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Deformation of thin plates subjected to impulsive loading—a review Part II: Experimental studies
Int. J. Impact Engng Vol. 8, No. 2, pp. 171-186, 1989 Printed in Great Britain
0734-743X/89 $3.00+ 0.00 © 1989 Pergamon Press plc
DEFORMATION OF THIN PLATES SUBJECTED IMPULSIVE LOADING--A REVIEW P A R T II: E X P E R I M E N T A L
TO
STUDIES
G. N. NURICK and J. B. MARTIN Faculty of Engineering, University of Cape Town, South Africa (Received 20 July 1988; and in revised form 1 December 1988)
Summry--This two-part article reviews the theoretical predictions and experimental work on the deformation of thin plates subjected to impulsive loading. Part I lint. J. Impact Enong 8, 159-170 (1989)] dealt with the theoretical predictions and focused on the comparisons of the theoretical studies. Experimental-theoretical correlations were also discussed. Part II discusses the experimental results, and presents an empirical relationship between the deflection-thickness ratio and a function of impulse, plate geometry, plate dimensions and material properties.
NOTATION A Ao B I L R t v /~ 2 p ao ad ~,
area of plate area of plate over which impulse is imparted breadth of quadrangular plate total impulse length of quadrangular plate circular plate radius plate thickness impact velocity Johnson's damage number geometry number aspect ratio density modified damage number static yield stress damage yield stress loading parameter geometrical damage number
INTRODUCTION
Most review papers on the dynamic plastic behaviour of structures, and in particular of plates, subjected to impulsive loading have focused mainly on the theoretical predictions. That is understandable, since the number of theoretical predictions far outnumber the experiments performed. This paper summarizes these experiments and in addition presents a dimensionless correlation between some of the experiments conducted under similar conditions. Measurements of the dynamic response of structures such as beams and plates using various testing techniques have been reported. These include situations where the structure is subjected to air pressure waves created by explosive devices, underwater explosive forming, direct impulsive loading using plastic sheet explosives and spring loaded arms. These investigations have primarily been concerned with the final deformed shape and the deflection-time history. Whereas it is simple to measure the deformed shape of a structure, the deflection-time history is more difficult to measure; methods used include high-speed photography, stereophotogrammetric techniques, strain gauges, condenser microphones and light interference techniques. 171
172
G . N . NURICK and J. B. MARTIN EXPERIMENTAL
DETAILS
There have been several experimental studies to measure large deformations of plates subjected to blast a~nd impact loading. These investigations have primarily been concerned with the measurement of the final deformed shape, while only measurements of the deflecti0n-time history and impulse have been reported. The magnitude and shape of the deformed plate depend on the form of impulsive loading. Whereas it is simple to measure this final deformed shape of the structure, the deflection-time history and, to a lesser degree, the impulse are more difficult. The earliest studies reported were mainly concerned with structures subjected to underwater explosive charges. Taylor [1] describes experiments of large steel plates, approximately 1.83 x 1.22 m, subjected to underwater explosive blasts fired at various distances from a position normal to the plate through its mid-point. The principal measurements made in cases where the plate did not burst were the volume contained between the dished plate and its original position, and the maximum displacement: Travis et al. [2] and Johnson et al. [3] studied the effect of underwater explosive forming on fully clamped circular discs of various thicknesses and different materials. Again, the maximum displacement was measured. In Refs [1-3], the response time and the impulse were not measured. Johnson et al. [4, 5] in other investigations measured the displacementtime history using pin-contactors developed by Minshall [6]. The pins were positioned at known intervals apart, and as the blank made contact with each pin a signal was generated and displayed on an oscilloscope. Finnie [7] investigated, using several plate materials and plate thicknesses, the experimental relationship between deformation and explosive parameters such as charge mass and stand off distance. Williams [8] used high-speed photography (15,000 frames per second) to photograph, against a graticule, the growing bulge of a plate specimen. Boyd [9] reported on underwater explosive tests on circular plates, in which the total applied impulse was determined using empirical formulae reported by Cole [10]. Bednarski [11] presented results of a circular membrane subjected to an abrupt pressure rise by the underwater detonation of explosives. The deformation of the membrane was measured with a high-speed camera kit equipped with a stereoscopic attachment. The filming speed was 6000 frames per second and the entire process was filmed with as few as ten frames. The deflection-time history was plotted for the entire plate. The range of deformations of the above experiments is 10-60 deflection-thickness ratios [1-3, 8, 11], and the plate specimens are deformed in approximately 1500/~s [1 I]. The second type of impulsive loading was reported by Witmer et al. [12], who described work by Hoffman [13]. Pressure waves were created by an air-blast from detonations of spherical charges of pentolite of various masses placed on the central axis and normal to, and at various distances from, the test specimen. The magnitude of the permanent deformations measured was up to 16 plate thicknesses. No impulse or response-time history measurements were reported. The third type of impulsive loading was first reported by Humphreys [14]; it involved the use of sheet explosive and a ballistic pendulum. Layers of Dupont sheet explosive of thickness 0.4 mm each were applied to a clamped beam, separated from the beam by a layer of sponge rubber to prevent spallation. The test specimens were rigidly clamped to a ballistic pendulum. The burn rate of the explosive was 6700 m s- 1. This is higher than the speed of sound in the materials used. Johnson [15] reports the speed of sound in the following materials: carbon steel, 5150 m s- 1; aluminium, 5700 m s- 1; copper, 3700 m s- 1; brass, 3350 m s-1. It was thus felt that a fair approximation to instantaneous uniform loading was obtained. The deformation-time history was measured by means of a high-speed camera modified for split frame use and capable of achieving effective speeds of 10,000-12,000 frames per second. The explosion was initiated at the centre of the test specimen by means of a pigtail of the same material leading back to a standard detonator inside a rigid container to shield the pendulum from the detonator blast. A Fastax camera was attached to the pendulum behind the test specimen and was shielded from the blast so that the light produced from
Deformation of impulsively loaded thin plates--II
173
the explosion was prevented from reaching the camera lens. No strain measurements were made, owing to the difficulty encountered in keeping strain gauges from being spalled off by the initial exPlosively produced transverse stress waves. Humphreys [14] reports that the two assumptions inherent in the successful use of this technique are verified in the first eight frames. The explosion starts to occur in the second frame and is completely over in the fourth frame (a time span of approximately 200-250 #s), before the beam had noticeably moved at all. Subsequent motion takes place under no load, purely as a result of the inertia. Thus the assumption of an initial velocity condition under impulsive loading is reasonable. The final plastic deformation was observed by the sixth frame--hence .deformation took place in approximately 150-200#s, which is extremely short compared to the natural period of the ballistic pendulum. (In Humphrey's experiments this period was 4.43 s.) Thus all plastic deformation is over well before the pendulum has moved at all. The recorded pendulum swing gives a direct indication of the maximum potential energy oflthe system after the dissipation of energy in plastic wOrk. This potential energy is used to calculate the maximum velocity of the whole pendulum system and hence gives an accurate measure of.the applied impulse. The sheet explosive/ballistic pendulum method has been used by several researchers during investigations of different types of specimen. The deflection-time history ,was not always measured.Florence and Firth [16] conducted experiments op pinned and clamped beams and separat!ng the sheet- explosive from the beam by a layer of Neoprene. The rig on which the beams were attached was fixed, and the impulse, was measured by calibration of the sheet explosive. Several of these experiments were photographed using a Beckman and Whitley framing camera. Florence and Wierzbicki [17; 18] performed similar experi, merits on fully edge-clamped circular plates in which only the ,final deflection of the plate profile was measured. Duffey and Key [19, 20] also calibrated the sheet explosive for experiments on fully edge-clamped circular plates. A high-speed camera was used to measure the displacement-time history, and strain gauges were used in some cases for strain measurements. Jones et al. [21] described experiments on end-clamped wide beams and rectangular plates in which the impulse was measured directly using a ballistie pendulum. Jones et al. [22] subsequently described experiments on fully clamped rectangular plates Using the same techniques. A significant result of this work was that it appears that the type of attenuator, i.e. foam or neoprene, did not influence the outcome of the test, except that the impulsive velocity varied according to the attenuator used. Further work by Jones and Baeder [23] investigated fully clamped rectangular plates of varying length-to-breadth ratios. Symonds and Jones [24] also reported on fully clamped beams attached to a ballistic pendulum. In the experiments described in Refs [21-24], only the final deformations were measured. Bodner and Symonds [25], in investigating the dynamic plastic loading of frames using a ballistic pendulum, measured the time-history response o f the frame by using wire resistance strain gauges placed at the top of the columns on both sides. Bodner and Symonds [,26] also investigated the response of fully clamped circular plates, but this time measured the deflection-time history by using a condenser microphone placed near the centre of the plate. Nuriek et a l . [27-31] described experiments using armular rings of sheet explosive to simulate impulsive loading and in which the magnitude of the impulse is measured by means of a ballistic pendulum. The deflection-time history was recorded using a light-interference technique in which photo-voltaic diodes were used t o measure the light interference'patterns obtained during deformation. Deflections of up to 20 mm during a time period of 200 #s were observed in over 100 experiments on fully clamped circular, square and rectangular plates. The range of deflection-thickness ratios for the plate experiments using sheet explosive is 0.4-12.0 [17-20,25,28,31] for circular plates and 0.2-12.0 [21-23,28,31] for quadrangular plates; this is significantly smaller than those due to underwater explosion tests. The deformed plates reach their maximum deflection in approximately 150-200/~s [20, 26, 28, 31], which is also significantly less than in underwater explosion tests.
