Experimental investigation of the scaling laws for metal plates struck by large masses

Int..I. Impact Engng Vol. 13, No. 3, pp. 485-505, 1993 Printed in Great Britain

0734-743X/93 $6.00 + 0.00 © 1993 Pergamon Press Ltd

E X P E R I M E N T A L I N V E S T I G A T I O N OF THE SCALING LAWS FOR METAL PLATES S T R U C K BY LARGE MASSES H E - M I N G WEN t a n d NORMAN JONES Impact Research Centre, Department of Mechanical Engineering, The University of Liverpool, PO Box 147, Liverpool L69 3BX, U.K. (Received 19 June 1992; and in revised form 30 October 1992)

Summary--An experimental investigation is presented into the geometrically similar scaling laws for circular plates impacted by cylindrical strikers with blunt ends travelling at velocities up to 5 m s - 1 which produce large permanent transverse displacements and perforation in some cases. The plate dimensions have a scale range of four for the mild steel (strain rate sensitive) specimens and approximately five for the aluminium alloy (strain rate insensitive) specimens. The experimental results obey the geometrically similar scaling laws within the accuracy expected for such tests. It is observed that the impossibility of geometrically similar scaling of the material strain rate sensitive effects appears not to influence the plate perforation energies, at least within the range of experimental parameters studied in the present investigation.

NOTATION d r ri t tr trp tO tOp D Do E Ek Ep Epp F F, Fap Fi Fm Fo Fop G H R l/crack

6 6" £f

Pi Pt

diameter of striker or projectile radial coordinate impact position along a radial line measured from the plate centre time response or impact time predicted value of t r according to the geometrically similar scaling laws from the smallest scale model time when impact velocity reaches zero during a test predicted value of t o according to the geometrically similar scaling laws from the smallest scale model inside diameter of a clamp outside diameter of a clamp impact energy; initial value of E k kinetic energy of a striker or projectile perforation energy predicted value of Ep according to the geometrically similar scaling laws from the smallest scale model impact force mean impact force during a test predicted value of F a according to the geometrically similar scaling laws from the smallest scale model interracial force between striker and plate maximum impact force during a test impact force when velocity reaches zero during a test predicted value of F o according to the geometrically similar scaling laws from the smallest scale model mass of striker or projectile plate thickness radius of the supports of a circular plate initial impact velocity velocity at which a visible crack forms on the distal side critical velocity; ballistic limit geometrical scale factor non-dimensiOnal maximum permanent transverse deflection (A/H) modified value of 6 with respect to yield stress for the smallest scale model (A*/H) percentage elongation over a 2" (50.8 mm) gauge length strain tensor, i,j = 1, 2, 3 strain rate tensor, i,j = 1, 2, 3 ratio of the ith material density to Pt target density

Research student, on leave from Taiyuan University of Technology, China. 485

486 17a O'il O"u O'y

X Z*

z.* A A.

Aap

A~

Acp Am Ap A*

Ep E,

H.-M. WEN and N. JONES average of try and a. stress tensor, i,j = 1, 2, 3 uniaxial ultimate tensile stress uniaxial yield stress non-dimensional impact energy (E/ayd 3) value of ~ modified with respect to flow stress for the smallest scale model non-dimensional impact energy (E/o.d 3) non-dimensional impact energy (E/aud a) ratio of striker diameter to plate thickness maximum permanent transverse deflection mean transverse deflection at the impact location during an impact test predicted value of A. according to the geometrically similar scaling laws from the smallest scale model critical maximum permanent transverse deflection at which a visible crack forms on the distal side predicted value of Ac according to the geometrically similar scaling laws from the smallest scale model maximum transverse deflection during an impact test predicted value of A according to the geometrically similar scaling laws from the smallest scale model modified value of A with respect to the yield stress for the smallest scale test specimen modified value of Ac with respect to the yield stress for the smallest scale model predicted value of A~*according to the geometrically similar scaling laws from the smallest scale model predicted value of A* according to the geometrically similar scaling laws from the smallest scale model characteristic material constant for projectile characteristic material constant for target

Superscripts p and a in Table 3 represent directions parallel to and across the rolling direction, respectively.

1. I N T R O D U C T I O N

It is becoming increasingly necessary to assess the safety, structural crashworthiness and perforation of many large complex structural systems, or components, which are subjected to dynamic loads causing large inelastic deformations. Computer codes and numerical schemes are sometimes used for design purposes but doubts often arise on the validity for these large complex problems. Thus, it is essential to calibrate the numerical schemes and to conduct experiments on small-scale models to avoid the high costs and size constraints associated with the testing of full-scale prototypes [ 1-3]. Early similitude studies were carried out by Goodier and Thomson [4] who developed the scaling laws for structural models, and Nevill [ 5 ] who investigated the dynamic response of re-entry vehicles to impulsive loading. Duffey [6] employed the well-known Buckingham H-theorem to generate 19 dimensionless parameters which govern the structural behaviour of fuel capsules subjected to blast, impact and thermal loading. The 19 dimensionless parameters must be equal for small-scale models and full-scale prototypes to have the same scaled response according to the geometrically similar scaling laws. Baker [7] examined the scaling laws for the impact puncturing of shipping casks, while Duffey et al. [8] developed scaling laws for punch-impact-loaded structures by following closely some previously published studies [5,6]. Booth et al. [9] carried out a series of 13 drop tests on mild steel and stainless steel thin-plated structures having various sizes between one-quarter scale and full-scale. The experimental results indicate that the post-impact deformations of the full-scale prototype mild steel specimens may be as much as 2.5 times greater than would be predicted from a one-quarter scale model test with the impact times larger and the accelerations smaller than expected. The deviation from the geometrically similar scaling laws is less for the stainless steel structures than for the mild steel structures, and the weld fracture and tearing are more pronounced in the full-scale eggbox and girder structures than in the small-scale ones. Reference [10] presents some test results of the dynamic cutting of mild steel plates having thicknesses between 1.5 and 6 mm, and which were struck on one edge by case-hardened heavy solid wedges travelling at impact velocities up to 11.1 m s-1. It was found that the deviation from the geometrically similar scaling laws was significant in terms of the penetration depth and the cutting energy, even when compensated for by the differences in yield and ultimate stresses for the various plates. However, this particular

