# The Effect of Viscous Drag on the Maximum Residual Stresses Achievable in High-Yield-Strength Materials in Laser Shock Processing

^{*}

## Abstract

**:**

^{4}s

^{−1}(viscous regime) is explicitly considered in laser shock processing (LSP) simulations. First, a detailed review of the most common high-strain-rate deformation models is presented, highlighting the expected strain rates in materials subject to LSP for a wide range of treatment conditions. Second, the abrupt yield stress increase presented beyond 10

^{4}s

^{−1}is explicitly considered in the material model of a titanium alloy subject to LSP. A combined numerical–analytical approach is used to predict the time evolution of the plastic strain. Finally, extended areas are irradiated covering a squared area of 25 × 25 mm

^{2}for numerical–experimental validation. The in-depth experimental residual stress profiles are obtained by means of the hole drilling method. Near-surface-temperature gradients are explicitly considered in simulations. In summary, the conventionally accepted strain rate range in LSP (10

^{6}–10

^{7}s

^{−1}) is challenged in this paper. Results show that the conventional high-strain-rate hardening models widely used in LSP simulations (i.e., Johnson Cook model) clearly overestimate the induced compressive residual stresses. Additionally, pressure decay, whose importance is usually neglected, has been found to play a significant role in the total plastic strain achieved by LSP treatments.

## 1. Introduction

^{2}) pulsed laser beam with a full width at half maximum (FWHM) lower than 50 ns [5,6]. The high-intensity irradiation forces a sudden vaporization of a thin layer, developing an ionized plasma at high pressures with the aid of a confining medium (typically water or quartz glass). Typically, the magnitude of the shockwave generated (about 5 GPa) is capable of deforming metallic alloys from the surface up to 1 mm depth [7], introducing in-depth compressive residual stresses. This generates a protective layer which finally develops a fatigue life improvement [8,9,10].

^{6}–10

^{7}s

^{−1}, which is inconsistent in general terms with the fact that an abrupt yield stress increase is experimentally observed beyond 10

^{4}s

^{−1}[20]. This is the main point discussed in this paper.

^{−3}s

^{−1}) to ultrahigh strain rates (10

^{8}s

^{−1}). The constitutive strain-rate-dependent models developed in the last years can be essentially divided into two groups: phenomenological models (the Johnson–Cook model [21], Khan–Huang–Liang (KHL) models [22,23,24,25] and a recent model presented by Kim [26]) and physically based models, for FCC (face-centered cubic), BC (centered cubic) and HCP (hexagonal close-packed) structures [27,28,29,30]. Phenomenological models are focused on achieving the best fit option between numerical predictions and experimental stress–strain curves, with no strict physical interpretation of the calibration constants. The material behavior is then modeled with a minimization of the required constants for calibration. On the other hand, the physically based models are focused on the strict physical interpretation of each calibration constant. However, a great number of constants are usually involved. In addition, the proper identification of them is often difficult to assess. Hence, very different sets of constants are reported in the literature for the same material, leading to inconsistent results. The recently proposed physical-based models consider explicitly the abrupt yield stress increase at 10

^{4}s

^{−1}. This noticeable yield stress has been documented by Couque [20], providing a precise description of the methods to characterize the strain rate behavior beyond 10

^{3}s

^{−1}with the direct impact Hopkinson pressure bar (DIHPB) technique [31] in copper, nickel, AISI 304L steel, Al 2017 aluminum alloy, tantalum and two tungsten alloys. Additionally, the strain rate threshold between the thermal activation regime and the viscous regime is identified for each material. The conventional Johnson–Cook model considers the yield stress as proportional to a strain rate term. It models the linear strain rate dependence experimentally observed from quasistatic conditions (${\dot{\epsilon}}_{p}\cong $ 10

^{−4}s

^{−1}) up to moderated dynamic conditions (${\dot{\epsilon}}_{p}\cong $ 10

^{3}s

^{−1}). However, beyond dynamic conditions, plasticity is governed by a viscous drag mechanism: the dislocation motion is slowed down by a viscous drag phenomenon. Hence, dislocation motion, with the required conditions for its thermal activation, becomes a minor effect in plasticity. This leads to a drastic increase in the yield stress at very high-strain-rate conditions (${\dot{\epsilon}}_{p}$ > 10

^{4}s

^{−1}) which can be hardly modeled by the Johnson–Cook formulation.

^{4}s

^{−1}. For higher strain rates in shock loading experiments, Swegle and Grady [35] proposed the exponential form $\sigma \cong K{\dot{\epsilon}}_{p}^{q}$ $\left(q=1/4\right)$, to achieve a best fit option for six different metals in the range 10

^{5}–10

^{8}s

^{−1}. An additional trend change is documented beyond 10

^{8}s

^{−1}, which can be properly simulated with dislocation dynamics [36]. The exponential factor is then recalibrated $\left(q=1/2\right)$.

