Dynamic response of an infinitely large rigid-plastic plate impacted by a rigid cylinder with transverse shear and rotatory inertia

Int. J. Impact Enon0 Vol. 7, No. 4, pp. 391-400, 1988

0734-743X/88 $3.00+ 0.00 © 1988 Pergamon Press plc

Printed in Great Britain

DYNAMIC RESPONSE OF AN INFINITELY LARGE RIGID-PLASTIC PLATE IMPACTED BY A RIGID CYLINDER WITH TRANSVERSE SHEAR AND ROTATORY INERTIA JIN QUANLIN Research Institute of Mechanical and Electrical Technology MMB, Beijing, China (Received 9 October 1987; and in revised form 13 M a y 1988)

Summary--A theoretical solution for the dynamic response of an infinitely large, rigid-perfectly plastic plate with transverse shear force and rotatory inertia under a normal impact of a rigid fiat-end cylinder is given herein. Embedding and perforation are investigated, and the numerical results indicate the influence of rotatory inertia and the transverse shear force on the dynamic plastic response of the plate.

NOTATION Eb

Ef E, G H Ko, K1

plate bending deformation energy residual kinetic energy of cylinder and plug plate shear deformation energy radius of gyration for plate plate thickness modified Bessel function of the second kind of order zero and order one, respectively

M, =M.I,oR Mr radial moment of plate cross-section Me fully plastic bending moment of plate cross-section Mo Q Q. Qo

R,L

U V

Vf

Vi Vo W X'

Yc tl

kr ko m

r, O, Z t tl tr Z z 1 "2o

circumferential moment of plate cross-section radial or transverse shear force

=QI,=R

fully plastic transverse shear force at plate cross-section radius and length of cylinder, respectively displacement of plug and cylinder = U[t=o initial velocity of plug cylinder residual velocity ballistic limit initial impact velocity of cylinder plate deflection = aX/Or = ~x/ot = H / [ R ( I + pL/m)]

radial curvature change circumferential curvature change mass of plate per unit area cylindrical coordinates time time at which the 1st phase of motion ceases time at which the 2nd phase of motion ceases radial coordinate of interface between two plastic regions radial coordinate of external boundary of second plastic region =

Zlt= 0

= G/H, dimensionless radius of gyration for plate

P

= Mo/(Qon) transverse shear strain mass density of cylinder material rotation of midplane due to bending

Other symbols are defined where they first appear. 391

392

J. QUANLIN INTRODUCTION

Some bending deformation and plugging failure can be caused when a thin plate is impacted by a flat-end rigid cylinder. The interaction of these two deformation modes is related to the prediction of the ballistic limit, but up to now there have been few studies of this problem. Previous theoretical analyses can be divided into two types [1]. One approach [2-4] is concerned only with bending, which allows an infinitely large transverse shear force to be transmitted through the plate. As a result, only plates which undergo deformation and no perforation can be dealt with by this type of analysis. In another approach [5] shear deformation but not bending is retained, so it is only applicable to perforation. Recently, some studies [6-9] have pointed out that the influence of rotatory inertia on the dynamic plastic response of structures is important. But it seems that the influence of rotatory inertia has been neglected in the previous studies on plates impacted by a projectile. A theoretical solution is presented for the dynamic response of an infinitely large plate made of rigid-perfectly plastic material under normal impact by a rigid flat-end cylinder. The plastic yield criterion for the plate retains transverse shear force as well as bending moments, and rotatory inertia is included in the equilibrium equations of the plate, so that it is possible to investigate both the perforation process of the plate and the embedding process of the projectile. Finally, some numerical results are presented in the various figures to show the influences of rotatory inertia and transverse shear force on the plate deflection and ballistic limit.

