Inertia and strain-rate effects in a simple plate-structure under impact loading

Int. J. Impact En~ln~I Vol. 11, No. 3, pp. 349 377, 1991

0734 743X/91 $3.00+0.00 !', 1991 Pergamon Press plc

Printed in Great Britain

INERTIA A N D STRAIN-RATE EFFECTS IN A SIMPLE PLATE-STRUCTURE U N D E R IMPACT LOADING L. L. TAM* and C. R. CALLADINE Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, U.K. (Received 30 November 1990; and in revised Jorm 5 dune 19913

Summary Previous studies have shown that the way in which metal structures absorb energy by gross distortion under impact conditions depends on the generic type of structure. In particular they have shown that structures which respond to quasi-static testing by means of an initial peak load followed by a falling load as deformation proceeds ('type If' response, corresponding broadly to plates loaded endwise) exhibit both inertia and strain-rate effects under impact loading from moving strikers. This paper describes a detailed study of these phenomena by means of experiment and theory. Experiments were conducted in a drop-hammer rig on a large number of specimens having the same general geometry, but made in two different sizes and of two different materials (mild steel, aluminium alloy) chosen for their different strain-rate characteristics in the plastic range. The experiments involved the overall measurement of final distortion of the specimens in relation to a wide range of testing conditions with moderate velocity; strain gauge studies, high-speed photography and other investigations of the detailed behaviour. The main emphasis of the various assays was to discover the way in which the initial kinetic energy of the striker was dissipated within the structure. During the course of the work, Zhang and Yu proposed a simple analysis of the same phenomena by means of a model based on the ideas of classical inelastic impact theory. According to their theory, a significant fraction of the incident kinetic energy of the striker is absorbed during the initial impact event; and this fraction depends only on the ratio of the mass of the striker to the mass of the specimen and the initial crookedness, but not on the velocity of impact. Our experiments agreed with this analysis in some overall respects, but were irreconcilable with it in several others, for which we had amassed substantial data. We therefore produced a revised analysis, which was less austere than that of Zhang and Yu but which nevertheless remained essentially simple. We show in the paper that this new theory agrees satisfactorily with all aspects of the experimental observations. The analysis reveals clearly the roles of inertia and strain rate in impact conditions. It also produces two new dimensionless groups, which together provide a key to classification of the various patterns of behaviour which are possible in the impact response of 'type IF specimens.

NOTATION

a b h ! m q t u v w y A D, d G KE

Lz. 3 M Q R SE

T~ T2 V

characteristic arc-length of hinge (Fig. 7) width of specimen thickness of plate (Fig. 2) length of specimen (Fig. 2) mass of specimen (Fig. 133 shortening of contour length of rod (Fig. 14t time overall shortening of specimen dimensionless velocity [Eq. (32)] transverse displacement (Fig. 14) reduction in vertical height of specimen (Fig. 14) combined cross-sectional area of the two plates of a specimen characteristic deformation (Section I ) mass of striker kinetic energy of striker immediately before impact dimensions of specimen IFig. 2) 'equivalent' mass of specimen [Fig. 13; Eq. (9)] parameter defined in Eq. (40) energy ratio [Eq. (6)] energy required in quasi-static test to produce observed deformation of specimen energy absorbed during phase 1 kinetic energy remaining at end of phase 1, and available for phase 2 velocity of striker

* Present address: Lubricants Department, Shell Developments (HK) Ltd, Shell House, Hong Kong. 349

751/ I* I|. If'

; tl J,* ,, y (7 I k A, 2 Z. (~

1 I . IAM and (

R ( xll~l~[',,J

,.clc,ciL,. ol upper end of specimen if chc hars arc assumed incxtcnsiomd ellc[gy deli,..cred (Section

II

parameler dclined in fiq. (26) inclination of plate in specimen ( F i g 14i peak curvature in hinge Wig, 7) mass ratio [Eq. 17i] dimensionless group [Eq. 134i] spccilic mass yield slress dimensionless time [Eq. {31}J dimcnsionless group [Eq. (37l] areas on Fig. 15 characteristic length {Section 1 } yield stress ISection 1 }

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

The ability of ductile metals to absorb a large amount of energy in the course of plastic straining makes these materials inherently suitable for use in vehicle and other structures which are subject to collision and impact [1 3]. An early example of the design of an energy-absorbing structure was the indoor 'Morrison' air-raid shelter of World War II. Baker [4] designed the shelter as a box or portal-frame which would absorb the energy of a collapsing house by the rotation of plastic hinges in the structure. The design calculation was extremely simple, since the mode of collapse of the frame was known and the potential energy of a collapsing house could be estimated: the energy was to be absorbed by a known number of hinges [5] rotating through known angles (which would leave room for the occupants not to be crushed); and so the required full-plastic moment of the steel beams was found immediately. The calculation was, in effect, a quasi-static one; and it was entirely satisfactory in view of the relatively low velocities involved. There are, of course, many situations in the design of collapsing structures in which a 'quasi-static' analysis is plainly not a valid basis for design, at least without some justification. Thus it is well known (e,g. [6,7]) that in the plastic straining of mild steel, is sensitive to the strain-rate--in contrast to various other ductile metals which exhibil much less 'strain-rate sensitivity'. Furthermore, inertia effects are also involved, prima Jacie. in structures whose components are subjected to rapid acceleration on account of a collision. In some situations it is desirable to assay a proposed design for an energy-absorbing structure by subjecting a sample structure to a specific test impact. And, of course, it would generally be convenient to be able to perform such tests on small-scale models of thc prototype structure, rather than on samples of the prototype itself (e.g. [8]). Now if the process of deformation were indeed 'quasi-static', the extrapolation of small-scale test results to predict the behaviour of the prototype would be very straightforward: "scaling' of the test results would present no problems. Thus, specifically, let us suppose that the structure takes a particular geometric form, which has a characteristic linear dimension A: that the structure is made of material having yield stress Z, and that it is required to deform by a linear distance D under a delivery of energy W. For some purposes it will be convenient also to denote the corresponding quantities for a geometrically similar small-scale model by the lower-case symbols 2, o-, d and w. Further, suppose that the geometric scale factor between model and prototype, i.e. )~/A, has been decided upon. The problem is then how to fix the value of w for the model so that the resulting deformation of the model will be geometrically similar to that of the prototype, i.e. diD = )./A. The answer to this question is revealed quickly by the technique of dimensional analysis. Thus, each of the four primary quantities A, Z, D, W which define the problem has specific dimensions: here the dimensions are [-L], [FL- 2], [L], [FL], respectively, where [L] and [F] stand for the dimensions of length and force, respectively. There are only two independent dimensionless groups for this particular problem, (by Buckingham's rule [9]: number of groups = number of variables - number of dimensions = 2, here) and these may

A simple plate-structure under impact loading

351

be written D/A,

W/~.,A 3 .

(1)

Thus (D/A) depends only on (W/32A3); and hence if we require (D/A) for the prototype and model to have the same value, then (W/Y~A3) must also have the same value for both prototype and model. Therefore, by arranging for W/G). 3 ~- W / Y ~ A 3

(2)

d/;. = D/A.

(3)

we ensure that

Equation (2) is thus the relation which is required for fixing the value of w. What is important for present purposes is that the above calculation has been made under the implicit assumption that no more variables, in addition to those listed above, are relevant to the problem. There are, of course, several other variables which could be relevant, including velocity of impact and specific mass of the material; but under the starting hypothesis that the structure behaves in a quasi-static manner these are neither necessary nor indeed appropriate. Now suppose that it turns out, when the prototype and model structures are tested experimentally under the constraint (2), that the values of D / A and d/2 are not equal. In this case we would be driven to the conclusion that some phenomenon (or phenomena) which we have disregarded by implication in our analysis--such as strain-rate and inertia effects--is (or are) indeed significant in the circumstances of the test. And in order to make further progress in such a case it would be necessary to postulate these other phenomena and associated variables; to construct more dimensionless groups; and to devise experiments in order to ascertain the significance of the various dimensionless groups. Now testing is particularly straightforward when there are only two independent dimensionless groups, as in the example above, since one of them must be a function of the other. But when there are three or more dimensionless groups, we have to deal instead with functions of two or more variables, which obviously requires much more extensive experimentation. The hypothetical situation described above applies exactly to a programme of testing which was undertaken by Booth et al. [10] in order to investigate the possibility of using small-scale dynamic tests together with a simple analytical formula in order to predict the behaviour of some full-scale prototype steel vehicle structures under dynamic conditions. They performed experiments at several values of the geometric scale factor 2/A, and they found that the satisfaction of relation (2) did not produce the desired similarity of distortion, i.e. d/). 4= O/A.

