Deformation of metallic components by explosive loads

Journal of OccupationaZ Accidents, 2 (1979) 99-112 0 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

DEFORMATION

OF METALLIC COMPONENTS BY EXPLOSIVE

99

LOADS

M.A. NETTLETON Central Electricity Research Laboratories, (U.K.)

Kelvin Avenue, Leatherhead,

Surrey, KT22

7SE

(Received June 7th, 1978)

ABSTRACT Nettleton, M.A., 1979. Deformation of metallic components by explosive loads. Journal of Occupational Accidents, 2: 99-112. It is not always realised that the relationship between deformation of a ductile material and load depends on the rate of loading. The present note compares the behaviour of discs of aluminium, copper and mild steel, when loaded by a slowly applied hydraulic load, with that obtained with rapidly applied loads of duration greater than the response time of the disc. The latter were achieved by allowing a shock in air to reflect from the disc. The results indicate that the yield stress of the metal is influenced by the rate of loading and by its history and nature. The implications to the assessment of intensity of load, by measuring deformations produced in the positive phase of a blast-wave, are discussed.

1. INTRODUCTION

During the early 1940’s it was widely realised that deformation of circumferentially-supported metallic discs by the loads produced by blast-waves in air and water was neither spherical nor was describable in terms of the yield stress determined under conventional conditions. Most of the references to this work are in classified documents and are extremely difficult to trace at the present time. However, Cole (1948) gives references suggesting that deformation produced by blast-waves is generally conical in nature and does not conform with that predicted from conventionally determined yield stresses. However, none of the references still available indicate what values to use for yield stress. Manjoine (1944) suggests that the yield stress of mild steel increases by a factor of 1.6 and Priest and May (1969) by a factor of 1.3 for strain rates of about 15 s-l. The importance of these observations lies in the fact that estimates of the intensities of explosions are frequently based on the deformation of metallic components produced by the dynamic explosive forces. In order to relate these dynamic loads with the design of structures which are based on static loads, it is necessary to know the relationship between the deformations produced by dynamic and static loads of the same magnitude. Conventional theories of elastic and elastic-plastic deformation predict that

100

the ratio of deformation produced by a dynamic load to that produced by a static load is a function of the duration of the dynamic load and the vibrational frequency of the deforming structure. For the dynamic loads which persist for much longer than the time constant of the structure, theory predicts that the ratio should be two. Furthermore, the value of the ratio should not be influenced by the properties of the material, provided that it behaves in a ductile fashion and that its yield stress is independent of the rate of loading. The present series of experiments has been carried out to measure the deformation of aluminium, copper and mild steel discs over a range of static and dynamic loads. Aluminium was chosen to represent metals which behave in an ideal ductile fashion, copper to represent those showing pronounced work-hardening effects and mild steel as the most widely used material. The dynamic loads were produced by reflecting a shock from the surface of the disc which formed the end plate of an 80 mm diameter shock tube. BEXPERIMENTAL

In the initial experiments, discs were cut from single sheets of material, avoiding material from the edges. This procedure was found to produce satisfactory results for aluminium. However, there was considerable scatter in the deformations of copper and steel discs under both static and dynamic loads. In an attempt to improve upon this scatter, Vickers hardness tests were carried out on individual discs and experiments were performed only with discs of similar hardness numbers. This produced little improvement in the scatter, so a further series of experiments was carried out with mild steel discs annealed at 883 K for two hours and allowed to cool slowly to room temperature and with copper discs annealed at 773 K in an atmosphere of argon for one hour and then allowed to cool in air. The annealing process reduced the spread of hardness numbers for mild steel discs from 100 f 5 to 85 t 1 and for copper discs, although not reducing the spread of results, dramatically decreased the hardness number from 73 + 3 to 31 f 1. The static pressure tests were carried out using water as the pressurising medium in a test rig constructed to test bursting pressures of materials intended for use as diaphragms in the 80 mm shock tube. Thus, similar clamping arrangements were used in the static and dynamic tests and the area of the disc pressurised was the same in both rigs. The pressure gauges were calibrated on a dead weight tester before their use with each of the metals tested. The rig was slowly pressurised (ca. 5 mins) to the required load and the disc allowed to set for five minutes before removing the load and measuring the central deflection of the disc with a depth micrometer. In a number of experiments both static and dynamic, deformations were measured at 5 mm intervals across the diameter of the disc in order to check whether the disc was deforming spherically. The dynamic tests were carried out in a 80 mm shock-tube using either helium or air as the high-pressure gas and air or argon as the test gas. The

