Int J Impact Enqnq Vol 12, No I, pp 21 36, 1992 0734 743X,92 $ 5 0 0 + 0 0 0 Printed m Great Britain ( 1992 Pergamon Press plc

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IMPACT RESPONSE OF A STRING-ON-PLASTIC FOUNDATION

TOMASZ WIERZBICKI a n d MICHELLE S. HOO FATT

Department of Ocean Engineering, Room 5-218, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A

(Received 8 February 1991, and in revlsed fi~rm 17 September 1991 )

Summary--A complete analysm of the large deformation response of a plastic string resting on a plastic foundation and undergoing localized mass impact, is presented. The formulation is based on the conservation of linear momentum and the kinematic and dynamic continmty conditions on a moving wave front. A closed-form solution ~s obtained for the instantaneous velocity and deflection profiles as well as for the final permanent deformed shape of the string and the magmtude of the maximum central deflection. An approximate expression for the central deflection amplitude ~s also derived using the solution of the corresponding static problem. Finally, the string-on-foundation problem is related to the large deflection dynamic plastic response of a cylindrical shell subjected to mass ~mpact. A parametric study ~s performed to assess the effect of the impacting mass, the impact velocity, and the shell parameters on the magnitude of the permanent displacements and deformed shape of the shell.

N O T A T I O N

plastic wave speed shell thickness normalized kinetic energy mass density per unit length equivalent ring resistance arc length time response time normalized time axial displacement transverse velocity impact velocity transverse displacement final displacement profile axial coordinate normalized axial coordinate external work internal energy dissipation applied axial force impacting mass bending moment per unit length fully plastic bending moment axial force per unit length fully plastic axial force equivalent shear force equivalent plastic axial resistance plastic tensile force in string load amplitude concentrated shear force shell radius cylindrical surface cyhndrical coordinate system central deflection final central deflection strain mass ratio circumferential coordinate shell curvature ratio of mass of deformed string to impacting mass

21

22 T. WIERZBICKI and M. S. H o o F A I r

v velocity ra t io

extent of plast ic zone ~r final extent of p lasnc zone

p mass densi ty

a stress

a o flow stress of mater ia l H( ) Heavis ide function

61 ) Dlrac del ta function

d ( X ) X.t

dt

d 2 ( X ) X .tf

dr 2

d { X ) X.,,

dx

d2(X) X .xx

dx 2

d2(X) X ,m

dt dx

] j u m p across wave front

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

The dynamics of thin elastic and plastic strings subjected to localized transverse loads has been studied by many investigators since the beginning of World War II. The first solution of an infinitely long elastic wire subjected to constant velocity resulting from a transverse impact was presented in 1942 by Housner (see ref. [1] ). The motivation for this research came from the practical need for describing an aircraft impact on a balloon barrage cable. Using conservation of linear momentum in the axial and transverse directions, Housner derived simple expressions for the strain in the cable and the contact force between the cable and the aircraft wing. Housner's solution was extended in 1945 by Rakhmatulin [2] to cover impact on strings made of an elastic-plastic rate independent material. The mechanics of wave propagation in extensible strings is discussed in great detail by Cristescu i-3].

Interest in one-dimensional transverse wave propagation in plastic beam/strings has been recently renewed because such structures can model the dynamic behavior of a cylindrical shell when the plastic foundation is added as a deformation resisting mechanism. This analogy was used by Calladine [4] to analyse local unsymmetric deformations of a statically loaded long cylindrical shell. It was argued that in the case of local indentation, the dominant deformation mechanisms are axial stretching of generators and circumferential bending of rings. In the string-on-foundation analogy these two mechanisms are present in the form of plastic axial resistance and foundation pressure, respectively. Applications of this to the solution of static pinching and local indentation of plastic cylinders were given by Reid [5] and Wierzbicki and Suh [6]. A more rational procedure of deriving equivalence parameters relating the strength of a cylindrical shell and a string on a plastic foundation was developed by Hoo Fatt and Wierzbicki [7]. Yu and Stronge 1-8] explored the same analogy in the dynamic problem of a localized mass impact. They derived a numerical solution valid in the range of deflections for which both bending and axial resistance are important.

This paper carries the ideas of Yu and Stronge into still larger deflections where the bending resistance of the beam becomes insignificant compared to axial resistance. In this case, considerable simplifications has been achieved in terms of the mechanics of the wave propagation process. The impact problem will be formulated in the next few sections where the concept of propagating "extensional" hinges is central to the solution methodology. A closed-form solution is then derived for the transient wave profiles, permanent deflection profiles, and maximum attainable displacements of the string under the impacting mass.