^{3}s

^{−1}). The trend of the curve is extrapolated for higher strain rates. This leads to predicted strain rates in LSP which are about 10

^{6}s

^{−1}, as documented in the literature [37,38,39,40]. However, this issue is challenged by solid theoretical arguments. Simon et al. [41] reported the limitations of the Johnson–Cook model to provide accurate stress–strain predictions over a wide range of temperatures and strain rates. The experimental results presented in their publication showed a significant increase in the stress rate sensitivity beyond 10

^{3}s

^{−1}for a high-strength steel. Furthermore, a recent publication explicitly considered the abrupt yield stress increase in LSP-treated OFHC (oxygen-free high thermal conductivity) copper specimens [42], which shows a good numerical–experimental agreement in the deformed profiles for the application of concentric shots. The in-depth Vickers hardness is also estimated with good precision. A later publication provided a numerical study of LSP in Ti6Al4V alloy, in which a dislocation density evolution model [43] was calibrated on the basis of the results provided by the Gao–Zhang model [27]. However, the original Gao–Zhang model [27] was updated in 2019 with the addition of a drag stress term since the abrupt yield stress increase was underestimated [29]. Consequently, a detailed description of the time evolution of the plastic strains and an experimental validation of the in-depth residual stresses are still required. This is especially relevant for the study of very high-density LSP treatments.

^{3}s

^{−1}, by means of the split-Hopkinson bar (SHB)). This significant difference was precisely attributed to the abrupt yield stress increase presented at 10

^{4}s

^{−1}. This last result may be interpreted as evidence of the existence of this phenomenon in specimens subject to LSP.

^{6}–10

^{7}s

^{−1}) is challenged in this paper. The results show that the conventional high-strain-rate hardening models widely used in LSP simulations (i.e., the Johnson–Cook model) clearly overestimate the induced compressive residual stresses, whereas the natural residual stress saturation widely reported in experiments can be properly modeled if viscous drag is considered. Furthermore, pressure decay, whose importance is usually neglected, has been found to play a significant role in the total plastic strain achieved by LSP treatments. Overall, it is expected that the present advances derived from the explicit consideration of viscous drag will represent a starting point of interest for future research.

## 2. Theoretical Basis: On the Consideration of the Viscous Drag Mechanism in Metal Material Alloys Subject to LSP

#### 2.1. Generalization of the Conventional Analytical Methods to Consider the Abrupt Yield Stress Increase in Materials Subject to LSP

^{6}–10

^{7}s

^{−1}). Therefore, the Hugoniot elastic limit, ${\sigma}_{H}$, remains practically constant during the deformation process. A generalized form of the original equation is proposed to compute the temporal significant variation in the Hugoniot elastic limit (Equation (1)): a functional form, ${\sigma}_{H}=f\left({\dot{\epsilon}}_{pz}\right)$, is considered in material modeling, where ${\dot{\epsilon}}_{pz}$ represents the axial plastic strain rate. Hence, the time evolution of the strain rate can be calculated. For relatively low strain rates, the axial plastic strain can be approximated using an integral expression (Equation (2)). If the viscous drag phenomenon is neglected, the Hugoniot elastic limit remains practically constant (${\dot{\sigma}}_{H}\cong 0$), and then the integration of Equation (1) leads to the conventional analytical expression widely used for plastic strain determination. The maximum plastic strain rate computed (${\dot{\epsilon}}_{p}\cong $ 10

^{6}–10

^{7}s

^{−1}) may be presented for alloys characterized by relatively low yield stress during the first nanoseconds, where $\dot{P}\left(t\right)$ reaches its maximum value (about 1 GPa/ns) and ${\dot{\sigma}}_{H}$ progressively decreases down to zero.

#### 2.2. Expected Response in Low/High-Yield-Stress Alloys Subject to LSP

^{6}s

^{−1}) for relatively low-yield-stress alloys (for instance, copper), in a consistent way with experimental results obtained using the SHB. In fact, a best fit in the range 10

^{6}–10

^{7}s

^{−1}with Swegle and Grady’s exponential form $\sigma \cong K{\dot{\epsilon}}_{p}^{q}$ proposed in [46] (where $q$ = 1/3 for aluminum and 1/2.32 for copper) shows that the Hugoniot elastic limit expected at 10

^{6}s

^{−1}is below the peak pressure, and, consequently, the strain rates in LSP may reach 10

^{6}s

^{−1}. On the other hand, the combination of a high yield stress (typical in titanium alloys) and the abrupt yield stress increase at the viscous regime (about 10

^{4}s

^{−1}) imposes a limited maximum achievable plastic strain rate (about 10

^{4}s

^{−1}) for the maximum shockwave pressures (about 5–6 GPa) generated by LSP. This limited plastic strain rate leads to a considerable reduction in the conventionally expected computed plastic strains. Nevertheless, this issue does not imply any inconsistency with the experimental results: very small plastic strains are required for generating notable in-depth residual stress profiles. In fact, plastic strains below 1% are enough to achieve significant compressive residual stresses according to the generalized Hooke’s law. Negligible hardening is expected for conventional low-density treatments and hence it may be suitable to adopt an elastic–perfectly plastic model for simulations.