Scaling laws for metal plates struck by large masses

487

problem has been further investigated by Lu and Calladine [11] who have used a dimensional analysis which indicates that the geometrically similar scaling laws are satisfied in the range of the test parameters investigated. The plate specimens were made from mild steel with thicknesses ranging from 0.72 to 2 mm and struck by wedges travelling at low velocities. Atkins 1,12] employed an energy balance approach to examine the scaling laws for combined plastic flow and fracture. The total kinetic energy is equated to the plastic work consumed in the material through structural deformations and the energy required for creating new fracture surfaces. The plastic and fracture energies scale as the cube and square of the scale factor (fl), respectively, as noted in Ref. [ 13 ] and elsewhere. In general, a practical static or dynamic problem involves fracture as well as plastic flow. Thus, the total energy absorbed would scale as fix, 2 ~< x ~< 3. Atkins [14] further discussed his theory with respect to the experimental results obtained for the double-shear specimens reported in Ref. [ 15]. Recent experiments on the tearing of thin aluminium alloy sheets I-16] reveal that the energies required to tear the specimens vary as fl2.61, which does not satisfy the elementary laws of similitude, although Mai 1,17] has commented on some non-scaling features of the experimental arrangement. It transpires that the departures from the geometrically similar scaling laws for this class of problem are non-conservative [3] and that the initiation of dynamic failure is extremely sensitive to small details of the modelling (e.g. Ref. 1,18]). More recently, Langseth and Larsen [ 19] performed an experimental investigation into the response of a simply supported rectangular plated structure struck by an idealized drill-collar. It was noted that the response was described as the global deformation of a plate combined with the possibility of a local failure (plugging) at the impact point. Langseth and Larsen [ 19] also conducted some scaling tests on plates having the same span. There is a paucity of experimental data to validate the geometrically similar scaling laws and to assess the influence of material strain rate sensitivity, fracture and the ductile-brittle fracture transition 1'1,2], or to explore the response of structures with a collapse strength which falls off sharply with increasing deflection 1"20]. It is likely that the scaling laws are problem-dependent with the geometrically similar scaling laws being satisfied in some cases but not in others. The dynamic response of some structures passes through several phases from elastic through elastic-plastic and large plastic strains to rupture with different scaling laws governing the principal phenomena during each phase. In this paper, an experimental investigation is reported into the scaling laws for plates struck by masses travelling at velocities up to 5 m s- 1. The model plates are made from BS4360-43A mild steel (strain rate sensitive) and BSL157-T6 aluminium alloy (strain rate insensitive). The test specimens and impact conditions were geometrically scaled as far as possible. The range of sizes in this study covered a scale factor of four for the mild steel specimens and approximately five for the aluminium alloy specimens. The tests are conducted with impact energies ranging from relatively low values which produce large plastic deformations in the plates up to high values which cause cracking and perforation. 2. SIMILITUDE STUDY The similitude analysis in this section follows closely the previous studies in Refs [ 1,6-8]. A replica scaling law requires a careful recognition and isolation of the pertinent input and output parameters by inspection and the listing of the associated physical dimensions. In the present study, the output or response parameters are expressed as functions of a set of physical input parameters. 2.1. Input parameters The geometry of the striker-target interaction problem is specified by a characteristic length, which is taken as the striker diameter (d), and the target thickness (H) and target radius (R), while the material input parameters include the densities and mechanical vroperties of the striker and target materials. The density of the ith material in the system

488

H.-M. WEN and N. JONES

is expressed as a ratio (Pl) by dividing it by the density of the target material (Pt). The mass of a striker (G) is selected as an independent input parameter since it plays an important role both in the loading and response aspects of the problem. In the present striker-target interaction problem, the striker is modelled as a block with a case-hardened tup head which remains undeformed during the tests, but the projectile or striker material behaviour can be expressed in terms of some characteristic material constant (Ep), such as the yield stress or the elastic modulus. The target response, on the other hand, is usually characterized by large elastic and plastic deformations with the possibility of a local shear failure at the impact point. Although the response time is short, material strain rate effects can be neglected when comparing the responses for structures which have relatively small differences in the scale factor (e.g. up to six) as discussed in Baker [7], Dallard and Miles [21], and in Ref. [1]. In the present investigation, the scale factor is 1:4 and 1:4.765 for the mild steel and aluminium alloy specimens, respectively. Thus, the material behaviour of the target can be approximated by some characteristic material constant (Et) such as the yield stress, or the ultimate tensile strength. The impact velocity of the striker or tup is V~which is also selected as an independent input parameter. Gravitational forces are usually ignored as they are negligibly small compared with the other dynamic forces. All the input parameters together with their associated physical dimensions are listed in Table 1. 2.2. Response parameters In the present problem, the response parameters are taken as the deflection (A) of the target, impact time (tr), the interfacial force (Fi) between the striker and a target, stresses (aifl, strains (eo) and the strain rates (~q) in a target. All the response parameters together with their associated physical dimensions are listed in Table 1. Table 1 contains 15 input and response parameters so that according to the Buckingham H-theorem, there are the following 12 independent but non-unique H terms:

l-It = Hid

I-I2 =

I-Is = A/d l.

H8 =

I-I6 = Rid I

dp¢%/Y.~

r l l o = p ~l v i / ~ . , l~

I

119 =

H 4 = Zp/T~t

H 3 = Pi

eij

I-I7 = t~/j/Z, l

Y~t~/dp~

I-Ill = Fi/d2y_, t

1-112 = G / P t d 3 "

Equality of the H terms for a small-scale model and a full-scale prototype gives the similarity requirements or the corresponding geometrically similar scaling laws. The similitude requirement is also represented in Table 1 for replica modelling with a scale factor (fl), TABLE 1. Parameter d G Input

Response

SCALINGPARAMETERS Dimension

Scale Factor

L M

fl [3~

H

L

[3

R

L

[3

Vi

LT-l

1

Pi

-

l

p, ~p Z,

ML -3 M T - 2 L -1 M T - 2 L -I

1 I 1

Fi t, eij eO %

MT-2L T

T- I M T - 2L - l

[3~ [3 1 [3- l 1

A

L

[3

Scaling laws for metal plates struck by large masses

489

FIG. I. Experimental arrangement. which is the ratio of a geometric quantity in a full-scale prototype to the corresponding value in a small-scale model (fl >/ 1). It is evident from the scaling laws developed above that the response of a small-scale model and a geometrically similar full-scale prototype when made from the same material are similar in the scaled sense, if (see Ref. I l l ) : (a) deflections and impact time scale as the geometrical scale factor (fl); (b) interfacial forces scale as the square of the scale factor (f12); (c) stresses and strains at scaled locations and times are identical, t It is also evident that combining l-Ilo and I-I12 and introducing E k = GV2/2 gives H13 = H2oH12 = 2Ek/Etd 3, which implies that: (d) impact energies scale as the cube of the scale factor (f13). The dimensionless term H, o shows that: (e) impact velocities are identical. An experimental p r o g r a m m e for assessing the above elementary geometrically similar scaling laws is described in the following sections. 3. EXPERIMENTAL PROGRAMME

3.1 General arrangement and equipment The experimental arrangement for the present scaling tests is shown in Fig. ! and some details of the drop h a m m e r rig are given in Ref. [ 2 2 ] . A square plate test specimen was held in a clamp with a circular aperture around the outer boundary, as shown in Fig. 2. A typical tup or striker consists of a mild steel block, a tup head connector and a tup head, as shown in Fig. 3. The 2 and 3 kg tups are similar except that the steel block is replaced by a square steel tube and by a wooden block, respectively. All the test parameters are listed in Table 2 and are geometrically scaled except for the bolt diameters which were selected from those readily available. t It is evident from (a) that the duration of impact scales as fl so that material strain rate effects do not satisfy the laws of geometrically similar scaling, as discussed further in Section 11.3.2 of Ref. [ 1].

