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Temperatures, pressures and stresses during laser shock processing
Optics and Lasers in Engineering 39 (2003) 51–71
Temperatures, pressures and stresses during laser shock processing Thord Thorslund, Franz-Josef Kahlen, Aravinda Kar* Laser-Aided Manufacturing, Materials and Microprocessing (LAMMMP) Laboratory, School of Optics and Center for Research and Education in Optics and Lasers (CREOL), Mechanical, Materials, and Aerospace Engineering Department, University of Central Florida, 4000 Central Florida Blvd., Orlando, FL 32816-2700, USA Received 1 November 2000; received in revised form 1 February 2002; accepted 1 March 2002
Abstract Mathematical models are developed to calculate the temperatures, pressures and stresses during laser shock processing for time-modulated (ramp-up, ramp-down and rectangular) laser pulses. Three different shock processing configurations are also considered: non-ablative exposure, ablative exposure and confined ablation with coating. The results for iron show that the plasma pressure reaches an average value of 9 GPa in direct ablation configuration and plays a dominant role for all three types of laser pulses. In the case of confined geometry, the plasma pressure reaches an average value of 20 GPa. All calculated pressures and stresses exceed the yield strength of the workpiece, indicating plastic deformation. It is also shown that pulses with short rise times yield higher plasma pressures. r 2002 Published by Elsevier Science Ltd. Keywords: Shock processing; Temperatures; Pressures; Stresses; Non-ablative; Ablative; Confined ablative
1. Introduction and background Laser shock processing (LSP) utilizes the recoil pressure of the rapidly vaporized atoms to improve the surface strength by compressing the material at the surface. LSP is very attractive because of its unique capability to process materials of complicated shapes and geometries. In comparison with other surface hardening *Corresponding author. Tel.: +1-407-823-6921; fax: +1-407-658-6880. E-mail address: [email protected] (A. Kar). 0143-8166/03/$ - see front matter r 2002 Published by Elsevier Science Ltd. PII: S 0 1 4 3 - 8 1 6 6 ( 0 2 ) 0 0 0 4 0 - 4
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Nomenclature A a0 a1 a2 E F ðtÞ fc I0 k kB Lv M mi n NA ne P Pave Patm pv R r rm r0 T Tatm Tp Tv T z
absorptivity absorptivity at T ¼ Tv absorptivity accounting for laser beam attenuation in the plasma plume absorptivity accounting for the increased laser beam attenuation in the plasma plume for Tp > Tv Young’s modulus (N m2) temporal shape of the laser pulse correction factor accounting for collisions in the plasma for confined geometry processing irradiance (W cm2) thermal conductivity (W m1 K1) Boltzmann constant (J K1) latent heat of vaporization (J kg1) molecular weight (kg kmol1) temporal pulse shape factor at t ¼ 0 (i ¼ 1; 2; 3) number density of atoms in the plume (cm3) Avogadro’s number (mol1) number density of electrons in the plume (cm3) power (W) average power per pulse (W) one atmospheric pressure (N m2) pressure at the vapor–liquid interface (N m2) gas constant (J kg1 K1) radius (m) melt pool radius (m) focal beam radius (m) temperature (K) boiling temperature at one atmospheric pressure (K) plasma plume temperature (K) boiling temperature at the vapor–liquid interface (K) excess temperature (T T0 ) (K) depth (m)
Greek characters aT thermal expansion coefficient (K1) a thermal diffusivity (m2 s1) n Poisson ratio y angle rl density of the liquid phase (kg m3) rv density of the vapor phase (kg m3) srr stress in the radial direction (GPa) szz stress in the axial direction (GPa)
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syy t
53
stress in the azimuthal direction (GPa) pulse width (ns)
techniques, such as cold rolling, shot peening, X-rays or high energy electron beams [1–3], LSP offers several advantages, such as a relatively easy process control through control of the laser beam, processing of only a thin surface layer, no potential dangers to the health of the operator such as X-rays, and no requirement to process the material in a vacuum. Disadvantages of LSP are (i) a potential for surface melting and resolidification which allows for surface wave formation and the introduction of surface roughness [4–9] and (ii) the small processing area (the laser beam is focused to a spot of 0.1–2 mm diameter) in comparison to the surface to be processed. LSP can be performed in a number of ways such as non-ablative exposure, ablative exposure, utilization of an energy absorptive layer and/or transparent overlay (confined ablation with coating) and combinations thereof (Fig. 1). Nonablative exposure is a process in which nearly instantaneous temperature rise in the material due to rapid energy deposition generates thermal stresses and shock waves in the material. In ablative exposure, the substrate surface is additionally subjected to recoil and plasma pressures. A considerable drawback of direct ablation is the possibility of surface damage due to melting and vaporization [10, 11]. For LSP in a confined geometry (confined ablation with coating), the substrate surface is typically coated with a 40 mm thin layer [12–18] and an overlay, which is transparent to the laser wavelength, on top of this coating. The pressure required to harden the substrate surface is generated by vaporizing the coating, and heating the vapor in a confined space. During LSP, the workpiece is exposed to high laser irradiance (108–1012 W cm2) for a few nanoseconds [19–23]. Short pulse (600 ps–100 ns) lasers ranging from KrF Excimer laser [24], Ruby laser [25] to solid-state (Nd:YAG and Nd:Glass) lasers [12, 13, 17, 18, 26] have been used for LSP. A wide variety of analytical and experimental investigations have been carried out since Densification of porous materials using LSP has been demonstrated by Darquey et al. [27], Dubrujeaud and Jeandin [13] successfully used LSP to perform
Shock beam
Shock beam Ejected particles
Shock beam Plasma Transparent Coating overlay
Material
Material
Material
(a)
(b)
(c)
Fig. 1. Geometries considered for laser shock processing: (a) non-ablative exposure, (b) ablative exposure, (c) confined ablation with coating.
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cladding of Al2024 alloy. Fabbro et al. [18] presented a one-dimensional analytical model to investigate the energy transfer during LSP and to derive a correlation between the laser beam parameters and the properties of the processed material. Verification of the model on a copper target with glass as transparent overlay, using a Nd:YAG laser with 2–30 ns pulse length showed that the energy absorption was higher in the case of a confined geometry than for direct ablation. They found that the model agrees with the experimental results for energy densities of 10–100 J cm2. Ballard et al. [28] presented a model for the plastic and elastic deformations of the processed material. Given the limited size and the temperature in the processed region, very few experimental investigations have been carried out to establish relationships between the laser irradiance and the stresses and residual stresses [18, 28] in the workpiece. Sarady et al. [25] published values about the pressures during LSP, measuring the displacements of a steel sphere suspended as a pendulum during the irradiation in vacuum. They measured mean pressures in the order of 6 and 11 GPa during the irradiation with a Ruby laser and a Nd:YAG laser, respectively. Their analytical calculations were based on Fabbro’s results [18]. Devaux et al. [29] successfully investigated the effect of the temporal pulse shape on the pressures due to breakdown of the transparent overlay experimentally, demonstrating a dependence of the pressures on the pulse rise time.
2. Approach In this paper, a mathematical model is developed to study the effect of laser irradiance and its time modulation on the temperature distributions in the workpiece and plasma plume. Thereby, this investigation extends previous studies on LSP of steels on short rise time pulses and connects the temperature distribution in the target with the recoil pressure on the material surface during the ablation process. The temperature fields are necessary to calculate the stresses generated during laser shock processing. The heat conduction equation is solved using the Hankel and LaPlace transforms based on the assumption that the surface absorptivity is temperature-dependant. This assumption is justified because of the short laser– material interaction times. During these interaction times, the surface will heat up and melt for sufficiently high laser intensities. It is established [30] that the surface absorptivity will increase when the surface temperature increases. A very limited number of models for the temperature dependence of thermophysical properties are available [31]. Their margins of error reach 50% for temperatures in excess of 2000 K; reliable models for even higher temperatures are not available. Due to lack of reliable data or models, thermophysical properties are considered constant in this investigation. It was expected that the plasma plume temperature remains below the second ionization temperature of steels (E17,000 K). Thus, the authors considered a single-ionized plasma in their model. The results presented later will confirm this assumption. The surface temperatures for ramp-up, ramp-down and rectangular laser pulses are calculated for direct irradiation/ablation and for a confined geometry. The stresses due to rapid energy deposition, the thermal shock, recoil and
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plasma pressures are considered for direct irradiation/ablation. The pressure related to the instantaneous energy deposition is assumed to be present during the entire duration of the laser pulse (30 ns). Thermal stresses are calculated by considering the thermophysical properties (Table 1) of stainless steel SS 304 (Table 2), the respective values for iron were used where data for SS 304 were not available. The recoil pressure is determined by applying the kinetic theory to the metal vapor. An expression for the plasma pressure is obtained by considering a freely expanding plasma, and the conservation of the thermal and kinetic energies. In the case of LSP with a confined geometry, recoil and plasma pressures are dominant. The plasma cannot expand freely. Its density is approximated as the liquid density due to the short laser–material interaction time. The recoil pressure is determined by applying a correction factor to the expression derived for the case of direct irradiation/ablation. High power laser beams typically are designed to produce a very good quality beam, approaching a near-Gaussian intensity distribution. While generally flat-top laser beam intensity distributions are used in LSP because of the larger surface areas which can be treated, they require additional modifications for the laser beam delivery and laser beam shaping. The authors considered a Gaussian laser beam in
Table 1 Thermophysical properties for iron [36–38] Thermophysical property
Symbol
Units
Value
Thermal diffussivity Thermal conductivity (at T ¼ 300 K) Initial temperature Melting temperature Vaporization temperature Latent heat of melting Latent heat of vaporization Molar mass Particle mass Poisson’s ratio Young’s modulus Heat capacity (at T0 ) Heat capacity (at Tm ) Heat capacity (at Tv ) First ionization potential Density (at T0 ) Density (at Tm ) Thermal expansion coefficient (at T ¼ 1000 K)
a k T0 Tm Tv Lm Lv M Mp n E cp cp;m cp;v EI r r aT
m2 s1 W m1 K1 K K K kJ kg1 kJ kg1 kg kmol1 kg — N m2 J kg1 K1 J kg1 K1 J kg1 K1 K kg m3 kg m3 K1
7.6 106 45 298 1810 3135 2.66 105 6.8 106 55.847 9.274 1026 0.27 2.1 1010 450 828 828 8500 7870 6660 15.5 106
Table 2 SS304 composition Powder
Fe
Ni
Cr
Si
Mg
P
C
S
SS 304
68.33
11.25
19.32
0.93
0.13
0.007
0.022
0.007
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this investigation, to analyze the effect of the highest irradiances on the mechanical properties of the material surface.
