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Surface Treatment by Laser with High Thermal Gradients
Surface Treatment by Laser with High Thermal Gradients G. Ricciardi (2).M. Cantello, C. Rivela; lstituto R. T. M., Vico Canavese/ltaly Received on January 20,1992
SUMMARY A theoretical study of h e a t i n t e r a c t i o n o f l a s e r r a d i a t i o n has b e e n carrird out. T h i s szudy b a s s u p p l i e d a n a t h e m a t i z a l a d i m e n s i o n a l n o d e l a b l e o f d e s c r i S i n g t h e t h e r m a l f i e l d v e r s u s time d e p t h i n a v e r y g e n e r a ; 'day. The m c e l h a s b e e n c n e c k e d w i t h e x p e r i m e n t a l t e s t s c n some f e r r o u s a l l o y s 3y s e a n s o f C32 l a s e r . S ? e c i f i c t e s t s u s i n g v e r y high Zradients, g o t from t h e m o d e l , k a v e shown t h e f o r m a t i o n og ir.novat i v e ne t a l Luraic a 1 struc t a r e s .
KEYWORDS: L a s e r , s u r f a c e t r e a t m e n t , m a t h e m a t i c a l model.
INTRODUCTION V a r i o u s c a l c u l u s programmes h a v e b e e n d e v e l o p e d i n t h e R.T.M I n s t i t u t e w i t h t h e aim o f produc:ng an i n s t r u m e n t f o r u n d e r s t a n d i n g t h e phenomena of h e a t t r a n s m i s s s i o n by c o n d u c t i o n i n material s u b j e c t t o laser radiation. These include simple ( n o n d i m e n s i o n a l ) programmes managed by a PC, and more complex o n e s ( t r i d i r n e n s i o n a i ) 'which c a n o n l y b e managed by a more powerfGl c o m p u t e r . Following t h e d e v e l o p m e n t p h a s e o f t h e m a t h e m a t i c a l model. t h e e x p e r i n e n t a l p h a s e commenced and aimed t o s t u d y t h e b e h a v i o u r and r e s u l t a n t m e t a l l u r g i c a l s t r u c t u r e i n common s t e e l s s u c h as C40 when s u b j e c t t o h i g h radiation. The thermal g r a d i e n t s from l a s e r v a l i d i t y o f t h e model and i t s c o r r e s p o n d e n c e t o experimental v a l u e s were t e s t e d by s e l e c t i n g a p p r o p r i a t e o p e r a t i v e p a r a m e t e r s which c a n p r o d u c e h i g h s p e c i f i c powers i n b r i e f t i m e i n t e r v a l s . MATHEMATICAL MODEL To o b t a i n t h e e q u a t i o n which g i v e s t h e t e m p e r a t u r e p a t t e r n a t v a r i o u s d e p t h s and as a f u n c t i o n o f time, t h e F o u r i e r l a w o f h e a t c o n d u c t i o n was u s e d as t h e s t a r t i n g p o i n t . With some s i m p l i f y i n g h y p o t h e s e s , i t t a k e s t h e f o l l o w i n g form:
= d e p t h a l o n g t h e d i r e c t i o n o f flow (cm) = t h e r m a l c o n d u c t i v i t y o f material (W/cm°C) = t e m p e r a t u r e ('C) A = heat generated i n t e r n a l l y (W/cm3) t = time ( s ) Cp = s p e c i f i c h e a t o f material ( J / g r ° C ) p = d e n s i t y (g/cm3 )
z k T
T h i s e q u a t i o n g i v e s s o l u t i o n s which v a r y , d e p e n d i n g on w h e t h e r t h e w o r k p i e c e S e i n g examined c a n b e c o n s i d e r e d a s e m i - i n f i n i t e or an i n f i n i t e t h i c k n e s s and i t c a n be o b t a i n e d using the following hypotheses: - c o n s t a n t h e a t p a r a m e t e r s o f t h e m a t e r i a l and c o e f f i c i e n t of a b s o r p t i o n - i n i t i a l t e m p e r a t u r e o f t h e workpiece O°C - n e g l i g i b l e phase changes - constant material density - homogenous and i s o t r o p i c material - u n i f o r m d i s t r i b u t i o n o f laser e n e r g y - monodimensional f l o w - no i r r a d i a t i o n or h e a t e x c h a n g e w i t h e x t e r i o r . T h e s e h y p o t h e s e s may be c o n s i d e r e d t o b e s a t i s f i e d i n v i e r o f t h e h i g h s p e e d w i t h which t h e t h e r m a l process takes p l a c e , with the exception of the first h y p o t h e s i s , s i n c e t h e s e parameters vary during the treatment. A further simplification d e r i v e s from t h e f a c t t h a t t h e l a s e r r a d i a t i o n is a b s o r b e d i n t o a t h i n l a y e r o f material so i t c a n be assumed t h a t t h e volume o f the heat source g e n e r a t e d i s z e r o , t h a t is A = O . By i n d i c a t i n g t h e i n c i d e n t laser f l o w on t h e t e s t p i e c e w i t h F, t h e s u r f a c e r e f l e c t i v i t y w i t h R and and c o n s i d e r i n g t h a t o n l y p a r t o f t h e f l o w Fo=F ( 1 4 ) p e n e t r a t e s the m a t e r i a l , the equation giving t h e heat f i e l d i n t h e m a t e r i a l as a f u n c t i o n o f time and d e p t h f o r a s e m i - i n f i n i t e t h i c k n e s s c a n be obtained:
