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3-D FEM simulation of laser shock processing
Surface & Coatings Technology 201 (2006) 1426 – 1435 www.elsevier.com/locate/surfcoat
3-D FEM simulation of laser shock processing Yongxiang Hu ⁎, Zhenqiang Yao, Jun Hu School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, 200030, China Received 27 October 2005; accepted in revised form 6 February 2006 Available online 6 March 2006
Abstract Laser shock processing is an innovative surface treatment technique similar to shot peening. It can impart compressive residual stresses in material for improving fatigue, corrosion and wear resistance of metals. FEM simulation is an effective method to predict mechanical effects induced in the material treated by laser shock processing. An analysis procedure including dynamic analysis performed by LS-DYNA and static analysis performed by ANSYS is presented in detail to attain the simulation of the single and multiple laser shock processing to predict the residual stress field and the surface deformation. History of the energies during dynamic analysis is analyzed and validated by the theoretical calculation. The predicted residual stress field for single laser shock processing is well correlated with the available experimental data and a homogeneous depression with little roughness modification in the action zone of the shock pressure is induced on the treated surface according to the simulation of surface deformation. Simulation of multiple laser shocks is also performed, which indicates that compressive residual stresses and plastically affected depth can be extensively increased and gradually reach the saturated state with the increase of laser shock number. © 2006 Elsevier B.V. All rights reserved. Keywords: Laser shock processing; Residual stress; Finite element method; Stress wave; Shock wave
1. Introduction More recently, surface treatment technologies have become more and more important in industry to cut costs and avoid the need for expensive materials. Demonstrated approximately 30 years ago, laser shock processing (LSP) is now emerging as a viable surface treatment technique. The compressive residual stresses in the metal material treated by LSP can extend deeper below the surface than those from shot peening. LSP is well suited for precisely controlled treatment of localized fatigue critical areas, such as holes, notches, fillets and welds. It has been proposed as a competitive alternative technology to classical treatments for improving fatigue, corrosion and wear resistance of metals [1]. Most investigations have been concentrated on experimentally determining mechanical effects including residual stress field and surface morphology induced in different LSP configurations for a number of industrial metals to get the optimized treatment results before. Finite Element Method (FEM) is first introduced by Braisted and Brockman to ⁎ Corresponding author. Tel.: +86 21 6293 3071; fax: +86 21 6293 2060. E-mail address: [email protected] (Y.X. Hu). 0257-8972/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.surfcoat.2006.02.018
investigate the mechanical behavior and predict the residual stresses from the laser shocked materials with software ABAQUES in 1999 [2]. It offered a new effective approach to optimize parameters of laser shock. From then on, several researchers have used ABAQUES to analyze the laser generated shock waves propagating into different metal materials in the past six years [2–7]. Some of these simulations have a close match with experimentally measured residual stresses. However, research on LSP simulation is still concentrated on accurately modeling the LSP process because of the complexity of LSP process. Work on the 3-D FEM simulation of the process to predict the residual stress field and the surface deformation is still too little now. To the best of our knowledge, no work has been done on the LSP FEM simulation with software besides ABAQUS so far. This paper proposes a new method to simulate shock wave propagation to predict residual stresses distribution and surface deformation of the material treated by LSP using LS-DYNA and ANSYS in three dimensions. FEM modeling strategy is presented in detail. And behaviors of the material subjected to single shock have been analyzed and compared with the experimental results. Simulation of successive shocks on the same position is also performed and discussed.
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2. Mechanism of LSP and experiment Unlike other laser application, LSP works by exerting a mechanical force on the surface of the treated material, while shocked surface is not affected thermally. The process involves a high-intensity laser beam (several GW/cm2) combined with suitable overlays to generate mechanical pressure waves up to several GPa. A schematic of how LSP works is shown in Fig. 1. The metallic surface to be treated is first locally coated with an overlay opaque to laser beam (typically a black paint) and then covered with a transparent overlay, which is a dielectric material transparent to laser beam (such as water). Opaque overlay acts as a sacrificial material, a thin layer of which vaporizes on absorption of laser energy. The transparent overlay confines the thermally expanding vapor and plasma against the surface of the material, thus generating higher pressure than with the direct ablation mode. The shock pressure can result in the plastic deformation and favorable compressive residual stresses on the surface layer of the material [8]. Ballard investigated into LSP effects on the residual stress field and fatigue behavior of steel in his doctoral dissertation [9]. A metal specimen (35CD4, 50HRC steel) was irradiated by a square spot 5 × 5 mm with a laser power density of 8 GW/cm2 and duration of 30 ns. The specimen surface was coated with black paint, and the water was used as the transparent overlay. Stresses generated on the specimen surface were measured by X-ray diffraction and the plastically affected depths were evaluated using successive electrolytical polishing and X-ray diffraction. As expected, the pressure profile of the induced plasma was a Gaussian temporal shape with a full width at half maximum (FWHM) of 50 ns, and its peak pressure was 3.0 GPa [5]. Ballard's experimental results are used as a reference to validate the FEM results of the simulation. 3. FEM simulation strategy 3.1. Analysis procedure Utilization of a very short interaction time and high power density laser is representative of LSP for material treatment. Shock waves induced by the pressure pulse propagate into the target material. If the pressure of the shock wave exceeds the dynamic yield strength of the material, the material will undergo extremely high strain-rate (106–108 s− 1) during a short period of time and be dynamically yielded [10].
Shock Wave
Focused pulse laser Trapped Plasma Transparent Overlay Absorbent Coating Metal Target
Fig. 1. Schematic of the LSP.
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Pre-processing ANSYS
Input ANSYS
Dynamic Analysis LS-DYNA Solver
Static Analysis ANSYS Solver
Transient Stress & Strain Deformation Output
Stable Stress & Strain Deformation Output
Post-processing ANSYS
Residual Stress Field & Surface Deformation Fig. 2. FEM analysis procedure.
