Dislocation density-based modeling of subsurface grain refinement with laser-induced shock compression

Computational Materials Science 53 (2012) 79–88

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Dislocation density-based modeling of subsurface grain refinement with laser-induced shock compression Hongtao Ding, Yung C. Shin ⇑ Center for Laser-based Manufacturing, School of Mechanical Engineering, 585 Purdue Mall, Purdue University, West Lafayette, IN 47907, USA

a r t i c l e

i n f o

Article history: Received 21 July 2011 Received in revised form 30 August 2011 Accepted 31 August 2011 Available online 22 October 2011 Keywords: Grain refinement Laser shock peening Dislocation density LY2 aluminum alloy Copper

a b s t r a c t Laser shock peening (LSP) is an innovative surface treatment technique applied to improve the mechanical properties and surface microstructures of metallic components. This paper is concerned with prediction of the microstructural evolution of metallic components subjected to single or multiple LSP impacts. A numerical framework is developed to model the evolution of dislocation density and dislocation cell size using a dislocation density-based material model. It is shown that the developed model captures the essential features of the material mechanical behaviors and predicts that the total dislocation density reaches the order of 1014 m2 and a minimum dislocation cell size is below 250 nm for LSP of monocrystalline coppers using the laser energy density on the order of 500 GW/cm2. It is further shown that the model is cable of predicting the material strengthening mechanism in terms of residual stress and microhardness of the LY2 aluminum alloy due to grain refinement in a LSP process with less laser energy densities on the order of several GW/cm2. Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction Laser shock peening (LSP) is an innovative surface treatment technique, which has been successfully applied to improve fatigue, corrosion and wear resistance as well as the mechanical properties of metallic components. During the LSP process, as the material surface is treated with a laser beam of high intensities above 1 GW/cm2 with pulse energies in the range of several Joules and pulse widths in the range of nanoseconds, a shock wave is generated due to the volume expansion of the plasma plume formed on the surface, which propagates into the substrate material. Although a significant amount of work from the experimental side has contributed to exploring the optimum LSP conditions and assessing their ultimate capability to provide compressive residual stresses beneath the treated surfaces of metallic materials, only limited attempts have been made on the full comprehension and predictive assessment of the characteristic mechanisms of the microstructural evolution and grain refinement. To characterize the deformation microstructures driven by the high pressure shock waves at high strain rates, Meyers et al. [1] conducted laser shock compression experiments on monocrystalline copper of [0 0 1] orientation in air using various laser powers. At resultant specimen surface pressures of 12 and 20 GPa, the transmission electron microscopy (TEM) showed that the specimen structure consists primarily of dislocation cells with a refined dislocation cell size

⇑ Corresponding author. Tel.: +1 765 494 9775; fax: +1 765 494 0539. E-mail address: [email protected] (Y.C. Shin). 0927-0256/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2011.08.038

of 200–500 nm; at 40 GPa, twinning and stacking-fault bundles are the principal defect structures; and at a pressure of 55– 60 GPa, the structure shows micro-twinning and elongated subgrains. In a follow-up study, Schneider et al. [2] reported the dislocation cell size as a function of distance from the laser irradiated surface for monocrystalline coppers and copper–aluminum specimens subjected to laser shock compression and experimentally determined that the threshold twinning stress for pure copper in  3 4 orithe [0 0 1] orientation is 25 GPa, whereas the one for the ½1 entation is between 40 and 60 GPa. Although the dislocation cell size of submicron was achieved in these aforementioned laser shock compression experiments, it is noted that the nanosecond laser facility used in their studies provides energy densities on the order of 500 GW/cm2, takes much more space, and is far more expensive to set up for a realistic LSP process used nowadays. Recently, the change of microstructure in the treated specimen by multiple LSP impacts of energy intensity of several GW/cm2 has gained more attention. Chu et al. [3] studied the effects of LSP on the microstructure, microhardness, and residual stress of low carbon SAE 1010 steel. They found that laser-shock induced plastic deformation caused the surface to be recessed by 1.5 lm and resulted in extensive formation of dislocations. Surface hardness increased by up to 80% after the LSP and the microstructure and mechanical properties were altered up to 100 lm in depth. Mordyuk et al. [4] conducted multiple LSP tests with the laser power density of 0.35 GW/cm2 on AISI 321 stainless steel and found the formation of dislocation-cell structures and highly tangled and dense dislocation arrangements in the austenitic surface layer of about 10 lm thick. Ye et al. [5] studied the dislocation structures

