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A split-platen pressure cell for the measurement of pressure distribution in upsetting operations
Int. J. mech. Sci. Pergamon Press. 1971. Vol. 13, pp. 355-371. Printed in Great Britain
A S P L I T - P L A T E N P R E S S U R E CELL FOR T H E M E A S U R E M E N T OF P R E S S U R E D I S T R I B U T I O N IN UPSETTING OPERATIONS G. H . DANESHI a n d J . B. HAWKYA~D Mechanical Engineering Department, University of Manchester Institute of Science and Technology
(Received 8 March 1971)
Summary--A description is given of a form of pressure cell by which the pressure distribution across a deforming billet can be measured. The cell performs essentially the same function as the pin loadcells which have been used by various workers in the field of plasticity research, but it is considered that this new device offers advantages in terms of simplicity of manufacture and maintenance. The construction and cahbration of two pressure cells are described and some preliminary results of friction hill measurements in upsetting tests axe given. Simple analyses are developed to indicate the probable magnitude of errors in the pressure measurements.
INTRODUCTION FOE THE measurement of pressures locally at the surface of metal forming tools the pin loadcell has found fairly extensive use, in rolling, forging, heading and upsetting operations. The pins are situated in closely fitting holes in the tool, the ends being flush with the surface and the forces on the ends are m e a s u r e d using resistance s t r a i n gauges or piezo electric crystals. T h e first i n v e s t i g a t o r to use t h e t e c h n i q u e a p p e a r s to h a v e b e e n Siebel, 1 for m e a s u r i n g roll pressure, a n d v a r i o u s o t h e r w o r k e r s 2,a h a v e d e v e l o p e d a n d e x t e n d e d its application. A c o m p r e h e n s i v e r e v i e w of its d e v e l o p m e n t has b e e n given r e c e n t l y b y Cole a n d Sansome. 4 T h e p r o p e r functioning o f t h e pin loadcell d e p e n d s on t h e close b u t free fit o f t h e pin, to p r e v e n t e x t r u s i o n o f m e t a l into t h e g a p a n d its levelness w i t h t h e s u r r o u n d i n g surface so as n o t to interfere w i t h m e t a l flow, a n d in p r a c t i c e t h e cell is s o m e w h a t difficult to c o n s t r u c t a n d m a i n t a i n . L o c a t i n g t h e device can also p r e s e n t difficulties, because of its size, a n d t h e r e is a l i m i t a t i o n to t h e closeness of spacing for this reason. I t was considered t h a t a d v a n t a g e s m i g h t be o b t a i n e d in some applications b y a n o t h e r f o r m o f pressure m e a s u r i n g device, using small resistance s t r a i n gauges positioned on a surface p e r p e n d i c u l a r to t h e die face a n d close u p to it. T h e m e t h o d offers t h e possibility o f closer spacing o f m e a s u r i n g points t h a n w i t h t h e pin loadcell a n d easier c o n s t r u c t i o n a n d m a i n t e n a n c e . This p a p e r describes e x p e r i m e n t s on such a "split p l a t e n " device, using resistance strain gauges n e a r t h e w o r k i n g surface. T w o pressure p l a t e n s h a v e 355
356 been constructed,
G . H . DA~ESHI a n d J . B. HAWKYARD one using ~ in. foil gauges positioned
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EQUIPMENT
Design of pressure cells A v i e w o f t h e split p l a t e n p r e s s u r e cell is s h o w n in Fig. 1. I t consists of t w o b l o c k s of h a r d e n e d tool steel, g r o u n d u p s q u a r e so t h a t t h e w o r k i n g faces are a t t h e s a m e level w h e n p o s i t i o n e d side b y side o n a firm s u p p o r t i n g surface. To t h e v e r t i c a l i n t e r f a c e of one b l o c k is a t t a c h e d a r o w of s m a l l s t r a i n gauges, close u p t o t h e w o r k i n g face a n d w i t h a x e s p e r p e n d i c u l a r t o it as i n d i c a t e d . T h e r e s p o n s e o f s u c h a g a u g e t o l o a d i n g d e p e n d s o n t h e d i s t a n c e f r o m t h e w o r k i n g face, ideally t h e g a u g e w o u l d b e i n f i n i t e s i m a l i n size a n d p o s i t i o n e d a t surface level, so t h a t i t r e s p o n d e d o n l y t o p r e s s u r e i m m e d i a t e l y a b o v e it. V e r y s m a l l g a u g e are o b t a i n a b l e n o w a d a y s , foil gauges are a v a i l a b l e d o w n t o ~ in. l o n g a n d s e m i - c o n d u c t o r gauges d o w n t o a b o u t ~ in. F r o m p r a c t i c a l c o n s i d e r a t i o n s it w a s d e c i d e d t o use a l a r g e r g a u g e t h a n t h e smallest, b e c a u s e of possible difficulties in a t t a c h m e n t a n d a c c o r d i n g l y t h e g a u g e size c h o s e n for p r e l i m i n a r y w o r k w a s ~ in. T h e d i s t a n c e f r o m t h e c e n t r e of t h e g a u g e t o t h e w o r k i n g surface is ~ in. a n d t h e gauges a r e a t t a c h e d w i t h 0.15 in. spacing, u s i n g E a s t m a n 110 a d h e s i v e . T h e t e s t s o n t h i s p l a t e n g a v e s a t i s f a c t o r y r e s u l t s b u t it was e v i d e n t t h a t b e t t e r r e s u l t s c o u l d b e o b t a i n e d w i t h s m a l l e r gauges f i t t e d closer to t h e w o r k i n g surfeme. A c c o r d i n g l y a s e c o n d p l a t e n w a s c o n s t r u c t e d w i t h gauges ~ in. long p o s i t i o n e d ~ in. f r o m t h e surface t o t h e c e n t r e of t h e gauge, w i t h a s p a c i n g of 0.1 in. T h e gauges a n d wiring a r e s i t u a t e d in grooves in t h e p l a t e n interfaces, as s h o w n in Figs. 2(a), (b), t h e g a u g e c o n n e c t i o n s b e i n g led o u t i n d i v i d u a l l y t o t e r m i n a l s o n t h e sides of t h e blocks. D u r i n g p r e l i m i n a r y c a l i b r a t i o n t e s t s o n t h e first p l a t e n it w a s o b s e r v e d t h a t t h e l o a d - s t r a i n r e l a t i o n s h i p s b e c a m e n o n - u n i f o r m a b o v e a c e r t a i n l o a d level, w h i c h v a r i e d w i t h t h e s p a c i n g b e t w e e n t h e p l a t e n s as d e t e r m i n e d b y shims. I t was d e d u c e d t h a t t h i s o c c u r r e d w h e n t h e t w o h a l v e s c a m e t o g e t h e r a t t h e w o r k i n g surface, d u e t o l a t e r a l e x p a n s i o n , c r e a t i n g b e n d i n g stresses a t t h e gauges. To a v o i d t h i s n o n - l i n e a r i t y w i t h i n t h e w o r k i n g r a n g e , a t p r e s e n t u p t o 36 t o n f / i n ~, it was n e c e s s a r y to h a v e a n initial g a p of 0.011 in. b e t w e e n t h e blocks. T h e s e c o n d p l a t e n was d e s i g n e d t o m i n i m i s e t h i s l a t e r a l s t r a i n i n g , b y m a k i n g t h e g r o o v e w h i c h a c c o m m o d a t e s t h e gauges a n d w i r i n g as shallow as possible i n t h e v e r t i c a l direction. I t was f o u n d t h a t a g a p of 0.006 in. was sufficient to p r e v e n t c o n t a c t , a n d n o n - l i n e a r i t y u p t o a p r e s s u r e of 36 t o n f / i n s. To p r e v e n t e x t r u s i o n i n t o t h e g a p b e t w e e n t h e blocks, a 0.010 in. s p r i n g steel s h i m is p l a c e d o v e r t h e p l a t e n . T h i s p r o v i d e s a c o n t i n u o u s w o r k i n g surface a n d c a l i b r a t i o n t e s t s i n d i c a t e t h a t i t does n o t a d v e r s e l y affect t h e o p e r a t i o n . I t p r o b a b l y does p r e v e n t t h e p l a t e n f r o m b e i n g u s e d t o m e a s u r e f r i c t i o n a l stresses, u s i n g gauges i n c l i n e d a t 45 ° t o t h e w o r k i n g surface, a l t h o u g h t h e g a p c a n b e r e d u c e d a n d t h e s h i m o m i t t e d in some c i r c u m s t a n c e s w h e r e t h e m a x i m u m p r e s s u r e is a p p r e c i a b l y less t h a n 36 t o n f / i n 2. I n u s i n g t h e s h i m i t is n e c e s s a r y to a v o i d l u b r i c a t i n g t h e i n t e r f a c e b e t w e e n t h e p l a t e n a n d s h i m , o t h e r w i s e t h e f r i c t i o n a l forces i m p o s e d b y t h e s p e c i m e n are t a k e n a l m o s t w h o l l y b y t h e s h i m , p o s s i b l y l e a d i n g t o v e r y h i g h t e n s i l e stresses. T h e t w o b l o c k s c o m p r i s i n g t h e first p l a t e n are h e l d t o g e t h e r b y a b a s e p l a t e w i t h s h o r t sidewalls t o p r e v e n t s e p a r a t i o n , b u t t h i s w a s f o u n d to b e u n n e c e s s a r y a n d t h e s e c o n d p l a t e n s i m p l y r e s t s o n a flat g r o u n d b l o c k w i t h o u t u n d e r g o i n g a n y r e l a t i v e m o t i o n . T h e g a p b e t w e e n t h e b l o c k s r e s u l t s in stress c o n c e n t r a t i o n s a t t h e edges, errors due t o t h i s b e i n g m i n i m i s e d b y u s i n g t h e s a m e a r r a n g e m e n t for c a l i b r a t i o n a n d e x p e r i m e n t .
