Lateral impact on a flexible viscoelastic tube

WAVE M O T I O N 8 (1986) 205-223 NORTH-HOLLAND

LATERAL IMPACT

205

ON A FLEXIBLE VISCOELASTIC

TUBE

Jurg DUAL* and Werner G O L D S M I T H Department of Mechanical Engineering, University of California, Berkeley, CA 94720, U.S.A. Received 9 May 1985

An experimental investigation was undertaken to examine the effects of transverse normal impact of an essentially rigid striker on flexible viscoelastic tubes with resultant large deformations and large local strains. The tests were conducted on a medical-grade ester-plasticized c o m p o u n d of polyvinyl chloride with nominal inner and outer diameters of 4.76 and 7.94 m m ( 3 in and ~ in), respectively, suspended vertically with a free length of 1.3 m. The specimens were dynamically loaded by two projectiles striking the center of the span: (i) a 6.35 m m (~ in) diameter steel ball bearing travelling at initial velocities between 40 and 70 m / s , and (ii) a special configuration composed of Lexan with a cylindrical striking surface of 3.97 m m ( ~ in) radius travelling between 4 and l0 m / s with the axis of the contact surface, the axis of the tube and the direction of motion forming an orthogonal triad. The impact p h e n o m e n a were observed by high-speed photography at a framing rate of 6500 pps. The displacement histories determined from the films were employed to compute an approximate force history using a nonlinear regression analysis. The pinching of the tube in the impact region, the velocity of propagation of the disturbance along the tube, the duration of contact and the rebound velocity of the striker were also measured. It was noted that the tube was completely closed by the striking sphere at higher velocities, while less pinching occurred when the plastic striker was utilized. It appears that the a m o u n t of pinching for a given tube is primarily determined by the initial energy of the striker and by the ratio of the initial velocity to the wave speed that characterizes the energy flow away from the impact point.

I. Introduction

The present work is concerned with an experimental examination of the phenomena that occur when an essentially rigid striker impinges transversely on the center of a flexible, viscoelastic tube in such a manner as to generate lateral deformations an order of magnitude larger than the tube diameter as well as large local strains. Such tubes are not only employed in a variety of life-support systems in space, under water and in hospital environments as well as in a variety of industrial applications, but also simulate quite well portions of the h u m a n circulatory system. In all instances, proper functioning demands the continuous transport of the enveloped fluid; an interruption of the flow due to temporary or permanent tube closure often results in life-threatening situations. * Current Address: Institute of Mechanics, ETH, Zurich, Switzerland.

In spite of its technical importance, there has apparently not been any previous work on this precise problem, although a variety of related or subsidiary problems have appeared in the literature. This includes an analysis of the transverse impact on the taut string, both linear and nonlinear [1]-[5] as well as some experimental verification of such a phenomenon [6] and more recent numerical studies of nonlinear onedimensional systems, such as submerged cables [7]. The force history for an infinite linear taut string with mass per length p', Young's modulus E and cross-sectional area A under constant tension T struck transversely by a mass m travelling with initial velocity Oo is given by [4] F(t) = (2 Tvo/ c) e -zr'/me) H(t) with cZ= T/p',

T = AE(1/2)2/3(0o/C2) 4/3,

0165-2125/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland)

where c 2 = EA/p'

(1)

206

J. Dual, W. Goldsmith / Impact on flexible tube

with H as the Heaviside function and t as time. The nonlinear string conforms to the study of the subject problem by inclusion of the nonlinearities due to large deflections and the propagation of two types of waves, albeit decoupled, but fails to include both such probable coupling and the effects of bending present in the tube. A substantial number of theoretical and experimental studies have been concerned with lateral impact on linearly elastic beams, including the modelling of a fluid-filled tube as a Timoshenko beam [8, 9]. In addition to a host of buckling problems, major attention has been devoted to the classical model of the slender beam subjected to large deformation, known as "elastica"; however, this is basically a static model. Investigations of large dynamic deformation have been generally confined to bars [10, 11] which may approximate the process of energy transport away from the impact point, but can not properly delineate the local contact phenomenon. For such a description, it is necessary to use an appropriate shell theory; while such nonlinear theories exist [12]-[14], they have not been applied to the present type of dynamic problem. Furthermore, the viscoelastic nature of the tube will inhibit the use of integral transform methods that are frequently used in conjunction with such materials; serious difficulties will be encountered during rebound when the contact area decreases [15]. Impact on fluid-filled tubes has also been studied in conjunction with the phenomenon o f w a t e r h a m mer [16] and blood flow, i.e. [17], but only [8] and [9] are specifically concerned with transverse impact, and that is limited to small strains and deformations of elastic hollow cylinders. It was found that, for sufficiently high velocities of an interior fluid, buckling of a conducting tube could occur [18, 19] which might be, but thus far has not been extended to include nonlinear effects. A variety of studies have been and currently appear to be in progress to ascertain the pinching or collapse of blood vessels, but no definitive publications in this area have come to the authors' attention.

2. Scope of the present investigation In view of the absence of reliable experimental information as well as a suitable theory covering the desired points, the present investigation was restricted to an experimental study of the subject problem so as to provide some sound data that could serve as the foundation for the construction of an analytical model. The present tests were confined to an air-filled tube both for the sake of initial simplicity of operation and validation of the methodology and measurement techniques as well as to serve as a basis for ascertaining the effect of the p r e s e n c e of the fluid. An initial step in this process consisted of the determination of the relevant properties of the tube material. The following aspects of the phenomenon were either measured directly or deduced by appropriate numerical techniques: (1) the history of the impact force, (2) the variation of the pinching process in the tube with time, (3) the speed of the front of the transverse deformation of the tube, (4) the duration of contact, and (5) the final velocity of the striker. Only selected combinations of the input parameters, including projectile mass and initial velocity, tube material and dimensions and initial and boundary conditions could be studied. The primary focus of the investigation concerned the influence of striker velocity, with a minor effort devoted to the determination of the effect of a slight tensile force in the reference configuration.