3. Mathematical modeling for laser ablation 3.1. Temperature modeling for the melt pool Laser ablation is a process during which the material is heated above its boiling temperature very rapidly. To model this rapid heating process, the absorptivity is assumed to depend on the temperature according to the following expression: Tðr; 0; tÞ for ToTv ; Tv Tv AðTðr; 0; tÞÞ ¼ a1 1 þ exp a2 Tðr; 0; tÞ
AðTðr; 0; tÞÞ ¼ a0
ð1Þ for T > Tv
ð2Þ
Eq. (1) describes a linear increase with temperature until the surface temperature reaches the vaporization temperature. The authors assumed an exponential decrease of the surface absorptivity for temperatures T > Tv ; to account for the laser beam attenuation in the plasma plume. Parametric studies are conducted by taking a0 ¼ 0:4; a1 ¼ 5:41 and a2 :¼ 105 in Eqs. (1) and (2). Defining T ðr; z; tÞ ¼ Tðr; z; tÞ T0 and assuming that the problem is symmetrical about the y direction (Fig. 2), the unsteady heat conduction equation and the boundary and initial conditions in cylindrical coordinates can be written as 1 q qT q2 T 1qT r ð3Þ ¼ þ r qr a qt qr qz2 and T ðr ¼ N; z; tÞ ¼ 0;
Fig. 2. Coordinate system for a cylindrical disk of radius r0 and depth Dz:
ð4Þ
T. Thorslund et al. / Optics and Lasers in Engineering 39 (2003) 51–71
T ðr; z ¼ N; tÞ ¼ 0; k
qT ðr; z ¼ 0; tÞ r2 ¼ AðTðr; z ¼ 0; tÞÞI exp 2 2 F ðtÞ; qz r0
57
ð5Þ ð6Þ
T ðr; z; t ¼ 0Þ ¼ 0;
ð7Þ
T ðr ¼ 0; z; tÞ must be finite:
ð8Þ
It is now possible to consider different temporal laser pulse shapes (Fig. 3) by properly selecting the function F ðtÞ such as F1 ðtÞ ¼ m1 t
for ramp-up pulses;
ð9Þ
F2 ðtÞ ¼ m2 ðt tÞ for ramp-down pulses;
ð10Þ
F3 ðtÞ ¼ m3
ð11Þ
for rectangular pulses;
Thorslund [32] solved Eqs. (3)–(8) using the Hankel and LaPlace transforms for a pulse width t and obtained T ðr; z; tÞ ¼
pffiffiffi Z I0 r20 a t A ðt t0 ÞF ðt t0 Þexp ð2r2 =r20 þ 8at0 Þ ðz2 =4at0 Þ pffiffiffi pffiffiffi dt0 : 2 þ 8at0 0 r k p 0 t 0
ð12Þ
For the operating conditions considered in this investigation (r0 ¼ 1 mm, aB104 m2 s1 tB30 ns), it can be shown that r20 þ 8at0 Er20 : The singularity at t0 is removed by substituting t0 ¼ t00 =t: Finally, the following expressions for the temperature distributions for three different types of laser pulses are derived: Ramp-up pulse: pffiffiffiffiffi pffiffiffi z2 t tt 2 Z 8P mt r 002 002 t t T ðr; z; tÞ ¼ 2 pffiffiffi exp 2 2 A t t t t xp 002 dt00 : 4at r0 0 pr0 k p ð13Þ
Fig. 3. Schematic of pulse shapes: (a) ramp-up pulse of pulse width t; (b) ramp-down pulse of pulse width t; (c) rectangular pulse of pulse width t:
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Ramp-down pulse:
pffiffiffiffiffi at r2 T ðr; z; tÞ ¼ 2 pffiffiffi exp 2 2 r0 pr0 k p pffiffiffi Z tt z2 t t002 t002 A t t t t xp 002 dt00 4at t 0
8P
and rectangular pulse: pffiffiffiffiffi pffiffiffi tt 2 Z 4P at r t002 z2 t T ðr; z; tÞ ¼ 2 pffiffiffi exp 2 2 A t xp 002 dt00 : 4at t r0 0 pr0 k p
ð14Þ
ð15Þ
3.2. Stress modeling during laser shock processing 3.2.1. Non-ablative exposure In this case, stresses may be generated in the material due to rapid energy deposition, resulting in thermal stresses. These thermal stresses are generally given by aT E T ðr; z; tÞ: If the laser beam diameter is much larger than the depth of heating (thermal energy transfer) (Dz=r51) the radial (r-direction) and azimuthal (y-direction) stresses can be calculated using the following expressions [32]: Z aT E r 0 srr ¼ 2 T ðr ; z; tÞr0 dr0 ; ð16Þ r 0 Z aT E r 0 syy ¼ 2 T ðr ; z; tÞr0 dr0 aT ET ðr; z; tÞ: ð17Þ r 0 The temperature T ðr0 ; z; tÞ can be calculated from Eqs. (13)–(15) for different pulse shapes. 3.2.2. Ablative exposure Besides the thermal stresses generated in direct irradiation, there are two additional effects to consider during material ablation which are the recoil pressure exerted by the ablated particles and the plasma pressure. The net mass flux due to vaporization can be calculated from the Hertz–Knudsen Equation [33] assuming Maxwellian temperature and density distributions. The pressure in the plasma plume is calculated from the kinetic energy of the evaporating particles [25,34,35] by applying the Saha Equation to singly ionized particles for the geometry shown in Fig. 4. The radial and axial (z-direction) stresses due to recoil and plasma pressure are then respectively expressed as n srr ¼ szz ; ð18Þ 1n 2 ! rv kB T ðr; z ¼ 0; tÞ ne 1:6652 rl 1þ þ szz ¼ 1 ; ð19Þ M n 2p rv
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Fig. 4. Model of surface layer in confined and direct ablation.
which need to be combined with the stresses obtained from Eqs. (16) and (17) to determine the total stress. The terms in Eq. (19) represent the ‘‘instantaneous’’ energy increase due to the energy deposition, the plasma pressure and the recoil pressure of the ablated particles. The factor 1.665 enters into Eq. (19) due to the Hertz–Knudsen equation for rarefied gas dynamics as described above. The Hertz– Knudsen equation contains a correction factor f which has been shown to be equal to 1.665 [33].
3.2.3. Confined ablation with coating Laser shock processing in a confined geometry is distinctly different from direct ablation because the coating layer is vaporized in the case of confined geometry [12] whereas the workpiece itself is vaporized in the case of direct ablation. The collisions between the vaporized particles in the vaporized layer must be considered in the case of confined geometry, and the recoil and plasma pressures are of interests to calculate the pressure on the substrate surface. The recoil pressure is determined using an approach similar to the case of direct ablation. A correction factor, fc is introduced to account for the collision of the particles in the confined geometry, and the coating layer is assumed to exist mainly in the liquid phase. The plasma pressure is calculated numerically from an energy balance for the plasma plume accounting for phase changes during the plasma formation. The radial and axial stresses can
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then be written as n szz ; srr ¼ 1n T ðr; z ¼ 0; tÞ T ðr; z ¼ 0; tÞ ne ð1:665fc Þ2 szz ¼ rl kB þNA 1 þ Tp : M 2p n
ð20Þ ð21Þ
The correction factor fc is introduced to account for the energy loss due to molecular and atomic collisions within the plasma. For the numerical evaluation it was assumed that 10% of the thermal energy in the plasma plume is lost due to internal collisions, i.e., fc ¼ 0:9: The numerical evaluations will be carried out for average pulse powers of 50 W, average pulse energies of 100 J, a laser beam radius of 1 mm, and pulse width of 30 ns.