i n which: 6 is t h e l e n g t h o f t h e r m a l d i f f u s i v i t y o f t h e material ( c m ) D * = 2 G F 5 i s t h e l e n g t h of thermal d i f f u s i v i t y i n t h e c o o l i n g p h a s e (cm). t = time r e q u i r e d t o c o n c i u d e t h e h e a t f l o w a r i s i n g from t h e l a s e r r a d i a t i o n d = k/pCp i s t h e t h e r m a l d i f f u s i v i t y ( c d / s ) The l e n g t h o f t h e t h e r m a l d i f f u s i v i t y C i n s e r t e d i n t o t h e e q u a t i o n i s a t y p e of h e a t h o r i z o n i n t h a t i t i n d i c a t e s t h e d e p t h a t which t h e t e m p e r a t u r e i s l e s s t h a n a t e n t h o f t h e s u r f a c e t e m p e r a t u r e and i f this i s much l o w e r t h a n t h e t h i c k n e s s o f t h e workpiece i n d i r e c t i o n z o f t h e h e a t f l o w , i t i n d i c a t e s t h a t t h e m a t e r i a l c a n be c o n s i d e r e d t o b e of a s e n i - i n f i n i t e t h i c k n e s s . The advantage of t h i s e x p r e s s i o n is t h a t one p a r a m e t e r u s e d is t h e l e n g t h o f i r r a d i a t i o n K t o which correspond well-defined values of t h e v a r i a b l e s o f f i e l d such a s t e m p e r a t u r e , l e n g t h of t h e r m a l d i f f u s i v i t y . e t c . ) which g i v e s a s i n g l e and unmistakable reference f o r the process. D = 2
Normalisation N o r m a l i s a t i o n w a s t h e n c a r r i e d o u t in o r d e r t o i d e n t i f y t h e p a r a m e t e r s w i t h a p h y s i c a l meaning a n d t o b r i n g t h e e q u a t i o n i n t o more g e n e r a l u s e . T h i s Nas a c h i e v e d by c o r r e l a t i n g t h e a v e r a g e t e m p e r a t u r e t o t h e maximum t e m p e r a t u r e which may be r e a c h e d d u r i n g t h e t h e r m a l p r o c e s s , and t h e t h i c k n e s s t o t h e l e n g t h o f thermal d i f f u s i v i t y . The n o r m a l i z e d e q u a t i o n is as f o l l o w s :
Tn = T/Tmax t n = t/? zn = z / C p represent the temperature, t h e time and t h e 1 normalized thickness (dimensionaless); Fig. shows a g r a p h o f t h e p a t t e r n o f t h e s e p a r a m e t e r s i n t h e n o r m a l i z e d p l a n e t n , Tn; t h e e q u a t i o n d e s c r i b e s t h e monodimensional h e a t f i e l d f o r any homogenous and i s o t r o p i c m a t e r i a l which m a i n t a i n s c o n s t a n t p h y s i c a l p r o p e r t i e s f o r each v a l u e o f t h e i n p u t flow; conversely, Fig. 2 i n t h e a p p e n d i x shows t h e b e h a v i o u r i n t h e p l a n e zn,Tn. By d i f f e r e n t i a t i n g (3) w i t h r e s p e c t to t h e n o r m a l i z e d t i m e and g i v i n g t h e d e r i v a t i v e t h e v a l u e of zero, a f u n c t i o n i s o b t a i n e d which g i v e s t h e curve o f maximum values of t h e normalized 1; t h i s thickness, shown by a d o t t e d l i n e i n F i g . c u r v e s e p a r a t e s t h e u p p e r a r e a , o f c o o l i n g , from t h e a r e a u n d e r n e a t h i n which h e a t i n g o c c u r s . i n which
The above r e f e r s t o s e m i - i n f i n i t e thicknesses but i t c a n n o t b e a p p l i e d when t h e l e n g t h o f t h e r m a l d i f f u s i v i t y c h a r a c t e r i s t i c o f t h e material b e i n g s t u d i e d i s c o m p a r a b l e t o or g r e a t e r t h a n t h e t h i c k n e s s o f t h e workpiece, because t h e lower s u r f a c e o f t h e workpiece reaches a c o n s i d e r a b l e t h e heat t e m p e r a t u r e and as no i o w e r l a y e r s e x i s t , d i f f u s e s towards the t o p , thereby modifying the temperature profile. By means o f boundary c o n d i t i o n s m o d i f i c a t i o n s and some s i m p l i f i c a t i o n s , t h e f o l l o w i n g e x p r e s s i o n i s produced:
(2)