(4)

Booth et al. had been aware that the constraint of equal velocity of impact, which was convenient for them to use in all of their tests, would result in higher strain-rates and thus higher material yield stress in their smaller specimens; but their attempts to account for this effect by regarding the yield stresses Z and a as the estimated dynamic rather than the static values were only partially successful in producing the desired result (3). The proper conclusion to be drawn from all of this work was that both strain-rate and inertia effects were significant in the problem under investigation. In an attempt to clarify the situation which Booth et al. had uncovered, Calladine [11] proposed a distinction between two generic types of plastically deforming structures in energy-absorbing situations. The two types are distinguished by the shape of their quasi-static plots of load against corresponding deflection, as shown in Fig. 1. In type I structures the load-deflection curve is flat-topped in the plastic range; just as it is, for example, in beams and portal frames loaded transvesely, and in arches and rings loaded radially. In type II structures the load-deflection curve has an initial peak followed by a rapidly descending portion; just as it does in columns loaded axially, and in buckling problems generally. Like many practically important distinctions, this particular one is

352

I

1.. T,~M itlltt (

R (',tl I A I ) I N I

¢

type ! type II

0 i

~.-

deflection

FI(;. 1. Schematic plot of load against corresponding deflection for type I and type II structures: see text for explanation.

not watertight, since the terms 'flat topped' and 'peak' are difficult to define unambiguously; for example, 'flat-topped' curves include those that rise or fall gently. Nevertheless, it is usually clear in practice into which of the two general types a given structure falls. The thinking behind this distinction between two types of structure is straightforward. In type I structures the absorbed energy increases linearly with deflection because the rotation of the plastic hinges is directly proportional to deflection, more or less. But in type II structures the shape of the curve indicates that a disproportionately large amount of energy is absorbed in the first small increment of displacement; which is a direct consequence of the geometric effect whereby the endwise shortening of an initially straight rigid rod containing a central hinge is proportional to the square of the angle of rotation of the hinge. The significance of this in dynamic conditions is that if an initial velocity is imposed on a type II structure--as it will be under impact conditions--then a disproportionately large initial rotation-rate is demanded of the plastic hinges. This in turn implies not only that there will be high initial strain-rates but also that there will be high transverse accelerations, and hence significant inertia loading. Thus the emergence of a 'type II' load-deflection curve in a quasi-static test acts as an early warning that both strain-rate and inertia effects will in principle be significant under dynamic test conditions. These ideas are, of course, non-specific in a variety of ways. Thus, they are not quantitiative in relation to the range of impact velocity, the overall size of structure, and the possibility that a type II structure may respond to an impact in part by undergoing some axial shortening. Nor, indeed, do they refer to the possibility that w a v e effects may be important. Nevertheless, they do point clearly to the general conclusion that a structure which in the quasi-static regime responds to load in the manner of a type II structure will be more strongly disposed to strain-rate and inertia effects under dynamic conditions than one which responds as a type I structure in the quasi-static regime. A preliminary experimental investigation of these matters was conducted by Calladine and English [-12]. In order to produce a manageable programme of testing for an undergraduate project they tested only two types of specimen under a single specific input of kinetic energy in a simple drop-hammer rig. Their tube and plate specimens (types l and It, respectively) were made from mild steel of approximately the same thickness, and matters were so arranged that under quasi-static tests up to the standard energy input, the two kinds of specimen shortened permanently by almost equal amounts. The various drop-hammer tests were performed with different values of mass and impact velocity; but it was so arranged that the kinetic energy of the striker just before impact was always the same.

The main experimental finding of that study was that the overall shortening of type I structures in dynamic conditions with impact velocity in the range 2-7 m s - l was only about 75% of the shortening which occurred under quasi-static conditions; whereas for

A simple plate-structure under impact loading

353

the type II structures the dynamic shortening was between about 30 and 10% of the corresponding quasi-static value. It was clear, therefore, that the type II specimens were considerably more sensitive to dynamic loading conditions than the type I specimens. The work of Calladine and English thus demonstrated an important point. But it was not entirely conclusive for a number of reasons, which may be listed as follows. (1) Although the experimental trends, as described above, were clear, there was a significant amount of scatter in the overall permanent shortening of identical specimens tested under nominally identical conditions. (2) The testing conditions involved only one independent variable. Thus, although both mass and velocity were altered, the product (mass) × (velocity) 2 was held constant. (3) There was no variation in the size of the specimens. (4) There was no variation in the material of the specimens. (5) Only a most rudimentary theoretical explanation was provided for the behaviour of type lI specimens. The programme of work to be described in the present paper was undertaken in order to remove the various limitations which applied to the previous study. The main aim of the project was to develop a satisfactory theory for the behaviour of a simple type lI plate specimen, and to provide full experimental confirmation of it. The layout of the paper is as follows. First we give an overall chronological survey of the project, and an explanation of how the various experimental and theoretical components of it came together. Then, under materials and methods we describe briefly the various experimental techniques which were used. In the next section we present some typical experimental observations, and explain how the results were presented in a more basic way than that which was used in the previous study. The following section develops a theoretical model, and finally the good correlation between this theory and the experimental results is demonstrated. The paper is devoted almost exclusively to the behaviour of type II specimens, since type I specimens pose few unresolved problems of understanding: see, for example, [12,13]. In the main, this paper is a condensation of a Ph.D. dissertation [14]. Full details of experimental procedures, results, etc., may be found therein. 2. CHRONOLOGICAL SYNOPSIS OF THE WORK Most of the experiments performed on type II specimens in the drop-hammer rig (see Table 1 for a summary) were of a simple kind, involving little instrumentation. The velocity of the falling striker just before impact was measured by timing accurately the interval between the breaking of two graphite rods; and the leading dimensions of the specimens

TABLE l . IN THE DYNAMIC TESTS THE MAIN VARIABLES WERE INITIAL CROOKEDNESS,

0o (FIG. 2); IMPACTVELOCITYVo; AND MASSRATIO, ~ [-EQ. (7)]. THE CHART INDICATES WHICH VARIABLES WERE ALTERED AND THE TOTAL NUMBER OF SPECIMENS IN A GIVEN TEST SERIES. DETAILS OF THE GEOMETRY OF THE SPECIMENS ARE GIVEN IN TABLE 2

Type of specimen

0o

Vo

/~

Number of tests

Sa Sa Sb Sc Aa Ad

× --x --

-× x x -x

-× x x -x

12 53 15 47 13 23

Here x indicatesthat the tests involvedchanges in a variable, while- - indicates that a quantity was fixed.

354

1., L |.,,,,1 a n d (

R (, x t t ,\l)l'-,i.

before and after impact were measured by micrometer, etc. The lirsl serics of lcsts ~ s aimed at tracking down and removing the "scatter" which had been a feature of the previou~ tests. It turned out that two factors were involved here. First it was found that scatter was reduced if the velocity of impact was measured directly, as described above, rather thal~ being estimated by conservation of energy from the known drop height, as in the previous work. Second, it was found that the final configuration of a specimen under given tcsting conditions is rather sensitive to the initial crookedness which was introduced deliberately to all specimens, but which had not been controlled carefully in the earlier study. We devised a simple and repeatable way of introducing accurately a given initial crookedness to the plates of the specimens. The next series of tests was aimed at investigating the detailed profiles of the plastic hinges in the individual plates of specimens after test. In principle both the yield stress and the strain-hardening modulus of the material depend on strain-rate; and so it is conceivable that the detailed profile of a hinge having a total angle of rotation of, say, 40 will be different in hinges formed under dynamic and quasi-static conditions, respectively. We developed techniques for surveying accurately the detailed profiles of hinges: but we found no significant differences in the profiles formed at different rates. The aim of the next phase of experimental work was to establish in detail the sequence of events in a typical test specimen when it receives a blow from the drop-hammer. To this end we instrumented the striker with accelerometers, and we mounted the specimcn on a load-cell. We also attached strain gauges on both sides of the component plates, and at various positions in relation to the plastic hinge locations. In general, we found that of all these schemes for assaying the forces in the specimen, the strain-gauge data was the most useful and reliable for our purposes. From these various studies we were able to obtain a fairly clear picture of the sequence of events following contact of the striker with the specimen, and in particular to determine key time parameters such as the rise-time, height and duration of the initial stress-pulse. We also discovered a number of unexpected elastic vibration phenomena: for example, we found that in the early stage of rapid rotation of the plastic hinges, the portions of plate lying between the successive hinges underwent rapidly damped elastic flexural vibrations. Some supplementary studies by high-speed photography at a frame rate of 5000 s L enabled us to obtain an independent measure of the rates of rotation of the plastic hinges in a particular specimen. At this stage in the testing program we had assembled a detailed and rather complicated picture of the sequence of events in a typical mild-steel type I1 specimen following impact. From this we had hoped to develop a theory which would fit all of the various events into a single overall rational framework. At this point we received, from Peking, unexpectedly, a draft manuscript of a paper by Zhang and Yu, which was subsequently published [15]. Dr T. X. Yu had been a visitor in our Department when our project began, and we discussed our various ideas with him. He had returned to Peking and had attempted to provide a detailed rational explanation for the experimental observations presented in ref. [12]. The central idea was to use the classical 'impulse' theory of the mechanics of inelastic impact of riqid bodies in order to study the state of the specimen immediately before lind after contact with the striker. According to this theory, the specimen is set in motion initially as if it were an assembly of rigid bars connected together by frictionless hinges. Some energy is 'lost' in such an instantaneous process, just as it is in the classical analysis of impact of rigid bodies: and the energy which remains is absorbed eventually by the rotation of plastic hinges. After a first reading of [15] we were inclined to dismiss this theory on the grounds that we knew from our experimental studies that the strong initial 'spike' on the load-time plot was indeed not instantaneous but of significant duration; and in particular it had not ended before the hinges began to rotate. Hence, we thought, the model in Zhang and Yu of an instantaneous impact event was not sufficiently realistic. Nevertheless this radical new theory certainly predicted correct orders of magnitude for overall hinge-rotation angles. On the other hand, the new analysis predicted a sensitivity of the final hinge rotation angles with the angle of initial crookedness to a degree which