3. Experimental arrangement Tubular specimens with a length of 1.34 m were suspended vertically from a metal frame, as shown in Fig. 1. Headless screws of 5 mm diameter were inserted into each end of the tube and connected to linear springs with a constant of 320 N / m that were attached to this frame. The masses of the screws and springs were each 3.81 and 2.19g,

J. Dual, W. Goldsmith / Impact on flexible tube

T

--J

Fig. I. Experimental arrangement for transverse impact on the tube by the steel sphere. respectively. The springs were also employed to measure the pretension in the tube within an accuracy of 5 per cent. The tube was struck transversely at the center of its span. As the result of its vertical position, the tube experienced a linearly increasing longitudinal stress by virtue of its own weight, amounting to a maximum of 15.1 kPa at the upper end and a value of 7.6 kPa at the center in the reference configuration. In Runs 59 and 60, an additional stress of 56 and 98 kPa, respectively, that produced an increment in engineering strain of up to 4.5 percent was added by adjustment of the upper screw to assess the influence of prestress on the force and deformation histories. The 10 s interval between the prestress measurement and the actual tests resulted in a small amount of stress relaxation. An ink grid with a line thickness of 0.3 mm was painted on the tube and a millimeter grid was affixed to a board serving as a background for the target to assist in the photographic analysis of the phenomenon. The basic propulsion unit consisted of a pneumatic gun with a 6.35 mm (~ in) diameter barrel and a reservoir capacity of 0.7 MPa (100 psi) that permits ejection of a 6.35 mm (~ in) diameter steel sphere at speeds up to 100 m/s. However, for the present tests, the maximum back pressure employed was limited to 0.14 MPa. A thick-walled tube with an inner and outer diameter of 4.76 and 7.94 mm (3 and ~6 in), respec-

207

tively, was chosen as the basic target. Two types of strikers were employed exhibiting contact diameters of 6.35 and 7.94 mm (¼and 5 in), respectively; thus, the tube and striker diameters were approximately the same. In the absence of any fracture during the production of the large deformations of the tube, its closure occurs as the result of the dual mechanisms of wave propagation and contact force action whenever the initial striker velocity was of the same order of magnitude as the tube wave speed. Since this domain was to be studied, an air gun rather than a pendulum had to be employed in order to achieve the necessary initial striker velocity. Two different experimental arrangements, each with slightly different boundary conditions were employed, both permitting a maximum transverse tube displacement of 11 cm. In the first set-up, the targets were struck by a 6.35 mm steel sphere with a mass of 1.052 g fired from an air gun at initial velocities between 40 and 70m/s. In order to restrict the motion of the tube to the plane formed by its initial position and the projectile trajectory, it was constrained between two parallel P M M A plates with dimensions of 285 x 45 x 8.75 mm as shown in Fig. 2; the impact point was located about 10 mm below the upper surfaces of the screens. An optimal separation distance of 8.65 mm was chosen as a compromise to avoid excessive friction

Fig. 2. Support arrangement for the front screen.

208

J. Dual, W. Goldsmith / Impact on flexible tube

on one hand and either a complete bypass of the tube by the striker, a noncentral impact or even significant and unwanted lateral tube motion on the other. In addition, friction was further reduced by forming a coherent film on the screens with a fine gun oil exhibiting a viscosity of 33.7 cp at 20 ° C in a rotating viscosimeter in the range from 12 to 60 rpm. Nevertheless, an ancillary test in which the tube, suitably located between spacing blocks was statically compressed by the 6.35 mm diameter steel ball indicated a sideways deflection even before the complete closure substantially greater than that allowed by the screen separation. Thus, the tube would distend further in that direction, at least statically and presumably also dynamically, if it were not constrained by the screens. Since a freelysuspended front screen was not effective in preventing the bypass of the tube by the b a l l - implying the exertion of substantial force between the striker-target combination and the s c r e e n - - a somewhat different arrangement was required to achieve the desired impact without excessive interference by the screens. This consisted of the suspension of the front screen by two steel wires of 0.254 mm diameter whose length could be varied by adjustment of their supporting screws that were mounted on the main frame of the system. Four pieces of 2 0 x 7 . 4 x 0 . 0 2 5 4 m m brass foil, pretensioned to prevent bending and attached by means of eight clamps to the two screens at the top and bottom of the left- and right-hand edges, provided the necessary separation (Fig. 2). In spite of these precautions, measurements of the forces exerted on the screens by means of strain gages mounted on one of the foils indicated an insufficient degree of reproducibility in the initial part of the signal, perhaps partly induced by wave propagation and repeated reflections in the screens themselves. An identical conclusion was reached by an examination of the deflection history of the tube using high-speed photography. Use of slower, heavier, flat-ended cylinders resulted in nonorthogonal impact that produced an undesired immediate rotation of the projectile.

In order to circumvent the drawbacks described above, a second set-up was devised where the tube was struck by a polycarbonate projectile with a total mass of 17.45 g consisting of a main body and a frontal section with a cylindrical impact surface of 3.97 mm ( ~ in) radius. The two parts were joined by a commercial epoxy; Fig. 3 depicts

Fig. 3. Polycarbonate (Lexan) striker with a ~ in. Diameter cylindrical frontal surface.

the unit. Such an arrangement permits the ready interchange of other types of impact geometries as well as the possibility of insertion of a forcemeasuring device between the two components. The latter was attempted without success; an accelerometer mounted on the main body of the projectile might prove a more effective technique for this type of data acquisition. The plastic striker was positioned between a series of four plastic running surfaces at a distance of about 0.15 mm by means of an air cushion provided by a reservoir maintained at 25 psi and fed to the clearance space by a series of 0.79 mm diameter holes in the running surface 8.53 mm apart. The heavier frontal part generated a frictional force measured to have at an average value of 0.07 N from photographic data when moving freely at a speed of 4.3 m/s. It is assumed that this friction force is not changed when a tube is actually struck. Forward projectile motion was produced

J. Dual, W. Goldsmith / Impact on flexible tube

by the impact on the rear surface of a blunt-faced steel cylinder fired by the pneumatic gun. It was necessary to attach a foam disk to the rear of the plastic striker in order to prevent its fracture during contact with the cylinder. A series of lines were marked on the striker tip to assist in the determination of its velocity from data provided by a framing camera. The specific shape of the striker and the four lucite supporting surfaces inhibited its rotation. However, a relatively large tolerance of 0.0150.020 in had to be provided between the striker sides and the still present P M M A screens since the latter were not totally fiat and could not be machined without impairing their transparency. The same tube size as employed in the first arrangement was also used here; however, due to the larger striker mass, the tube could not be arrested within the m a x i m u m transverse displacment bound of 11 cm above initial striker speeds of 4.5 m / s which, consequently, represents the upper velocity bound for the measurement of contact time and rebound velocity for this configuration. A lower velocity limit of about 2.5 m / s was determined for this arrangement from the degree of scatter in measurements of the contact time and rebound velocity.

4. Instrumentation and procedure The principal mechanism of data acquisitior, consisted of photographic observation of the event by means of a Photec IVA 1 framing camera that could be operated at preset intervals between 100 and 10 000 frames per second. The usual operation occurred at 6500 pps representing the m a x i m u m speed that could accommodate the use of 100 foot long rolls of Kodak Tri-X reversal 7278 black and white film with an ASA rating of 200. Actual film speed was ascertained from the exposure and precise measurement of the trace produced by a timing light generator flashing at a rate of 1 kHz. Photot Redlake Corporation, Santa Clara, CA.