490

H.-M. WEN and N. JONES

1 L--.

3 ',~

et,, Flo. 2.

I

Specimen clamp: ( 1) upper annular support; (2) plate specimen; (3) lower annular support; and (4) base plate.

•

2

1

FIG. 3. Tup or striker: ( 1) mild steel sheets; (2) mild steel stock; (3) connector and (4) tup head.

TABLE 2.

TEST PARAMETERS

Plate thickness n (mm)

Bolt diameter (mm)

D (ram)

DO (ram)

G (kg)

d (ram)

Mild steel

50.8 101.6 152.4 203.2

76.2 152.4 228.6 304.8

3 24 81 192

5.95 11.90 17.85 23.80

2 4 6 8

5 8 16 20

Aluminium alloy

42.6 101.6 135.6 203.2

64.0 152.4 203.2 304.8

2 27 64 216

8.00 19.05 25.40 38.10

2 4.76 6.35 9.53

5 8 16 20

Material

Scaling laws for metal plates struck by large masses

lili

t,

!II/---IH

I

FIG. 4.

491

i

Plate specimen.

600o(MPo)

t:'f

200I

o. o

0 0.2°/.0-05

E

o. s

(n)

I

'°°F o; (b)

o'o E'o:,.s

FIG. 5. Stress-strain curves: (a) aluminium alloy; (b) mild steel (50.8 mm gauge length).

3.2. Specimen and material properties Square plate specimens were cut directly from the metal sheets, as shown in Fig. 4. Eight equally spaced holes with the diameters indicated in the last column of Table 2 were drilled in each plate to secure the plate in the clamping device in Fig. 2. The plate thicknesses varied from 2 to 8 m m for the mild steel specimens and from 2 to 9.53 m m for the aluminium alloy specimens. The dimensions and other details of the specimens are also given in Table 2. The specimens were made from BS4360-43A mild steel and BSL157-T6 aluminium alloy sheets; at least four static tensile tests were conducted on each plate thickness in order to obtain the mechanical properties. The static tensile tests were carried out on a Dartec testing machine at average strain rates of approximately 5 x 10 -4 s-1 with at least two being cut parallel to the rolling direction and at least another two cut across the rolling direction. Typical engineering stress-engineering strain curves for the aluminium alloy and mild steel materials are shown in Fig. 5 and a summary of the tensile test data is given in Table 3.

Aluminium alloy

Mild steel

Material

ALl-1 ~24

ALII- 1 ~ 28

ALIII- 1 ~ 29

ALIV- I ~ 24

4.76

6.35

9.53

STIV- 1 ~ 20

STill- 1 ~ 28

STI I- 1 ~ 24

STI- 1 ~ 24

Specimen number

2

Plate thickness (mm)

TABLE 3.

Aluminium sheet Aluminium sheet Aluminium sheet Aluminium sheet

Steel sheet Steel sheet Steel sheet Steel sheet

Original material

447.0

393.0

432.3

461.5

243.9

261.6

317,0

243.2

a~ (MPa)

438.3

386.5

429.8

448.5

249,4

261.7

303.5

266.6

a~ (MPa)

Yield stress

442.6

389.8

431.0

455.0

246.6

261.7

310.3

254,9

ar (MPa)

490.2

445.2

474.2

500.8

409.4

427.9

442.8

339.9

a~ (MPa)

SPECIMEN DETAILS AND TENSILE MECHANICAL PROPERTIES OF THE MATERIALS

486.3

444.9

484.1

493.2

405.3

401.5

440.8

341.2

a~ (MPa)

U.T.S.

488.3

445.0

479.2

497.0

407.3

417.7

44 1.8

340,6

a~ (MPa)

(50.8 m m GAUGE LENGTH)

12.84

11.10

11.42

11.80

11.54

10.24

10.12

12.30

11.30

10.80

10.00

40.70

40.54 40.78

9.96

31.90 27.82

33.50

30.00 37,00 35.86

36.10 31.66

40.46

Ef (%)

e~ (%)

E~' (%)

Elongation

b"

7'