Strain gauge circuit T h e electrical circuit for t h e s t r a i n gauges is s h o w n in Fig. 3. E a c h g a u g e is c o n n e c t e d i n t u r n t o a b r i d g e b y a m a n u a l selector switch, t h e o u t p u t b e i n g fed to a P e e k e l a u t o m a t i c digital s t r a i n m e a s u r i n g s y s t e m a n d r e c o r d e d b y a n a u t o m a t i c s t r a i n g a u g e logging s y s t e m . R e a d i n g s c a n b e t a k e n a t i n t e r v a l s of 1 sec so t h a t t h e a r r a y of gauges c a n b e c o v e r e d i n 20 sec for t h e first p l a t e n a n d 31 sec for t h e second.
A s p l i t - p l a t e n p r e s s u r e cell for t h e m e a s u r e m e n t of p r e s s u r e d i s t r i b u $ i o n
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CALIBRATION
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Because t h e strain gauges are n o t s i t u a t e d a c t u a l l y a t surface level t h e o u t p u t s will be influenced b y loading of t h e area surrounding each gauge position. The gauge will react correctly in this respect to a u n i f o r m pressure and it can be reasoned t h a t the pressure on a plane g r a d i e n t should be also tm~ly represented. Non-linear slopes a n d conical pressure hills are subject to some error, however, also t h e edges of pressure hills. The calibration procedure and analysis which follow give some indication of possible errors i n v o l v e d in using t h e calibrations directly and deduce a p p r o p r i a t e corrections. T h e errors a p p e a r to be small for m o s t practical situations, using t h e specimen sizes envisaged for t h e experiments. A n o t h e r source of error in t h e strain gauge response arises due to variations in lateral constraint of the region u n d e r the specimen. The constraint depends on t h e size and shape of the specimen in relation to t h e platen, for a relatively small specimen the surrounding region n o t subject to axial loading imposes considerable lateral constraint, resulting in a reduced axial strain as c o m p a r e d w i t h the condition of u n i f o r m pressure o v e r the whole of t h e platen.
Calibration under uniform pressure Calibration is carried out b y a p p l y i n g u n i f o r m pressure to the p l a t e n face, using t h e e q u i p m e n t shown schematically in Fig. 4. The pressure cylinder is positioned on t h e p l a t e n a n d load is applied, via t h e piston, to t h e plasticine c o n t a i n e d in t h e cylinder. Calibration pressures are in t h e range of 1 to 36 t o n f / i n 2 so t h a t the yield s t r e n g t h of t h e plasticine, a t a b o u t 20 lbf/in 2, produces negligible error. Plasticine was chosen for its ease and cleanliness in h a n d l i n g a n d w i t h t h e close fitting pressure disc and r u b b e r seal u n d e r t h e r a m and t h e differential area sealing b e t w e e n t h e cylinder and t h e platen, there is no leakage during t h e operation. E a c h gauge is calibrated in loading steps corresponding to pressure increments of 1.2 t o n / i n ~. A typical calibration curve is shown in Fig. 5, for each of the platens the overall v a r i a t i o n in slope b e t w e e n gauges is a b o u t 7 per cent. The response will be c o m p a r e d w i t h theoretical values in t h e following section.
Response to point loading The response of a strain gauge to a n o n - u n i f o r m pressure distribution o v e r the p l a t e n can be deduced b y d e t e r m i n i n g the gauge o u t p u t due to loading on an elemental area, defined in relation to the gauge position, and t h e n integrating o v e r t h e surface area. The situation is represented in Fig. 6, the gauge being positioned at m e a n d e p t h h below t h e surface, and the load i n c r e m e n t at distance r f r o m t h e axis of the gauge. I t is assumed t h a t t h e gauge response is i n d e p e n d e n t of angular position 0, this would be so if the p l a t e n
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e x t e n d e d c o n t i n u o u s l y in all d i r e c t i o n s a b o u t t h e g a u g e p o s i t i o n a n d t h e g a u g e was i n f i n i t e s i m a l in size. T h e r e will b e some d i s c r e p a n c y b e c a u s e t h e s e c o n d i t i o n s do n o t hold. T h e i n c r e m e n t a l response, de of t h e g a u g e ( m e c h a n i c a l s t r a i n ) t o p r e s s u r e p o v e r e l e m e n t a l a r e a d A c a n b e e x p r e s s e d in t h e f o r m d~
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w h e r e E is Y o u n g ' s m o d u l u s a n d ~ P o i s s o n ' s ratio, g(v) a c c o u n t s for t h e effect of t h e l a t e r a l c o n s t r a i n t i m p o s e d b y s u r r o u n d i n g regions o f t h e p l a t e n n o t s u b j e c t e d t o a x i a l loading. T h e possible influence of t h i s c o n s t r a i n t c a n b e assessed b y c o n s i d e r i n g a n idealised s i t u a t i o n w i t h a n e x t r e m e c o n d i t i o n of c o n s t r a i n t s u c h t h a t s t r a i n e~, p a r a l l e l t o t h e p l a t e n interface, is zero (see Fig. 6). T h e a x i a l s t r a i n e~ is t h e n r e a d i l y s h o w n t o b e a z ( 1 - v 2 ) / E . F o r steel, w i t h v = 0.29, t h i s b e c o m e s 0.916aJE, so t h a t t h e s t r a i n c a n v a r y b e t w e e n t h e l i m i t s (0"916-1"0) a~/E. I n o r d e r t o e v a l u a t e e q u a t i o n (1) it is c o n v e n i e n t t o d e t e r m i n e t h e f u n c t i o n ](r) a t c o n s t a n t pressure, a n d t h e n a p p l y t h e a p p r o p r i a t e p r e s s u r e d i s t r i b u t i o n r e l a t i o n s h i p . F o r t h e p r e s e n t w o r k t h e f u n c t i o n is d e t e r m i n e d e x p e r i m e n t a l l y b y m e a s u r i n g t h e g a u g e o u t p u t d u e t o a p o i n t l o a d of c o n s t a n t m a g n i t u d e , a p p l i e d a t v a r i o u s d i s t a n c e s f r o m t h e g a u g e position. T h e e x p e r i m e n t a l a r r a n g e m e n t u s e d for t h i s c a l i b r a t i o n t e s t is s h o w n i n Fig. 7. I t was c a r r i e d o u t in a B r i n e l l h a r d n e s s t e s t i n g m a c h i n e , w i t h a s p r i n g steel s h i m i n s e r t e d b e t w e e n t h e b a l l a n d t h e p l a t e n surface t o a v o i d a n y i n d e n t a t i o n . Two t r a v e r s e s were m a d e across t h e p l a t e n face, o n e p a r a l l e l t o t h e g a u g e face a n d as close to it as possible a n d t h e o t h e r p e r p e n d i c u l a r t o t h e face, as i n d i c a t e d . R e s u l t s of t h e s e t w o l o a d i n g t e s t s are s h o w n i n Figs. 8(a), (b) for t h e t w o p l a t e n s . I t is e v i d e n t t h a t t h e s e c o n d p l a t e n , w i t h t h e s m a l l e r g a u g e s i t u a t e d closer to t h e surface, shows a g r e a t e r s e n s i t i v i t y for l o a d i n g d i r e c t l y o v e r t h e g a u g e a n d a v e r y r a p i d fall in r e s p o n s e w i t h d i s t a n c e . I t is t o b e o b s e r v e d t h a t t h i s g a u g e shows negligible r e s p o n s e to l o a d i n g b e y o n d 0.15 in. T h e c u r v e s of Fig. 8(a) are n o t i n fact t r u l y i n d i c a t i v e of t h e a c t u a l r e s p o n s e g r a d i e n t s , b e c a u s e t h e d i s t a n c e m e a s u r e m e n t s x are n o t t a k e n f r o m t h e a c t u a l g a u g e centres, b u t t h e s e r e s u l t s a r e s h o w n r e p l o t t e d in Fig. 8(b), t o g e t h e r w i t h t h o s e t a k e n a l o n g t h e p e r p e n d i c u l a r axes, in t e r m s of a c t u a l