4. Results and discussions The radial temperature distributions during direct irradiation/ablation of iron [31, 36, 37] subjected to three different pulse shapes (ramp-up, ramp-down and constant energy (rectangular) pulses) are shown in Fig. 5. In each case, the pulse width t is 30 ns and the energy contained in each pulse is kept constant. It can be seen that the ramp-down pulse yields the highest surface temperatures over the entire irradiated area. This result agrees with the expectation that the laser irradiance rate (dIðr; tÞ=dt) determines the heating rate in the irradiated material. The surface temperature for a ramp-up pulse is considerably lower then the surface temperature for a ramp-down pulse, but is still higher than the rectangular pulse (Fig. 5). This behavior of the
Fig. 5. Radial temperature distribution in direct irradiation/ablation for ramp-up, ramp-down and rectangular laser pulses at z ¼ 0 and z ¼ 100 nm, beam radius r0 ¼ 1 mm, t ¼ 30 ns, Pave ¼ 50 W.
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surface temperature is due to the irradiance rate and, therefore, an indication that this irradiance rate can have stronger influence on the material temperature than the actual irradiance Iðr; tÞ: If the irradiance rate dIðr; tÞ=dt > 0; more energy may be coupled into a control volume (disc) of depth Dz and cross-section pr2 than is lost due to conduction into the surrounding material. In this case, the rate of change of internal energy in this control volume would be positive and the temperature in the control volume would rise. It is, therefore, possible that the surface temperature of a material which is exposed to a ramp-up laser pulse exceeds the surface temperature produced by a rectangular pulse containing the same energy as the ramp-up pulse. The model shows that the surface temperature of the irradiated material (14,500 K) exceeds its boiling temperature (3000 K). A surface temperature of 14,500 K appears to be very large. It is possible that the model’s assumptions oversimplified the physics and that the true surface temperature is below 14,500 K. However, this temperature must be evaluated in the context of the material’s exposure to the very high laser irradiance. For an ideal gas, the Clausius–Clapeyron equation (Eq. (22)) establishes a link between a reference evaporation temperature (typically Tatm ¼ 3000 K at one atmosphere patm ) and the evaporation pressure at
Pressures and stresses [GPa]
5
0
-5
-10 Thermal stresses
-15
Energy deposition
Axial
Axial
r = 1 mm
Azimuthal
Radial
z= 0
0
Radial 0
5
10
15
20
25
30
Irradiation time [ns] Fig. 6. Temporal tensile (positive) and compressive (negative) stresses in direct irradiation/ablation for a ramp-up laser pulse at z ¼ 0; beam radius r0 ¼ 1 mm, t ¼ 30 ns, Pave ¼ 50 W.
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higher surface temperatures: Lv 1 1 pv ¼ patm exp : R Tv Tatm
ð22Þ
A surface temperature Tv exceeding 3000 K is an indication that the surface pressure pv also exceeds one atmospheric pressure. For laser irradiance lower than 106 W cm2, Kahlen et al. [38] showed that the surface pressure reaches a value of 2 105 N m2 (2 atmospheres), employing an analytical model. This value was verified experimentally by Fabbro [39]. In their analytical model, Kahlen et al. [38] also calculated the plasma plume temperature for short laser pulses (pulse width tB80 ns, FWHM). For laser irradiances of 106 W cm2 they found plume temperatures of Tp B7000 K which is just below the first ionization temperature (8500 K [31]). In comparison, the laser irradiance in the current investigation reaches 108 W cm2. It is expected that the higher laser irradiance will increase the plume temperature during LSP above the first ionization temperature. As a result, a single-ionized plasma plume is formed above the irradiated surface [40]. This
2
Thermal stresses [GPa]
0
t = 30 ns p
-2
z = 100 nm
-4 Axial thermal stress Radial thermal stress Azimuthal thermal stress
-6
-8 0
0.2
0.4
0.6
0.8
1
Radius [mm] Fig. 7. Radial tensile (positive) and compressive (negative) stresses in direct irradiation/ablation for a ramp-up laser pulse at z ¼ 100 nm, beam radius r0 ¼ 1 mm, t ¼ 30 ns, Pave ¼ 50 W.
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plasma plume attenuates the energy transfer from the laser beam to the substrate surface. Also, Fig. 5 identifies three concentric thermal regions: (i) a central region ranging from the depth z ¼ 0 to 100 nm where the temperature exceeds the boiling temperature of stainless steel, (ii) a middle region in which the temperature is between the melting and boiling temperatures surrounding the central regions, and (iii) a peripheral region where the ambient material temperature does not exceed the meting temperature. The cross-sections of the two inner regions decrease in the positive z-direction. The material in the central region is removed by ablation. Following the Hertz–Knudsen formula, some material will be evaporated from the surface of the liquid pool in the middle region. Since the evaporation rate is proportional to the surface temperature, the evaporation rate is higher for smaller values of r: Fig. 6 shows the thermal stresses at the substrate surface at the radius r ¼ r0 ¼ 1 mm for a ramp-up pulse. It can be seen that a shock wave is generated during the first nanosecond. As a result, the radial, axial and azimuthal thermal stresses in the material rise sharply during this first nanosecond. The stresses in the radial and axial directions are compressive (negative) whereas the azimuthal stress is tensile (positive). After approx. 26 ns the material has been melted and the stresses vanish. At a depth of z ¼ 100 nm, the thermal stresses appear at the melt pool radius
4 r = 1 mm
Pressures and stresses (GPa)
2
z=0
0
0 -2 -4
Thermal stresses
-6 -8
Energy deposition
Axial
Axial
Azimuthal
Radial
Radial -10 -12
0
5
10
15
20
25
30
Irradiation time [ns] Fig. 8. Temporal tensile (positive) and compressive (negative) stresses in direct irradiation/ablation for a ramp-down laser pulse at z ¼ 0; beam radius r0 ¼ 1 mm, t ¼ 30 ns, Pave ¼ 50 W.
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rm ¼ 0:6 mm (Fig. 7). It can also be seen that the azimuthal stress changes from compressive (negative) to tensile (positive) for increasing radius. In the case of a ramp-down pulse, the irradiated surface is melted after 8 ns (Fig. 8). The maximum values of the radial, axial and azimuthal stresses are very close to the values for the ramp-up pulse; also, the radial and axial stresses are compressive and the azimuthal stress is tensile. Thermal stresses appear at a depth of z ¼ 100 nm at the radius rm ¼ 0:8 mm (Fig. 9). The radial and axial stresses are compressive and the azimuthal stress is tensile throughout the remaining irradiated spot (0:8oro1 mm). No surface melting occurs for the rectangular laser pulse (Fig. 10) at the laser beam radius after 30 ns. As in the case of the ramp-up laser pulse, a shock wave is generated during the first nanosecond of irradiation. The radial, axial and azimuthal stresses assume the same values and directions as in the case of the other pulse shapes. At the end of the laser pulse, the melt pool radius at a depth of z ¼ 100 nm is rm ¼ 0:63 mm (Fig. 11). The axial, radial and azimuthal thermal stresses exhibit similar trends as in the case of the ramp-up laser pulse. The axial and radial thermal stresses are compressive and the azimuthal thermal stress changes from compressive to tensile for rm oror0 : It is noteworthy that the behavior of the thermal stresses for the ramp-up and rectangular pulses at a depth of z ¼ 100 nm appears to be very similar. In both cases the melt pool extends to about the same value of rm (0.63 mm for the rectangular pulse and 0.6 mm for the ramp-up pulse), the peak values of the axial and radial 2
Thermal stresses [GPa]
0
= 30 ns -2 z = 100 nm -4 Azimuthal Axial Radial
-6
-8 0
0.2
0.4
0.6
0.8
1
Radius [mm]
Fig. 9. Radial tensile (positive) and compressive (negative) stresses in direct irradiation/ablation for a ramp-down laser pulse at z ¼ 100 nm, beam radius r0 ¼ 1 mm, t ¼ 30 ns, Pave ¼ 50 W.
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Pressures and stresses (GPa)
5
0
-5
-10 Thermal stresses Axial Azimuthal Radial
-15 0
5
10
Energy deposition Axial Radial
15
20
25
30
Irradiation time [ns] Fig. 10. Temporal tensile (positive) and compressive (negative) stresses in direct irradiation/ablation for a rectangular laser pulse at z ¼ 0; beam radius r0 ¼ 1 mm, t ¼ 30 ns, Pave ¼ 50 W.
thermal stresses are 6.5 and 5 GPa, respectively, and the value of the axial thermal stress decreases more rapidly than the value of the radial stress. Also the azimuthal thermal stress first increases to 1.5 GPa (compressive) and then decreases with approximately the same gradient, as does the axial thermal stress. The azimuthal thermal stress changes its sign from negative (compressive) to positive (tensile) at r ¼ 0:8 mm and reaches a value of 1.2 GPa at r ¼ r0 : In the case of the ramp-down pulse, the azimuthal thermal stress is always tensile and the radial thermal stress always exceeds the axial thermal stress for rm oror0 : Also, the melt pool radius is larger in this case (rm ¼ 0:8 mm) than for the ramp-up or rectangular pulse. This distinctly different trend in the results can be explained from the point of view of the control volume introduced earlier (a disc of depth Dz and cross-section pr2 ). For a positive rate of change of internal energy, the temperature inside the control volume will increase. Therefore, the possibility for material removal due to heating and ablation are much larger for the ramp-down pulse than for the other two cases, as illustrated earlier. This explains why the melt pool radius is larger for the ramp-down pulse. The radial and axial thermal stresses should be decreasing quickly for a related reason, that is, when the internal energy in the control volume is increased, less energy is conducted radially and axially into the material. Therefore, the temperature gradients in the r- and z-directions are small and the resulting axial thermal stress is smaller than in the case of the ramp-down pulse (Figs. 7, 9 and 11).