-v

ix2

Scaling laws for metal plates struck by large masses

493

The aluminium alloy material strain hardens, while a distinct yield flow region appears in the stress-strain curves for the mild steel except for the 4 mm thick mild steel. The yield stresses are obtained from the average values in the yield flow region of the stress-strain curves for the mild steel, while, for the aluminium alloy and the 4 mm thick mild steel, the yield stresses are obtained from the stress-strain curves using an offset method with a 0.2% strain (proof stress). The superscripts p and a in Table 3 represent directions parallel to and across the rolling direction, respectively, and ay, au and ef are mean values with respect to these two directions. 3.3. Loading and instrumentation A type of test on a circular plate is defined as one made from the same material and having the same plate thickness, striker diameter and impact position. Eight types of test were undertaken in the present experimental programme and at least four specimens of each type were tested with different impact velocities. A failure (cracked or perforated plate) was sought for each type of test in order to examine the phenomenon of plate failure. The external kinetic energy, therefore, ranges from a relatively low value, which produces small plastic deformations of a plate, to high values which cause cracking or perforation. The velocity of the striker or the velocity at the impact point of a plate was measured by means of a laser Doppler velocimeter [-23]. It is assumed that the striker is in continuous contact with a plate prior to any rebound. Transient signals were stored on mini-cassettes in a DL1080 programmable transient recorder, which has a resolution of 20 MHz. The initial impact velocity was also measured using a photocell system which consists of a photodiode and an internal counter. The values of the initial impact velocities obtained in this way have been compared with those given by the laser Doppler velocimeter. It was found that the difference is generally within 1%. The maximum permanent transverse deflections of the plates were measured in situ (i.e. immediately after a test and before a plate was removed from the clamps) and the values reported in this paper were determined by averaging several separate recordings. 4. EXPERIMENTAL RESULTS The circular plate test results are summarized in Tables 4 and 5, together with some additional results in Table 6 which were obtained using the laser Doppler velocimeter. The impact or response time (tr) and the time (to) when the impact velocity reaches zero, which corresponds to the maximum deflection, are estimated from the velocity-time traces as shown in Fig. 6. It is found that filtering cut-off frequencies in the range from 315 up to 2000 Hz has virtually no influence on the mean forces, and on the velocity-time and displacement-time histories, but does influence significantly the maximum accelerations and the maximum impact forces [23-1. In the present study, the cut-off frequencies have also been scaled* when processing the transient data, as indicated in Table 6. Strictly speaking, the sampling rate should also have been scaled but it was kept constant at 20/as (i.e. 50,000 sample points per second) for all of the tests. The maximum and mean transverse deflections, and the maximum and mean impact forces during impact were obtained in Table 6 from the deflection-time and force-time traces, respectively. The difference between in Tables 4 and 5 and A m / H in Table 6 is due mainly to the elastic recovery of the plates after a test. The difference between Aa and Am is simply because Aa is the average value of plastic deformations at the impact point throughout a test and Am is the largest value. Figure 7 shows some selected photographs of the specimens after impact. In particular, Fig. 7(b) shows a typical perforated mild steel plate specimen which was struck transversely by a mass with a velocity of 4.98 m s - 1, while Figs 7(c) and (d) show typical cracked and *It is evident from Section 2.2 that when using F~ = Gg the dimensionless number FIg(FII t/Yl~2){= tr(g/d)X2 = t,f2c, where f2c is a frequency. It is necessaryfor this dimensionlessnumber to have the same value for a small-scale model (to, t) and a full-scale prototype (f/., T), i.e. co = f~T/t = fir2 since the equality of I-I 9 for a model and prototype made from the same material gives t = T/ft.

3.10 3.70 4.11 4.50 4.76 5.00 3.38 3.78 4.52 4.73 5.04 3.28 3.87 4.43 4.80 4.98 3.52 4.02 4.50 4.81 5.01

STI-20 STI-25 STI-26 STI-27 STI-28 STI-29 STII-21 STII-20 STII-22 STII-23 STII-24 STII1-24 STIII-25 STII 1-26 STIII-27 STIII-28 STIV- 15 STI V- 14 STIV- 16 STI V- 17 STIV- 18

v~

(ms -j )

Specimen number

3 3 3 3 3 3 24 24 24 24 24 81 81 81 81 81 192 192 192 192 192

G (kg)

5.95 5.95 5.95 5.95 5.95 5.95 11.90 11.90 11.90 11.90 11.90 17.85 17.85 17.85 17.85 17.85 23.80 23.80 23.80 23.80 23.80

d (mm)

TABLE 4.

D (mm)

50.8 50.8 50.8 50.8 50.8 50.8 101.6 101.6 101.6 101.6 101.6 152.4 152.4 152.4 152.4 152.4 203.2 203.2 203.2 203.2 203.2

H (mm)

2 2 2 2 2 2 4 4 4 4 4 6 6 6 6 6 8 8 8 8 8

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Before test r i (mm) 3.5 0 1.0 1.0 3.5 1.0 2.0 2.0 1.5 0 0 5.5 2.5 2.5 4 3.5 4.0 4.0 2.0 0 0

After test r I (mm)

Impact point

2 2 2 2 2 3 3 3 3 3 4 4 4 4 4

fl

0.269 0.382 0.472 0.566 0.633 0.698 0.319 0.399 0.571 0.624 0.738 0.301 0.419 0.548 0.644 0.693 0.346 0.452 0.566 0.646 0.701

0.269 0.382 0.472 0.566 0.633 0.698 0.262 0.328 0.469 0.513 0.606 0.293 0.408 0.534 0.627 0.675 0.358 0.467 0.585 0.668 0.725 1.589 1.919 2.208 2.418

1.407 1.643 1.980 2.250

1.390 1.665 2.053 2.155

X* 1.220 1.640 1.960 2.190 2.450

2.975)

;(

E X P E R I M E N T A L R E S U L T S F O R T H E S C A L I N G T E S T S O N M I L D S T E E L C I R C U L A R PLATES ( ~ =

1.570 1.850 2.152 2.365

1.403 1.701 2.030 2.272

1.595 1.834 2.347 2.509

1.220 1.640 1.960 2.190 2.450

Perforated

Perforated

Perforated

Perforated

Comment

Z

e-,

.f

4~ %0 4~

v, (ms -1)

2.90 3.70 4.07 4.27 4.80 3.06 3.70 4.20 4.72 2.96 3.73 3.95 4.20 4.70 3.15 3.70 4.27 4.75

(ms-')

4.50 4.52 4.43 4.50 3.70 3.70 3.73

Specimen number

ALI-20 ALI-21 ALI-22 ALI-23 ALI-24 ALII-24 ALII-25 ALII-26 ALII-27 ALIII-25 ALIII-26 ALIII-27 ALIII-28 ALIII-29 ALIV-21 ALIV-19 ALIV-20 ALIV-22

Specimen number

STI-27 STII-22 STIII-26 STIV- 16 ALI-21 ALII-25 ALIII-26

3 24 81 192 2 27 64

(kg)

G

2 2 2 2 2 27 27 27 27 64 64 64 64 64 216 216 216 216

G (kg)

5.95 11.90 17.85 23.80 8.00 19.05 25.40

d (ram)

TABLE 6.

8.00 8.00 8.00 8.00 8.00 19.05 19.05 19.05 19.05 25.40 25.40 25.40 25.40 25.40 38.10 38.10 38.10 38.10

d (mm)

TABLE 5.