Since the pressure pulse duration is very short and the load is very severe, the response of the material changes rapidly. Although the loading time of the shock pressure is very short, response of the target material will take much longer to become stable owing to the reflection and interaction of various stress waves produced in the target material. Accurate tracking of the front of various stress waves is important to capture the dynamic response of the material. The explicit algorithm is especially well-suited to solving this high-speed dynamic event efficiently. Therefore, it is adopted to simulate the propagation of the shock wave induced by the short-time pressure and solve the dynamic response of the material. On the other hand, the high-speed dynamic deformation processes tend to generate a large amount of elastic strain energy in the material. The elastic strain energy, which has been stored in the material after performing the explicit dynamic analysis, is subsequently released if the solution time of the explicit analysis is long enough, but it will be time costly. Performing an explicit-to-implicit sequential solution will be necessary to release all the elastic strain energy efficiently and the final stable desired residual stress field can be obtained. Hence, FEM analysis procedure of laser shock processing should be composed of two distinct parts including dynamic analysis and static analysis to obtain an absolutely stable residual stress field and surface deformation. Dynamic analysis is adopted to simulate the propagation of the shock wave and obtain the dynamic response of the material. When the dynamic stress state of the target material becomes approximately stable, all transient stress will be imported into implicit FEM codes to perform static analysis to obtain the residual stress field and the spring-back deformation in static equilibrium. By simulating the dynamic pressure shock explicitly, and then modeling the elastic energy release implicitly, stringent tolerances can be attained for the distribution of the residual stresses and the surface deformation. Both ANSYS and LS-DYNA are adopted to attain full FEM simulation of laser shock processing. LS-DYNA codes are used for the dynamic analysis. The explicit solution method used by LS-DYNA provides fast solutions for short-time, high speed
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nonlinearity problems. After the dynamic analysis is completed, stress field of the model calculated by the dynamic analysis will be read into ANSYS codes, which integrate LS-DYNA codes with the powerful pre- and post-processing capabilities; so we can model the structure in ANSYS, obtain the explicit dynamic solution via LS-DYNA, and review results using standard ANSYS post-processing tools. The LSP FEM analysis procedure can be depicted in Fig. 2. The combined procedure takes advantage of the best feature of each code to perform explicit and implicit analysis. 3.2. Governing equation 3.2.1. Explicit finite element algorithm In order to analyze the response of the material to the shock pressure with the explicit finite element method, the dynamic equation at time t can be expressed as follows, according to the virtual work principle: Z Z :: : qu i dui dX þ qui dui dX X Z XZ Z ¼ fi dui dX þ TC dui dC− rij dDij dX ð1Þ X
C
X
where ρ is mass density, ü and u˙ nodal acceleration and velocity, respectively, υ the damping coefficient, δu the virtual displacement, fi the body force density, TΓ the boundary force applied on the boundary Γ, σij the Cauchy stress tensors, Dij ¼ 1 dui;j þ duj;i is deformation rate tensors [11]. 2 Discretization of this problem is accomplished by means of the standard finite element procedure. After aggregation, we have the dynamic equilibrium equation as follows:
::
:
MU þ CU þ Fint ¼ Fext
ð2Þ
where M is the diagonal mass matrix, Ü and U˙ the nodal acceleration and velocity vector, respectively, C diagonal damping matrix, Fint the internal element force vector, Fext accounts for external and body force loads. To advance to time tn+1, central difference time integration is usually used as follows:
::
:
U ¼ M−1 ðFext −Fint −CUÞ
: nþ
U
1 2
1
::n
¼ Un−2 þ U Dt n
: nþ Dtnþ
Unþ1 ¼ Un þ U
1 2
ð3Þ ð4Þ
1 2
ð5Þ
ðDt n þ Dt nþ1 Þ 1 . Then we update the geometry by where Dt nþ2 ¼ 2 adding the displacement increments to the initial geometry Xnþ1 ¼ X0 þ Unþ1
ð6Þ
The central difference scheme is conditionally stable. The stability limit can be defined using the element length Le, and the wave speed of the material Cd: Dtstable ¼
Le Cd
ð7Þ
3.2.2. Implicit finite element algorithm According to the virtual work principle, the equilibrium equation of the implicit algorithm can be expressed as follows: Z Z Z rddeddX ¼ f dduddX þ TC dduddC ð8Þ X
X
C
where σ and ε are stress and strain tensors, respectively, f and TΓ body force density and boundary force, respectively, and δu the virtual displacement. Dicretization of this problem is accomplished by means of the standard finite element procedure. After aggregation, we have a group of nonlinear equations as follows that require the Newton–Raphson method to linearize them: KDU ¼ Fext −Fint
ð9Þ
where K = ∫ΩB DBdΩ is the tangential stiffness matrix, B the general geometric matrix, D the stress–strain matrix, ΔU the displacement incremental at the element nodes, Fint = ∫ΩBTσdΩ Newton–Raphson restored force vector and Fext external and body force load that equals zero for the elastic energy release problem [12]. After the static analysis is performed, a stable residual stress distribution and a spring-back deformation field will be dyn sta obtained. Supposing Usur and Usur are the node displacement on top surface obtained from the dynamic analysis and static analysis, respectively. Then the deformation on top surface can be calculated as follows: T
dyn sta Ures sur ¼ Usur þ Usur
where LSP.
res Usur
ð10Þ
is the displacement of each node on top surface for
3.3. LSP model 3.3.1. Loading The high amplitude, short duration pressure pulse of the shock wave generated by laser shock is one of key parameters related to the effect of laser shock processing. Significant experimental efforts have been performed to determine the magnitude and duration of the pressure upon the shock surface [13–16]. The pressure time history is assumed to be known in advance because the simulation procedure does not explicitly model the physical process of the pressure pulse formation. Generally, the pressure pulse is relatively uniform over the entire surface of the laser spot regardless whether the spot is circular or rectangular. And because of the narrow duration of the pressure pulse induced by LSP, the pressure time history is modeled as a triangular ramp in the simulation, in which the pressure rises linearly to the peak value over the time of the pressure pulse FWHM and then decays linearly to zero during the following FWHM [5]. 3.3.2. Constitutive model LSP generates strain-rate exceeding 106s− 1 within the target material. As the strain-rate increases, materials typically exhibit little change in elastic modulus, but an increase in yield strength and the event becomes a shock wave phenomenon. Under uni-
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axial strain conditions, the highest elastic stress level in the shock wave propagation is defined as the Hugoniot Elastic Limit (HEL) for the semi-infinite model. When the pressure is greater than HEL, permanent deformation occurs. Assuming that the yielding occurs when the stress in the direction of the wave propagation reaches the HEL, the dynamic yield strength σydyn under uni-axial strain conditions can be defined in terms of the HEL by: rdyn y
¼ HELð1−2mÞ=ð1−mÞ
Non-reflecting Boundary Surface
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z
3.3.3. Finite element model A three-dimensional semi-infinite finite element model with size 20 × 20 × 10 mm has been developed to simulate one-sided laser shock solid surface. Since the model is symmetric and subjected to a symmetric uniform square pressure pulse, only a quarter of the configuration is used to perform the finite element calculation instead of a full one to ensure computational efficiency. A schematic configuration of this model including boundary conditions is shown in Fig. 3. The symmetric boundary conditions are employed on the x-0-z and y-0-z planes. Nonreflecting boundaries, which will prevent stress wave reflections generated at the boundary from re-entering the model and contaminating the results, are applied to the exterior surfaces of the model to simulate the semi-infinite model. Solid 164 explicit elements, defined by eight nodes having the following degrees of freedom at each node – translations, velocities, and accelerations in the nodal x, y, and z directions – are chosen for dynamic analysis. They will be converted into corresponding companion implicit elements Solid 185 for static analysis [12]. Results of FEM analysis are sensitive to the mesh density [5]. It is easy to obtain more accurate results with dense meshes, but higher computational cost. To accurately capture mechanical effects induced by laser shock, it is necessary to have sufficient mesh density about the region subjected to the laser shock, which we are most interested in. As shown in Fig. 3, a densely meshed region extends two times the spot size in three directions with the element edge length on the order of 2.5% of the spot size. For the rest region, mesh density is four times as Table 1 Mechanical properties of 35CD4 50HRC steel [5] Properties
Value
Unit
Density, ρ Poisson's ratio, ν Elastic modulus, E Hugoniot elastic limit, HEL
7800 0.29 210 2.1
kg m− 3 GPa GPa
x
y Symmetric Boundary Surface Symmetric Boundary Surface
ð11Þ
where ν represents Poisson's Ratio [2]. Also, the target material is assumed to be perfectly elasticplastic with isotropic and homogeneous characteristics for the analysis. The plastic strain is assumed to follow the Von Mises yielding criterion and the dynamic yield strength σydyn is based on Eq. (11). Given above assumptions, the material properties of 35CD4 50HRC steel required for the FEM simulation are shown in Table 1.