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d i s t a n c e r t o t h e gauge eentres. T h e results follow t h e s a m e f o r m of curve, for t h e t w o d i r e c t i o n s of m e a s u r e m e n t . T h e r e s u l t s of Fig. 8(b) are s h o w n p l o t t e d to a l o g a r i t h m i c scale i n Fig. 8(c), f r o m w h i c h it is e v i d e n t t h a t a t p o i n t s o t h e r t h a n p e r h a p s in t h e i m m e d i a t e v i c i n i t y of t h e g a u g e t h e y c a n b e r e p r e s e n t e d r e a s o n a b l y b y e q u a t i o n s of t h e f o r m Ae A-L = s . e - ~
(2)
w h e r e As is t h e s t r a i n d u e to p o i n t load AL. F o r s i m p l i c i t y t h e r e s p o n s e c u r v e s a r e idealised t o t h e f o r m s h o w n b y t h e d o t t e d lines in Fig. 8(b), c o n s i s t i n g of a region of c o n s t a n t
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(a) FIG. 8. R e s p o n s e of g a u g e t o p o i n t loading. L e g e n d s for e x p e r i m e n t a l p o i n t s a n d d i s t a n c e s are as s h o w n in Fig. 7. (a) R e s u l t s t a k e n parallel t o s t r a i n g a u g e face. (b) R e s u l t s t a k e n parallel a n d p e r p e n d i c u l a r t o s t r a i n g a u g e face, a t c o r r e c t e d d i s t a n c e r. (e) S h o w i n g l o g a r i t h m i c f o r m of gauge response.
s e n s i t i v i t y , e x t e n d i n g to r a d i u s A a b o u t t h e g a u g e axis, a n d a s u r r o u n d i n g g r a d i e n t h a v i n g t h e f o r m of e q u a t i o n (2). ~ c a n b e c o n v e n i e n t l y m a d e e q u a l t o t h e h a l f - w i d t h o f t h e s t r a i n gauge, w i t h i n t h i s r a d i u s t h e f o r m of t h e r e s p o n s e c u r v e h a s n o t b e e n d e t e r m i n e d a n d t h e c o n s t a n t v a l u e a s s u m e d p r o b a b l y gives a n u n d e r e s t i m a t e of t h e a c t u a l response. The calibration measurements involving concentrated loading can be simply related t o a u n i f o r m p r e s s u r e d i s t r i b u t i o n , if it is a s s u m e d t h a t t h e c o n c e n t r a t e d load AL is e q u i v a l e n t to a p r e s s u r e p a c t i n g o v e r a finite a r e a AA. T h e s t r a i n m e a s u r e m e n t s f r o m t h e t e s t s
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- - = 2"34 x 10 -s in~/lb for p l a t e n 2. P0 I n T a b l e 1 t h e s e r e s u l t s a r e c o m p a r e d w i t h t h e o r e t i c a l v a l u e s a n d r e s u l t s of c a l i b r a t i o n s c a r r i e d o u t u n d e r u n i f o r m p r e s s u r e u s i n g t h e p l a s t i e i n e filled cylinder. TABLE 1 Theoretical, no lateral constraint Platen 1 2
Theoretical, with lateral constraint
e
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1 --v ~
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2.9 x 10 -s 2-34 × 10 -8
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Derived from point loading tests
T h e v a l u e s for p l a t e n No. I s h o w r e a s o n a b l e a g r e e m e n t b u t t h e p o i n t l o a d i n g r e s u l t o n No. 2 p l a t e n is low, s u g g e s t i n g t h a t t h e r e g i o n close t o t h e g a u g e a n d n o t c o v e r e d b y measurements might be more sensitive than assumed. EXPERIMENTAL
RESULTS
T h e p l a t e n s w e r e u s e d t o m e a s u r e a x i a l stress d i s t r i b u t i o n s o v e r c o m p r e s s i o n s p e c i m e n s o f c i r c u l a r a n d r e c t a n g u l a r forms. T h e s p e c i m e n s were m a d e i n a n a l u m i n i u m a l l o y (HE19V~P) a n d t o a p p r o a c h a n ideal m a t e r i a l c o n d i t i o n , w i t h n o s t r a i n h a r d e n i n g , t h e m a t e r i a l w a s p r e s t r a i n e d i n c o m p r e s s i o n b y a b o u t 20 p e r c e n t before t h e final s h a p i n g . A t y p i c a l stress s t r a i n c u r v e for t h e m a t e r i a l i n t h i s c o n d i t i o n is s h o w n i n Fig. 9. 2S
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Three batches of materials were used, in fact, w i t h s o m e w h a t different yield strengths. I t was considered desirable to conduct p r e l i m i n a r y tests w i t h the friction conditions controlled a n d reproducible, as far as possible, and accordingly t h e lubricant used was pure lead, in t h e f o r m of foil a b o u t 0.004 in. thick, interposed b e t w e e n t h e specimen and platens. The experiments were p e r f o r m e d in a 300 t o n h y d r a u l i c press, w i t h t h e axes of t h e platens and specimen carefully aligned w i t h i n it. Compression was indicated b y dial gauges a n d w i t h the specimen fully plastic and u n d e r a v e r y slowly increasing load t h e strain gauge m e a s u r e m e n t s were recorded in quick succession, at a r a t e of a b o u t 1 sec per reading. Pressure values were t h e n deduced f r o m t h e calibration curves. Results of m e a s u r e m e n t s on circular a n d rectangular compression specimens arc shown in Figs. 10 and 11. 28
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(c) FIG. 11. Pressure distribution on m a j o r axes of a r e c t a n g u l a r specimen 3 in. b y 1 in. a n d 0.2 in. thick, lubricated w i t h lead foil, (a, b) using p l a t e n No. 1 (e) using p l a t e n No. 2.
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368
G . H . DA~ESttI and J . B. HAWKYARD
Discussion of results The pressure m e a s u r e m e n t s shown in Figs. 10 and 11 are disposed fairly s m o o t h l y a n d s y m m e t r i c a l l y o v e r the sections, t h e points being positioned generally quite closely a b o u t m e a n "friction hill" curves, w i t h i n a b o u t 2 per cent in t e r m s of axial pressure. A few of t h e results show a spread of up to a b o u t 1 tonf/in ~ or 4 per cent. These m a y be due to errors in calibration or recording, or otherwise local variations in yield stress or friction m a y be responsible. I t is possible to m a k e estimates of t h e gradients of the friction hills for t h e circular specimens, Fig. 10, assuming t h a t the frictional stress is equal to t h e yield shear stress of t h e lead lubricant. The simple analysis of the friction hill, for a c o n s t a n t frictional stress Ts, provides the expression daz _ - 2~s dr h
(5)
where h is current height and doz/dr is the radial g r a d i e n t of axial pressure. F o r pure lead t h e yield stress Y is a b o u t 1 tonf/in ~, so t h a t t a k i n g the yield shear stress Vs as Y/2 = ½t o n f / i n ~ a n d p u t t i n g h = 0.2 in. gives daz dr
5 tonf/in~/in.