BASIC EQUATIONS

A rigid-perfectly plastic infinitely large thin plate is impacted by a rigid flat-end cylinder with velocity V, as illustrated in Fig. 1. The thickness of the plate is H, while the radius, length and density of the cylinder are R, L and p respectively. In the cylindrical coordinate system (r, 0, Z) shown in Fig. 1, the lateral displacement of a material particle of the plate, radial moment, circumferential moment and radial shear force are represented by W, Mr, Me and Q respectively, and their directions are shown in Figs 1 and 2. Let the mass per unit area of the plate be m and the plate's radius of gyration be G, then the equilibrium equations of the plate under infinitesimally small deformation can be written as [8] (rQ)' = r(m~¢ - p) 1 (rMr)' - Me = r(Q - raG2~),,

(1)

where ~b is rotation of the midplane due to bending and = W'-

2R

~k,

vo

FIG. 1. Plate impacted by a rigid cylinder.

kr = ~k',

ko = - ~ k / r

~

rt

FIG. 2. Element of plate.

dMr

Dynamic response of an infinitelylarge plate

393

are transverse shear strain, radial curvature change and circumferential curvature change respectively. The lateral pressure p = 0 in the region r > R. For a thin plate, the process of forming a plug is so short that the error due to assumed initial conditions ul,=o = w l , = o = 0,

o1,=o = v,

Vl,:o = 0

is very small. Here we suppose that a plug is formed at the moment t = O, and the plug velocity is the same as the cylinder velocity when t > O. Let the velocity be O. If Qa = QI,=R, the equilibrium equation of the plug is

0 = 2QJ[R(pL + m)].

(2)

By momentum conservation we obtain V = 1Io/[1 + m/(pL)]. Let the fully plastic bending moment and the fully plastic transverse shear force of the cross-section be Mo and Qo respectively. Let Mo, Qo, H, ~ , n/v/~o/m and QoH2 be characteristic parameters of moments, transverse shear force, dimension, velocity, time and energy respectively. Define

a = G/H, fl = Mo/(QoH)~., a H/[R(1 +pL/m)] J

(3)

Then equations (1) and (2) may be written in dimensionless form as

(rQ)' = rCV (flrMr)'

-

).

M o = r(Q

-

~t2ff)

(4) (5)

U= OI,= 2a. o = Qa v = (1 _aR)Vo}.

Note that all variables below have been dimensionalized; their symbols have not been changed for simplicity. The simplified yield surface in Fig. 3 was used to study the dynamic plastic response of circular plates [8]. The discontinuity behaviour of the interface between a rigid region and a plastic region associated with dynamics has been discussed [10]. Theorem 1 is valid for the interface between plastic regions, which will be met in the following analysis. Theorem 1 is written in a form appropriate for the plate as follows. The continuity of tractions:

[Mr] = [Q] = 0.

MO/MO

Mr/M 0

FIG. 3. Plastic yieldsurface.

J. QUANLIN

394

Here [X] = X2 - X~ means the difference in X on either side of an interface which travels with a velocity ~ from region 1 to region 2. The continuity of velocity for case ~ # 0 is

[~]

= [¢v]

= 0.

According to kinematic continuity [ I l l we have [If'] = -~[~]. If there is no shear sliding on the interface with ~ = 0, the continuity of lateral displacement requires that

[P¢] = [¢¢] = 0. For an interface where

> 0, the equation

[ w ' ] = I¢)] + [~]

= o

is valid [11]. To sum up, the discontinuity conditions [Mr] = [Q] = [p/z] = [~3 = 0 may be used for both moving and stationary interfaces, where

(6) > 0 and [~] = 0.

DYNAMIC ANALYSIS

A velocity field based on some experimental results is shown in Fig. 4. It may be divided into four regions: the rigid plug in the center (r < R) with a velocity 0 , a rigid stationary region (r > zl, z 1 ~ oo), two plastic regions (R ~
0~
Mo=l

(7)

in the region R ~
Mr=0 ,

M0=I

(8)

in the region z ~
I;V = A l r + A2

ff"= Axr + A2 rQ = RQa + (r a

-

R3),41/3 + (r 2 -

R2)A2/2

flrMr = [fir. Ma + (fl + RQ~)(r - R)] + (r 4 - 6ct2r2 - 4Rar + 3R 4 + 60t2R2) , A1/12 + (r 3 - 3R2r + 2Ra)A2/6 Mo= l

~=0

z FIG. 4. Admissible velocity field of plate.