42.6 42.6 42.6 42.6 42.6 101.6 101.6 101.6 101.6 135.6 135.6 135.6 135.6 135.6 203.2 203.2 203.2 203.2

D (mm) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Before test rj (mm) 3.0 0 3.0 3.0 0 0 2.5 2.5 0 3.5 3.0 5.0 1.5 1.5 4.0 0 1.5 1.5

After test rl (mm) 1 1 1 1 1 2.380 2.380 2.380 2.380 3,175 3.175 3.175 3.175 3.175 4.765 4.765 4.765 4.765

fl 0.036 0.058 0.071 0.078 0.099 0.042 0.062 0.080 0.101 0.044 0.070 0.078 0.088 0.111 0.044 0.060 0.080 0.100

X

0.632 0.850 1.208

0.558 0.866 1.024 1.247

0.677 0.924 1.219

0.460 0,850 1.030 1.150

(Ip = 4)

2 4 6 8 2 4.76 6.35

H (mm) 50.8 101.6 152.4 203.2 42.6 101.6 135.6

D (ram) 4.85 10.15 15.56 21.11 3.19 8.57 10.10

Am (ram) 3.55 7.30 10.94 15.19 2.33 6.23 7.40

Aa (mm)

2.42 4.72 6.96 10.02 2.52 5.94 7.08

tr (ms)

1.85 3.63 5.80 7.80 1.43 3.50 4.25

to (ms)

10.95 47.17 93.39 155.4 7.00 47.99 85.01

Fm (kN)

EXPERIMENTAL RESULTS FOR THE SCALING TESTS OBTAINED BY THE LASER DOPPLER VELOCIMETER

2 2 2 2 2 4.76 4.76 4.76 4.76 6.35 6.35 6.35 6.35 6.35 9.53 9.53 9.53 9,53

H (mm)

Impact point

EXPERIMENTAL RESULTS FOR THE SCALING TESTS ON ALUMINIUM ALLOY CIRCULAR PLATES

8.88 37.14 74.28 128.57 7.00 43.71 81.71

Fo (kN)

0.036 0.058 0.071 0.078 0.099 0.040 0.059 0.076 0.096 0.038 0.060 0.067 0.075 0.095 0.043 0.058 0.078 0.097

X*

6.78 28.65 60.99 100.7 4.55 25.03 49.2

F, (kN)

0.597 0.853 1.161

0.520 0.868 0.983 1.120

0.633 0.901 1.145

0.460 0.850 1.030 1.150

2 1 0.667 0.5 1 0.42 0.315

Cut-off frequency (kHz)

Cracked Perforated

Cracked Perforated

Cracked Perforated

Cracked Perforated

Comment

oQ

496

H.-M. WEN and N. JoNES

6

120~

,

z~
"~ 2 O

0 -2

,oi- ,,7" "\~

o,IX,,

',,~

Oa

6_~ r'-i tel

0n

2

4

-40F

to/'~ 8 Time t(ms)

-6

f:

6~

--6 E


F

E t=a

~2

2 _mc"l

0

0 -2

31-

.2...... Timetiros)

-2

(b) FIG. 6.

Typical traces obtained by the laser Doppler velocimeter: (a) specimen No. STIII-26 (b) specimen No. ALI-21.

(a)

(b)

(c)

(d)

Fio. 7. (a) Specimen No. STIV-16, G = 192kg, Vi = 4 . 5 0 m s - I ; (b) specimen No. STIII-28, G = 81 kg, V i = 4.98 m s- 1; (c) specimen No. ALII-26, G = 27 kg, V, = 4.20 m s - i; (d) specimen No. ALII-27, G = 27 kg, Vi = 4.72 m s- i.

497

Scaling laws for metal plates struck by large masses

perforated aluminium alloy plate specimens which were struck transversely by masses travelling at velocities of 4.20 and 4.72 m s - l, respectively. In the cracked case, a visible crack can be seen on the distal side of the plate, as shown in Fig. 7(c), while in the perforated cases in Figs 7(b) and (d), a plug was obtained. It was observed that the plug has approximately the same diameter as the striker for all of the specimens except for the thickest aluminium alloy plate which failed in a mixed mode of plugging and discing [24] due to the competition between the transverse shear and bending resistance. 5. DISCUSSION Some of the experimental data in Tables 4 and 5, and Table 6 are presented graphically in Figs 8-11 and Figs 12-13, respectively. It is evident from Figs 8(a) and (b) that the non-dimensional m a x i m u m permanent transverse deflections (&), which includes the critical values for the aluminium alloy plates at which a visible crack can be seen on the distal side, is related linearly to the non-dimensional impact energy (Z) within the range of the experimental test results. The solid lines in Figs 8(a) and (b), which were obtained by the least square method, represent the variation of the non-dimensional m a x i m u m permanent transverse deflection with the non-dimensional impact energy for the smallest scale mild steel and aluminium alloy test specimens, respectively. It is evident that the test data for the larger and full-scale test specimens have the same trend as those for the smallest scale test specimens with the largest deviation from the solid lines lying within a range of approximately 10%. It is observed that extensions of the solid lines in Figs 8(a) and (b) would intercept the positive a n d negative values of the respective ordinates. However, the ranges of the dimensionless values for the aluminium alloy plates in Fig. 8(b) are smaller than those in Fig. 8(a) for the mild steel plates. The larger slope of the line in Fig. 8(b) is, therefore, consistent with the trends of the curves for the m a x i m u m permanent transverse displacements of rigid-plastic plates subjected to large dynamic loads, as shown in Fig. 7.21 of Ref. [ 1 ], for example, together with a finite value of )~for 6 = 0 due to recoverable elastic strain energy effects. Figures 9(a) and (b) show the ratios of the m a x i m u m permanent transverse deflection and the corresponding predicted value from the smallest scale model versus the scale factor (fl) for the geometrically scaled impact tests on the mild steel and aluminium alloy

0

2"0

9

/

12

o

8

o

1-0

8 ].c

/

I!

t~

o/ AV

O8 o

/

0.6

1'0

/

O.C, 05

0-2

0 0'I (a)

/

Z~

012

0'-3

01t~

0"5

016

017 X

o (b)

0"02

fib&

0 " 0 6 0"08

X

FIG.8. Variation of the non-dimensional permanent transverse deflection (6) with non-dimensional impact energy (Z) for geometrically scaled impact tests. (a) Mild steel. A, O, f-I, and V represent the experimental data on 2, 4, 6 and 8 mm thick plates, respectively. - - : Least square line for the smallest scale specimens. (b) Aluminium alloy. A, ©, [] and V represent the experimental data on 2, 4.76, 6.35 and 9.53 mm thick plates, respectively. - - : Least square line for the smallest scale specimens..4,, ~, [] and W indicate cracking conditions for 2, 4.76, 6.35 and 9.53 mm thick plates, respectively.

0"10

498

H.-M. WEN and N. JONES

A/Ap 1'0 075

(a)

0"5 0

1

½

3

¼

[3

....

1.0 A/Ap 075 0.5

FIG. 9. Ratio of the permanent transverse deflection (A) and the corresponding predicted value (Ap) versus fl for geometrically scaled impact tests. (a) Mild steel specimens with ~, = 2.975 for an impact velocity of 4.5 m s -~, approximately. Ap f o r / k , ©, [] and W are based on the experimental results for 2 (fl = 1 ), 4 (fl = 2), 6 (fl = 3) and 8 m m (fl = 4) thick plates, respectively. - - : A/Ap = 1 for geometrically similar scaling. (b) Aluminium alloy specimens with ff = 4 for an impact velocity of 3.7 m s - ~, approximately. Ap for A , C), O and ~ are based on the experimental results for 2 (fl = I), 4.76 (fl = 2.38), 6.35 (fl = 3.175) and 9.53 m m (fl = 4.765) thick plates, respectively. - - : A/Ap = 1 for geometrically similar scaling.