200 200
Boyd, 1966 [9]
Bednarski, 1969 [11]
Ghosh et al., 1976 [32]; 1979 [35]; 1984 [34]
Inertial forming machine
Witmer, Balmer, Leach, Piaan, 1963 [12]
120
610
140
Finnie, 1962 [7]
Air blast
150
Diameter
Travis & Johnson, 1962 [2]
Underwater blast
Dimensions (mm)
Lead Aluminium
Aluminium alloy
Mild steel
Aluminium
Titanium and alloys
Steel
Copper
Titanium Aluminium
Brass
Stainless steel Mild steel
CIRCULAR PLATES
Specimen type
0.61 0.31
3.2 6.4 9.6
0.83
1.3
1.2 1.6 2.7 3.3 6.4 1.3 3.2 6.2
0.9 0.9 1.2 1.6 0.9 2.0 0.9 0.9 2.0 0.9 2.0
Plate thickness (ram)
44 42
9
16
60
53
47 36-38 18 17 8 38-48 9-18 4-9
35-54 20--45 16-36 17-24 31-48 15-23 22-37 22-35 11-20 27-48 16-24
Deflection-thickness ratio
TABLE |. RI~SUM[~AND RESULTSOF EXPERIMENTALTECHNIQUESOF IMPULSIVELYLOADEDPLATES
1700
Response time (#s)
64 100
Bodner & Symonds, 1979 [26]
Nurick, 1986 [29-31]
Nurick, 1986 129-31]
128 x 32
Jones&Baeder, 1972 [23]
113 x 70 89 x 89
128 x 128
128 x 96
128 x 6 4
129 x 76
Jones, Uran & Tekin, 1970 [22]
Sheet explosive blast
Taylor, 1942 [1]
1830 x 1220
150
Duffey& Key, 1967 [19]; 1968 [20]
Underwater blast
100
FIorence&Werzbicki, 1966 [17]; 1970 [18]
Sheet explosive blast
Mild steel Mild steel
Mild steel Aluminium Mild steel Aluminium Mild steel Aluminium Mild steel Aluminium
Aluminium
Mild steel
Steel
NON-CIRCULAR PLATES
Mild steel
Titanium Mild steel
Mild steel
Aluminium
Aluminium Mild steel
1.6 1.6
2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7
1.6 2.5 4.4 3.1 4.8 6.2
3.1 4.4 5.9-6.4 9.3
1.6
2.3 1.9
1.6 3.2 1.6
6.3 6.3
3-12 6-12
0.3-I .2 0.5-1.2 0.8-4 1.2-3 0.7-7.5 2.0-4.5 1.6-9.5 3.0-6.0
3.5-7.0 1-4.5 0.3-1.7 1.8-3.5 0.8-2.4 0.2-1.4
51 6-25 12-35 2-21
4-12
0.9-6 0.5-7
4-9 1.5-1.7 3-4
1.6-7 0.4-4
120-200 90-180
140-190
60-75 100-120
200-250
o<
Q
176
G . N . NuRIc~ and J. B. MARTIN
Ghosh et al. [32-34] used another method, described in detail by Ghosh and Travis [35], to subject a membrane to an impulsive load. This method made use of an inertial forming machine which comprises a spring-loaded arm which carries a pair of clamping rings holding the membrane at its end. The arm is drawn back against the resistance of the springs by a winch arrangement. After release, the arm accelerates the membrane and clamping ring assembly to a maximum speed of 55 m s- 1, and is brought to rest when the clamping ring strikes an anvil and separates from the arm following the failure of a set of shear pins. The arrest of the clamping ring is followed by the deformation of the membrane under its own inertia. Strain gauges are used to measure the strain-time histories, but at higher initial velocities these gauges spall from the specimens. Deflection-thickness ratios of up to 45 were measured. Table 1 summarizes the plate test results of all the methods described above. COMPARISON
OF VARIOUS
DATA
In an attempt to compare experimental results with other experiments, which have different target dimensions, use is made of a dimensionless damage number defined by Johnson [15] as
(1)
= pv2/~rd,
which is used as a guide for assessing the behaviour of metals in impact situations. Here p is the material density, v the impact velocity and a d the damage stress. Table 2 shows the damage regime as a function of damage number, and also for comparison shows the results of some circular plate experiments. It can be seen that ~ predicts an order of magnitude deformation, and, in doing so, does not consider the method of impact, the interpretation of ad, the target geometry and boundary conditions or the target dimensions. Since we wish to compare results of deformed plates of similar geometries, boundary conditions and loading, it seems reasonable to introduce factors which would allow all other variables to be normalized into dimensionless groups. Johnson's damage number can be written in terms of the impulse as 12
102
% - Ao2t2pad
tZpa d'
(2)
where I is the total impulse, A o is that area of the plate over which the impulse is imparted, I o is the impulse per area and t is the plate thickness. Figure 1 shows a plot of deflection-thickness ratio versus So for circular plates of varying dimensions and material properties. This indicates the order of magnitude reliability of Johnson's damage number, but, as can be seen, the variations for similar loading conditions are a factor of up to 2.5. For dissimilar loading conditions, the variation has a factor of up to 8.5. In the case of quadrangular plates, we introduce a geometry number defined as L fl = - ,
(3)
B
where L are B are the plate length and breadth respectively. A geometrical damage number is then written as F
I A \2-1x/2
=ta ot ) j
,
(4,
where A is the area of the plate. A relationship between the distance from the plate centre to the nearest boundary and the plate thickness is introduced as the aspect ratio 2, where 2 = R / t for circular plates
(5a)
2 = B/2t for quadrangular plates.
(5b)
0.0316
1 x 10 - 3
31.622
10a
x
3.162
1 x 101 (10)
1
0.316
1 x10-1 (0.1)
(0.001)
0.0032
Gt½
1 x 10 - s
Damage No.
Hypervelocity impact
Extensive plastic deformation (ordinary bullet speeds)
Moderate plastic behaviour (slow bullet speeds)
Plastic behaviour starts
Quasi-static elastic
Regime
0.61 1.6 1.9 6.2 1.6 1.9
0.61 6.22
2.95 1.17 1.17
(nun)
t
TABLE 2. REGIME OF DAMAGE (FROM JOHNSON
1
1.36 5.7 3.1 138 14 7
0.28 35
0.2 0.1 0.15
98 31 32 16 31 32
98 16
25 64 64
t
R
Plate example
AND SOME RELATED EXAMPLES
(N s)
[15])
46 3.8 2.6 4.6 11.1 6.4
11.5 0.6
0.1 0.5 0.7
t
6
0.09 0.10 0.16 0.20 0.60 0.77
0.004 0.013
6.6 x 10- 6 1 x 10 - s 2.4 x 10 -5
[34] [28] [26] [18] [28] [26]
[34] [18]
[36] [36] [36]
Ref.
/
~
~.
e~
3" "~
~'
0
178
G . N . NURICK and J. B. MARTIN 12
-/ iO
¢,
o P
,/I
8
i~
•
30
o
I
4
/ .-
/
/-
3d
o..~ofl
¢-1 0
i 05
i I
I 15
I
(oolg
I 2
I
= z 0/tlpo.012
FIG. 1. Graph of deflection-thickness ratio vs Johnson's damage number for different plate geometries and loading conditions (the lines represent the least squares fit to the respective data). 1. Nurick et al. [28]; R = 50 ram, t = 1.6 ram, mild steel uniformly loaded. 2. Wierzbicki & Florence [18]; R = 50 mm, t = 6.3 mm, mild steel uniformly loaded. 3. Bodner & Symonds [26]; R = 32 ram, a, c, d - - t = 1.90mm; mild steel, I ~ - t = 2 . 3 m m ; titanium a, b---uniformly loaded; c--loaded Ro/ R = ½; d--loaded Ro/ R = I.