209

graphic quality of the moving film was enhanced by the additional operation of a rotating prism that reduced the angular sweep per frame to 45 ° . The camera shutter ratio of 1/2.5 corresponded to a frame exposure time of 1/16 000 s at the operating speed, and its 45 mm f/2.8 lens produced a fiat field with a resolution of 68 lines/mm. Illumination was provided by a photoflood-type of lamp (Pallite VIII) consisting of eight 300 watt bulbs mounted in a circular arrangement. Since approximately 90 percent of the film is consumed during its acceleration period, the event had to be triggered with an accuracy of 5 ms. The pneumatic gun is actuated by the depression of a button which releases a firing pin corlstraining the motion of the projectile due to the action of controlled pneumatic pressure on its rearward surface. The camera is also equipped with a switch that opens an electric circuit at a specified footage of the film; this was employed to deactivate an electromagnet holding a weight above the gun button whose impact on this trigger mechanism removed the pin and initiated the bullet movement. The time delay between the signal from the camera and the impact on the target was then measured as a function of projectile type and air pressure for a fixed drop height of the weight and voltage of the magnet. An approximate relation was established between the chamber pressure and the striker impact velocity; a more exact value was established from the photographic data. Frame-by-frame analysis was performed by means of a Vanguard motion picture analyzer providing a magnification of about 18. Any desired position on a particular frame was digitized and recorded by alignment of a cursor and activation of a button. Each frame exhibited a total of 7800 and 5500 screen units in the x- and y-directions, respectively. For a fixed frame, the cursor could be positioned with an accuracy of 2 screen units, but this accuracy was reduced to a standard deviation of 15 such units for a comparison of ycoordinates in different frames due to inaccuracies induced by film advancement solely by means of gears, requiring resetting of the reference position

210

J. Dual, W. Goldsmith / Impact on flexible tube

for this direction. Most of the data involved only measurements in the x-direction which did not require such a time-consuming task. However, a further diminution of the measurement accuracy, whose magnitude depended on the quantity examined, was induced by shadows and fuzziness caused by rapid motion. After intermediate storage of the data on floppy disks, the files were transferred to a VAX-11/750 computer that performed the data analysis. The force transducer in the first arrangement mounted on the upper right-hand brass foil used for the screen positioning consisted of a BLH FAE-03G-12-S9L foil strain gage with a gage length of 0.79mm, a gage factor of 1.85 and a resistance of 120 ohms. It formed one leg of a Wheatstone bridge and its output was recorded on a Tektronix 5103 N oscilloscope with a differential amplifier flat to 1 MHz. The separation force was calculated from the measured strain and the crosssectional area of 0.188 mm 2 using a literature value of Young's modulus for brass of 104 GPa. For the second arrangement, contact durations and rebound velocities of the striker were also

measured directly by use of an electric circuit as indicated by the schematic presented in Fig. 4. This was necessary since film analysis posed several problems, including attainment of the stop at the end of the permitted transverse deflection by the tube and the limited film length. The circuit is closed when either a wire brush E attached to the striker touches wires A or B, or when the striker tip touches the tube D which was painted with a trace of a conducting glue. A typical record is shown in Fig. 5; irregularities in the data obtained by this device indicated the lower velocity bound for this set-up. It was found that the tube and striker separate for a finite time interval before the striker changes direction. Below an initial speed of 2.5 m/s, the random forces exerted by the screens on the striker become comparable to those produced by the tube; at a velocity of 1.1 m/s, the average total force on the striker during impact was found to be only 0.3 N and the corresponding friction force might be in excess of that previously quoted for a speed of 4.3 m/s. The stiffening of the tube as the result of the application of the conducting paint was considered to be negligible.

Tube E

C

I

B

D

.J

\ 2.94cm

147

batt~ery ~

200k~

DF

1 m~_~

Fig. 4. Schematic of the contact duration measurement. A, B: Wires, C: Striker surface, D: Tube surface, E: Brush. Contacts A - E and B - E give velocities. Contact C - D measures contact time between tube and striker.

J. Dual, W. Goldsmith / Impact on flexible tube

211

0

3

~

2

.

LI

;

A,

i

;

Time

..A~_

[ms]

Fig. 5. Typical contact duration record. A signal of about 3.6 V signifies contact.

5. Target material characteristics The vast majority of the experiments were conducted on basically thick-walled tubes with an inner diameter and a thickness of 4.76 mm (~6 in) and 1.59 m m (~6 in), respectively; a few of the tests utilized tubes with inner and outer diameters of 9.53 mm and 14.29 mm (3 in and 9 in), respectively. Variations in these dimensions were found to be within a 2 percent limit from the mean. The target material was Nalgene 8000, 2 a commercial ester plasticized compound of polyvinyl-chloride (PVC); however, its properties could not be obtained from either the manufacturer or literature sources. Thus, a variety of initial tests were conducted in order to obtain relevant material characteristics. The density of the material was measured as 1.15 x 103 k g / m 3 with a 1.5 percent margin of error, and its glass transition temperature was estimated 2 Nalge Company, a Division of Sybron Corporation, Rochester, NY.

to lie below -40°C. Although the extrusion process employed in the manufacture of the tube might result in some anisotropy, the material will be considered as isotropic in accordance with material tests. Since an examination of the specimens under a microscope with a magnification of 50 failed to reveal any discontinuities, homogeneity for wave lengths greater than 1 mm will be assumed. Its viscoelastic nature required the conduct of dynamic property tests under circumstances that would generate both large and small strains, the former occurring in the vicinity of the impact point. The material properties of the sample for large strains were found by means of creep and constant strain rate extension tests. The results of the first set are presented in Table I, with tests 1 and 2 involving the manual attachment and 3 and 4 the sudden release of 0.53 and 2.22 kg masses, respectively, straining vertically-suspended tube specimens initially loaded only by their own weight. Strains were measured from the resistance change in a Wheatstone bridge with the active element

212

J. Dual, W. Goldsmith / Impact on flexible tube

Table 1 Results of creep tests in uniaxial extension Test

m

1 2 3 4

o"

0.526 2.219 0.526 2.219

t = 1s

0.163 0.687 0.163 0.687

t = 14.7 s

t =29.4 s

e

E

e

E

e

E

3.11 15.00 * *

5.24 4.58 * *

3.92 17.30 4.32 17.40

4.16 3.97 3.77 3.95

4.00 17.80 4.40 17.80

4.08 3.86 3.70 3.86

m: applied mass [kg]. o-: engineering stress [MPa]. e: engineering strain [% ].