A simple plate-structure under impact loading

355

was not matched by the experimental observations in a special study of this effect, that we had by then conducted. The upshot of all this was that we decided to try to modify the analysis of [15] in a way that would make it more realistic both in relation to the timing of the initial short 'spike' of loading, and also in relation to the effect of initial crookedness; but which would retain to a large extent the simplicity of the original in relation to the complicated details of behaviour which our strain-gauge studies had uncovered. In fact our final theory--to be described below--has a strong resemblance to the early, crude attempt at a theory made in [12], and which suffered both from inadequacy of supporting experimental data and also from lack of attention to the details of initial conditions, etc. Our final theory produced clearly two new dimensionless groups whose values characterized the behaviour of the specimen under impact. The remaining phase of our experimental program, which returned once more to simple testing without sophisticated instrumentation, was concerned with the gathering of data along lines suggested by our theory. At this stage we were concerned to obtain data for specimens of two different sizes in both steel and aluminium; for a comparison of the two materials would enable us to differentiate between the effects of inertia and strain-rate. The broad thrust of our theory is that the first phase of behaviour, immediately following impact, is dominated by inertia effects. In the idealization of Zhang and Yu, this period is instantaneous; and energy is 'lost' in the impact to an extent which is closely related to the initial crookedness of the specimen. In our theory the first phase is of finite duration, and the specimen responds by a small overall shortening, during which the initial crookedness increases somewhat in accordance with the value of two new dimensionless groups. In this way we account directly for the energy 'lost' in the initial impact event and our analysis reduces precisely to that of [15] if we take a very large value of the relevant dimensionless group. Overall, as we have said, inertia is the dominant effect in the first phase of behaviour. In contrast, the second phase of behaviour, in which energy is absorbed by the rotation of plastic hinges, is more sensitive to strain-rate effects. All of the above relates, of course, to impact of type II specimens. Type I specimens, in contrast, are at most only marginally involved in anything analogous to the initial impact event; and almost all of the energy supplied is absorbed in a straightforward manner by rotation of the plastic hinges.

3. M A T E R I A L S AND M E T H O D S

Specimens All of the specimens were made in accordance with the drawing of Fig. 2. The two plates were clamped together at the base and bolted together at the top. Before assembly the plates were pre-bent in a specially designed rig which inserted three plastic hinges with prescribed small rotations 0o, 200 and 0o, respectively. Steel plate of two thicknesses and aluminium plate of two thicknesses were used, and all dimensions in the view of Fig. 2 were made proportional to the thickness h, thus preserving complete geometric similarity in this elevation. For specimens made from the thicker steel plate, two different widths b (perpendicular to the view of Fig. 2) were used, since it was clear that the mechanics of the various impact processes, etc., would not be affected by the overall width. Specimens were denoted by letters S (steel) or A (aluminium) according to material, followed by a lower-case letter which identifies the pattern of dimensions which is detailed in Table 2. After assembly of each specimen, the value of 0 o was determined from micrometer measurement of the overall thickness of the specimen at the central hinge. For most series of tests the angle 0o was specified in advance. Specimens for which 0 o as measured differed from the intended value by + 0.1 ° were rejected. The upper and lower edges of the two plates of each specimen was trimmed on a milling cutter.

~56

L . I . . "I'A~,I and (.. R. (,~l 1AI)INI / h I.

~.,

BA bolt

2

j

Oo 14 '5

[ L2

6

q// "/ /

, //

F/:/:/ f/: :-/t22 FIG. 2. Sketch drawing of the 'type I1" specimen used throughout this investigation, and defining the leading dimensions. The drawing is approximately to scale, but the plate thickness, h, and initial crookedness 0o have been exaggerated. The two plates are held together by a number of bolts at the top, and are clamped between steel blocks at the base. Details of dimensions and bolt arrangements are given in Table 2. For an explanation of strain-gauge positions, here marked 2 7. see Section 4.

TABLE 2. GEOMETRICAL DETAILS OF SPECIMENS. THE ELEVATION OF ALL SPECIMENS IS SHOWN IN FIG. 2. FOR BOTH STEEL (S) AND ALUMINIUM (A) SPECIMENS TWO DIFFERENT THICKNESSES, h, OF PLATE ARE USED. LENGTH, l, IS ALWAYS CLOSE TO A FIXED MULTIPLE ( ~ 3 1 )

OF h, s o THA'I SPECIMENS OF DIFFERENT SIZE ARE GEOMETRICALLY

SIMILAR IN ELEVATION; AND THE SIZE OF THE BOLTS IS SCALED ACCORDINGLY. LENGFHS L 2 AND g 3 NOT ENTER THE ANALYSIS

WHICH I)O

ARE DEFINED IN F I G . 2. THE WIDTH OF THE SPECIMEN, b, WHICH IS PERPENDICULAR

TO THE PLANE OF FIG. 2, IS EQUAL TO I IN SOME SPECIMENS, BUT LESS THAN / IN OTHERS; AND THE NUMBER OE BOLTS VARIES WITH b/I. THE REASON FOR VARYING b WAS T O CONTROL THE MASS m OF THE SPECIMEN. THE SECOND, LOWER-CASE, LETTER IN THE SPECIMEN DESIGNATION {So/, etc.) IS u FOR THE THINNER AND b, c OR d FOR ]'HE I HICKER ONES

Type of specimen

h (mmt

/ (mm)

(mm)

L3 (mm)

h tmm)

m (kg)

Size of bolt (BA)

Number of bolts

Sa Sh Sc Aa Ad

1.6 2.5 2.5 1.6 2.66

50 78 78 50 78

109.5 170.8 170.8 109.5 170.8

45 70 70 45 70

50 78 20.5 50 60

0.062 0.236 0.062 0.022 0.067

6 2 2 6 2

5 5 I 5 4

L2

Material properties T h e u n i a x i a l tensile p r o p e r t i e s of the steel a n d a l u m i n i u m sheet m a t e r i a l w e r e d e t e r m i n e d by m e a n s of c o u p o n tests. T y p i c a l s t r e s s - s t r a i n curves, o b t a i n e d f r o m a H o w d e n E U - 5 0 0 D B S t e s t i n g m a c h i n e , are s h o w n in Fig. 3. In o r d e r to i n c r e a s e the d u c t i l i t y of the m i l d steel, the plates a n d c o u p o n s w e r e all h e a t - t r e a t e d in a v a c u u m f u r n a c e at 9 0 0 ° C for 35 m i n , f o l l o w e d by c o o l i n g in the f u r n a c e . T h e a l u m i n i u m s p e c i m e n s w e r e h e a t - t r e a t e d in the f u r n a c e at 3 6 0 ° C for 2.5 h, f o l l o w e d by f u r n a c e c o o l i n g . M e a n v a l u e s of yield stress are g i v e n in T a b l e 3: for m i l d steel ao was d e f i n e d as the l o w e r y i e l d - p o i n t stress, while for a l u m i n i u m a l l o y ao was d e f i n e d as the 0 . 2 % p r o o f stress.

Test machines (I) A c o m m e r c i a l H o w d e n test m a c h i n e was used for c o u p o n testing, as n o t e d a b o v e . It was also u s e d for p e r f o r m i n g q u a s i - s t a t i c tests o n the t y p e II s p e c i m e n s .

A simple plate-structure under impact loading

357

400

300 E z 200

100

~ A l u m i n i u m

HP 30

I

0.1

nominalstrain

0!2

i

0.3

FIG. 3. Typical nominal tensile stress/strain curves for coupons of the two materials.

TABLE 3. MATERIAL YIELD STRESS (7o FROM COUPON TESTS

Material

h (ram)

oo (N m m -2)

Average a o (N m m -2)

1.6

239 231 227 225

235

58 62

60

Mild steel 2.5

AI HP-30

2.66

226

(2) A drop-hammer apparatus In the mechanics laboratory of our department was used for dynamic tests. The same apparatus had been used for the earlier experiments [12]. In that study, the velocity of impact Vo was calculated from the height of release of the striker. However, on account of friction from the guide rails, the striker is not under perfect free-fall conditions. Therefore the velocity of the striker just before impact was measured directly from the interval of time between the breaking of two graphite rods mounted 50 mm apart, measured by a Racal digital clock. In our experiments, Vo was in the range 1.6-6.2 m s- 1. The striker itself consisted of a stack of mild steel blocks sandwiched between two aluminium guide plates. The total mass, G, of the striker could be changed by altering the number of blocks. In all, seven different values of G were used, and details are given in Table 4. It turns out, as we shall see, that the ratio of the mass of the striker to the mass of the specimen is an important parameter in tests on type II specimens; and so values of this parameter are also given in Table 4, as appropriate. Quasi-static tests Quasi-static tests were performed on the various different kinds of type II specimen in the Howden machine at a crosshead speed of 0.05 m s- 1. Plots of load against shortening deflection u for Sa specimens having two different values of 0o are shown in Fig. 4, together with the corresponding energy~lisplacement plots. The initial peak load depends strongly on 0o, as expected; but the falling part of the curve does not. For 0 o = 1.6 ° the peak load is 18.2 kN (by interpolation from data of the kind shown in Fig. 4), which is about one half of the static 'squash' load of 37.5 kN. The total energy absorbed when the plastic hinges are well-developed is clearly indifferent to 0o, to first-order. As will be explained in Section 5, we decided to describe the final, deflected shape of a

~5S

t. I. I~xxuand t

R ( " , , u ~p,xJ

] ABLt 4. VAI [:IS ()J }.lASS R~II() I~,ESI!I lING I R(IM IIII1 [5,1 ()1 SlRIKItRN (Jl (x

I4A\I'-,(.