## 3. Materials and Methods

^{2}. This surface is enough to measure thein-depth residual stress profiles by means of the hole drilling method.

#### 3.1. Experimental Set-Up for LSP Irradiation and Measurement of In-Depth Residual Stresses

^{3}(Table 1). The specimen is thick enough ($\cong $7 mm) to prevent it from severe residual stress redistribution due to specimen bending. All the specimens were subjected previously to a thermal relaxation cycle (710 °C for 2 h) to suppress any tensile residual stress derived from the manufacturing process. Additionally, this leads to a reduction in the stress–strain asymmetry.

^{2}and a peak pressure of 5.3 GPa, calculated with the aid of HELIOS code and the methods presented in [47]. The overlapping distance is set to $d=0.14\mathrm{m}\mathrm{m}$, leading to an equivalent overlapping density [48] ($EOD)=$ 5000 pp/cm

^{2}. A schematic representation of the treatment strategy and a picture of the result after irradiation is represented in Figure 1. The laser spot diameter, $\varphi $, and overlapping distance between successive pulses, $d$, are not represented at scale to provide a clearer understanding of the treatment strategy.

#### 3.2. FEM Model

#### 3.2.1. Model Definition for LSP Simulation

^{2}forces a sudden vaporization of the irradiated surface, leading to a very high-temperature plasma (about 30,000 K). This causes additional local near-surface plastic strains. The time–heat flux exchange between the confined plasma and the irradiated surface can be calculated with the aid of the HELIOS code. The time irradiation pulse, $I\left(t\right)$, the heat flux, $q\left(t\right)/A$, and the corresponding pressure, $P\left(t\right)$, are represented in Table 2. Both the heat flux, $q\left(t\right)/A$, and the pressure, $P\left(t\right)$, are plotted together in Figure 2, in which a slight delay is observed between both profiles. This is consistent with the results reported by Morales et al. [50]. The pressure pulse is set in ABAQUS with the aid of user subroutine VDLOAD.

^{3}hexahedral elements with a reduced integration scheme (C3D8R), in which the minimum dimension corresponds to the in-depth dimension (10 μm). The mesh is surrounded by infinite elements (CIN3D8) covering a total depth of 7 mm. The minimum step time provided by the FEM code that ensures stable solutions in the explicit algorithm is ${t}_{s}\cong $ 1.6 ns. This is the result of the application of the Courant–Friedrichs–Lewy (CFL) condition and a safety factor which is set automatically by the FEM code. However, ${t}_{s}$ has been reduced to 0.1 ns. This way, a precise characterization of the deformations in the range 10

^{−4}to 10

^{4}s

^{−1}is computed, although the computational cost rises. The laser device operates at 10 Hz, leading to a step time of ${t}_{p}=$ 0.1 s between consecutive pulses. Nevertheless, postprocessing results analysis shows that the stresses are balanced beyond 2 μs. Hence, the time between consecutive pulses is set to ${t}_{p}=2\mathsf{\mu}\mathrm{s}$ to reduce the computational cost.

^{2}is finally used for the simulations, which is enough to ensure a sufficient overlapped area. Results may differ slightly from a complete 25 × 25 mm

^{2}treatment simulation. Nevertheless, minor differences are expected considering the small magnitude of predicted plastic strains. Considering both the treatment density (5000 pp/cm

^{2}) and the representative patch area (3 × 3 mm

^{2}), a total amount of ${n}_{p}=$ 450 shots need to be simulated. Consequently, the simulation time responds to ${t}_{sim}=$ ${n}_{p}{t}_{p}=$ 0.0009 s and the number of step increments for simulation is ${n}_{sim}=9000$.

#### 3.2.2. Near-Surface Thermal Effect Simulation (Implicit Analysis)

## 4. Results

^{4}s

^{−1}). Therefore, a Hugoniot elastic limit for the achievable pressures in LSP may be presented precisely near this strain rate threshold. Consequently, significant differences are expected with respect to the results provided by conventional modeling. Setting a small spot diameter, $\varphi =1.5\mathrm{m}\mathrm{m}$, ensures the highest pressures, which may be necessary considering the high yield stress of Ti6Al4V alloy.

#### 4.1. Analytical–Numerical Results for LSP Single Shots in Ti6Al4V

^{−4}to 10

^{4}s

^{−1}. As discussed previously, the material is not expected to reach the typical reported strain rates in LSP (10

^{6}s

^{−1}) for Ti6Al4V alloy.

^{6}s

^{−1}is computed using Equation (1), setting ${\dot{\sigma}}_{H}=0$. This scenario corresponds to the conventionally reported strain rates in LSP reported in the literature.