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induced by overlapping warm laser shock peening (WLSP) impacts on AISI 4140 steel and showed that increasing the workpiece temperature to 250 °C produced fewer dislocation pile-ups but more dislocation tangles (DT) and a higher dislocation density. Lu et al. [6] investigated the plastic deformation behaviors of the LY2 aluminum alloy subjected to multiple LSP impacts of laser pulse duration of 20 ns and energy of 5 J in the water confinement regime and studied the material microstructural strengthening mechanisms due to grain refinement induced by high pressure laser shock waves. Their study showed that the original coarse LY2 aluminum alloy grains were subdivided into many subgrains or refined grains with an average size of 3–5 lm with a minimum dislocation cell size of 100–200 nm. Based on the microstructure studies, they further proposed the grain refinement mechanisms subjected to multiple LSP impacts as illustrated in Fig. 1: (i) the formation and development of dislocation lines in original grains; (ii) DTs and the formations of dense dislocation walls (DDWs); (iii) transformation of DTs and DDWs into subgrain boundaries; and (iv) evolution of the continuous dynamic recrystallization (DRX) in subgrain boundaries to refined grain boundaries. In the sequential studies, Lu et al. experimentally studied the microstructural evolution and grain refinement in AISI 304 stainless steel [7] and AISI 8620 steel [8] subjected to multiple LSP impacts by means of cross-sectional optical microscopy and transmission electron microscopy observations. Their studies showed that a novel structure with submicron triangular blocks was developed at the top surface of AISI 304 stainless steel mainly due to dislocations and mechanical twin interactions, while for the case of AISI 8620 steel the coarse grain was refined to an average grain size of about 220 nm by dislocation movement near the surface region and gradually increased with an increase of depth from the treated surface. As discussed above, the majority of the reports on microstructural changes induced by LSP have been qualitative with few quantitative details such as dislocation density and grain size refinement. Shehadeh et al. [9] investigated the ultra-short pulse shock wave propagation, plastic deformation and evolution of dislocations in monocrystalline coppers with [0 0 1], [0 1 1] and [1 1 1] orientations using multi-scale dislocation dynamics-based plasticity analyses. Their results showed that the dislocation density has a power law dependence on pressure with a power exponent of 1.70 and that the dislocation density is proportional to pulse duration and sensitive to crystal orientation. Cheng and Shehadeh [10] studied the induced dislocation activity in silicon crystal by integrated shock dynamics and multi-scale dislocation dynamics-based plasticity simulations. Their models were limited to a workpiece computation domain of several micrometers in length and hence are not

suitable to simulate the microstructural evolution of a specimen size typical of hundreds of micrometers or more. Meyers et al. [1] investigated the dislocation generation using a computation method for dislocation densities based on nucleation of loops at the shock front and their extension due to the residual shear stresses behind the front. It was shown the calculated dislocation densities compare favorably with experimentally observed results. Schneider et al. [2] investigated the threshold stress for deformation twinning in shock compression using the constitutive equations for slip, twinning, and the Swegle–Grady relationship. They calculated thresholds twinning stress in pure copper: 18 GPa for [0 0 1] and 25 GPa for  3 4, which are in reasonable good agreement with the experimen½1 tal observation. Their proposed model was useful in assessing the dislocation densities under different surface pressures, but was not suitable for modeling the significant variance of grain sizes or microstructure textures in the workpiece subjected to multiple LSP impacts. Several theoretical attempts have been made to quantitatively analyze the microstructural evolution during severe plastic deformation (SPD) processes with strain rates generally less than 105 s1 based on material dislocation models. Estrin and other researchers presented a set of differential equations to evaluate the dislocation density evolution rates and applied the dislocation density-based material model to grain refinement in the equal channel angular processing (ECAP) processes of various materials such as aluminum [11], copper [12] and interstitial-free (IF) steel [13]. The nucleation of dislocations due to deformation, annihilation of dislocations due to dynamic recovery, and interaction of dislocations between the dislocation cell interiors and cell walls are evaluated based on the deformation process state variables. Their proposed dislocation density-based material model is compatible with the material constitutive models developed under varying conditions of strains, strain rates and temperatures and has been adapted to model other high-strain rate deformation processes such as a high speed impact test of copper with an average strain rate of 2500 s1 [14] and a machining process of copper with a strain rate close to 105 s1 [15]. Hence their dislocation density-based material model is adapted in this study for modeling the microstructural evolution by the LSP process with a typical maximum strain rate of 107 s1. In the present paper, the suitability of the dislocation densitybased model is investigated for predicting the microstructural evolution of metallic components subjected to LSP. The dislocation density-based material models are calibrated to replicate the observed material constitutive mechanical behaviors under the high-strain rate LSP process. A hydrodynamics model is employed to calculate the shock pressure induced by the interaction of high