075 05 FIG. 10. Ratio of the critical permanent transverse deflection (Ac) and the corresponding predicted value (Acp) versus fl for geometrically scaled impact tests on a l u m i n i u m alloy plates with ~ = 4 for an impact velocity of 4.2 m s - I approximately. Acp for .,~, (It, [] and ~ are based on the experimental results for 2 (fl = 1), 4.76 (fl = 2.38), 6.35 (fl = 3.175) and 9.53 m m (fl = 4.765) thick plates, respectively. - - : Ac/Acv = l for geometrically similar scaling.

circular plates, respectively. Figure 10 presents the ratio of the critical maximum permanent transverse deflection and the corresponding value predicted from a small-scale aluminium alloy circular plate model. It is found that the largest deviation from the geometrically similar scaling laws is generally within + 10% with the exception of the 6.35 mm thick aluminium alloy plate for which the deviation is about 20%. However, the 6.35 mm thick aluminium alloy plate has lower yield and ultimate stresses than the other aluminium alloy plates, as shown in Table 3. The ratio of the perforation energy to the predicted value from a small-scale model test is shown in Figs 11 (a) and (b)for the mild steel and aluminium alloy circular plates. The largest deviation from the geometrically similar scaling laws is within a range of approximately + 7%. It is evident from Figs 12(a) and (b) that the largest deviation from the law of similitude for the impact times is generally within + 10%. The ratio of the average impact force and the corresponding predicted value from a small-scale impact test is shown in Figs 13(a) and (b) for the mild steel and aluminium

Scaling laws for metal plates struck by large masses

499

1.0

Ep/Epp 075

05 0

(a)

1.0 Ep/Epp

. . . . . .

--L- . . . .

.-'t:---------i

075

0.5

(bj

0

h 13

FIG. 11. Ratio of the perforation energy (Ep) and the corresponding predicted value (Ep,~) versus for geometrically scaled impact tests. (a) Mild steel specimens with @ = 2.975. Epp for A, O, • and • are based on the experimental results for 2 (/~ = 1), 4 (8 = 2), 6 (8 = 3) and 8 m m (8 = 4) thick plates, respectively. - - : Ep/Epp = 1 for geometrically similar scaling. (b) Aluminium alloy specimens with @ = 4. Epp for A , O, • and • are based on the experimental results for 2 (8 = 1), 4.76 (,8 = 2.38), 6.35 (/~ = 3.175) and 9.53 m m (8 = 4.765) thick plates, respectively. - - : Er,/Epp = 1 for geometrically similar scaling.

1.0

~__===_~ ==----~-=_._=-.~

to/top

1.0

:

/ top

075

o.s

(a) 0

.

.

.

.

.

°

075

I

½

3

~3 l+

o.si

(b) 0

I

½

FIG. 12. Ratio of the zero-velocity time (to) and the corresponding predicted value

3 13 ¼ (top) for

geometrically scaled impact tests. (a) Mild steel plates with ¢, = 2.975. top for A , O, [] and V are based on the experimental results for 2 (8 = 1 ), 4 (8 = 2), 6 (8 = 3) and 8 m m (8 = 4) thick plates, respectively. - - : to~/top = 1 for geometrically similar scaling. (b) Aluminium alloy plates with @ = 4. top for 6, O, and [] are based on the experimental results for 2 (,8 = 1 ), 4.76 (8 = 2.38) and 6.35 m m (8 = 3.175) thick plates, r e s p e c t i v e l y . - - : to/t.p = I for geometrically similar scaling.

alloy specimens with ~, = 2.975 and @ = 4, respectively. It is clear for the mild steel plates that the largest deviation from the elementary geometrically similar scaling laws is within + 10%, approximately, when the 2 mm thick plate is taken as the smallest-scale model. The deviation becomes larger if the 4 mm thick steel plate is taken as the smallest-scale model but is still less than 20% for the average impact forces. It is observed that the impact forces are largest for the 4 mm thick steel plates, which have the highest yield and ultimate tensile stresses, as shown in Table 3. This suggests that the violation of the scaling laws by the mechanical properties, such as the yield stress, may cause the impact forces to deviate from the predictions of the geometrically similar scaling laws. The largest deviation of the average impact force for the aluminium alloy results is about 20% when the 2 mm thick plate is taken as the smallest-scale model. Scaled impact load-transverse deflection comparisons, which are obtained from the laser Doppler velocimeter, are shown in Fig. 14(a) for some mild steel plates struck at a velocity

500

H.-M. WEN and N. JONES

.0o /s..A

,.A

.....

~- . . . .

Fo/Fo,

.~----Z~ ~ --o

1.0

FJFap

075,

~ - -

0.75

0.5 (a)

_~-

......

o

3

ib)

0.5

0

FIG. 13. Ratio of the average impact force (F=) and the corresponding predicted value (F=p) versus for geometrically scaled impact tests. (a) Mild steel plates with ~ = 2.975. F=p for A , O, [] and are based on the experimental results for 2 (fl = 1 ), 4 (fl = 2), 6 (fl = 3) and 8 m m (fl = 4) thick plates, respectively. - - : F.//Fop = 1 for geometrically similar scaling. (b) Aluminium alloy plates with ~, = 4. F.p for A , ©, and [] are based on the experimental results for 2 (fl = l), 4.76 (fl = 2.38) and 6.35 m m (fl = 3.175) thick plates, respectively. - - : Fa/F~p= I for geometrically similar scaling.

