κt π = (5) T surrface k BASELINE CASE

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1 II. BASELINE CASE PRACICAL CONSIDERAIONS FOR HERMAL SRESSES INDUCED BY SURFACE HEAING James P. Blanhard Universi of Wisonsin Madison 15 Engineering Dr. Madison, WI ABSRAC Rapid surfae heaing an indue large sresses in solids. A relaivel simple model, assuming full onsrain in o dimensions and no onsrain in he hird dimension, an adequael model sresses in a ide varie of siuaions. his paper derives his simple model, and suppors i ih rieria for is validi. Phenomena ha are onsidered inlude non-zero peneraion dephs for he hea deposiion, spaial non-uniformi in he surfae heaing, and elasi aves. Models for eah of hese ases, using simplified geomeries, are used o develop quaniaive limis for heir appliabili. he base ase onsiders a solid resrained from deformaion in o dimensions and ihou onsrain in he hird dimension. he lak of onsrain in he hird dimension is a resul of he free surfae. he full onsrain in he oher dimensions assumes ha he hermal field is shallo relaive o he deph of he sruure, so ha old maerial belo he surfae resrains moion of he heaed maerial in ha shallo laer. o develop a model for sresses in his siuaion, e begin ih he sress-srain relaions: λ λ λ ( ε ε ε µε ( 3λ µ α ( ε ε ε µε ( 3λ µ α ( ε ε ε µε ( 3λ µ α here λ is he Lame onsan, µ is he shear modulus, and α is he hermal epansion oeffiien. his equaion assumes ha is he emperaure differene from a sress-free referene emperaure. Under he assumpions disussed above, e impose ε ε ( From he las of hese ondiions, ombined ih Eq. 1, e find he srain in he direion o be ε 3λ µ λ µ α (3 (1 I. INRODUCION his an hen be subsiued ino Eq. 1 o find he ransverse sresses, hih an be rien as: Rapid surfae heaing an indue large sresses in Eα solids, possibl leading o surfae roughening, ielding, or fraure. he deerminaion of he sresses for a given maerial and se of loads an be quie diffiul, requiring a ime-dependen, hree dimensional analsis. For man ases, hough, a relaivel simple model, assuming full onsrain in o dimensions and no onsrain in he hird dimension, an adequael model he peak sress. his paper derives suh a model, and suppors i ih rieria for is validi. Phenomena ha are onsidered inlude non-zero peneraion dephs for he hea deposiion, spaial non-uniformi in he surfae heaing, and elasi aves. Models for eah of hese ases, using simplified geomeries, is used o develop quaniaive limis for heir appliabili. hermal aves are an addiional phenomenon ha an be of onern for ver shor pulses, bu his effe is lef for fuure ork. (4 1ν here E is he elasi modulus and ν is Poisson's raio. o omplee he model e need an esimae of he surfae emperaure indued b he surfae heaing. Assuming uniform heaing applied on a half-spae, he surfae emperaure is 1 q surrfae k κ (5 here q is he surfae hea flu, k is he hermal onduivi, and κ is he hermal diffusivi. Combining

2 Eqs. 4 and 5 provides he folloing model for sresses indued b rapid surfae heaing: qeα κ 1ν k (6 ( his resul provides a baseline esimaion of he surfae sresses indued b rapid surfae heaing. I assumes spaial uniformi of he applied hea, no volumeri heaing belo he surfae, and i ignores boh elasi and hermal aves. Fig. 1 provides a plo of his raio as a funion of ζ. he orresponding sresses ould follo he same urve. From his urve one an see ha he effe is less han 1% for ζ>8 and less han 1% for ζ>6. his laer resul as deermined using he asmpoi resul: 1 R ~ 1 (11 ζ ζ III. DEPOSIION BELOW HE SURFACE Mos surfae heaing auall deposis hea as volumeri heaing ihin a hin laer near he surfae. A pial model for volumeri heaing resuling from energ impinging on a surfae is Q γ (7 Ae Where A is a onsan and γ is he aenuaion oeffiien. o provide he same oal hea inpu as a rue surfae heaing flu q, e mus enfore Aqγ. he emperaure disribuion resuling from volumeri heaing of his pe is 1 q η ζ ierf e kγ ζ e ζ η η erfζ e ζ η ζ η η erfζ ζ here ζ γ κ, represening he raio of he diffusion lengh in ime o he haraerisi deposiion lengh, and ηγ. he surfae emperaure resuling from his soluion is q ζ 1 e ζ surfae erf( ζ kγ (9 he raio of he surfae emperaure from Eq. 9 o he surfae emperaure due o surfae heaing (Eq. 5 is (8 Figure 1: Raio of surfae emperaures due o volumeri heaing and equivalen surfae heaing IV. Spaial Non-Uniformi in Surfae Heaing Quie ofen he heaing disribuion over he surfae is non-uniform. For eample, man lasers produe a gaussian heaing disribuion hen he laser is normall iniden on a fla surfae. o eplore his effe, e onsider he soluion b Heor and Henarski. his gives he peak sress due o laser heaing ih a gaussian shape on a half-spae as (a rz: rr here G h( G 1 ν 1 erf erf 1 z G d d (1 (13 R 1 [ 1 e ζ erf( ζ ] (1 ζ