E: apparent Young's modulus [MPa]. *: could not be measured due to oscillations.

c o n s i s t i n g o f a 300 m m l o n g m e r c u r y - f i l l e d t u b e

f o r m e r o c c u r r e d a l m o s t e n t i r e l y w i t h i n t h e first 3

t a p e d to t h e t u b u l a r s p e c i m e n . C a l i b r a t i o n indi-

cycles, w h i l e t h a t f o r t h e l a t t e r e v o l v e d m o r e u n i -

c a t e d t h a t this d e v i c e was l i n e a r u p to at least 35

f o r m l y d u r i n g the six r e p e a t e d l o a d i n g p r o g r a m s .

percent

engineering

strain.

Unfortunately,

this

Results from small-strain experiments can be

g a g e c o u l d n o t be e m p l o y e d in the i m p a c t tests

a n a l y z e d by m e a n s o f t h e l i n e a r t h e o r y o f v i s c o e l a s -

d u e to s e p a r a t i o n o f the m e r c u r y c o l u m n u n d e r these conditions. E x t e n s i o n tests w e r e c a r r i e d o u t on an I n s t r o n

' 6 Cm~.)

m a c h i n e u s i n g 100 m m l o n g s p e c i m e n s cut f r o m a tube.

The

ultimate

strength

was

found

to

be

6.2 M P a at a strain o f a b o u t 200 p e r c e n t r e l a t i v e to the r e f e r e n c e c o n f i g u r a t i o n . S t r e s s - s t r a i n c u r v e s o b t a i n e d f o r the first a n d sixth l o a d i n g c y c l e o f t w o s p e c i m e n s t e s t e d u p to an e n g i n e e r i n g s t r a i n o f 50 p e r c e n t at t w o d i f f e r e n t rates are p r e s e n t e d in Fig. 6; b o t h h e r e a n d in T a b l e 1, the s o f t e n i n g effect o f p r e v i o u s strain

h i s t o r y can be n o t e d .

/

/

H o w e v e r , n o a d d i t i o n a l c h a n g e s a p p e a r e d to o c c u r a f t e r t h e s i x t h r e p e t i t i v e l o a d i n g . T h e v a r i a t i o n in Y o u n g ' s m o d u l u s a n d p e a k stress for this e l o n g a t i o n are i n d i c a t e d in T a b l e 2; t h e r e d u c t i o n in t h e Table 2 Results of extension tests at constant strain rate

0.586 5.84

El

O'ma x ~

E6

Oemax6

4.0 4.9

2.3 3.0

2.3 2.6

1.9 2.4

i: Engineering strain rate; e = ( l - Io)ll o [10 -2 s-t]. E,: Young's modulus at the beginning of the ith cycle [MPa]. O'ma~,: Maximum stress reached during the ith cycle at e = 0.5 [MPa].

Fig. 6. Stress-strain curves for Nalgene 8000 in uniaxial extension. The engineering values are plotted for the first (1) and the sixth (6) cycle. Solid line: Strain rate=0.0584s -t, dashed line: Strain rate = 0.00586 s -t.

J. Dual, W. Goldsmith / Impact on flexible tube

ticity [15]. An examination of the force histories in the principal experiments indicated that the behavior of the material should be examined in the frequency range up to 3000 s -t. Free torsional vibration tests, with the period recorded by a stop watch, were conducted by release of an inertial mass twisting the specimen by an initial angle of less than 2 °. Two different masses, with moments of inertia of 2.6 and 0.0917 g-m 2, respectively, were utilized on three tubular specimens with polar moments of inertia of 0.370, 2.86 and 3.36 × 1 0 - 9 m 4 and lengths ranging from 12.9 mm to 1.36 m, resulting in a frequency band from 0.4 to 9.7 s -~. An axial load induced by means of a counterweight was applied so that no net tension prevailed in the samples. The measured shear storage modulus was found to vary between 1.42 and 1.55 MPa, constant within experimental error at about 1.48 MPa, and t h e corresponding Young's modulus was computed as 4.26 to 4.65 MPa upon use of the incompressibility hypothesis. D a m p e d longitudinal vibration tests were also produced with the same arrangement using a specimen length of 1.4 m, and also by means of impact tests where a much shorter sample was sandwiched between a cylindro-conical steel head and a quartz crystal attached on its distal side to an inertial mass o f about 1 g [20, 21]. The results are presented in Table 3; since tests 2 and 3 exhibited a damping ratio of 0.3, the values of the storage Young's modulus calculated from linear theory are likely

213

to be inaccurate. In all experiments, considerable care was taken to avoid the introduction of modes of loading other than that desired. Wave propagation tests 3 conducted according to the method presented in [22] using 500 and 700 K H z pulses yielded values of the Lam6 constants of the order of GPa, but are probably inapplicable for the present, lower frequency tests. A normalized plot of Young's modulus and frequency relative to standard values of 1 MPa and 1 s -l, respectively, obtained from various tests is shown in Fig. 7.

4D+ r~ ?j,

\ ~ t~ _o

E+

3-

4-

7'-. Ct ~4-+)

I-

A i

i+ B

0-

I o

I

i

I

I

I

I

1

2

3

4

5

6

I o g ( co / LO,)

Table 3 Results of longitudinal vibration experiments Test

L

m

to

E'

b/to

1 2 3 4 5

140 0.61 0.61 0.15 0.13

1050 8.6 2.75 2.75 0.75

16.5 2600 4600 13 000 61 000

4.51 (11.0) (11.0) 22.5 114.0

0.07 0.3 0.3 0.1 0.1

Test 1 was performed with a Mercury vessel strain gage. Tests 2-5 used an X-cut quartz crystal as transducer. L: sample length [cm]. E': storage Young's m o d u l u s [MPa]. m: inertial mass [g]. b/to: d a m p i n g ratio. to: circular frequency [s-t].

Fig. 7. S u m m a r y of small strain tests on Nalgene: Normalized Young's m o d u l u s vs. normalized frequency. A: Torsional vibrations, B: Longitudinal vibration, C: Impact, D, E: Wave propagation. Reference modulus: 1 MPa. Reference frequency: 1 s -t.

6. Data evaluation The digitized position data were associated with the appropriate time as determined from the trace of the timing light. A small computer program then 3 The authors are grateful to Prof. M. King for the conduct of these tests.