This g r a d i e n t is c o m p a r e d w i t h the e x p e r i m e n t a l results in Fig. 10. E q u a t i o n (5) can be shown to a p p l y also to t h e case of plane strain compression, daz/dr t h e n being t h e axial pressure gradient in the transverse direction and in Fig. 11 t h e calculated slopes are c o m p a r e d w i t h e x p e r i m e n t a l gradients deterrained for t h e short axis of r e c t a n g u l a r specimens, where plane strain conditions are approached. Along t h e m a j o r axis t h e g r a d i e n t is smaller towards t h e centre of t h e specimen, since t h e comp o n e n t of sliding v e l o c i t y in t h a t direction is small. F r o m t h e results shown in Figs. 10 and 11 it is e v i d e n t t h a t the lead lubricant is reasonably effective in controlling the frictional stress a n d t h e e x p e r i m e n t a l gradients agree quite well w i t h t h e theoretical values. The influence of friction in this p a r t i c u l a r operation is, in fact, relatively small in relation to u p s e t t i n g operations generally, and if a rough e s t i m a t e is m a d e of the e q u i v a l e n t Coulomb friction coefficient #~ b y p u t t i n g frictional stress Ts = #~a~-/~ Y we h a v e Vs /x=Y=
__0"5 = 0.023. 22
I f the a l u m i n i u m specimens were compressed w i t h no lubricant, the friction coefficient would be e x p e c t e d to be in t h e range 0-2-0.5 so t h a t m u c h steeper gradients w o u l d occur, for a given specimen configuration.
Estimation of errors As discussed in the previous section, t h e pressure distribution derived directly from t h e strain gauge m e a s u r e m e n t s will be s o m e w h a t in error in certain circumstances, and p a r t i c u l a r l y at the a p e x or edge of a friction hill, where t h e greatest error would be expected. The probable errors at these positions will be e s t i m a t e d for the present tests, using e q u a t i o n (3) w i t h t h e a p p r o p r i a t e pressure relationships inserted. F o r t h e conical hill o v e r the circular specimen t h e pressure relationship is p -- Pnmx - mr where Pmax occurs at the a p e x and m is t h e gradient. Combining this w i t h e q u a t i o n (3) and integrating to radius a gives =
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A s p l i t - p l a t e n p r e s s u r e cell for t h e m e a s u r e m e n t o f p r e s s u r e d i s t r i b u t i o n
369
a n d t h e p r e d i c t e d p e r c e n t a g e e r r o r A for a g a u g e s i t u a t e d a t t h e a p e x is t h e n A
=
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F o r v a l u e s of a, g r e a t e r t h a n s a y 6A, e - ~ c a n b e n e g l e c t e d a n d t h e e r r o r is = (A3/3 + 22/~ + 2 2 / ~ + 2/~3) m × 100. ( ~ / 2 + ~/a + l/a 3) Pma, F o r p l a t e n No. 1 w i t h A = 0.063 in., a = 20, t h i s gives A = 9m/pma x per cent
a n d for p l a t e n No. 2 w i t h A = 0-032 in., a = 33 A = 6 " 4 m / p m a x p e r cent.
F o r t h e r e s u l t s i n Fig. 10 v a l u e s of m = 5 t o n f / i n 2 a n d Pmax = 25 t o n f / i n ~ a r e r e a s o n a b l y r e p r e s e n t a t i v e , g i v i n g A = 1.8 p e r c e n t for p l a t e n No. 1 a n d 1.3 p e r c e n t for No. 2. A s i m i l a r p r o c e d u r e c a n b e a p p l i e d t o d e t e r m i n e t h e e r r o r a t t h e ridge o f a p l a n e s t r a i n p r e s s u r e hill, w h i c h gives A
=
2(23/3 + A 2 / a +
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x
100.
~(A2/2 + A/a + 1/a 2) Pmax T h e e r r o r is less t h a n for t h e conical hill b y t h e f a c t o r 2/~r. To o b t a i n a n e s t i m a t e o f t h e e r r o r n e a r t h e edge of a s p e c i m e n w h e r e t h e r e is a n a b r u p t fall o f p r e s s u r e t o zero, t h e s i t u a t i o n is simplified as i n d i c a t e d i n Fig. 12, o t h e r w i s e
STRAIN GAUGE
FIG.
12. S h o w i n g simplified g e o m e t r y for d e t e r m i n i n g t h e r e s p o n s e o f a g a u g e n e a r a s p e c i m e n edge.
t h e a n a l y s i s is v e r y c o m p l e x . I t is a s s u m e d t h a t t h e p r e s s u r e is c o n s t a n t i n t h e v i c i n i t y of a g a u g e n e a r t h e edge a n d t h e edge is c o n s i d e r e d t o b e s t r a i g h t , t h e s e a s s u m p t i o n s a r e g e n e r a l l y r e a s o n a b l e if t h e p r e s s u r e zone i n f l u e n c i n g t h e g a u g e is s m a l l i n r e l a t i o n t o t h e s p e c i m e n size. F u r t h e r simplifications a r e i n t r o d u c e d b y p u t t i n g a l i m i t o n r a d i u s a e q u a l t o 6A, since b e y o n d t h i s r a d i u s t h e influence is negligible (less t h a n 1 p e r cent), a n d b y r a t i o n a l i s i n g
370
G . H . DANESHI and J. B. HAWKYARD
the areas affecting the gauge according to Fig. 12 to simplify the integration. Using equation (5) the response for the case represented is
f)t f~t
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--I
J b JO
( t1 1 I n Fig. 13 this equation is evaluated for the two platens, with gauge output expressed as a fraction of the full response. When the gauge centre coincides with the specimen edge the response is 50 per cent, and 90 per cent, response occurs at a distance from the edge of the specimen to the gauge centre of 0.19 in. for platen )70. 1 and 0.10 in. for No. 2. The superior response of platen No. 2 is clearly evident. I n Fig. 13 are shown also some
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F i e . 13. Experimental and theoretical gauge response across a specimen edge. [] Platen No. 1. • Platen No. 2. corresponding strain gauge measurements taken across the edge of specimens. These suggest t h a t the theoretical curves overestimate errors near the edge. F o r platen No. 2 an error of less than 5 per cent is indicated at 0.1 in. from the edge. CONCLUDING
REMARKS
The analytical and experimental work has indicated that this device for m e a s u r i n g p r e s s u r e d i s t r i b u t i o n is r e l a t i v e l y s i m p l e t o m a n u f a c t u r e a n d s h o u l d g i v e a n a c c u r a c y t o w i t h i n a few p e r c e n t . T h e s u r f a c e s h i m u s e d t o p r e v e n t e x t r u s i o n i n t o t h e g a p p r e s e n t s a disadvantage and limits the possible range of application, but further developm e n t s , i n r e d u c i n g t h e g r o o v e size n e c e s s a r y f o r t h e g a u g e s , f o r e x a m p l e , m a y e l i m i n a t e t h e n e e d f o r t h e s h i m . T h e g a p o f 0.006 in. i n t h e e x i s t i n g equipment can be reduced in proportion to the reduction in the working p r e s s u r e r a n g e , a t p r e s e n t 36 t o n f / i n . P i n l o a d c e l l s h a v e b e e n u s e d b y a f e w
A split-platen pressure cell for the measurement of pressure distribution
371
workers 5 up to 50 tonf/in2 but generally the working pressure appears to have been limited to about 5 tonf/in. 2 Electric resistance strain gauges are available down to ~ in. in length so t h a t in principle the sensitivity and accuracy can be further improved. Acknowledgements--The authors wish to t h a n k Professor W. Johnson for his interest and encouragement in this work. The assistance of the laboratory staff of the Applied Mechanic Division is also gratefully acknowledged. REFERENCES 1. E. SZEBEL and W. LtrEG, mitt Kais--Wilh. Inst. Eisenforgh, Dusseldorf, 15 hfg 1, 1 (1933). 2. E. P. U~xsov, A n Engineering Theory of Plasticity, p. 112, Butterworth, London ( 1961). 3. G. T. V ~ O O Y E ~ and N. I. BACXO~TEN, Int. J. mech. Sci. 1, 1 (1960). 4. I. M. COLE and D. H. SANSOM~, 9th Int. M.T.D.R. Conf., Birmingham (1968). 5. I. GOKYU, T. KISHI and J. NX~OMIYA,Japan Inst. Metals J. 31, Part 1, 83 (1967).