(9)

Dynamic response of an infinitely large plate

395

where M, = Mr[r = R. N o t e stationary conditions at r = z 1; in the region z ~< r ~< z I the solution is if" = BKo(r/~t) r. Q = - fl - B a r K l i f / a ) [

Mr = 0

~,

(10)

|

M°= l

I

~=0

/

and, when ~/> O, we have I~ = B K o ( r / ~ ). Here K o and K 1 are the modified Bessel functions of the second kind of order zero and order one, respectively. According to the n o r m a l flow rule we m a y predict that [/~0//~,[ > 0 at r = z and that the discontinuity conditions (6) are valid for the interface r = z. Then, using equation (6), the following equations can be obtained: A l Z + A 2 - BKo(z/ot ) = 0

~1

a I + (B/ot)Kx(z/ot) = 0

(z 3 -- R3)A1/3 + (z 2 -- RZ)A2/2 + ~tBzKl(z/ot ) = - f l - RQa

.

(11)

(z* - 6~t2z 2 - 4 R 3 z + 3R 4 + 6~t2R2).'il/12 + (z 3 - 3R2z + 2R3)A2 = - - f l R M ~ - (fl + R Q , ) ( z - R )

The initial conditions are UI,= o = wl,ffi o = 0 ul,= o = V

J .

(12)

¢¢1,=o=0 W h e n t/> 0, the dynamic response of the plate is divided into two phases of motion. First phase (0 <<.t <<.t l)

During this phase, the interface r = R is a plastic hinge circle with shear sliding. The interface r = z remains stationary [ 7 - 9 ] . N o t e that if ~ = 0, Q, = - 1 and Ma = 1, we can give the solution of equation (11) as follows: A 1 = - (R - f l ) K l ( z / a ) / [ E ( z ) S ( z ) ] A 2 = (R -- t i f f S ( z ) = (R -- f l ) a / [ E ( z ) S ( z ) ]

'

(13)

(z -- R) 2 = [ 1 2 f l R S ( z ) / ( R - f l ) ] / F ( z ) where E(z) = aKo(z/ct ) + z K l (z/Ct), S(z) = [3~t(z 2 - R2)Ko(z/oO "% {6~2z q- (z - R)2(z q- 2 R ) } K I ( z / c t ) ] / [ 6 E ( z ) ] , F ( z ) = - ( 3 z 2 + 2 R z + R 2 - 6~te)Kl(Z/~)/E(z) + 4z + 2 R .

It is clear that A1, A2, B and z are independent of time. Since E(z) > 0 and S(z) > 0, an inequality R t> fl

(14)

must be satisfied in order to give W 1> 0 in the first phase. W h e n the impact velocity of the cylinder is equal to the ballistic limit ~ , the following equations are required at t = tl: v0

I2¢1,=e = 0 J"

(15)

396

J.

QUANLIN

Combining equations (5), (9), (13 ) and (12), the ballistic limit Vi can be expressed as follows: Vi = 2(a + 0.5I;f'l,=R)~/2/(1 - aR),

(16)

where lTv'l,= R = (R - fl)[gKo(z/ct) + (z - R)KI(Z/~)]/[S(z)E(z)]. If Vo ~< V~, this phase of motion is terminated by embedding the plug and the cylinder in the plate at t = t~. The first part of equation (15) gives tx = Vo(1 - aR)/(2a + ITv'l,=R).

(17)

If Vo > Vi, the plate is perforated and the plug and cylinder are separated from the plate at t = t~. The second part of equation (15) gives tl = [1 - w/1

-

(Vi/Vo)2](l

-

aR)Vo/(2a + I7¢'1,=R).