Sl!VI N I )11 f.FRI/N I MAS,SI S I O ( i l ]tlER W1 lit [ I {l: VARI( 11_5, I)IFFI Rt N I 5,PI('IMtLNS Nil, I I ( . SI I TABI.li 2 F()R ] t n MASS 177 1)1 111t SPI!('IMt NS,

Striker

M a s s r a d o for s p e c i m e n (see ] a b l e

It

Mass Symbol

(kgl

Sa

(;~

6.4

103

Sh

6.55

(L

10.01

G3

15.21

245

G4

20.29

327

G,

25.39

409

G~,

30.48

492

G~

34.90

Aa

106

298

Ad

g8

162

9.33

150 245 86

227

327 409

129

492

455

148

2°-[I Pl /11

Sc

//

eo = 1.o7o

....

%= ~.~°

~,.-

120

~-~--" -100

-8o ~Z

/.ff

-60 g

-40

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v a l u e s o f 0 o,

specimen tested dynamically by reference to the energy which it would have received in a quasi-static test providing the same overall shortening, u. Since a certain amount of elastic 'spring-back' occurred on unloading in both quasi-static and dynamic tests, it was necessary to check whether there was a significant difference between the energy absorbed by a specimen under load up to, say, u = 30 mm, and then released (see Fig. 4) and the energy absorbed by a specimen loaded a little further and then unloaded to u = 30 ram. But in general the difference was negligible. (Note, in Fig. 4, that the elastic stiffness of heavily-deformed specimens is significantly lower than of those having smaller values of 0.)

Pr(~l~le survey o/plastic hinges Detailed profiles of plastic hinges were measured by means of a 3D Omicron machine for Sa specimens which had been shortened by ~ 5, 10, 20 and 30 mm, respectively, under both quasi-static and dynamic conditions. The quasi-static tests were performed in the Howden machine, as described above, and the dynamic tests were conducted in the drop-hammer rig with G = 10 kg and Vo = 5.4 m s - 1. The dynamic tests were 'interrupted" at the various different levels of shortening by means of hardwood blocks of the appropriate height placed on either side of the specimen.

A simple plate-structure under impact loading

359

In order to measure a profile, one plate from each specimen was mounted horizontally and traversed longitudinally by the vertical probe of the Omicron machine, which was tipped by a sphere of diameter 2 mm. Readings were taken at 1 mm traverse intervals, approximately. The machine measures, of course, the coordinates of the centre of the probe tip. A computer program was written in order to compute first the coordinates of the point of contact of the probe tip with the surface of the plate, and then the coordinates of the corresponding point on the mid-surface of the plate, on the assumption that the thickness of the plate was uniform. Finally, the mean curvature of the central surface was computed from the coordinates of four successive data-points.

High-speed photograph)' A commercial high-speed camera (Hyspeed, John Hadland Photographic Instrumental Ltd), set at a rate of 5000 frame s- 1, was used to study the history of deformation in two Sa specimens having 0o = 1.6°, under impact with G = 10.3 kg and Vo = 4.8 m s- 1. The near edges of the specimens were painted white. Sample frames are displayed in Fig. 5. Plastic deformation of the specimen was practically complete within about 30 frames after the instant of contact; and significant angular rotations were observed within the first few frames. The resolution of the photographs was not good enough to enable the vertical shortening, u, of the specimen to be measured with precision; but angular positions were well defined. In particular the photographs showed that the striker remained in contact with the top of the specimen from the moment of contact until the plastic hinges were fully formed. Strain measurements Strain gauge positions are indicated by numbers 2, 4, 5, 6 and 7 in Fig. 2. Gauges were attached back-to-back on the inner and outer faces of one plate of the specimen, in the longitudinal direction, and in a typical test gauges were located at two positions. Position 4 coincides with the central hinge, and in general the number assigned to a position is proportional to the distance of the position from the upper hinge: thus positions 2 and 6 are mid-way between hinges. At even-numbered positions the gauges were of type FLA-3-11 (Tokyo Sokki Kenkyuko Co. Ltd), with gauge-length 3 mm, while at positons 5 and 7 the gauges were type FLA-I-11, with gauge-length 1 mm. For all gauges the resistance was 120 f~ and the gauge factor was 2.12. These gauges are claimed to be capable of measurement up to 5% strain, but in our experience they become detached at smaller strain than this: see below. Gauges at positions 5 and 7 were intended to assay the bending moment at two locations in the elastic region, so that the bending moment at the adjacent plastic hinges could be determined by extrapolation. Short gauge lengths were used in order to avoid problems arising from the possibility of zones of plastic straining at the hinges spreading as far as the gauges. For every active strain gauge, a dummy gauge (FLA-3-11), attached to a separate mild steel plate, was used for temperature compensation. A pair of remote gauges of the output amplifier (FYLDE 154 ABS, 100 kHz frequency response) was used to complete the bridge. Four signals from four amplifiers were logged by two Hameg oscilloscopes and by a data logger (CED 1401,25 kHz frequency response per channel) separately. An external resistor was used to calibrate every strain gauge bridge. The total time bases of the oscilloscopes and of the data logger were set at 10 ms and 20 ms, respectively. The oscilloscopes were intended to record at high resolution signals appearing immediately after impact, whereas the data logger was used to capture the whole deformation process. The recording equipment was triggered externally by a 9 V d.c. signal produced by a microswitch which was actuated by the striker shortly before it hit the specimen. Signals recorded by the oscilloscopes and the data-logger were practically identical, with the exception that the reflection of elastic waves in the early stages of impact could be detected only by the oscilloscopes.

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FJc;. 5. Assembly of sclectcd p h m o g r a p h s s h o w i n g thc history of d e f o r m a t i o n m a Sa spccm~ci~ having 0,, 1.6 . and subjected to impact loading with I,, 4.8 m s ~. (; 10.3 kg. F r a m c n u m b e r s are indicated: the interval between successive frames is 0.2 ms. and frame 1 was the firs! Io sho'~ ~.tny displacement from the original configuration. 4. E X P E R I M E N T A L

RESULTS

For purposes of comparison with the theory which will be developed in Section 6, we are interested mostly in measurements of the 'overall' energy absorption parameters of the various kinds of specimen under different impact conditions. Before presenting these data,

A simple plate-structure under impact loading

361

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however, we shall give a brief s u m m a r y of the results of the various other assays w h i c h have been outlined in Section 3.

Hinge profiles Figure 6 shows the detailed measurements of curvature against centre-line arc length for the central hinge under (a) quasi-static and (b) d y n a m i c conditions, respectively, at four stages of deformation. The area under each curve gives the total hinge rotation in radians; and this agrees closely with the angular rotations measured independently. The

3~2

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pattern of straining is somewhat more regular under dynamic than under static loading: but the differences are not striking. Figure 7 (a) shows how a peak curwtture, I,-*, and a characteristic arc length of hinge, a, may be delined: the areas enclosed under tile curve and the rectangle are equal. Figure 7tb) plots measurements of hi,*2 where h is lhc thickness of the plate deduced from the data of Fig. 6 and corresponding data (not shm~ n) for the upper and lower hinges, against the total angle of hinge rotation. The dimensionless form of ~c* used here corresponds to the level of surface straining on account of curvalure according to the Kirchhoff hypothesis. The general pattern of these results is the same for both quasi-static and dynamic hinges. The data for the central and outer hinges are in fact self-consistent. In general the peak curvature J,* rises linearly with the total angle of rotation. Also shown in Fig. 7(b) are lines corresponding to hinge-length, a, equal to several different multiples of thickness. It is clear that as the hinge angle increases so also does the characteristic length of the hinge: in our experiments the value of a/h varies from 3 to 4.5.

Hi qlt-st~eed photo qraplty Figure 8 shows the angle 0 measured frame-by-frame. The frame interval is also indicated: and it is clear that the camera speed was not high enough to be able to capture in detail the initial angular response of the plates. In spite of this deficiency, it is of interest to make a n estimate of the strain-rate at the "extreme fibre' of the central hinge, by assuming a hinge-length of 4h (see above), ignoring any axial straining of the centre-line and measuring rotation-rates by computing the slope between successive points of Fig. 8. This calculation gives an early strain-rate of ~ 200 s ~, but the actual initial rate could well be higher.