The loading area per total plate area must also be considered; this is accounted for by a loading parameter assumed to be ( = 1 + ln(R/Ro)
(6)
for circular plates only, where Ro is the radius of the loaded area. Equation (6) implies that when R0 ~ R, ( ~ 1 and the plate is uniformly loaded over the full area, and that when R0 is very small, the loading tends to a point-loaded situation and is more concentrated, resulting in a larger mid-point deflection. The effect of partial loading on quadrangular plates has not been included. Combining equations (4)-(6) results in a modified damage number that incorporates dimensions and loading: 4' = ~b2~.
(7)
4,c = I(1 + In R/Ro) rrRt2(ptro) 1/2
(8a)
For circular plates then,
and for quadrangular plates I 4,q -- Z
[DL
"1/2
(8b)
Table 3 summarizes all the dimensionless parameters, and Table 4 summarizes the experimental data. The geometry and material density variables in equation (8) are easily obtainable. The variables I and aa are measured experimentally--/from impulse tests, and, for convenience, ~d is given the value of ~r0, the static yield stress. The value of 0~o is determined from the results of quasi-static uniaxial tensile tests (see for example Nurick et al. [28], Bodner and Symonds [26]). Figures 2 and 3 show dimensionless plots of deflection-thickness ratio versus 4, for the experimental results on circular and quadrangular plates respectively. Also shown are dimensionless plots of theoretical predictions described in part I. Least squares correlation analysis of the circular plate experimental data yields a correlation coefficient of 0.973,
Deformation of impulsively loaded thin plates--II
179
TABLE 3. DIr,~NS[ONLESSPAI~A~m~I~S Johnson's damage No. (eq. 1)
c¢
pv 2
--
ad
12
or
A2t2pffd
Circular geometry Geometry No. (eq. 3)
fl
I/~ =
L
--
= ~/~
Geometrical damage No.
Quadrangular geometry
12
12
A2t2pff d
B3Lt2pff d
I
I
At(pad) 1/2
Bt(BLp¢o)l/2
(~/)1/2
(eq. 4) Aspect No. (eq. 5)
2
-t
R
B
Loading parameter (eq. 6)
~
1 + ln(R/Ro)
--
(eq. 8) ~b, = ¢~(
I
I
7tRt2(pff d)l/2
2 t 2 ( B L p f d)l/2
I(1 + l n ~o ) 7tRt2(ptrd) 1/2
Dimensionless numbers reported by others Guedes Soares [37]
412R 2
4,~2ct
po'oA2t't =
4#p2 [2L2
Jones [22, 23]
- 4,~2fl20[
paoA2t * = 4,8~b~
and these experimental results are almost all within one deflection-thickness ratio of the least squares line, as shown in Fig. 4. For the quadrangular plates (Fig. 5), the experimental results are also all mostly within one deflection-thickness ratio of the least squares correlation which in this case has a correlation coefficient of 0.980. All readings thus far reported were for deformations in which no tearing of the plate was observed. However, a few experiments were performed by Nurick 1-38] in which, in a small proportion (approximately one-tenth) of the boundary perimeter, tearing was observed. This partial tearing of the plate was observed at the clamped boundaries at impulses of 17.7 N s and 19.0 N s for the square and rectangular plates. This translates to ~b values of 26 and 28 respectively. For the circular plates, tearing occurred at an impulse of 15.6 N s, which translates to a ~b value of 26. Insufficient data on impulse parameters preclude an attempt being made to perform a similar detailed analysis for the other forms of impulsive loading, i.e. underwater blast pressure, air blast pressure and inertial frame loading. However, it is noted that Ghosh et al. [32] have used Johnson's damage number in their analysis of the comparison between their experimental results and same theoretical predictions. DISCUSSION
In the attempt to compare experimental results presented by different researchers using different plate dimensions and different plate materials, it became obvious that Johnson's
Nurick, 1986 [28-31]
128 x 32
Jones & Baeder, 1972 [23]
113 x 70 89 x 89
128 x 128
128 x 96
128 x 64
129 x 76
Jones, U r a n & Tekin, 1970 [22]
120
100
Nurick, 1986 [28-31] 40.3 162
282
Mild steel Mild steel
Mild steel Aluminium Mild steel Aluminium Mild steel Aluminium Mild steel Aluminium
Aluminium
Mild steel
282 296
251 279 251 279 251 279 251 279
248 233 254 284 281 286
NON-CIRCULAR PLATES
Lead Aluminium
Mild steel
255 223
540
Mild steel Titanium Mild steel
280
Aluminium
150
290 283
Aluminium Mild steel
64
Inertial formino machine Ghosh et al., 1976 [27] 1979 [30] 1984 [29]
Static yield stress (MPa)
CIRCULAR PLATES
Specimen type
100
Diameter
Dimensions (nun)
Bodner & Symonds, 1979 [26]
Florence & Werzbicki, 1966 [17] 1970 [18] Duffey & Key, 1967 [19] 1968 [20]
Sheet explosive blast
1.6 1.6
2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7
1.6 2.5 4.4 3.1 4.8 6.2
0.61 0.31
1.6
2.3 1.9
1.6 3.2 1.6
6.3 6.3
Plate thickness (mm)
TABLE 4. SUMMARY OF COMPARATIVE EXPERIMENTAL DATA
3-12 6-12
0.3-1.2 0.5-1.2 0.8-4 1.2-3 0.7-7.5 2.1)-4.5 1.6-9.5 3.0-6.0
3.5-7.0 1-4.5 0.3-1.7 1.8-3.5 0.8-2.4 02.-1.4
44 42
4-12
0.9-6 0.5-7
4-9 1.5-1.7 3-4
1.6-7 0.4-4
Deflection-thickness ratio
120-200 90-180
140-190
65 75 100-120
200-250
Response time (/zs)
Z
~r
¢3
Z z
Deformation of impulsively loaded thin plates--II 12
3181" la/'~bll 10'9/4 7 G/5~;4
a
/;/F'/;'
: ,'
i
iJ
IO'
.o
181
8
I I
f "i
il"
U"'//
., :11..
II/f/I
I
~ ~I <~1
I .'". / .
A,~lx I
.
I
~)
I
3o
.
I"
..i,
I
r'~ 4
/
I 2 3 4 5
I0
15
2O l
25
~c "I(l+tnR/Ro)/rRt2 (P~lg
FIG. 2. Graph of deflection-thickness ratio vs dimensionless No. ~bo. EXPERIMENTAL DATA
Material
Loading
Lx Nurick
A
D
50
1.6
• Bodner & Symonds []
B A
D D
•
B
E
O
A
E
32 32 22 32
2.3 1.9 2.3 1.9
A
D
C
D
50 50
6.3 6.3
•
Wierzbicki & Florence
•
Radius
Thickness
Notes
A--steel; B--titanium; C--aluminium; D--uniform loading; E--partial loading. Predictions (see part I for cited references)
1--Lippmann [33]; 2--Perrone & Bhadra [47]; 3--Noble & Oxley [31], a--circular profile, b--conical profile; 4--Calladine [52]; 5--Duffey [51]; 6-43uedes Soares [34]; 7--Westine & Baker [44]; 8--Wierzbicki & Florence [56], a--large deflections, b---small deflections; 9--Hudson [8]; lO--Batra & Dubey [32]; 1l--Ghosh & Weber [34]; 12--Wang & Hopkins [14]; 13--Wang [15]; 14--Jones [41].