(18)

The residual velocity of the plug and the cylinder is Vf = [1 - { 1 - x / 1

-

(Vi/Vo)2}/{1 "4- ff'l,=R/(2a)}](1 --aR)Vo.

(19)

Second phase (tl <<.t <<.tf) During this phase, the sliding at the interface r = R ceases and the interface r = z is likely to move outward [71. N o t e that if ~ ~ 0, the solutions of equation (1 I) in this case are A 1 = - K 1(z/g)A2/E(z)

'~

B = gA2/E(z ) fl + RQ, = - [ S ( z ) A 2 + i~SI(z)Az]

,

(20)

(z - R) 2 = 12flR(M.t - [M.]"tx)/[F(z)A2] = 12flR(M, EA2 - ,42t] - tx [M.]t~A2)/[Y(z)(A2) 2] where Sl(Z ) = [(z 3 - R3)Ko(z/g) + 3gzZKl(z/g)]El(z)/[3zE2(z)] E1 (z) = zKo(z/~t) + gK 1(z/g) Y(z) = (1 - R/z)El(z)[{(z 3 - R 3) + z(z z - R e) + z2(z - R) + 6g2(z + R)}Ko(z/g) + 12gzZKl(z/g)]/E2(z)

[Md" = M.l,=,,+o-- M.t,=.-o. In the case Vo ~< Vi, the embedding of the plug and the cylinder gives I)¢'1,=a = 0 , then A 2 and ~ can be expressed as follows. A 2 = [ R V / ( 2 a ) -- flt]/Fl(z )

"~

= ( 6 f l R 2 V / a ) [ { Y ( z ) / V t ( z ) } { R V / ( Z a ) - f l t } { R V / ( 2 a ) - [1 + l Z R F 2 ( z ) / Y ( z ) ] } ] ) , tf =

(21)

R V /(2afl) where Fx(z ) = S(z) + R/(2a)[1 - RKx(z/g)/E(z)] F2(z) = Sx (z) + [ R2 / (2a)]Ex (z)Ko(z/~)/[zE2(z)].

I n the case V o > Vi, note that the plug has separated from the plate in this p h a s e - - t h e plastic hinge circle r = R has become a free surface (i.e. Q, = Ma = 0). Using this comlition, A2 and $ can be expressed as follows:

a~ = (Rtx - flt)/S(z)

)

i = 12fl2Rq/[{Y(z)/S(z)}(Rt~ - f l t ) ( R q { l - 12flSl(z)/Y(z)} - f i t ) ] i "

(22)

tf = R t l / f l When A 2 = 0 , the second phase of motion ends with z ~ ov and reaches the m a x i m u m deflection, which shows ~ > 0 during the second phase of motion. The numerical calculation of energy shows that the above dynamic analysis satisfies the

Dynamic response of an infinitely large plate

397

energy conservation requirement when t i> 0. From equations (16)-(19), we can give the plate shear deformation energy Es, the plate bending deformation energy Eb and the residual energy Ef of the cylinder and the plug. In the case of no perforation: E s = 2~R(Vo/Vi) 2

"~

(23)

Eb = [rcR/(2a)][{ (1 -- a R )Vo} 2 - 4a](Vo/V,) ) . E b / E ~ = [(1 - aR)Vi]2/(4a) - 1

In the case of perforation: Es = 2rrR

]

Ef = [rrR/(2a)][1 - 4a{1 - x/1 - (Vi/Vo)2}/{(1 - aR)Vi}2][(1 - aR)Vo] 2

"

i . (24)

E b = [nR/(2a)][(1 - aR)Vo]2[1 - {4a(1 - x/1 - (Vi/1/o)2)/[(1 - aR)Vi]2} 2] - 2 n R RESULTS