A S P L I T - P L A T E N P R E S S U R E CELL FOR T H E M E A S U R E M E N T OF P R E S S U R E D I S T R I B U T I O N IN UPSETTING OPERATIONS G. H . DANESHI a n d J . B. HAWKYA~D Mechanical Engineering Department, University of Manchester Institute of Science and Technology
(Received 8 March 1971)
Summary--A description is given of a form of pressure cell by which the pressure distribution across a deforming billet can be measured. The cell performs essentially the same function as the pin loadcells which have been used by various workers in the field of plasticity research, but it is considered that this new device offers advantages in terms of simplicity of manufacture and maintenance. The construction and cahbration of two pressure cells are described and some preliminary results of friction hill measurements in upsetting tests axe given. Simple analyses are developed to indicate the probable magnitude of errors in the pressure measurements.
INTRODUCTION FOE THE measurement of pressures locally at the surface of metal forming tools the pin loadcell has found fairly extensive use, in rolling, forging, heading and upsetting operations. The pins are situated in closely fitting holes in the tool, the ends being flush with the surface and the forces on the ends are m e a s u r e d using resistance s t r a i n gauges or piezo electric crystals. T h e first i n v e s t i g a t o r to use t h e t e c h n i q u e a p p e a r s to h a v e b e e n Siebel, 1 for m e a s u r i n g roll pressure, a n d v a r i o u s o t h e r w o r k e r s 2,a h a v e d e v e l o p e d a n d e x t e n d e d its application. A c o m p r e h e n s i v e r e v i e w of its d e v e l o p m e n t has b e e n given r e c e n t l y b y Cole a n d Sansome. 4 T h e p r o p e r functioning o f t h e pin loadcell d e p e n d s on t h e close b u t free fit o f t h e pin, to p r e v e n t e x t r u s i o n o f m e t a l into t h e g a p a n d its levelness w i t h t h e s u r r o u n d i n g surface so as n o t to interfere w i t h m e t a l flow, a n d in p r a c t i c e t h e cell is s o m e w h a t difficult to c o n s t r u c t a n d m a i n t a i n . L o c a t i n g t h e device can also p r e s e n t difficulties, because of its size, a n d t h e r e is a l i m i t a t i o n to t h e closeness of spacing for this reason. I t was considered t h a t a d v a n t a g e s m i g h t be o b t a i n e d in some applications b y a n o t h e r f o r m o f pressure m e a s u r i n g device, using small resistance s t r a i n gauges positioned on a surface p e r p e n d i c u l a r to t h e die face a n d close u p to it. T h e m e t h o d offers t h e possibility o f closer spacing o f m e a s u r i n g points t h a n w i t h t h e pin loadcell a n d easier c o n s t r u c t i o n a n d m a i n t e n a n c e . This p a p e r describes e x p e r i m e n t s on such a "split p l a t e n " device, using resistance strain gauges n e a r t h e w o r k i n g surface. T w o pressure p l a t e n s h a v e 355
356 been constructed,
G . H . DA~ESHI a n d J . B. HAWKYARD one using ~ in. foil gauges positioned
from the surface and the second with
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EQUIPMENT
Design of pressure cells A v i e w o f t h e split p l a t e n p r e s s u r e cell is s h o w n in Fig. 1. I t consists of t w o b l o c k s of h a r d e n e d tool steel, g r o u n d u p s q u a r e so t h a t t h e w o r k i n g faces are a t t h e s a m e level w h e n p o s i t i o n e d side b y side o n a firm s u p p o r t i n g surface. To t h e v e r t i c a l i n t e r f a c e of one b l o c k is a t t a c h e d a r o w of s m a l l s t r a i n gauges, close u p t o t h e w o r k i n g face a n d w i t h a x e s p e r p e n d i c u l a r t o it as i n d i c a t e d . T h e r e s p o n s e o f s u c h a g a u g e t o l o a d i n g d e p e n d s o n t h e d i s t a n c e f r o m t h e w o r k i n g face, ideally t h e g a u g e w o u l d b e i n f i n i t e s i m a l i n size a n d p o s i t i o n e d a t surface level, so t h a t i t r e s p o n d e d o n l y t o p r e s s u r e i m m e d i a t e l y a b o v e it. V e r y s m a l l g a u g e are o b t a i n a b l e n o w a d a y s , foil gauges are a v a i l a b l e d o w n t o ~ in. l o n g a n d s e m i - c o n d u c t o r gauges d o w n t o a b o u t ~ in. F r o m p r a c t i c a l c o n s i d e r a t i o n s it w a s d e c i d e d t o use a l a r g e r g a u g e t h a n t h e smallest, b e c a u s e of possible difficulties in a t t a c h m e n t a n d a c c o r d i n g l y t h e g a u g e size c h o s e n for p r e l i m i n a r y w o r k w a s ~ in. T h e d i s t a n c e f r o m t h e c e n t r e of t h e g a u g e t o t h e w o r k i n g surface is ~ in. a n d t h e gauges a r e a t t a c h e d w i t h 0.15 in. spacing, u s i n g E a s t m a n 110 a d h e s i v e . T h e t e s t s o n t h i s p l a t e n g a v e s a t i s f a c t o r y r e s u l t s b u t it was e v i d e n t t h a t b e t t e r r e s u l t s c o u l d b e o b t a i n e d w i t h s m a l l e r gauges f i t t e d closer to t h e w o r k i n g surfeme. A c c o r d i n g l y a s e c o n d p l a t e n w a s c o n s t r u c t e d w i t h gauges ~ in. long p o s i t i o n e d ~ in. f r o m t h e surface t o t h e c e n t r e of t h e gauge, w i t h a s p a c i n g of 0.1 in. T h e gauges a n d wiring a r e s i t u a t e d in grooves in t h e p l a t e n interfaces, as s h o w n in Figs. 2(a), (b), t h e g a u g e c o n n e c t i o n s b e i n g led o u t i n d i v i d u a l l y t o t e r m i n a l s o n t h e sides of t h e blocks. D u r i n g p r e l i m i n a r y c a l i b r a t i o n t e s t s o n t h e first p l a t e n it w a s o b s e r v e d t h a t t h e l o a d - s t r a i n r e l a t i o n s h i p s b e c a m e n o n - u n i f o r m a b o v e a c e r t a i n l o a d level, w h i c h v a r i e d w i t h t h e s p a c i n g b e t w e e n t h e p l a t e n s as d e t e r m i n e d b y shims. I t was d e d u c e d t h a t t h i s o c c u r r e d w h e n t h e t w o h a l v e s c a m e t o g e t h e r a t t h e w o r k i n g surface, d u e t o l a t e r a l e x p a n s i o n , c r e a t i n g b e n d i n g stresses a t t h e gauges. To a v o i d t h i s n o n - l i n e a r i t y w i t h i n t h e w o r k i n g r a n g e , a t p r e s e n t u p t o 36 t o n f / i n ~, it was n e c e s s a r y to h a v e a n initial g a p of 0.011 in. b e t w e e n t h e blocks. T h e s e c o n d p l a t e n was d e s i g n e d t o m i n i m i s e t h i s l a t e r a l s t r a i n i n g , b y m a k i n g t h e g r o o v e w h i c h a c c o m m o d a t e s t h e gauges a n d w i r i n g as shallow as possible i n t h e v e r t i c a l direction. I t was f o u n d t h a t a g a p of 0.006 in. was sufficient to p r e v e n t c o n t a c t , a n d n o n - l i n e a r i t y u p t o a p r e s s u r e of 36 t o n f / i n s. To p r e v e n t e x t r u s i o n i n t o t h e g a p b e t w e e n t h e blocks, a 0.010 in. s p r i n g steel s h i m is p l a c e d o v e r t h e p l a t e n . T h i s p r o v i d e s a c o n t i n u o u s w o r k i n g surface a n d c a l i b r a t i o n t e s t s i n d i c a t e t h a t i t does n o t a d v e r s e l y affect t h e o p e r a t i o n . I t p r o b a b l y does p r e v e n t t h e p l a t e n f r o m b e i n g u s e d t o m e a s u r e f r i c t i o n a l stresses, u s i n g gauges i n c l i n e d a t 45 ° t o t h e w o r k i n g surface, a l t h o u g h t h e g a p c a n b e r e d u c e d a n d t h e s h i m o m i t t e d in some c i r c u m s t a n c e s w h e r e t h e m a x i m u m p r e s s u r e is a p p r e c i a b l y less t h a n 36 t o n f / i n 2. I n u s i n g t h e s h i m i t is n e c e s s a r y to a v o i d l u b r i c a t i n g t h e i n t e r f a c e b e t w e e n t h e p l a t e n a n d s h i m , o t h e r w i s e t h e f r i c t i o n a l forces i m p o s e d b y t h e s p e c i m e n are t a k e n a l m o s t w h o l l y b y t h e s h i m , p o s s i b l y l e a d i n g t o v e r y h i g h t e n s i l e stresses. T h e t w o b l o c k s c o m p r i s i n g t h e first p l a t e n are h e l d t o g e t h e r b y a b a s e p l a t e w i t h s h o r t sidewalls t o p r e v e n t s e p a r a t i o n , b u t t h i s w a s f o u n d to b e u n n e c e s s a r y a n d t h e s e c o n d p l a t e n s i m p l y r e s t s o n a flat g r o u n d b l o c k w i t h o u t u n d e r g o i n g a n y r e l a t i v e m o t i o n . T h e g a p b e t w e e n t h e b l o c k s r e s u l t s in stress c o n c e n t r a t i o n s a t t h e edges, errors due t o t h i s b e i n g m i n i m i s e d b y u s i n g t h e s a m e a r r a n g e m e n t for c a l i b r a t i o n a n d e x p e r i m e n t .