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w i t h I = t h i c k n e s s o f t t e f i n i s h e d s h e e t , i n which of t h e t h e f i r s t term i s t h e sane as t h a t semi-infinite thick sheet whilst the second, e x p r e s s e d as an i n f i n i t e and i n t e r r u p t a b l e s e r i e s zt an a p p r o p r i a t e l y selected value, c a n be i n t e r p r e t e d as t h e c o n t r i b u t i o ? t o t h e t e m p e r a t u r e by t h e s u b s e q u e n t r e f l e c t i o n s o f h e a t b e t w e e n t h e u p p e r and l o w e r f a c e o f t h e w o r k p i e c e . The p r o f i l e of t h e f i n a l t e n p e r a t u r e i s t h e r e s u l t of ',he sum of t5.e i n d i v i d u a l t e m p e r a t u r e p r o f i l e s o b t a i n e d from t h e s i i b s e q u e n t heat r e f l e c t i o n s ; i d e a l l y i t i s as i f t h e temperature p r o f i l e o b t a i n a b l e f o r a s e m i - i n f i n i t e t.*,ic;cness, s i t u a t e d an t h e o u t s i d e of t h e s h e e t . x a s f l i p p e d over t o t h e i n s i d e o f t n i s and t h e i n d i , / i d u a l c o n t r i b u t i o n s 'were t l - e r e f o r e ai,?eS t o g s ' h e r . Even in t h i s c a s e n o r m a l i z a t i o n was c a r r i e d o u t Sy c o r r s l a t i n g the average surface temperature t o t h e however, the ec,uations maximurn temperature ; i l l u s t r a t e d io t h e n o r m a l i s e d p l a n e g i v e a f a m i l y of p a r a m e t r i c a l c u r v e s i n the normalized t h i c k n e s s r a z h e r tilan a s i n g l a Zraph. As c a n b e seen i n Fig. 3 t h e p a t t e r n i n time O f t h e s u r f a c e temperature i n ?he f i n i s h e d s h e e t is similiar t o t h a t o f t h e s e m i - i n f i n i t e t h i c k n e s s f o r a section, c h a r a c t e r i s t i c o f e a c h norn;alized thickness , than it straightens out. 411 t h e s e f a c t o r s are v a l i d i f t h e m a t e r i a l d o e s n o t undergo s o l i d t o l i q u i d phase t r a n s i t i o n s t h a t i s , as l o n g a s t h e r e is no m e l t i n g b u t , when t h e material mel:s and t h e c o n v e c t i v e m o t i o n of t h e molten phase takes o v e r w i t h s u b s e q u e n t r e m i x i n g , In this t h e c o c d i t i o n s assumed p r e v i o u s l y Change. c a s e t h e h e a t i n 3 a n d c o o l i n g g r a d i e n t s above t h e m e l t i n g t e m p e r a t u r e are l o w e r and i t i s n e c e s s a y t o m o d i f y t h e model u s e d a c c o r d i n g l y . EXPERIMENTAL PHASE To v e r i f y the v a l i d i t y o f t h e model and i t s c 3 r r e s p o n b e n c e t o t h e e x p e r i m e n t a l v a l u e s a Series of tssts were c a r r i e d o u t on a commonly u s e d S t e e l , C40, d u r i n g w h i c h t h e o p e r a t i n g p a r a m e t e r s were c s l c u l a t e d t o o b t a i n an i n c r e a s i c g h e a t g r a d i e n t . TP.e s t e e l 'was h a r r ' e n e d and tempered i n a f u r n a c e t o proriuce a more homogenous s t r u c t u r e and t e s t p i e c e s w i t h d i m e n s i o n s 5 0 x 2 0 and i O m m t h i c k were u s e d . S i n c e t > e l e n g t h of t h e r a a l d i f f u s i v i t y k ( w h i c n i s c l o s e l y c o r r e l a t e d t o t h e i n t e r a c t i o n t i m e . which i n t h i s c a s e i s r a t h e r l o w ) i s much l o w e r t h a n t h e t h i c k n e s s of t h e m a t e r i a l t h e model c a n be u s e d f o r 3 semi-infinite thickness. The p a t t e r n of t h e thernal gradient was also studied Sy di'fersntiating t h e n0rma:ized e q u a t i o n ( 3 ) . set o c t a b o v e , w i t h r e s p e c t t o t h e time and s p a c e . The t e s t s were c a r r i e d o u t o n t'wo d i f f e r e n t C02 lasers: t k e C o h e r e n t laser ' w i t h a maxi:nuin power of SOOW, which & i v e s minisum d i m e n s i o n s p o t s ( u p t o 0 . 2 mnin i n d i a m e t e r ) a n d t h e AVCO l a s e r w i t h a maximum power o f 15 kV. The u s e o f two l a s e r s w i t h s u c h d i f f e r e n t power l e v e l s is due to the extreme e x p e r i m e n t a l c o n d i t i o n s d u r i n g which h i g h s p e c i f i c p o w e r s had t o be p r o d u c e d which is o n l y p o s s i b l e by means o f v e r y tiny spots. The C o h e r e n t l a s e r , u r , l i k e t h e AVCO e n a b l e s small s p o t s t o b e p r o d u c e d , up t o 0 . 2 mm i n dianeter. F u r t h e r m o r e some of t h e t e s t s rust be c a r r i e d o u t a t h i g h s p e e d s w h i c h c a n n o t be a c h i e v e d w i t h t h e It was t h u s ?ranslating tables available. 