J. Dual, W. Goldsmith / Impact on flexible tube

214

converted the information to physical dimensions and calculated the positions and velocities at times t = 0, 6 and 20 ms. These correspond to (i) the initial conditions, (ii) the longest time for which an empirical equation for the contact force history on the tube, based on a combination of physical assumptions and statistical data treatment and also handled by the program is valid, and (iii) the longest interval permitted by the constraint of a maximum excursion of 115 mm for any of the runs. The program also provides an analytical expression for the position and velocity of the striker, the time during which contact exists between striker and target, and the lateral deformation ratio of the tube which describes the pinching effect that is shown in Fig. 8. Finally, this code evaluates the position of the first cross-over point of the deflected tube, Xo, for comparison with the theory for the transverse impact on an infinite elastic beam [23] as Xo = 2.13 (EI/pA)~/4t 1/2

(2)

where I and p are the moment of inertia and the mass density of the member, respectively. Striker position was always ascertained by delineation of two separate points on this member. Data were recorded from the instant the striker entered the picture up to the instant of contact, and for the 6 ms values, information was gathered for 4 frames prior and 4 frames subsequent to this instant. The beginning of impact was delineated with an accuracy of 0.005 ms by extrapolation of the initial striker motion to the instant where the tip touched the tube. Striker velocity was determined from these data by a linear regression formula; its accuracy is estimated within 1 percent for the initial value, but only within 2 percent for the very low speeds at t = 20 ms. The principal objective of the tests was the delineation of the contact force history. By considering the striker as rigid, this force can be evaluated from its motion; however, a dual numerical differentiation of the measured displacement his-

Vo

r

8

a .~

Striker

X

(Ball)

f//S//Y//////A (c)

~

j

bail

block

Fig. 8. Tube pinching (a), (b) and static compression test (c). Line CABD is the intersection of the striker trajectory and the tube surface. During extreme pinching, points A and B are in contact.

tory leads to large errors. In consequence, a physically reasonable, analytically integrable force-time expression was used as a basis for a fit of the experimental displacement data, using a nonlinear regression process [24]. Since the most significant pinching occurs at the beginning of the impact, and since the accuracy of the regression at a specific point will decrease with an increasing length of the time interval, only the first 45 frames corresponding to the initial 6 ms of the impact event were utilized in the regression. During this period, no reflections from the tube ends will return to the impact point to disturb the basic phenomenon;

215

J. Dual, W. Goldsmith / Impact on flexible tube

based on bar speed, the earliest arrival of such reflected waves ensues after 20 ms. The accuracy of the stipulated histories were checked according to the following criteria: (i) the standard deviation inside the interval, (ii) satisfaction of the boundary conditions, based on the impulse-momentum equation evaluated between initial contact and t = 6 ms, and (iii) the number of coefficients required to obtain a satisfactory standard deviation of no more than 0.2 mm for the matched positions. The greater the number of parameters required, the less significant will be their values. The initial use of polynomial regressions demonstrated their deficiencies for the portrayal of the present force curves because of their failure to meet the criteria indicated above. In consequence, a nonlinear regression process was utilized; since no information for this procedure had been considered by other investigators for depicting impact processes, no hint as to the most appropriate nature of such a relation was available. Three different functions were utilized, based on force histories given by F , ( t ) = C~ e x p ( - C 2 t ) ,

(3a)

F2( t) = C~ t e x p ( - C 2 t ) ,

(3b)

F3(t) = C i t 2 e x p ( - C E t ) .

(3c)

All three exhibit physically reasonable behavior based on observations from the results of force history development using polynomial regressions, and are readily integrable, requiring the determination of a total of five constants in the displacement expressions. These were determined by a least mean square method using the values of position and velocity at the ends of the interval as constraints. The weighting functions were changed until the regression yielded sufficient accuracy for the boundary values within their previously determined standard deviations. The constant C was added to allow for the possibility of a constant force during part or all of the initial 6 ms interval. Figure 9 presents the fitted and measured data points for two of the runs. On the average, 90 data

8

.o. o~ 8

o. 4

2--

e

•

I

I

I

L

I

I

2

3

4

S

TImQ Cm=]

Fig. 9. Comparison of measured data (triangles) and fitted curves (solid and dashed lines). A: Run 12--Dashed line, eq. (3b), solid line, eq. (3a). B: Run 42--Solid line, eq. (3b).

points were collected for each run, and the position on every frame was measured twice, the two sets being in excellent accord as indicated by the coincidence of the sets of triangles representing the two. On the other hand, while the curve fitted to eq. (3b) employed for Run 42 is in very good accord with the experimental data, eq. (3a) used for Run 12 shows some deviations with the measured values, and these would be even greater if eqs. (3b) or (3c) were to be used. The five nonlinear equations for the constants Cj were solved by an iteration process. No problems of convergence were encountered with the results from the first experimental arrangement; however, a few of the runs from the second set-up exhibited such problems due to the lower change in the striker velocity as the result of the impact, but only when a regression of degree 5 was utilized. This problem was solved by suitably constraining the value of the initial velocity of the regression in an absolute sense [24].

216

J. Dual, W. Goldsmith / Impact on flexible tube

The d i s t a n c e C D s h o w n in Fig. 8 was c o n s i d e r e d to be the q u a n t i t a t i v e m e a s u r e o f tube p i n c h i n g in the i m p a c t region; c h a n g e s in this d i m e n s i o n were o b s e r v e d o n l y d u r i n g the first 45 frames after impact. This d i s t a n c e d i v i d e d by the initial t u b e thickness is defined as the d e f o r m a t i o n ratio a ; this t h i c k n e s s was m e a s u r e d on the film to i m p r o v e c o n s i s t e n c y o f the results a n d a t h r e e - p o i n t s m o o t h i n g t e c h n i q u e was a p p l i e d at all times o t h e r t h a n the initial i m p a c t to m i n i m i z e errors d u e to p o o r visibility. The errors in the value o f a are e s t i m a t e d as less t h a n +0.04 c o r r e s p o n d i n g to an error in the c o o r d i n a t e o f 0.3 mm. If the wall thickness r e m a i n s constant, a value a = 0.4 r e p r e s e n t s the s u p e r p o s i t i o n o f p o i n t s A a n d B. A c o r r e s p o n d ing static p i n c h i n g e x p e r i m e n t was also p e r f o r m e d with the L e x a n striker. This resulted in a h y p e r b o l i c d e c r e a s e f r o m an initial value a = 1 at zero force to a value o f a = 0.73 for a force o f 5.34 N, f o l l o w e d b y a l i n e a r d e c r e a s e to a value o f a = 0 . 4 8 at a force level o f 12.5 N. The p o s i t i o n Xo r e p r e s e n t i n g the first intersection o f the d e f o r m e d tube centerline b e y o n d the i m p a c t p o i n t with its initial configuration c o u l d be m e a s u r e d , with s o m e difficulty, only for the first 20 frames; b e y o n d , this p o i n t m o v e d o u t s i d e the field o f view. A linear fit for the d i s p l a c e m e n t history p r o v i d e s an average velocity for the m o t i o n o f this point, while a fit a c c o r d i n g to eq. (2) allows the c o m p u t a t i o n o f an a p p a r e n t b e a m c o n s t a n t a n d storage m o d u l u s E ' .

