On the cutting of a plate by a wedge

Int. J. Mech. Sci. Vol. 32, No. 4, pp. 293-313, 1990

0020-7403/90 $3.00 + .00 © 1990 Pergamon Press plc

Printed in Great Britain.

ON THE CUTTING OF A PLATE BY A WEDGE G. L u a n d C. R. CALLADINE Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, U.K.

(Received 8 August 1989) A~traet--The paper describes a study of the cutting of a metal plate by a rigid wedge which is pushed into it in a direction almost parallel to the plate. In the main series of tests, wedges were pushed slowly into mild-st~l plates having thickness in the range 0.7-2 ram; and various angular parameters~.g, the included angle of the wedge--were varied. On the hypothesis that the yield stress is the only relevant property of the material, the theory of dimensional analysis gives only two dimensionless groups for a given set of angular parameters; and when these are plotted against each other the curves corresponding to plates of different thickness come together. From this emerges a simple empirical formula for the energy absorbed (W) in terms of the yield stress (~y), the length of cut (l) and thickness (t): W = C1.3~Trll"3t 1"7,

5 < l/t < 150.

Here, C1.3 is a purely numerical constant whose value depends somewhat on the angular parameters. We argue that the exponent of t in this formula is not related to the mechanics of cutting in the vicinity of the wedge tip, but to the mechanics of rolling and folding of "the flaps" which are formed in the wake of the cut, and which affects the exponent of t via the exponent of I. We show that the results of other workers obtained by use of drop-hammer tests may be correlated in the same way, though with somewhat different values of C1.3; and we discuss qualitatively the effects of inertia, variation of friction with sliding velocity and material strain-rate effects in order to resolve these differences. By means of a subsidiary study of the cutting process in plates made of four other materials, we show that friction has a significant effect upon the value of CL3.

NOTATION l n t u B C F H W W~ ct fl r/ 0 au ar

length of cut empirical constant (equation 15) thickness of plate vertical displacement of crosshead of the testing machine width of "shoulder" of wedge, if any (Fig. la) empirical constant (equation 15) cutting force, as recorded by a load-cell on the crosshead of the testing machine hardness of material energy absorbed in the cutting process energy absorbed in the vicinity of the tip of the wedge angle of inclination of plate to vertical (Fig. la) angle between normal to plate and edge of wedge (Fig. la) empirical constant ratio of withdrawing force to cutting force (equation 16; Fig. 7) semi-angle of wedge (Fig. la) ultimate tensile strength of plate material characteristic yield stress of plate material in simple tension

1. I N T R O D U C T I O N W h e n vehicles s u c h as a u t o m o b i l e s a n d ships collide, e n e r g y is a b s o r b e d by t h e i r sheetm e t a l c o m p o n e n t s in a v a r i e t y o f c r u m p l i n g , f o l d i n g a n d t e a r i n g processes. T h i s p a p e r is c o n c e r n e d w i t h o n e s u c h process, in w h i c h a rigid w e d g e cuts i n t o a s t a t i o n a r y m e t a l p l a t e w h i c h is a l m o s t p a r a l l e l to the d i r e c t i o n o f m o t i o n . T h i s process, w h i c h is a n i d e a l i s a t i o n of the c u t t i n g by the b o w s o f o n e ship i n t o the d e c k - p l a t i n g o f a n o t h e r , has b e e n s t u d i e d p r e v i o u s l y b y several w o r k e r s i n c l u d i n g , n o t a b l y , V a u g h a n [1] a n d J o n e s et al. [2]. T h e p r o c e s s is, u n d o u b t e d l y , c o m p l e x . It i n v o l v e s c o m p l i c a t e d plastic flow in the p l a t e in the v i c i n i t y o f the tip o f the w e d g e (and, s o m e w o u l d a r g u e , f r a c t u r e p r o c e s s e s there) as well as l a r g e - s c a l e b e n d i n g o f t h e " f l a p s " o f p l a t e r e m o t e f r o m t h e tip o f 293

294

G. Lu and C. R. CALLADINE

the wedge into various scroll motifs and other folding patterns. Our reason for taking up this problem was a desire to disentangle and understand the various aspects of the behaviour of the plate by combining experimentation with plasticity theory, and by making the smallest number of presuppositions in our thinking about the behaviour. Most of the previous experimental work has been done by dropping a heavy wedge into a vertical or near-vertical steel plate in a drop-hammer rig. We decided instead to conduct experiments by pushing the wedge into the plate in a quasi-static fashion. An obvious disadvantage of this scheme is that it removes the dynamic aspects of actual collision processes; but we believe that this disadvantage is far outweighed by the possibility of recording continuously various features of the cutting process, and thereby obtaining far more information from each individual specimen than is possible in a drop-hammer test. Most of our experiments have been conducted on mild-steel plates; but we have also tested a few specimens of other ductile metals. In this paper we report on the first part of our study, in which we seek gross relationships between the main variables of the problem by means of simple experiments. In a subsequent paper we shall describe further investigations of various kinds into the detailed mechanics of the cutting process. The layout of the paper is as follows. In section 2 the work of previous investigators is summarised, and in section 3 certain shortcomings of their methods of analysis are pointed out. In section 4 our apparatus and experimental methods are described. In sections 5 and 6 some of our "raw" experimental observations are presented, and it is shown how dimensional analysis may be used to extract empirical formulae from them. In section 7 a preliminary assessment of the role of friction in the cutting process is made, by comparing results obtained from specimens of different materials. Finally, in section 8 the relationship between static and dynamic testing is briefly discussed. 2. SUMMARY OF PREVIOUS WORK Figure l(a) shows the basic geometrical features of the cutting set-up, which are common to all previous work and to our own. The notation which we use is indicated on the diagram, and we shall also use it for the purposes of discussing the work of others. For the sake of convenience we shall always describe the wedge as descending vertically into a fixed plate. Akita, Ando, Fujita and Kitamura [3-1 first conducted quasi-static tests with a rigid round-nosed (r = 15mm) wedge (20 = 60 °) which was pushed a distance l into a vertical steel plate of thickness t, and made of material with yield stress ar in simple tension. They proposed a simple conceptual model (Fig. 2) in which the plate exerts a normal compressive stress of ay onto the plate over a nominal area; and thus they obtained, by means of a straightforward equilibrium calculation, an expression for F in terms of the leading variables: F = 2cryt/tan 0.