4) with a n e c e s s a r y t o p l a c e a motor (Fig. r o t a t i n g t a b l e on t h e t r a n s l a t i n g t a b l e so as t o g i v e a maximum s p e e d o f r o t a t i o n o f 3000 rpm. which i n V i e w o f t h e d i m e n s i o n s of t h e r o t a t i n g t a b l e means a maximuia s u r f a c e v e l o c i t y o f 40 m / s e c . Before u n d e r g o i n g iaser r a d i a t i o n t h e t e s t p i e c e s were c o a t e d w i t h a l a y e r o f c o l l o i d a l g r a p h i t e : p r e l i m i n a r y t e s t s have p o i n t e d t o t h e use of an O p t i m a l C o a t i n g o f 2 9 t m and c a l o r i m e t r i c t e s t s h a v e ( i n d i c a t e d by Fo) shown an e n e r g y a b s o r p t i o n r a n g i n g from 60 t o 65%. T a b l e 1 shows t h e o p e r a t i v e p a r a m e t e r s which g a v e t h e m o s t S i E n i f i C & ? t r e s u l t s o n t h e materials b e i n g studied. Gradient
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A s s e s s m e n t of r e s u l t s a c h i e v e d
For e a c h t e s t c a r r i e d o u t , a n a l y s e s by o p t i c a l m i c r o s c o p e were made to show the d i f f e r e n t metallographic structure which result and microhardness p r o f i l e s . During t h e e x p er i m en t al near the p h a s e a maximum s u r f a c e t e m p e r a t u r e , m e l t i n g p o ? n t , w a s a p p l i e d ir. o r d e r t o o b t a i n homogenous s u r f a c e conditi0r.s and a p p r o p r i a t e p a r a m e t e r s were u s e d , c a l c u l a t e d on t h e b a s i s of pre-selec'ed aodelling. Ey means of o p t i e a l a n a l y s i s 20-30,km o f m o l t e n t h i c k n e s s c a n be s e e n , which c m f i r m s t h e t h e o r e t i c a l v a l u e s . For e a c h t h e r m a l g r a d i e n t s t u d i e d , a m a c r o g r a p h o f t h e area which h a s u n d e r g o n e l a s e r Paeat t r e a t m e n t . t h e h e a t p r o f i l e as a f > J n c t i o n of d e p t h and t h e S , 6 , 7 , and hardness p a t t e r n a r e provibed, Figs. 8. I n trie n i c r o h a r d n e s s g r a p h s t n e h a r c n e s s v a l u e of 525 :3V 0 . 3 is a l s o i n d i c a t e d , which , J s u a l l j . c o r r e s p o n d s t o t h e e f f e c t i v e hardening oepth i n i n d u s t r i a l heat treatments. i t c a n be s e e n t h a t i n each t e a t c a r r i e s o u t , i n correspondence t o t h i s value, s i m i l i a r t e m p e r a t u r e v a l u e s are o b t a i n e d on the upper curve: t h e s e v a i u e s a r e b e t w e e n 850 and 950 oc I t c a n c o n s e q u e n t l y be s e e n t h a t t h e t r a n s f o r m a t i o n t e m p e r a t u r e is h i g h e r t h a n t h e t e m p e r a t u r e s u s u a l l y found i n t h e phase diagram o f a s t e e l w i t h around 0.4% carbon. T h i s c a n be e x p l a i n e d by t h e f a c t t h a t t 3 e p h a s e rliagram is c a l c u l a t e d i n e q u i l i b r i u m conditions, b u t l a s e r h e a t treatments do n o t t a k e place i n equilibrium conditions i n that the process is c o m p l e t e d i n a v e r y s h o r t t i m e . This f a c t has b e e n c o n f i r m e d by e x p e r i m e r . t a t i o n u n d e r way i n other research centres. F u r t h e r m o r e t h e d e p t h o f t r e a t m e n t is s t i l l a g i v e n p e r c e n t a g e o f t h e Gy , a r o u n d 20 %. Regarding h a r d n e s s , i t w a s f o u n d t h a t a maxisum v a l u e b e t w e e n 550 and 700 HV 3 . 3 is reaci-.ed, w i t h a c l e a r d e c r e a s e i n t h e p a s s a g e from t h e t r e a t e d zone t o t h a t o f t h e b a s e material, which c a n a l s o b e s e e n i n t h e macrographs. The h i g h v a l u e r e a c h e d , g r e a t e r than t h a t u s u a l l y obtained w i t h t r a d i t i o n a l is related to the specific treatments c h a r a c t e r i s t i c s of l a s e r t r e a t m e n t s .
.
CONCLUSION On t h e b a s i s o f t e s t s c a r r i e d o u t i t c a n be s e e n t h a t the experimental values closely follow the m o n o d i m e n s i o n a l model d e v e l o p e d . The c o r r e s p o n d e n c e of t h e model t o t h e e x p e r i m e n t a l values obtained during the t e s t s carried out w i t h v a r y i n g t h e r m a l g r a d i e n t s and t h e r e f o r e v a r i o u s s p e c i f i c p o w e r s a n d d i f f e r e n t i n t e r a c t i o n times. i n d i c a t e s t h a t t h e many s i m p l i f y i n g h y p o t h e s e s made i n d e v e l o p i n g t h e model d o e s n o t i n v a l i d a t e i t and i t c a n be S t a t e d t h a t t h e p o s s i b l e e r r o r s a r i s i n g from t h e i r u s e c r e a t e e f f e c t s which a r e s e l f compensating
.
