WAVE-SOLID INTERACTIONS IN SHOCK INDUCED DEFORMATION PROCESSES

three conservation equations, equation of state thatcan be expressed in terms of specific internal energyas a function of pressure and density forhydrodynamic behavior of material, and the elasticplastic constitutive relation for deviatoric behavior.The calculation of mechanical behavior of solidsunder shock loading is usually made using the threeconservation equations in integral forms:d(1)㺦dV 0dt Vd(2)㺦u dV 㺦 ij u j dSdt V iS1d(3)(E u 2 )dV 㺦ij n jk u k dS㺦2dt VSwhere V is the volume of a cell, S is the surface thatcovers this volume, nij is the unit vector normal to thissurface, ui are the velocity components, and ıij are thestress components. The elasto-plastic behavior shouldalso be considered except the hydrodynamic change ofvolume or density. The stress-tensor components aredivided into a hydrostatic equation of state and anelastic-plastic constitutive model. The stresscomponents ij can be written as:ij P s ij(4)where P is the hydrostatic pressure and sij is thedeviatoric stress components.A commonly used equation of state for solids is theMie-Grüneisen equation of state. The Mie-Grüneisenequation of state which establishes relationshipbetween pressure P and internal energy E withreference to the material Hugoniot curve, was used:(5)P -P H 0 0 ( E - E H )where PH and E H are the Hugonoit pressure andinternal energy,00is a material constant andrepresents the initial state density.The Hugonoit curve is described by the linear relationbetween the shock velocity U and particle velocity uwith coefficients from experimental data:(6)U C 0 Suwhere the constant C0 is the sound speed at zeropressure, and the material constant s has a valuebetween 1.0 and 1.7 for most metals.Combining equation (6) with the Rankine-Hugonoitjump conditions [9], the Hugonoit pressure andinternal energy can be obtained as:C0 0PH (7)(1 - S ) 2EH PH2(8)0and substituting equation (7) and (8) into equation (5)yields:C0 00) 0 0E(9)P (1 2(1 - S ) 2where 1-0, andis density. The equation (9) isthe final form of equation of state to be used in thissimulation. In the following numerical modeling ofshock-solid interactions, work hardening, strain rateand pressure effects on yield strength are consideredwhile temperature is taken as room temperature. Thisis reasonable because only the coating is vaporized andminimal thermal effects are felt by the sample. Thesolid target is assumed to be isotropic.Numerical ModelingFEM Explicit & Implicit plicit and ABAQUS/Standard, werecombined to perform the LSP and LPF simulation.These two solvers accomplish different calculationsduring this simulation. The ABAQUS/Explicit is anon-linear explicit time integration finite element code,which is especially well suited for solving high speed,short duration, highly dynamic events that requiremany small time step increments to obtain a highresolution solution. One important issue about thesimulation of LPF and LSP is the balance between ashort time for dynamic shock-solid interaction (2 3times of the laser pulse duration) and a much longerrelaxation time (up to 1 second) to reach a stabilizedmechanical state. So the ABAQUS/Explicit code isfirst applied to simulate the dynamic shock-solidinteraction process. But the ABAQUS/Explicit methodis only conditionally stable and very small time step isrequired. Therefore, the second step is to simulatematerial relaxation in ABAQUS/Standard. As soon asthe calculation of the highly dynamic shock-solidinteraction process is completed in ABAQUS/Explicit,the obtained intermediate stress and strain state istransferred into ABAQUS/Standard to simulate thematerial relaxation and get the residual stress filed instatic equilibrium. In the ABAQUS/Explicit step, asmall amount of damping in a form of bulk viscosity( 0.06) is included in the calculation to limit numericaloscillations.Strain Rate Dependent ConsiderationsLaser Materials Processing ConferenceICALEO 2005 Congress ProceedingsPage 481