damage number was a good guide, as shown in Fig. 1. However, this did not provide a suitable means for comparison, and hence an extension to Johnson's damage number was formulated. Figures 2-5 show plots of deflection-thickness ratio vs this new dimensionless number $. Also shown in Figs 2 and 3 are plots of predictions that do not consider strain rate effects, and therefore the predictions of Nurick et al. [31] and those of Symonds and Wierzbicki are omitted. It will be noted that there are similarities between Fig. 2 and Fig. 1 of part I and between Fig. 3 and Figs 2 and 3 of part I. However, the important issue here is illustrated in Figs 4 and 5, in which the experimental data results are bounded on either side of the least squares fit by a one deflection-
182
G . N . NURICK and J. B. MARTIN :
32
._0
/
8
.g: I
6
•
/
/
8 o
~ A
a 4
I, zx
t'• I 2 3 4 5
I0
15
¢q I/2tPXBLp~o)~
20
25
=
FIG. 3. Graph of deflection-thickness ratio vs dimensionless No. ~bq. EXPERIMENTALDATA Material
fl = L / B
Thickness
[] A
Nurick
A A
1.0 1.62
1.6 1.6
•
Jones & Baeder [23]
A
4.0 2.0 1.33
2.69 2.69 2.69
1.0
2.69
•
•
B
Jones, Uran & Tekin [22]
•
A B
1.69
1.63
2.49 4.39
Notes: A--steel; B--aluminium. All uniformly loaded. Predictions (see part I for cited references) 1--Baker [49]; 2--Jones et al. [50, 58], fl = 1.6; 3--Jones et al. [50, 58], fl = 1.0; 4--Jones [41, 59], fl = 1.0; 5--Jones [41], fl = 1.6; ~ - J o n e s [59], fl = 1.6.
thickness ratio confidence limit. It can be seen that in both cases most of the data points lie within these bounds. Figure 6 shows both the least squares fit and bounds of Figs 4 and 5, from which guidelines for a design code may be considered. The least squares analysis yields from the circular plate experimental data ( 6 ) = 0.425q~c + 0.277,
io
where the number of data points was 109, with a correlation coefficient of 0.974.
(9)
Deformation of impulsively loaded thin plates--II 12'
,/ I0 •
8"
/
/ /~/ / /
/// /
6"
,/ 4"
/
0
I 2 3 4 5
I0
15
20
25
$c PiG. 4. Graph of deflection-thickness ratio vs dimensionless No. @~, showing least squares correlation ( ) bounded by +_! deflection-thickness ratio ( - - - - - - ) .
/ /o
.o a m
I
~6
~ m2
i i 3 4 5
I 10
E
I
I
15
20
25
~q FIG. 5. Graph of deflection-thickness ratio vs dimensionless No. ~q, showing least squares correlation ( ) bounded by + 1 deflection thickness ratio ( - - - - - - ) .
183
184
G . N . NURICK and J. B. MARTIN
,2t
Circutapl r ates
/ .
I0
8
/~
~/
~O~uadrangut prat clres
6-
4-
2-
0
I0
34
20
40
50
~c,q FIG. 6. Graph of deflection-thickness ratio vs 4~4.
From the quadrangular plate experimental data, the analysis results are ( 6 ) =0.471~bq+O.O01,
Tq
(10)
using 156 data points, with a correlation coefficient of 0.984, and recalling that 1(1 + ln[R/Ro] ) ~P~= rcgt2(ptro)X/2 C~q --
I 2t2(BLptro )l/2 "
(1 la)
(1 lb)
Hence it may now be possible to determine the deflection-thickness ratio of a circular or quadrangular plate subjected to an impulsive load, by using equations (9) and (10). There appears to be some justification of equations (9) and (10). Consider two plates, one circular the other quadrangular, of equal area, thickness and material properties, subjected to an impulse over the entire plate area; then from equation (11) q~¢
4'q
2 -
-
1.128,
(12)
and substituting into equation (9), ( ~ ) = 0.480q~q + 0.277.
(13)
A comparison of equations (10) and (13) shows similar gradients.
CONCLUDING
REMARKS
An analysis of the experimental results provides a useful guideline for predicting maximum central deflection of impulsively loaded plates. An indication of when tearing might occur is presented, but more results are required for clarification. This is currently being investigated.
Deformation of impulsively loaded thin plates--II
185
REFERENCES 1. G. I. TAYLOR,The distortion under pressure of a diaphragm which is clamped along its edge and stressed beyond its elastic limit. Underwater Explosion Research, Voi. 3, The Damage Process, pp. 107-121. Office of Naval Research (1950, originally written 1942). 2. F. W. TRAVISand W. JOHNSON, Experiments in the dynamic deformation of clamped circular sheets of various metals subject to an underwater explosive charge. Sheet Metal lndust. 39, 456-474 (1961). 3. W. JOHNSON, A. POYNTON, H. SINGH and F. W. TRAVlS, Experiments in the underwater explosive stretch forming of clamped circular blanks. Int. J. Mech. Sci. 8, 237-270 (1966). 4. W. JOHNSON, K. KORMI and F. W. TRAVIS,An investigation into the explosive deep drawing of circular blanks using the plug cushion technique. Int. J. Mech. Sci. 6, 293-328 (1966). 5. W. JOHNSON, K. KORMIand F. W. TRAVlS,The explosive drawing of square and flat-bottomed circular cups and bubble pulsation phenomena. Adv. Mach. Tool Des. Res. 293-328 (1964). 6. S. MINSHALL,Properties of elastic and plastic waves determined by pin contactors and crystals. J. Appl. Phys. 26, 463-469 (1955). 7. T. M. FINNm, Explosive forming of circular diaphragms. Sheet Metal Indust. 39, 391-398 (1962). 8. T. WILLIAMS,Some metallurgical aspects of metal forming. Sheet Metal Indust. 39, 487-494 (1962). 9. D. E. BOYD, Dynamic deformation of circular membranes. ASCE 92 (EM3), 1-15 (1966). 10. R. H. COLE, Underwater Explosions. Princeton University Press (1948). 11. T. BEDNARSKI,The dynamic deformation of a circular membrane. Int. 3. Mech. Sci. 11, 949-959 (1969). 12. E. A. WIT~R, N. A. BALr~ER,J. W. LEECHand T. N. N. PLAN,Large dynamic deformations of beam, rings, plates and shells. AIAA 1, 1848-1857 (1963). 13. A. J. HOFFMAN,The plastic response of circular plates to air blast. Masters Thesis, University of Delaware (1955). 14. J. S. HUMPHREYS,Plastic deformation of impulsively loaded straight clamped beams. 3. appl. Mech. 32, 7-10 (1965). 15. W. JOHNSON,Impact Strength of Materials. Edward Arnold, London (1972). 16. A. L. FLORENCEand R. D. FIRTH, Rigid-plastic beams under uniformly distributed impulses. 3. appl. Mech. 32, 481-488 (1965). 17. A. L. FLORENCE,Circular plates under a uniformly distributed impulse. Int. 3. Solids Struct. 2, 37-47 (1966). 18. T. WmszmcKI and A. L. FLORENCE,A theoretical and experimental investigation of impulsively loaded clamped circular viscoplastic plates. Int. 3. Solids Struct. 6, 555-568 (1970). 19. T. A. DUFFEY,The large deflection dynamic response of clamped circular plates subject to explosive loading. Sandia Laboratories Research Report No. SC-RR-67-532 (1967). 20. T. A. Dur~Y and S. W. KEY, Experimental-theoretical correlation of impulsively loaded clamped circular plates. Sandia Laboratories Research Report SC-RR-68-210 (1968). 21. N. JONES, R. N. GRI~riN and R. E. VAN DUZEg, An experimental study into the dynamic plastic behaviour of wide beams and rectangular plates. Int. J. Mech. Sci. 13, 721-735 (1971). 22. N. JONES,T. URAN and S. A. TEKIN, The dynamic plastic behaviour of fully clamped rectangular plates. Int. 3. Solids Struct. 6, 1499-1512 (1970). 23. N. JONESand R. A. BAEDER,An experimental study of the dynamic plastic behaviour of rectangular plates. Symp. Plastic Analysis of Structures. Published by Ministry of Education, Polytechnic Institute of Jassy, Civil Engineering Faculty, Rumania, Vol. 1, pp. 476-497 (1972). 24. P. S. SYMONDSand N. JONES, Impulsive loading of fully clamped beams with finite plastic deflections and strain-rate sensitivity. Int. J. Mech. Sci. 14, 49-69 (1972). 25. S. R. BODNERand P. S. SYMONDS,Experiments on dynamic plastic loading of frames. Int. 3. Solids Struct. 15, 1-13 (1979). 26. S. R. BODNER and P. S. SYMONDS,Experiments on viscoplastic response of circular plates to impulsive loading. J. Mech. Phys. Solids 27, 91-113 (1979). 27. G. N. NURICK,A new technique to measure the deflection-time history of a material subjected to high strain rates. Int. 3. Impact Engng 3, 17-26 (1985). 28. G. N. NURICK, H. T. PEARCEand J. B. MARTIN, The deformation of thin plates subjected to impulsive loading. In Inelastic Behaciour of Plates and Shells (Edited by L. BEVILACQUA).Springer-Verlag, Berlin (1986). 29. G. N. NuRICg, The measurement of the deformation response of a structure subjected to an explosive load using a light interference technique. Prec. 1986 SEM Spring Conf. Exp. Mech., pp. 105-114. The Society for Experimental Mechanics (1986). 30. G. N. NURICK, Using photo-voltaic diodes to measure the deformation response of a structure subjected to an explosive load. Prec. 17th Int. Cong. High Speed Photography and Photonics, pp. 215-226. International Society for Optical Engineering (1986). 31. G. N. NURICK, H. T. PEARCEand J. B. MARTIN, Predictions of transverse deflections and in-plane strains in impulsively loaded thin plates. Int. J. Mech. Sci. 29, 435-442 (1987). 32. S. K. GHOSH and H. WEBER, Experimental-theoretical correlation of impulsively loaded axisymmetric rigid-plastic membrane. Mech. Res. Comm. 3, 423-428 (1976). 33. S. K. GHOSH, R. BALENDRAand F. W. TRAVlS,Inertial forming of circular and annular diaphragms. Adv. Mach. Tool Des. Res. 845-854 (1978). 34. S. K. GHOSH, S. R. REID and W. JOHNSON, The large deflection impulsive loading of clamped circular membranes. Structural Impact and Crashworthiness Conf., pp. 471-481. London (1984).