AND DISCUSSION

All the numerical results below (including all parameters in the figures) are dimensionless. The curve in Fig. 5 shows different dynamic plastic response characteristics between perforation and no perforation. If V0 ~< V~, the plate deflection W increases with the impact velocity Vo until reaching its maximum at Vo = Vi. If Vo > Vi, W decreases with Vo, and the value of W is much lower when Vo > Vi than when Vo ~< Vi. Therefore, the W-Vo curve is discontinuous at the ballistic limit. This phenomenon may be explained by a discussion of energy. In the no-perforation case, equation (23) shows that both the shear energy Es and the bending energy Eb are directly proportional to the impact velocity Vo. That is why the plate deflection increases with the impact velocity. The ratio Eb/Es between bending energy and shear energy is independent of the impact velocity, and is a quadratic function of the ballistic limit Vi. From the variations of the ballistic limit (see Figs 6 and 8), it can be stated that the ratio decreases with rotatory inertia and the parameter ft. In the perforation case, because the cylinder and the plug are projected out from the plate at a higher velocity than the plate, only a part of the initial impact kinetic energy is converted into the plate deformation energy. Equation (24) shows that the shear energy is independent of the impact velocity, while the bending energy decreases with the impact velocity and tends to zero when Vo ~ ~ . From equation (24) it can also be seen that the bending energy is discontinuous at the ballistic limit, and the difference in the values on either side of the ballistic limit is just equal to the residual kinetic energy of the cylinder and the plug. For example, in Fig. 5, the ratio between the difference and the shear energy is about 57.3. For this reason, the W - V o curve in Fig. 5 is discontinuous at the ballistic limit and the plate deflection decreases with the impact velocity. In Fig. 6 the variations of the ballistic limit Vi (curve 1) and the initial coordinate z o of the interface r = z (curve 2) with the parameter fl are shown, where the Zo-fl curve for u =0.289 200 - ; 9 = 0 . 5

L = 25 R=5

0 = 1/130 150

W Joo

50

o 1.0

I I,I

1.2

1.3

1.4

Vo

FIG. 5. Maximum deflection W versus impact velocity Vo.

398

QUANLIN

J. o:

z

,q:5

1/130

5~L=2 5 I

~

,

--25

a

= 028g

-- ZO

'°

lO

o.5[-

~

I

0

0.5

5

i

1

I

1

I0

15

20

2.5

B FIG. 6. Ballistic limit 1{ (curve 1) a n d initial c o o r d i n a t e z 0 of interface r = z (curve 2) versus ~.

= 0 coincides with that for • = 0.289. This figure shows that the primary factor that controls the ballistic limit is the parameter/~ = Mo/(QoH), which correctly represents the plate resistance to plugging failure. Figure 7 shows that under the same impact velocity, different values of rotatory inertia can give rise to different plate deformations with or without perforation. In this figure the critical value = = 0.289 corresponds to a plate with a uniform distribution of mass along the thickness direction. For a different impact velocity, the critical value of rotatory inertia is different. For the same reason that there is a discontinuity in the W-V curve at V~, W-= is also discontinuous at the critical value of =. The influence of rotatory inertia is considerable for this load state and boundary conditions. When ¢ < 0.289, the plate cannot be perforated and the smaller the value of =, the smaller the value of W, which is just opposite to the behaviour of simply supported beams [7], fully clamped beams [9] and simply supported circular plates [8] loaded inpulsively. When = > 0.289, the plate is perforated and a decrease of = leads to a rise of W. If we take other impact velocities, then the influence of rotatory inertia on W is not shown so clearly as when Vo = 1.4, because the ballistic limit is 1.4 and Zo reaches its minimum when ~ = 0.289 (see curve 2 in Fig. 8). From this example it can be concluded that the influence of rotatory inertia on the dynamic plastic response of a structure is not always to decrease the deflection. The influence of rotatory inertia on the ballistic limit is shown in Fig. 8, Fig. 6 and Fig. 9 (curve I). These figures demonstrate that the ballistic limit is smaller when ¢ > 0 than when ~ = 0. The curve 1 in Fig. 8 indicates that the ballistic limit increases monotonically with decreasing =; finally it tends to a constant as = --, 0. For the case o f R = 5 and 1/= 0.5, the ballistic limit is 7% larger for = = 0 than for • = 0.289. This means that the influence of rotatory inertia is not negligible unless ¢ < 0.1 or ~ > 1.0 or R < I[H/D = 1/(2R) > 0.5], as shown in these figures. The variations of the initial coordinate Zo of the interface r = z with = , / / a n d H/D are represented by curve 2 in Figs 8, 6 and 9 respectively. It is clear that the influence of rotatory inertia on the variations of Zo is very small, so that the curves for ¢ = 0 coincide with those for = = 0.289 in Figs 6 and 9. L=25 200