T. Thorslund et al. / Optics and Lasers in Engineering 39 (2003) 51–71
66 2
Thermal stresses [GPa]
0
-2
Azimuthal Radial Axial
-4
-6
-8 0
0.2
0.4
0.6
0.8
1
Radius [mm]
Fig. 11. Radial tensile (positive) and compressive (negative) stresses in direct irradiation/ablation for a rectangular laser pulse at z ¼ 100 nm, beam radius r0 ¼ 1 mm, t ¼ 30 ns, Pave ¼ 50 W.
During the laser–material interaction, the recoil pressure on the molten material surface displaces the melt away from the center of the laser beam, in a radial direction [41]. The pressure in the liquid phase exerts a radial force on the surrounding solid material. This radial force is balanced by a compressive (negative) azimuthal thermal stress in the solid phase. In the solid phase, a large stress gradient ds=dr balances the radial pressure in the liquid pool. If the laser irradiance is high towards the end of the laser pulse (as the case for the ramp-up and rectangular pulses), a large compressive azimuthal thermal stress is also developed. However, at farther radial distances from the laser beam center, this compressive azimuthal thermal stress turns into a tensile azimuthal thermal stress, as shown in Figs. 7 and 11. This azimuthal stress syy is a combination of radial and thermal stresses as shown in Eq. (17). The radial stress (which decreases as 1=r2 ) at any point in the medium may be assumed to be due to a hot core of materials. When r is small both radial and thermal stresses contribute to the azimuthal stress resulting in its negative value (which corresponds to the compressive stress). For large values of r; that is when one is outside of the hot core, the contribution of the radial stress decreases significantly, resulting in the negative value of the azimuthal stress (which corresponds to the compressive stress in the material). From a physical point of view, for small values of r the atoms are displaced by Dr which is a combination of radial and thermal expansions. As the value of r increases, the radial expansion due to radial stress diminishes. The tensile thermal expansion becomes dominant in this case and the
T. Thorslund et al. / Optics and Lasers in Engineering 39 (2003) 51–71
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0
Pressure (GPa)
-10
-20 z=0
τ = 30 ns -30 Ablation -40
Plasma
-50 0
0.2
0.4
0.6
0.8
1
Radius [mm] Fig. 12. Axial compressive (negative) stresses for confined geometry ablation for a ramp-up laser pulse at z ¼ 0; beam radius r0 ¼ 1 mm, t ¼ 30 ns, Pave ¼ 50 W.
resulting azimuthal stress is tensile. At large distances from the laser beam center, the temperature T ðr; z; tÞ decreases to zero, resulting in zero thermal stress. The plasma pressures are considerably larger for ablation occurring under a confined geometry. Figs. 12–14 shows the compressive axial thermal stress on the substrate surface for a ramp-up pulse. As expected the maximum surface pressure is much higher (45 GPa) than in the case of direct irradiation/ablation (11 GPa). No ablation takes place for r > ra ¼ 0:85 mm. Similar result is found for the rectangular pulse, however, the maximum surface pressure is much smaller (19.5 GPa). The radial limit for the occurrence of ablation take place is ra ¼ 0:84 mm. In the case of the ramp-down pulse, the maximum plasma pressure is of the same magnitude (46 GPa) as in the case of the ramp-up pulse. Also, the material ablation is a much more confined effect, extending to a radius of ra ¼ 0:41 mm for the ramp-down laser pulse.
5. Conclusions Mathematical models have been developed to calculate the temperature distributions and the thermal stresses in three dimensions for direct laser
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0
Pressure (GPa)
-5
z=0
-10
τ = 30 ns
Ablation
-15
Plasma
-20 0
0.2
0.4
0.6
0.8
1
Radius [mm] Fig. 13. Axial compressive (negative) stresses for confined geometry ablation for a rectangular laser pulse at z ¼ 0; beam radius r0 ¼ 1 mm, t ¼ 30 ns, Pave ¼ 50 W.
irradiation/ablation and ablation in a confined geometry. In the case of direct irradiation/ablation, the ramp-down pulses generate higher mean temperatures in the workpiece than in the cases of ramp-up and rectangular pulses for the same energy per pulse and pulse length. The temperatures for all pulse shapes and shock geometries are strongly dependent on the absorption of the laser energy. Pulses with short rise are favorable for laser shock processing because higher surface pressures can be obtained. The models confirm that higher pressures are achieved for laser shock processing under a confined geometry.
Acknowledgements Thord Thorslund visited The University of Central Florida as an exchange student during his MS study at the University of Uppsala, Sweden. He also acknowledges the assistance received from the University of Dalarna and would like to thank everybody who supported this project in any way.
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0
Pressure (GPa)
-10
-20 z=0
τ = 30 ns -30 Ablation -40
Plasma
-50 0
0.2
0.4
0.6
0.8
1
Radius [mm] Fig. 14. Axial compressive (negative) stresses for confined geometry ablation for a ramp-down laser pulse at z ¼ 0; beam radius r0 ¼ 1 mm, t ¼ 30 ns, Pave ¼ 50 W.
References [1] Roberts LW, editor. Cold rolling of steel. New York: Marcel Dekker, 1978. [2] Niku-Lari A. Shot peening. First International Conference on Shot Peening, Paris, 14–17 September 1981, 1st ed. Oxford, New York: Pergamon Press, 1982. p. 3–22. [3] Dav!e V, Matz J, Eagar T, Goodman D. High energy electron beam (HEEB)—solid interaction model for EB deposition and shock processing. In: Singh J, Copley S, Mazumder J, editors. Proceedings of Second International Conference of Advanced Materials. Materials Park, OH: ASM, 1996. [4] Clauer AH, Fairand BP, Wilcox BA. Pulsed laser induced deformation in an Fe-3 wt pct Si alloy. Metal Mater Trans A 1977;8A:119–25. [5] Fairand BP, Clauer AH. In: Ferris S, Leamy H, Poate J, editors. Laser–solid interactions and laser processing. New York, NY: AIP, 1979. p. 27–42. [6] Dupont A, Caminat P, Bournot PC. Efficiency of metallic materials ablation using impulsional laser with several wavelengths. In: Fotakis C, Kalpouzos C, Papazoglou Th, editors. Proceedings SPIE Vol. 1810, Ninth International Symposium on Gas Flow and Chemical Lasers, 1993. p. 696–9. [7] Phipps Jr, C, Harrison R, Turner T, Anderson G, Haynes L. Impulse coupling to targets in vacuum by KrF, HF, and CO2 single pulse laser, CLEO 1988, Technical Digest Series, vol. 7. Piscataway, NJ: IEEE, 1988. p. 2, 3. [8] Hoffman C. Laser-target interactions. J Appl Phys 1974;45(5):2125–8. [9] Vohra A, Bansal SK, Sharma RK, Srivastava GP, Grag CL, Mull RP. Surface effects on laserinduced damage in Si. J Phys D 1990;23(1):56–66. [10] Fairand B, Wilcox B. Laser shock-induced microstructural and mechanical property changes in 7075 Al. J Appl Phys 1972;43:3893–6.