Strain-gauge as.s'ay.~" The aim of the tests conducted with strain-gauge instrumentation was to build up a picture of the history of straining in various positions on a specimen Sa having 0o = 1.6'. In each individual test four gauges were deployed. The location of gauges (see Fig. 2) was planned to include one set on the central hinge, two sets midway between plastic hinges and other sets in intermediate positions. We shall present here an overall synopsis of the results of this study: full details may be found in [14]. Most of the quantities quoted below are given to two significant figures. Figure 9 presents data obtained from gauges mounted at location 2 in a typical test. The thicker curve shows the mean compressive strain, and the thinner curve shows half 60

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A simple plate-structure under impact loading

363

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the difference in strain between gauges mounted on the two faces of the plate, that is the 'bending strain' as usually defined. If the straining were entirely elastic, the thicker curve would also represent the total axial force, at least in the early stages of deformation, when 0 is small. There is an initial peak load, with a rise-time of ~0.12 ms. Then the load falls rapidly, and by about 1 ms after contact it has fallen to about one quarter of its peak value. (Henceforth, all times are given with respect to zero at the instant of contact.) Thereafter the load falls steadily. The most striking feature of the bending strain is a set of violent oscillations between ~0.12 and 1 ms. These may be identified (see below) as highly-damped flexural vibration of the plate segments between pairs of hinges. It is possible that the axial load-time trace should actually be smooth between about 0.15 and 1 ms, and that the observed oscillations in the thicker curve are some sort of contamination from the bending-strain trace. By 1 ms five frames on the high-speed film have elapsed and the value of 0 is about 20 °. Between 1 and 9 ms after contact the bending strain remains more or less constant. This corresponds to the period during which the hinges are rotating plastically, and there is a non-zero signal because the gauges are not located precisely at the point of contraflexture between neighbouring hinges. The hinge rotation reaches a peak value at about 9 ms (Fig. 8; but note that the strain-gauged and filmed specimens were physically different, and the times do not match exactly) and from then until 13 ms there is some elastic recoil. The entire picture of straining as described above was reproducible, as shown by tests on 10 specimens. From the hypothesis that the material is elastic, the dynamic peak load for a mean compressive strain of 2.2 x 10-3 is 74 kN, which is almost exactly two times the static squash load. As we have seen, the static peak load was about 0.5 of the static squash load, so the dynamic peak load is about four times the static peak load. Measured compressive strain-rates are about 20 s - 1 in the rising portion of the initial peak, while the strain rate estimated by dividing the impact velocity Vo by the overall length of the specimen comes out at about 40 s - 1. If we take the widely used Cowper-Symonds relation [6,16] fly/O"o = 1 + (t~t40) °'2,

(5)

as an approximate indication of the raising of the yield stress try of mild steel above its static value a o on account of strain rate ~ (s- 1), we expect the full-plastic dynamic squash load to be ~ 7 3 kN. It thus appears from these calculations that the peak load in the

364

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dynamic test is controlled by full plastic squashing over the entire cross-section. This conclusion should be regarded with some caution, however, because there are clear signs in Fig. 9 of bending strain during the 0.1 ms rise-time of the initial load peak. To what are we to attribute the violent but strongly bending oscillations immediately after the peak load is reached'? The well-defined period of these three cycles corresponds closely to the vibration of a simply supported elastic plate of the same material over a span of 25 mm; and indeed in tests on an Sc specimen the period was also that of a simply supported but correspondingly larger plate. It seems clear that these oscillations are initiated by the sudden appearance of plastic hinges at about 0.12 ms. The 'simple supports' for the vibrating plates at the two hinges and the strong damping effect are both attributable to the rotating plastic hinges at either end; these provide both constant bending moment and a highly dissipative environment at the ends of the elastic plate. Figure 10 shows more strain-gauge readings, from similar specimens under nominally the same dynamic test conditions, with gauges mounted in positions 2 and 4. The diagram looks different because the time scale has been stretched and the strain scale contracted in comparison with Fig. 9. The readings for gauges 2 show the same features as before, with the strong bending-strain oscillations commencing at about 0.12 ms. The readings from gauges 4 which are located on the central plastic hinge--show a quite different pattern. During the first 0.12 ms the mean strain level is the same as for gauges 2, but there is clearly more bending strain. At about 0.12 ms the strain-rate on both inner and outer gauges increases dramatically. It is, of course, unfortunate that the inner and outer gauges became detached at a strain of 4% (compressive) and 2% (tensile), respectively: several attempts which were made to secure better adhesion of the gauges were unsuccessful. The mean strain of gauges 4 is also plotted. On the Kirchhoff hypothesis this represents the mid-surface strain; and so the hinge-is shortening as it rotates. There are two reasons for believing that the material at gauges 4 enters the plastic range at a time of 0.12 ms after contact. The first is the magnitude of the measured strains. The second is more subtle. The plate is evidently moving transversely at time 0.1 ms, as evidenced by the increase of 'bending strain' with time. The total transverse m o m e n t u m cannot change abruptly at 0.12 ms (or indeed at any other time); and so we must attribute the enhanced strain rates for gauges 4 as indications of localization of deformation in a plastic hinge. We believe that it is this localization of straining and the associated change of mode-form which initiates the elastic vibrations described above. In this connection we

A simple plate-structure under impact loading

365

have found that the bending strains measured at gauge positions 2 and 6 are in phase, indicating that the mode of vibration is symmetric about the central hinge.

5. MAIN TEST RESULTS In the previous paper [12] the results of dynamic tests on type II specimens were presented in terms of the shortening of a specimen under the application of a specific amount of kinetic energy. And the measured shortening was compared with the shortening of a nominally identical specimen under a quasi-static test taken to the point where the energy absorbed by the specimen was equal to the specific energy mentioned above. For present purposes, in which both impact velocity and mass of striker were variable, and different materials and sizes of specimen were to be used, it seemed desirable to employ some other measure of the level of distortion of a specimen. For this purpose it is convenient to determine an energy ratio R = KE/SE

(6)

where KE is the kinetic energy of the striker immediately before contact with the specimen, and SE is the 'static energy' which would be required to produce the final deflected shape of the dynamic specimen in a quasi-static test. Such a measure would not be appropriate, of course, if the mode of deformation under a static test were appreciably different from that in a dynamic test; but in the present study we have demonstrated not only that the general pattern of deformation is the same, but also that the detailed geometry of the plastic hinges is very similar in the two kinds of test. An obvious advantage of the energy-ratio approach is that it automatically takes account of any strain-hardening effects; and in particular it is not necessary to assign a value to the yield stress of the material. The energy ratio, R, is not only a rational way of describing the level of deformation of a given specimen, but it is also a useful factor from the point of view of impact design. For, if we could somehow calculate theoretically the value of R in some particular impact problem, we should be able to use a quasi-static test for the purposes of design. But we must not run ahead of our place in the story; and the present task is now to find out, by experiment, how R varies with such factors as the mass and velocity of the striker. Our main programme of experiments involved testing various specimens made from mild steel and aluminium, and in geometrical patterns a-d (Table 2) under a range of values of G and Vo. We found it convenient to plot R against Fo, using different types of point to represent different values of G; and indeed it became convenient to quote the mass of the striker in terms of a mass ratio ~: p = (mass of striker)/(mass of specimen) = G/m.

(7)

Table 4 gives the various mass ratios resulting from the use of seven different versions of the striker with the various specimens. Figure 11 gives plots of this kind for data from steel (Sa, Sc) and aluminium (Ad) specimens. Each plot also shows some theoretical curves, in accordance with the analysis to be presented in Section 6, below. Several points emerge immediately from this experimental data, as follows. First, the data points for a particular mass ratio lie on a fairly well defined curve; but there is a certain amount of scatter, particularly for the aluminium specimens. Second, in general the value of R decreases as the value of Vo increases. Third, in general the value of R decreases as the value of/~ increases. Fourth, corresponding points for the larger steel specimens (b) fall somewhat lower in R values than for the smaller steel specimens (a); but points for aluminium specimens (c) lie lower than those for steel, particularly for ~t~ 100. In Section 7 we shall discuss the relationship between the experimental and theoretical results. For the present we note simply that in general the experimental points lie above the theoretical curves for steel specimens, but not for aluminium specimens. These results are entirely consistent with those presented by Calladine and English [12], and described briefly above. In those experiments the kinetic energy delivered was constant,

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so that an increase in velocity was compensated by a decrease in mass of striker. It is clear from Fig, I l that an increase in velocity at constant mass would decrease the value of R: and it is in fact the decrease in mass of the striker which is mainly responsible for the increase in value of R as Vo increases with KE held constant. In the explanation put forward in [12], the problem is seen as one in which the effective yield stress is raised on account of strain-rate effects. In the present study we find, paradoxically, that the mass of the striker plays an important part in determining the effective raising of the yield stress. It was mentioned in Section 2 that we made a study of the effect which the initial crookedness 0o had upon the behaviour of the specimens. Results from such tests are presented in Fig. 12 for: (a) steel; and (b) aluminium specimens. Both the impact velocity and the mass of the striker were held constant for these tests; and it is seen that the measured value of the energy ratio R increases significantly as 0o decreases. It is also clear that the values of R for steel are higher than those for aluminium,just as in Fig. 11. Curves corresponding to the theory to be developed below are also given in Fig. 12. 6. A T H E O R E T I C A L

MODEL

Zhang and Yu [15] proposed a simple theory for explaining the results presented in [l 2]. Their key idea was to treat the initial impact as one of dassical inelastic impact [ 17] between ideal, rigid bodies. Thus, the specimen is taken as consisting of four rigid links of mass m/4, hinged together and arranged at 0 = 0o originally, When the descending

367

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striker of mass G makes contact with the specimen there is an instantaneous impulse, J, which acts upwards on the striker and downwards on the specimen; and its value is such that immediately after impact the striker and the top of the specimen have a common velocity. This classical problem has a simple, unique solution; and in particular it is possible to find the kinetic energy, say T2, which remains immediately after impact. Zhang and Yu [15] give an expression for T2 which may be simplified by the omission of negligible terms, as follows, without essential loss when 0 o << 1 : K E / T 2 = 1 + m/12OZoG = 1 + 1/12#0o2.