Fig. 1. Schematic illustration showing microstructural evolution process of LY2 aluminum alloy induced by multiple LSP impacts [6].

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power nanosecond laser pulses with the target in air [16] or water [17]. A novel finite element (FE) model embedded with a dislocation density subroutine is developed to model the high-strain rate plastic deformation and evolution of grain refinement over the repetitive passes of the LSP process. To assess the validity of the numerical solution, the dislocation density and dislocation cell size are first modeled for monocrystalline coppers treated by the laser shock waves of high pressures of 12–40 GPa induced by energy densities on the order of 500 GW/cm2 and compared with the TEM measurement conducted by Meyers et al. [18]. The dislocation structure and material strengthening mechanism due to grain refinement in the LSP process are further studied for multiple LSP impacts of LY2 aluminum alloy with a laser energy density of 3–5 GW/cm2. The model-predicted residual stress and microhardness are compared with those measured by Lu et al. [6], Zhang et al. [19] and Luo et al. [20], respectively.

location cell structure is assumed to form during deformation, which consists of dislocation cell walls and cell interiors, and to obey a rule of mixtures. Different types of dislocation densities are distinguished in the model: the cell interior dislocation density (qc) and the cell wall dislocation density (qw), which is further divided into two distinct groups of statistical dislocation density (qws) and geometrically necessary dislocation density (qwg). The evolutions of the dislocation densities qc, qws and qwg follow different routes and are governed by the following equations:

1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 6 qws þ qwg c_ w  b c_ rc 3b bdð1  f Þ1=3  r 1=n c_  ko c qc c_ rc c_ o

q_ c ¼ a pffiffiffi

q_ ws

2. Experiments Two configurations of LSP are investigated in this work as illustrated in Fig. 2. The experiments under setup A were conducted for monocrystalline coppers in air with high laser energy densities on the order of 500 GW/cm2 and high laser-induced shock pressures of 12–40 GPa for the case of [0 0 1] copper by Meyers et al. [18]  3 4 copper by Schneider et al. [2]. Table 1 and for the case of ½1 shows the laser shock peening test conditions for monocrystalline  3 4 and LY2 alumicoppers with the orientations of [0 0 1] and ½1 num alloy. Tests A1 and A2 were conducted for [0 0 1] copper,  3 4 copper with varying lawhile A3 and A4 were conducted for ½1 ser power levels. Cylindrical workpieces with no coating or absorption layer were used in setup A with the dimension specified in Fig. 2a. One atmosphere of clean air breaks down at the threshold laser power intensity of about 1554 GW/cm2 [21], and hence air would not break down for tests conducted under setup A. The experiments using setup B were conducted on LY2 aluminum alloy in the water confinement regime with laser energy densities on the order of several GW/cm2 and high laser-induced shock pressures of 3–5 GPa by Lu et al. [6], Zhang et al. [19] and Luo et al. [20]. The LY2 aluminum alloy has a chemical composition similar to the AlCu4Mg1 alloy in ISO standard or aluminum 2024 [6,19,20,22]. Plate workpieces were used in setup B with a 7075 aluminum foil of 100 lm thickness as the absorption layer and running water as the plasma confining medium. Single impact tests were conducted for setup A, while multiple impact tests were conducted for setup B with repetition rates of 0.5–1 Hz.

pffiffiffi 2 6ð1  f Þ3 r 3ð1  f Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼b qws þ qwg c_ c þ ð1  nÞb c_ c bdf fb  r 1=n c_  ko w qws c_ rw c_ o