(1)

This is the earliest recorded analysis of our problem. The formula does not agree at all well with experimental observations which have been made subsequently; and indeed this is not surprising, since the scheme of analysis takes no account of the observed rolling-up of the sheet material in the wake of the cut. Vaughan I-1] studied the behaviour of a plate penetrated by a rigid wedge in order to estimate the damage which would be suffered by a ship's bottom plate if it were cut by a sharp reef or ice projection. He assumed that the work done, W, by a wedge with an included angle 20 when it penetrates a length l into a plate of thickness t is made up of two parts, as follows. First, a certain energy of fracture is required to create a unit area of cut surface. Second, a certain energy of plastic deformation is required per unit volume of displaced, "flap" material. Using the experimental data of Akita and Kitamura [4], together with an empirical formula obtained by Minorsky [5] for the energy absorbed in ship collisions, Vaughan proposed the following formula for the energy absorbed in the cutting

On the cutting of a plate by a wedge

295

F,u

\--.~

\

(a)

IO

(b) FIG. I. (a) Schematic view of a plate specimen about to be cut by a wedge, fl is the angle of rotation of the wedge, about the vertical axis, relative to the mirror symmetric position. Holes used for clamping three of the edges of the plate are not shown. (b) Schematic view of the support for the plate, which allows angle 7 (see (a)) to be adjusted. of a plate by a wedge: W = 33.9 It + 0.095/2ttan 0.

(2)

Here, l a n d t are m e a s u r e d in mm, a n d W is in N m . V a u g h a n [6, 7] subsequently p e r f o r m e d his own e x p e r i m e n t s in o r d e r to investigate the m a t t e r further. H e c o n d u c t e d 64 tests by d r o p - h a m m e r on mild-steel specimens with :~ = 10 °, t in the range 0.75 to 1 . 9 m m a n d 20 having values of 10 °, 30 ° a n d 60 °, a n d he l u b r i c a t e d the wedge. H e m e a s u r e d the length of cut for each specimen, a n d a s s u m e d that the kinetic energy of the falling mass j u s t before i m p a c t was a b s o r b e d by the specimen.

296

G. Lu and C. R. CALLADINE

~F

FIG. 2. Forces acting on a wedge during the cutting process, according to Akita et al. [3]. The plate is assumed to exert a normal pressure, over its nominal cut area, equal to the yield stress or of the material; so F = 21taytan O.

Using the same fundamental idea that the cutting and bending of the plate were separate energy-absorbing processes, he found that in order to fit his own data and Minorski's to a two-term formula for W, it was necessary to allow for the thickness t in each term to be raised to an arbitrary exponent. In this way he obtained the following formula, expressed in the same units as before: W = 5.5 It 15 + 0.004412t 2 tan 0.

(3)

Vaughan chose ~ = 10 ° for his tests because he discovered that with this orientation the bending of the plates occurred by rolling up on one side, in contrast to the more complicated back-and-forth bending which is a characteristic of plates tested with e = 0°: see Fig. 5(b), below. Woisin [8] conducted d r o p - h a m m e r tests with ~ = 0 ° and 20 = 30 °, 70 ° and 100 °. He reported that in 7 out of 19 tests there was no cutting, even though in some cases there was deflection up to 30 thicknesses. O n the basis of a further 13 tests in which two equal plates having thickness in the range 2 to 1 0 m m cut into each other, Woisin gave the following formula for the energy of cutting mild-steel plates: Wc = 4.8 It l"v.

(4)

Here, also, l and t are in m m and W in Nm. Since 1984 Jones and his colleagues have made several experimental studies of the cutting of a plate by a wedge. Thus Jones [9] reports a set of 19 tests on mild-steel plates cut by a wedge in a d r o p - h a m m e r rig with ~t = 0 ° and 20 = 15 °. The thickness of the plates was in the range from 1.5 to 6.1 mm. Jones, Jouri and Birch [10] conducted another 34 tests on plates of the same thicknesses, but with 20 = 15 °, 30 °, 45 ° and 60 °. Further, Jones and Jouri [2] conducted similar tests but on thicker specimens, in the range from 3.2 to 6.0 mm. These tests had 20 = 30 °, 45 ° and 60 °, and again c~ = 0 °. In these various studies, which are summarised in [2], Jones has reported on the testing of a total of 89 specimens. Jones has identified several distinct mechanisms by which energy may be a b s o r b e d - cutting, bending, friction--and he has attempted to partition the energy delivered between them. In particular, he has given the following formulas in the same units as before - f o r the energy absorbed in cutting: Wc = 3.9 It 1"44

for a~. = 255 MPa;

(5)

W~ = 7.2/t 131

for cry = 399 MPa.

(6)

Jones has also estimated the energy of plate-bending by measuring curvatures, etc. of the distorted specimens; and he has concluded that this moiety of the energy absorbed amounts to only 10% of the total. He has also concluded that friction accounts for an a m o u n t of energy of the same order as the energy of distortion. All of the experimental work reported so far has been conducted on plates made from mild steel--which is, of course, the material used most widely in ship-building. U n d e r the supervision of S. C. Palmer, two third-year Cambridge students, Goldfinch [11] and

On the cutting of a plate by a wedge

297

Prentice [12], performed wedge-cutting tests in a drop-hammer rig on plates made from five different materials. Altogether 50 tests were performed at ~ = 10 ° and with 20 = 20 ° and 40 °. The five materials were aluminium alloy, copper, dural, mild steel and brass. The aim was to find out more about the way in which the material properties affect the cutting process. Goldfinch concluded that the total energy absorbed was given by W = Cault q,