186
G . N . NURICK and J. B. MARTIN
35. S. K. GHOSH and F. W. TRAVIS, A unique experimental high rate forming machine. Int. J. Mech. Engng Educ. 7, 69-74 (1979) 36. G. N. NURICK, R. S. HAMILTONand D. PENN1NGTON, A simple method to predict the initial and subsequent deflections of a clamped circular plate subjected to repeated impacts. Proc. Soc. Exp. Mech. Spring Conf. Exp. Mech., New Orleans (1986). 37. C, GUEDES SOARES,A mode solution for the finite deflections of a circular plate loaded impulsively. Rozprawy lnzynierskie Engng Trans. 29, 99-114 (1987). 38. G. N. NURICK, Large deformations of thin plates subjected to impulsive loading. Ph.D. Thesis, University of Cape Town (1987). 39. N. JONES, Structural Impact. Cambridge University Press (in press).
0734-743X/89 $3.00+ 0.00 © 1989 Pergamon Press plc
DEFORMATION OF THIN PLATES SUBJECTED IMPULSIVE LOADING--A REVIEW P A R T II: E X P E R I M E N T A L
TO
STUDIES
G. N. NURICK and J. B. MARTIN Faculty of Engineering, University of Cape Town, South Africa (Received 20 July 1988; and in revised form 1 December 1988)
Summry--This two-part article reviews the theoretical predictions and experimental work on the deformation of thin plates subjected to impulsive loading. Part I lint. J. Impact Enong 8, 159-170 (1989)] dealt with the theoretical predictions and focused on the comparisons of the theoretical studies. Experimental-theoretical correlations were also discussed. Part II discusses the experimental results, and presents an empirical relationship between the deflection-thickness ratio and a function of impulse, plate geometry, plate dimensions and material properties.
NOTATION A Ao B I L R t v /~ 2 p ao ad ~,
area of plate area of plate over which impulse is imparted breadth of quadrangular plate total impulse length of quadrangular plate circular plate radius plate thickness impact velocity Johnson's damage number geometry number aspect ratio density modified damage number static yield stress damage yield stress loading parameter geometrical damage number
INTRODUCTION
Most review papers on the dynamic plastic behaviour of structures, and in particular of plates, subjected to impulsive loading have focused mainly on the theoretical predictions. That is understandable, since the number of theoretical predictions far outnumber the experiments performed. This paper summarizes these experiments and in addition presents a dimensionless correlation between some of the experiments conducted under similar conditions. Measurements of the dynamic response of structures such as beams and plates using various testing techniques have been reported. These include situations where the structure is subjected to air pressure waves created by explosive devices, underwater explosive forming, direct impulsive loading using plastic sheet explosives and spring loaded arms. These investigations have primarily been concerned with the final deformed shape and the deflection-time history. Whereas it is simple to measure the deformed shape of a structure, the deflection-time history is more difficult to measure; methods used include high-speed photography, stereophotogrammetric techniques, strain gauges, condenser microphones and light interference techniques. 171
172
G . N . NURICK and J. B. MARTIN EXPERIMENTAL
DETAILS
There have been several experimental studies to measure large deformations of plates subjected to blast a~nd impact loading. These investigations have primarily been concerned with the measurement of the final deformed shape, while only measurements of the deflecti0n-time history and impulse have been reported. The magnitude and shape of the deformed plate depend on the form of impulsive loading. Whereas it is simple to measure this final deformed shape of the structure, the deflection-time history and, to a lesser degree, the impulse are more difficult. The earliest studies reported were mainly concerned with structures subjected to underwater explosive charges. Taylor [1] describes experiments of large steel plates, approximately 1.83 x 1.22 m, subjected to underwater explosive blasts fired at various distances from a position normal to the plate through its mid-point. The principal measurements made in cases where the plate did not burst were the volume contained between the dished plate and its original position, and the maximum displacement: Travis et al. [2] and Johnson et al. [3] studied the effect of underwater explosive forming on fully clamped circular discs of various thicknesses and different materials. Again, the maximum displacement was measured. In Refs [1-3], the response time and the impulse were not measured. Johnson et al. [4, 5] in other investigations measured the displacementtime history using pin-contactors developed by Minshall [6]. The pins were positioned at known intervals apart, and as the blank made contact with each pin a signal was generated and displayed on an oscilloscope. Finnie [7] investigated, using several plate materials and plate thicknesses, the experimental relationship between deformation and explosive parameters such as charge mass and stand off distance. Williams [8] used high-speed photography (15,000 frames per second) to photograph, against a graticule, the growing bulge of a plate specimen. Boyd [9] reported on underwater explosive tests on circular plates, in which the total applied impulse was determined using empirical formulae reported by Cole [10]. Bednarski [11] presented results of a circular membrane subjected to an abrupt pressure rise by the underwater detonation of explosives. The deformation of the membrane was measured with a high-speed camera kit equipped with a stereoscopic attachment. The filming speed was 6000 frames per second and the entire process was filmed with as few as ten frames. The deflection-time history was plotted for the entire plate. The range of deformations of the above experiments is 10-60 deflection-thickness ratios [1-3, 8, 11], and the plate specimens are deformed in approximately 1500/~s [1 I]. The second type of impulsive loading was reported by Witmer et al. [12], who described work by Hoffman [13]. Pressure waves were created by an air-blast from detonations of spherical charges of pentolite of various masses placed on the central axis and normal to, and at various distances from, the test specimen. The magnitude of the permanent deformations measured was up to 16 plate thicknesses. No impulse or response-time history measurements were reported. The third type of impulsive loading was first reported by Humphreys [14]; it involved the use of sheet explosive and a ballistic pendulum. Layers of Dupont sheet explosive of thickness 0.4 mm each were applied to a clamped beam, separated from the beam by a layer of sponge rubber to prevent spallation. The test specimens were rigidly clamped to a ballistic pendulum. The burn rate of the explosive was 6700 m s- 1. This is higher than the speed of sound in the materials used. Johnson [15] reports the speed of sound in the following materials: carbon steel, 5150 m s- 1; aluminium, 5700 m s- 1; copper, 3700 m s- 1; brass, 3350 m s-1. It was thus felt that a fair approximation to instantaneous uniform loading was obtained. The deformation-time history was measured by means of a high-speed camera modified for split frame use and capable of achieving effective speeds of 10,000-12,000 frames per second. The explosion was initiated at the centre of the test specimen by means of a pigtail of the same material leading back to a standard detonator inside a rigid container to shield the pendulum from the detonator blast. A Fastax camera was attached to the pendulum behind the test specimen and was shielded from the blast so that the light produced from
Deformation of impulsively loaded thin plates--II
173