REFERENCES L a Rocca, A . , Criteri d i normalizzazione n e l l a modellistica della diffusione del calore per spessori f i n i t i e semiinfiniti.
van S p r a n g . 1 . . 1991, Optimization of Yeijer, J., l a s e r beam t r a n s f o r m a t i o n h a r d n e n i n g by o n e s i n g l e param5 t e r . ?licodemi, W . ,
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SUMMARY A theoretical study of h e a t i n t e r a c t i o n o f l a s e r r a d i a t i o n has b e e n carrird out. T h i s szudy b a s s u p p l i e d a n a t h e m a t i z a l a d i m e n s i o n a l n o d e l a b l e o f d e s c r i S i n g t h e t h e r m a l f i e l d v e r s u s time d e p t h i n a v e r y g e n e r a ; 'day. The m c e l h a s b e e n c n e c k e d w i t h e x p e r i m e n t a l t e s t s c n some f e r r o u s a l l o y s 3y s e a n s o f C32 l a s e r . S ? e c i f i c t e s t s u s i n g v e r y high Zradients, g o t from t h e m o d e l , k a v e shown t h e f o r m a t i o n og ir.novat i v e ne t a l Luraic a 1 struc t a r e s .
KEYWORDS: L a s e r , s u r f a c e t r e a t m e n t , m a t h e m a t i c a l model.
INTRODUCTION V a r i o u s c a l c u l u s programmes h a v e b e e n d e v e l o p e d i n t h e R.T.M I n s t i t u t e w i t h t h e aim o f produc:ng an i n s t r u m e n t f o r u n d e r s t a n d i n g t h e phenomena of h e a t t r a n s m i s s s i o n by c o n d u c t i o n i n material s u b j e c t t o laser radiation. These include simple ( n o n d i m e n s i o n a l ) programmes managed by a PC, and more complex o n e s ( t r i d i r n e n s i o n a i ) 'which c a n o n l y b e managed by a more powerfGl c o m p u t e r . Following t h e d e v e l o p m e n t p h a s e o f t h e m a t h e m a t i c a l model. t h e e x p e r i n e n t a l p h a s e commenced and aimed t o s t u d y t h e b e h a v i o u r and r e s u l t a n t m e t a l l u r g i c a l s t r u c t u r e i n common s t e e l s s u c h as C40 when s u b j e c t t o h i g h radiation. The thermal g r a d i e n t s from l a s e r v a l i d i t y o f t h e model and i t s c o r r e s p o n d e n c e t o experimental v a l u e s were t e s t e d by s e l e c t i n g a p p r o p r i a t e o p e r a t i v e p a r a m e t e r s which c a n p r o d u c e h i g h s p e c i f i c powers i n b r i e f t i m e i n t e r v a l s . MATHEMATICAL MODEL To o b t a i n t h e e q u a t i o n which g i v e s t h e t e m p e r a t u r e p a t t e r n a t v a r i o u s d e p t h s and as a f u n c t i o n o f time, t h e F o u r i e r l a w o f h e a t c o n d u c t i o n was u s e d as t h e s t a r t i n g p o i n t . With some s i m p l i f y i n g h y p o t h e s e s , i t t a k e s t h e f o l l o w i n g form:
= d e p t h a l o n g t h e d i r e c t i o n o f flow (cm) = t h e r m a l c o n d u c t i v i t y o f material (W/cm°C) = t e m p e r a t u r e ('C) A = heat generated i n t e r n a l l y (W/cm3) t = time ( s ) Cp = s p e c i f i c h e a t o f material ( J / g r ° C ) p = d e n s i t y (g/cm3 )
z k T
T h i s e q u a t i o n g i v e s s o l u t i o n s which v a r y , d e p e n d i n g on w h e t h e r t h e w o r k p i e c e S e i n g examined c a n b e c o n s i d e r e d a s e m i - i n f i n i t e or an i n f i n i t e t h i c k n e s s and i t c a n be o b t a i n e d using the following hypotheses: - c o n s t a n t h e a t p a r a m e t e r s o f t h e m a t e r i a l and c o e f f i c i e n t of a b s o r p t i o n - i n i t i a l t e m p e r a t u r e o f t h e workpiece O°C - n e g l i g i b l e phase changes - constant material density - homogenous and i s o t r o p i c material - u n i f o r m d i s t r i b u t i o n o f laser e n e r g y - monodimensional f l o w - no i r r a d i a t i o n or h e a t e x c h a n g e w i t h e x t e r i o r . T h e s e h y p o t h e s e s may be c o n s i d e r e d t o b e s a t i s f i e d i n v i e r o f t h e h i g h s p e e d w i t h which t h e t h e r m a l process takes p l a c e , with the exception of the first h y p o t h e s i s , s i n c e t h e s e parameters vary during the treatment. A further simplification d e r i v e s from t h e f a c t t h a t t h e l a s e r r a d i a t i o n is a b s o r b e d i n t o a t h i n l a y e r o f material so i t c a n be assumed t h a t t h e volume o f the heat source g e n e r a t e d i s z e r o , t h a t is A = O . By i n d i c a t i n g t h e i n c i d e n t laser f l o w on t h e t e s t p i e c e w i t h F, t h e s u r f a c e r e f l e c t i v i t y w i t h R and and c o n s i d e r i n g t h a t o n l y p a r t o f t h e f l o w Fo=F ( 1 4 ) p e n e t r a t e s the m a t e r i a l , the equation giving t h e heat f i e l d i n t h e m a t e r i a l as a f u n c t i o n o f time and d e p t h f o r a s e m i - i n f i n i t e t h i c k n e s s c a n be obtained:
i n which: 6 is t h e l e n g t h o f t h e r m a l d i f f u s i v i t y o f t h e material ( c m ) D * = 2 G F 5 i s t h e l e n g t h of thermal d i f f u s i v i t y i n t h e c o o l i n g p h a s e (cm). t = time r e q u i r e d t o c o n c i u d e t h e h e a t f l o w a r i s i n g from t h e l a s e r r a d i a t i o n d = k/pCp i s t h e t h e r m a l d i f f u s i v i t y ( c d / s ) The l e n g t h o f t h e t h e r m a l d i f f u s i v i t y C i n s e r t e d i n t o t h e e q u a t i o n i s a t y p e of h e a t h o r i z o n i n t h a t i t i n d i c a t e s t h e d e p t h a t which t h e t e m p e r a t u r e i s l e s s t h a n a t e n t h o f t h e s u r f a c e t e m p e r a t u r e and i f this i s much l o w e r t h a n t h e t h i c k n e s s o f t h e workpiece i n d i r e c t i o n z o f t h e h e a t f l o w , i t i n d i c a t e s t h a t t h e m a t e r i a l c a n be c o n s i d e r e d t o b e of a s e n i - i n f i n i t e t h i c k n e s s . The advantage of t h i s e x p r e s s i o n is t h a t one p a r a m e t e r u s e d is t h e l e n g t h o f i r r a d i a t i o n K t o which correspond well-defined values of t h e v a r i a b l e s o f f i e l d such a s t e m p e r a t u r e , l e n g t h of t h e r m a l d i f f u s i v i t y . e t c . ) which g i v e s a s i n g l e and unmistakable reference f o r the process. D = 2
Normalisation N o r m a l i s a t i o n w a s t h e n c a r r i e d o u t in o r d e r t o i d e n t i f y t h e p a r a m e t e r s w i t h a p h y s i c a l meaning a n d t o b r i n g t h e e q u a t i o n i n t o more g e n e r a l u s e . T h i s Nas a c h i e v e d by c o r r e l a t i n g t h e a v e r a g e t e m p e r a t u r e t o t h e maximum t e m p e r a t u r e which may be r e a c h e d d u r i n g t h e t h e r m a l p r o c e s s , and t h e t h i c k n e s s t o t h e l e n g t h o f thermal d i f f u s i v i t y . The n o r m a l i z e d e q u a t i o n is as f o l l o w s :
Tn = T/Tmax t n = t/? zn = z / C p represent the temperature, t h e time and t h e 1 normalized thickness (dimensionaless); Fig. shows a g r a p h o f t h e p a t t e r n o f t h e s e p a r a m e t e r s i n t h e n o r m a l i z e d p l a n e t n , Tn; t h e e q u a t i o n d e s c r i b e s t h e monodimensional h e a t f i e l d f o r any homogenous and i s o t r o p i c m a t e r i a l which m a i n t a i n s c o n s t a n t p h y s i c a l p r o p e r t i e s f o r each v a l u e o f t h e i n p u t flow; conversely, Fig. 2 i n t h e a p p e n d i x shows t h e b e h a v i o u r i n t h e p l a n e zn,Tn. By d i f f e r e n t i a t i n g (3) w i t h r e s p e c t to t h e n o r m a l i z e d t i m e and g i v i n g t h e d e r i v a t i v e t h e v a l u e of zero, a f u n c t i o n i s o b t a i n e d which g i v e s t h e curve o f maximum values of t h e normalized 1; t h i s thickness, shown by a d o t t e d l i n e i n F i g . c u r v e s e p a r a t e s t h e u p p e r a r e a , o f c o o l i n g , from t h e a r e a u n d e r n e a t h i n which h e a t i n g o c c u r s . i n which
The above r e f e r s t o s e m i - i n f i n i t e thicknesses but i t c a n n o t b e a p p l i e d when t h e l e n g t h o f t h e r m a l d i f f u s i v i t y c h a r a c t e r i s t i c o f t h e material b e i n g s t u d i e d i s c o m p a r a b l e t o or g r e a t e r t h a n t h e t h i c k n e s s o f t h e workpiece, because t h e lower s u r f a c e o f t h e workpiece reaches a c o n s i d e r a b l e t h e heat t e m p e r a t u r e and as no i o w e r l a y e r s e x i s t , d i f f u s e s towards the t o p , thereby modifying the temperature profile. By means o f boundary c o n d i t i o n s m o d i f i c a t i o n s and some s i m p l i f i c a t i o n s , t h e f o l l o w i n g e x p r e s s i o n i s produced:
(2)