^{4}s

^{−1}in experimental curves of Ti6Al4V suggests that a Hugoniot elastic limit about the maximum pressure (${\sigma}_{H}={P}_{max}\cong $ 5.3 GPa) may be presented approximately about 2 × 10

^{4}s

^{−1}. Consequently, this imposes a limit on the plastic strain rate, and much higher pressures would be required to achieve $\left|{\dot{\epsilon}}_{pzmax}\right|$. The computed axial plastic strain by means of the Johnson–Cook model, ${\epsilon}_{pzfree}$, and Gao–Zhang model, ${\epsilon}_{pzVd}$, is presented in Equations (5) and (6). As expected, significant differences are computed between both calculations.

^{6}s

^{−1}). Consequently, the proper simulation of plasma dynamics is essential to identify both the maximum pressure, and the corresponding nature of the time profile beyond this maximum. The plastic strains achieved are proportional to the difference ${P}_{max}-{\sigma}_{H0}$ and the time in which the pressure decay is presented.

^{4}s

^{−1}take place. This fact justifies that reasonable deformations are obtained (with their corresponding conventional residual stresses) although much lower deformation rates are involved.

#### 4.2. Realistic Thermomechanical Modeling of Extended Surface High-Coverage LSP Treatments with Explicit Consideration of the Viscous Drag Mechanism

^{2}) is suitable to evidence the overestimated in-depth compressive residual stresses obtained using conventional modeling.

#### 4.2.1. Near-Surface Residual Stress Calculation

#### 4.2.2. In-Depth Numerical–Experimental Residual Stress Determination

^{4}s

^{−1}against conventionally reported 10

^{7}s

^{−1}), the time window where plastic strain takes place is extended (during pressure decay, as plotted in Figure 3b), resulting in relatively precise in-depth compressive residual stress calculations, as demonstrated in the results plotted in Figure 7.