Impact response of a strmg-on-plastic foundation 23

The solution is related to the damage sustained by a plastic cylindrical shell locally deformed by an impacting mass. Finally, an approximate solution based on a corresponding static analysis of the shell is compared to the dynamic solution.

2. P R O B L E M F O R M U L A T I O N

Consider a very long but finite plastic string resting on an ideal plastic foundation with resistance ?/as shown in Fig. 1. The string has a mass per unit length ff~ and is capable of producing an ideal plastic axial tensile force of N when it undergoes impact by a rigid mass M o, moving at a velocity V o.

The impact of the rigid mass on the plastic string gives rise to transverse disturbances that propagate away from the impact point with a constant wave speed c 2 = ao/p, where a o is the material flow stress and p is the mass density of the string. At the wave front the velocities and accelerations suffer jumps while the transverse deflections are continuous. It is assumed that active plastic deformations are concentrated at the front of a wave of strong discontinuity or at a propagating extensional "hinge". Behind the wave front the already deformed part of the string is assumed to move as a rigid body. In other words, stretching deformation instantaneously occurs in the string on crossing the wave front. This model is developed on the premise that the most important information about the solution is carried by the wave front. Accordingly, the solution is constructed in a form that satisfies kinematic and dynamic continuity conditions on the wave front. In addition, the solution is required to satisfy equilibrium in a global sense. Satisfying global equilibrium entails integration of the equation of motion over the entire length of the deforming region. Global equilibrium is believed to capture the physics involved during the impact while preserving the appealing simplicity of a closed-form solution. A solution that satisfies local equilibrium at every material point of the string for any time will be derived in future research [9] .

C • = C

FIG. 1. Parameters characterizing the impact problem of a string-on-plastic foundation.

24 T. WIERZBtCKI and M. S. Hoo FATT

3. ANALYSIS OF DISCONTINUITIES

The conditions on moving discontinuities in rigid plastic beams and plates characterized by bending and shear resistance were first formulated by Hopkins and Prager [10,11]. However, in the case of a string with only axial resistance, the condition of kinematic continuity takes the following form:

w.,] + ~.,w.x] = 0 ( 1 )

where ¢.t = d~/dt is the velocity of the moving discontinuity and the symbol ] denotes a jump of a given quantity across the wave front; for example, A] = A + - A- . Following Eqn ( 1 ), two situations may arise. At a stationary extensional "hinge," ~.t = 0 and therefore, w.,] = 0. At a moving extensional "hinge," which is considered in the present theory, ¢,t :/: 0 and both velocity and slope can suffer a jump:

w x] :~ 0 (2)

w.,] ~ 0 (3)

on the moving wave front. However, the magnitude of the jumps must be such that Eqn ( 1 ) is always satisfied.

In addition to the condition of kinematic continuity, momentum must also be conserved at the moving wave front. In one-dimensional wave propagation in bars, this requirement is expressed via the condition of dynamic continuity:

~ ] = pC,iv] (4)

where a is the stress in the bar, p is the mass density, and v is the particle velocity. In plastic string problems, a counterpart of the axial stress is the shear force. The corresponding velocity is then the transverse velocity of the string :

¢r ~ - R w , x (5)

v ~ w., (6)

where ~7 is the plastic tensile force in the string. The condition of dynamic continuity for the string problem can then be re-written in the form:

Nw.x] + rh#.tw.t] = 0 (7)

where rh is the mass density of the string per unit length. Here, the assumption that transverse deflections are moderately large has been made, w.~ = tan 0 ~ sin 0.

By comparing Eqns (1) and (7), we find that the velocity of the wave propagation in the plastic string is constant:

~., = c= J ~ (8)

and independent of the foundation parameter ~. In the analysis of beams, including Yu and Stronge's analysis [8] of the beam/string, the velocity of the moving disturbance ~.t was variable and was considered to be as an unknown parameter of the solution. By contrast, in the present formulation, the velocity of the transverse wave is known and constant.