(7)

where au is the ultimate tensile strength of the material and C, q are experimentally determined constants. The value of q ranged from 1.4 to 1.7 for the different materials. Inspection of the specimens after testing revealed no signs of cracking of the material ahead of the tip of the wedge, apart from a single dural specimen. Except for the work of Akita et al. [3], all of the work mentioned above has been done by means of a drop-hammer. Shu [13] performed some quasi-static tests on steel and aluminium alloy specimens with thickness in the range 0.9 to 2.4mm. He attempted to calculate the various components of energy absorbed, and in particular he made a study of the rolling-up of the flaps by regarding the process as one of cylindrical bending. Atkins 1-14] has analysed the experimental results of Jones by decomposing the energy absorbed into a fracture component and a plate-bending component, and comparing the performance of plates of different thickness from the point of view of "scaling" theory. This involves making assumptions about modes of deformation which are almost certainly oversimplifications of the actual modes. 3. COMMENTS ON PREVIOUS WORK It is clear from the above outline that although it is a relatively straightforward matter to perform experiments on the cutting of a plate by a wedge in a drop-hammer rig, it is by no means straightforward to understand the results in physical terms. Thus, for example, the apparently simple process of rolling-up the "flaps" in the wake of the cut actually involves complex patterns of incremental flow; and hence the energy absorbed in bending cannot be evaluated merely from an inspection of the final deformed shape. At a lower level, even the process of extracting parameters in hypothetical formulas by means of "curve-fitting" techniques is fraught with difficulty, since there is evidently a lot of experimental "scatter" to be contended with in all data from drop-hammer tests. Two features stand out from the union of the various studies reported above. First, there is a widespread belief that while the yield stress is the appropriate material parameter for a proper discussion of the energy absorbed in the bending which occurs when a wedge penetrates a plate, it is also necessary to think in terms of a fracture parameter in order to account for events in the vicinity of the tip of the wedge. Second, it is remarkable that none of the empirical formulas proposed (apart from (1)) is dimensionally consistent. Thus, there are numerical constants in the formulas whose value would have to be altered if the system of units were changed. This is contrary to the accepted view of Dimensional Analysis that "any properly posed physical law can be expressed in a form which is independent of the arbitrary system of units which is being used": see, e.g. Sedov [15]. In relation to the first point, we observe that it is not necessary to involve the idea of fracture at all in the present investigation. Thus, our problem has some similarity with that of machining, which may be analysed satisfactorily as a process of steady plastic flow. In particular, there is no need in that cutting process for energy to be associated specifically with the creation of new surfaces. The only evidence of which we are aware which supports the idea of a fracture process in the vicinity of the tip of the wedge is the crack seen extending ahead of the tip in a single Dural specimen in Goldfinch's series. In none of our mild-steel test specimens have we seen cracks of this sort, even after careful inspection. In relation to the second point, we shall take it as axiomatic that in developing theories and in plotting results we must never depart from the constraints required for dimensional consistency. As we shall see in section 6, all experimental observations of the kind which are reported in the literature may be plotted in a dimensionless form without any need for hypothetical partitioning of the energy into various components. We shall discover that the MS 32:4-B

298

G. Lu and C. R. CALLADINE

energy absorbed is generally proportional to t r, where y ~ 1.7, much as in formulas (4-7); but we shall find that dimensional analysis demands that the energy is simultaneously proportional to It3-r). The reason why 1 is not raised to fractional powers in various formulas quoted above (2-7) is that most previous workers have a s s u m e d - - o n the basis of quite insufficient evidence--that Wis directly proportional to l. We shall discover that this view is not tenable; this emerges clearly when the data are plotted in appropriate dimensionless terms. 4. APPARATUS AND SPECIMENS The tests were performed in a H o w d e n EU 500 BS universal machine normally running at a crosshead speed of 10 mm/min. Traces of load against crosshead movement were recorded automatically. F o u r different wedges, whose geometrical particulars are given in Table 1, were made from case-hardened mild steel. They were m o u n t e d onto the crosshead of the machine by means of a screw thread on a protruding bolt and an adaptor-piece. The penetrating length of each wedge was 100 ram. Two of the wedges were plain, and two had shoulders. The thinking behind the wedges with shoulders was that they would eventually reach steady-state conditions as the cutting proceeded, in contrast to the plain wedges for which there is no steady state: see section 5, below. N o attempt was made to lubricate the wedges. The clamping-frame (Fig. l(b)) was cut from mild-steel plate of thickness 25 and 13 mm. It was made to hold specimens of plate 275 x 225mm, leaving an exposed area of 215 x 200 mm. Plate specimens were punched with 13 m m diameter holes to register with holes on the frame, through which 12 m m diameter bolts were passed and tightened. The clamping-frame proper was m o u n t e d in a support which allowed the angle ~t to be varied at will between 0 ° and 20 °. The plate specimens were cut by guillotine from 8 ft x 4 ft sheets of mild steel. Specimens from the same "parent sheet" were marked by the same letter. Letters used for this purpose are given in Table 2. Batches of specimens were heat treated by holding them in vacuum at 900°C for 1.5h, and cooling them in the furnace. The thickness of each specimen was measured by micrometer. A tensile-test c o u p o n with a parallel section ~ 13 m m wide and 65 m m long was cut from one specimen of each parent plate. This was fitted with an extensometer and tested in the H o w d e n machine. In addition, Vickers hardness tests were performed on each specimen. The original intention was to use the initial yield stress from the tensile test in order to characterise the material of all specimens cut from the same parent plate; hardness tests were originally intended as a check on the uniformity of the material. It was discovered, however, that the hardness H varied from specimen to specimen with a range of + 5%. It was then decided to use try = H/3

(8)

as the relevant measure of the yield stress for each specimen. It is well k n o w n 1-16] that H/3 is equal to the true yield stress which would be measured in a tensile or compressive test at a strain of about 0.07; and it was t h o u g h t that a strain of this level might well be

TABLE I. PARAMETERS OF THE FOUR WEDGES USED

No.