ð1Þ



ð2Þ

2

q_ wg ¼ nb

6ð1  f Þ3 r c_ c bdf

ð3Þ

The first terms of Eqs. (1) and (2) on the right side correspond to the generation of dislocations due to the activation of Frank–Read sources; the second terms account for the transfer of cell interior dislocations to cell walls where they are woven in; the last terms represent the annihilation of dislocations leading to dynamic recovery in the course of straining. The density of geometrically necessary dislocations is assumed to arise from a fraction (n) of the dislocations incoming into cell walls from the cell interiors as shown in Eq. (3). a⁄, b⁄ and ko are dislocation evolution rate control parameters for the material, n is a temperature sensitivity parameter, f is the volume fraction of the dislocation cell wall, b is the magnitude of the Burgers vector of the material, d is the dislocation cell size, c_ rw and c_ rc are the resolved shear strain rates for the cell walls and interiors, respectively, and c_ o is the reference resolved shear strain rate. It is assumed that the resolved shear strain rates across the cell walls and cell interiors are equal, c_ rw ¼ c_ rc ¼ c_ r , which satisfies the strain compatibility along the interface between interiors and boundaries. The resolved shear strain rate c_ r can be calculated by the von Mises strain rate e_ by using c_ r ¼ Me_ , where M is the Taylor factor. To obtain the resolved shear stress sr, the resolved shear strain rate c_ r is integrated with the dislocation densities as follows:

 1=m

3. Models

pffiffiffiffiffi c_ r src ¼ aGb qc _ c co

3.1. Dislocation density-based material model The dislocation density-based model is briefly presented in this section and more detailed descriptions can be found from the work of Lemiale et al. [14] and Estrin and Kim [23]. In the model, a dis-

ð4Þ 

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c_ r srw ¼ aGb qws þ qwg _w co sr ¼ f srw þ ð1  f Þsrc

Fig. 2. LSP experimental setups.

1=m ð5Þ ð6Þ

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Table 1 LSP tests of monocrystalline coppers and LY2 aluminum alloy. Setup

Test

Material

Plaser (J)

Dlaser (mm)

tpulse (ns)

Ilaser (GW/cm2)

A

A1 A2 A3 A4

[0 0 1] Cu [0 0 1] Cu  3 4 Cu ½1  3 4 Cu ½1

40 70 70 200

2.5 2.5 3 3

3 3 2.5 2.5

272 475 396 1132

351 351 351 351

B

B1 B2 B3

LY2 LY2 LY2

3 5 3

20 20 20

3.54 6.37 3.54

1064 1064 1064

5 25 5

where G is the shear modulus and m is the strain rate sensitivity of the material. Volume fraction f of the dislocation cell wall, total dislocation density qtot and dislocation cell size d are given as follows:

 f ¼ f1 þ ðfo  f1 Þe

cr

c~r

 ð7Þ

qtot ¼ f ðqws þ qwg Þ þ ð1  f Þqc

ð8Þ

K d ¼ pffiffiffiffiffiffiffiffi

ð9Þ

qtot

where fo and f1 are the initial and saturation volume fractions of cell walls, respectively, and the average cell size d is assumed to scale with the inverse of the square root of total dislocation density qtot. Dependence of the material microhardness as well as the flow stress on the refined microstructure in terms of dislocation density can be added into the current model, and the relationship between the strengthening of the microhardness (Dh) is given as

pffiffiffiffiffiffiffiffi Dh ¼ kh MaGb qtot

ð10Þ

where kh is a constant slope adopted to be 0.5 [24,25]. The dislocation density-based material model parameters need to be determined to replicate the observed material constitutive mechanical behaviors under the high-strain rate LSP process. The constitutive response for slip in face-centered cubic (fcc) metals is well-modeled by the Zerilli–Armstrong (ZA) constitutive description [26]. The material constitutive plastic models of slip stress for monocrystalline copper are shown as follows:

rS ¼ rG þ C 2 f ðeÞeC3 TþC4 Tlnðe_ Þ þ kS d1=2

ð11Þ

Meyers et al. [18] showed that for [0 0 1] copper,

f ðeÞ ¼ 19466:2e6  18522:2e5 þ 7332e4  1582e3 þ 189:5e2  2:4e þ 0:07

Pmax (GPa)

Impacts

Refs.