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w i t h I = t h i c k n e s s o f t t e f i n i s h e d s h e e t , i n which of t h e t h e f i r s t term i s t h e sane as t h a t semi-infinite thick sheet whilst the second, e x p r e s s e d as an i n f i n i t e and i n t e r r u p t a b l e s e r i e s zt an a p p r o p r i a t e l y selected value, c a n be i n t e r p r e t e d as t h e c o n t r i b u t i o ? t o t h e t e m p e r a t u r e by t h e s u b s e q u e n t r e f l e c t i o n s o f h e a t b e t w e e n t h e u p p e r and l o w e r f a c e o f t h e w o r k p i e c e . The p r o f i l e of t h e f i n a l t e n p e r a t u r e i s t h e r e s u l t of ',he sum of t5.e i n d i v i d u a l t e m p e r a t u r e p r o f i l e s o b t a i n e d from t h e s i i b s e q u e n t heat r e f l e c t i o n s ; i d e a l l y i t i s as i f t h e temperature p r o f i l e o b t a i n a b l e f o r a s e m i - i n f i n i t e t.*,ic;cness, s i t u a t e d an t h e o u t s i d e of t h e s h e e t . x a s f l i p p e d over t o t h e i n s i d e o f t n i s and t h e i n d i , / i d u a l c o n t r i b u t i o n s 'were t l - e r e f o r e ai,?eS t o g s ' h e r . Even in t h i s c a s e n o r m a l i z a t i o n was c a r r i e d o u t Sy c o r r s l a t i n g the average surface temperature t o t h e however, the ec,uations maximurn temperature ; i l l u s t r a t e d io t h e n o r m a l i s e d p l a n e g i v e a f a m i l y of p a r a m e t r i c a l c u r v e s i n the normalized t h i c k n e s s r a z h e r tilan a s i n g l a Zraph. As c a n b e seen i n Fig. 3 t h e p a t t e r n i n time O f t h e s u r f a c e temperature i n ?he f i n i s h e d s h e e t is similiar t o t h a t o f t h e s e m i - i n f i n i t e t h i c k n e s s f o r a section, c h a r a c t e r i s t i c o f e a c h norn;alized thickness , than it straightens out. 411 t h e s e f a c t o r s are v a l i d i f t h e m a t e r i a l d o e s n o t undergo s o l i d t o l i q u i d phase t r a n s i t i o n s t h a t i s , as l o n g a s t h e r e is no m e l t i n g b u t , when t h e material mel:s and t h e c o n v e c t i v e m o t i o n of t h e molten phase takes o v e r w i t h s u b s e q u e n t r e m i x i n g , In this t h e c o c d i t i o n s assumed p r e v i o u s l y Change. c a s e t h e h e a t i n 3 a n d c o o l i n g g r a d i e n t s above t h e m e l t i n g t e m p e r a t u r e are l o w e r and i t i s n e c e s s a y t o m o d i f y t h e model u s e d a c c o r d i n g l y . EXPERIMENTAL PHASE To v e r i f y the v a l i d i t y o f t h e model and i t s c 3 r r e s p o n b e n c e t o t h e e x p e r i m e n t a l v a l u e s a Series of tssts were c a r r i e d o u t on a commonly u s e d S t e e l , C40, d u r i n g w h i c h t h e o p e r a t i n g p a r a m e t e r s were c s l c u l a t e d t o o b t a i n an i n c r e a s i c g h e a t g r a d i e n t . TP.e s t e e l 'was h a r r ' e n e d and tempered i n a f u r n a c e t o proriuce a more homogenous s t r u c t u r e and t e s t p i e c e s w i t h d i m e n s i o n s 5 0 x 2 0 and i O m m t h i c k were u s e d . S i n c e t > e l e n g t h of t h e r a a l d i f f u s i v i t y k ( w h i c n i s c l o s e l y c o r r e l a t e d t o t h e i n t e r a c t i o n t i m e . which i n t h i s c a s e i s r a t h e r l o w ) i s much l o w e r t h a n t h e t h i c k n e s s of t h e m a t e r i a l t h e model c a n be u s e d f o r 3 semi-infinite thickness. The p a t t e r n of t h e thernal gradient was also studied Sy di'fersntiating t h e n0rma:ized e q u a t i o n ( 3 ) . set o c t a b o v e , w i t h r e s p e c t t o t h e time and s p a c e . The t e s t s were c a r r i e d o u t o n t'wo d i f f e r e n t C02 lasers: t k e C o h e r e n t laser ' w i t h a maxi:nuin power of SOOW, which & i v e s minisum d i m e n s i o n s p o t s ( u p t o 0 . 2 mnin i n d i a m e t e r ) a n d t h e AVCO l a s e r w i t h a maximum power o f 15 kV. The u s e o f two l a s e r s w i t h s u c h d i f f e r e n t power l e v e l s is due to the extreme e x p e r i m e n t a l c o n d i t i o n s d u r i n g which h i g h s p e c i f i c p o w e r s had t o be p r o d u c e d which is o n l y p o s s i b l e by means o f v e r y tiny spots. The C o h e r e n t l a s e r , u r , l i k e t h e AVCO e n a b l e s small s p o t s t o b e p r o d u c e d , up t o 0 . 2 mm i n dianeter. F u r t h e r m o r e some of t h e t e s t s rust be c a r r i e d o u t a t h i g h s p e e d s w h i c h c a n n o t be a c h i e v e d w i t h t h e It was t h u s ?ranslating tables available. 4) with a n e c e s s a r y t o p l a c e a motor (Fig. r o t a t i n g t a b l e on t h e t r a n s l a t i n g t a b l e so as t o g i v e a maximum s p e e d o f r o t a t i o n o f 3000 rpm. which i n V i e w o f t h e d i m e n s i o n s of t h e r o t a t i n g t a b l e means a maximuia s u r f a c e v e l o c i t y o f 40 m / s e c . Before u n d e r g o i n g iaser r a d i a t i o n t h e t e s t p i e c e s were c o a t e d w i t h a l a y e r o f c o l l o i d a l g r a p h i t e : p r e l i m i n a r y t e s t s have p o i n t e d t o t h e use of an O p t i m a l C o a t i n g o f 2 9 t m and c a l o r i m e t r i c t e s t s h a v e ( i n d i c a t e d by Fo) shown an e n e r g y a b s o r p t i o n r a n g i n g from 60 t o 65%. T a b l e 1 shows t h e o p e r a t i v e p a r a m e t e r s which g a v e t h e m o s t S i E n i f i C & ? t r e s u l t s o n t h e materials b e i n g studied. Gradient