Shocked Region
Non-reflecting Boundary Surface Fig. 3. The schematic configuration of 3-D model.
coarse as the densely meshed one. The number of total elements in model is 125,000. Region of the refined element mesh needs to be extended or adjusted if the calculated plastic deformed region is close to the boundary of two regions with different mesh density. 3.3.4. Solution time choosing To accurately capture the dynamic response of the material, the solution time must take much longer than the duration of the pressure pulse, owing to the reflection and interaction of the multiple stress waves propagating in the target. Ding suggested that the solution time could be set as two orders of the magnitude longer than the pressure pulse duration to ensure that the saturation of the plastic deformations occurred in the target material [5]. But it is not easy to operate in practice because the saturation of the plastic deformations is difficult to distinguish from other states of the material in the post-processing. Stress waves induced by the short duration pressure pulse cause the movement of the material particles and deform the target material, which are demonstrated by the kinetic energy and the internal energy including elastic strain energy and plastic strain energy, respectively. When the kinetic energy tends to zero and the internal energy tends to constant simultaneously, it indicates that the interaction of the stress waves in the material is quite weak and the dynamic stress state is driven to be stable. Certainly, no plastic deformation will occur in the material after that time. Hence, it can be chosen as the solution time for the dynamic analysis. A detail discussion will be given in the Section 4.1 based on the dynamic analysis results. 4. Results of single LSP and discussion 4.1. Dynamic stress state The response of the material is a process of propagation, reflection and interaction of stress waves. The total energy Et from the external work of the pressure pulse is mainly converted into the kinetic energy Ek, internal energy Ei and hourglass energy Eh. Hourglass energy is usually used to demonstrate the effect of zero energy modes, which is also called hourglass modes arising because of shortcoming in the element
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0.05 0.04
Energy /J
It is easy to find that the elastic wave velocity is always faster than the plastic one. Then time needed for the first two types of stress wave propagating to the bottom surface of the target material is shown as follows:
a b c d
0.03 0.02 0.01
te ¼
lthick Cle
ð14Þ
tp ¼
lthick Clp
ð15Þ
0 0.1
1
1.7 2
3
4
5
time /μs Fig. 4. History of energies (a—internal energy, b—kinetic energy, c—hourglass energy, d—total energy).
formulation for the under-integrated solid elements in explicit algorithm. A notable hourglass energy mode will result in a state that cannot exist in practice, but is mathematically stable, which usually exists for problems that deform with high velocities [11]. According to the stress wave theory, the elastic wave velocity and the plastic wave velocity can be calculated as follows [17]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1−mÞE 1 e Cl ¼ d ð12Þ ð1 þ mÞð1−2mÞ q Clp
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E 1 d ¼ 3ð1−2mÞ q
ð13Þ
where Cle is the elastic wave velocity, Clp the plastic wave velocity, ρ the material density, E the Elastic modulus, ν the Poisson's Ratio.
where lthick is the thick of target material. Boundary condition for the bottom surface of the FEM model is nonreflecting to model the semi-infinite solid, so stress waves propagating to the bottom surface will be completely transmitted, which will definitely cause a dramatically decreasing of the total energy. How these energy changes as the shock waves propagate through the target material by FEM simulation is shown in Fig. 4. The total energy of the model is rapidly increased to the peak in the initial period of 100 ns, and then remains approximately invariable during the period from 100 ns to 1700 ns, just with a little decreasing for hourglass energy defined in the LS-DYNA. A sudden declination of the total energy occurred at the time about 1700 ns, which is well consistent with the theoretical result calculated by Eq. (14). It is the time that first elastic stress wave propagates to the bottom surface of the material. According to Fig. 4, the total energy and the internal energy remain stable after the time 4500 ns while the kinetic energy finally diminishes to zero, which means that the response of the target material is driven to be stable. The surface stress σx along the x-axis varies with the solution time as described in Fig. 5. It can be seen that the dynamic stress profile changes quite clearly in the solution period from 1000 ns to 2000 ns but is oscillating slightly and tends to stable after 4000 ns, which is consistent
200 100
Residual stressσx /MPa
0 -100 -200 -300 -400
0997ns 1998ns 3996ns 4499ns 4997ns
-500 -600 -700 -800 0
0.5
1
1.5
2
2.5
3
3.5
Distance from centerline along x-axis /mm Fig. 5. Stress state on top surface at different time.
4
4.5
5
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with the history of the energies. So 5000 ns is enough for dynamic stress state to become stable and no further interaction of different stress waves will occur in the target later. It can be chosen as the solution time for the dynamic analysis. The method for solution time choosing based on whether history of internal energy remains stable or history of kinetic energy diminishes to zero is distinct and easy to select compared with that based on the saturation of the plastic deformation. 4.2. Static analysis result 4.2.1. Residual stress field Stresses of dynamic analysis results at 5000 ns are read into ANSYS codes to perform the static analysis. Contours of the Von Mises stress distribution of the dynamic analysis at 5000 ns and of static analysis are shown in Fig. 6. Maximum Von Mises stress has decreased from 394.5 MPa to 349.2 MPa because of the elastic strain release. Contours of three components of the residual stress field from FEM simulation are shown in Fig. 7. The distribution of
Fig. 7. Contour of residual stresses field for three components.
surface residual stresses along the x-axis and in depth compared with experimental results is shown in Fig. 8. The following observations can be made from these figures:
Fig. 6. Contour of Von Mises stress distribution of single LSP.
(1) Simulated results in Fig. 7 show that x and y components of residual stresses are only produced in the layer
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Residual Stress σx /MPa
0 -100 -200 -300
FEM σx Experiment σx FEM σy Experiment σy
-400 -500 -600 0
1
2
3
4
5
Distance from centerline along x-axis/mm
(a) On top surface along x-axis
Residual Stress σx /MPa
100
Lp
0 -100 -200
FEM σx Experiment σx FEM σy Experiment σy
-300 -400 -500 0
0.2
0.4
0.6
0.8
4.2.2. Surface deformation The surface deformation of the material has a great effect on fatigue behavior. FEM simulation in full 3-D geometry is an effective tool for the assessment of the surface deformation after the treatment, which should be based on the assumptions that the material surface is an ideal plane in geometry and thermal effect induced by laser beam is totally avoided. Z component of displacement on the top surface simulated by the dynamic analysis is shown in Fig. 9(a) and that for elastic energy release by static analysis is shown in Fig. 9(b). Simulation of the surface deterioration of the material treated by LSP is calculated according to the Eq. (10) and the result is shown in Fig. 9(c). From the Fig. 9(c), a homogeneous depression less than 1 μm with little roughness modification in the action zone of the shock pressure is induced on top surface of the material treated by single LSP for the compressive residual stress. Similar results were also observed in the experiment results, which presented a comparison of the roughening effects of laser shock processing and shot peening on A356 and 7075 aluminum alloys. In both cases, while the laser shocked surface remains partially unchanged, shot peening generates a detrimental roughened surface with large increases in the mean and peak roughness Ra and Rt [18]. 5. Multiple LSP analysis
1
Depth along z-axis /mm
(b) In depth along z-axis Fig. 8. Residual stress distribution of single LSP.
about 1 mm from the top surface after laser shock processing. The stress distributions between x and y components are similar. And z component of stress is nearly zero on the top surface. A tensile residual stress is found in the z component of stress of the model as shown in Fig. 7(c), which is generated by the reversing yielding effect due to the interaction of the various stress waves in the material. (2) As shown in Fig. 8, the experimental data show that the maximum compressive residual stress σx is about 355 MPa, with a plastically affected depth of 0.80 mm. FEM simulation predicted maximum compressive residual stress σx is about 349 MPa, 1.7% smaller than that from experiments and the plastically affected depth is 0.67 mm, 16.3% smaller than that from experiments. Hence, the predicted residual stress field is well consistent with those experimental results obtained by Ballard [9] and the FEM simulation method used for LSP is validated. (3) The simulated results also indicate that compressive residual stress of the material is approximately uniform on the shocked region. And the residual stress is highest at the surface decreasing gradually with distance below the surface, i.e., the stress gradient in depth is small, which is useful to get uniform residual stress field in the surface layer.