12 20 20 40

1 1 1 1

[18] [18] [2] [2]

3.25 4 3.25

5 3 1

[6] [19] [20]

dislocation annihilation (ko), were calibrated for monocrystalline coppers and LY2 aluminum alloy according to the flow stress data given by the material ZA and JC constitutive models, respectively. The dislocation annihilation rate parameter ko increases as the workpiece temperature increases. For the temperature range from ambient to 300 °C, ko linearly increases from 4.4 to 5.4 for [0 0 1]  3 4 copper, while from 4.3 to 8.6 for copper, from 5.8 to 6.7 for ½1 LY2 aluminum alloy. The temperature sensitivity coefficients m and n are given as follows:

A T

ð15Þ

B T

ð16Þ

m¼

n¼

where A and B were adopted as 30,000 and 14,900 for monocrystalline coppers, respectively, while A and B were adopted as 50,000 and 14,900 for LY2 aluminum alloy, respectively [15]. A constant fraction (n) of 0.1 was used for the evolution of the geometrically necessary dislocations, which was adopted from the modeling work of copper by Tóth et al. [12]. The other non-mentioned material constants were generally derived from an earlier study of ECAP process of pure aluminum [11] and copper [27]. The material dislocation density model parameters are given in Table 4. Figs. 3 and 4 compare the dislocation density-based material model predictions of the stress–strain relationships with the conventional constitutive model predicted flow stress data at different temperatures and strain rates for monocrystalline coppers and LY2 aluminum alloy, respectively. The identified model parameters have been proved accurate enough to predict the stress–strain relationships under different temperatures and strain rates, which validates the dislocation density-based plasticity models for the materials.

ð12Þ 3.2. Laser-induced shock pressure calculation

 3 4 copper, while for ½1

f ðeÞ ¼ 6293e6 þ 7441:4e5  3163e4 þ 515:65e3  4e2 þ 0:13e þ 0:059

ð13Þ

Since the properties of LY2 aluminum alloy are very similar to those of aluminum 2024-T3, the material Johnson–Cook (JC) constitutive plastic model for the latter is used in the calibration as follows:



klaser (lm)

   mJC  T  T ref 1 T m  T ref

e_ r ¼ ðAJC þ BJC enJC Þ 1 þ C al ln _ eo

ð14Þ

The constitutive model parameters for the materials are given in Table 2. The material mechanical and thermal properties are given in Table 3. The dislocation density-based material models were developed using MATLAB to replicate the material constitutive behaviors modeled by the material constitutive models. The reference strain rate was set as 107 s1. The dislocation density evolution rate control parameters, i.e., the dynamic coefficients of dislocation generation (a⁄), interaction between the cell walls and interiors (b⁄) and

Numerous FE analyses have been conducted on different aspects of LSP processes such as predictions of indentation profile and residual stresses [28–32]. However, the inputs of these FE analyses, the laser-induced shock pressure, mostly come from a simple analytical model proposed by Fabbro et al. [33], which has two free variables. Wu and Shin [34] have developed a selfclosed thermal model for the confined plasma in the LSP process without using any free variables but considering most of the important relevant physical processes. By applying the confined plasma model to the FE analysis of LSP, Cao et al. [35] accurately modeled the indentation profile and residual stresses of various metallic specimens such as 4140 steel, 316L stainless steel and Ti6Al4V alloy under different LSP conditions. Recently, Wu and Shin [17] have developed a numerical model with more strict physics consideration, where the one dimensional hydrodynamic equations, governing the conservation of mass, momentum, and energy for the confining medium and metal targets, are supplemented with appropriate equations of state. For nanosecond laser pulses with irradiances of several GW/cm2, the plasma induced by

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H. Ding, Y.C. Shin / Computational Materials Science 53 (2012) 79–88 Table 2 Constitutive model parameters. C3 (K1)

kS (MPa mm1/2)

C4 (K1)

Material

rG (MPa)

Cu [18,26]

46.5

115

0.0028

0.000115

5

Material

AJC (MPa)

BJC (MPa)

nJC

CJC

mJC

LY2 [42]

265

426

0.34

0.015

1.0

C2 (MPa)

Table 3 Material properties. Material

E (GPa)

G (GPa)

m

b (nm)

q (kg/m3)