Strain gauge circuit T h e electrical circuit for t h e s t r a i n gauges is s h o w n in Fig. 3. E a c h g a u g e is c o n n e c t e d i n t u r n t o a b r i d g e b y a m a n u a l selector switch, t h e o u t p u t b e i n g fed to a P e e k e l a u t o m a t i c digital s t r a i n m e a s u r i n g s y s t e m a n d r e c o r d e d b y a n a u t o m a t i c s t r a i n g a u g e logging s y s t e m . R e a d i n g s c a n b e t a k e n a t i n t e r v a l s of 1 sec so t h a t t h e a r r a y of gauges c a n b e c o v e r e d i n 20 sec for t h e first p l a t e n a n d 31 sec for t h e second.
A s p l i t - p l a t e n p r e s s u r e cell for t h e m e a s u r e m e n t of p r e s s u r e d i s t r i b u $ i o n
S T R A I N GAUGE
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CALIBRATION
AND
ANALYSIS
OF
RESPONSE
Because t h e strain gauges are n o t s i t u a t e d a c t u a l l y a t surface level t h e o u t p u t s will be influenced b y loading of t h e area surrounding each gauge position. The gauge will react correctly in this respect to a u n i f o r m pressure and it can be reasoned t h a t the pressure on a plane g r a d i e n t should be also tm~ly represented. Non-linear slopes a n d conical pressure hills are subject to some error, however, also t h e edges of pressure hills. The calibration procedure and analysis which follow give some indication of possible errors i n v o l v e d in using t h e calibrations directly and deduce a p p r o p r i a t e corrections. T h e errors a p p e a r to be small for m o s t practical situations, using t h e specimen sizes envisaged for t h e experiments. A n o t h e r source of error in t h e strain gauge response arises due to variations in lateral constraint of the region u n d e r the specimen. The constraint depends on t h e size and shape of the specimen in relation to t h e platen, for a relatively small specimen the surrounding region n o t subject to axial loading imposes considerable lateral constraint, resulting in a reduced axial strain as c o m p a r e d w i t h the condition of u n i f o r m pressure o v e r the whole of t h e platen.
Calibration under uniform pressure Calibration is carried out b y a p p l y i n g u n i f o r m pressure to the p l a t e n face, using t h e e q u i p m e n t shown schematically in Fig. 4. The pressure cylinder is positioned on t h e p l a t e n a n d load is applied, via t h e piston, to t h e plasticine c o n t a i n e d in t h e cylinder. Calibration pressures are in t h e range of 1 to 36 t o n f / i n 2 so t h a t the yield s t r e n g t h of t h e plasticine, a t a b o u t 20 lbf/in 2, produces negligible error. Plasticine was chosen for its ease and cleanliness in h a n d l i n g a n d w i t h t h e close fitting pressure disc and r u b b e r seal u n d e r t h e r a m and t h e differential area sealing b e t w e e n t h e cylinder and t h e platen, there is no leakage during t h e operation. E a c h gauge is calibrated in loading steps corresponding to pressure increments of 1.2 t o n / i n ~. A typical calibration curve is shown in Fig. 5, for each of the platens the overall v a r i a t i o n in slope b e t w e e n gauges is a b o u t 7 per cent. The response will be c o m p a r e d w i t h theoretical values in t h e following section.
Response to point loading The response of a strain gauge to a n o n - u n i f o r m pressure distribution o v e r the p l a t e n can be deduced b y d e t e r m i n i n g the gauge o u t p u t due to loading on an elemental area, defined in relation to the gauge position, and t h e n integrating o v e r t h e surface area. The situation is represented in Fig. 6, the gauge being positioned at m e a n d e p t h h below t h e surface, and the load i n c r e m e n t at distance r f r o m t h e axis of the gauge. I t is assumed t h a t t h e gauge response is i n d e p e n d e n t of angular position 0, this would be so if the p l a t e n
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e x t e n d e d c o n t i n u o u s l y in all d i r e c t i o n s a b o u t t h e g a u g e p o s i t i o n a n d t h e g a u g e was i n f i n i t e s i m a l in size. T h e r e will b e some d i s c r e p a n c y b e c a u s e t h e s e c o n d i t i o n s do n o t hold. T h e i n c r e m e n t a l response, de of t h e g a u g e ( m e c h a n i c a l s t r a i n ) t o p r e s s u r e p o v e r e l e m e n t a l a r e a d A c a n b e e x p r e s s e d in t h e f o r m d~
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w h e r e E is Y o u n g ' s m o d u l u s a n d ~ P o i s s o n ' s ratio, g(v) a c c o u n t s for t h e effect of t h e l a t e r a l c o n s t r a i n t i m p o s e d b y s u r r o u n d i n g regions o f t h e p l a t e n n o t s u b j e c t e d t o a x i a l loading. T h e possible influence of t h i s c o n s t r a i n t c a n b e assessed b y c o n s i d e r i n g a n idealised s i t u a t i o n w i t h a n e x t r e m e c o n d i t i o n of c o n s t r a i n t s u c h t h a t s t r a i n e~, p a r a l l e l t o t h e p l a t e n interface, is zero (see Fig. 6). T h e a x i a l s t r a i n e~ is t h e n r e a d i l y s h o w n t o b e a z ( 1 - v 2 ) / E . F o r steel, w i t h v = 0.29, t h i s b e c o m e s 0.916aJE, so t h a t t h e s t r a i n c a n v a r y b e t w e e n t h e l i m i t s (0"916-1"0) a~/E. I n o r d e r t o e v a l u a t e e q u a t i o n (1) it is c o n v e n i e n t t o d e t e r m i n e t h e f u n c t i o n ](r) a t c o n s t a n t pressure, a n d t h e n a p p l y t h e a p p r o p r i a t e p r e s s u r e d i s t r i b u t i o n r e l a t i o n s h i p . F o r t h e p r e s e n t w o r k t h e f u n c t i o n is d e t e r m i n e d e x p e r i m e n t a l l y b y m e a s u r i n g t h e g a u g e o u t p u t d u e t o a p o i n t l o a d of c o n s t a n t m a g n i t u d e , a p p l i e d a t v a r i o u s d i s t a n c e s f r o m t h e g a u g e position. T h e e x p e r i m e n t a l a r r a n g e m e n t u s e d for t h i s c a l i b r a t i o n t e s t is s h o w n i n Fig. 7. I t was c a r r i e d o u t in a B r i n e l l h a r d n e s s t e s t i n g m a c h i n e , w i t h a s p r i n g steel s h i m i n s e r t e d b e t w e e n t h e b a l l a n d t h e p l a t e n surface t o a v o i d a n y i n d e n t a t i o n . Two t r a v e r s e s were m a d e across t h e p l a t e n face, o n e p a r a l l e l t o t h e g a u g e face a n d as close to it as possible a n d t h e o t h e r p e r p e n d i c u l a r t o t h e face, as i n d i c a t e d . R e s u l t s of t h e s e t w o l o a d i n g t e s t s are s h o w n i n Figs. 8(a), (b) for t h e t w o p l a t e n s . I t is e v i d e n t t h a t t h e s e c o n d p l a t e n , w i t h t h e s m a l l e r g a u g e s i t u a t e d closer to t h e surface, shows a g r e a t e r s e n s i t i v i t y for l o a d i n g d i r e c t l y o v e r t h e g a u g e a n d a v e r y r a p i d fall in r e s p o n s e w i t h d i s t a n c e . I t is t o b e o b s e r v e d t h a t t h i s g a u g e shows negligible r e s p o n s e to l o a d i n g b e y o n d 0.15 in. T h e c u r v e s of Fig. 8(a) are n o t i n fact t r u l y i n d i c a t i v e of t h e a c t u a l r e s p o n s e g r a d i e n t s , b e c a u s e t h e d i s t a n c e m e a s u r e m e n t s x are n o t t a k e n f r o m t h e a c t u a l g a u g e centres, b u t t h e s e r e s u l t s a r e s h o w n r e p l o t t e d in Fig. 8(b), t o g e t h e r w i t h t h o s e t a k e n a l o n g t h e p e r p e n d i c u l a r axes, in t e r m s of a c t u a l