The intensity of laser shock processing is related to the laser intensity, i.e., the power per unit area applied to the laser shocked spot. As the power density is increased, the peak shock pressure increases, and this increases both the surface compressive residual stress and the depth of the compressive residual stress. The ability to continue to increase the intensity to increase the magnitude and the depth of the compressive residual stress is limited, however, by the onset of dielectric breakdown in the transparent overlay above a certain power density. This limitation is overcome by applying successive shocks to a spot [10]. An attempt to model the residual stresses for multiple onesided laser shock processing at the same location using the same model parameters as for the single shock predictions above is investigated according to the procedure shown in Fig. 2. The successive laser shocks are modeled by applying an interval pressure pulse to the shocked surface of the material for dynamic analysis. The loading curve for the multiple LSP is shown in Fig. 10. The surface and in depth residual stress field resulting from multiple LSP analysis are shown in Fig. 11 and the contour of Von Mises residual stress field after five laser shocks is shown in Fig. 12. Compressive residual stress and plastically affected depth are significantly increased by multiple LSP. The maximum compressive residual stress σx and plastically affected depth are increased by 40.0% from 349 MPa to 485 MPa and 35.8% from 0.67 mm to 0.91 mm after the second shock on the same position, respectively. By five laser shocks at the same position, the maximum compressive residual stress σx increases to 604 MPa and the plastically affected depth is driven to 1.05 mm. The saturation of the residual stress and plastically
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z displacement /μm
0.5 0 -0.5 -1 -1.5 5 4 3
4 2 1 0
y axis /mm
2
1 0
5
3
x axis /mm
(a) Explicit analysis
z displacement /μm
0.5
0
-0.5 5 4 3
4 2 1 0
y axis /mm
1
2
0
5
3
x axis /mm
(b) Implicit analysis
z displacement /μm
0.5
0
-0.5
-1 5 4 3
4 2 1
y axis /mm
0
1
2
0
3
x axis /mm
(c) Calculated results Fig. 9. Surface deformation on top surface of single LSP.
5
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affected depth is observed in the material treated after five laser shocks on the same position. The predicted residual stresses induced by multiple LSP are similar to what is observed experimentally: for a 7075 aluminum alloy, a 4 GW/cm2 treatment generates − 170 MPa at first impact, − 240 MPa with a second impact and − 340 MPa with a third; for a 0.55% carbon steel, as the number of shocks on the surface increased from one to three, the depth of compressive residual stresses increased from 0.9 mm to 1.8 mm. Hence, multiple LSP increases the residual stresses on the surface and drives them deeper into the material. This relates to work hardening of the material incurred from the previous shocks on the surface layer, decreasing the rate of attenuation of
Fig. 10. Load curve for multiple LSP simulation.
0
Residential stress σx/MPa
-100
-200
-300
-400 1 shock 2 shocks 3 shocks 4 shocks 5 shocks
-500
-600 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Distance from centerline along x-axis /mm
(a) On top surface along x-axis 100
Residential stress σx/MPa
0 -100 -200 -300 1 shock 2 shocks 3 shocks 4 shocks 5 shocks
-400 -500 -600 0
0.2
0.4
0.6
0.8
1
1.2
Depth along z-axis /mm
(b) In depth along z-axis Fig. 11. Residual stress distribution of multiple LSP.
1.4
1.6
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Fig. 12. Contour of Von Mises stress distribution of multiple LSP.
the subsequent pressure pulse, and enabling a higher peak pressure to persist deeper into the material on successive shocks. Opaque overlay applied on the top surface of the material only one time should be thicker enough to avoid thermal effects on the material treated by successive laser shock. Meanwhile, too thick opaque overlay will be disadvantageous for the interaction of a local-induced stress wave with material. More recently, there has been some work with small spot sizes of nominally 1 mm diameter or less. The effect of treatment must be compensated by laser shocking the same area a large number of times to drive the residual stresses deeper. So, FEM simulation of multiple LSP is becoming much more necessary to optimize the treatment parameters. 6. Conclusions A finite element analysis method adopting LS-DYNA and ANSYS is described in detail to attain the simulation of single and multiple laser shock processing. The simulation is composed of two distinct parts including dynamic analysis performed by LS-DYNA and static analysis performed by ANSYS. The method is used to simulate shock wave propagation and predict the distribution of the residual stresses in the metal alloys of 35CD4 50HRC steel treated by LSP with square laser spot. The following conclusions can be made according to the simulated results:
for dynamic analysis based on whether history of internal energy remains steady or history of kinetic energy diminishes to zero is easy to distinguish. (2) The simulation of residual stress field for single laser shock is found to be well correlated with the experimental data and indicates that compressive residual stress of the material is approximately uniform on shocked region and the stress gradient in depth is small. (3) According to the simulation of surface deformation, a homogeneous depression with little roughness modification in the action zone of the shock pressure is induced on top surface of the material treated by single LSP. (4) Simulation of multiple LSP shows that the compressive residual stresses and plastically affected depth can be extensively increased and gradually reach the saturated state with the increase of laser shock number. However, the FEM modeling strategy proposed in this paper should be integrated with an accurately laser–material interaction model to achieve full numerical simulation of laser shock processing, which will be invaluable in enabling close process control in production laser shock processing. This is obviously a long way from being realized because of the complexity of laser–matter interaction during the process. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
(1) History of multiple energies during the dynamic analysis of single LSP is analyzed and validated by the theoretical analysis. The method for determining the solution time
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[17] [18]
D.W. See, J.L. Dulaney, A.H. Clauer, et al., Surf. Eng. 18 (2002) 32. W. Braisted, R. Brockman, Int. J. Fatigue 21 (1999) 719. K. Ding, L. Ye, Surf. Eng. 19 (2003) 127. P. Peyre, A. Sollier, I. Chaieb, et al., EPJ Appl. Phys. 23 (2003) 83. K. Ding, L. Ye, Surf. Eng. 19 (2003) 351. J.L. Ocana, M. Morales, C. Molpeceres, et al., Appl. Surf. Sci. 238 (2004) 242. W. Zhang, Y.L. Yao, I.C. Noyan, J. Manuf. Sci. Eng., Trans. ASME 126 (2004) 8. C.S. Montross, T. Wei, L. Ye, et al., Int. J. Fatigue 24 (2002) 1021. P. Ballard, Residual stressed induced by rapid impact-applications of laser shocking: [Ph.D Thesis]. Palaiseau, France: Ecole Poly technique, 1991. A.H. Clauer, D.F. Lahrman, Key Eng. Mater. 197 (2001) 121. LS-DYNA user's manual version 960 (2001). ANSYS user's manual version 8.1 (2003). B.P. Fairand, A.H. Clauer, J. Appl. Phys. 50 (1979) 1497. C. Oros, Shock Waves 11 (2002) 393. J.P. Romain, F. Bonneau, G. Dayma, et al., J. Phys., Condens. Matter 14 (2002) 10793. M.R. Hill, A.T. DeWald, J.E. Rankin, et al., Mater. Sci. Technol. 21 (2005) 3. L. Wang, Stress Wave Theory, 1 ed. National Defense Industry Publisher, Beijing, 1985. P. Peyre, R. Fabbro, P. Merrien, et al., Mater. Sci. Eng., A 210 (1996) 102.