Tm (°C)

Thermal expansion (106/°C)

Thermal conductivity (W/m K)

Specific heat (J/kg K)

Cu [15] LY2 [20,22,43]

116 73.1

48 28

0.34 0.34

0.256 0.286

8960 2770

1083 502

16.6 23.2

400 121

385 875

Table 4 Dislocation density-model parameters. Material

a⁄

b⁄

ko

A (K)

B (K)

c_ o

fo

f1

K

M

c~r

qwo (mm2)

qco (mm2)

b (mm)

a

[0 0 1] Cu  3 4 Cu ½1

0.04 0.12 0.06

0.01 0.006 0.014

4.4–5.4 5.8–6.7 4.3–8.6

30,000 30,000 50,000

14,900 14,900 14,900

1E7 1E7 1E7

0.25 0.25 0.25

0.077 0.077 0.06

10 10 30

3.06 3.06 3.06

3.2 3.2 3.2

1E7 1E7 1E7

1E7 1E7 1E7

2.56E7 2.56E7 2.86E7

0.25 0.25 0.25

LY2

Fig. 3. Dislocation density-based plasticity model predictions for monocrystalline coppers.

Fig. 4. Dislocation density-based plasticity model predictions for LY2 aluminum alloy.

laser ablation of metal targets could be described by the hydrodynamic equations for the whole physical domain, where the condensed phase contributes mass to the plasma region mainly through hydrodynamic expansion. Therefore, this hydrodynamics model is adopted in this study to calculate the laser-induced shock pressure.

According to the criteria developed by Wu and Shin [36], the 1D plasma expansion assumption is valid when the laser beam diameter is equal or larger than 300 lm. Since the laser beam diameter used in this work is around 3 mm, it is sufficient to use the 1D model in this work to describe the confined plasma behavior under air or water. Fig. 5 shows the standardized laser-induced shock

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pressure durations in air and water medium for monocrystalline coppers in setup A and LY2 aluminum alloy in setup B calculated using the hydrodynamics model using the laser processing conditions given in Table 1. It is noted that the normalized pressure profiles for different laser power densities using the same experiment setup are very similar: the pressure increases drastically during the laser pulse duration, and then gradually drops to below 15% of the peak value after 40 ns and 200 ns for experiment setup A and B, respectively. The calculated peak pressures for all the tests are given in Table 1. 3.3. Finite element analysis FE models were developed to simulate the high-strain rate plastic deformation and grain refinement in single or multiple LSP impacts by using the commercial software ABAQUS 6.10.1. Fig. 6 shows the 2D FE model setups for the LSP process, in which axisymmetric models were used for the workpieces. FE analyses for setup A use the same workpiece size as that in the experiments, while a 10 (width) by 4 (height) mm workpiece size is modeled for setup B, which has the same height as the actual workpiece and large enough width to approximate the actual one. The modeled axisymmetric parts are constrained in the X direction along their center axes and in the Y direction on the bottoms. The workpiece domain is meshed using 4-node thermally coupled axisymmetric quadrilateral, bilinear displacement and temperature elements using reduced integration and enhanced hourglass control. The laser-induced shock wave pressures predicted using the hydrodynamics model were applied as user-defined distributed time-dependent pressure loads in ABAQUS, as illustrated in Fig. 6. For each LSP impact, the pressure loads are reduced to zero after 40 ns and 200 ns pressure durations for setup A and B, respectively. To capture the dynamic nature of the shock wave in each LSP impact, fully coupled thermo–mechanical ABAQUS/explicit analysis was first carried out for a time step of 1000 or 2000 ns, which is at least 10 times longer than the shock pressure duration. The resultant solution was then imported to the following implicit step, which calculates the steady-state deformation and dislocation structure by the ABAQUS/standard solver to save computation cost. The dislocation density-based material model subroutines defined earlier are incorporated in the FE models to calculate the dislocation fields in the workpiece over single or repetitive LSP impacts. Fig. 7 shows the flow chart for modeling using the dislocation density-based material models in the FE analyses. Laser-induced shock pressures calculated by the hydrodynamics model are input as pressure loading in the beginning of each LSP impact simulation. A uniform initial state of dislocation model parameters as given in Table 4 is defined for the first LSP impact simulation, while the dislocation model parameters are imported from the earlier impacts for the following ones, as illustrated in Fig. 7.