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d i s t a n c e r t o t h e gauge eentres. T h e results follow t h e s a m e f o r m of curve, for t h e t w o d i r e c t i o n s of m e a s u r e m e n t . T h e r e s u l t s of Fig. 8(b) are s h o w n p l o t t e d to a l o g a r i t h m i c scale i n Fig. 8(c), f r o m w h i c h it is e v i d e n t t h a t a t p o i n t s o t h e r t h a n p e r h a p s in t h e i m m e d i a t e v i c i n i t y of t h e g a u g e t h e y c a n b e r e p r e s e n t e d r e a s o n a b l y b y e q u a t i o n s of t h e f o r m Ae A-L = s . e - ~
(2)
w h e r e As is t h e s t r a i n d u e to p o i n t load AL. F o r s i m p l i c i t y t h e r e s p o n s e c u r v e s a r e idealised t o t h e f o r m s h o w n b y t h e d o t t e d lines in Fig. 8(b), c o n s i s t i n g of a region of c o n s t a n t
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(a) FIG. 8. R e s p o n s e of g a u g e t o p o i n t loading. L e g e n d s for e x p e r i m e n t a l p o i n t s a n d d i s t a n c e s are as s h o w n in Fig. 7. (a) R e s u l t s t a k e n parallel t o s t r a i n g a u g e face. (b) R e s u l t s t a k e n parallel a n d p e r p e n d i c u l a r t o s t r a i n g a u g e face, a t c o r r e c t e d d i s t a n c e r. (e) S h o w i n g l o g a r i t h m i c f o r m of gauge response.
s e n s i t i v i t y , e x t e n d i n g to r a d i u s A a b o u t t h e g a u g e axis, a n d a s u r r o u n d i n g g r a d i e n t h a v i n g t h e f o r m of e q u a t i o n (2). ~ c a n b e c o n v e n i e n t l y m a d e e q u a l t o t h e h a l f - w i d t h o f t h e s t r a i n gauge, w i t h i n t h i s r a d i u s t h e f o r m of t h e r e s p o n s e c u r v e h a s n o t b e e n d e t e r m i n e d a n d t h e c o n s t a n t v a l u e a s s u m e d p r o b a b l y gives a n u n d e r e s t i m a t e of t h e a c t u a l response. The calibration measurements involving concentrated loading can be simply related t o a u n i f o r m p r e s s u r e d i s t r i b u t i o n , if it is a s s u m e d t h a t t h e c o n c e n t r a t e d load AL is e q u i v a l e n t to a p r e s s u r e p a c t i n g o v e r a finite a r e a AA. T h e s t r a i n m e a s u r e m e n t s f r o m t h e t e s t s
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and E
- - = 2"34 x 10 -s in~/lb for p l a t e n 2. P0 I n T a b l e 1 t h e s e r e s u l t s a r e c o m p a r e d w i t h t h e o r e t i c a l v a l u e s a n d r e s u l t s of c a l i b r a t i o n s c a r r i e d o u t u n d e r u n i f o r m p r e s s u r e u s i n g t h e p l a s t i e i n e filled cylinder. TABLE 1 Theoretical, no lateral constraint Platen 1 2
Theoretical, with lateral constraint
e
1
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3.33 x 10 -8 3.33 x 10 -8
Calibration under uniform pressure
1 --v ~
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2.9 x 10 -s 2-34 × 10 -8
E
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Derived from point loading tests
T h e v a l u e s for p l a t e n No. I s h o w r e a s o n a b l e a g r e e m e n t b u t t h e p o i n t l o a d i n g r e s u l t o n No. 2 p l a t e n is low, s u g g e s t i n g t h a t t h e r e g i o n close t o t h e g a u g e a n d n o t c o v e r e d b y measurements might be more sensitive than assumed. EXPERIMENTAL
RESULTS
T h e p l a t e n s w e r e u s e d t o m e a s u r e a x i a l stress d i s t r i b u t i o n s o v e r c o m p r e s s i o n s p e c i m e n s o f c i r c u l a r a n d r e c t a n g u l a r forms. T h e s p e c i m e n s were m a d e i n a n a l u m i n i u m a l l o y (HE19V~P) a n d t o a p p r o a c h a n ideal m a t e r i a l c o n d i t i o n , w i t h n o s t r a i n h a r d e n i n g , t h e m a t e r i a l w a s p r e s t r a i n e d i n c o m p r e s s i o n b y a b o u t 20 p e r c e n t before t h e final s h a p i n g . A t y p i c a l stress s t r a i n c u r v e for t h e m a t e r i a l i n t h i s c o n d i t i o n is s h o w n i n Fig. 9. 2S
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Three batches of materials were used, in fact, w i t h s o m e w h a t different yield strengths. I t was considered desirable to conduct p r e l i m i n a r y tests w i t h the friction conditions controlled a n d reproducible, as far as possible, and accordingly t h e lubricant used was pure lead, in t h e f o r m of foil a b o u t 0.004 in. thick, interposed b e t w e e n t h e specimen and platens. The experiments were p e r f o r m e d in a 300 t o n h y d r a u l i c press, w i t h t h e axes of t h e platens and specimen carefully aligned w i t h i n it. Compression was indicated b y dial gauges a n d w i t h the specimen fully plastic and u n d e r a v e r y slowly increasing load t h e strain gauge m e a s u r e m e n t s were recorded in quick succession, at a r a t e of a b o u t 1 sec per reading. Pressure values were t h e n deduced f r o m t h e calibration curves. Results of m e a s u r e m e n t s on circular a n d rectangular compression specimens arc shown in Figs. 10 and 11. 28
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(c) FIG. 11. Pressure distribution on m a j o r axes of a r e c t a n g u l a r specimen 3 in. b y 1 in. a n d 0.2 in. thick, lubricated w i t h lead foil, (a, b) using p l a t e n No. 1 (e) using p l a t e n No. 2.
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368
G . H . DA~ESttI and J . B. HAWKYARD
Discussion of results The pressure m e a s u r e m e n t s shown in Figs. 10 and 11 are disposed fairly s m o o t h l y a n d s y m m e t r i c a l l y o v e r the sections, t h e points being positioned generally quite closely a b o u t m e a n "friction hill" curves, w i t h i n a b o u t 2 per cent in t e r m s of axial pressure. A few of t h e results show a spread of up to a b o u t 1 tonf/in ~ or 4 per cent. These m a y be due to errors in calibration or recording, or otherwise local variations in yield stress or friction m a y be responsible. I t is possible to m a k e estimates of t h e gradients of the friction hills for t h e circular specimens, Fig. 10, assuming t h a t the frictional stress is equal to t h e yield shear stress of t h e lead lubricant. The simple analysis of the friction hill, for a c o n s t a n t frictional stress Ts, provides the expression daz _ - 2~s dr h
(5)
where h is current height and doz/dr is the radial g r a d i e n t of axial pressure. F o r pure lead t h e yield stress Y is a b o u t 1 tonf/in ~, so t h a t t a k i n g the yield shear stress Vs as Y/2 = ½t o n f / i n ~ a n d p u t t i n g h = 0.2 in. gives daz dr
5 tonf/in~/in.