3-D FEM simulation of laser shock processing Yongxiang Hu ⁎, Zhenqiang Yao, Jun Hu School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, 200030, China Received 27 October 2005; accepted in revised form 6 February 2006 Available online 6 March 2006
Abstract Laser shock processing is an innovative surface treatment technique similar to shot peening. It can impart compressive residual stresses in material for improving fatigue, corrosion and wear resistance of metals. FEM simulation is an effective method to predict mechanical effects induced in the material treated by laser shock processing. An analysis procedure including dynamic analysis performed by LS-DYNA and static analysis performed by ANSYS is presented in detail to attain the simulation of the single and multiple laser shock processing to predict the residual stress field and the surface deformation. History of the energies during dynamic analysis is analyzed and validated by the theoretical calculation. The predicted residual stress field for single laser shock processing is well correlated with the available experimental data and a homogeneous depression with little roughness modification in the action zone of the shock pressure is induced on the treated surface according to the simulation of surface deformation. Simulation of multiple laser shocks is also performed, which indicates that compressive residual stresses and plastically affected depth can be extensively increased and gradually reach the saturated state with the increase of laser shock number. © 2006 Elsevier B.V. All rights reserved. Keywords: Laser shock processing; Residual stress; Finite element method; Stress wave; Shock wave
1. Introduction More recently, surface treatment technologies have become more and more important in industry to cut costs and avoid the need for expensive materials. Demonstrated approximately 30 years ago, laser shock processing (LSP) is now emerging as a viable surface treatment technique. The compressive residual stresses in the metal material treated by LSP can extend deeper below the surface than those from shot peening. LSP is well suited for precisely controlled treatment of localized fatigue critical areas, such as holes, notches, fillets and welds. It has been proposed as a competitive alternative technology to classical treatments for improving fatigue, corrosion and wear resistance of metals [1]. Most investigations have been concentrated on experimentally determining mechanical effects including residual stress field and surface morphology induced in different LSP configurations for a number of industrial metals to get the optimized treatment results before. Finite Element Method (FEM) is first introduced by Braisted and Brockman to ⁎ Corresponding author. Tel.: +86 21 6293 3071; fax: +86 21 6293 2060. E-mail address: [email protected] (Y.X. Hu). 0257-8972/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.surfcoat.2006.02.018
investigate the mechanical behavior and predict the residual stresses from the laser shocked materials with software ABAQUES in 1999 [2]. It offered a new effective approach to optimize parameters of laser shock. From then on, several researchers have used ABAQUES to analyze the laser generated shock waves propagating into different metal materials in the past six years [2–7]. Some of these simulations have a close match with experimentally measured residual stresses. However, research on LSP simulation is still concentrated on accurately modeling the LSP process because of the complexity of LSP process. Work on the 3-D FEM simulation of the process to predict the residual stress field and the surface deformation is still too little now. To the best of our knowledge, no work has been done on the LSP FEM simulation with software besides ABAQUS so far. This paper proposes a new method to simulate shock wave propagation to predict residual stresses distribution and surface deformation of the material treated by LSP using LS-DYNA and ANSYS in three dimensions. FEM modeling strategy is presented in detail. And behaviors of the material subjected to single shock have been analyzed and compared with the experimental results. Simulation of successive shocks on the same position is also performed and discussed.
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2. Mechanism of LSP and experiment Unlike other laser application, LSP works by exerting a mechanical force on the surface of the treated material, while shocked surface is not affected thermally. The process involves a high-intensity laser beam (several GW/cm2) combined with suitable overlays to generate mechanical pressure waves up to several GPa. A schematic of how LSP works is shown in Fig. 1. The metallic surface to be treated is first locally coated with an overlay opaque to laser beam (typically a black paint) and then covered with a transparent overlay, which is a dielectric material transparent to laser beam (such as water). Opaque overlay acts as a sacrificial material, a thin layer of which vaporizes on absorption of laser energy. The transparent overlay confines the thermally expanding vapor and plasma against the surface of the material, thus generating higher pressure than with the direct ablation mode. The shock pressure can result in the plastic deformation and favorable compressive residual stresses on the surface layer of the material [8]. Ballard investigated into LSP effects on the residual stress field and fatigue behavior of steel in his doctoral dissertation [9]. A metal specimen (35CD4, 50HRC steel) was irradiated by a square spot 5 × 5 mm with a laser power density of 8 GW/cm2 and duration of 30 ns. The specimen surface was coated with black paint, and the water was used as the transparent overlay. Stresses generated on the specimen surface were measured by X-ray diffraction and the plastically affected depths were evaluated using successive electrolytical polishing and X-ray diffraction. As expected, the pressure profile of the induced plasma was a Gaussian temporal shape with a full width at half maximum (FWHM) of 50 ns, and its peak pressure was 3.0 GPa [5]. Ballard's experimental results are used as a reference to validate the FEM results of the simulation. 3. FEM simulation strategy 3.1. Analysis procedure Utilization of a very short interaction time and high power density laser is representative of LSP for material treatment. Shock waves induced by the pressure pulse propagate into the target material. If the pressure of the shock wave exceeds the dynamic yield strength of the material, the material will undergo extremely high strain-rate (106–108 s− 1) during a short period of time and be dynamically yielded [10].
Shock Wave
Focused pulse laser Trapped Plasma Transparent Overlay Absorbent Coating Metal Target
Fig. 1. Schematic of the LSP.
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Pre-processing ANSYS
Input ANSYS
Dynamic Analysis LS-DYNA Solver
Static Analysis ANSYS Solver
Transient Stress & Strain Deformation Output
Stable Stress & Strain Deformation Output
Post-processing ANSYS
Residual Stress Field & Surface Deformation Fig. 2. FEM analysis procedure.