7. R e s u l t s and d i s c u s s i o n

A t y p i c a l p h o t o g r a p h i c s e q u e n c e at a f r a m i n g rate o f 6500 p p s s h o w i n g the i m p a c t o f the steel s p h e r e on the tube at a s p e e d o f 40.4 m / s is p r e s e n ted in Fig. 10. This s e q u e n c e consecutively exhibits frames 0 - 6 , 8, 10, a n d 12. In view o f the i m p a c t p o s i t i o n , 1 cm b e l o w the u p p e r screen edges, at least o n e - h a l f o f the t u b e s h o u l d not be d i s t u r b e d by the e v e n t u a l c o n t a c t o f t u b e a n d screens at l o c a t i o n s o t h e r t h a n the i m p a c t point. N e v e r t h e less, in a n u m b e r o f runs, a t o r s i o n a l m o t i o n was

o b s e r v e d with the t u b e rotating a b o u t its axis, p r o b a b l y c a u s e d b y o n e - s i d e d t o u c h i n g o f the screens. T h e forces a s s o c i a t e d with this c o n t a c t were m e a s u r e d by m e a n s o f strain gages [24]. Figure 11 p o r t r a y s the first 15 frames at a rate o f 6500 p p s o f an i m p a c t w h e r e the p l a s t i c striker with a c y l i n d r i c a l c o n t a c t surface i m p i n g e d on the target at a s p e e d o f 4.95 m / s . It m a y be n o t e d that the tube d e f o r m a t i o n is m u c h s m a l l e r for the corres p o n d i n g t i m e interval t h a n in the case o f steels p h e r e contact. A s u m m a r y o f the results o b t a i n e d from p h o t o g r a p h i c a n a l y s i s is p r e s e n t e d in T a b l e 4; a n u m b e r o f tests, p r i m a r i l y t h o s e involving the first experim e n t a l a r r a n g e m e n t were not r e d u c e d b e c a u s e i m p a c t c o n d i t i o n s o t h e r t h a n those d e s i r e d were f o u n d to have p r e v a i l e d . Runs 59 a n d 60 were e x e c u t e d with t u b e prestresses o f 56 a n d 98 kPa, respectively. T h e a v e r a g e forces e x e r t e d on the striker d u r i n g the time intervals from 0 to 6 ms, a n d from this instant to 20 ms, respectively, are

Table 4 Summary of the runs analyzed Run No.

Experimental arrangement

Uo

12

1

67.6

13 14 24 41 42 45 51 54 55 56 57 58 59 60

1 1 1 2 2 2 2 2 2 2 2 2 2 2

66.8 52.7 40.4 4.82 4.95 4.32 1.13 4.66 7.46 7.65 9.83 9.54 4.82 4.93

tro

0

0 0 0 0 0 -0 0 0 0 0 0 56 98

Arrangements No. 1 and No. 2 involve the use of a ball (m = 1.052 g) and a striker with a cylindrical contact surface (m = 17.45 g), respectively. Vo: initial velocity [m/s] %: prestress [kPa] Run 45 was done without a tube to obtain the friction exerted on the striker.

J. Dual, W. Goldsmith / Impact on .flexible tube

217

4

,4

%

Fig. 10. Framing sequence for a steel ball impact at 40 m / s , Run 24. Framing rate: 6500 pps. Frames shown are 0-6, 8, 10, 12.

indicated in Fig. 12 for both experimental arrangements. The results differ sharply for the second time span where there is virtually no force acting on the steel sphere. The force histories obtained by nonlinear regression and the corresponding lateral deformation ratios a of the tube are presented for a sequence of runs in Figs. 13 and 14 for the first experimental set-up and in Figs. 15 and 16 for the second. The first pair represent regressions of the

type given by eqs. (3a) and (3b), respectively. The first relation provides lower residuals, whereas eq. (3b) is physically more acceptable as it avoids a discontinuous jump in the contact force at time t--0. However, the actual rise time appears to be too fast for this type of regression. The results corresponding to eq. (3c) which has been suggested as a suitable empirical equation for the perforation of metallic plates by sharp-pointed projectiles [25], is even less satisfactory and will

218

J. Dual, W. Goldsmith / Impact on flexible tube

..-__

Fig. 11. Framing sequence for the impact of the polycarbonate projectile with a cylindrical contact surface striking at 4.95 m / s , Run 42. Framing rate: 6500 pps. Frames shown are 0-14.

not be presented. The smaller the initial velocity, the more accurate is the use of an exponential forcing function for a description of the phenomenon. Only regressions using eq. (3b) were used in the data evaluation for the second arrangement because no convergence could be obtained with eq. (3a) and the standard deviation for eq. (3c) was again substantially greater. However, the

results for this case exhibited much smaller standard deviations than for the first set-up when the same relation was employed, albeit being numerically less stable, which is attributed to the lesser clarity of the films that prevented, in some instances, the satisfaction of all boundary conditions. As shown in Figs. 15 and 16, the peak force increases and the rise time decreases both with initial velocity and prestress. The more rapid force

J. Dual, W. Goldsmith / Impact on flexible tube

219

- -

I"

+

18--

.....

+ 4"

12 13 14 24

Iv o vo vo v o

= 67.6 = 66.g = 52.7 = 40.4

m/s) m/s) m/s) m/S)

~_ ~0~ t_l

5--

b _'-~

Run Run Run Run

+

I ~

40

~J u k. 0 b..

I

S0

60

Veloci £y Cm/~] SO ~'~ a) S e t - u P

?,

No.1

.',,~

+

lw

0

+

1

2

I..

I

t

3

4

5

7.54÷

5.0-

1

¢ 4"

2.S+

.