(8)

Here, 0o is measured in radians. In the case where there is no strain-rate effect on the yield stress, the subsequent motion involves the dissipation of energy T 2 by simple rotation of plastic hinges at constant bending moment. In such a case, of course, T 2 is equal to SE, i.e. energy absorbed in a quasi-static test and so (8) provides a way of predicting the behaviour of a type II specimen under impact. An attractive feature of this model in relation to the experimental results of Fig. 11 is the importance attached to the mass ratio/~. But on the other hand the model gives no indication that R will depend on Vo as it evidently does in the experiments described in Fig. 11. Nor can it be argued that the Vo effect is associated entirely with strain-rate effects, since it is clearly present in aluminium specimens. Formula (8) also gives a clear prediction in relation to the experiments displayed in Fig. 12. Here, however, it predicts a greater sensitivity of R to 0o than is found in practice: see the broken curves drawn in the diagrams. (We should remark here that the results shown in Figs 11 and 12 were not available to Zhang and Yu when they wrote their paper.) The most straightforward way of understanding formula (8) is as follows. Over a very short period of time the striker and the specimen exert large, opposed forces on each other. The outcome of the classical impact situation is determined by the downwards acceleration of the top of the specimen under the force applied by the striker. Figure 13(a) shows the specimen idealized as four rigid links freely hinged to each other under an applied force F, and it is a straightforward matter to compute the corresponding instantaneous acceleration of the upper end. Figures 13(b)-(d) show an array of assemblies which are all equivalent to the linkage of (a) in the sense that equal applied forces give equal acceleration of the upper end. These include: (b) a linkage of light, rigid rods with a 'lumped' mass of m/6 at each of the two central nodes; (c) a 'one-sided' version of (b) with a single lumped mass m/3--for which, of course, horizontal reactions must be supplied at the upper and lower ends, in the absence of the 'back-to-back' effect of (b); and (d) a single mass M, where M = m/1202.

(9)

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It follows from this that the result of a classical, instantaneous impact between the moving striker and any of the four systems in Fig. 13 will be just the same, in the sense that the impulse J and the kinetic energies immediately after impact will be equal. The calculation giving the equivalence (9) for case (d) is a straightforward one in elementary mechanics, and it is based on the assumption that 0o is sufficiently small for the approximation sin 0,, ~ 0 o

i10)

to be valid. Consider now the impact indicated in Fig. 13(e) between the falling striker of mass G and the compact 'equivalent' mass M as described above. Further, let the two masses adhere to each other immediately after impact, and move off with a common velocity V1. Conservation of momentum provides the relation GVo = (G + M ) V 1.

(11)

The initial kinetic energy is given by K E = ~ G V o~

it2)

while the kinetic energy immediately after impact is given by T2= ~2tG+ M)V~.

(13)

Eliminating Vo and Vl from these three equations, we have KE T2

= 1+

M

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And on substituting for M from (9), we obtain (8), precisely. In this calculation we thus find that the energy ratio K E / T 2 depends only on the mass ratio, and neither on the impact velocity nor the mechanical properties of the material, except in that the collision must be 'perfectly inelastic', i.e. the two bodies must adhere to each other after impact. This single, idealized, aspect of the material/contact properties is sufficient to determine the outcome uniquely. It is the fortunate fact that there is no elastic rebound immediately after contact in our experiments which makes this extremely simple analysis valid as a first approximation. All of the above analysis has been conducted, in accordance with the scheme of Zhang and Yu [15], in the context of classical, instantaneous impact between rigid bodies. Let us now seek a modification of the above model which will enable us to have a force-interaction of finite duration between the striker and the top of the specimen. For

A simple plate-structure under impact loading

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Ft(L 14. Geometry of the specimen in: [a) original and; (b) deformed configuration, showing the definitions of y and q. Angles and axial shortening are here shown larger than in practice, for the sake of clarity.

this purpose the version of the specimen shown in Fig. 13(c) is most convenient. As described above, this consists of a central mass and two light but rigid links. We now propose to replace these links by rods which are axially deformable; and we shall further assume for the sake of simplicity that the rods are rigid until a yield stress trr is reached, whereupon they yield under constant compressive force Arty, where A is the cross-sectional area of the rods in the model: so A is equal to the combined cross-sectional area of the two plates of the specimen. We shall also assume that the forces and axial shortening are equal in the upper and lower rods. Figure 14 shows the outline geometry of the specimen (a) in its initial condition and (b) at a typical stage after impact. In (a) the angle is 0o and the overall contour length is I, while in (b) the angle is 0 and the overall length is l - q, where q is the shortening of the contour length of the rod on account of plastic squashing. Between (a) and (b) the vertical height of the specimen is reduced by y, which is computed geometrically, as follows. y = lcos 0 o - ( l - q) cos 0 =/(cos 0 o - cos 0) + q cos 0.

(15)

We now argue that during phase 1 of impact the angle 0 remains small--although it may become large in the later stages of deformation, of course--and so we shall employ the truncated Taylor-series expansion cos 0 = 1 -

02/2+

• • • .

(16)

Using this, we find y = (1/2)(0 z -

02) + q

(17)

after higher-order terms have been discarded. Now it will be convenient in the following analysis to use the transverse displacement, w (see Fig. 14) as a variable instead of 0. Accordingly we put w = 01/2

(18)

and so (17) becomes y = (2/I)(w 2 -

Wo 2) + q.

(19)

Next we need an expression for the vertical velocity of the top of the specimen. We find this by differentiating (19) w.r.t, time; and since there is no rebound between the top of the specimen and the striker, we write £ = V,

(20)

where ! is the v e l o c i t v o l t h c s t r i k e r . Thus ~c obtain l ' = (4 /Iw~i' + ~)

= l * + ,) where V* = 14 /)w~i'.

f221

In Eq. (21), the first terms on the r.h.s, gives the vertical velocity, 1'*, which the upper end of the specimen would have in terms of w and fi' ([ the bars q] the .~pecimen were inextensional: while the second term is the additive contribution to V of the shortening of the specimen on account of plastic squashing. Suppose that plastic squashing is taking place a short time after the striker has made contact. There is an axial force of Aar in each of the inclined bars: and these exert a lateral force on the lumped mass, which in turn provides a lateral acceleration. Hence we lind the equation of motion for the specimen {m/3)/i, = (4a~A/I)w.

f23)

Solving this subject to the initial condition w=wo

at t = 0

t24)

we obtain w = Wocosh ",'t,

(25)

where 7 = {12avA/hn)l'2.

26 }

Hence by substitution we find (27%//) sinh(27t).

1271

Turning now to the striker, we find that the equation of motion is very simple, because the plastically shortening specimen exerts a constant retarding force on it: thus we have

(/= -- Arry/G.

(18)

Integrating this and using the initial condition V = 1/',, at t = 0,

(29)

V = V , , - lAay/G)t.

1301

we obtain

Figure 15 shows a schematic plot of both Vand V* against t. The rising curve is for V*, and the descending straight line for V. The vertical intercept between the two curves is equal to //, by (21); and since / / > 0 by hypothesis (as the specimen is assumed to be yielding plastically in compression) the solution ceases to be valid beyond the time t = t~ at which the two curves intersect. Thus, the intersection of the two curves marks the end of the first phase and the beginning of the second phase of behaviour, i.e. the phase in which the remaining energy is absorbed only by the rotation of simple plastic hinges against their full-plastic moment. There is, of course, some rotation of plastic hinges during phase 1; but since it occurs under conditions of full plastic 'squashing', no bending moment is developed. Thus the energy absorbed is associated directly with the shortening, as in the analysis above. A convenient feature of Fig. 15 is that the shaded area, A 1, is equal to the total axial shortening of the specimen in phase 1 of the behaviour. Further, since the compressive force in the specimen is constant, this area is also proportional to the energy absorbed by the specimen by axial shortening during phase 1. And indeed this is the only mechanism

A simple plate-structure under impact loading

Vo(~)

371

A

(slooe t)

, 0

h('q)

"C time

(3)

FIG. 15. Graphical representation of Eqs (27j and (30) by means of a plot of velocity against time. The plot is labelled both in the original variables and also in dimensionless variables, the latter being enclosed in parentheses. The descending line, AB, represents the motion of the falling mass IV, (t,)], while the rising curve OB represents the motion which the top of the specimen [see Fig. 14(b)] would have if the rods were inextensional [V*, (v*)]. The slope of curve OB is inversely proportional to the current "effective mass' of the specimen; and according to the idealization of Zhang and Yu [12] this curve is a straight line. Phase 1 ends, at time t ~, when the curves intersect.

of energy absorption during phase 1 in the present model: the energy absorbed in this way is precisely the energy 'lost' during phase 1, in the sense that it is not available subsequently to produce rotation of plastic hinges in phase 2. We shall presently consider the dimensionless groups which determine the leading parameters of the behaviour in phase 1. But before we do this it is useful to examine the form of Eq. (27). The key point is that the equation describes the motion of the top end of the specimen under the action of a constant downward force. The situation is similar to that depicted in Fig. 13(c), where the specimen was replaced for this purpose by a single, compact mass. In that case, however, the specimen was in its original geometry, whereas in (27) the geometry of the specimen is changing continuously while the force acts. In fact the slope of the curve V*(t) in Fig. 15 is inversely proportional to the 'equivalent mass' of the specimen in its current configuration, which of course decreases as 0 increases, by (9). In particular it is straightforward to verify that the initial slope corresponds precisely to the mass, M, defined in Eq. (9). (For further plots of this kind, see [-17].) At this stage it is convenient to. re-write our equations in dimensionless form. There are many possible ways of doing this, and the following scheme seems as good as any. The overall form of Fig. 15 is preserved, of course, when we change the variables in this way. First we define a dimensionless time r tVo ( a y A l ~

I \GV o}

I31)