101

combination of helium and argon was used to produce the highest shock pressures and air the lowest. The metal discs were clamped firmly between O-rings at the flanges between the end of the shock-tube and the start of the window section. The window section served to protect the operator in the event of the disc rupturing. Since the volume of the window section is about low3 m, the pressure in this section did not rise appreciably above atmospheric when the diaphra~ deformed. Shock pressures were recorded photo~aphic~ly on a Tektronix 556 oscilloscope, using Kistler 603A pressure transducers and 5001 amphfiers. Shock velocities were determined from the time of passage of the front as it approached the disc using platinum film gauges spaced at 607 mm intervals. Incident and reflected shock pressures were determined from the velocities, appropriately corrected for any deceleration before the shock reached the disc, from the standard Rankine-Hugo~liot relationships. The later method gave the most accurate value of the load on the disc (f 1%). 3. RESULTS

3.1. General Table 1 gives the properties of the metals used to derive the frequency of response of the disc together with its time constant. Typically the duration of the load was 2 to 10 ms (see Fig. 4), so that the condition that the duration of the load must be much greater than the time constant of the disc is amply fulfilled. TABLE 1 Time constants of the discs used _____-. -Disc radius, a (m) Disc thickness, h (m ) Disc density, p (kg m-‘) Young’s Modulus, E (N m-*) Poission’s ratio, v Time constant, t (jds)

3.2. Aluminium

Aluminium

Copper

Mild steel

4 x 10-2 7.1 x lo-’ 2.70 x 10) 7.05 x 10”’ 0.345 89 -

4 x lo-” 9.6 x lo-’ 8.96 x lo3 12.98 x 1O’O 0.343 87

4 x 10-z 5.6 x lo-* 7.90 x 103 21.00 x 10’0 0.293 114

discs

Figure la shows stress-strain curves obtained in a conventional test of samples of aluminium cut from the sheet which was used to prepare the test discs, The properties of the ~uminium sheet evidently vary with the direction of rolling. However, averaging the results from the two samples indicates gross yielding at 90 MN m-‘, with some yielding occurring at about 75 MN m-‘. The results of the static tests on the discs are shown in Fig. 2. There is a

f_!c l(aL STRESS-STRAIN PLOT_F~f?_A?MlNlUM SHEET

well-established linear relationship between extension and load for loads of up to 1.4 MN me2 and this extrapolates back through the origin. The line drawn through the experimental points comes from the relationship between load Ap, radius of the disc a, thickness of the disc h, yield stress up and deformation d, with a2 >> d2 suggested by Cole (1948). d = a2Ap/40ph The value of up was obtained from a plot of Ap versus 4hd/(d2 derived from eqn. (1) without the restriction a2 >> cl2 aP = Ap r/2h

(1) + a’)

(2)

where r is the radius of curvature of the deformed disc. This value is 99.1 MN m-‘, in reasonable agreement with the value for gross yielding from the conventional test. The use of eqns. (1) and (2) is fully justifiable, only if the disc deforms spherically. Figure 3 shows the measured deformations along a chord for discs

103

11.0

0

IO 0

9.0 ~

0

0

STATIC

X

DYNAMIC

LOAD LOAD

a = 397mm h= 07 ” ep= 99 I MN I,-*

02

FIG.Z

04

DEFORMATION

06

OFALUMINIUM

oiscs B~STATICANO

DYNAMIC LOADS

STATIC

LOAD

DYNAMIC

8

0

5

10

15

20 DISTANCE

FIG. 3

TEST

OF SPljERlCAL

ALONG

7.5 CHORD.

DEFORMAT{ON

30

35

mm

FOR ALUMINIUM

DISCS

40

LOAD

104

loaded statically and dynamically to the maximum levels tested. The line represents the equation of circle determined from the first two points. Thus, even under the most severe test conditions, the disc deforms spherically when loaded statically. However, Fig. 3 demonstrates that dynamic loads producing similar deflections to higher static loads result in the centre of the disc deforming beyond the bounds of spherical distension. The degree of excess deformation at the centre increases with increasing deflection. Deformations under dynamic loads are given in Fig. 2 also, together with a full line representing twice the deformation occurring under static loads. Although there is some scatter in the results, it is evident that the dynamically loaded discs distend more than twice those statically loaded. The best straight line through the results is d = 15.32