I 2

1 4 Veloel£y

W

I 6

VV

I 8 Em/E]

2 e 50 <0 0 2--

o

L

10

I

20

I

I

30

40

Time

b) 5 e t - u p

I

50

Ema]

No.7

Fig. 12. Average force exerted on the striker for two time intervals for both the first and the second experimental arrangement. ( + ) : Time interval 0-6 ms, Triangle: Time interval 6-20 ms, Diamond: Relate to runs with initial prestress.

decay calculated for the first arrangement is in qualitative agreement with the predictions of a linear string model, eq. (1), where the decay constant is inversely proportional to striker mass. If the tube can be regarded as a taut extensible string subjected only to small strains, the maximum tension occurring during the impact can be approximated from eq. (1). Since the measured properties of the material show that the longitudinal wave travels more rapidly than the transverse one, it may be assumed that this tension is nearly constant for the time interval under consideration as long as transverse displacements remain small. The impact force is then given by F = 2 Tvo/c. Tension, string wave speed, and peak

Fig. 13. Force history and lateral deformation for the first experimental arrangement using a regression for eq. (3a).

force obtained in this manner are presented in Table 5 together with measured values of the peak and average force during the initial phase of the impact process. Although grossly oversimplified, this model should provide at least an order of magnitude for the forces generated. The pinching of the tube, portrayed in Figs. 13-16, can be represented for the two arrangements by a linear relation between the minimum deformation ratio atom and the initial velocity for the limited speed range examined here. The corresponding equations are: Set-up No. 1 ami, = 0.63 -0.0046Vo, Set-up No. 2 amin =0.98-0.02100,

40< Vo< 70 m/s,

(4)

0 < Vo< 10 m/s.

220

J. Dual, W. Goldsmith / Impact on flexible tube

Table 5 Comparison of peak forces obtained with eq. (1) and measured values for the peak and the average force during the initial phase of the impact

Run 51 30 -•.

Run

T

c

2T/mc #

Pm:~

12 24 51 42 56 58

99.7 50.2 0.42 3.05 5.46 7.33

52.3 37.1 3.42 9.14 12.22 14.17

3624 255 10.9 2572 109 6.4 14.1 0.28 0.91 38.2 3.3 3.7 51.2 6.8 5.8 59.3 9.9 8.9

139 79 2.94 8.5 12.8 20.7

u L 0

o ~ I

I .0~

~

(v o = 67.6 m/s) !v o 66.9 m/s) !v o 52.7 m/s) tv o a O . a m/s)

m/s~

2

i

)

)

3

4

s

~

.......................

~06-04-02-O

100

u

9.54

\

o.8 ~ • . .

-----

7.46 m/s

)

2~ --'".

10 - I

12 13 14 2a

= 1.13 m/a) )

#',-....

T: tension in the tube [N]. c: string wave speed [m/s]. 2T/mc: exponential decay constant of the impact force in the taut string model Is-t]. •~: peak force obtained using eq. (1) [N]. Pme~: average force measured IN]. ?~m=a,:peak force measured [N I.

Run Run Run Run

(v

Run 55 Run 58

I

I

1

I

1.0

2.0

3.0

4.0

Time

h

I 5.0

[m~]

I.

b_

Fig. 15. Force history and lateral deformation for the second experimental arrangement using a regression for eq. (3b). The influence of the initial velocity is clearly indicated.

50

~

I

L

I

2

3

4

I 1.0

1 2.0

I 3.0

8

I S

1.0 .~0.8 .__&0 . 6 44:0. 4

0.2 0

Time

I 4 0

I 5.0

[mm]

Fig. 14. Force history and lateral deformation for the first experimental arrangement using a regression for eq. (3b).

In the first case, the t u b e closes t o t a l l y at high velocities a n d d o e s not r e o p e n s i m u l t a n e o u s l y with force r e m o v a l ; in the s e c o n d a r r a n g e m e n t , the p i n c h i n g is m o r e g r a d u a l . This difference in b e h a v i o r is d u e b o t h to the difference in striker c o n f i g u r a t i o n s a n d the initial velocities utilized, since the t u b e wave s p e e d is p r o b a b l y l o w e r t h a n the i m p a c t velocity e m p l o y e d in the first a n d h i g h e r than that u s e d in the s e c o n d set-up. A l t h o u g h n o t a n a l y t i c a l l y verified, this p r o b a b l y signifies that, for low i m p a c t s p e e d s , the distal t u b e side is accele r a t e d b y stress waves, w h e r e a s in the h i g h e r initial velocity d o m a i n , p h y s i c a l c o n t a c t b e t w e e n p o i n t s A a n d B in Fig, 8 occurs first. T h e v a r i a t i o n in the t u b e w a v e s p e e d as a result o f s t r a i n - r a t e effects p r o b a b l y also c o n t r i b u t e s to this p h e n o m e n o n .

J. Dual, W. Goldsmith / Impact on flexible tube

3l~-.....

Run Run Run Run

al 42 5g 60

(v = a.82 m/s) ( v ) = 4.g5 m/s) (v: 4.82 m/s) (v) 4.g3 m/s)

t-~ Z u @

uL 0

20--

b_

I 0 -/-

221

ments and strains are invalid. Hence, the increase in the apparent modulus up to values of 15 MPa for the second arrangement for velocities below 10m/s, and up to values of about 3 0 M P a for set-up No. 1 at speeds of about 7 0 m / s is not unreasonable. Thus, because all force curves exhibit about the same frequency content, the material can be considered to a first approximation as strain-rate independent in the velocity range employed, at least within the domain of small tube strains.

8. Conclusions

o

i

I

I

I

i

2

3

4

1 0~.~.~ . . . . . . . . . . . .

I

---.~

.-- __~ . . . . .

co0 8 - o-06_ 0 4-0 2--

I 1.0

2 0

3.0 Time

I

I

40

5O

[ms]

Fig. 16. Force history and lateral deformation for the second experimental arrangement using a regression for eq. (3b). This diagram illustrates the etiect of prestress: Run 59, 56kPa; Run 60, 98 kPa.

The average propagation speeds of the point of zero lateral deflection nearest to the impact position, Xo, for the two different experimental arrangements, amounting to about 21 m/s, are relatively close. An additional applied prestress significantly increases this speed. The fit according to eq. (2) exhibited considerably lower standard deviations than the approximation using a constant propagation speed. As expected, the value of the corresponding apparent (storage) modulus E' for the run with the lowest impact velocity, 1.13 m/s provided a value of about 5 MPa, closest to that of 4.5 MPa obtained from the material property tests in the low frequency range. At higher speeds, the assumptions in the theory [23] of small displace-