4. ENERGY DISSIPATION

The rate of internal energy dissipation consists of the energy dissipated by the string and by the foundation. Consider half of the length of the string. The rate of internal energy dissipation is defined as:

L L F.i~t-= l~i(x,t)dx + gl~(x,t)dx (9)

Impact response of a string-on-plastic foundation 25

where the strain rate ~ in the material description is given by:

~ ( x , t ) = w .xw x ,. (10)

Note that in this expression for the strain rate, the contribution of the axial displacement is disregarded. The assumption made in Section 2 that the string moves as a rigid body behind the wave front implies that i = 0. Furthermore, slopes are zero ahead of the wave front so that w..~] = W.x, and once the wave front passes a particular point, the slope never vanishes but remains at a constant value. Substituting these relations into Eqn (10), it may be shown that w x, = 0 behind the wave front. Therefore, a form of the velocity field that is constant behind and suffers a jump on the wave front is assumed:

w , ] = w.,(x, t ) [ 1 - H ( x - ~)] ( l l )

where H is a Heaviside function and w ( x , t ) is a continuous function. A gradient of the velocity field is:

w.,x = W.,x[1 - H ( x - ¢)] - w . , 6 ( x - ~) (12)

where 6 is a Dirac delta function. Substituting Eqn (12) into Eqn (9) yields the following :

fo ;o F",,t = N w x , w . x d x -- Nw.~w.,~l~= ~ + ctw, d x . (13)

The first term in this equation represents the continuous energy dissipation in the string and the second term, the discontinuous part of the moving wave front. In particular, for a rectangular velocity:

w , = w.° , ( t ) [ 1 - H ( x - ¢ ) ] . ( 1 4 )

The spatial gradient of the velocity field vanishes within the region 0 < x < ¢ and the rate of energy dissipation reduces to the following:

E i o , = -Nw.xW.,lx=~ + Ow.,~. (15)

The first term represents the product of a plastic resisting force due to instantaneous rotations Q = ~Twx and the corresponding transverse velocity w.tl~=~. The incoming material elements are subjected to instantaneous stretching at x = ¢. Behind the wave front, no active plastic flow takes place and the deformed string moves as a rigid body. Such discontinuous deformation fields are commonly accepted in the theory of rigid-plastic beams where the energy is usually dissipated at the stationary or moving plastic hinges. By analogy, the discontinuous plastic deformation in strings will be referred to as a propagating "extensional hinge." It should be mentioned that the famous Taylor and Wiffin [12] solution for a right circular cylinder impacting a rigid target involves the assumption that the material is rigid-plastic and all deformations are concentrated at the moving wave front.

5. EQUILIBRIUM

The global equations of equilibrium of a string can be expressed through the conservation of linear momentum or the principal of virtual velocities. Both formulations are equivalent but we have chosen the momentum conservation approach to retain simplicity of the original Housner's solution for the elastic string.

The mechanical properties of the string are described by the plastic tensile force /q, foundation constant g/, and mass density per unit length fit. At x = 0 and t = 0, the string is impacted by a rigid mass Mo moving with velocity Vo. After impact the mass maintains contact with the string until the velocity of the system is brought to rest.

An instantaneous displacement and velocity profile of the string is shown in Fig. 1. The plastic wave front, traveling with the constant velocity c would move a distance ~ = ct on both sides of the impact point. The total instantaneous linear momentum of the string

26 T. WIERZBICKI and M. S. Hoo FATT

consists of the momentum of the rigid mass and the momentum of the moving part of the string. For equilibrium, the rate of change of the momentum must be equal to the resisting force. In the present model the resisting force is provided by the foundation. Hence:

d l ; 2 " ' 1 f J M o v + 2 r h v ( x , t ) d x = - 2 0dx (16)

where v = w, is the downward velocity. Because the upper limit of the integral is time-dependent, the time differentiation of the integral gives rise to two terms :

I,) d~ vhvlx = < = 2 0 dx. ( 17 ) Mov., + 2 rhv., dx + 2 d t

According to the discussion in the preceding section, it is assumed that the deformed region of the string moves as a rigid body so that:

v(x , t) = v( t ) . (18)

The integration with respect to x in Eqn (16) can now be readily performed to give:

[ m o + 2rh~]v., + 2crhv = -2~g/. (19)

An alternative interpretation of Eqn (19) can be given by introducing the condition of dynamic continuity, Eqn (7), and the expression for the rate of internal energy dissipation, Eqn ( 15 ). After some re-arrangement :

-2Nwxlx=~ + 2g/~ = - [ M o + 2r~]v. , . (20)

The left-hand side of this equation represents the plastic resisting forces due to axial stretching at the extensional hinge and compression of the plastic foundation. The right-hand side is the d'Alembert inertial force of the system.