20 (°)

1 2 3 4

40 20 20 20

B (mm) Type 20 10

plain plain shoulder shoulder

The included angle is 20, and B is the overall width of the shoulders on wedges 3 and 4: see Fig. l(a). The shape of wedge 4 can be seen in Fig. 4(b).

On the cutting of a plate by a wedge

299

TABLE 2. CODING OF THE STEEL SHEET FROM WHICH THE SPECIMENS WERE CUT

Thickness of sheet (mm) 0.72 0.90 1.6 2.0

Code F A, B, C D E

Each sheet was assigned a unique letter. representative of strain levels in the plastically deforming regions of the plate specimens. For all thicknesses of material it was found that the mean value of HI3 for all specimens exceeds the initial try measured in the tensile test by 40%. Subsequent examination of the test-data both by means of try as defined in (8), and by try taken as the initial yield stress in a tensile-coupon test, showed neither better nor worse empirical correlations: see section 8, below. 5. TEST RESULTS: TYPICAL CASES The first seven columns of Table 3 give the conditions under which we tested the various plate specimens. In the following discussion we shall be concerned mainly with results for fl = 0 °, and so we have not listed tests for fl ~ 0 in the table. Here and elsewhere the absence of a value for B indicates that a plain wedge was used.

Test condition ~ = 10 °, 20 = 40 °, fl = 0 ° This may be regarded as the most conventional test condition, because it has been used by previous workers, e.g. Vaughan [7]. A typical F,u curve (for t = 0.9mm) is shown in Fig. 3, together with a photograph of the final deformed shape of the specimen. The initial part, OA, of the trace corresponds to a process of plane-stress wedge indentation: the plate remains flat and the wedge cuts into it. At point A the plate is pushed over sideways, in a kind of buckling process. The falling trace AB involves no further cutting: the advance u of the crosshead is associated with out-of-plane bending of the plate. Cutting eventually re-commences, and the cutting force then increases as the cut lengthens. The trace rises steadily with u; and it seems clear that this steady increase of cutting force corresponds broadly to the enlargement of the zone of rolled-up material. Just as in Vaughan's tests, the inclination of the plate at ct = 10 ° guarantees a one-sided rolling up of the plate material as shown in Fig. 3(b). Test condition ~ = 10 °, 20 = 20 °, fl = 0 °, B = 10 mm When the wedge is truncated the main features of the physical behaviour, and of the F,u trace, are of the same kind as in the preceding case until the upper edge of the plate reaches the shoulder of the wedge, whereupon the force F remains practically constant as the crosshead descends further. Figure 4(a) shows the F,u curve for t = 1.6 mm, and (b) is a photograph of the final state of the plate. Test condition ~ = 0 °, 20 = 40 °, fl = 0 ° The main difference between this test and the first one is that here the plate is vertical, = 0 °. The shape of the F, u curve and the final deformed shape of the specimen, shown in Fig. 5 for t = 1.6 mm, are both quite different from those of Fig. 3. The initial stages are rather similar; but whereas in Fig. 3 the rising F, u curve BC becomes progressively less steep as cutting proceeds, here the curve rises to a peak. What happens is that, as already described, the sheet material buckles at the first peak and bends over in one direction. Since the wedge is now cutting through inclined material, the region of the plate which is currently being cut moves back along the tip-edge of the wedge. The process is a stable one with ct = 10°; but with ct = 0 the plate moves progressively into a position where it again buckles, but in the opposite direction. Thus, the successive peaks of the F,u curve correspond to the periodic changes of direction of bending of the plate. As mentioned above, ~ = 0 ° is the test

300

G. Lu and C. R. CALLADINE TABLE 3. SUMMARYOF THE RESULTSOF OUR TESTS,AND OF THOSE OF OTHER WORKERS

B (mm)

Static (S) or Dynamic (D)

No. of Specimens

C

n

C1.3

20 10 10 10

S S S S S S S S

2 6 5 5 2 4 8 3

3.5 1.9 3.1 1.9 2.8 2.5 2.1 0.9

1.2 1.3 1.2 1.3 1.2 1.2 1.2 1.4

2.4 2.0 2.3 1.9 2.2 1.9 1.9 l.l

(°)

20 (°)

0 10 0 10 0 0 10 20

40 40 20 20 20 20 20 20

Aluminium Brass Copper Dural

10 10 10 10

20 20 20 20

S S S S

1 1 1 1

1.0 1.4 1.4 1.6

1.5 1.3 1.4 1.2

2.2 1.4 2.2 1.2

Goldfinch [1 I] and Prentice [12]

Mild steel Aluminium Brass Copper Dural

10 10 10 10 10

20 20 20 20 40

D D D D D

11 13 13 7 3

0.9 1.9 0.6 2.2 2.8

1.5 1.2 1.4 1.2 1.0

2.0 1.5 1.0 1.4 0.8

Jones

Mild steel

0 0 0 0

15 30 45 60

D D D D

5 25 29 27

3.8 3.9 4.8 4.3

1.4 1.3 1.3 1.3

5.4 4.6 4.5 4.6

Workers

Material

Present Studies

Mild steel

eta/. [2]

"Static" tests were conducted slowly in the Howden machine, as described, while "dynamic" tests were conducted in drop-hammer rigs. Least-squares methods were used to determine values of C, n and C 1.3 (see section 6) for data with l/t > 5: values are quoted here to two significant figures.

condition preferred by Jones. Force-deflection curves of the general type of Fig. 5(a) were sometimes found for ~ = 10 °, even though the plate did not actually bend back and forth; but for ~ = 20 ° the F,u curves were always of the kind shown in Fig. 3(a). Test condition ~ = O, fl 4: 0

In all of the preceding cases the mode of deformation of a plate is symmetrical, or nearly so, about the cutting line when viewed along the plane in which the tip of the wedge moves. However, this is obviously not so when fl 4= 0, as shown in Fig. 6. The F,u curve is broadly similar to that for c~, fl = 10 °, 0 °, although the descending portion AB is absent when > 30 °, approximately. Nevertheless, the cutting mechanism is very different from cases in which fl = 0, and it seems clear that elastic deformation begins to play an important part in the bending of the plate; indeed the plastic "rolling up" of the plate is almost entirely absent.