0

2

t~

0

6 (a)

L~/~(mm)

1

2

3

(b)

t~

5

Alp(ram)

FIG. 14. Impact force-transverse displacement scaling comparisons. (a) Mild steel: STI-27(fl = 1 ), S T I I - 2 2 ( f l = 2 ) , S T I I I - 2 6 ( f l = 3 ) and S T I V - 1 6 ( f l = 4 ) ; (b) aluminium alloy: A L I - 2 1 ( f l = 1), ALlI-25(fl = 2.38) and ALIII-26(fl = 3.175).

of 4.5 m s - l , approximately, while similar comparisons for the aluminium alloy plates struck at a velocity of 3.7 m s - 1, approximately, are shown in Fig. 14(b). The quite different characteristics for the mild steel plates in Fig. 14(a) and the aluminium alloy plates in Fig. 14(b) are due partly to the different elastic characteristics of the two materials. To facilitate the comparisons in Figs 14(a) and (b) all the impact load values have been divided by f12, while all the transverse deflections have been divided by fl, which are consistent with the geometrically similar scaling laws given in Section 2.2 and Table 1. Notwithstanding the reasonable agreement with the geometrically similar scaling laws in Figs 8-14, it was noted earlier that the mechanical properties of the test specimens are different for each plate thickness. In order to assess the influence of the mechanical properties of the materials, such as the yield stress, on the deviation from the elementary geometrically similar scaling laws, all the test data in Tables 4 and 5 may be modified by taking as a basis the yield stresses for the smallest-scale mild steel and aluminium alloy test specimens, respectively. As stated earlier and shown in Fig. 8, the non-dimensional permanent transverse deflections (6) are related linearly to the non-dimensional impact energies (X), or 6 = a'x + b' with a' and b' being the slope and intercept, respectively. The values of a' and b' for the mild steel plates with different thicknesses have been obtained using the least squares procedures and are presented in Table 7. The modified values of the non-dimensional impact energies (X*) for all the test specimens are obtained by replacing the yield stress

0.269 0.382 0.472 0.566 0.633 0.698

0.262 0.328 0.469 0.513 0.606

0.293 0.408 0.534 0.627 0.675

0.358 0.467 0.585 0.668 0.725

STII-2J STII-20 STII-22 STII-23 STII-24

STIII-24 STIII-25 STIII-26 STIII-27 STIII-28

STIV-15 STIV- 14 STIV-16 STIV-17 STIV- 18

Z

STI-20 STI-25 STI-26 STI-27 STI-28 STI-29

Specimen number

t~

0.270 0.352

0.441

0.504 0.547

2.208

2.418

0.226 0.314 0.411 0.483 0.520

0.216 0.271 0.387 0.423 0.500

1.589 1.919

1.407 1.643 1.980 2.250

1.390 1.665 2.053 2.155

0.230 0.327 0.404 0.485 0.542 0.597

Z*

0.217 0.283 0.354 0.404 0.439

0.184 0.256 0.335 0.393 0.423

0.184 0.230 0.329 0.360 0.426

0.201 0.286 0.353 0.424 0.474 0.522

Z*~

4 4 4 4 4

3 3 3 3 3

2 2 2 2 2

1 I 1 1 1 1

fl

2.651

2.532

2.988

3.305

a'

0.652

0.642

0.642

0.358

b'

3.513

3.295

3.630

3.855

a'.

0.657

0.639

0.640

0.359

b'.

4.397

4.051

4.194

4.407

a'~

COMPARISON BETWEEN THE SLOPES AND INTERCEPTS OBTAINED BY NORMALIZING IMPACT ENERGY WITH DIFFERENT FLOW STRESSES

1.220 1.640 1.960 2.190 2.450

TABLE 7.

0.651

0.637

0.658

0.360

b '~

Perforated

Perforated

Perforated

Perforated

Comment

502

H.-M. WEN and N. JONES

in the dimensionless energy X = E/ar d3 with either (a r + au)/2 or o-u for the smallest-scale model. The coefficients of the least mean square lines for the experimental results, which are plotted using X*, are given in Table 7, where the subscripts a and u represent the dimensionless values of X using (ay + au)/2 and cru, respectively. It is evident from the results in Table 7 t that by normalizing the impact energy with different flow stresses the slope for a given value of fl is changed, while the intercept remains virtually constant. It is also noted that apart from the H = 2 mm plates, the intercepts are very similar for all of the mild steel plates, vis. b ~ 0.65 which accounts for a significant contribution to the non-dimensional permanent transverse deflections (,:5). On the other hand, it was found that the intercept is less consistent for the aluminium alloy plates, but in contradistinction to the mild steel plates, accounts for less than 10% of the non-dimensional permanent transverse deflections (6) at large plastic deformations (6 >t 0.8). Thus, it is assumed that modified with respect to the yield stress for the smallest scale test specimen the non-dimensional maximum permanent transverse deflections (6") can now be obtained from the relations 6* = a'x* + b'. This procedure implies that the slope depends on the flow stress, while the intercept does not. The elementary theoretical predictions for the dimensionless maximum permanent transverse displacements of rigid, perfectly plastic circular plates subjected to impulsive velocity loadings are of the form `5 = kx {e.g. Eqns (4.29), (4.93) and (4.100)in Ref. [1]} where k is a constant coefficient which depends on the boundary conditions and the characteristics of the dynamic loading. Thus, it is evident that the dimensionless variable Z and hence 6 would have different values for plates with different yield stresses, but which are otherwise geometrically similar and, therefore, the laws of geometrically similar scaling would not be satisfied. However, if the value of O-yfor the full-scale prototype was replaced by the corresponding value for the smallest scale model, then it is evident that the same values of,5 and X would be obtained and, therefore, the laws of geometrically similar scaling would be satisfied. Thus, the correction procedure discussed above agrees with these elementary equations when b ' = 0 since 6 is related linearly to X over the ranges of parameters examined in the present study. Atkins [ 12] has also suggested an alternative procedure that compensates for the different mechanical properties in experimental scaling studies on small-scale models and full-scale prototypes. The modified data are given in Tables 4 and 5 and presented in Figs 15-17. It is evident when comparing Figs 15(a) and 16(a) with Figs 8(a) and 9(a) that only a slight improvement is obtained for the mild steel specimens. However, a significant improvement is obtained for the permanent transverse deformations of the aluminium alloy specimens in Figs 16(b) and 17 when compared with Figs 9(b) and 10. The conclusions of the modified procedures for the mild steel specimens are consistent with those obtained previously [2,9] which also show no significant improvement, but no similar observations have been reported for an aluminium alloy. It is well known that the plastic flow stresses of many metals increase with an increase in the strain rate, with mild steel showing a particularly marked increase, while aluminium alloy is relatively insensitive [ 1]. Although the impact velocities and scale factors in the present study are relatively small, the local strain rates within any shear zones may be large and could influence the material properties. However, notwithstanding this observation, it is clear from Figs 10, 11 (b) and 17 that the perforation or critical impact energies of the aluminium alloy plates almost conform to the geometrically similar laws. Similar remarks apply to the perforation energies for the mild steel plates in Fig. 11 (a). It is interesting to note that the experimental results in Table 1 of Neilson [25] also indicate that the laws of geometrically similar scaling were almost obeyed for the perforation energies of full-scale prototypes and half-scale mild steel test specimens ~:subjected to impact velocities of lOOms -1, approximately. The present experimental results and the *It may also be shown that the proposed modification for the aluminium alloy plates implies that the slope depends on the flow stress, while the interceptdoes not. : The mild steel panels were scaled except for the span which was constant.