In LPF and LSP, the target is subjected to very strongshock pressures ( 1 GPa), the interaction time is veryshort ( 200 ns), and the strain rate is very high( 100,000 s-1). It is necessary to consider the effect ofhigh strain rate on the flow behavior of metals.Johnson, et al.[10] first included the influence of strainrate H into their working hardening model. ButJohnson's model22 could not cover the high strain rate(greater than 10-6 s-1) in LPF and LSP. It did not alsoconsider pressure effects, which are very important inlaser shock processing. Steinberg's model [11] isapplicable to ultrahigh pressures but it did not considerrate dependent effects. It was found that the ratedependent effects cannot be neglected for shockpressures below 10 GPa. In laser shock processing, thepressure involved is fairly high ( 1 GPa) but less than10 GPa.For laser shock processing, therefore, both the strainrate effects and ultrahigh pressure effects on materialyield stress need to be considered. A prior research5has included the strain rate (even above 106 s-1) effectsand ultrahigh pressure effects on material yield stress,and the obtained dynamic yield stress data were usedin this research.ExperimentsLaser and Sample PreparationsAll experiments were made by a frequency tripled Qswitched Nd:YAG laser with a wavelength of 355nmin TEM00 mode. The pulse duration was 50 ns, andpulse repetition rate could vary between 1 KHz to 20KHz. Laser beam diameter is 12 microns and laserintensity was varied from 2 to 6 GW/cm2.For LPF, copper stripes with thickness of 100μm wereused as samples. These stripes were cut to20mmu3mm using a wire electric discharge machine(EDM) to avoid inducing stress and strain, and thenheat treated and electro-polished to relieve residualstress. Then, a thin layer of high vacuum grease (about10 microns) was spread evenly on the polished samplesurface, and the coating material, aluminum foil of 16microns thick, which was chosen for its relatively lowthreshold of vaporization, was tightly pressed onto thegrease. These stripes were clamped at both ends,leaving 10 mm length in the middle unsupported forLPF experiments. Caution was exercised to prebending effects and to ensure these stripes remain flatduring these steps .For LSP, well-annealed pure aluminum samples in thedimension of 15mmu10mmu5mm were used. Thesample preparation was the same as introducedbefore3.For both LPF and LSP, the laser process procedure issimilar. The samples were placed in a shallowcontainer filled with distilled water around 3 mmabove the sample top surface. A series of laser pulseswere applied along the width direction (the dimensionof 3 mm for LPF and the dimension of 10 mm forLSP) with 25 Pm spacing between adjacent pulses.This forms a straight shocked line. Pulse energies, 226and 280PJ, corresponding to laser intensities of 4.0 and4.95GW/cm2, were used for LPF and LSP,respectively. After shock processing, the coating layerand the vacuum grease were solved in Acetonesolution, and shock induced deformation and residualstresses on the samples were measured. The conditionsare summarized in Table 1.Table. 1 Samples and Experimental Conditions forLPF and LSP.LaserPulseSizeIntensityMaterial3Energy(mm )2(GW/cm )4.95LPFCu280 PJ10u3u0.14.0LSPAl226 PJ15u10u5Deformation & Residual Stresses MeasurementsBefore and after LPF, the curvatures of the stripeswere measured by a profilometer, and the bendingcaused by LPF is the net effect, as shown in Fig. 3.After LSP, the dented surface was measured usingAtomic Force Microscopy (AFM).The residual stresses were measured by synchrotron Xray diffraction. Synchrotron X-ray diffraction canmake accurate residual stress measurement of a highspatial resolution because it provides high brightnessX-ray beams. The extremely high brightness X-raybeams from synchrotron radiation sources arenarrowed down and then focused to micron orsubmicron spot sizes using X-ray optics such asFresnel Zone Plates (FZP) or tapered glass capillaries,and either white beam or monochromatic X-rays canbe used. The tapered capillary tube is aligned to takein the X-ray beam from the synchrotron beamline, andsuccessively focuses the beam to a small spot size bytotal external reflection. Both small spot size andincreased intensity are desired in X-ray diffraction witha micron-level spatial resolution.The samples are mounted on a translation stage withpositioning accuracy of 1Pm in the x and y directionsin the sample surface. Monochromatic synchrotron radiation at 8.0 KeV ( 1.54024 A ) is used, since itis smaller than the K absorption edge for Al and CuLaser Materials Processing ConferenceICALEO 2005 Congress ProceedingsPage 482

which are 8.98KeV and 8.3KeV so that thefluorescence radiation would not be excited.Multiple measurement points were chosen along a lineperpendicular to the shocked line. The spacingbetween adjacent measurement points starts from 20Pm (when 100 Pm away from the center of theshocked line) and reduces to 5 Pm within 20 Pm fromthe center of the shocked line in order to spatiallyresolve the residual stress. At each position, thecorresponding X-ray diffraction profile is recorded andrepeated for each shocked line. For LSP, only theshocked surface was measured while for LPF, theresidual stresses measurements on both top and bottomsurfaces were conducted.Results and DiscussionsModel ValidationComparison with experimental resultsThe comparison between the measured deformation ofthe copper stripe after LPF and the numerical predictedis shown in Fig. 3. Before laser peen forming, thestripe slightly curves upward with the center of thestripe up by about 5 microns. After LPF, the stripebended upward further, and the shocked area wasraised by up to 10 microns. The numerically predicteddeformation and the experimental are in goodagreement.pops up towards larger diffraction angle. The fullwidth at half maximum (FWHM) of the profile aroundthe shocked line is up to 3 times greater than that of theline profile at 100Pm away from the center. It isknown that when both elastic and plastic strains aresuperposed in plastically deformed metals, diffractionis both shifted and broadened. It is the superpositionthat makes it difficult to evaluate the local strain andresidual stress distribution. However, on the basis of acomposite model [12], local strain and residual stresscan be evaluated for metals under plastic deformationby recognizing that the crystal dislocations oftenarrange themselves in a cell structure after beingsubjected to a shock loading. Following the analysismethod above for each measurement point, thespatially resolved residual stress distributions on bothtop and bottom are shown in Fig. 5. The comparisonsof the residual stress distributions on both top andbottom show the numerically predicted distributionmatches the experimental results very well. Themodeled residual stress distribution and bendinginduced by LPF of copper stripe are also shown in Fig.6. When high pressure was applied to the copper, theshocked material tended to flow away from theshocked center and caused elongation of the top layerof the stripe, which led the stripe to bend up, andmeanwhile induced compressive residual stress on thebottom surface and, because of spring back and shockcompression, the top surface.16Before LPFAfter LPFNetNumericalZ (um)12Shocked line840012345678X (mm)Fig. 3 Comparisons of deformation after LPF betweenthe experimental and the numerical results.Fig. 4

utilizing high power pulsed lasers to generate shock waves in solid targets, the laser shock technique has led to many investigations, including laser peen-forming (LPF) and laser shock peening (LSP), shown as in Fig. 1. Laser-generated shock waves result from the expansion of a high pressure plasma caused by a pulsed laser.