3 G z and 4 ( erf ( ep 4 ep 4 ( (14 h. (15 In hese equaions, he sarred quaniies are all dimensionless, aording o: K 4κK 4κK 4 1 ij ( ν ( 1 ν k K q αµ q k K ij (16 Here and are inegraion variables, K is a measure of he idh of he gaussian laser profile on he surfae, and q is he peak surfae heaing. Puing Eqs ogeher gives rr ep 4 ( 1 ν ( erf ep 4 Carring ou his inegraion provides: rr (1 ν 4 4(1 ν ( d d 1 [ 1 (1 ν ] an ( (17 o ompare his o our simple analial soluion, e an rie he sress from Eq. 6 using he dimensionless variables in Eq. 16, giving base 8 Hene, he raio of he sress due o a gaussian heaing profile o ha of he uniform heaing profile is R (1 ν 4 1 (1 ν 1 an [ ] ( 1 (1 ν (19 ( his raio is ploed in Fig. as a funion of he dimensionless ime for several values of he Poisson raio. A similar approah an be aken ih he emperaure. Heor and Henarski give he emperaure as 1 d 4 (1 Carring ou his inegral gives ( 1 an (1 ( In he dimensionless unis given in Eq. 16, he onedimensional surfae emperaure from Eq. 5 beomes base (3 Hene, he raio of he emperaure due o he gaussian heaing profile o ha of he uniform profile is R ( 1 an (4 (18 his raio is ploed in Fig. 3.

4 Here is he ave speed and ρ is he densi of he solid. Wih hese definiions, he governing equaions hen beome: φ φ ξ ˆ ξ ˆ φ (6 Figure : Raio of sresses due o gaussian and uniform heaing profiles I is assumed here ha he onl non-zero displaemen is perpendiular o he surfae of he half-spae. ha is, u u z.he iniial ondiions are suh ha all emperaures, sresses, and ime derivaives are noneisen. he boundar ondiions are ha he emperaures and sresses vanish a equals infini, hile a he surfae φ 1 ξ ˆ (7 he soluion for he dimensionless emperaure is given b 1 φ ξ ep 4 ξ ξ erf (8 Figure 3: Raio of sresses due o gaussian and uniform heaing profiles V. Elasi Waves o model elasi aves, e mus inlude inerial erms in he sress equaions. o esimae heir effes, onsider hermoelasi deformaion of a half-spae, ih denoing he perpendiular disane from he surfae. Folloing Sernberg and Chakravor 3, one an define he folloing dimensionless variables ξ a κ a κ a φ (1 ν k ˆ (1 ν α qaµ k qa (5 (1 ν µ (1 ν ρ and he sresses an be found o be 1 ξ ˆ ep( ξ 1 ep(ξ erf and ξ erf erf ˆ ˆ ν ˆ (1 ν φ z ( ξ H ( ξ (9 (3 his omplees our soluion for he sresses indued b surfae heaing on a half-spae. Sine he longiudinal sress ( ˆ is zero a he surfae, he ransverse sress ( ˆ a he surfae is given b ˆ ˆ (1 ν φ z (31 or

5 ˆ ˆ ν z (1 (3 sress a he same dimensionless ime. I an be seen ha beond a dimensionless ime of approimael 4, he surfae sress eeeds he sress a he ave peak. he peak sress in he ave ours a ξ. Subsiuing his ino Eqs. (9 and (3 gives ˆ ep( erf ν 3 ˆ 1 ep( erf (1 ν ep 4 erf erf erf (33 For long imes he longiudinal sress approahes 1, hile he ransverse sress approahes (-ν. pial ave shapes are shon in Fig. 4, hih plos he o dimensionless sresses as a funion of disane from he surfae. hese resuls are given for dimensionless imes of.5, 1, and 1. As one ould epe, he sresses are all ompressive, and he peak sress in he ave ours a ξ. Eep a earl imes, he ransverse sress peaks a he surfae beause ha's here he emperaure peaks. A earl imes, here is a loal peak in he ransverse sress here he ave fron lies, and a his poin he ransverse sress is less han he longiudinal sress. Figure 5: Sresses a surfae and a ave fron he raio of he longiudinal sress a he ave peak o he surfae sress is given b: R 3 1 ep( erf erf ν 4(1 (34 his raio is ploed in Fig. 6. For large imes, his raio is. As approahes, his raio approahes 1.1/(1-ν. Figure 4: Dimensionless sresses as a funion of deph for differen imes. he peak in he longiudinal sress ours a ξ, hile he peak ransverse sress ours a he surfae (eep a shor imes. hese peaks are ploed in Fig 5, hih gives boh sresses a ξ along ih he surfae Figure 6: Raio of peak longiudinal sress o surfae sress

6 VI. Conlusions For mos siuaions, a simple formula provides an adequae represenaion of he hermal sress indued in a rapidl heaed solid. When his formula is no valid, here are ofen simple analial represenaions of hese sresses. his paper provides hese formulas, along ih heir regions of validi. Aknoledgemens his ork as sponsored b he Naval Researh Laboraor in suppor of he High Average Poer Laser program. Referenes 1. H. Carsla, and J. Jaeger, 1959, Conduion of Hea in Solids, Oford Clarendon Press, p L. Heor and R. Henarski, 1996, "hermal Sresses Due o a Laser Pulse: Elasi Soluion," J. Appl. Meh., 63, E. Sernberg, and J. Chakravor, 1959, "On Ineria Effes in a ransien hermoelasi Problem," J. Appl. Meh., 6, 53.