## 5. Conclusions

- a.
- The conventional models used in LSP modeling (i.e., the Johnson–Cook model) predict plastic strain rates of about 10
^{7}s^{−1}. However, the solid theoretical argument supports the idea of a notable reduction in the plastic strain rate to 10^{4}s^{−1}in alloys characterized by high yield stresses, for instance, Ti6Al4V alloy. The numerical–experimental results presented in this paper are consistent with this hypothesis. - b.
- The time pressure decrease plays an extremely important role in the deformation mechanism. In fact, the significant increase in the yield stress at the viscous regime ($\cong $10
^{4}s^{−1}) imposes a threshold on the maximum plastic strain rate, and hence most of the deformation takes place during pressure decay, where two simultaneous conditions are presented: relatively high pressures with a smooth time evolution, which is responsible for extending the time during which deformation rates of about 10^{4}s^{−1}take place. This fact justifies that reasonable deformations are obtained (with their corresponding conventional residual stresses), although much lower deformation rates are involved. The numerical–experimental results confirm that notable overestimations in the in-depth compressions are predicted by means of conventional modeling in very high-density treatments, whereas good agreement is obtained when viscous drag is explicitly considered. - c.
- The calculated thermal effects in LSP induce significant reverse plastic straining through a very thin layer of material (about 10 μm is estimated for Ti6Al4V). This justifies the abrupt near-surface stress gradients: from very high tensile stresses (about the yield stress) at the surface to relatively high compressions (about −600 MPa) at 10 μm. Therefore, near-surface numerical–experimental validation can hardly be approached using the usual semi-destructive experimental residual stress measurements (i.e., the hole drilling method). X-ray diffraction methods may be suitable for this purpose.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Li, Y.; Gan, W.; Zhou, W.; Li, D. Review on residual stress and its effects on manufacturing of aluminium alloy structural panels with typical multi-processes. Chin. J. Aeronaut.
**2023**, 36, 96–124. [Google Scholar] [CrossRef] - Li, C.; Liu, Z.Y.; Fang, X.Y.; Guo, Y.B. Residual stress in metal additive manufacturing. Procedia CIRP
**2018**, 71, 348–353. [Google Scholar] [CrossRef] - Zhu, Q.; Chen, J.; Gou, G.; Chen, H.; Li, P. Ameliorated longitudinal critically refracted—Attenuation velocity method for welding residual stress measurement. J. Mater. Process. Technol.
**2017**, 246, 267–275. [Google Scholar] [CrossRef] - Chen, Y.; Sun, S.; Zhang, T.; Zhou, X.; Li, S. Effects of post-weld heat treatment on the microstructure and mechanical properties of laser-welded NiTi/304SS joint with Ni filler. Mater. Sci. Eng. A
**2020**, 771, 138545. [Google Scholar] [CrossRef] - Morales, M.; Ocaña, J.; Molpeceres, C.; Porro, J.; García-Beltrán, A. Model based optimization criteria for the generation of deep compressive residual stress fields in high elastic limit metallic alloys by ns-laser shock processing. Surf. Coat. Technol.
**2008**, 202, 2257–2262. [Google Scholar] [CrossRef] - Morales, M.; Porro, J.A.; Blasco, M.; Molpeceres, C.; Ocaña, J.L. Numerical simulation of plasma dynamics in laser shock processing experiments. Appl. Surf. Sci.
**2009**, 255, 5181–5185. [Google Scholar] [CrossRef] - Angulo, I.; Cordovilla, F.; García-Beltrán, A.; Smyth, N.S.; Langer, K.; Fitzpatrick, M.E.; Ocaña, J.L. The effect of material cyclic deformation properties on residual stress generation by laser shock processing. Int. J. Mech. Sci.
**2019**, 156, 370–381. [Google Scholar] [CrossRef] - Correa, C.; de Lara, L.R.; Díaz, M.; Porro, J.A.; García-Beltrán, A.; Ocaña, J.L. Influence of pulse sequence and edge material effect on fatigue life of Al2024-T351 specimens treated by laser shock processing. Int. J. Fatigue
**2015**, 70, 196–204. [Google Scholar] [CrossRef] - Zhang, L.; Lu, J.Z.; Zhang, Y.K.; Luo, K.Y.; Zhong, J.W.; Cui, C.Y.; Kong, D.J.; Guan, H.B.; Qian, X.M. Effects of different shocked paths on fatigue property of 7050-T7451 aluminum alloy during two-sided laser shock processing. Mater. Des.
**2011**, 32, 480–486. [Google Scholar] [CrossRef] - Zhang, X.Q.; Li, H.; Yu, X.L.; Zhou, Y.; Duan, S.W.; Li, S.Z.; Huang, Z.L.; Zuo, L.S. Investigation on effect of laser shock processing on fatigue crack initiation and its growth in aluminum alloy plate. Mater. Des.