Equation (19) furnishes a linear, first order ordinary differential equation for v, which can be re-arranged in standard form:

2rhc 2ct?t v t + v - ( 2 1 )

Mo + 2rhct Mo + 2rhct

subject to the initial condition: v(0) = V°. (22)

The solution of the string impact problem resulting from global equilibrium should be considered as an approximation to the exact solution of the wave equation.

6. SOLUTION FOR VELOCITIES

The solution of the initial value problem, defined by Eqns (21) and (22) is:

Mo Vo Ct t2

2rhc 2rh v( t ) - (23)

Mo - - + t 2rhc

The velocity amplitude is diminishing with time and reaches zero at:

tf = . I v ° M° (24)

where tf is the so-called response time of the string. At t = t e the entire string is brought to rest. For t > t e, the string remains rigid and motionless. The maximum extent of the deformation is,

Cf = ctf. (25)

Impact response of a string-on-plastic foundation 27

I

08

~ ° 0.6

0.4

0,2

1, I 1.2 p=O equation (29)

=1~0 equation (28)

~=2.0 p=5.0

0.2 0 4 0.6 0.8 1 ?

FIG 2. Variation of the norma]]zed velocity w,th t,me for varying parameter it.

l v(x,O

Vo

p=l

C ~ V(t ) ~ C

FIG. 3. Instantaneous velocity profiles of the string.

| t is convenient to write down the solution in the dimensionless form:

v(t) 1 - ~2 - ( 2 6 )

Vo 1 + W

where t = t/tf is a dimensionless time and # can be interpreted as the ratio of the total mass of the deformed part of the string to the mass of the projectile :

2th~f _ 2m/VoC ~ - Mo - ~ / M ~ (27)

A plot of the dimensionless velocity versus dimensionless time for several values of the mass parameters is shown in Fig. 2. In the case of small projectiles or large impact velocities (large/~), the string velocity diminishes rapidly with time in the initial phase of the string motion. At an intermediate value of the parameter/a,/~ = 1, the velocity becomes a linear function of time :

v(t) _ 1 - t. (28) Vo

The corresponding acceleration is constant and equal to v.t = -Vo/ t f . Finally, for large impacting mass or small velocities,/~ ~ 0, the velocity is described by a parabolic function :

v(t) _ 1 - fz. (29) Vo

Figure 3 shows instantaneous velocity profiles in the string for a chosen value of the parameter # = 1.

28 T. WIERZBICKI and M. SHoo FA'Fr

7. SOLUTION FOR DISPLACEMENTS

The transverse displacements are obtained by integrating the velocity of each deforming point on the string with respect to time:

w(x,t) =- v(t)dt. (30) I = x / C

A point on the string located at a distance x from the impact site starts to acquire a displacement after the wave front arrives at that location at time tl = x/c. Substituting Eqn (26) into Eqn (30), the transverse displacement is given by:

w(x , t )=go t fF( l12 - - l ) ln ( l+# t ~ (t---~) (~2 -- F(2)]

L \ l (31)

where .~ = x/if is the dimensionless axial coordinate. The permanent displacement is reached at t = tf of t = 1 :

Vot f r ( l~2 -1 ) ln ( l+p '~ ( 1 - - 2 ) ( 1 -- .~2)1 wf(x) = W(X, tf) = I / L p 2 . (32)

Of particular interest are the two cases considered in the preceding section, # = 1 and p = 0. Substituting/~ = 1 into Eqn (31), the following displacement profile is obtained:

Votf Wr(X) = ~ - (1 - . £ )2 . (33)

Similarly taking p --* 0 and calculating appropriate limits in Eqn (31):

Wf(X) = ~Votf(l -- 2)2(2 + .~). (34)

Similar results are obtained by integrating Eqns (28) and (29), respectively. Plots of the normalized deflection profiles wf(x)/6 f are shown in Fig. 4 for various values of the parameter #. Again, small impacting masses or large velocities give rise to more localized deflection in terms of the steepness of the resulting cusp. It transpires from Fig. 4 that for # ~ oo, slopes near the impacting edge become exceedingly high, rendering moderately large deflection theory invalid. A variation of the tensile force /V with distance x should be taken into account in a more exact approach. However, using the above theory, it may be possible to predict the onset of perforation of the string by the projectile because the axial strain in the string is related to the square of the gradient of the deflection profile. The problem of localization of plastic flow and rupture will be pursued in future research.

