | unshocked region |

Subscript ’p’ | plasma region |

Subscript ’w’ | water region |

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**Figure 2.**Schematic of the applied LSP simulation scheme, including the comparison of predicted and experimentally determined RS. The global LSP model yields the plasma pressure. This pressure is used in FE simulations to determine the RS numerically. The predicted RS are compared with measurements to determine unknown plasma parameters for the case of a 5 J energy pulse without coating. Based on this, the global model is adjusted to allow for predictions of RS state at different laser intensities and coating materials.

**Figure 3.**Non-symmetric temporal profile of the laser pulse used in the current research with FWHM $T=20\phantom{\rule{3.33333pt}{0ex}}\mathrm{ns}$ and the total duration ${T}_{\mathrm{L}}=80\phantom{\rule{3.33333pt}{0ex}}\mathrm{ns}$.

**Figure 4.**Schematic of the global LSP model with water curtain. The region of water–metal–plasma interaction is separated into five parts: unshocked and shocked water, unshocked metal, shocked metal coupled with a coating material and a plasma of thickness L. Expansion velocities of plasma in water ${U}_{\mathrm{pw}}$ and metal ${U}_{\mathrm{pm}}$ directions are depicted.

**Figure 5.**FE model of the LSP process for square laser focus. The borders in x- and y-direction were fixed in all degrees of freedom, where the bottom surface was modeled as a free surface. The laser pulses were modeled as pressure loading, obtained from the global model, in the shown order, corresponding to the applied laser pulse sequence.

**Figure 6.**Sensitivity analysis of the maximum plasma pressure dependent on initial plasma density ${\rho}_{\mathrm{p}0}$ for different absorption coefficients ${A}_{\mathrm{p}}$. LSP of aluminum with a laser pulse energy of 5 J without coating is simulated. The points P1, P2, P3 and P4 correspond to the borders of the parameter space and represent pairs of values ${\rho}_{\mathrm{p}0}$ and ${A}_{\mathrm{p}}$, which were used within the sensitivity study. The star corresponds to identified simulation parameters based on the comparison to experimental RS results; see Section 2.3.3.

**Figure 7.**Temporal distributions of particle velocities in water (

**a**) and metal (

**c**); the expansion speed of plasma in water (

**b**) and metal (

**d**) directions for the four points P1–P4; see Figure 6. The particle velocity of water is approximately five times larger than the velocity of metal particles. The expansion speed of plasma in the direction of water is one order higher than the expansion speed in the direction of metal.

**Figure 8.**Time-dependent mass flows from water region (

**a**) and metal region (

**b**) to plasma for the four border points P1–P4; see Figure 6. The major mass flow from water indicates that the plasma mostly consists of water particles.

**Figure 9.**Plasma pressure distributions for the four border points P1–P4 defined in Figure 6 without relaxation (

**a**) and with relaxation (

**b**), which happens after laser pulse terminates and results in a pressure drop.

**Figure 10.**(

**a**) Resulting RS distribution after the application of the four pressure pulses (P1–P4) shown in Figure 9b, which were defined for the four border points P1–P4; see Figure 6. (

**b**) Sensitivity study of the maximum plasma pressure regarding the calculated maximum plasma density ${\rho}_{\mathrm{pmax}}$ at different absorption coefficients ${A}_{\mathrm{P}}$. The star corresponds to identified simulation parameters based on the comparison to experimental RS; see Section 2.3.3.

**Figure 11.**Temporal pressure distributions for the cases with aluminum ablative surface (solid black line) and with steel foil as a coating material (dashed-dotted red line) in comparison with laser intensity profiles (dashed blue line) for laser pulse energy of 3 J (

**a**) and 5 J (

**b**).

**Figure 12.**Temporal distribution of the expansion speed of plasma in water direction for the cases with aluminum ablative surface (solid black line) and with steel foil as a coating material (red line) for laser pulse energy of 3 (

**a**) and 5 J (

**b**).

**Figure 13.**Time-dependent mass flows from metal to plasma for the cases with aluminum ablative surface (solid black line) and with steel foil as a coating material (red line) for laser pulse energy of 3 (

**a**) and 5 J (

**b**).

**Figure 14.**Comparison between calculated (${\sigma}_{xx}^{Sim}={\sigma}_{yy}^{Sim}$) and experimentally determined (${\sigma}_{xx}^{M}$, ${\sigma}_{yy}^{M}$) RS in AA2198-T3 after LSP treatment for different laser pulse energies and coating materials. (

**a**) 3 J pulse energy, no coating. (

**b**) 5 J pulse energy, no coating. (

**c**) 3 J pulse energy, aluminum foil. (

**d**) 5 J pulse energy, aluminum foil. (

**e**) 3 J pulse energy, steel foil. (

**f**) 5 J pulse energy, steel foil.

**Table 1.**Material parameters employed, assuming pressure of unshocked regions ${P}_{0}$ of ${10}^{5}$ Pa for all materials.

Parameter | Water | Aluminum | Steel |
---|---|---|---|

Density ${\rho}_{0}$, $\phantom{\rule{0.166667em}{0ex}}{\mathrm{kg}/\mathrm{m}}^{3}$ | 1000 | 2800 | 7900 |

Sound velocity ${U}_{s}$ [43], m/s | 2393 | 5328 | - |

Coefficient S [43] | 1.333 | 1.338 | - |

Specific phase change energy Q [44,45,46,47], $\phantom{\rule{0.166667em}{0ex}}\mathrm{MJ}/\mathrm{kg}$ | ≈3 | ≈15 | ≈9 |

Parameter | AA2198-T3 |
---|---|

Young’s modulus ${E}_{\mathrm{Y}}$, GPa | 78 |

Poisson’s ratio $\nu $ | 0.33 |

Quasi-static yield strength A, MPa | 310 |

Strengthening coefficient B, MPa | 1177 |

Strain hardening exponent n | 0.894 |

Reference plastic strain rate ${\dot{\epsilon}}_{P,0}$, s${}^{-1}$ | 1.8 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{-4}$ |

Dynamic strain hardening coefficient C | 0.01 |

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

Expansion velocities ${U}_{\mathrm{pw}}$ and ${U}_{\mathrm{pm}}$, m/s | 0 |

Plasma density ${\rho}_{\mathrm{p}0}$, kg/m${}^{3}$ | 0.1–10 |

Plasma pressure ${P}_{\mathrm{p}0}$, Pa | ${10}^{5}$ |

Absorption coefficient ${A}_{\mathrm{p}},\%$ | 10–50 |

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**MDPI and ACS Style**

Pozdnyakov, V.; Keller, S.; Kashaev, N.; Klusemann, B.; Oberrath, J.
Coupled Modeling Approach for Laser Shock Peening of AA2198-T3: From Plasma and Shock Wave Simulation to Residual Stress Prediction. *Metals* **2022**, *12*, 107.
https://doi.org/10.3390/met12010107

**AMA Style**

Pozdnyakov V, Keller S, Kashaev N, Klusemann B, Oberrath J.
Coupled Modeling Approach for Laser Shock Peening of AA2198-T3: From Plasma and Shock Wave Simulation to Residual Stress Prediction. *Metals*. 2022; 12(1):107.
https://doi.org/10.3390/met12010107

**Chicago/Turabian Style**

Pozdnyakov, Vasily, Sören Keller, Nikolai Kashaev, Benjamin Klusemann, and Jens Oberrath.
2022. "Coupled Modeling Approach for Laser Shock Peening of AA2198-T3: From Plasma and Shock Wave Simulation to Residual Stress Prediction" *Metals* 12, no. 1: 107.
https://doi.org/10.3390/met12010107