Scaling laws for metal plates struck by large masses

£

503

2.5 2.0 A/O

0

1'5

o.7

J

1"0 0"5 0 (a) 0

I

o'.i o12 o!3

I

0.s

0.6

0'7 X"

/

p,

5" 121 1.0

A 0

08 (}6 O.t,

/

O~, / O/ A

0.2 0 (b)

o.b2

0b6 0bB X

FIG. 15. Variation of the non-dimensional permanent transverse deflection (6*) with nondimensional impact energy (Z*) for geometrically scaled impact tests. (a) Mild steel plates. 6* and Z* are corrected with respect to the yield stress for the smallest scale test specimens, A, O, [] and represent the experimental data on the 2, 4, 6 and 8 mm thick plates, respectively. - - : Least square line for the smallest-scale test specimens. (b) Aluminium alloy plates. 6* and X* are corrected with respect to the yield stress for the smallest scale test specimens. A, O, [] and V represent the experimental data on the 2, 4.76, 6.35 and 9.53 mm thick plates, respectively. - - : Least square line for the smallest scale test specimens. observations of Neilson suggest that the lack of geometrically similar scaling of m a t e r i a l strain rate effects m a y not be i m p o r t a n t for the p e n e t r a t i o n a n d p e r f o r a t i o n of thin plates with impact velocities up to at least 100 m s - 1. It is not suggested that m a t e r i a l strain rate sensitivity effects would not be i m p o r t a n t , but, as noted in Section 11.3.2 of Ref. [ 1 ], the change in the d y n a m i c flow stress of mild steel is a b o u t 16% over a scale range of four, whereas the absolute values of the d y n a m i c flow stresses m a y be e n h a n c e d considerably over the c o r r e s p o n d i n g static values. However, it is possible that the energy a b s o r b i n g mechanism m a y change for higher i m p a c t velocities a n d thereby cause a violation of the elementary geometrically similar scaling laws. 6. C O N C L U S I O N S An experimental investigation is r e p o r t e d into the scaling laws for fully c l a m p e d circular plates struck by blunt projectiles travelling at relatively low i m p a c t velocities. The test m o d e l plates were m a d e from either BS4360-43A mild steel (strain rate sensitive) or BSL157-T6 a l u m i n i u m alloy (strain rate insensitive) materials. The test specimens a n d

504

H.-M. WEN and N. JONES

i

1 1"0

/~/Ap

0.75 0.5 (a)

0

i1"0

A/a n 0.75 05

(b) FIG. 16. Ratio of the permanent transverse deflection (A*) and the corresponding predicted value (A*) versus fl for the geometrically scaled impact tests. (a) Mild steel plates with ~, = 2.975 for an impact velocity of 4.5 m s - 1, approximately. A~ for A, O, [] and V are based on the experimental results, which are corrected with respect to the yield stress for the smallest scale specimens, for 2 (fl = I), 4 (fl = 2), 6 (fl = 3) and 8 mm (fl = 4) thick plates, respectively. - - : A*/A* = I for geometrically similar scaling. (b) Aluminium alloy plates with ~, = 4 for an impact velocity of 3.7 m s- I, approximately. A* for A, O, [] and ~ are based on the experimental results, which are corrected with respect to the yield stress for the smallest scale specimens, for 2 (fl = 1), 4.76 (fl = 2.38), 6.35 (fl = 3.175) and 9.53 mm (fl = 4.765) thick plates, respectively. - - : A*/A* = 1 for geometrically similar scaling.

i

t

1"0

A~/A~p

A.

O75

°50

4

FIG. 17. Ratio of the critical permanent transverse deflection (A*) and the corresponding predicted value (A~*p)versus fl for the geometrically scaled impact tests on aluminium alloy plates with ~, = 4 for an impact velocity of 4.2 m s - 1, approximately. A¢* for ,i, ~, [] and ql' are based on the experimental results, which are corrected with respect to the yield stress for the smallest scale specimens, for 2 (fl = 1), 4.76 (fl = 2.38), 6.35 (fl = 3.175) and 9.53 mm (fl = 4.765) thick plates, respectively. - - : A,/Acp - 1 for geometrically similar scaling.

i m p a c t c o n d i t i o n s were g e o m e t r i c a l l y scaled as far as possible a n d scale factors of f o u r for the m i l d steel plates a n d a p p r o x i m a t e l y five for the a l u m i n i u m alloy s p e c i m e n s were e x a m i n e d in this study. T h e e x t e r n a l kinetic energies r a n g e d f r o m relatively low values giving small plastic d e f o r m a t i o n s in the plates up to high values which p r o d u c e d c r a c k i n g o r perforation. T h e e x p e r i m e n t a l results for the transverse d e f o r m a t i o n s , interracial forces, r e s p o n s e times and p e r f o r a t i o n energies o b e y the g e o m e t r i c a l l y similar scaling laws w i t h i n a r a n g e of a c c u r a c y which is e x p e c t e d for d y n a m i c tests of the type r e p o r t e d here. It is o b s e r v e d that the influence of the m e c h a n i c a l p r o p e r t i e s of the m a t e r i a l s causes a slight d e v i a t i o n f r o m the g e o m e t r i c a l l y scaling laws for the a l u m i n i u m alloy w h i c h is less p r o n o u n c e d for the p e r m a n e n t t r a n s v e r s e d e f o r m a t i o n s of the mild steel specimens. It also a p p e a r s that

Scaling laws for metal plates struck by large masses

505

t h e lack of g e o m e t r i c a l l y s i m i l a r s c a l i n g o f m a t e r i a l s t r a i n r a t e s e n s i t i v e effects is n o t i m p o r t a n t for t h e p e r f o r a t i o n of p l a t e s a t least w i t h i n t h e r a n g e of t h e e x p e r i m e n t a l p a r a m e t e r s s t u d i e d in t h e p r e s e n t i n v e s t i g a t i o n . Acknowledgements--The authors wish to take this opportunity to express their thanks to the Impact Research Centre in the Department of Mechanical Engineering at the University of Liverpool, especially to Dr R. S. Birch and Mr G. Swallow for their assistance with the experimental work. Thanks are also due to Mr F. J. Cummins for tracing the figures and Mrs M. White for assistance with the preparation of the manuscript. The first author is indebted to the British Council for the payment of his tuition fees and to the Chinese Government for further financial support. He also wishes to express his gratitude to Taiyuan University of Technology for granting a leave of absence.

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