Ap - 0.4

(3) PRESSURE TRANSDUCER

R FlLM GAUGE

CONTACT

SURFACE

%REFLECTED

SHOCK

SHOCK

I DISTANCE

-

I 0

I

/

I2

I

34

TIME HELIUM

5

ms DRIVING

AIR

TIME. mr

FIG. 4

TYPICAL

LOAD HISTORIJS

FOR AIR DRIVING

AIR

WITH A SKETCH

OF WAVE

INTERACTIONS

105

The ratio of dynamic to static deformation is a function of Ap and (with Ap = 0.6 MN m-*) h = 2.6, as opposed to the theoretical value. Figure 4 shows typical load profiles for these tests. For air driving shocks into the air, the interaction of the reflected shock with the contact surface, separating the expanded driver gas from the compressed test gas, results in a series of weak shocks which slowly raise the pressure at the end of the reflectedshock plateau. The effect is most marked with the strongest shocks, occurring earlier and resulting in a higher intensity relative to the reflected shock. With helium driving shocks into air, the interaction results in an expansion wave which merges with the one reflected from the high-pressure end of the tube, producing a rapid fall in pressure. The results in Fig. 2 are for reflected shock pressures. Similar tests using the peak pressure levels produced much greater scatter about the best straight line.

I 031

I 0 62

I 0 93 STRAIN

FIG.lfb)

I

I 14

(ARBITRARY

STRESS-STRAIN PLOTFOR

I

I 55

5)

COPPERSHEET

PARALLEL PERPENDICULAR

TO ROLLINGopc254 TO

MN m

ROLLING

q-248

’ MN 11, ~’

PARALLEL

AND

ROLLING

PERPENDICULAR op

3.3. Copper

TO

146 7 MN cr

TO

ROLLlNG

2

discs

Figure lb shows the results of a conventional measurement of the stress-strain relationship for samples of copper sheet in the “received” and annealed conditions. The yield stresses obtained from an extrapolation of the initial and final slopes are 160 MN m-’ for “as-received” and 110 MN me2 for the annealed material. These values are in reasonable agreement with the results obtained from the gradient of deformation versus static load curves illustrated in Fig. 5 and 6, viz. 152.4 and 134.4 MN m-* for “as-received” and annealed discs. Figure 5 shows the results for the deformation of “as-received” copper discs by dynamic loads. There is considerable scatter at the largest deformation. However, the results are tolerably well represented by the line d = 5.67 Ap + 0.6

The theoretical

line representing

d = 5.74 Ap - 0.4

(4) twice the static deformation (5)

107 a = 39.1 mm t = OPSmm ‘Jp= 1524MNn1-~ VICKERS HARDNESS 0 STATIC X CYNAWC

NO.

= 73 0:

3

LOAD LOAD

/” =5b’Ap ’ O6

x

_O

/ /

/ /’ / /

‘/,

02

FIG.5

d = 272~~

-

02

0’

h O/

/

o/o 0

06

I Q

08

DEFORMATIONOF'ASRECEIVED' ANDDYNAMIC ___

/

1

I

04

14

12

c

16

COPPERDISCSBYSTATIC

LOADS

a = 39 7 “un t = 0 95 ml” s,, = 134 3 MNm-’

X

d=

103EA,p

+ 25

0<3:5,+ 3

01

0-I

0

/

0’

0’ 0

STATIC

X

DYNAMIC

VICKERS

LOAD LOAD

HARDNESS

31 : I

108

is included in Fig. 5. It is interesting to note that the theory predicts the gradient of the plot but not the magnitude of the deformation. With a load of 0.6 MN m-*, the theoretical approach predicts a deformation of 3 mm and the experimental value is 4 mm. As with aluminium discs which were dynamically stressed, “as-received” copper discs similarly stressed deformed non-spherically. Deformation beyond a spherical shape occurred at the centre of the disc and the degree of non-sphericity increased with increasing deformation. Figure 6 gives the static and dynamic deformations of annealed copper discs. Again there is considerable scatter in the results for the largest deformations under dynamic loads. The best straight line through the data for static loads is d = 3.15