Other types of test We have supplemented the main series of tests, as described above, by a number of tests of different kinds, aimed at elucidating particular aspects of behaviour. Thus, some specimens have been monitored by strain gauges, some have been coated with brittle lacquer, and others have been inscribed with regular grids. These and other tests will be described in detail in a subsequent paper. For present purposes only one kind of subsidiary test is relevant. This was a test designed to find out something about friction at the interface between the wedge and the plate. A test of this sort began in the usual way, with ~ = 10 ° and 20 = 20 °, and the wedge was pushed into the plate until a pair of"flaps" had developed. The testing machine was then stopped and set into reverse, and the wedge was slowly withdrawn. It was found that the wedge pulled itself free from the plate only after a significant tension had been applied; and that the disengagement occurred suddenly, without warning. Figure 7 shows a typical force/displacement trace from a test of this sort, with a total of four of these pulling-out assays. It seems clear that the amount of force

On the cutting of a plate by a wedge

3.0

i,,,,,,ILi1,1,,i,,i

301

[III,,L,LL

F(kN) 2.5

2.0

1.5

1.0

0.5

u(mm)

(a) 0.0

50

I O0

150

FIG. 3. (a) Plot of cutting force F against crosshead displacement u for a steel specimen B11 having t = 0.9 mm, ct = 10 °, ,8 = 0 °, 20 = 40 °. Letters against the curve are referred to in the text. (b) Photograph of the specimen at the end of the test, at l ~ 80 mm. The spacing of the holes is 75 mm.

302

G. Lu and C. R. CALLADINE

F(kN)

u(mm)

(a) 0

.

0

.

.

.

.

.

.

.

.

l

50

.

.

.

.

.

.

'

'

'

I

IO0

.

.

.

.

.

.

.

.

.

150

FIG. 4. (a) F,u plot for steel specimen I)08 having t = 1.6 ram, ~t = 10°, fl = 0 °, 20 = 20 °, B = 10 mm. (b) Photograph of the specimen at the end of the test: note the profile of the shouldered wedge.

On the cutting of a plate by a wedge

303

14

12

10

8

6

4

2

(a)

u(mrn) 0

50

1O0

150

FIG. 5. (a) F,u plot for steel specimen D02 having t = 1.6mm, c t = 0 °, f l = 0 °, 2 0 = 4 0 °. (b) Photograph of the specimen at the end of the test, showing the back-and-forth folding. The peaks of the curve in (a) correspond to changes in direction of folding.

304

G. Lu and C. R. CALLAD1NE

F(kN)

u(mm)

(a) .

0

FIG. 6. (a)

F,u

.

.

.

'

.

.

.

.

J

50

[

I00

150

plot for steel specimen DI1 having t = 1.6mm, : t = 0 °, f l = 4 5 °, 2 0 = 2 0 °. (b) Photograph of the specimen at end of the test.

On the cutting of a plate by a wedge i

i

i

~

]

i

L

J

,

I

,

,

L

,

I

,

305 J

,

]

i

L

_Y

f -2

-4

u(rnm) I 40

60

80

.

.

.

.

I

.

.

1 O0

.

.

I 120

.

.

.

. 140

FIG. 7. F,u plot for a copper specimen with four interruptions for pulling-out assays; t = 1.6mm, ct= 10°,/3=0 °,20=20 °.

needed for disengagement is related to the force required for cutting immediately before unloading; and indeed the ratio of the two forces is found to be roughly constant. Tests of this sort were conducted on steel specimens, and also on specimens of aluminium alloy, brass, copper and dural which were left over from the study by Goldfinch and Prentice, mentioned earlier. An "ordinary" static test was also made on a fresh specimen of each of these materials. 6. TEST RESULTS: USE OF DIMENSIONAL ANALYSIS It is clear that for a given plate specimen the F,u trace depends on the angles ~,/3, 0 which define the test condition and, of course, on the thickness of the plate. The yield stress of the material is also relevant, since there is evidently considerable plastic deformation in the process of penetration. Thus, for a test in which the angular parameters ct,/3, 0 are prescribed, we may state: F depends on u, t, a r.

(9)

Other factors which might be invoked as being relevant to the process are E, the Young's modulus of elasticity of the material (and perhaps even a second elastic measure) and the linear dimensions of the specimen support frame. But we shall reject these for the present, on the grounds that (except as noted above) there is little sign of elastic deformation; and there is no case in our testing program (except again for/3 :~ 0) in which the enlarging zone of plastic deformation reaches the edges of the frame. We explicitly exclude fracture toughness as a relevant parameter, since, as already mentioned, there is no evidence of cracking ahead of the tip of the wedge. The dimensions of the problem may conveniently be taken as Force and Length: a total of two. Four variables are involved in (9), and Buckingham's rule (e.g. 1-17]) tells us that there will be 2( = 4 - 2 ) independent dimensionless groups. It is obviously convenient to have F and u in different dimensionless groups, and so we find, by the usual procedures, that F u depends on - . art 2 t

(10)

This immediately suggests that we should plot one of these groups against the other; and this is what has been done in Fig. 8 for ~ = 10 °, 20 = 20 °, for specimens of four different thickness. Although these curves do not lie perfectly on top of each o t h e r - - a n d indeed three curves are of the rising-and-falling kind mentioned a b o v e - - t h e agreement is sufficiently good to warrant further exploration along these lines. Note, incidentally, that the initial peaks fit less well than the other parts of the curves. This suggests that the initial "buckling"

306

G. L u and C. R. CALLAD1NE

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of the plate depends partly on elastic effects. We shall not pursue this further, since our main aim is to investigate the energy-absorbing aspects of deep penetration. Now it will in general be more useful to work in terms of W, the energy absorbed, instead of F, the force. Thus we define

W=

Fdu.