This g r a d i e n t is c o m p a r e d w i t h the e x p e r i m e n t a l results in Fig. 10. E q u a t i o n (5) can be shown to a p p l y also to t h e case of plane strain compression, daz/dr t h e n being t h e axial pressure gradient in the transverse direction and in Fig. 11 t h e calculated slopes are c o m p a r e d w i t h e x p e r i m e n t a l gradients deterrained for t h e short axis of r e c t a n g u l a r specimens, where plane strain conditions are approached. Along t h e m a j o r axis t h e g r a d i e n t is smaller towards t h e centre of t h e specimen, since t h e comp o n e n t of sliding v e l o c i t y in t h a t direction is small. F r o m t h e results shown in Figs. 10 and 11 it is e v i d e n t t h a t the lead lubricant is reasonably effective in controlling the frictional stress a n d t h e e x p e r i m e n t a l gradients agree quite well w i t h t h e theoretical values. The influence of friction in this p a r t i c u l a r operation is, in fact, relatively small in relation to u p s e t t i n g operations generally, and if a rough e s t i m a t e is m a d e of the e q u i v a l e n t Coulomb friction coefficient #~ b y p u t t i n g frictional stress Ts = #~a~-/~ Y we h a v e Vs /x=Y=
__0"5 = 0.023. 22
I f the a l u m i n i u m specimens were compressed w i t h no lubricant, the friction coefficient would be e x p e c t e d to be in t h e range 0-2-0.5 so t h a t m u c h steeper gradients w o u l d occur, for a given specimen configuration.
Estimation of errors As discussed in the previous section, t h e pressure distribution derived directly from t h e strain gauge m e a s u r e m e n t s will be s o m e w h a t in error in certain circumstances, and p a r t i c u l a r l y at the a p e x or edge of a friction hill, where t h e greatest error would be expected. The probable errors at these positions will be e s t i m a t e d for the present tests, using e q u a t i o n (3) w i t h t h e a p p r o p r i a t e pressure relationships inserted. F o r t h e conical hill o v e r the circular specimen t h e pressure relationship is p -- Pnmx - mr where Pmax occurs at the a p e x and m is t h e gradient. Combining this w i t h e q u a t i o n (3) and integrating to radius a gives =
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A s p l i t - p l a t e n p r e s s u r e cell for t h e m e a s u r e m e n t o f p r e s s u r e d i s t r i b u t i o n
369
a n d t h e p r e d i c t e d p e r c e n t a g e e r r o r A for a g a u g e s i t u a t e d a t t h e a p e x is t h e n A
=
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F o r v a l u e s of a, g r e a t e r t h a n s a y 6A, e - ~ c a n b e n e g l e c t e d a n d t h e e r r o r is = (A3/3 + 22/~ + 2 2 / ~ + 2/~3) m × 100. ( ~ / 2 + ~/a + l/a 3) Pma, F o r p l a t e n No. 1 w i t h A = 0.063 in., a = 20, t h i s gives A = 9m/pma x per cent
a n d for p l a t e n No. 2 w i t h A = 0-032 in., a = 33 A = 6 " 4 m / p m a x p e r cent.
F o r t h e r e s u l t s i n Fig. 10 v a l u e s of m = 5 t o n f / i n 2 a n d Pmax = 25 t o n f / i n ~ a r e r e a s o n a b l y r e p r e s e n t a t i v e , g i v i n g A = 1.8 p e r c e n t for p l a t e n No. 1 a n d 1.3 p e r c e n t for No. 2. A s i m i l a r p r o c e d u r e c a n b e a p p l i e d t o d e t e r m i n e t h e e r r o r a t t h e ridge o f a p l a n e s t r a i n p r e s s u r e hill, w h i c h gives A
=
2(23/3 + A 2 / a +
2A/a2+ 2/aa) m
x
100.
~(A2/2 + A/a + 1/a 2) Pmax T h e e r r o r is less t h a n for t h e conical hill b y t h e f a c t o r 2/~r. To o b t a i n a n e s t i m a t e o f t h e e r r o r n e a r t h e edge of a s p e c i m e n w h e r e t h e r e is a n a b r u p t fall o f p r e s s u r e t o zero, t h e s i t u a t i o n is simplified as i n d i c a t e d i n Fig. 12, o t h e r w i s e
STRAIN GAUGE
FIG.
12. S h o w i n g simplified g e o m e t r y for d e t e r m i n i n g t h e r e s p o n s e o f a g a u g e n e a r a s p e c i m e n edge.
t h e a n a l y s i s is v e r y c o m p l e x . I t is a s s u m e d t h a t t h e p r e s s u r e is c o n s t a n t i n t h e v i c i n i t y of a g a u g e n e a r t h e edge a n d t h e edge is c o n s i d e r e d t o b e s t r a i g h t , t h e s e a s s u m p t i o n s a r e g e n e r a l l y r e a s o n a b l e if t h e p r e s s u r e zone i n f l u e n c i n g t h e g a u g e is s m a l l i n r e l a t i o n t o t h e s p e c i m e n size. F u r t h e r simplifications a r e i n t r o d u c e d b y p u t t i n g a l i m i t o n r a d i u s a e q u a l t o 6A, since b e y o n d t h i s r a d i u s t h e influence is negligible (less t h a n 1 p e r cent), a n d b y r a t i o n a l i s i n g
370
G . H . DANESHI and J. B. HAWKYARD
the areas affecting the gauge according to Fig. 12 to simplify the integration. Using equation (5) the response for the case represented is
f)t f~t
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--I
J b JO
( t1 1 I n Fig. 13 this equation is evaluated for the two platens, with gauge output expressed as a fraction of the full response. When the gauge centre coincides with the specimen edge the response is 50 per cent, and 90 per cent, response occurs at a distance from the edge of the specimen to the gauge centre of 0.19 in. for platen )70. 1 and 0.10 in. for No. 2. The superior response of platen No. 2 is clearly evident. I n Fig. 13 are shown also some
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F i e . 13. Experimental and theoretical gauge response across a specimen edge. [] Platen No. 1. • Platen No. 2. corresponding strain gauge measurements taken across the edge of specimens. These suggest t h a t the theoretical curves overestimate errors near the edge. F o r platen No. 2 an error of less than 5 per cent is indicated at 0.1 in. from the edge. CONCLUDING
REMARKS
The analytical and experimental work has indicated that this device for m e a s u r i n g p r e s s u r e d i s t r i b u t i o n is r e l a t i v e l y s i m p l e t o m a n u f a c t u r e a n d s h o u l d g i v e a n a c c u r a c y t o w i t h i n a few p e r c e n t . T h e s u r f a c e s h i m u s e d t o p r e v e n t e x t r u s i o n i n t o t h e g a p p r e s e n t s a disadvantage and limits the possible range of application, but further developm e n t s , i n r e d u c i n g t h e g r o o v e size n e c e s s a r y f o r t h e g a u g e s , f o r e x a m p l e , m a y e l i m i n a t e t h e n e e d f o r t h e s h i m . T h e g a p o f 0.006 in. i n t h e e x i s t i n g equipment can be reduced in proportion to the reduction in the working p r e s s u r e r a n g e , a t p r e s e n t 36 t o n f / i n . P i n l o a d c e l l s h a v e b e e n u s e d b y a f e w
A split-platen pressure cell for the measurement of pressure distribution
371
workers 5 up to 50 tonf/in2 but generally the working pressure appears to have been limited to about 5 tonf/in. 2 Electric resistance strain gauges are available down to ~ in. in length so t h a t in principle the sensitivity and accuracy can be further improved. Acknowledgements--The authors wish to t h a n k Professor W. Johnson for his interest and encouragement in this work. The assistance of the laboratory staff of the Applied Mechanic Division is also gratefully acknowledged. REFERENCES 1. E. SZEBEL and W. LtrEG, mitt Kais--Wilh. Inst. Eisenforgh, Dusseldorf, 15 hfg 1, 1 (1933). 2. E. P. U~xsov, A n Engineering Theory of Plasticity, p. 112, Butterworth, London ( 1961). 3. G. T. V ~ O O Y E ~ and N. I. BACXO~TEN, Int. J. mech. Sci. 1, 1 (1960). 4. I. M. COLE and D. H. SANSOM~, 9th Int. M.T.D.R. Conf., Birmingham (1968). 5. I. GOKYU, T. KISHI and J. NX~OMIYA,Japan Inst. Metals J. 31, Part 1, 83 (1967).