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[11] Clauer A, Holbrook J, Fairand B. In: Meyers M, Murr L, editors. Shock waves and high strain rate phenomena in metals. New York: Plenum Press, 1981. p. 675–702. [12] Chu J, Rigsbee JM, Banas G, Lawrence FV, Elsayed-Ali H. Effects of laser-shock processing on the microstructure and surface mechanical properties of Hadfield manganese steel. Metal Mater Trans A 1995;26A(6):1507–17. [13] Dubrujeaud B, Jeandin M. Cladding by laser shock processing. J Mater Sci Lett 1994;13(11):773–5. [14] Anderholm N. Laser-generated pressure waves. Bull Am Phys Soc 1968;13:388. [15] Anderholm N. Laser generated stress-waves. Appl Phys Lett 1970;16(3):113–5. [16] O’Keefe J, Skeen C. Laser-induced stress wave and impulse augmentation. Appl Phys Lett 1972;21:464–6. [17] O’Keefe J, Skeen C. Laser-induced deformation modes in thin metal targets. J Appl Phys 1973;44(10):4622–6. [18] Fabbro R, Faral B, Virmont J, Pepin H, Cottet F, Romain J. Experimental evidence of the generation of multi-hundred Megabar pressures in 0.26 mm wavelength laser experiments. Laser Part Beams 1985;4(3–4):413–9. [19] Singh J. Laser-beam and photon-assisted processed materials and their microstructures. J Mater Sci 1994;29(20):5232–58. [20] Ready JF. Industrial applications of lasers. New York: Academic Press, 1978. p. 356. [21] Roessler DM. The industrial laser annual handbook. In: Belforte D, Levitt M, editors. Proceedings of SPIE, vol. 629. Tulsa, OK: PennWell Books, 1986. p. 16. [22] Mazumder J. Overview of melt dynamics in laser processing. Opt Eng 1991;30(8):1208–19. [23] Kar A, Langlais MD. Optothermal effects of laser modes in laser materials processing. Opt Quantum Electron 1995;27(12):1165–80. [24] Dyer PE, Key PH, Sands D, Snelling HV, Wagner FX. Blast-wave studies of excimer laser ablation of ZnS. Appl Surf Sci 1995;86(1):18–23. [25] S!arady I, Magnusson CF, Wei L-Y, Meijer J. Phase transformation by high intensity submicrosecond laser pulses. LIA. In: Farson D, Steen W, Miyamoto I, editors. Proceedings of ICALEO ’92, Orlando, Florida, USA, 1992. [26] Forget P, Strudel JL, Jeandin M, Lu J, Castex L. Laser shock surface treatment of Ni-based superalloys. Mater Manuf Process 1990;5(4):501–28. [27] Dufour J, Gerland M, Darquey P. Traitements de surface d’un alliage d’aluminium par des implusions laser (Surface treatments of an aluminum alloy by laser pulses). Scr Metall 1989;23(4):483–8. [28] Fabbro R, Faral B. Temperature measurements of very high pressure laser shocks above 100 Mbars using time-resolved X-UV spectroscopy. Laser Part Beams 1991;9(2):397. [29] Devaux D, Fabbro R, Tollier L, Bartnicki E. J Appl Phys 1993;74:2268–73. [30] Duley W. Co2 lasers: effects and applications. Academic Press: New York, 1976. [31] Iida T, Guthrie R. The physical properties of liquid metals. Oxford: Clarendon Press, 1988. p. 6, 91, 240, 241. [32] Thorslund Th. Mathematical models for temperatures, pressures and stresses during shock processing by utilization of a laser source. M.S. Thesis, University of Uppsala, Sweden, 1996. [33] Koffman LD, Plesset MS, Lees L. Theory of evaporation and condensation. Phys Fluids 1984;27:876–80. [34] Frank-Kamenetskii DA. Plasma–the fourth state of matter. New York: Plenum Press, 1972. [35] Rosenbluth MN, Sagdeev R. Handbook of plasma physics, vol. 3. Amsterdam: North-Holland, 1991. p. 69. [36] Touloukian Y, editor. Thermophysical properties of high temperature solid materials, vols. 1, 3, 10. New York: Macmillan, 1967. [37] Smithells C. In: Brandes E, Brook G, editors. Metals reference book, 6th ed. Oxford, Boston: Butterworth-Heinemann, 1988. [38] Kahlen F-J, Emminger H, Sankaranarayanan S, Kar A. Plasma instability and optimum utilization of laser energy in laser materials processing. SPIE conference Advanced High Power Lasers and Applications, Osaka, Japan 1–5 November 1999.
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[39] Fabbro R, Chouf K, Sabatier L, Coste F. Dynamical interpretation of deep penetration cw laser welding. In: Beyer E, Chen X, Miyamoto I, editors. Proceedings of the Laser Institute of America (ICALEO98), 1998. p. F179–F186. [40] Herziger G, Kreutz E, Wissenbach K. In: Duley W, Weeks R, editors. Proceedings of the SPIE, Laser Processing: Fundamentals, Applications, and System Engineering, vol. 668, 1986. p. 2–10. [41] Kahlen F-J. The development of a laser reflectometer diagnostic technique for laser keyhole welding studies of stainless steel SS 304. M.S. thesis, University of Tennessee Space Institute, 1994.
Temperatures, pressures and stresses during laser shock processing Thord Thorslund, Franz-Josef Kahlen, Aravinda Kar* Laser-Aided Manufacturing, Materials and Microprocessing (LAMMMP) Laboratory, School of Optics and Center for Research and Education in Optics and Lasers (CREOL), Mechanical, Materials, and Aerospace Engineering Department, University of Central Florida, 4000 Central Florida Blvd., Orlando, FL 32816-2700, USA Received 1 November 2000; received in revised form 1 February 2002; accepted 1 March 2002
Abstract Mathematical models are developed to calculate the temperatures, pressures and stresses during laser shock processing for time-modulated (ramp-up, ramp-down and rectangular) laser pulses. Three different shock processing configurations are also considered: non-ablative exposure, ablative exposure and confined ablation with coating. The results for iron show that the plasma pressure reaches an average value of 9 GPa in direct ablation configuration and plays a dominant role for all three types of laser pulses. In the case of confined geometry, the plasma pressure reaches an average value of 20 GPa. All calculated pressures and stresses exceed the yield strength of the workpiece, indicating plastic deformation. It is also shown that pulses with short rise times yield higher plasma pressures. r 2002 Published by Elsevier Science Ltd. Keywords: Shock processing; Temperatures; Pressures; Stresses; Non-ablative; Ablative; Confined ablative
1. Introduction and background Laser shock processing (LSP) utilizes the recoil pressure of the rapidly vaporized atoms to improve the surface strength by compressing the material at the surface. LSP is very attractive because of its unique capability to process materials of complicated shapes and geometries. In comparison with other surface hardening *Corresponding author. Tel.: +1-407-823-6921; fax: +1-407-658-6880. E-mail address: [email protected] (A. Kar). 0143-8166/03/$ - see front matter r 2002 Published by Elsevier Science Ltd. PII: S 0 1 4 3 - 8 1 6 6 ( 0 2 ) 0 0 0 4 0 - 4
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Nomenclature A a0 a1 a2 E F ðtÞ fc I0 k kB Lv M mi n NA ne P Pave Patm pv R r rm r0 T Tatm Tp Tv T z
absorptivity absorptivity at T ¼ Tv absorptivity accounting for laser beam attenuation in the plasma plume absorptivity accounting for the increased laser beam attenuation in the plasma plume for Tp > Tv Young’s modulus (N m2) temporal shape of the laser pulse correction factor accounting for collisions in the plasma for confined geometry processing irradiance (W cm2) thermal conductivity (W m1 K1) Boltzmann constant (J K1) latent heat of vaporization (J kg1) molecular weight (kg kmol1) temporal pulse shape factor at t ¼ 0 (i ¼ 1; 2; 3) number density of atoms in the plume (cm3) Avogadro’s number (mol1) number density of electrons in the plume (cm3) power (W) average power per pulse (W) one atmospheric pressure (N m2) pressure at the vapor–liquid interface (N m2) gas constant (J kg1 K1) radius (m) melt pool radius (m) focal beam radius (m) temperature (K) boiling temperature at one atmospheric pressure (K) plasma plume temperature (K) boiling temperature at the vapor–liquid interface (K) excess temperature (T T0 ) (K) depth (m)
Greek characters aT thermal expansion coefficient (K1) a thermal diffusivity (m2 s1) n Poisson ratio y angle rl density of the liquid phase (kg m3) rv density of the vapor phase (kg m3) srr stress in the radial direction (GPa) szz stress in the axial direction (GPa)
T. Thorslund et al. / Optics and Lasers in Engineering 39 (2003) 51–71
syy t
53
stress in the azimuthal direction (GPa) pulse width (ns)
techniques, such as cold rolling, shot peening, X-rays or high energy electron beams [1–3], LSP offers several advantages, such as a relatively easy process control through control of the laser beam, processing of only a thin surface layer, no potential dangers to the health of the operator such as X-rays, and no requirement to process the material in a vacuum. Disadvantages of LSP are (i) a potential for surface melting and resolidification which allows for surface wave formation and the introduction of surface roughness [4–9] and (ii) the small processing area (the laser beam is focused to a spot of 0.1–2 mm diameter) in comparison to the surface to be processed. LSP can be performed in a number of ways such as non-ablative exposure, ablative exposure, utilization of an energy absorptive layer and/or transparent overlay (confined ablation with coating) and combinations thereof (Fig. 1). Nonablative exposure is a process in which nearly instantaneous temperature rise in the material due to rapid energy deposition generates thermal stresses and shock waves in the material. In ablative exposure, the substrate surface is additionally subjected to recoil and plasma pressures. A considerable drawback of direct ablation is the possibility of surface damage due to melting and vaporization [10, 11]. For LSP in a confined geometry (confined ablation with coating), the substrate surface is typically coated with a 40 mm thin layer [12–18] and an overlay, which is transparent to the laser wavelength, on top of this coating. The pressure required to harden the substrate surface is generated by vaporizing the coating, and heating the vapor in a confined space. During LSP, the workpiece is exposed to high laser irradiance (108–1012 W cm2) for a few nanoseconds [19–23]. Short pulse (600 ps–100 ns) lasers ranging from KrF Excimer laser [24], Ruby laser [25] to solid-state (Nd:YAG and Nd:Glass) lasers [12, 13, 17, 18, 26] have been used for LSP. A wide variety of analytical and experimental investigations have been carried out since Densification of porous materials using LSP has been demonstrated by Darquey et al. [27], Dubrujeaud and Jeandin [13] successfully used LSP to perform
Shock beam
Shock beam Ejected particles
Shock beam Plasma Transparent Coating overlay
Material
Material
Material
(a)
(b)
(c)
Fig. 1. Geometries considered for laser shock processing: (a) non-ablative exposure, (b) ablative exposure, (c) confined ablation with coating.