(11)

0

Hence W can be evaluated from an F,u curve at any point; and here again we shall ignore elastic recovery effects. Furthermore, it will be useful to change from variable u, the crosshead displacement, to l, the length of the cut, by means of the trigonometrical formula I = u sec ~.

(12)

W depends on l, t, ay;

(13)

In this way we may write, in general,

and in dimensionless terms this becomes W o'yt 3

depends on

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(14)

Figure 9 shows plots of these groups against each other for two different testing conditions, and with different curves for different values of t. These plots are logarithmic. It is clear that the curves corresponding to different thickness lie close to each other. It is this observation which justifies the hypothetical exclusion of all other factors such as fracture toughness, elastic modulus and frame size under the present test conditions. Various comments are relevant at this point. First, the use of an integrated variable, W, and also of a double-logarithmic plot tend to reduce apparent differences between curves: compare Figs 8 and 9. Nevertheless, the agreement is very good overall, and there are no systematic trends which warrant the addition of another variable to the analysis. Second, the way in which the curves for different values of t fall onto a single curve justifies the scheme. The fact that the various curves are nearly straight on the double logarthmic plots is of no consequence in this connection: note in particular that peaked curves for ~ = 0 such as Fig. 5 become wavy lines in Fig. 9(b). Third, the fact that the various curves in Fig. 9 are nearly straight from l/t ~ 5 to l/t ~ 150--a factor of 30 in l/t--will be useful in obtaining empirical formulas, even though

On the cutting of a plate by a wedge

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FIG. 9. (a) The data of Fig. 8 re-plotted as dimensionless energy absorbed by means of (11) and (14), against dimensionless length of cut, on logarithmic scales. It is hard to distinguish between some of the curves: a = 10 °, fl = 0 °, 20 = 20 °. (b) As for (a), but with ~t = 0°: curves like that of Fig. 5(a) come out wavy in this plot.

it is irrelevant to the justification of the dimensional analysis. In general the curves do not follow the same plan for lit < 5, approximately; but this is not surprising in view of the rather complicated F,u traces seen in the early stages of cutting. The obvious next step in correlating the results is to find the equations of lines which give a good fit to the data of Fig. 9. This has been done by use of the least-squares technique; and the results are shown in the last three columns of Table 3. In general we find

W_c

O'vt3

(~)"

/ t

150,

(15)

where n and C are (purely numerical) constants to be determined. The values of C and n vary according to the test condition, with C in the range 0.9 to 3.5 and with n lying in the range 1.2 to 1.4, approximately. We shall argue below that since energy is absorbed by two different processes, the value of n in the single-term empirical formula (15) is not of fundamental significance. Accordingly, it is reasonable to simplify the matter of the constants by introducing a single, standard value of n = 1.3, and then work out the bestfitting value o f C = C1. 3. In this way we obtain the results shown in the last column of Table 3.

308

G. Lu and C. R. CALLADINE

Inspection of the values of C1. 3 given for our tests in Table 3 leads directly to a major conclusion: the energy absorbed is sensitive to the value of ct, but insensitive to the value of 0. Thus, a change of ~ from 0 ° to 10° lowers the energy absorbed for a given value of l/t by around 17%, while the larger change from ct = 0 to 20 ° reduces the energy absorbed by 40%. In contrast, a change of 20 from 20 ° to 40 ° increases the energy absorbed by only about 5% on the same basis. It is perhaps not surprising that ct = 0 requires more energy than ~ = 10°: the succession of peaks in F as the plate bends back and forth (e.g. Fig. 5) must contribute to this effect, even though it has little influence on the value of n. But there is no obvious reason why a change from ct = 10° to 20 ° should make so much difference. The insensitivity of the energy absorbed to the angle 0 - - a t least, within the range of values tested here--is not straightforward to explain, either. 7. THE EFFECT OF FRICTION It is clear from the "pull-out" tests described above in section 5 that there is significant friction and/or adhesion between wedge and plate in a typical test. It is also clear from a detailed examination of the data that the ratio ~/= (force of withdrawal)/(force of penetration)

(16)

in a test such as that of Fig. 7 has different values for tests on different materials: see Table 4. Now the values of constant C1.3 listed in Table 3 for static tests on different materials are also different. Thus it is possible that these differences are related to differences in friction and/or adhesion between the steel wedge and the different plate materials. The solid points in Fig. 10 are a plot of C1.3 against ~/for different materials. Although there is a fair amount of scatter, and although we have no definite scheme for interpreting values of r/in terms of a "coefficient of friction", this plot suggests fairly strongly that frictional effects contribute significantly to the value of C1.3. Another possible explanation for the differences in the value of CL3 for different materials, is that they are somehow attributable to differences in the strain-hardening characteristics of the materials. To examine this hypothesis we have re-interpreted the raw test data on the different materials by using for try the initial yield stress, as determined in a tensile test on a coupon from the plate, in calculating the value of C~. 3. When values of C a.3 computed in this way are also plotted against r/, (the open points in Fig. 10) the general features of the plot are preserved. Thus we may conclude that frictional effects are mainly responsible for the different values of CL3 for the different materials. 8. COMPARISON BETWEEN QUASI-STATIC AND DROP-HAMMER TESTS It is plain that our policy of investigating the phenomena of cutting a plate by a wedge by means of quasi-static testing has enabled us to clarify a number of aspects of the behaviour. But we also need to relate our work to the results of dynamic testing, since in general the motivation for our study is the energy absorbed in collisions. First we shall examine the relationship between the drop-hammer tests of Goldfinch [11] and Prentice [12] and our own quasi-static tests. Then we shall compare the drop-hammer tests of Jones with our work.

TABLE 4. VALUES OF r/FOR DIFFERENT MATERIALS Material

Mild Steel Aluminium Brass Copper Dural

~/ m e a n

~/max

i/min

N o . of a s s a y s

0.40 0.64 0.18 0.32 0.13

0.53 0.71 0.21 0.36 0.14

0.24 0.61 0.13 0.25 0.11

4 4 3 4 2

This measure of friction/adhesion, defined in (16), was assayedin pull-out tests of the kind shown in Fig. 7. Mean, maximumand minimum values are quoted.