**2015**, 65, 425–431. [Google Scholar] [CrossRef] - Tang, Z.; Gao, J.; Xu, Z.; Guo, B.; Jiang, Q.; Chen, X.; Weng, J.; Li, B.; Chen, J.; Zhao, Z. Effect of Laser Shock Peening on the Fatigue Life of 1Cr12Ni3Mo2VN Steel for Steam Turbine Blades. Coatings
**2023**, 13, 1524. [Google Scholar] [CrossRef] - Ganesh, P.; Sundar, R.; Kumar, H.; Kaul, R.; Ranganathan, K.; Hedaoo, P.; Raghavendra, G.; Kumar, S.A.; Tiwari, P.; Nagpure, D.; et al. Studies on fatigue life enhancement of pre-fatigued spring steel specimens using laser shock peening. Mater. Des.
**2014**, 54, 734–741. [Google Scholar] [CrossRef] - Liu, Q.C.; Baburamani, P.; Zhuang, W.; Gerrard, D.; Hinton, B.; Janardhana, M.; Sharp, K. Surface modification and repair for aircraft life enhancement and structural restoration. In Materials Science Forum. Mater. Sci. Forum
**2010**, 654–656, 763–766. [Google Scholar] [CrossRef] - Sharp, P.K.; Liu, Q.; Barter, S.A.; Baburamani, P.; Clark, G. Fatigue life recovery in aluminium alloy aircraft structure. Fatigue Fract. Eng. Mater. Struct.
**2002**, 25, 99–110. [Google Scholar] [CrossRef] - Correa, C.; de Lara, L.R.; Díaz, M.; Gil-Santos, A.; Porro, J.A.; Ocaña, J.L. Effect of advancing direction on fatigue life of 316L stainless steel specimens treated by double-sided laser shock peening. Int. J. Fatigue
**2015**, 79, 1–9. [Google Scholar] [CrossRef] - Nikitin, I.; Scholtes, B.; Maier, H.J.; Altenberger, I. High temperature fatigue behavior and residual stress stability of laser-shock peened and deep rolled austenitic steel AISI 304. Scr. Mater.
**2004**, 50, 1345–1350. [Google Scholar] [CrossRef] - Prime, M.B.; Dewald, A.T. The contour method. In Practical Residual Stress Measurement Methods; Wiley: Hoboken, NJ, USA, 2013; pp. 109–138. [Google Scholar] [CrossRef]
- ASTM E837-13a; Standard Test Method for Determining Residual Stresses by the Hole-Drilling Strain-Gage Method. ASTM International: West Conshohocken, PA, USA, 2013.
- Shao, Z.; Zhang, C.; Li, Y.; Shen, H.; Zhang, D.; Yu, X.; Zhang, Y. A Review of Non-Destructive Evaluation (NDE) Techniques for Residual Stress Profiling of Metallic Components in Aircraft Engines. Aerospace
**2022**, 9, 534. [Google Scholar] [CrossRef] - Couque, H. The use of the direct impact Hopkinson pressure bar technique to describe thermally activated and viscous regimes of metallic materials. Philos. Trans. R. Soc. A
**2014**, 372, 20130218. [Google Scholar] [CrossRef] - Johnson, G.R.; Cook, W.H. Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures. Eng. Fract. Mech.
**1985**, 21, 31–48. [Google Scholar] [CrossRef] - Khan, A.S.; Huang, S. Experimental and theoretical study of mechanical behavior of 1100 aluminum in the strain rate range 10
^{−5}–10^{4}s^{−1}. Int. J. Plast.**1992**, 8, 397–424. [Google Scholar] [CrossRef] - Khan, A.S.; Liang, R. Behaviors of three BCC metal over a wide range of strain rates and temperatures: Experiments and modeling. Int. J. Plast.
**1999**, 15, 1089–1109. [Google Scholar] [CrossRef] - Khan, A.S.; Zhang, H. Mechanically alloyed nanocrystalline iron and copper mixture: Behavior and constitutive modeling over a wide range of strain rates. Int. J. Plast.
**2000**, 16, 1477–1492. [Google Scholar] [CrossRef] - Khan, A.S.; Zhang, H. Finite deformation of a polymer: Experiments and modeling. Int. J. Plast.
**2001**, 17, 1167–1188. [Google Scholar] [CrossRef] - Kim, H.; Yoon, J.W.; Chung, K.; Lee, M.-G. A multiplicative plastic hardening model in consideration of strain softening and strain rate: Theoretical derivation and characterization of model parameters with simple tension and creep test. Int. J. Mech. Sci.
**2020**, 187, 105913. [Google Scholar] [CrossRef] - Gao, C.Y.; Zhang, L.C.; Yan, H.X. A new constitutive model for HCP metals. Mater. Sci. Eng. A
**2011**, 528, 4445–4452. [Google Scholar] [CrossRef] - Gao, C.Y.; Zhang, L.C. Constitutive modelling of plasticity of fcc metals under extremely high strain rates. Int. J. Plast.
**2012**, 32–33, 121–133. [Google Scholar] [CrossRef] - Sedaghat, H.; Xu, W.; Zhang, L.; Liu, W. On the constitutive models for ultra-high strain rate deformation of metals. Int. J. Automot. Technol.
**2019**, 20, 31–37. [Google Scholar] [CrossRef] - Zerilli, F.J.; Armstrong, R.W. Dislocation-mechanics-based constitutive relations for material dynamics calculations. J. Appl. Phys.
**1987**, 61, 1816–1825. [Google Scholar] [CrossRef] - Dharan, C.K.H.; Hauser, F.E. Determination of stress-strain characteristics at very high strain rates. Exp. Mech.
**1970**, 10, 370–376. [Google Scholar] [CrossRef] - Couque, H.; Boulanger, R.; Bornet, F. A modified Johnson-Cook model for strain rates ranging from 10
^{−3}to 10^{5}s^{−1}. J. Phys. IV Fr.**2006**, 134, 87–93. [Google Scholar] [CrossRef] - Nabarro, F.R.N. Dislocations in a simple cubic lattice. Proc. Phys. Soc.