the explosion was prevented from reaching the camera lens. No strain measurements were made, owing to the difficulty encountered in keeping strain gauges from being spalled off by the initial exPlosively produced transverse stress waves. Humphreys [14] reports that the two assumptions inherent in the successful use of this technique are verified in the first eight frames. The explosion starts to occur in the second frame and is completely over in the fourth frame (a time span of approximately 200-250 #s), before the beam had noticeably moved at all. Subsequent motion takes place under no load, purely as a result of the inertia. Thus the assumption of an initial velocity condition under impulsive loading is reasonable. The final plastic deformation was observed by the sixth frame--hence .deformation took place in approximately 150-200#s, which is extremely short compared to the natural period of the ballistic pendulum. (In Humphrey's experiments this period was 4.43 s.) Thus all plastic deformation is over well before the pendulum has moved at all. The recorded pendulum swing gives a direct indication of the maximum potential energy oflthe system after the dissipation of energy in plastic wOrk. This potential energy is used to calculate the maximum velocity of the whole pendulum system and hence gives an accurate measure of.the applied impulse. The sheet explosive/ballistic pendulum method has been used by several researchers during investigations of different types of specimen. The deflection-time history ,was not always measured.Florence and Firth [16] conducted experiments op pinned and clamped beams and separat!ng the sheet- explosive from the beam by a layer of Neoprene. The rig on which the beams were attached was fixed, and the impulse, was measured by calibration of the sheet explosive. Several of these experiments were photographed using a Beckman and Whitley framing camera. Florence and Wierzbicki [17; 18] performed similar experi, merits on fully edge-clamped circular plates in which only the ,final deflection of the plate profile was measured. Duffey and Key [19, 20] also calibrated the sheet explosive for experiments on fully edge-clamped circular plates. A high-speed camera was used to measure the displacement-time history, and strain gauges were used in some cases for strain measurements. Jones et al. [21] described experiments on end-clamped wide beams and rectangular plates in which the impulse was measured directly using a ballistie pendulum. Jones et al. [22] subsequently described experiments on fully clamped rectangular plates Using the same techniques. A significant result of this work was that it appears that the type of attenuator, i.e. foam or neoprene, did not influence the outcome of the test, except that the impulsive velocity varied according to the attenuator used. Further work by Jones and Baeder [23] investigated fully clamped rectangular plates of varying length-to-breadth ratios. Symonds and Jones [24] also reported on fully clamped beams attached to a ballistic pendulum. In the experiments described in Refs [21-24], only the final deformations were measured. Bodner and Symonds [25], in investigating the dynamic plastic loading of frames using a ballistic pendulum, measured the time-history response o f the frame by using wire resistance strain gauges placed at the top of the columns on both sides. Bodner and Symonds [,26] also investigated the response of fully clamped circular plates, but this time measured the deflection-time history by using a condenser microphone placed near the centre of the plate. Nuriek et a l . [27-31] described experiments using armular rings of sheet explosive to simulate impulsive loading and in which the magnitude of the impulse is measured by means of a ballistic pendulum. The deflection-time history was recorded using a light-interference technique in which photo-voltaic diodes were used t o measure the light interference'patterns obtained during deformation. Deflections of up to 20 mm during a time period of 200 #s were observed in over 100 experiments on fully clamped circular, square and rectangular plates. The range of deflection-thickness ratios for the plate experiments using sheet explosive is 0.4-12.0 [17-20,25,28,31] for circular plates and 0.2-12.0 [21-23,28,31] for quadrangular plates; this is significantly smaller than those due to underwater explosion tests. The deformed plates reach their maximum deflection in approximately 150-200/~s [20, 26, 28, 31], which is also significantly less than in underwater explosion tests.
200 200
Boyd, 1966 [9]
Bednarski, 1969 [11]
Ghosh et al., 1976 [32]; 1979 [35]; 1984 [34]
Inertial forming machine
Witmer, Balmer, Leach, Piaan, 1963 [12]
120
610
140
Finnie, 1962 [7]
Air blast
150
Diameter
Travis & Johnson, 1962 [2]
Underwater blast
Dimensions (mm)
Lead Aluminium
Aluminium alloy
Mild steel
Aluminium
Titanium and alloys
Steel
Copper
Titanium Aluminium
Brass
Stainless steel Mild steel
CIRCULAR PLATES
Specimen type
0.61 0.31
3.2 6.4 9.6
0.83
1.3
1.2 1.6 2.7 3.3 6.4 1.3 3.2 6.2
0.9 0.9 1.2 1.6 0.9 2.0 0.9 0.9 2.0 0.9 2.0
Plate thickness (ram)
44 42
9
16
60
53
47 36-38 18 17 8 38-48 9-18 4-9
35-54 20--45 16-36 17-24 31-48 15-23 22-37 22-35 11-20 27-48 16-24
Deflection-thickness ratio
TABLE |. RI~SUM[~AND RESULTSOF EXPERIMENTALTECHNIQUESOF IMPULSIVELYLOADEDPLATES
1700
Response time (#s)
64 100
Bodner & Symonds, 1979 [26]
Nurick, 1986 [29-31]
Nurick, 1986 129-31]
128 x 32
Jones&Baeder, 1972 [23]
113 x 70 89 x 89
128 x 128
128 x 96
128 x 6 4
129 x 76
Jones, Uran & Tekin, 1970 [22]
Sheet explosive blast
Taylor, 1942 [1]
1830 x 1220
150
Duffey& Key, 1967 [19]; 1968 [20]
Underwater blast
100
FIorence&Werzbicki, 1966 [17]; 1970 [18]
Sheet explosive blast
Mild steel Mild steel
Mild steel Aluminium Mild steel Aluminium Mild steel Aluminium Mild steel Aluminium
Aluminium
Mild steel
Steel
NON-CIRCULAR PLATES
Mild steel
Titanium Mild steel
Mild steel
Aluminium
Aluminium Mild steel
1.6 1.6
2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7
1.6 2.5 4.4 3.1 4.8 6.2
3.1 4.4 5.9-6.4 9.3
1.6
2.3 1.9
1.6 3.2 1.6
6.3 6.3
3-12 6-12
0.3-I .2 0.5-1.2 0.8-4 1.2-3 0.7-7.5 2.0-4.5 1.6-9.5 3.0-6.0
3.5-7.0 1-4.5 0.3-1.7 1.8-3.5 0.8-2.4 0.2-1.4
51 6-25 12-35 2-21
4-12
0.9-6 0.5-7
4-9 1.5-1.7 3-4
1.6-7 0.4-4
120-200 90-180
140-190
60-75 100-120
200-250
o<
Q
176
G . N . NuRIc~ and J. B. MARTIN
Ghosh et al. [32-34] used another method, described in detail by Ghosh and Travis [35], to subject a membrane to an impulsive load. This method made use of an inertial forming machine which comprises a spring-loaded arm which carries a pair of clamping rings holding the membrane at its end. The arm is drawn back against the resistance of the springs by a winch arrangement. After release, the arm accelerates the membrane and clamping ring assembly to a maximum speed of 55 m s- 1, and is brought to rest when the clamping ring strikes an anvil and separates from the arm following the failure of a set of shear pins. The arrest of the clamping ring is followed by the deformation of the membrane under its own inertia. Strain gauges are used to measure the strain-time histories, but at higher initial velocities these gauges spall from the specimens. Deflection-thickness ratios of up to 45 were measured. Table 1 summarizes the plate test results of all the methods described above. COMPARISON
OF VARIOUS
DATA
In an attempt to compare experimental results with other experiments, which have different target dimensions, use is made of a dimensionless damage number defined by Johnson [15] as
(1)
= pv2/~rd,
which is used as a guide for assessing the behaviour of metals in impact situations. Here p is the material density, v the impact velocity and a d the damage stress. Table 2 shows the damage regime as a function of damage number, and also for comparison shows the results of some circular plate experiments. It can be seen that ~ predicts an order of magnitude deformation, and, in doing so, does not consider the method of impact, the interpretation of ad, the target geometry and boundary conditions or the target dimensions. Since we wish to compare results of deformed plates of similar geometries, boundary conditions and loading, it seems reasonable to introduce factors which would allow all other variables to be normalized into dimensionless groups. Johnson's damage number can be written in terms of the impulse as 12
102
% - Ao2t2pad
tZpa d'
(2)
where I is the total impulse, A o is that area of the plate over which the impulse is imparted, I o is the impulse per area and t is the plate thickness. Figure 1 shows a plot of deflection-thickness ratio versus So for circular plates of varying dimensions and material properties. This indicates the order of magnitude reliability of Johnson's damage number, but, as can be seen, the variations for similar loading conditions are a factor of up to 2.5. For dissimilar loading conditions, the variation has a factor of up to 8.5. In the case of quadrangular plates, we introduce a geometry number defined as L fl = - ,
(3)
B
where L are B are the plate length and breadth respectively. A geometrical damage number is then written as F
I A \2-1x/2
=ta ot ) j
,
(4,
where A is the area of the plate. A relationship between the distance from the plate centre to the nearest boundary and the plate thickness is introduced as the aspect ratio 2, where 2 = R / t for circular plates
(5a)
2 = B/2t for quadrangular plates.