1.2

1 tJ=O equation (34)

06 ,~ ~ N ~ p=l equations (33) and (46)

0 . 2 ~

I I 0.2 0.4 016 0.8

FIG. 4. Normalized permanent deflection profiles for varying parameter p.

Impac t response of a s t r ing-on-plas t ic founda t ion

]

29

0.8

0.6

0(

-1 . '*"-0.5 ,

t ' = 1 . 0

j = 0 . 7 5

~=o.5

, 0 5 ~',,. 1

E

FIG. 5. I n s t a n t a n e o u s deflection profiles of the string.

It can be observed that the final deformation profiles do not suffer slope discontinuities at x = ~r. This is explained by the fact that the jump in the velocity diminishes at the wave front with time and vanishes at t = tf. However, instantaneous deflection profiles do exhibit discontinuities at the moving wave front. This is illustrated in Fig. 5 which shows instantaneous deflection profiles calculated from Eqn (31) at several times.

According to the condition of dynamic continuity, the jump in the deflection profile must be related to the jump of the velocity at the wave front. For the sake of illustration consider a particular case/~ = 1. The slope of the displacement field, calculated from Eqn (31)is:

V°tf( ~ - - 1) V°(t - 1). (35) Cr = c

The jump in the velocity field can be found from Eqn (28) (see Fig. 3). It can be easily checked that with the above expressions the condition of dynamic continuity, Eqn (4), is satisfied.

The maximum central displacement is obtained from Eqn (33) by setting ~ = 0:

6 f = wr(O) = V°tf F(/~2---- 1)ln(1 +/~) + - - . (36)

In particular, the permanent deflection corresponding to/~ = 1 is:

5f = ½Vot f. (37)

In another limiting case /~ ~ 0 (large impacting masses or low velocities) the central deflection becomes"

5f = 2Vot f. (38)

A plot of normalized central deflection versus mass/velocity parameter is shown in Fig. 6. The curve 5f/Votf is monotonically decreasing and tends asymptotically to zero for /.t ---~ oO.

30 T. WIERZBICKI and M. S. Hoo FATT

°8 I 0.6

0.4

0.2

tJ=2m / _ - - ~ q ~M o

""-.. / p p r o x i m a t e . . . . . .

Dynamic

I I I I I 10 20 30 40 50

P

FIG. 6. Variatmn of the normalized central deflection with parameter It.

8. S T A T I C S O L U T I O N

A static solution corresponding to the mass impact problem can be obtained by neglecting the string's lateral inertia and considering a concentrated load, P, on the string. The local equilibrium equations for the string on the plastic foundation is:

_ d2w N dx 2 - ?/ 0 (39)

subjected to the boundary conditions"

P at x = 0 (40)

2

and

w = 0 a t x = ¢ .

A straightforward solution to this boundary-value problem is:

P ( ~ - x ) c/(~ 2 - x 2) w ( x ) - - -

21~ 21~

(41)

(42)

The solution involves an unknown length of the deformation zone ~. A unique relationship between P and ~ will be obtained from the condition of kinematic continuity [see Eqn ( 1 )].

In static problems, differentiation with respect to time can be replaced by differentiation with respect to any time-like parameter, such as the load P. Because the shell is undisturbed ahead of the deformation front, the condition of kinematic continuity reduces to:

d d~ w' - - w + - - = 0 at x = ¢. (43) dP dP

Calculating the slope w' from Eqn (42) and substituting into Eqn (43) it is found that:

e = 2[/~. (44)

Therefore, the deformation zone increases linearly with the applied load. Substituting this result back into Eqn (42), one obtains the load-central deflection relationship,

P = 2~/~h76 (45)

Impact response of a string-on-plastic foundation 31

and the normalized deflection profile,

w ( x ) = 6 1-- (46)

It should be noted that the normalized static deflection profile is identical to the dynamic profile calculated for ~ = 1 [-see Eqn (33)]. The normalized static deflection profile is compared in Fig. 4. with dynamic profiles, showing good agreement for a wide range of parameter/~. This suggests that the static solution for the point loaded strings can be used to generate approximate dynamic solutions in the mass impact problem.