**1947**, 59, 256. [Google Scholar] [CrossRef] - Wickham, L.K.; Schwarz, K.W.; Stölken, J.S. Rules for Forest Interactions between Dislocations. Phys. Rev. Lett.
**1999**, 83, 4574. [Google Scholar] [CrossRef] - Swegle, J.W.; Grady, D.E. Shock viscosity and the prediction of shock wave rise times. J. Appl. Phys.
**1985**, 58, 692–701. [Google Scholar] [CrossRef] - Kattoura, M.; Shehadeh, M.A. On the ultra-high-strain rate shock deformation in copper single crystals: Multiscale dislocation dynamics simulations. Philos. Mag. Lett.
**2014**, 94, 415–423. [Google Scholar] [CrossRef] - Adu-Gyamfi, S.; Ren, X.D.; Larson, E.A.; Ren, Y.; Tong, Z. The effects of laser shock peening scanning patterns on residual stress distribution and fatigue life of AA2024 aluminium alloy. Opt. Laser Technol.
**2018**, 108, 177–185. [Google Scholar] [CrossRef] - Bikdeloo, R.; Farrahi, G.H.; Mehmanparast, A.; Mahdavi, S.M. Multiple laser shock peening effects on residual stress distribution and fatigue crack growth behaviour of 316L stainless steel. Theor. Appl. Fract. Mech.
**2020**, 105, 102429. [Google Scholar] [CrossRef] - Zhang, X.; Huang, Z.; Chen, B.; Zhang, Y.; Tong, J.; Fang, G.; Duan, S. Investigation on residual stress distribution in thin plate subjected to two sided laser shock processing. Opt. Laser Technol.
**2019**, 111, 146–155. [Google Scholar] [CrossRef] - Zhou, W.; Ren, X.; Yang, Y.; Tong, Z.; Larson, E.A. Finite element analysis of laser shock peening induced near-surface deformation in engineering metals. Opt. Laser Technol.
**2019**, 119, 105608. [Google Scholar] [CrossRef] - Simon, P.; Demarty, Y.; Rusinek, A.; Voyiadjis, G.Z. Material behavior description for a large range of strain rates from low to high temperatures: Application to high strength steel. Metals
**2018**, 8, 795. [Google Scholar] [CrossRef] - Wang, C.; Wang, X.; Xu, Y.; Gao, Z. Numerical modeling of the confined laser shock peening of the OFHC copper. Int. J. Mech. Sci.
**2016**, 108–109, 104–114. [Google Scholar] [CrossRef] - Wang, C.; Li, K.; Hu, X.; Yang, H.; Zhou, Y. Numerical study on laser shock peening of TC4 titanium alloy based on the plate and blade model. Opt. Laser Technol.
**2021**, 142, 107163. [Google Scholar] [CrossRef] - Zhou, Z.; Bhamare, S.; Ramakrishnan, G.; Mannava, S.R.; Langer, K.; Wen, Y.; Qian, D.; Vasudevan, V.K. Thermal relaxation of residual stress in laser shock peened Ti–6Al–4V alloy. Surf. Coat. Technol.
**2012**, 206, 4619–4627. [Google Scholar] [CrossRef] - Ballard, P.; Fournier, J.; Fabbro, R.; Frelat, J. Residual stresses induced by laser-shocks. J. Phys. IV Fr.
**1991**, 1, C3-487–C3-494. [Google Scholar] [CrossRef] - Yao, S.; Yu, J.; Cui, Y.; Pei, X.; Yu, Y.; Wu, Q. Revisiting the power law characteristics of the plastic shock front under shock loading. Phys. Rev. Lett.
**2021**, 126, 085503. [Google Scholar] [CrossRef] [PubMed] - Angulo, I.; Cordovilla, F.; García-Beltrán, A.; Porro, J.A.; Díaz, M.; Ocaña, J.L. Integrated Numerical-Experimental Assessment of the Effect of the AZ31B Anisotropic Behaviour in Extended-Surface Treatments by Laser Shock Processing. Metals
**2020**, 10, 195. [Google Scholar] [CrossRef] - Ocaña, J.; Morales, M.; Porro, J.; Blasco, M.; Molpeceres, C.; Iordachescu, D.; Gómez-Rosas, G.; Rubio-González, C. Induction of engineered residual stresses fields and associate surface properties modification by short pulse laser shock processing. Mater. Sci. Forum
**2010**, 638–642, 2446–2451. [Google Scholar] [CrossRef] - Peral, D.; de Vicente, J.; Porro, J.A.; Ocaña, J.L. Uncertainty analysis for non-uniform residual stresses determined by the hole drilling strain gauge method. Measurement
**2017**, 97, 51–63. [Google Scholar] [CrossRef] - Morales, M.; Correa, C.; Porro, J.A.; Molpeceres, C.; Ocaña, J.L. Thermomechanical modelling of stress fields in metallic targets subject to laser shock processing. Int. J. Struct. Integr.
**2011**, 2, 51–61. [Google Scholar] [CrossRef] - Peyre, P.; Carboni, C.; Forget, P.; Beranger, G.; Lemaitre, C.; Stuart, D. Influence of thermal and mechanical surface modifications induced by laser shock processing on the initiation of corrosion pits in 316L stainless steel. J. Mater. Sci.
**2007**, 42, 6866–6877. [Google Scholar] [CrossRef] - Lesuer, D.R. Experimental Investigations of Material Models for Ti-6A1-4V Titanium and 2024-T3 Aluminum; No. DOT/FAA/AR-00/25; U.S. Department of Transportation: Washington, DC, USA, 2000. [Google Scholar]