(5b)
0.0316
1 x 10 - 3
31.622
10a
x
3.162
1 x 101 (10)
1
0.316
1 x10-1 (0.1)
(0.001)
0.0032
Gt½
1 x 10 - s
Damage No.
Hypervelocity impact
Extensive plastic deformation (ordinary bullet speeds)
Moderate plastic behaviour (slow bullet speeds)
Plastic behaviour starts
Quasi-static elastic
Regime
0.61 1.6 1.9 6.2 1.6 1.9
0.61 6.22
2.95 1.17 1.17
(nun)
t
TABLE 2. REGIME OF DAMAGE (FROM JOHNSON
1
1.36 5.7 3.1 138 14 7
0.28 35
0.2 0.1 0.15
98 31 32 16 31 32
98 16
25 64 64
t
R
Plate example
AND SOME RELATED EXAMPLES
(N s)
[15])
46 3.8 2.6 4.6 11.1 6.4
11.5 0.6
0.1 0.5 0.7
t
6
0.09 0.10 0.16 0.20 0.60 0.77
0.004 0.013
6.6 x 10- 6 1 x 10 - s 2.4 x 10 -5
[34] [28] [26] [18] [28] [26]
[34] [18]
[36] [36] [36]
Ref.
/
~
~.
e~
3" "~
~'
0
178
G . N . NURICK and J. B. MARTIN 12
-/ iO
¢,
o P
,/I
8
i~
•
30
o
I
4
/ .-
/
/-
3d
o..~ofl
¢-1 0
i 05
i I
I 15
I
(oolg
I 2
I
= z 0/tlpo.012
FIG. 1. Graph of deflection-thickness ratio vs Johnson's damage number for different plate geometries and loading conditions (the lines represent the least squares fit to the respective data). 1. Nurick et al. [28]; R = 50 ram, t = 1.6 ram, mild steel uniformly loaded. 2. Wierzbicki & Florence [18]; R = 50 mm, t = 6.3 mm, mild steel uniformly loaded. 3. Bodner & Symonds [26]; R = 32 ram, a, c, d - - t = 1.90mm; mild steel, I ~ - t = 2 . 3 m m ; titanium a, b---uniformly loaded; c--loaded Ro/ R = ½; d--loaded Ro/ R = I.
The loading area per total plate area must also be considered; this is accounted for by a loading parameter assumed to be ( = 1 + ln(R/Ro)
(6)
for circular plates only, where Ro is the radius of the loaded area. Equation (6) implies that when R0 ~ R, ( ~ 1 and the plate is uniformly loaded over the full area, and that when R0 is very small, the loading tends to a point-loaded situation and is more concentrated, resulting in a larger mid-point deflection. The effect of partial loading on quadrangular plates has not been included. Combining equations (4)-(6) results in a modified damage number that incorporates dimensions and loading: 4' = ~b2~.
(7)
4,c = I(1 + In R/Ro) rrRt2(ptro) 1/2
(8a)
For circular plates then,
and for quadrangular plates I 4,q -- Z
[DL
"1/2
(8b)
Table 3 summarizes all the dimensionless parameters, and Table 4 summarizes the experimental data. The geometry and material density variables in equation (8) are easily obtainable. The variables I and aa are measured experimentally--/from impulse tests, and, for convenience, ~d is given the value of ~r0, the static yield stress. The value of 0~o is determined from the results of quasi-static uniaxial tensile tests (see for example Nurick et al. [28], Bodner and Symonds [26]). Figures 2 and 3 show dimensionless plots of deflection-thickness ratio versus 4, for the experimental results on circular and quadrangular plates respectively. Also shown are dimensionless plots of theoretical predictions described in part I. Least squares correlation analysis of the circular plate experimental data yields a correlation coefficient of 0.973,
Deformation of impulsively loaded thin plates--II
179
TABLE 3. DIr,~NS[ONLESSPAI~A~m~I~S Johnson's damage No. (eq. 1)
c¢
pv 2
--
ad
12
or
A2t2pffd
Circular geometry Geometry No. (eq. 3)
fl
I/~ =
L
--
= ~/~
Geometrical damage No.
Quadrangular geometry
12
12
A2t2pff d
B3Lt2pff d
I
I
At(pad) 1/2
Bt(BLp¢o)l/2
(~/)1/2
(eq. 4) Aspect No. (eq. 5)
2
-t
R
B
Loading parameter (eq. 6)
~
1 + ln(R/Ro)
--
(eq. 8) ~b, = ¢~(
I
I
7tRt2(pff d)l/2
2 t 2 ( B L p f d)l/2
I(1 + l n ~o ) 7tRt2(ptrd) 1/2
Dimensionless numbers reported by others Guedes Soares [37]
412R 2
4,~2ct
po'oA2t't =
4#p2 [2L2
Jones [22, 23]
- 4,~2fl20[
paoA2t * = 4,8~b~
and these experimental results are almost all within one deflection-thickness ratio of the least squares line, as shown in Fig. 4. For the quadrangular plates (Fig. 5), the experimental results are also all mostly within one deflection-thickness ratio of the least squares correlation which in this case has a correlation coefficient of 0.980. All readings thus far reported were for deformations in which no tearing of the plate was observed. However, a few experiments were performed by Nurick 1-38] in which, in a small proportion (approximately one-tenth) of the boundary perimeter, tearing was observed. This partial tearing of the plate was observed at the clamped boundaries at impulses of 17.7 N s and 19.0 N s for the square and rectangular plates. This translates to ~b values of 26 and 28 respectively. For the circular plates, tearing occurred at an impulse of 15.6 N s, which translates to a ~b value of 26. Insufficient data on impulse parameters preclude an attempt being made to perform a similar detailed analysis for the other forms of impulsive loading, i.e. underwater blast pressure, air blast pressure and inertial frame loading. However, it is noted that Ghosh et al. [32] have used Johnson's damage number in their analysis of the comparison between their experimental results and same theoretical predictions. DISCUSSION
In the attempt to compare experimental results presented by different researchers using different plate dimensions and different plate materials, it became obvious that Johnson's
Nurick, 1986 [28-31]
128 x 32
Jones & Baeder, 1972 [23]
113 x 70 89 x 89
128 x 128
128 x 96
128 x 64
129 x 76
Jones, U r a n & Tekin, 1970 [22]
120
100
Nurick, 1986 [28-31] 40.3 162
282
Mild steel Mild steel
Mild steel Aluminium Mild steel Aluminium Mild steel Aluminium Mild steel Aluminium
Aluminium
Mild steel
282 296
251 279 251 279 251 279 251 279
248 233 254 284 281 286
NON-CIRCULAR PLATES
Lead Aluminium
Mild steel
255 223
540
Mild steel Titanium Mild steel
280
Aluminium
150
290 283
Aluminium Mild steel
64
Inertial formino machine Ghosh et al., 1976 [27] 1979 [30] 1984 [29]
Static yield stress (MPa)
CIRCULAR PLATES
Specimen type
100
Diameter
Dimensions (nun)
Bodner & Symonds, 1979 [26]
Florence & Werzbicki, 1966 [17] 1970 [18] Duffey & Key, 1967 [19] 1968 [20]
Sheet explosive blast
1.6 1.6
2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7
1.6 2.5 4.4 3.1 4.8 6.2
0.61 0.31
1.6
2.3 1.9
1.6 3.2 1.6
6.3 6.3
Plate thickness (mm)
TABLE 4. SUMMARY OF COMPARATIVE EXPERIMENTAL DATA
3-12 6-12
0.3-1.2 0.5-1.2 0.8-4 1.2-3 0.7-7.5 2.1)-4.5 1.6-9.5 3.0-6.0
3.5-7.0 1-4.5 0.3-1.7 1.8-3.5 0.8-2.4 02.-1.4
44 42
4-12
0.9-6 0.5-7
4-9 1.5-1.7 3-4
1.6-7 0.4-4
Deflection-thickness ratio
120-200 90-180
140-190
65 75 100-120
200-250
Response time (/zs)
Z
~r
¢3
Z z
Deformation of impulsively loaded thin plates--II 12
3181" la/'~bll 10'9/4 7 G/5~;4
a
/;/F'/;'
: ,'
i
iJ
IO'
.o
181
8
I I
f "i
il"
U"'//
., :11..