Lateral impact experiments with two types of strikers were conducted against mildly viscoelastic tubes composed of a commercial PVC substance, Nalgene 8000. The two strikers employed consisted of a 1 g, 6,35 mm diameter steel sphere and a 17 g polycarbonate block with a cylindrical contact surface fired pneumatically within the ranges of 4070 m/s and up to 10 m/s, respectively. A series of small- and large-strain dynamic tests were conducted to ascertain the characteristics of the material which was found to be homogeneous, isotropic and incompressible. Its tubular form exhibited a Young's modulus of 4.5 MPa. The density of the material was found to be 1.15 x 103 kg/m 3. No significant viscoelastic effects of the material were found during the conduct of the tests. The primary experiments consisted of the transverse impingement of the strikers against a 1.3 m long vertically-suspended tube with inside and outside diameters of 4.76 and 7.94 mm, respectively; some specimens were placed in a condition of prestress. Striker and target motion were obtained from the film of a framing camera operating at a speed of 6500 pps. Nonlinear regression methods were employed to determine the motion of the strikers and tubes, the pinching effect on the tube, and the history of the force acting on the striker. The tube was closed by the impact of the sphere travelling at its lowest speed and did not immediately reopen after cessation of contact. On the other

222

J. Dual, W. Goldsmith / Impact on flexible tube

h a n d , the plastic striker p r o d u c e d a m u c h lower p i n c h i n g effect. In the first case, where the tube particles travel faster t h a n its wave speed, the tube d e f o r m a t i o n occurs primarily as the result of the forward m o t i o n of the striker, whereas in the second case, the d e f o r m a t i o n is p r i n c i p a l l y the result of stress wave p r o p a g a t i o n . These conclusions are reinforced by the more a b r u p t rise of the force curve for the smaller, faster striker, compared to the more massive P M M A block for force histories calculated from the data on the basis of satisfaction of the b o u n d a r y c o n d i t i o n s a n d m i n i m u m residual in the n o n l i n e a r regression analysis. C o m p a r i s o n

with

presently available

e l e m e n t a r y theories for a c o r r e s p o n d i n g idealized system subject to small strains a n d d e f o r m a t i o n s do not provide good c o r r e s p o n d e n c e , primarily due to the failure of the test c o n d i t i o n s to meet the k i n e m a t i c hypotheses. F u r t h e r experiments involving different tube materials a n d diameters, different ratios of striker to tube mass a n d i n c l u s i o n of liquids within the c o n t a i n e r need be c o n d u c t e d to further explore this type o f p h e n o m e n o n . In addition, a theoretical investigation of the p r o b l e m should be initiated; however, the extremely high degree of analytical complexity i n h e r e n t in the p r o b l e m will r e n d e r a solution other than a completely n u m e r i c a l procedure quite difficult.

Acknowledgment The authors w o u l d like to express their appreciation to Mr. S t e p h e n Virostek for i n v a l u a b l e assistance d u r i n g the c o n d u c t of the investigation. The first a u t h o r was partly s u p p o r t e d by a Fulbright grant, a n d the work was performed u n d e r the s p o n s o r s h i p of the N a t i o n a l Institutes of Health by m e a n s of G r a n t No. 5RO1AM18184.

References [1] N. Cristescu, Dynamic Plasticity, North-Holland, Amsterdam (1967) 181-291.

[2] H.A. Rakhmatulin, "Oblique impact upon an elastic fibre with large velocity in the presence of friction", Prikl. Mat. Mekh. 9, 449-462 (1945). [3] E.V. Riabova, "Normal impact with varying velocity upon a flexible fiber", Moskva Universitet, Vestnik IO, 85-91 (1953). [4] F.O. Ringleb, "Motion and stress of an elastic cable due to impact", 3. Appl. Mech. 24, 417-425 (1957). [5] D.-N. Fan and J.F. McGarvey, "Nonlinear stress waves in a perfectly flexible string", 3". Astron. Sci. X X V , 20-33 (1977). [6] D.R. Petterson et al., "Dynamic distribution of strain in textile materials under high speed impact", Textile Res. J. 30, 411-431 (1960). [7] J.W. Kamman and R.L. Huston, "Modelling of submerged cable dynamics", ONR Technical Report, Contract No. N00014-76C-0139, 1983.. [8] F. Katsamanis and W. Goldsmith, "Transverse impact on fluid-filled cylindrical tubes", J. Appl. Mech. 49, 149-156 (1982). [9] F. Katsamanis and W. Goldsmith, "Fluid effects and response in transverse impact on liquid-filled tubes", Exp. Mech. 22, 245-255 (1982). [10] A.C. Eringen, "On the nonlinear vibrations of elastic bars", Q. Appl. Math., Ser. 2, 9, 361-369 (1952). [11] S.S. Antman, "The theory of rods", in: C. Truesdell, ed., Handbuch der Physik VI a/2, Springer, Berlin (1972). [12] P.M. Naghdi, "The theory of shells and plates", in : C. Truesdell, ed., Handbuch der Physik VI a/2, Springer, Berlin (1972). [13] P.M. Naghdi, "Finite deformations of elastic rods and shells", Proc. I U T A M Symposium on Finite Elasticity, The Hague, 47-103 (1982). [14] P.M. Naghdi, "On the formulation of contact problems of plates and shells", J. Elasticity 5, 379-398 (1975). [15] R.M. Christensen, Theory of Viscoelasticity, Academic Press, New York (1971). [16] R. Skalak, "An extension of the theory of water hammer", Technical Report No. 15, Dept. of Civil Eng. and Eng. Mech., Columbia University, New York, AFSWP No. 922, N-ONR-266(08), 1955. [ 17] R.J. Tait and T.B. Moodie, "'Wavesin nonlinearfluid-filled tubes", Waoe Motion 6, 197-203 (1984). [18] G.W. Housner, "Bending vibrations of a pipeline containing flowing fluid", J. Appl. Mech. 19, 205-208 (1952). [ 19] T.B. Benjamin, "'Dynamics of a system of articulated pipes conveying fluid--I. Theory", Proc. Roy. Soc. London, Ser. A 261, 457-486 (1961). [20] J. Liss and W. Goldsmith, "Plate perforation phenomena due to normal impact by blunt cylinders", Internat. J. Impact Engrg. 2, 37-64 (1984). [21] N. Levy and W. Goldsmith, "Normal impact and perforation of thin plates by hemispherically-tipped projectiles, Ii: Experimental results", Internat. J. Impact Engrg. 2, 299-324 (1984). [22] M.S. King, "'Static and dynamic elastic properties of rocks from the Canadian shield", Internat. J. Rock Mech. Min. Sci. S Geomech. Abstr. 20, 237-241 (1983).

J. Dual, W. Goldsmith / Impact on flexible tube [23] P.E. Duwez, D.S. Clark and H.F. Bohnenblust, "The behavior of long beams under impact loading", J. Appl. Mech. 17, 27-34 (1950). [24] J. Dual,"Lateral impact on a flexible, viscoelastic tube with

223

resultant large deformations", Thesis (M.S.), University of California, Berkeley, 1984. [25] A.V. Masket, "The measurement of forces resisting armor penetration", J. AppL Phys. 20, 132-140 (1949).