9. A P P R O X I M A T E S O L U T I O N

An estimate of the maximum central deflection of the impacted string can be obtained by equating the kinetic energy of the impacting mass to the plastic work of the resisting transverse force :

1 2 IO rf ~MoV o = P(6)d6 (47)

where the function P(6) is given by Eqn (45). Upon integration:

= F 3MoV2 ~2/3

(6r)ap p L 8 2x/2x/2x/2x/2x/2x/2~ J . (48)

This equation can be expressed in terms of parameters tf and/a, defined by the dynamic solution :

(6r)aoo = 0"52Vot:lt- 1/3. (49)

Because the deflection profiles of the static and dynamic solution coincide for /~ = 1, it should be expected that the corresponding predictions of the central deflection would be similar. Indeed, from Eqn (49), one finds that fir = 0.52Votr- This value differs only by 4% from the dynamic solution, Eqn (37). Comparison of both solutions in a wide range of the parameter /a is shown in Fig. 6. For /~---, 0, the approximate solution approaches infinity, while the dynamic solution converges to 6f = ~Votr [see Eqn (38)]. Hence the approximate solution is invalid for/a < 0.2 (very large masses or small impact velocities). Apart from this range, the formula given by Eqn (49) follows the general trend of the dynamic expression, Eqn (36), and gives good results in the range 0.5 </~ < 5.0.

10. LOCAL DAMAGE TO C Y L I N D R I C A L SHELLS

The strength of the string on a plastic foundation can be related to the local strength of a cylindrical shell through the so-called equivalence parameters. The interested reader is referred to refs I-7] and [13] where the procedure was first developed and applied to the solution of static problems. The equivalence parameters are derived by integrating the rate of energy dissipation in the circumferential direction for an assumed deformation mode of the shell in the cross-sectional plane. This procedure effectively reduces the two- dimensional shell problem to a one-dimensional string problem.

Consider a cylindrical shell of radius R, thickness h, flow stress ao, and mass density p, as shown in Fig. 7. The equivalent plastic axial force in the string assumed to be related to the shell parameter by:

1~ - 2~RNpl (50) 3

where Npl = troh is the fully plastic axial force. This assumption is valid only within the theory of moderately large rotations. Similarly, the equivalent foundation constant is

32

i

C

T. WIERZBICKI and M. S. Hoo FATT

R

q (t) (t)

( lit

FIG. 7. Cylindrical shell locally deforming under impacting mass.

X

expressed as :

-- 8Mpl (51)

R

where Mp) = aoh2/4 is the fully plastic bending moment. The above relationship for ?/has been derived under the assumption that the portion of the shell under the impacted mass is flat in the circumferential direction [6]. This is true for impact masses that are relatively wide in the cross-sectional plane but narrow in the axial direction as shown in Fig. 7. A generalization of the solution to round-off cylindrical projectiles is the subject of ongoing research. Stronge [14] and more recently Corbett et al. [15] performed a series of tests on deformation and perforation of cylinders by blunt and hemispherically nosed projectiles. However, the results of these experiments cannot be compared directly to the present theory because of a different cross-sectional deformations shape of the cylinders. Further- more, the validity of Eqn (51) was discussed in ref. [6] and more recently by Moussouros [ 16]. In general, there is a tendency for the circumferential bending resistance to increase with deflections. However, Eqn (51) is a good approximation for deflection in the range 0 < 5 < R .

The equivalent mass of the string r~ is calculated from the requirement that the plastic wave speed in the string be equal to a uniaxial plastic wave propagation in a bar:

C ~ ~ . (52)

Using Eqn (50), the mass equivalent parameter becomes:

rh = 2npRh. (53)

This expression can be substituted into the previously derived formulae for the string to yield similar results for a cylindrical shell.

The parameter /~ in the string problem can now be expressed in terms of a new

Impact response of a string-on-plastic foundation 33

dimensionless mass parameter q and velocity parameter v. These are defined by"

Mo (54) r/ rhR

and

vo v = - - . ( 5 5 )

c

The parameter/a can then be expressed in terms of r/and v :

2~/~v~Rv (56) P = ~ 3hq

Of particular interest is the maximum permanent displacement of the shell under the impacted mass 6f.

The exact solution of the problem, Eqn (36), takes on new variables in the following form :

6f 1 F(IA2 - 1) 1 11 ~ = ~ v r / t ~ T ~ l n ( 1 + p ) + - - - / ~ (57)

where /~ is defined by Eqn (56). The approximate solution, expressed in terms of the parameter q and v is:

( 6 f ) =0.52r/v 1 (58) a p p 2~ 1/3"

Again, both solutions almost coincide at/~ = 1. Plots of the normalized central deflection, given by Eqn (57), over a wide range of mass

and velocity parameters are shown in Figs 8 and 9. The present solution can be replotted for a constant value of the dimensionless kinetic

energy :

K E = vq 2. ( 59 )

Now one can eliminate from the solution the impact velocity and express the results in terms of the mass ratio alone. A plot of the dimensionless maximum shell displacement

0.6

0.4

0.2

/ , / /~=10.