The point of this particular definition is that the striker would come to rest at r = 1 if it were acted upon by the constant force AO-yuntil it stopped: see (30). We also define dimensionless velocity v, v* corresponding to V, V* as follows v = V~/V o

(32)

v*= V * ~ / V o

(33)

___1 m(w/o)Z_ 1 m 1 ~=48G 12 G 002.

(34)

where

The parameter ~ is closely related to the equivalent mass M(9), and in effect it has already appeared in (8): it depends purely on mass and geometry. The point behind these

172

l.. I.. F,~,Mand ( R. ('.,'dlADl",l: .

substitutions is that the initial slope of t,*(r) is unity; and this in turn makes the initial value of t~ equal to 4. Substituting for V, I * in {32) and (33) from (30) and 127) we have t,= (1 -- v)4

(35)

t,* = (I/F) sinh Vt

(36)

/ 48 ,X1.'2 F = GVo[ / \GAIm]

1371

where

The second dimensionless group, F, has been introduced in order to give a simple argument for the sinh function in (27). It involves mass ratio, velocity and yield stress, but not the initial crookedness of the specimen. Figure 15 shows a re-labelling appropriate to the dimensionless version. It is straightforward to show that the marked areas A~ and A2 are proportional to the energy absorbed in phase ! and the kinetic energy delivered by the striker, respectively, according to either the original or the dimensionless variables. Let us now redefine T z as follows for present purposes [cf. (13)]: T 2 = kinetic energy remaining in the striker and specimen at t = t 1.

(38)

Then we find A2 -

KE

T2

A2 - -

= ¢/(2-4Q)

(39)

A 1

where Q=r~

v~

l-coshFt

- -2 +

FQ

"

(40)

It follows from this that the theoretical ratio K E / ~ can be evaluated in principle in any given test on a specimen as soon as the values of the dimensionless parameters F and have been found. In practice, the evaluation of rl is not straightforward, and requires some iteration, A systematic study of a large set of numerical solutions reveals that a relatively simple, approximate empirical formula for r~ can be found. For this purpose it is most convenient to focus attention on a new energy quantity, T~: T1 = energy absorbed in phase 1

(41)

T I is, of course, directly related to T2 and KE: Tl + T2 = KE,

(42)

The empirical approximate formula, which is accurate to within a few per cent over the working range of the variables, is: KE 1 0.1F T1 ~ 1 + c" + ~0J2"

(43)

T a m [14] explains how he found this approximation. There seems to be no simple explanation for the form of (43). The first two terms on the r.h.s, of (43) correspond exactly to Eq. (8), which is due to Zhang and Yu. In their analysis the fraction of the initial kinetic energy which is absorbed in phase 1 depends only on the mass ratio and 0o. The additional term in (43) brings into the calculation, via F, the velocity Vo, the yield stress Cry, and the volume of the specimen Al, in addition to the mass ratio. This additional term is needed whenever the quantity 0 , 1 F ~ °'8

(44)

A simple plate-structure under impact loading

373

is not negligible in comparison to unity. Substituting from (37) and (34) we find that if 0.1

° %Ala~yf

001"6

(45)

is small in comparison with 1, then the formula (8) of Zhang and Yu is satisfactory; but otherwise (43) should be used instead. Here, of course, 0o is measured in radians. There is one specially simple case of the analysis, which in fact corresponds exactly to the analysis of Zhang and Yu. Thus, suppose that we assign a very large value to ay, so that the interactive force between the striker and the specimen is very large. In this case phase 1 finishes quickly, and in particular before the initial angle 0o has changed appreciably. The dimensionless parameter F takes a small value, so (36) reduces to v* = r.

(46)

Thus in Fig. 15 the curved lower edge of the area A1 is now replaced by a straight line--corresponding to a constant 'effective mass', as mentioned a b o v e - - and we find, finally

KE/T2 = 1 + ~

(47)

in exact accordance with (8); this justifies the extended definition of T2. This result can also be seen as an extreme simplification of (43): the last term on the r.h.s, of (43) brings in the effects which were omitted by Zhang and Yu.

7. COMPARISON OF THEORY WITH EXPERIMENT The most direct way of comparing this theory with our experimental results is to determine the values of the dimensionless parameters F and ~ for a given test and compute the value of K E / T 2 accordingly, by means of Eqs (42) and (43)--or (39) and (40). We shall describe comparisons of this sort in due course; but before we do so it is helpful to study qualitatively the effect of various changes on the expression (43). First, suppose that 0o decreases, while all other quantities remain unchanged. The 'effective mass ratio' parameter ~ increases, but F does not change. From Fig. 15 we can see that an increase in ~ raises the inclined line AC and leads to an increase in A1/A2, i.e. an increase in the fraction of energy 'lost' in phase 1. But this is not such a strong increase as in the version of Zhang and Yu, which corresponds to F ~ 0, and the straight lower boundary of area A1. Second, suppose instead that Vo increases, all other quantities remaining unchanged. In this case ~ remains constant, but F increases. The effect of this is to make the curve v*(r) bend up more steeply, and so also to diminish the ratio A~/A2. In any given testing situation the leading variables are known, so the dimensionless parameters F and ~ can be evaluated. In practice, the only problem is to decide upon a suitable value of O'y, the assumed yield stress in axial squashing. It needs to be computed with regard to the strain-rate expected in phase 1 (which is of order Vo/l) and the strain-rate sensitivity of the material. In general this causes no difficulty. But in any case the ratio KE/T z is insensitive to ay in practice: note that Cry only appears in expression (37) for F raised to the power 0.5. We have already mentioned that the theoretical curves included in the plots of Fig. 11 were computed from (39) and (40) in accordance with our theory. It is convenient to begin an examination of these by looking at the plots for aluminium specimens, since the strain-rate insensitivity of that material enables us to make a direct comparison between R ( = KE/SE) as measured in the experiments and KE/Tz as evaluated theoretically. There are two sets of data. First, Fig. 12(b) shows tests on smaller-size aluminium specimens in which only the initial crookedness measure 0o was varied, together with theoretical curves according to our analysis and that of Zhang and Yu, respectively. Second, Fig. 1l(c) shows tests on larger-size specimens in which both the impact velocity and the mass of the striker were changed. There is some scatter in both plots, of course, but overall

~74

I

I..

2ol--__ . . . . . . . . . . . . . . . . . . • ,

IA'.,1 a l l d

(

R

( ,\ll,~,l)l",t

F ........ IKEy

.

,,u

1.81

= 103

2 o ~r . . . . . . . . . . . . . . . .

I 1[ = ~ 06 ' 4 150 I '45 ~2:' : 409 ~ 492

1 8!

162. 245 * 327 409 492 i. . . . . .

i d

i i

• ,

. . . . . . [ KEY i

[

1.6-

~r

j -

16 ~

F2 1.4-

_+012

"~ - *

~'

'

-....

}. . . .

meanq

l

12.

1.o---~-

~

--'

i

4

(a)

1.4-

6

+0,09

i!

1.0

~

~

r

1.2

1. 2

*

1.1"

KEY !.1 = 9 8 + 227 455 *

k!~ KEY = 298 × 1.1

+

~2

f

1.0-

_+0.09--

1.0 --mean

-

×

~

--

mean --

--

0.9-

0.9 ¸

08

:0.06 -

, , ~

,

A Vo(mS -1)

,

~

06 (d)

,

fi

,

1

8

Vo(m s 1)

(b)

(c)

m e a n ---, I

i

a

V°(ms-1)

--

1.2

,; Oo (o)

F'lc;. t6. (a) tc) Replots of Fig. II(a) (c) as T2SE [=(KE/SE)+iKE:T2)J against F.. Note the enlarged vertical scale of (c). (d) Corresponding replot of Fig. 12(b). In each case the mean ordinate and + standard deviation are shown,

the agreement between our theory and the experiments in both is encouraging. Re-plots of these data in the form of T2/SE against 0 o or Vo as appropriate are shown in Fig. 16(c) and (d), and it can be seen that the mean ordinate for both plots is around 1.0. Figure 11 (a) and (b) refer to tests on steel specimens under various test conditions. Here the general trend of the experimental points and theoretical points is the same, but the experimental values of KE/SE exceed the theoretical values of KE/T 2. Ratios T2/SE are plotted from the same data in Fig. 16(a) and (b). These have a mean value of around 1.4, and we attribute this to a raising of the yield stress in phase 2 of the tests on account of strain-rate sensitivity in mild steel. In the development of our theory we have assumed explicitly that the energy lost in phase 1 i.e. lost in the sense of not being available for absorption by rotation of plastic hinge action is all consumed in axial straining of the specimen in compressive yield. Can we find any direct evidence of this? One possible approach would be to measure the overall reduction in contour-length of a specimen. Unfortunately, we did not measure the overall length of our specimens with sufficient accuracy before testing to be able to do a direct comparison; but in any case it would not be straightforward to measure the contour length in the final configuration. Strain-gauge data might be useful here. The particular results presented in Fig. 9 suggest a mean permanent strain of the order of 70 x 10 - 6 , which is considerably below what is needed. But those gauges were attached mid-way between plastic hinges; and perhaps the major shortening occurs in the hinges themselves. Figure l0 is more helpful, for it shows a mean strain i.e. a strain at mid-thickness according to Kirchhoff's hypothesis of 1.4% and still rising at the point where the first gauge became detached. In an attempt to obtain an estimate of permanent straining we did a survey of thickness over the surface of a large aluminium specimen (Ad) which had been subjected