Ap + 3.0

(61

The best straight line for the deformation d = 10.38

by dynamic

load is

Ap + 2.5

(71

with Ap = 0.6 MN m-*, h = 1.8. Deformations of the annealed copper discs of less than 5 mm were approximately spherical. As the deformation increased, the shape became increasingly less spherical. With deformations of greater than 10 mm, lobes appeared in the disc. With deformations of 15 mm and greater the deflection of the lobes was as large or larger than the central deflection. 3.4. Mild steel discs Figures 7 and 8 illustrate

FIG.7

the influence

DEFORMATION ____-OF ‘AS RECEIVED’

MILD STEEL

of static loads on the deformation

DISCS BY STATIC AND DYNAMIC LOADS

of

109

FIG. 8

DEFORMATIONOFANNEALEDMILDSTEELDISCS

EYSTATICANDDYNAMIC

LOA@

mild steel discs in the “as-received” and annealed conditions respectively. The gradient of the linear portion of the plot for “as-received” steel leads to a yield stress of 210 MN m-’ compared with 250 MN m-’ determined by a conventional test and that for annealed 202 MN mm2compared with 147 MN mm2for a conventional test. Although Figs. 7 and 8 indicated that there is only a small change in the yield stress, the Vickers Hardness No. was reduced from 100 f 5 to 85 f 1 by annealing. The results for the dynamic deformation of “as-received” mild steel are represented by the best straight line (Fig. 6). d = 5.53 Ap - 1.7

(3)

and for annealed mild steel (Fig. 7) d = 4.46 - 0.7

(9)

A higher load must be chosen to compare dynamic and static deformations of mild steel in the region in which deformation is linearly related to load. At 1.2 MN m-‘, X = 1.22 for “as-received” mild steel and for annealed mild steel h = 1.13. As with aluminium and copper discs, there were indications of nonsphericity in the largest deformations produced dynamically. 4. DISCUSSION

The Cole (1948) model of large-deflection perfectly plastic analysis can be derived as follows. Consider a rigidly clamped plate in a uniformly stressed state

110

of the magnitude of the yield stress. This level is assumed to be independent of the strains in the disc, which is deformed with a radius of curvature r. The load gives a normal force acting on the circular element of the disc of Ap(rd8 )“. By equating the vertical component of this force with the normal applied force, the radius of curvature is given by

r = 2ophlAp

(2)

Since the same arguments apply to every point on the plate, the radius of curvature is constant. Thus the central deflection of the plate is given by

d = a2Ap/4aph

(1)

Thus the model predicts that strain and stress are linearly related. A simple model for comparing the effects of dynamic and static stressing of a disc can be based on the response of a mass m at the end of a spring stiffness k fixed at the opposite end to the mass. The restoring force on the mass F displaced by a distance u is given by

F=ku

(10)

With the mass subjected

to a force Ap the equation

of motion

m d2u -+ku=Ap dt2

is

(11)

With applied force o
AP = APO ; and Ap = 0;

t < 0, tl < t

eqn. (10) has the solution F=ku=Acoswt+Bsinwt+Ap

(12)

where w = (k/m)% and A and B are integration constants. With the boundary conditions u = li = 0; t = 0 and u and D continuous at t = t,

F = Apo(l

- coswt);

o
(13)

and

F = Ape [sinwt,

sinwt -

Thus the maximum

F,

(1 - cosw It)cosw t] ;

force F,

= 2Ap, ; = Ape [2(1=Apo

wt,;

cosw tl)] %;

t, < t

is given by t1>

n/w

(14)

t1<

71/w

(15)

t1-C < It/w

W-3)