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It is convenient to begin with the work of Goldfinch and Prentice because their specimens had been tested in our laboratory, and it was a straightforward matter for us to perform quasi-static tests on specimens nominally identical to theirs. Results from the drop-hammer tests have been plotted in Fig. 12(a). Portions of Table 3 present values of the parameter C1.3 as deduced by drop-hammer tests and quasi-static tests, respectively, on specimens of the five different materials. Figure 11 plots the values of C~.3 from the two sorts of experiment against each other. The most obvious feature of this plot is that, with the single exception of mild steel, the value of C1.3 from drop-hammer tests is about 70% of the value from quasi-static tests. How are we to interpret these results? There are, in principle, three factors in which there may be differences between the two tests. The first is inertia: rapid deformation may involve accelerations high enough to alter the pattern of loading on the specimens. The second is strain-rate effects: for some materials--and notably, of course, mild steel--the plastic flow stress is sensitive to the strain-rate [18]. The third is friction, for it is well-known that the coefficient of friction in dynamic conditions is lower than that in static and quasi-static sliding [ 191. Of these three factors, the first two tend to raise the load in a dynamic test, while the third tends to lower it. In the present experiments there is no evidence for different patterns of deformation between static and dynamic conditions, and so it seems likely that inertia effects are not significant. Thus the obvious explanation of Fig. 11 is that for the aluminium alloy, brass, copper and dural specimens the value of C1.3 is lower in the drop-hammer tests by virtue of lower frictional forces; and that, while the same is true for the mild-steel specimens, there is a compensating effect on account of strain-rate. The magnitudes of these effects are not unreasonable. Thus, from the (admittedly sparse) data in Bowden and T a b o r [19] we may perhaps take as ~ 0.7 the factor by which the coefficient of friction of a "typical metal" is reduced when sliding is speeded up from quasi-static conditions to the order of 1 m/s; and this could well correspond to a factor of around 0.7 in the value of C~.3. We can make a rough estimate of strain-rate effects as follows. Strain gauges attached to the surface of one of our mild steel specimens of thickness 2 mm, at approximately 5 m m from the line of cutting [20], recorded a maximum spatial rate of strain of about 1% per m m penetration in a quasi-static test. Taking a typical cutting speed as 4 m/s in a drop-hammer test, and assuming that the quasi-static straining pattern is preserved, we would obtain a peak strain-rate of about 40 s - ~ in the plastically deforming region in the vicinity of the tip of the wedge (in a drop-hammer test). The Cowper-Symonds relationship (e.g. [18]) indicates that the static yield stress of mild steel is doubled at this strain rate. We need to

310

G. Lu and C. R. CALLADINE

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On the cutting of a plate by a wedge

311

know, of course, the dynamically raised yield stress at a typical point in the plastically deforming region, rather than at the point which experiences the highest strain rate. Perhaps a 30% increase in yield stress overall on account of strain-rate effects is not unreasonable. We have collected data from drop-hammer tests performed on mild-steel plates by Jones and his colleagues [2] and have re-plotted it in the manner of Fig. 9, as shown in Fig. 12 (b). Plots of this kind done separately for the different values of wedge angle 20 yield a range of values of C1. 3 in the region of 4.6 + 0.8 for data having I/t > 5: see Table 3 for details. The best fitting value of C~.3 for the union of all the data is 4.6. The most obvious feature of these results is that the values of C1.3 obtained by Jones are about double those found by quasi-static testing in our experimental program on mild steel plates, and those obtained by Goldfinch and Prentice in their drop-hammer tests. Reflection on various points enables us to reduce the gap between the two sets of results; but there still remains a substantial difference. Thus, the fact that Jones quotes material yield stress for initial yielding of coupons in tension, whereas we have used H/3 as a measure of yield stress, makes a difference in the value of C1.3. Measurements to which we have already referred indicate that for mild steel the ratio of these two quantities is about 1.4; and this would lower Jones' value of C 1.3 to 3.3 on our scale. Also, of course, Jones used ~ = 0 ° for his tests, where Goldfinch and Prentice had ~ = 10 °. We have already stated that a change in ~ from 0 ° to 10 ° lowers the value of C~. 3 by about 1 7 o in our static tests. If we applied this factor to Jones' data, in order to "convert" his results to the conditions used by Goldfinch and Prentice, we would obtain C1.3 = 2.7. This is, of course, still substantially greater than the value of 2.0 obtained by these workers. A possible explanation of this discrepancy is that the change in value of C~.3 in going from ~t = 0 ° to ~ = 10° is different in dynamic and static tests. In this connection we note that our colleague L. L. Tam has discovered an apparent loss of energy on impact in drophammer testing of strut-like specimens which is not found in tests on beam-like specimens. The key to the situation is that energy appears to be lost in circumstances where there is a high initial peak in the quasi-static load-deflection curve. Our quasi-static tests certainly reveal a higher initial peak load for ct = 0 ° than for ct = 10 °. Further work will be required in order to resolve the remaining difference between the results of Jones and ourselves. The most effective start on this would be to conduct some quasi-static tests on specimens having the parameters of Jones' program. This would also shed some light on the claim of Jones (private communication) that thickness of plate per se--rather than thickness incorporated into a dimensionless group--makes a difference to the behaviour.

9. D I S C U S S I O N

Our general empirical formula (15), with n = 1.3, may be re-arranged as follows: I,V = C1.3~Tyll'3t 1"7.

(17)

Here, the value of C1.3 depends on testing conditions. This version of the formula brings out the relationship, which is required by dimensional analysis, between the powers to which l and t are raised: their sum must be 3.0. Previous investigators have overlooked this constraint and thus have not been able to extract as much from their experimental data as they could have done. Since dW/du = F, and d(..)/du = d(..)/d/sec ~, we may obtain from (17) F = 1.3 (sect) C1.3tryl°'3t 1"7.