4. Validation of dislocation model on monocrystalline coppers Fig. 8 shows the total dislocation densities and dislocation cell sizes predicted along the center axis of the [0 0 1] copper workpiece after single LSP impact. The average values of the total dislocation densities achieved are simulated to be 2.7  1014 m2 and 6  1014 m2 under the laser pulse energy of 40 J in test A1 and 70 J in tests A2, respectively. Compared with those experimentally measured by Murr and Kuhlmann-Wilsdorf [37] and Meyers [38] at various pressures, the model predicted dislocation densities agree well with the trend of pressure-dislocation density as can be seen in Fig. 8a. The variations in the total dislocation densities come from the high-strain rate deformation heterogeneity from the top of the workpiece to the bottom predicted by the model for the LSP process. The shock pressure load propagates from the top of the workpiece to the bottom, and the decay of the shock pressure has been well simulated in the FE analyses. For test A1 using laser pulse energy of 40 J, the predicted dislocation cell size is refined to a range from 550 nm under a peak shock pressure of 10.5 GPa to 900 nm under a peak shock pressure of 7.9 GPa, while for test A2 using laser pulse energy of 70 J, the predicted dislocation cell size is refined to a range from 250 nm under a peak shock pressure of 18 GPa to 500 nm under a peak shock pressure of 13.5 GPa. The dislocation cell sizes predicted by the model along the [0 0 1] copper workpiece center axes are compared with the measurements by Murr [39], Gray [40] and Meyers et al. [1] at various pressures as shown in Fig. 8b. It can be seen that the predictions of the dislocation sizes are very close to the measurements at various pressure and the model predictions catch the exponential decay of the cell size well with increasing pressure. Fig. 9 plots the dislocation cell sizes predicted against (a) distance from the LSP driven surface and (b) the peak pressures along  3 4 copper workpiece after single LSP imthe center axes of the ½1 pact under various laser pulse energies. The dislocation cell size distributions in the depth direction predicted by the model are very close to the TEM measurement taken by Schneider et al. [2]. The model-predicted minimum dislocation cell size drops from 220 nm to 70 nm as the laser pulse energy increased from 70 J to 200 J and matched well with the trends observed in the measurements. For instance, for test A4 under 200 J laser pulse energy, the model-predicted dislocation cell sizes of 116, 220, 860, 1140 nm at depths of 0.25, 0.75, 1.75 and 2.25 mm from the top surface are quite close to the measured sizes of 160, 230, 770, 1400 nm taken at the same depths. The cell size-pressure profiles predicted using the model are in reasonably good agreement with the measurements, but there is some discrepancy as can be seen in Fig. 9b. It is mainly because that the peak pressures calculated in the FE analyses are slightly different from those given by Schneider et al. [2]. Nonetheless, the predicted dislocation sizes are very similar to the measurements and it further validates the necessity of developing such a numerical model as in this study, which captures the under-

Fig. 5. Normalized laser-induced pressure.

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Fig. 6. FE setups for LSP.

Fig. 7. Flow chart for model calculations.

Fig. 8. Total dislocation density and dislocation cell size predictions for [0 0 1] copper.

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 3 4 copper in comparison with TEM measurement by [2]. Fig. 9. Dislocation cell size predictions for ½1

Fig. 10. Dislocation predictions for the first LSP impact of test B1 of LY2 aluminum alloy.

lying physics during the high-strain rate LSP process by considering all the important process parameters.

5. Simulations of material strengthening under multiple LSP impacts of LY2 aluminum alloy FE analyses were conducted for tests B1–B3 of LY2 aluminum alloy using setup B. These simulations were aimed at modeling the material strengthening mechanism due to grain refinement in a usual LSP process in a water confinement regime with laser energy densities on the order of several GW/cm2. Fig. 10 shows the total dislocation density and dislocation cell size distribution predicted by the model for the LY2 aluminum alloy specimen subjected to the first LSP impact of test B1. The highest total dislocation density was modeled to reach 1013 m2 after the first LSP impact, and the minimum dislocation cell size or subgrain size was modeled to be refined to about 9.5 lm from an initial grain size of 50 lm after the first LSP impact. Fig. 11 shows the total dislocation densities and dislocation cell size profiles of the LY2 aluminum alloy specimen subjected to multiple LSP impacts during test B1 predicted along the center path by the model. It can be seen that the shock wave-induced dislocation structure exists at the depth of about 0.4 mm after the first LSP impact, while it reaches about 0.8 mm below the top surface after 5 LSP impacts. The model predicted that the total dislocation density increased and the dislocation cell size decreased with every more LSP impact. The highest total dislocation density reached about 6.6  1013 m2, while the minimum dislocation cell or subgrain size was predicted to be refined to about 3.8 lm after 5 LSP impacts, which is close to the TEM observation of an average grain size of 3–5 lm near the top surface by Lu et al. [6].