Since the pressure pulse duration is very short and the load is very severe, the response of the material changes rapidly. Although the loading time of the shock pressure is very short, response of the target material will take much longer to become stable owing to the reflection and interaction of various stress waves produced in the target material. Accurate tracking of the front of various stress waves is important to capture the dynamic response of the material. The explicit algorithm is especially well-suited to solving this high-speed dynamic event efficiently. Therefore, it is adopted to simulate the propagation of the shock wave induced by the short-time pressure and solve the dynamic response of the material. On the other hand, the high-speed dynamic deformation processes tend to generate a large amount of elastic strain energy in the material. The elastic strain energy, which has been stored in the material after performing the explicit dynamic analysis, is subsequently released if the solution time of the explicit analysis is long enough, but it will be time costly. Performing an explicit-to-implicit sequential solution will be necessary to release all the elastic strain energy efficiently and the final stable desired residual stress field can be obtained. Hence, FEM analysis procedure of laser shock processing should be composed of two distinct parts including dynamic analysis and static analysis to obtain an absolutely stable residual stress field and surface deformation. Dynamic analysis is adopted to simulate the propagation of the shock wave and obtain the dynamic response of the material. When the dynamic stress state of the target material becomes approximately stable, all transient stress will be imported into implicit FEM codes to perform static analysis to obtain the residual stress field and the spring-back deformation in static equilibrium. By simulating the dynamic pressure shock explicitly, and then modeling the elastic energy release implicitly, stringent tolerances can be attained for the distribution of the residual stresses and the surface deformation. Both ANSYS and LS-DYNA are adopted to attain full FEM simulation of laser shock processing. LS-DYNA codes are used for the dynamic analysis. The explicit solution method used by LS-DYNA provides fast solutions for short-time, high speed
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nonlinearity problems. After the dynamic analysis is completed, stress field of the model calculated by the dynamic analysis will be read into ANSYS codes, which integrate LS-DYNA codes with the powerful pre- and post-processing capabilities; so we can model the structure in ANSYS, obtain the explicit dynamic solution via LS-DYNA, and review results using standard ANSYS post-processing tools. The LSP FEM analysis procedure can be depicted in Fig. 2. The combined procedure takes advantage of the best feature of each code to perform explicit and implicit analysis. 3.2. Governing equation 3.2.1. Explicit finite element algorithm In order to analyze the response of the material to the shock pressure with the explicit finite element method, the dynamic equation at time t can be expressed as follows, according to the virtual work principle: Z Z :: : qu i dui dX þ qui dui dX X Z XZ Z ¼ fi dui dX þ TC dui dC− rij dDij dX ð1Þ X
C
X
where ρ is mass density, ü and u˙ nodal acceleration and velocity, respectively, υ the damping coefficient, δu the virtual displacement, fi the body force density, TΓ the boundary force applied on the boundary Γ, σij the Cauchy stress tensors, Dij ¼ 1 dui;j þ duj;i is deformation rate tensors [11]. 2 Discretization of this problem is accomplished by means of the standard finite element procedure. After aggregation, we have the dynamic equilibrium equation as follows:
::
:
MU þ CU þ Fint ¼ Fext
ð2Þ
where M is the diagonal mass matrix, Ü and U˙ the nodal acceleration and velocity vector, respectively, C diagonal damping matrix, Fint the internal element force vector, Fext accounts for external and body force loads. To advance to time tn+1, central difference time integration is usually used as follows:
::
:
U ¼ M−1 ðFext −Fint −CUÞ
: nþ
U
1 2
1
::n
¼ Un−2 þ U Dt n
: nþ Dtnþ
Unþ1 ¼ Un þ U
1 2
ð3Þ ð4Þ
1 2
ð5Þ
ðDt n þ Dt nþ1 Þ 1 . Then we update the geometry by where Dt nþ2 ¼ 2 adding the displacement increments to the initial geometry Xnþ1 ¼ X0 þ Unþ1
ð6Þ
The central difference scheme is conditionally stable. The stability limit can be defined using the element length Le, and the wave speed of the material Cd: Dtstable ¼
Le Cd
ð7Þ
3.2.2. Implicit finite element algorithm According to the virtual work principle, the equilibrium equation of the implicit algorithm can be expressed as follows: Z Z Z rddeddX ¼ f dduddX þ TC dduddC ð8Þ X
X
C
where σ and ε are stress and strain tensors, respectively, f and TΓ body force density and boundary force, respectively, and δu the virtual displacement. Dicretization of this problem is accomplished by means of the standard finite element procedure. After aggregation, we have a group of nonlinear equations as follows that require the Newton–Raphson method to linearize them: KDU ¼ Fext −Fint
ð9Þ
where K = ∫ΩB DBdΩ is the tangential stiffness matrix, B the general geometric matrix, D the stress–strain matrix, ΔU the displacement incremental at the element nodes, Fint = ∫ΩBTσdΩ Newton–Raphson restored force vector and Fext external and body force load that equals zero for the elastic energy release problem [12]. After the static analysis is performed, a stable residual stress distribution and a spring-back deformation field will be dyn sta obtained. Supposing Usur and Usur are the node displacement on top surface obtained from the dynamic analysis and static analysis, respectively. Then the deformation on top surface can be calculated as follows: T
dyn sta Ures sur ¼ Usur þ Usur
where LSP.
res Usur
ð10Þ
is the displacement of each node on top surface for
3.3. LSP model 3.3.1. Loading The high amplitude, short duration pressure pulse of the shock wave generated by laser shock is one of key parameters related to the effect of laser shock processing. Significant experimental efforts have been performed to determine the magnitude and duration of the pressure upon the shock surface [13–16]. The pressure time history is assumed to be known in advance because the simulation procedure does not explicitly model the physical process of the pressure pulse formation. Generally, the pressure pulse is relatively uniform over the entire surface of the laser spot regardless whether the spot is circular or rectangular. And because of the narrow duration of the pressure pulse induced by LSP, the pressure time history is modeled as a triangular ramp in the simulation, in which the pressure rises linearly to the peak value over the time of the pressure pulse FWHM and then decays linearly to zero during the following FWHM [5]. 3.3.2. Constitutive model LSP generates strain-rate exceeding 106s− 1 within the target material. As the strain-rate increases, materials typically exhibit little change in elastic modulus, but an increase in yield strength and the event becomes a shock wave phenomenon. Under uni-
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axial strain conditions, the highest elastic stress level in the shock wave propagation is defined as the Hugoniot Elastic Limit (HEL) for the semi-infinite model. When the pressure is greater than HEL, permanent deformation occurs. Assuming that the yielding occurs when the stress in the direction of the wave propagation reaches the HEL, the dynamic yield strength σydyn under uni-axial strain conditions can be defined in terms of the HEL by: rdyn y
¼ HELð1−2mÞ=ð1−mÞ
Non-reflecting Boundary Surface
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z
3.3.3. Finite element model A three-dimensional semi-infinite finite element model with size 20 × 20 × 10 mm has been developed to simulate one-sided laser shock solid surface. Since the model is symmetric and subjected to a symmetric uniform square pressure pulse, only a quarter of the configuration is used to perform the finite element calculation instead of a full one to ensure computational efficiency. A schematic configuration of this model including boundary conditions is shown in Fig. 3. The symmetric boundary conditions are employed on the x-0-z and y-0-z planes. Nonreflecting boundaries, which will prevent stress wave reflections generated at the boundary from re-entering the model and contaminating the results, are applied to the exterior surfaces of the model to simulate the semi-infinite model. Solid 164 explicit elements, defined by eight nodes having the following degrees of freedom at each node – translations, velocities, and accelerations in the nodal x, y, and z directions – are chosen for dynamic analysis. They will be converted into corresponding companion implicit elements Solid 185 for static analysis [12]. Results of FEM analysis are sensitive to the mesh density [5]. It is easy to obtain more accurate results with dense meshes, but higher computational cost. To accurately capture mechanical effects induced by laser shock, it is necessary to have sufficient mesh density about the region subjected to the laser shock, which we are most interested in. As shown in Fig. 3, a densely meshed region extends two times the spot size in three directions with the element edge length on the order of 2.5% of the spot size. For the rest region, mesh density is four times as Table 1 Mechanical properties of 35CD4 50HRC steel [5] Properties
Value
Unit
Density, ρ Poisson's ratio, ν Elastic modulus, E Hugoniot elastic limit, HEL
7800 0.29 210 2.1
kg m− 3 GPa GPa
x
y Symmetric Boundary Surface Symmetric Boundary Surface
ð11Þ
where ν represents Poisson's Ratio [2]. Also, the target material is assumed to be perfectly elasticplastic with isotropic and homogeneous characteristics for the analysis. The plastic strain is assumed to follow the Von Mises yielding criterion and the dynamic yield strength σydyn is based on Eq. (11). Given above assumptions, the material properties of 35CD4 50HRC steel required for the FEM simulation are shown in Table 1.