0.2 0.4 0.6 0.8 1

FIG. 8. Effect of the mass ratio on the permanent central deflection of the shell.

34 T. WIERZ~CKI and M. S. Hoo FATT

0.8

0,6

0.4

0.2

5 10 15 20 25 30

FIG. 9. Effect of the velocity ratio on the permanent central deflecuon of the shell

02

o.15

0.1

0.05

Approximate

R =100 h

KE=0.025

c=315m/sec

Dynamic

I L , I I I 10 20 30 40 50

"q

Fx6. 10. Comparison of ~he dynamic and approximate solution under constant kinetic energy.

versus the parameter r/is shown in Fig. 10. The graph has been constructed for fixed values of dimensionless kinetic energy KE = 0.025 and radius to thickness ratio R/h = 100. The approximate solution, based on the energy balance is represented in Fig. 10 by a horizontal line. Note that the central deflection attains a maximum at ~/~ 7.0. In the range of the parameter r/shown in Fig. 10, the dynamic and approximate solutions are close to each other.

1I. D I S C U S S I O N A N D C O N C L U D I N G R E M A R K S

The problem of impact on a plastic string resting on a plastic foundation has been formulated as a transient wave propagation problem. It was shown that a transverse wave of strong discontinuity propagates with constant speed away from the impacted point. All

Impact response of a string-on-plastic foundation 35

plastic deformations of the string are concentrated at the moving "extensional" hinge. Beyond the wave front no active plastic stretching takes place and the string moves as a rigid body causing the foundation to be continuously compressed. Transient wave profiles were determined and a closed form solution was obtained for the permanent deflection profile and maximum deflection amplitude. It was shown that the normalized deflection shape depends on a single dimensionless parameter p. This parameter involves the ratio of the impact velocity to the impact mass. Therefore, for constant values of p, increasing the mass of the projectile has a similar effect as decreasing the impact velocity.

The string solution was applied to predict the amount of local damage of cylindrical shells caused by impact. Impacts can be caused by vehicle collisions, accidentally dropped objects, or missile impacts. A link between the formulation for a string and a cylindrical shell is provided by means of the equivalence parameters. Furthermore, the possibility of predicting the onset of rupture of the shell by the projectile was briefly mentioned. An approximate solution to the mass impact of the shell was also developed based on a static solution for the same structure acted upon by point load. The approximate solution was shown to correlate well with the dynamic solution in a wide range of input parameters. This encouraging result can form a basis for deriving approximate dynamic solutions for more complex structures such as stiffened shells.

Various assumptions made in the problem formulation raise the question of the range of applicability of the proposed analysis. The neglect of elastic deformations and axial bending resistance is valid only for deflections greater than the thickness of the shell. Furthermore, final slopes cannot be exceedingly high because the formulation was based on moderately large deflection theory. Another limitation is that rupture of the string caused by large tensile strains was not considered. Therefore, impact of small masses moving with high velocity, projectile impacts, must be excluded from the analysis because they produce highly localized deflection profiles and large strains. At the other end of the spectrum of impact conditions lie the quasi-static loading characterized by large masses and low impact velocities. The accuracy of the solution should deteriorate as the condition of static loading is approached because the string acquires less deformations at the wave front and more active plastic flow occurs in the region between the propagating extensional hinges. As mentioned earlier, continuous plastic deformation was neglected. However, the present computational model does represent some realism to the behavior of a rigid-plastic cylinder under localized impact loading in the intermediate range of impact, such as those caused by accidently dropped objects.

Acknowledgements--The research presented in this paper was sponsored by the Office of Naval Research under contract No. N00014-89-C-0301. The authors would like to thank Professor W. J. Stronge of Cambridge Umverslty, Professor T. X. Yu of Peking University, and Dr S. Bhat of Amoco Production Company for their critical comments which significantly contributed to the clarity of the paper. Also helpful discussions with Mr Wdliam McDonald and Dr Minos Moussouros of the Naval Surface Warfare Center are gratefully acknowledged.

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