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cladding of Al2024 alloy. Fabbro et al. [18] presented a one-dimensional analytical model to investigate the energy transfer during LSP and to derive a correlation between the laser beam parameters and the properties of the processed material. Verification of the model on a copper target with glass as transparent overlay, using a Nd:YAG laser with 2–30 ns pulse length showed that the energy absorption was higher in the case of a confined geometry than for direct ablation. They found that the model agrees with the experimental results for energy densities of 10–100 J cm2. Ballard et al. [28] presented a model for the plastic and elastic deformations of the processed material. Given the limited size and the temperature in the processed region, very few experimental investigations have been carried out to establish relationships between the laser irradiance and the stresses and residual stresses [18, 28] in the workpiece. Sarady et al. [25] published values about the pressures during LSP, measuring the displacements of a steel sphere suspended as a pendulum during the irradiation in vacuum. They measured mean pressures in the order of 6 and 11 GPa during the irradiation with a Ruby laser and a Nd:YAG laser, respectively. Their analytical calculations were based on Fabbro’s results [18]. Devaux et al. [29] successfully investigated the effect of the temporal pulse shape on the pressures due to breakdown of the transparent overlay experimentally, demonstrating a dependence of the pressures on the pulse rise time.
2. Approach In this paper, a mathematical model is developed to study the effect of laser irradiance and its time modulation on the temperature distributions in the workpiece and plasma plume. Thereby, this investigation extends previous studies on LSP of steels on short rise time pulses and connects the temperature distribution in the target with the recoil pressure on the material surface during the ablation process. The temperature fields are necessary to calculate the stresses generated during laser shock processing. The heat conduction equation is solved using the Hankel and LaPlace transforms based on the assumption that the surface absorptivity is temperature-dependant. This assumption is justified because of the short laser– material interaction times. During these interaction times, the surface will heat up and melt for sufficiently high laser intensities. It is established [30] that the surface absorptivity will increase when the surface temperature increases. A very limited number of models for the temperature dependence of thermophysical properties are available [31]. Their margins of error reach 50% for temperatures in excess of 2000 K; reliable models for even higher temperatures are not available. Due to lack of reliable data or models, thermophysical properties are considered constant in this investigation. It was expected that the plasma plume temperature remains below the second ionization temperature of steels (E17,000 K). Thus, the authors considered a single-ionized plasma in their model. The results presented later will confirm this assumption. The surface temperatures for ramp-up, ramp-down and rectangular laser pulses are calculated for direct irradiation/ablation and for a confined geometry. The stresses due to rapid energy deposition, the thermal shock, recoil and
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55
plasma pressures are considered for direct irradiation/ablation. The pressure related to the instantaneous energy deposition is assumed to be present during the entire duration of the laser pulse (30 ns). Thermal stresses are calculated by considering the thermophysical properties (Table 1) of stainless steel SS 304 (Table 2), the respective values for iron were used where data for SS 304 were not available. The recoil pressure is determined by applying the kinetic theory to the metal vapor. An expression for the plasma pressure is obtained by considering a freely expanding plasma, and the conservation of the thermal and kinetic energies. In the case of LSP with a confined geometry, recoil and plasma pressures are dominant. The plasma cannot expand freely. Its density is approximated as the liquid density due to the short laser–material interaction time. The recoil pressure is determined by applying a correction factor to the expression derived for the case of direct irradiation/ablation. High power laser beams typically are designed to produce a very good quality beam, approaching a near-Gaussian intensity distribution. While generally flat-top laser beam intensity distributions are used in LSP because of the larger surface areas which can be treated, they require additional modifications for the laser beam delivery and laser beam shaping. The authors considered a Gaussian laser beam in
Table 1 Thermophysical properties for iron [36–38] Thermophysical property
Symbol
Units
Value
Thermal diffussivity Thermal conductivity (at T ¼ 300 K) Initial temperature Melting temperature Vaporization temperature Latent heat of melting Latent heat of vaporization Molar mass Particle mass Poisson’s ratio Young’s modulus Heat capacity (at T0 ) Heat capacity (at Tm ) Heat capacity (at Tv ) First ionization potential Density (at T0 ) Density (at Tm ) Thermal expansion coefficient (at T ¼ 1000 K)
a k T0 Tm Tv Lm Lv M Mp n E cp cp;m cp;v EI r r aT
m2 s1 W m1 K1 K K K kJ kg1 kJ kg1 kg kmol1 kg — N m2 J kg1 K1 J kg1 K1 J kg1 K1 K kg m3 kg m3 K1
7.6 106 45 298 1810 3135 2.66 105 6.8 106 55.847 9.274 1026 0.27 2.1 1010 450 828 828 8500 7870 6660 15.5 106
Table 2 SS304 composition Powder
Fe
Ni
Cr
Si
Mg
P
C
S
SS 304
68.33
11.25
19.32
0.93
0.13
0.007
0.022
0.007
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this investigation, to analyze the effect of the highest irradiances on the mechanical properties of the material surface.
3. Mathematical modeling for laser ablation 3.1. Temperature modeling for the melt pool Laser ablation is a process during which the material is heated above its boiling temperature very rapidly. To model this rapid heating process, the absorptivity is assumed to depend on the temperature according to the following expression: Tðr; 0; tÞ for ToTv ; Tv Tv AðTðr; 0; tÞÞ ¼ a1 1 þ exp a2 Tðr; 0; tÞ
AðTðr; 0; tÞÞ ¼ a0
ð1Þ for T > Tv
ð2Þ
Eq. (1) describes a linear increase with temperature until the surface temperature reaches the vaporization temperature. The authors assumed an exponential decrease of the surface absorptivity for temperatures T > Tv ; to account for the laser beam attenuation in the plasma plume. Parametric studies are conducted by taking a0 ¼ 0:4; a1 ¼ 5:41 and a2 :¼ 105 in Eqs. (1) and (2). Defining T ðr; z; tÞ ¼ Tðr; z; tÞ T0 and assuming that the problem is symmetrical about the y direction (Fig. 2), the unsteady heat conduction equation and the boundary and initial conditions in cylindrical coordinates can be written as 1 q qT q2 T 1qT r ð3Þ ¼ þ r qr a qt qr qz2 and T ðr ¼ N; z; tÞ ¼ 0;
Fig. 2. Coordinate system for a cylindrical disk of radius r0 and depth Dz:
ð4Þ
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T ðr; z ¼ N; tÞ ¼ 0; k
qT ðr; z ¼ 0; tÞ r2 ¼ AðTðr; z ¼ 0; tÞÞI exp 2 2 F ðtÞ; qz r0
57
ð5Þ ð6Þ
T ðr; z; t ¼ 0Þ ¼ 0;
ð7Þ
T ðr ¼ 0; z; tÞ must be finite:
ð8Þ
It is now possible to consider different temporal laser pulse shapes (Fig. 3) by properly selecting the function F ðtÞ such as F1 ðtÞ ¼ m1 t
for ramp-up pulses;
ð9Þ
F2 ðtÞ ¼ m2 ðt tÞ for ramp-down pulses;
ð10Þ
F3 ðtÞ ¼ m3
ð11Þ
for rectangular pulses;
Thorslund [32] solved Eqs. (3)–(8) using the Hankel and LaPlace transforms for a pulse width t and obtained T ðr; z; tÞ ¼
pffiffiffi Z I0 r20 a t A ðt t0 ÞF ðt t0 Þexp ð2r2 =r20 þ 8at0 Þ ðz2 =4at0 Þ pffiffiffi pffiffiffi dt0 : 2 þ 8at0 0 r k p 0 t 0
ð12Þ
For the operating conditions considered in this investigation (r0 ¼ 1 mm, aB104 m2 s1 tB30 ns), it can be shown that r20 þ 8at0 Er20 : The singularity at t0 is removed by substituting t0 ¼ t00 =t: Finally, the following expressions for the temperature distributions for three different types of laser pulses are derived: Ramp-up pulse: pffiffiffiffiffi pffiffiffi z2 t tt 2 Z 8P mt r 002 002 t t T ðr; z; tÞ ¼ 2 pffiffiffi exp 2 2 A t t t t xp 002 dt00 : 4at r0 0 pr0 k p ð13Þ
Fig. 3. Schematic of pulse shapes: (a) ramp-up pulse of pulse width t; (b) ramp-down pulse of pulse width t; (c) rectangular pulse of pulse width t:
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Ramp-down pulse:
pffiffiffiffiffi at r2 T ðr; z; tÞ ¼ 2 pffiffiffi exp 2 2 r0 pr0 k p pffiffiffi Z tt z2 t t002 t002 A t t t t xp 002 dt00 4at t 0
8P
and rectangular pulse: pffiffiffiffiffi pffiffiffi tt 2 Z 4P at r t002 z2 t T ðr; z; tÞ ¼ 2 pffiffiffi exp 2 2 A t xp 002 dt00 : 4at t r0 0 pr0 k p
ð14Þ
ð15Þ
3.2. Stress modeling during laser shock processing 3.2.1. Non-ablative exposure In this case, stresses may be generated in the material due to rapid energy deposition, resulting in thermal stresses. These thermal stresses are generally given by aT E T ðr; z; tÞ: If the laser beam diameter is much larger than the depth of heating (thermal energy transfer) (Dz=r51) the radial (r-direction) and azimuthal (y-direction) stresses can be calculated using the following expressions [32]: Z aT E r 0 srr ¼ 2 T ðr ; z; tÞr0 dr0 ; ð16Þ r 0 Z aT E r 0 syy ¼ 2 T ðr ; z; tÞr0 dr0 aT ET ðr; z; tÞ: ð17Þ r 0 The temperature T ðr0 ; z; tÞ can be calculated from Eqs. (13)–(15) for different pulse shapes. 3.2.2. Ablative exposure Besides the thermal stresses generated in direct irradiation, there are two additional effects to consider during material ablation which are the recoil pressure exerted by the ablated particles and the plasma pressure. The net mass flux due to vaporization can be calculated from the Hertz–Knudsen Equation [33] assuming Maxwellian temperature and density distributions. The pressure in the plasma plume is calculated from the kinetic energy of the evaporating particles [25,34,35] by applying the Saha Equation to singly ionized particles for the geometry shown in Fig. 4. The radial and axial (z-direction) stresses due to recoil and plasma pressure are then respectively expressed as n srr ¼ szz ; ð18Þ 1n 2 ! rv kB T ðr; z ¼ 0; tÞ ne 1:6652 rl 1þ þ szz ¼ 1 ; ð19Þ M n 2p rv
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59
Fig. 4. Model of surface layer in confined and direct ablation.