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Interaction
Soecif i c
-10.E5
O.OC38
45000
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0.038
14000
14
-10.E3
0.3
4000
-10. €2
1.2
3000
Table 1
252
A s s e s s m e n t of r e s u l t s a c h i e v e d
For e a c h t e s t c a r r i e d o u t , a n a l y s e s by o p t i c a l m i c r o s c o p e were made to show the d i f f e r e n t metallographic structure which result and microhardness p r o f i l e s . During t h e e x p er i m en t al near the p h a s e a maximum s u r f a c e t e m p e r a t u r e , m e l t i n g p o ? n t , w a s a p p l i e d ir. o r d e r t o o b t a i n homogenous s u r f a c e conditi0r.s and a p p r o p r i a t e p a r a m e t e r s were u s e d , c a l c u l a t e d on t h e b a s i s of pre-selec'ed aodelling. Ey means of o p t i e a l a n a l y s i s 20-30,km o f m o l t e n t h i c k n e s s c a n be s e e n , which c m f i r m s t h e t h e o r e t i c a l v a l u e s . For e a c h t h e r m a l g r a d i e n t s t u d i e d , a m a c r o g r a p h o f t h e area which h a s u n d e r g o n e l a s e r Paeat t r e a t m e n t . t h e h e a t p r o f i l e as a f > J n c t i o n of d e p t h and t h e S , 6 , 7 , and hardness p a t t e r n a r e provibed, Figs. 8. I n trie n i c r o h a r d n e s s g r a p h s t n e h a r c n e s s v a l u e of 525 :3V 0 . 3 is a l s o i n d i c a t e d , which , J s u a l l j . c o r r e s p o n d s t o t h e e f f e c t i v e hardening oepth i n i n d u s t r i a l heat treatments. i t c a n be s e e n t h a t i n each t e a t c a r r i e s o u t , i n correspondence t o t h i s value, s i m i l i a r t e m p e r a t u r e v a l u e s are o b t a i n e d on the upper curve: t h e s e v a i u e s a r e b e t w e e n 850 and 950 oc I t c a n c o n s e q u e n t l y be s e e n t h a t t h e t r a n s f o r m a t i o n t e m p e r a t u r e is h i g h e r t h a n t h e t e m p e r a t u r e s u s u a l l y found i n t h e phase diagram o f a s t e e l w i t h around 0.4% carbon. T h i s c a n be e x p l a i n e d by t h e f a c t t h a t t 3 e p h a s e rliagram is c a l c u l a t e d i n e q u i l i b r i u m conditions, b u t l a s e r h e a t treatments do n o t t a k e place i n equilibrium conditions i n that the process is c o m p l e t e d i n a v e r y s h o r t t i m e . This f a c t has b e e n c o n f i r m e d by e x p e r i m e r . t a t i o n u n d e r way i n other research centres. F u r t h e r m o r e t h e d e p t h o f t r e a t m e n t is s t i l l a g i v e n p e r c e n t a g e o f t h e Gy , a r o u n d 20 %. Regarding h a r d n e s s , i t w a s f o u n d t h a t a maxisum v a l u e b e t w e e n 550 and 700 HV 3 . 3 is reaci-.ed, w i t h a c l e a r d e c r e a s e i n t h e p a s s a g e from t h e t r e a t e d zone t o t h a t o f t h e b a s e material, which c a n a l s o b e s e e n i n t h e macrographs. The h i g h v a l u e r e a c h e d , g r e a t e r than t h a t u s u a l l y obtained w i t h t r a d i t i o n a l is related to the specific treatments c h a r a c t e r i s t i c s of l a s e r t r e a t m e n t s .
.
CONCLUSION On t h e b a s i s o f t e s t s c a r r i e d o u t i t c a n be s e e n t h a t the experimental values closely follow the m o n o d i m e n s i o n a l model d e v e l o p e d . The c o r r e s p o n d e n c e of t h e model t o t h e e x p e r i m e n t a l values obtained during the t e s t s carried out w i t h v a r y i n g t h e r m a l g r a d i e n t s and t h e r e f o r e v a r i o u s s p e c i f i c p o w e r s a n d d i f f e r e n t i n t e r a c t i o n times. i n d i c a t e s t h a t t h e many s i m p l i f y i n g h y p o t h e s e s made i n d e v e l o p i n g t h e model d o e s n o t i n v a l i d a t e i t and i t c a n be S t a t e d t h a t t h e p o s s i b l e e r r o r s a r i s i n g from t h e i r u s e c r e a t e e f f e c t s which a r e s e l f compensating
.
REFERENCES L a Rocca, A . , Criteri d i normalizzazione n e l l a modellistica della diffusione del calore per spessori f i n i t i e semiinfiniti.
van S p r a n g . 1 . . 1991, Optimization of Yeijer, J., l a s e r beam t r a n s f o r m a t i o n h a r d n e n i n g by o n e s i n g l e param5 t e r . ?licodemi, W . ,
Metallurgia applicata.
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1200
600
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800
600
400
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