Shocked Region
Non-reflecting Boundary Surface Fig. 3. The schematic configuration of 3-D model.
coarse as the densely meshed one. The number of total elements in model is 125,000. Region of the refined element mesh needs to be extended or adjusted if the calculated plastic deformed region is close to the boundary of two regions with different mesh density. 3.3.4. Solution time choosing To accurately capture the dynamic response of the material, the solution time must take much longer than the duration of the pressure pulse, owing to the reflection and interaction of the multiple stress waves propagating in the target. Ding suggested that the solution time could be set as two orders of the magnitude longer than the pressure pulse duration to ensure that the saturation of the plastic deformations occurred in the target material [5]. But it is not easy to operate in practice because the saturation of the plastic deformations is difficult to distinguish from other states of the material in the post-processing. Stress waves induced by the short duration pressure pulse cause the movement of the material particles and deform the target material, which are demonstrated by the kinetic energy and the internal energy including elastic strain energy and plastic strain energy, respectively. When the kinetic energy tends to zero and the internal energy tends to constant simultaneously, it indicates that the interaction of the stress waves in the material is quite weak and the dynamic stress state is driven to be stable. Certainly, no plastic deformation will occur in the material after that time. Hence, it can be chosen as the solution time for the dynamic analysis. A detail discussion will be given in the Section 4.1 based on the dynamic analysis results. 4. Results of single LSP and discussion 4.1. Dynamic stress state The response of the material is a process of propagation, reflection and interaction of stress waves. The total energy Et from the external work of the pressure pulse is mainly converted into the kinetic energy Ek, internal energy Ei and hourglass energy Eh. Hourglass energy is usually used to demonstrate the effect of zero energy modes, which is also called hourglass modes arising because of shortcoming in the element
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0.05 0.04
Energy /J
It is easy to find that the elastic wave velocity is always faster than the plastic one. Then time needed for the first two types of stress wave propagating to the bottom surface of the target material is shown as follows:
a b c d
0.03 0.02 0.01
te ¼
lthick Cle
ð14Þ
tp ¼
lthick Clp
ð15Þ
0 0.1
1
1.7 2
3
4
5
time /μs Fig. 4. History of energies (a—internal energy, b—kinetic energy, c—hourglass energy, d—total energy).
formulation for the under-integrated solid elements in explicit algorithm. A notable hourglass energy mode will result in a state that cannot exist in practice, but is mathematically stable, which usually exists for problems that deform with high velocities [11]. According to the stress wave theory, the elastic wave velocity and the plastic wave velocity can be calculated as follows [17]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1−mÞE 1 e Cl ¼ d ð12Þ ð1 þ mÞð1−2mÞ q Clp
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E 1 d ¼ 3ð1−2mÞ q
ð13Þ
where Cle is the elastic wave velocity, Clp the plastic wave velocity, ρ the material density, E the Elastic modulus, ν the Poisson's Ratio.
where lthick is the thick of target material. Boundary condition for the bottom surface of the FEM model is nonreflecting to model the semi-infinite solid, so stress waves propagating to the bottom surface will be completely transmitted, which will definitely cause a dramatically decreasing of the total energy. How these energy changes as the shock waves propagate through the target material by FEM simulation is shown in Fig. 4. The total energy of the model is rapidly increased to the peak in the initial period of 100 ns, and then remains approximately invariable during the period from 100 ns to 1700 ns, just with a little decreasing for hourglass energy defined in the LS-DYNA. A sudden declination of the total energy occurred at the time about 1700 ns, which is well consistent with the theoretical result calculated by Eq. (14). It is the time that first elastic stress wave propagates to the bottom surface of the material. According to Fig. 4, the total energy and the internal energy remain stable after the time 4500 ns while the kinetic energy finally diminishes to zero, which means that the response of the target material is driven to be stable. The surface stress σx along the x-axis varies with the solution time as described in Fig. 5. It can be seen that the dynamic stress profile changes quite clearly in the solution period from 1000 ns to 2000 ns but is oscillating slightly and tends to stable after 4000 ns, which is consistent
200 100
Residual stressσx /MPa
0 -100 -200 -300 -400
0997ns 1998ns 3996ns 4499ns 4997ns
-500 -600 -700 -800 0
0.5
1
1.5
2
2.5
3
3.5
Distance from centerline along x-axis /mm Fig. 5. Stress state on top surface at different time.
4
4.5
5
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with the history of the energies. So 5000 ns is enough for dynamic stress state to become stable and no further interaction of different stress waves will occur in the target later. It can be chosen as the solution time for the dynamic analysis. The method for solution time choosing based on whether history of internal energy remains stable or history of kinetic energy diminishes to zero is distinct and easy to select compared with that based on the saturation of the plastic deformation. 4.2. Static analysis result 4.2.1. Residual stress field Stresses of dynamic analysis results at 5000 ns are read into ANSYS codes to perform the static analysis. Contours of the Von Mises stress distribution of the dynamic analysis at 5000 ns and of static analysis are shown in Fig. 6. Maximum Von Mises stress has decreased from 394.5 MPa to 349.2 MPa because of the elastic strain release. Contours of three components of the residual stress field from FEM simulation are shown in Fig. 7. The distribution of
Fig. 7. Contour of residual stresses field for three components.
surface residual stresses along the x-axis and in depth compared with experimental results is shown in Fig. 8. The following observations can be made from these figures:
Fig. 6. Contour of Von Mises stress distribution of single LSP.