which need to be combined with the stresses obtained from Eqs. (16) and (17) to determine the total stress. The terms in Eq. (19) represent the ‘‘instantaneous’’ energy increase due to the energy deposition, the plasma pressure and the recoil pressure of the ablated particles. The factor 1.665 enters into Eq. (19) due to the Hertz–Knudsen equation for rarefied gas dynamics as described above. The Hertz– Knudsen equation contains a correction factor f which has been shown to be equal to 1.665 [33].
3.2.3. Confined ablation with coating Laser shock processing in a confined geometry is distinctly different from direct ablation because the coating layer is vaporized in the case of confined geometry [12] whereas the workpiece itself is vaporized in the case of direct ablation. The collisions between the vaporized particles in the vaporized layer must be considered in the case of confined geometry, and the recoil and plasma pressures are of interests to calculate the pressure on the substrate surface. The recoil pressure is determined using an approach similar to the case of direct ablation. A correction factor, fc is introduced to account for the collision of the particles in the confined geometry, and the coating layer is assumed to exist mainly in the liquid phase. The plasma pressure is calculated numerically from an energy balance for the plasma plume accounting for phase changes during the plasma formation. The radial and axial stresses can
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then be written as n szz ; srr ¼ 1n T ðr; z ¼ 0; tÞ T ðr; z ¼ 0; tÞ ne ð1:665fc Þ2 szz ¼ rl kB þNA 1 þ Tp : M 2p n
ð20Þ ð21Þ
The correction factor fc is introduced to account for the energy loss due to molecular and atomic collisions within the plasma. For the numerical evaluation it was assumed that 10% of the thermal energy in the plasma plume is lost due to internal collisions, i.e., fc ¼ 0:9: The numerical evaluations will be carried out for average pulse powers of 50 W, average pulse energies of 100 J, a laser beam radius of 1 mm, and pulse width of 30 ns.
4. Results and discussions The radial temperature distributions during direct irradiation/ablation of iron [31, 36, 37] subjected to three different pulse shapes (ramp-up, ramp-down and constant energy (rectangular) pulses) are shown in Fig. 5. In each case, the pulse width t is 30 ns and the energy contained in each pulse is kept constant. It can be seen that the ramp-down pulse yields the highest surface temperatures over the entire irradiated area. This result agrees with the expectation that the laser irradiance rate (dIðr; tÞ=dt) determines the heating rate in the irradiated material. The surface temperature for a ramp-up pulse is considerably lower then the surface temperature for a ramp-down pulse, but is still higher than the rectangular pulse (Fig. 5). This behavior of the
Fig. 5. Radial temperature distribution in direct irradiation/ablation for ramp-up, ramp-down and rectangular laser pulses at z ¼ 0 and z ¼ 100 nm, beam radius r0 ¼ 1 mm, t ¼ 30 ns, Pave ¼ 50 W.
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surface temperature is due to the irradiance rate and, therefore, an indication that this irradiance rate can have stronger influence on the material temperature than the actual irradiance Iðr; tÞ: If the irradiance rate dIðr; tÞ=dt > 0; more energy may be coupled into a control volume (disc) of depth Dz and cross-section pr2 than is lost due to conduction into the surrounding material. In this case, the rate of change of internal energy in this control volume would be positive and the temperature in the control volume would rise. It is, therefore, possible that the surface temperature of a material which is exposed to a ramp-up laser pulse exceeds the surface temperature produced by a rectangular pulse containing the same energy as the ramp-up pulse. The model shows that the surface temperature of the irradiated material (14,500 K) exceeds its boiling temperature (3000 K). A surface temperature of 14,500 K appears to be very large. It is possible that the model’s assumptions oversimplified the physics and that the true surface temperature is below 14,500 K. However, this temperature must be evaluated in the context of the material’s exposure to the very high laser irradiance. For an ideal gas, the Clausius–Clapeyron equation (Eq. (22)) establishes a link between a reference evaporation temperature (typically Tatm ¼ 3000 K at one atmosphere patm ) and the evaporation pressure at
Pressures and stresses [GPa]
5
0
-5
-10 Thermal stresses
-15
Energy deposition
Axial
Axial
r = 1 mm
Azimuthal
Radial
z= 0
0
Radial 0
5
10
15
20
25
30
Irradiation time [ns] Fig. 6. Temporal tensile (positive) and compressive (negative) stresses in direct irradiation/ablation for a ramp-up laser pulse at z ¼ 0; beam radius r0 ¼ 1 mm, t ¼ 30 ns, Pave ¼ 50 W.
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higher surface temperatures: Lv 1 1 pv ¼ patm exp : R Tv Tatm
ð22Þ
A surface temperature Tv exceeding 3000 K is an indication that the surface pressure pv also exceeds one atmospheric pressure. For laser irradiance lower than 106 W cm2, Kahlen et al. [38] showed that the surface pressure reaches a value of 2 105 N m2 (2 atmospheres), employing an analytical model. This value was verified experimentally by Fabbro [39]. In their analytical model, Kahlen et al. [38] also calculated the plasma plume temperature for short laser pulses (pulse width tB80 ns, FWHM). For laser irradiances of 106 W cm2 they found plume temperatures of Tp B7000 K which is just below the first ionization temperature (8500 K [31]). In comparison, the laser irradiance in the current investigation reaches 108 W cm2. It is expected that the higher laser irradiance will increase the plume temperature during LSP above the first ionization temperature. As a result, a single-ionized plasma plume is formed above the irradiated surface [40]. This
2
Thermal stresses [GPa]
0
t = 30 ns p
-2
z = 100 nm
-4 Axial thermal stress Radial thermal stress Azimuthal thermal stress
-6
-8 0
0.2
0.4
0.6
0.8
1
Radius [mm] Fig. 7. Radial tensile (positive) and compressive (negative) stresses in direct irradiation/ablation for a ramp-up laser pulse at z ¼ 100 nm, beam radius r0 ¼ 1 mm, t ¼ 30 ns, Pave ¼ 50 W.
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plasma plume attenuates the energy transfer from the laser beam to the substrate surface. Also, Fig. 5 identifies three concentric thermal regions: (i) a central region ranging from the depth z ¼ 0 to 100 nm where the temperature exceeds the boiling temperature of stainless steel, (ii) a middle region in which the temperature is between the melting and boiling temperatures surrounding the central regions, and (iii) a peripheral region where the ambient material temperature does not exceed the meting temperature. The cross-sections of the two inner regions decrease in the positive z-direction. The material in the central region is removed by ablation. Following the Hertz–Knudsen formula, some material will be evaporated from the surface of the liquid pool in the middle region. Since the evaporation rate is proportional to the surface temperature, the evaporation rate is higher for smaller values of r: Fig. 6 shows the thermal stresses at the substrate surface at the radius r ¼ r0 ¼ 1 mm for a ramp-up pulse. It can be seen that a shock wave is generated during the first nanosecond. As a result, the radial, axial and azimuthal thermal stresses in the material rise sharply during this first nanosecond. The stresses in the radial and axial directions are compressive (negative) whereas the azimuthal stress is tensile (positive). After approx. 26 ns the material has been melted and the stresses vanish. At a depth of z ¼ 100 nm, the thermal stresses appear at the melt pool radius
4 r = 1 mm
Pressures and stresses (GPa)
2
z=0
0
0 -2 -4
Thermal stresses
-6 -8
Energy deposition
Axial
Axial
Azimuthal
Radial
Radial -10 -12
0
5
10
15
20
25
30
Irradiation time [ns] Fig. 8. Temporal tensile (positive) and compressive (negative) stresses in direct irradiation/ablation for a ramp-down laser pulse at z ¼ 0; beam radius r0 ¼ 1 mm, t ¼ 30 ns, Pave ¼ 50 W.