(18)

We can now begin to see where the non-integral powers of I and t in various empirical formulas come from. In effect, the empirical straight line on the plot as in Fig. 9 is equivalent to a single-term power-law approximation of curves such as those in Fig. 8 by F oc /o.3.

(19)

312

G. Lu and C. R. CALLAD|NE

When lit > 10, say, the difference between the actual curve and its one-term approximation becomes negligible in its effect on W. In this connection it is important to realize that we do not proceed to argue that the entire operation of cutting a plate by a wedge may be reduced to a single process. Thus, although we deny that a fracture parameter is necessary for correlation of the experimental observations, we believe that there are nevertheless two essentially distinct processes of plastic deformation involved in the near-tip and far-field regions of the plate, respectively. O u r view at present is that the force associated with the near-tip process is practically constant and, in particular, independent of l; while that which is associated with the rolling-up of the plate remote from the tip increases with I in a rather complicated way. It turns out that the sum of the two effects may be approximated empirically, for sufficiently large values of l/t, by a single term; and it is this which enables us to obtain single-term empirical formulas (17) or (18). In other words, we have not assumed a single-term formula, but have discovered one after having used the due processes of dimensional analYsis. We have pointed out above that the exponent 1.7 to which thickness t is raised in (17) is a direct reflection of the fact that the cutting force increases with l as in (18) on account of the formation of flaps. In particular, it is not related to anything fundamental about the behaviour of the plate in the immediate vicinity of the tip of the wedge. In this connection we note that Yu et al. [21] have claimed that the energy required to tear a ductile plate in a steady-state "trousers test" is proportional to t ~ where 7 is an empirically determined exponent whose value is in the region of 1.5. Thus in our view there is no fundamental connection between these two superficially similar results. In the present report we have described a study which has involved the assembly of data from a large n u m b e r of relatively simple tests, and the extraction of empirical formulas by the use of dimensional analysis and curve-fitting techniques. We believe that our use of quasi-static testing has eliminated a major source of scatter and indeed other, u n k n o w n factors associated with drop-weight testing; and this has enabled us to show convincingly that fracture toughness is not a relevant parameter for quasi-static testing, at least. The other assays which we have made on the details of straining in the material in the vicinity of the tip of the wedge [20], and which we shall describe elsewhere, will, we hope, enable us to achieve a more incisive understanding of the processes which are involved when a wedge cuts into a plate. Acknowledgements--G. Lu acknowledges, with thanks, financial support from the Government of the People's

Republic of China and the British Council. The work was done in connection with SERC Research Grant GR/D 7684.4 on Dynamic collapse and tearing of structures. We th~nk Mr S. C. Palmer for continued help and interest, and Professor D. Tabor for a helpful discussion. REFERENCES 1. H. VAUGHAN,Bending and tearing of plate with application to ship-bottom damage. The Naval Architect 97 (May 1978). 2. N. JONESand W. S. JOURI,A study of plate tearing for ship collision and grounding damage. J. Ship Research 31,253 (1987). 3. Y. AKITA,N. ANDO,Y. FUJITAand K. KITAMURA,Studies on collision-protective structures in nuclearpowered ships. Nucl. Engn9 Design 19, 365 (1972). 4. Y. AKITAand K. KITAMURA,A study on collision by an elastic stem to a side structure of ships. J. Soc. Naval Architects of Jap. 131, 307 (1972). 5. V. U. MINORSKY,An analysis of ship collisions with reference to protection of nuclear power plants. J. Ship Research 3, 1 (1959). 6. H. VAUGHAN,The tearing and cutting of mild steel plate with application to ship grounding damage, in Proc. 3rd Int. Conf. Mechanical Behaviour of Materials (edited by K. J. M1LLERand R. F. SMITH),Vol. 3, pp. 479-487 Pergamon Press, Oxford (1979). 7. H. VAUGHAN,The tearing of mild steel plate. J. Ship Research 24, 96 (1980). 8. G. WOISIN,Comments on Vaughan: the tearing strength of mild steel plate. J. Ship Research 26, 50 (1982). 9. N. JONES,Scaling of inelastic structures loaded dynamically. In Structural Impact and Crashworthiness (edited by G. A. O. DAVIES),Vol. 1, pp. 45-74. Elsevier Applied Science Publishers, Amsterdam (1984). |0. N. JONES,W. S. JOURIand R. S. BIRCH,On the scaling of ship collision damage. In Proc., 3rd Int. Congress on Marine Technology, 287-294 Athens International Maritime Association of East Mediterranean, Phivos Publishing Co., Greece (1984).

On the cutting of a plate by a wedge

313

11. A. C. GOLDFINCH,Plate tearing energies. Part II Project, Engineering Department, Cambridge University (1986). 12. J. PRENTICE, Wedge drop tests to investigate plate tearing characteristics. Part II Project, Engineering Department~ Cambridge University (1986). 13. D. SHU, Energy dissipation in cutting plates and splitting tubes, (in Chinese). M.Sc. Thesis, Peking University (1986). 14. A. G. ATKINS,Scaling in combined plastic flow and fracture. Int. J. Mech. Sci. 30, 173 (1988). 15. L. I. SEDOV, Similarity and Dimensional Methods in Mechanics. Infosearch Ltd. (1959). 16. D. TABOR, The Hardness of Metals. University Press, Oxford (1951). 17. E. BUCKINGHAM,On physically similar systems: illustrations of the use of dimensional equations. Physics Ret'iew 4, 347 (1914), 18. N. JONES, Structural aspects of ship collisions. Structural Crashworthiness (edited by N. JONES and T. WIERZr~ICK1),pp. 308--338. Butterworth Press, London and Boston (1983). 19. F. P. BOWDENand D. TABOR, The Friction and Lubrication of Solids. Oxford University Press, Oxford (1950). 20. G. Lu, Cutting of a plate by a wedge. Ph.D. dissertation, University of Cambridge (March 1989). 21. T. X. Yu, D. J. ZHANG,Y. ZHANGand Q. ZHOU, A study of the quasi-static tearing of thin metal sheets. Int. J. Mech. Sei. 30, 193 (1988).