**Figure 1.**(

**a**) Schematic representation of the treatment strategy. (

**b**) Experimental result after irradiation.

**Figure 3.**(

**a**) FEM results predicted using conventional model. (

**b**) FEM results predicted with explicit consideration of viscous drag.

**Figure 4.**(

**a**) Time shockwave evolution for representative depths computed by conventional model. (

**b**) Results with explicit consideration of viscous drag.

**Figure 7.**(

**a**) Experimental vs. simulated in-depth residual stress profile predicted by conventional modeling. (

**b**) Experimental vs. simulated in-depth residual stress profile with explicit consideration of viscous drag phenomenon.

Element | Al | V | C | O | N | Ti |
---|---|---|---|---|---|---|

Weight % | 6.1 | 4.2 | 0.01 | 0.12 | 0.006 | Bal. |

**Table 2.**Time evolution of the irradiance, $I\left(t\right)$, heat flux, $q\left(t\right)/A$, and pressure, $P\left(t\right)$.

Time (ns) | Irradiation, $\mathit{I}\left(\mathit{t}\right)$ (GW/cm^{2}) | Heat Flux, $\mathit{q}\left(\mathit{t}\right)/\mathit{A}$ (MW/cm^{2}) | Pressure, $\mathit{P}\left(\mathit{t}\right)$ (MPa) |
---|---|---|---|

0 | 0 | 0 | 0 |

2 | 8 | 20 | 3370 |

5 | 20 | 50 | 5350 |

6 | 18 | 60 | 5183 |

10 | 11 | 110 | 4514 |

16 | 3 | 63 | 3511 |

23 | 0 | 42 | 2340 |

25 | 0 | 36 | 2177 |

30 | 0 | 32 | 1770 |

50 | 0 | 18 | 1160 |

100 | 0 | 9 | 840 |

300 | 0 | 3 | 617 |

400 | 0 | 2 | 506 |

770 | 0 | 0 | 0 |

Depth (nm) | ${\mathit{S}}_{\mathit{m}\mathit{i}\mathit{n}}$ (MPa) | ${\mathit{S}}_{\mathit{m}\mathit{a}\mathit{x}}$ (MPa) |
---|---|---|

0 | 1254 | 1398 |

870 | 849 | 1071 |

1740 | 473 | 657 |

2610 | 153 | 328 |

3480 | −48 | 102 |

4350 | −190 | −44 |

5220 | −308 | −178 |

6090 | −408 | −296 |

6960 | −504 | −410 |

7830 | −576 | −499 |

8700 | −609 | −541 |

9570 | −613 | −544 |

Experimental Results | Free Deformation | Explicit Viscous Drag | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

Depth (µm) | ${\mathit{S}}_{\mathit{m}\mathit{i}\mathit{n}\mathit{e}\mathit{x}\mathit{p}}$ (MPa) | ${\mathit{S}}_{\mathit{m}\mathit{a}\mathit{x}\mathit{e}\mathit{x}\mathit{p}}$ (MPa) | ${\mathit{S}}_{\mathit{m}\mathit{i}\mathit{n}\mathit{s}\mathit{i}\mathit{m}}$ (MPa) | ${\mathit{S}}_{\mathit{m}\mathit{a}\mathit{x}\mathit{s}\mathit{i}\mathit{m}}$ (MPa) | ${\mathit{e}\mathit{r}\mathit{r}}_{\mathit{m}\mathit{i}\mathit{n}}$ $\mathit{\%}$ | ${\mathit{e}\mathit{r}\mathit{r}}_{\mathit{m}\mathit{a}\mathit{x}}$ $\mathit{\%}$ | ${\mathit{S}}_{\mathit{m}\mathit{i}\mathit{n}\mathit{s}\mathit{i}\mathit{m}}$ (MPa) | ${\mathit{S}}_{\mathit{m}\mathit{a}\mathit{x}\mathit{s}\mathit{i}\mathit{m}}$ (MPa) | ${\mathit{e}\mathit{r}\mathit{r}}_{\mathit{m}\mathit{i}\mathit{n}}$ $\mathit{\%}$ | ${\mathit{e}\mathit{r}\mathit{r}}_{\mathit{m}\mathit{a}\mathit{x}}$ $\mathit{\%}$ |

10 | 41 | 104 | −405 | −276 | 1088 | 365 | −283 | −181 | 790 | 274 |

60 | −488 | −392 | −795 | −732 | 63 | 87 | −711 | −619 | 46 | 58 |

120 | −541 | −336 | −812 | −723 | 50 | 115 | −675 | −566 | 25 | 68 |

240 | −571 | −379 | −852 | −717 | 49 | 89 | −533 | −438 | 7 | 16 |

400 | −426 | −218 | −803 | −678 | 88 | 211 | −388 | −316 | 9 | 45 |

620 | −335 | −176 | −579 | −434 | 73 | 147 | −254 | −188 | 24 | 7 |

910 | −184 | −48 | −336 | −227 | 83 | 373 | −142 | −86 | 23 | 79 |

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## Share and Cite

**MDPI and ACS Style**

Angulo, I.; Warzanskyj, W.; Cordovilla, F.; Díaz, M.; Porro, J.A.; García-Beltrán, Á.; Ocaña, J.L.
The Effect of Viscous Drag on the Maximum Residual Stresses Achievable in High-Yield-Strength Materials in Laser Shock Processing. *Materials* **2023**, *16*, 6858.
https://doi.org/10.3390/ma16216858

**AMA Style**

Angulo I, Warzanskyj W, Cordovilla F, Díaz M, Porro JA, García-Beltrán Á, Ocaña JL.
The Effect of Viscous Drag on the Maximum Residual Stresses Achievable in High-Yield-Strength Materials in Laser Shock Processing. *Materials*. 2023; 16(21):6858.
https://doi.org/10.3390/ma16216858

**Chicago/Turabian Style**

Angulo, Ignacio, Wsewolod Warzanskyj, Francisco Cordovilla, Marcos Díaz, Juan Antonio Porro, Ángel García-Beltrán, and José Luis Ocaña.
2023. "The Effect of Viscous Drag on the Maximum Residual Stresses Achievable in High-Yield-Strength Materials in Laser Shock Processing" *Materials* 16, no. 21: 6858.
https://doi.org/10.3390/ma16216858