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rm ¼ 0:6 mm (Fig. 7). It can also be seen that the azimuthal stress changes from compressive (negative) to tensile (positive) for increasing radius. In the case of a ramp-down pulse, the irradiated surface is melted after 8 ns (Fig. 8). The maximum values of the radial, axial and azimuthal stresses are very close to the values for the ramp-up pulse; also, the radial and axial stresses are compressive and the azimuthal stress is tensile. Thermal stresses appear at a depth of z ¼ 100 nm at the radius rm ¼ 0:8 mm (Fig. 9). The radial and axial stresses are compressive and the azimuthal stress is tensile throughout the remaining irradiated spot (0:8oro1 mm). No surface melting occurs for the rectangular laser pulse (Fig. 10) at the laser beam radius after 30 ns. As in the case of the ramp-up laser pulse, a shock wave is generated during the first nanosecond of irradiation. The radial, axial and azimuthal stresses assume the same values and directions as in the case of the other pulse shapes. At the end of the laser pulse, the melt pool radius at a depth of z ¼ 100 nm is rm ¼ 0:63 mm (Fig. 11). The axial, radial and azimuthal thermal stresses exhibit similar trends as in the case of the ramp-up laser pulse. The axial and radial thermal stresses are compressive and the azimuthal thermal stress changes from compressive to tensile for rm oror0 : It is noteworthy that the behavior of the thermal stresses for the ramp-up and rectangular pulses at a depth of z ¼ 100 nm appears to be very similar. In both cases the melt pool extends to about the same value of rm (0.63 mm for the rectangular pulse and 0.6 mm for the ramp-up pulse), the peak values of the axial and radial 2
Thermal stresses [GPa]
0
= 30 ns -2 z = 100 nm -4 Azimuthal Axial Radial
-6
-8 0
0.2
0.4
0.6
0.8
1
Radius [mm]
Fig. 9. Radial tensile (positive) and compressive (negative) stresses in direct irradiation/ablation for a ramp-down laser pulse at z ¼ 100 nm, beam radius r0 ¼ 1 mm, t ¼ 30 ns, Pave ¼ 50 W.
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Pressures and stresses (GPa)
5
0
-5
-10 Thermal stresses Axial Azimuthal Radial
-15 0
5
10
Energy deposition Axial Radial
15
20
25
30
Irradiation time [ns] Fig. 10. Temporal tensile (positive) and compressive (negative) stresses in direct irradiation/ablation for a rectangular laser pulse at z ¼ 0; beam radius r0 ¼ 1 mm, t ¼ 30 ns, Pave ¼ 50 W.
thermal stresses are 6.5 and 5 GPa, respectively, and the value of the axial thermal stress decreases more rapidly than the value of the radial stress. Also the azimuthal thermal stress first increases to 1.5 GPa (compressive) and then decreases with approximately the same gradient, as does the axial thermal stress. The azimuthal thermal stress changes its sign from negative (compressive) to positive (tensile) at r ¼ 0:8 mm and reaches a value of 1.2 GPa at r ¼ r0 : In the case of the ramp-down pulse, the azimuthal thermal stress is always tensile and the radial thermal stress always exceeds the axial thermal stress for rm oror0 : Also, the melt pool radius is larger in this case (rm ¼ 0:8 mm) than for the ramp-up or rectangular pulse. This distinctly different trend in the results can be explained from the point of view of the control volume introduced earlier (a disc of depth Dz and cross-section pr2 ). For a positive rate of change of internal energy, the temperature inside the control volume will increase. Therefore, the possibility for material removal due to heating and ablation are much larger for the ramp-down pulse than for the other two cases, as illustrated earlier. This explains why the melt pool radius is larger for the ramp-down pulse. The radial and axial thermal stresses should be decreasing quickly for a related reason, that is, when the internal energy in the control volume is increased, less energy is conducted radially and axially into the material. Therefore, the temperature gradients in the r- and z-directions are small and the resulting axial thermal stress is smaller than in the case of the ramp-down pulse (Figs. 7, 9 and 11).
T. Thorslund et al. / Optics and Lasers in Engineering 39 (2003) 51–71
66 2
Thermal stresses [GPa]
0
-2
Azimuthal Radial Axial
-4
-6
-8 0
0.2
0.4
0.6
0.8
1
Radius [mm]
Fig. 11. Radial tensile (positive) and compressive (negative) stresses in direct irradiation/ablation for a rectangular laser pulse at z ¼ 100 nm, beam radius r0 ¼ 1 mm, t ¼ 30 ns, Pave ¼ 50 W.
During the laser–material interaction, the recoil pressure on the molten material surface displaces the melt away from the center of the laser beam, in a radial direction [41]. The pressure in the liquid phase exerts a radial force on the surrounding solid material. This radial force is balanced by a compressive (negative) azimuthal thermal stress in the solid phase. In the solid phase, a large stress gradient ds=dr balances the radial pressure in the liquid pool. If the laser irradiance is high towards the end of the laser pulse (as the case for the ramp-up and rectangular pulses), a large compressive azimuthal thermal stress is also developed. However, at farther radial distances from the laser beam center, this compressive azimuthal thermal stress turns into a tensile azimuthal thermal stress, as shown in Figs. 7 and 11. This azimuthal stress syy is a combination of radial and thermal stresses as shown in Eq. (17). The radial stress (which decreases as 1=r2 ) at any point in the medium may be assumed to be due to a hot core of materials. When r is small both radial and thermal stresses contribute to the azimuthal stress resulting in its negative value (which corresponds to the compressive stress). For large values of r; that is when one is outside of the hot core, the contribution of the radial stress decreases significantly, resulting in the negative value of the azimuthal stress (which corresponds to the compressive stress in the material). From a physical point of view, for small values of r the atoms are displaced by Dr which is a combination of radial and thermal expansions. As the value of r increases, the radial expansion due to radial stress diminishes. The tensile thermal expansion becomes dominant in this case and the
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0
Pressure (GPa)
-10
-20 z=0
τ = 30 ns -30 Ablation -40
Plasma
-50 0
0.2
0.4
0.6
0.8
1
Radius [mm] Fig. 12. Axial compressive (negative) stresses for confined geometry ablation for a ramp-up laser pulse at z ¼ 0; beam radius r0 ¼ 1 mm, t ¼ 30 ns, Pave ¼ 50 W.
resulting azimuthal stress is tensile. At large distances from the laser beam center, the temperature T ðr; z; tÞ decreases to zero, resulting in zero thermal stress. The plasma pressures are considerably larger for ablation occurring under a confined geometry. Figs. 12–14 shows the compressive axial thermal stress on the substrate surface for a ramp-up pulse. As expected the maximum surface pressure is much higher (45 GPa) than in the case of direct irradiation/ablation (11 GPa). No ablation takes place for r > ra ¼ 0:85 mm. Similar result is found for the rectangular pulse, however, the maximum surface pressure is much smaller (19.5 GPa). The radial limit for the occurrence of ablation take place is ra ¼ 0:84 mm. In the case of the ramp-down pulse, the maximum plasma pressure is of the same magnitude (46 GPa) as in the case of the ramp-up pulse. Also, the material ablation is a much more confined effect, extending to a radius of ra ¼ 0:41 mm for the ramp-down laser pulse.
5. Conclusions Mathematical models have been developed to calculate the temperature distributions and the thermal stresses in three dimensions for direct laser
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0
Pressure (GPa)
-5
z=0
-10
τ = 30 ns
Ablation
-15
Plasma
-20 0
0.2
0.4
0.6
0.8
1
Radius [mm] Fig. 13. Axial compressive (negative) stresses for confined geometry ablation for a rectangular laser pulse at z ¼ 0; beam radius r0 ¼ 1 mm, t ¼ 30 ns, Pave ¼ 50 W.
irradiation/ablation and ablation in a confined geometry. In the case of direct irradiation/ablation, the ramp-down pulses generate higher mean temperatures in the workpiece than in the cases of ramp-up and rectangular pulses for the same energy per pulse and pulse length. The temperatures for all pulse shapes and shock geometries are strongly dependent on the absorption of the laser energy. Pulses with short rise are favorable for laser shock processing because higher surface pressures can be obtained. The models confirm that higher pressures are achieved for laser shock processing under a confined geometry.
Acknowledgements Thord Thorslund visited The University of Central Florida as an exchange student during his MS study at the University of Uppsala, Sweden. He also acknowledges the assistance received from the University of Dalarna and would like to thank everybody who supported this project in any way.
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0
Pressure (GPa)
-10
-20 z=0
τ = 30 ns -30 Ablation -40
Plasma
-50 0
0.2
0.4
0.6
0.8
1
Radius [mm] Fig. 14. Axial compressive (negative) stresses for confined geometry ablation for a ramp-down laser pulse at z ¼ 0; beam radius r0 ¼ 1 mm, t ¼ 30 ns, Pave ¼ 50 W.
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