II/f/I
I
~ ~I <~1
I .'". / .
A,~lx I
.
I
~)
I
3o
.
I"
..i,
I
r'~ 4
/
I 2 3 4 5
I0
15
2O l
25
~c "I(l+tnR/Ro)/rRt2 (P~lg
FIG. 2. Graph of deflection-thickness ratio vs dimensionless No. ~bo. EXPERIMENTAL DATA
Material
Loading
Lx Nurick
A
D
50
1.6
• Bodner & Symonds []
B A
D D
•
B
E
O
A
E
32 32 22 32
2.3 1.9 2.3 1.9
A
D
C
D
50 50
6.3 6.3
•
Wierzbicki & Florence
•
Radius
Thickness
Notes
A--steel; B--titanium; C--aluminium; D--uniform loading; E--partial loading. Predictions (see part I for cited references)
1--Lippmann [33]; 2--Perrone & Bhadra [47]; 3--Noble & Oxley [31], a--circular profile, b--conical profile; 4--Calladine [52]; 5--Duffey [51]; 6-43uedes Soares [34]; 7--Westine & Baker [44]; 8--Wierzbicki & Florence [56], a--large deflections, b---small deflections; 9--Hudson [8]; lO--Batra & Dubey [32]; 1l--Ghosh & Weber [34]; 12--Wang & Hopkins [14]; 13--Wang [15]; 14--Jones [41].
damage number was a good guide, as shown in Fig. 1. However, this did not provide a suitable means for comparison, and hence an extension to Johnson's damage number was formulated. Figures 2-5 show plots of deflection-thickness ratio vs this new dimensionless number $. Also shown in Figs 2 and 3 are plots of predictions that do not consider strain rate effects, and therefore the predictions of Nurick et al. [31] and those of Symonds and Wierzbicki are omitted. It will be noted that there are similarities between Fig. 2 and Fig. 1 of part I and between Fig. 3 and Figs 2 and 3 of part I. However, the important issue here is illustrated in Figs 4 and 5, in which the experimental data results are bounded on either side of the least squares fit by a one deflection-
182
G . N . NURICK and J. B. MARTIN :
32
._0
/
8
.g: I
6
•
/
/
8 o
~ A
a 4
I, zx
t'• I 2 3 4 5
I0
15
¢q I/2tPXBLp~o)~
20
25
=
FIG. 3. Graph of deflection-thickness ratio vs dimensionless No. ~bq. EXPERIMENTALDATA Material
fl = L / B
Thickness
[] A
Nurick
A A
1.0 1.62
1.6 1.6
•
Jones & Baeder [23]
A
4.0 2.0 1.33
2.69 2.69 2.69
1.0
2.69
•
•
B
Jones, Uran & Tekin [22]
•
A B
1.69
1.63
2.49 4.39
Notes: A--steel; B--aluminium. All uniformly loaded. Predictions (see part I for cited references) 1--Baker [49]; 2--Jones et al. [50, 58], fl = 1.6; 3--Jones et al. [50, 58], fl = 1.0; 4--Jones [41, 59], fl = 1.0; 5--Jones [41], fl = 1.6; ~ - J o n e s [59], fl = 1.6.
thickness ratio confidence limit. It can be seen that in both cases most of the data points lie within these bounds. Figure 6 shows both the least squares fit and bounds of Figs 4 and 5, from which guidelines for a design code may be considered. The least squares analysis yields from the circular plate experimental data ( 6 ) = 0.425q~c + 0.277,
io
where the number of data points was 109, with a correlation coefficient of 0.974.
(9)
Deformation of impulsively loaded thin plates--II 12'
,/ I0 •
8"
/
/ /~/ / /
/// /
6"
,/ 4"
/
0
I 2 3 4 5
I0
15
20
25
$c PiG. 4. Graph of deflection-thickness ratio vs dimensionless No. @~, showing least squares correlation ( ) bounded by +_! deflection-thickness ratio ( - - - - - - ) .
/ /o
.o a m
I
~6
~ m2
i i 3 4 5
I 10
E
I
I
15
20
25
~q FIG. 5. Graph of deflection-thickness ratio vs dimensionless No. ~q, showing least squares correlation ( ) bounded by + 1 deflection thickness ratio ( - - - - - - ) .
183
184
G . N . NURICK and J. B. MARTIN
,2t
Circutapl r ates
/ .
I0
8
/~
~/
~O~uadrangut prat clres
6-
4-
2-
0
I0
34
20
40
50
~c,q FIG. 6. Graph of deflection-thickness ratio vs 4~4.
From the quadrangular plate experimental data, the analysis results are ( 6 ) =0.471~bq+O.O01,
Tq
(10)
using 156 data points, with a correlation coefficient of 0.984, and recalling that 1(1 + ln[R/Ro] ) ~P~= rcgt2(ptro)X/2 C~q --
I 2t2(BLptro )l/2 "
(1 la)
(1 lb)
Hence it may now be possible to determine the deflection-thickness ratio of a circular or quadrangular plate subjected to an impulsive load, by using equations (9) and (10). There appears to be some justification of equations (9) and (10). Consider two plates, one circular the other quadrangular, of equal area, thickness and material properties, subjected to an impulse over the entire plate area; then from equation (11) q~¢
4'q
2 -
-
1.128,
(12)
and substituting into equation (9), ( ~ ) = 0.480q~q + 0.277.
(13)
A comparison of equations (10) and (13) shows similar gradients.
CONCLUDING
REMARKS
An analysis of the experimental results provides a useful guideline for predicting maximum central deflection of impulsively loaded plates. An indication of when tearing might occur is presented, but more results are required for clarification. This is currently being investigated.
Deformation of impulsively loaded thin plates--II
185
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Solids 27, 91-113 (1979). 27. G. N. NURICK,A new technique to measure the deflection-time history of a material subjected to high strain rates. Int. 3. Impact Engng 3, 17-26 (1985). 28. G. N. NURICK, H. T. PEARCEand J. B. MARTIN, The deformation of thin plates subjected to impulsive loading. In Inelastic Behaciour of Plates and Shells (Edited by L. BEVILACQUA).Springer-Verlag, Berlin (1986). 29. G. N. NuRICg, The measurement of the deformation response of a structure subjected to an explosive load using a light interference technique. Prec. 1986 SEM Spring Conf. Exp. Mech., pp. 105-114. The Society for Experimental Mechanics (1986). 30. G. N. NURICK, Using photo-voltaic diodes to measure the deformation response of a structure subjected to an explosive load. Prec. 17th Int. Cong. High Speed Photography and Photonics, pp. 215-226. International Society for Optical Engineering (1986). 31. G. N. NURICK, H. T. PEARCEand J. B. MARTIN, Predictions of transverse deflections and in-plane strains in impulsively loaded thin plates. Int. J. Mech. Sci. 29, 435-442 (1987). 32. S. K. GHOSH and H. WEBER, Experimental-theoretical correlation of impulsively loaded axisymmetric rigid-plastic membrane. Mech. Res. Comm. 3, 423-428 (1976). 33. S. K. GHOSH, R. BALENDRAand F. W. TRAVlS,Inertial forming of circular and annular diaphragms. Adv. Mach. Tool Des. Res. 845-854 (1978). 34. S. K. GHOSH, S. R. REID and W. JOHNSON, The large deflection impulsive loading of clamped circular membranes. Structural Impact and Crashworthiness Conf., pp. 471-481. London (1984).
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G . N . NURICK and J. B. MARTIN
35. S. K. GHOSH and F. W. TRAVIS, A unique experimental high rate forming machine. Int. J. Mech. Engng Educ. 7, 69-74 (1979) 36. G. N. NURICK, R. S. HAMILTONand D. PENN1NGTON, A simple method to predict the initial and subsequent deflections of a clamped circular plate subjected to repeated impacts. Proc. Soc. Exp. Mech. Spring Conf. Exp. Mech., New Orleans (1986). 37. C, GUEDES SOARES,A mode solution for the finite deflections of a circular plate loaded impulsively. Rozprawy lnzynierskie Engng Trans. 29, 99-114 (1987). 38. G. N. NURICK, Large deformations of thin plates subjected to impulsive loading. Ph.D. Thesis, University of Cape Town (1987). 39. N. JONES, Structural Impact. Cambridge University Press (in press).