FE analyses using the dislocation density-based material models were also used to predict the residual stress and microhardness after multiple LSP impacts of LY2 aluminum alloy. Fig. 12a shows the distribution of the residual stress in the horizontal direction of LY2 aluminum alloy specimen after 5 LSP impacts predicted by the model. An evident compressive residual stress profile with the maximum compressive stress of 140 MPa was predicted beneath the laser treated area. The comparison of the simulated residual stress and experimental data are shown in Fig. 12b, where the error bar of the measured residual stress using the X-ray diffraction method was estimated to be 15 MPa [41]. It can be seen that the simulation results from the current model are very close to the experimental data in both magnitude and trend in the region of depth less than 0.7 mm. Overall, reasonable prediction of residual stresses was obtained for the multiple LSP impact test of LY2 aluminum alloy, which further validated the efficacy of the numerical model. Surface and subsurface Vickers microhardness (HV0.2) were simulated for the LY2 aluminum alloy samples subjected to multiple LSP impacts. Fig. 13a and b shows the predicted subsurface microhardness in depth from the LSP driven surface of test B2 and surface microhardness from the LSP impact center of test B3, respectively. The microhardness data measured by Zhang et al. [19] and Luo et al. [20] are shown in Fig. 13a and b, respectively, for a comparison. It can be seen in Fig. 13a and b that surface strengthening in microhardness was predicted to reach 154 HV0.2 after the first LSP impact and 166 HV0.2 after 3 LSP impacts, as compared with the average microhardness of 138 HV0.2 of the nontreated specimen. It is also shown that the shock wave-hardened surface layer has a depth of about 0.6 mm after the first LSP impact, while it increases to above 1 mm after 3 LSP impacts. The simulation results of the microhardness profile from the current model are very close to the experimental data in both magnitude and

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Fig. 11. Dislocation predictions for multiple LSP impacts of test B1 of LY2 aluminum alloy.

Fig. 12. Residual stresses after 5 LSP impacts of test B1 of LY2 aluminum alloy (experimental data from [41]).

Fig. 13. Microhardness for multiple LSP impacts of LY2 aluminum alloy.

trend on the surface and in the subsurface region to a depth of 2 mm.

6. Conclusions In this paper, a dislocation density-based predictive model was developed to simulate the high-strain rate plastic deformation and microstructural evolution during LSP. Novel FE models embedded with the dislocation density subroutine were used to model the grain refinement under the repetitive passes of the LSP process. The validity of this approach was assessed through a direct comparison with the experimental data on dislocation density and dislocation cell size achieved by LSP of monocrystalline coppers. The model predicted that the total dislocation densities reach the order

of 1014 m2, and the minimum dislocation cell size is refined to below 250 nm for monocrystalline coppers under laser energy density on the order of 500 GW/cm2, which are in good agreement with those experimentally measured. The model was also applied to study the material strengthening mechanism of the LY2 aluminum alloy due to grain refinement in the usual LSP process in the water confinement regime with laser energy densities on the order of several GW/cm2. The simulation results showed the evident subsurface compressive residual stresses induced by multiple LSP impacts with a maximum compressive surface stress of 140 MPa, which are very close to the experimental data in the region of depth less than 0.7 mm. The microhardness profiles predicted from the current model are also close to the experimental data with the surface microhardness increased to above 160 HV0.2 after 3 LSP impacts from 138 HV0.2 for the non-treated specimen. The dislocation

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density-based framework developed in this study has been shown to be a useful tool to model the microstructural evolution during the single or multiple LSP process and can be used to better design process parameters to achieve optimum mechanical properties of the refined microstructure.

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