(1) Simulated results in Fig. 7 show that x and y components of residual stresses are only produced in the layer
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Residual Stress σx /MPa
0 -100 -200 -300
FEM σx Experiment σx FEM σy Experiment σy
-400 -500 -600 0
1
2
3
4
5
Distance from centerline along x-axis/mm
(a) On top surface along x-axis
Residual Stress σx /MPa
100
Lp
0 -100 -200
FEM σx Experiment σx FEM σy Experiment σy
-300 -400 -500 0
0.2
0.4
0.6
0.8
4.2.2. Surface deformation The surface deformation of the material has a great effect on fatigue behavior. FEM simulation in full 3-D geometry is an effective tool for the assessment of the surface deformation after the treatment, which should be based on the assumptions that the material surface is an ideal plane in geometry and thermal effect induced by laser beam is totally avoided. Z component of displacement on the top surface simulated by the dynamic analysis is shown in Fig. 9(a) and that for elastic energy release by static analysis is shown in Fig. 9(b). Simulation of the surface deterioration of the material treated by LSP is calculated according to the Eq. (10) and the result is shown in Fig. 9(c). From the Fig. 9(c), a homogeneous depression less than 1 μm with little roughness modification in the action zone of the shock pressure is induced on top surface of the material treated by single LSP for the compressive residual stress. Similar results were also observed in the experiment results, which presented a comparison of the roughening effects of laser shock processing and shot peening on A356 and 7075 aluminum alloys. In both cases, while the laser shocked surface remains partially unchanged, shot peening generates a detrimental roughened surface with large increases in the mean and peak roughness Ra and Rt [18]. 5. Multiple LSP analysis
1
Depth along z-axis /mm
(b) In depth along z-axis Fig. 8. Residual stress distribution of single LSP.
about 1 mm from the top surface after laser shock processing. The stress distributions between x and y components are similar. And z component of stress is nearly zero on the top surface. A tensile residual stress is found in the z component of stress of the model as shown in Fig. 7(c), which is generated by the reversing yielding effect due to the interaction of the various stress waves in the material. (2) As shown in Fig. 8, the experimental data show that the maximum compressive residual stress σx is about 355 MPa, with a plastically affected depth of 0.80 mm. FEM simulation predicted maximum compressive residual stress σx is about 349 MPa, 1.7% smaller than that from experiments and the plastically affected depth is 0.67 mm, 16.3% smaller than that from experiments. Hence, the predicted residual stress field is well consistent with those experimental results obtained by Ballard [9] and the FEM simulation method used for LSP is validated. (3) The simulated results also indicate that compressive residual stress of the material is approximately uniform on the shocked region. And the residual stress is highest at the surface decreasing gradually with distance below the surface, i.e., the stress gradient in depth is small, which is useful to get uniform residual stress field in the surface layer.
The intensity of laser shock processing is related to the laser intensity, i.e., the power per unit area applied to the laser shocked spot. As the power density is increased, the peak shock pressure increases, and this increases both the surface compressive residual stress and the depth of the compressive residual stress. The ability to continue to increase the intensity to increase the magnitude and the depth of the compressive residual stress is limited, however, by the onset of dielectric breakdown in the transparent overlay above a certain power density. This limitation is overcome by applying successive shocks to a spot [10]. An attempt to model the residual stresses for multiple onesided laser shock processing at the same location using the same model parameters as for the single shock predictions above is investigated according to the procedure shown in Fig. 2. The successive laser shocks are modeled by applying an interval pressure pulse to the shocked surface of the material for dynamic analysis. The loading curve for the multiple LSP is shown in Fig. 10. The surface and in depth residual stress field resulting from multiple LSP analysis are shown in Fig. 11 and the contour of Von Mises residual stress field after five laser shocks is shown in Fig. 12. Compressive residual stress and plastically affected depth are significantly increased by multiple LSP. The maximum compressive residual stress σx and plastically affected depth are increased by 40.0% from 349 MPa to 485 MPa and 35.8% from 0.67 mm to 0.91 mm after the second shock on the same position, respectively. By five laser shocks at the same position, the maximum compressive residual stress σx increases to 604 MPa and the plastically affected depth is driven to 1.05 mm. The saturation of the residual stress and plastically
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z displacement /μm
0.5 0 -0.5 -1 -1.5 5 4 3
4 2 1 0
y axis /mm
2
1 0
5
3
x axis /mm
(a) Explicit analysis
z displacement /μm
0.5
0
-0.5 5 4 3
4 2 1 0
y axis /mm
1
2
0
5
3
x axis /mm
(b) Implicit analysis
z displacement /μm
0.5
0
-0.5
-1 5 4 3
4 2 1
y axis /mm
0
1
2
0
3
x axis /mm
(c) Calculated results Fig. 9. Surface deformation on top surface of single LSP.
5
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affected depth is observed in the material treated after five laser shocks on the same position. The predicted residual stresses induced by multiple LSP are similar to what is observed experimentally: for a 7075 aluminum alloy, a 4 GW/cm2 treatment generates − 170 MPa at first impact, − 240 MPa with a second impact and − 340 MPa with a third; for a 0.55% carbon steel, as the number of shocks on the surface increased from one to three, the depth of compressive residual stresses increased from 0.9 mm to 1.8 mm. Hence, multiple LSP increases the residual stresses on the surface and drives them deeper into the material. This relates to work hardening of the material incurred from the previous shocks on the surface layer, decreasing the rate of attenuation of
Fig. 10. Load curve for multiple LSP simulation.
0
Residential stress σx/MPa
-100
-200
-300
-400 1 shock 2 shocks 3 shocks 4 shocks 5 shocks
-500
-600 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Distance from centerline along x-axis /mm
(a) On top surface along x-axis 100
Residential stress σx/MPa
0 -100 -200 -300 1 shock 2 shocks 3 shocks 4 shocks 5 shocks
-400 -500 -600 0
0.2
0.4
0.6
0.8
1
1.2
Depth along z-axis /mm
(b) In depth along z-axis Fig. 11. Residual stress distribution of multiple LSP.
1.4
1.6
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Fig. 12. Contour of Von Mises stress distribution of multiple LSP.
the subsequent pressure pulse, and enabling a higher peak pressure to persist deeper into the material on successive shocks. Opaque overlay applied on the top surface of the material only one time should be thicker enough to avoid thermal effects on the material treated by successive laser shock. Meanwhile, too thick opaque overlay will be disadvantageous for the interaction of a local-induced stress wave with material. More recently, there has been some work with small spot sizes of nominally 1 mm diameter or less. The effect of treatment must be compensated by laser shocking the same area a large number of times to drive the residual stresses deeper. So, FEM simulation of multiple LSP is becoming much more necessary to optimize the treatment parameters. 6. Conclusions A finite element analysis method adopting LS-DYNA and ANSYS is described in detail to attain the simulation of single and multiple laser shock processing. The simulation is composed of two distinct parts including dynamic analysis performed by LS-DYNA and static analysis performed by ANSYS. The method is used to simulate shock wave propagation and predict the distribution of the residual stresses in the metal alloys of 35CD4 50HRC steel treated by LSP with square laser spot. The following conclusions can be made according to the simulated results:
for dynamic analysis based on whether history of internal energy remains steady or history of kinetic energy diminishes to zero is easy to distinguish. (2) The simulation of residual stress field for single laser shock is found to be well correlated with the experimental data and indicates that compressive residual stress of the material is approximately uniform on shocked region and the stress gradient in depth is small. (3) According to the simulation of surface deformation, a homogeneous depression with little roughness modification in the action zone of the shock pressure is induced on top surface of the material treated by single LSP. (4) Simulation of multiple LSP shows that the compressive residual stresses and plastically affected depth can be extensively increased and gradually reach the saturated state with the increase of laser shock number. However, the FEM modeling strategy proposed in this paper should be integrated with an accurately laser–material interaction model to achieve full numerical simulation of laser shock processing, which will be invaluable in enabling close process control in production laser shock processing. This is obviously a long way from being realized because of the complexity of laser–matter interaction during the process. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
(1) History of multiple energies during the dynamic analysis of single LSP is analyzed and validated by the theoretical analysis. The method for determining the solution time
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