111

i.e. the maximum force for a pulse of long duration is twice that for a slowly applied force of the same intensity. The model on which the analysis of the results is based involves the assumption that the yield stress is uniformly achieved throughout the area of the disc up to the rigidly clamped boundary, Thus the disc deforms as a section of a sphere. The assumption apparently holds for an intermediate range of deflections. For small detections, i.e. deflections not much greater than the thickness of the disc, bending stresses become important and the experimental deflections fall below values predicted from the linear portion of the strain-stress curves. For the largest deflections obtained by dynamic stressing, ~though not for static stressing, excess deformation occurs at the centre of the disc. However, the scatter in the experimental results apparently conceals any systematic departure from a linear strain-stress relationship It is difficult to account for the pronounced scatter in the results for dynamic deflections. The results for static deflections of similar magnitudes, with the exception of those for annealed mild steel, show very little scatter. Thus, it is tempting to speculate that the effect is related to the dynamic stress wave propagated through the disc. However, the time for the incident wave to propagate through the disc and reflect back from the metal-air interface (t - 2h/c) is only a fraction of a microsecond, of the same order as the rise time of the load. Movement of the disc in this interval must be infinitesim~ (Time const~ts - 100 ,us). However, it is possible to visualise the mi~ation of defects within the metal during the passage of the stress wave. Thus, Leiber (1975) has shown that carbon granules in cast-iron and minute bubbles in cast explosives are moved by strong shocks transmitted through the material. Furthermore, he suggests that the interaction of the wave with these inhomogeneities affects the strength of the tr~smitt~ shock. Such an effect might account for the scatter in the results. The annealing procedures used are unlikely to produce a uniform distribution of inhomogeneities within the disc, and the hardness tests employed would not detect them. The present experiments were not designed to investigate the reasons for the departure from the predicted ratio of dynamic to static deformation. However, the results do indicate that the ratio is not only a function of the magnitude of load (or degree of defo~ation), but also of the material deformed. A tenable explanation would be that the yield stress depends on the rate of loading and the dependency varies with different metals and with the physical state of a particular metal. Finally, it may be of some interest to compare the results over which the strain-stress relationship is linear with the stresses at which failure of the discs might be expected to occur. Mild steel discs were statically loaded up to 2.76 MN mm2resulting in deflections of 8.5 mm, annealed copper discs similarly loaded deformed by 10.5 mm and “as-received” copper discs loaded to 2.96 me2 deformed by 7.6 mm. The high values of load required to produce these deflections indicated that the hydraulic testing facility would not be capable of producing a failure of the disc. However, lower loads were required to produce

112

deformation of aluminium and a hydraulic test to destruction produced a deflection of 25 mm at 3.40 MN m-‘. Thus is would appear that loads of two to three times those quoted for copper and mild steel would be necessary for failure of the disc. 5. CONCLUSIONS

The present experiments show that the presently-accepted theory for plastic deformation of materials is unsatisfactory. Thus, the theory for step loads of longer duration than the time constant of the structure predicts that the ratio of dynamic to static deformations should be two and should be independent of the magnitude of the deformations and the properties of the deforming material. Experimentally, the ratio is a function of the degree of deformation and the properties of the material deformed. It is concluded that the discrepancy between experiment and theory may be accounted for, if the yield stress depends on the rate of loading and the dependency is different for different materials. The ratios of dynamic to static deformation are for aluminium h = (15.32

Ap - 0.4)/5.68

for “as-received” h = (5.67

X = (10.38

Ap + 2.5)/(3.15

Ap + 3.0)

mild steel

Ap - 1.7)/3.36

for annealed X = (4.46

Ap - 0.2)

copper

for “as-received” X = (5.53

copper

Ap + 0.6)/(2.72

for annealed

Ap

Ap

mild steel

Ap - 0.7)/(3.49

Ap - 0.1)

ACKNOWLEDGEMENTS

The author is particularly grateful to A. Hemmings for his assistance with the experimental work which was carried out at the Central Electricity Research Laboratories and is published by permission of the Central Electricity Generating Board.

REFERENCES Cole, R.H., 1948. Underwater Explosions. Princeton University Press, Princeton, N.J., p. 409. Leiber, C.O., 1975. Alterations of the Hugoniots by bubble flow. J. App. Phys., 46: 2306. Manjoine, M.J., 1944. Influence of rate of strain and temperature on yield stresses of mild steel. J. App. Mech., A211. Priest, A.H. and May, M.J., 1969. Strain rate transitions in structural steels, B.I.S.R.A. Industry Rept. MG/C/95/69.