A simple plate-structure under impact loading

375

to impact at Vo ~ 6 m s -1, G = 6.55 kg, so /~ = 98. The striker delivered 120 J, of which approximately 30 J are 'lost' in phase 1, according to both theory and experiment [-see Fig. 1 l(c)]. Changes of thickness were measured by means of an ordinary micrometer at about 20 positions along the working length of the specimen. Such measurements are at best only crude, and they have a mean increase in thickness of 0.5% with a suggestion of higher-than-mean thickening at the top and centre hinges, but not at the bottom hinge. On the assumption that plastic shortening will be accompanied not only by increase of thickness but also by increase of width, we may suppose that there is a mean axial strain of ~ 1% over the top 100 mm of length of the specimen, i.e. of the material standing proud of the clamps. This suggests an axial shortening of 1 mm. Taking a dynamic yield stress for aluminium as 70 M Nm -2 and the cross-sectional area of the specimen as 320 mm 2 (see Table 2), we obtain a 'squash load' of 22 kN. This force would absorb the 30 J over a distance of 1.3 mm; which is close to the value deduced by measurements. It would be unwise to attach much weight to the consistency of the results shown by a simple calculation. It would perhaps be worthwhile to mount an experimental 'energy audit' for phase 1. It is well known, of course, that some energy can be radiated away in the form of elastic waves after an impact [18]. Perhaps the most remarkable feature of the analysis of Zhang and Yu is that, whatever the mechanisms by which energy is 'lost' in phase 1, the a m o u n t is determined directly from momentum considerations. This calculation depends only on the assumption that there is no elastic rebound during phase 1; which appears to be amply justified in the present context.

8. G E N E R A L D I S C U S S I O N

The main point established by the earlier paper [12] was the qualitative difference in performance between specimens of type I and type II under impact from a moving striker. The present paper is a continuation of this earlier work, with particular reference to the behaviour of type II specimens, since these are the ones which pose a challenge in understanding. As a result of our experimental and analytical studies, we can now present a clear and concise picture of the behaviour of type II specimens. The key point is that the response of a specimen to impact involves two separate phases of behaviour. In phase 1 a high initial axial force is set up in the specimen. This phase ends when the specimen has accelerated laterally sufficiently to enable the motion of the striker to be accommodated without further axial shortening of the specimen. During phase 1 energy is absorbed in much the same way as it is during the collision of two compact masses which adhere to each other after impact, and the fraction of energy 'lost' depends strongly on the ratio between the mass of the striker and the mass of the specimen. Phase 2 of the behaviour is more straightforward to understand. The kinetic energy which remains in the striker and specimen at the end of phase 1 is absorbed in phase 2 by the rotation of plastic hinges. The relevant yield stress in this process is that corresponding to the material of the specimen at the appropriate strain-rate. The model which we have developed for the impact behaviour of type II specimens involves several simplifications which are justified ultimately by the agreement between experimental observations and the predictions of the model. One such simplification is the neglect of elastic straining in our work. Thus in our model it is assumed that the axial compressive stress rises instantly to the yield value upon contact with the striker, although it is known from the experimental studies (see Figs 9 and 10) that there is a finite 'rise-time' for the compressive pulse. In fact the actual rise-time is such that elastic compression waves traverse the specimen several times before the peak load is reached in our specimens. It is the absence of discrete elastic wave effects which justifies the simplified view of the compression pulse which we have made in our model. In this connection it is worthwhile to pursue an argument based on dimensional analysis. If we ignore for the present the fact that yield stress is strain-rate sensitive, we may state

376

l.. L.'IAM and ('. R. ( ,XILAI)INJ-

that the dimensionless energy ratio R = K E S E which define the test conditions. Thus

is a function of the various parameters

K E / S E depends on G, 1~,, l, cry, 0,, and ?

f48)

where l) is the specific mass of the material. Assembling dimensionless groups, wc m;tv conclude in general that a R = K E / S E depends on

a~l 3 " p l 3"

0,,.

(49)

Thus, dimensional analysis indicates that R depends on three dimensionless groups, rather than on the t w o which emerge from our solution of the equation of motion. It is straightforward to express both of our key parameters F and ~ in terms of the three groups given above. The absence of a third group in our detailed analysis is attributable, we believe, to our implicit assumption that the mode of collapse in dynamic conditions is the same as the mode under quasi-static conditions. In our experiments there was no evidence to support a contrary view; but it seems likely that if the impact velocity V,, were increased substantially, a stage would be reached where the buckling mode occupies less than the full length 1 of the specimen 1-19]. In such a circumstance a third dimensionless group among the same variables might be relevant. Finally, we note that Grzebieta and Murray [20] have tested plate specimens in a drop-hammer under conditions somewhat similar to ours. They tested single steel plates 150 mm square, each with a central bend of a few degrees. The striker loaded the upper edge of the plate through a cross-bar, and the lower edge of the plate was supported on a load cell over a relatively small fraction of its width. Some of the specimens suffered from 'bearing' failure at the lower edge. In order to damp down oscillations in the instrumentation, rubber pads were inserted in the loading path at various places. It is clear that the testing conditions were significantly different from ours in several different respects, in a way which is difficult to assess quantitatively. Nevertheless the results show, broadly, trends similar to ours in that the ratio K E / S E (which, in effect, they plot in Fig. 9) decreases as both Vo and 0 o increase. Ackm~wledyements L.L. Tam acknowledges, with thanks, financial support from the Cambridge Commonwealth Trust, the Committee of Vice Chancellors and Principals (O.R.S. award), Darwin College, The University of Cambridge Department of Engineering, the Taylor Woodrow Charity Trust and Ove Arup and Partners. The work was done in connection with SERC Research Grant GR/D 7684.4. We thank Mr S. C. Palmer for continued help and interest, Professor T. X. Yu for an advance copy of ref. [.15], and Dr J. Miles and Professor S. R. Reid for helpful advice.

REFERENCES 1. W. JOHNSON and S. R. REID, Metallic energy dissipating systems. Appl. Mech. Rev. 31,277-288 (1978). 2. W. JOHNSON and S. R. REID, Update to 'Metallic energy dissipating systems' [1], Appl. Mech. Rec. 39, 315 319 (1986). 3. N. JONES, Recent studies on the dynamic plastic behaviour of structures. Appl. Mech. Ret,. 42, 95 115 (1989). 4. J. F. BAKER, Plasticity as a factor in the design of war-time structures, The Civil Enyineer in l~lr, Vol. 3, pp. 30 52. Institution of Civil Engineers, London (1948). 5. J. F. BAKER, M. R. HORNE and J. HEYMAN, The Steel Skeh, ton, Plastic Behaviour and Design, Vol. 2. Cambridge University Press (1956). 6. N. JONES, Structural aspects of ship collisions, Structural Crashworthiness (Edited by N. JONES and T. WIERZBICKI), pp. 308-337. Butterworths, London (1983). 7. K. J. MARSH and J. D. CAMPBELL, The effect of strain rate on the post-yield flow of mild steel. J. Mech. Phys. Solids 11, 49-63 (1963). 8. M. A. MACAULAYand R. G. REDWOOD, Small scale model railway coaches under impact. The Enqineer 218, 1041-1063 (1966). 9. E. BUCKINGHAM, Model experiments and the forms of empirical equations. Trans. Am. Sot'. Mech. En(crs 37, 263-297 (1915). 10. E. BOOTH, D. COLLIER and J. MILES, Impact scalability of plated steel structures, Structural Crashworthiness (Edited by N. JONES and T. WIERZBICKI), pp. 136-175. Butterworths, London (1983).

A simple plate-structure under impact loading

377

11. C. R. CALLADINE,An investigation of impact scaling theory, Struclural Crashworthiness (Edited by N. JONES and T. WIERZmCKI), pp. 169--174. Butterworths, London (1983). 12. C. R. CALLADINE and R. W. ENGLISH, Strain-rate and inertia effects in the collapse of two types of energy-absorbing structure. Int. J. Mech. Sci. 26, 689-701 (1984). 13. S. R, REID, Laterally compressed metal tubes as impact energy absorbers, Structural Crashworthiness (Edited by N. JONES and T. WIERZBICKI),pp. 1--42. Butterworths, London (1983). 14. L. L. TAM, Strain-rate and inertia effects in the collapse of energy-absorbing structures, Ph.D. dissertation, University of Cambridge (1990). 15. T. G. ZHANG and T. X. Yu, A note on a 'velocity sensitive' energy-absorbing structure. Int. J. Impact En~tng 8, 43-51 (1989). 16. P. S, SYMONDS,Viscoplastic behaviour in response of structures to dynamic loading, in Behaviour of Materials Under Dynamic Loading (Edited by N. J. HUFFINGTON), pp. 106 124. ASME, New York (1965). 17. C. R. CALLADINE, The teaching of some aspects of the theory of inelastic collisions. Int. J. Mech. Engng Educ. 18, 301-310 (1990). 18. R. S. BIRCH, N. JONES and W. S. Joum, Performance assessment of an impact rig. Proc. Inst. Mech. Engrs 202, 275-285 (1988). 19. G. R. ABRAHAMSONand J. N. GOODIER, Dynamic flexural buckling of rods within an axial compression wave. J. Appl. Mech. 33, 241-247 (1966). 20. R. H. GRZEmETA and N. W. MURRAY, Energy absorption of an initially imperfect strut subjected to an impact load. Int. J. Impact Engng 4, 147-159 (1986).