--

V =1.4

R=5 a=l/130

/3=Q5

150

W

IO0

50-------.-0

--0.5

I

I

I

-I.0

-I.5

--2.0

L.og(a ) FIO. 7. M a x i m u m deflection W versus =.

Dynamic response of an infinitely large plate R=5

~

1.5

a>O

L=25 . . . . - -

9=0.5

399

a= i/130

o=0

6.60

6,55 r3

6

50

6,45

I.i

I

0

i

--OS --I,0

I

i

--I.5 --2.0

Log ((~)

FIG. 8. Ballistic limit Vi (curve 1) and initial coordinate Zo of interface r = z (curve 2) versus a.

,e:o5

L = 25

2.2 --

o = 1/130

~--o

a :0 289

2.0 --

/

1.8 ->-

/ /3~o

I

-

/

-- 20

,.6-

,5

\

/

1.2 -

,.o

25

-

/-,o

~

J

0

I

0.2.

04

N°

--5

I

i

0.6

08

!

1.0

H /D

FIG. 9. Ballistic limit Vi (curve 1) and initial coordinate Zo of interface r--z (curve 2) versus H/D (D = 2r).

p r a t e rni~ 12560 2 3 0 BHN

6 "-cj,Ymder ( f l a t - enc~)

2 3 0 BHN

?

6

0 ~c 3

i

2 --

O

[ o

I

0.2

I

J__J

0.4

0.6

08

H/D

FIG. 10. Experimental data about ballistic limit versus H/D (from Fig. 24 in Ref. [1]). A qualitative c o m p a r i s o n of Figs 9 and 10 (from Fig. 24 in Ref. [1]) indicates that both curve 1 in Fig. 9 and the experimental result in Fig. 10 have the same tendency. W h e n the plate thickness decreases, the ballistic limit decreases m o n o t o n i c a l l y from a higher value to a m i n i m u m value and then increases. In the meantime, the main deformation pattern of the plate changes from plugging to dishing. In the vicinity of the m i n i m u m neither bending nor shear deformation is negligible. In Fig. 9 the ballistic limit V~ ~ oo w h e n

400

J. QUANLIN

H / D ~ O, which is not true. This may be due to assuming infinitesimally small deformations

and neglecting the tensile failure of the plate in the present analysis. CONCLUSIONS

1. Assuming a plugging failure mode for a plate impacted normally by a rigid flat-end cylinder, the present theoretical solution can be used to investigate both perforation and the case without perforation. 2. Based on the numerical results obtained in this article, two experimental phenomena reported previously are explained theoretically: in the vicinity of the ballistic limit, the deflection of a perforated plate is less than that of a plate without perforation; a decrement of plate resistance to bending may cause an increment of the ballistic limit. 3. The influence of rotatory inertia is not negligible unless g < 0.1 or fl > 1 or R < 1. Neglecting rotatory inertia may result in a larger predicted ballistic limit. For a plate with uniform mass distribution in the direction of thickness, the difference is about 7%. 4. In the case of perforation, the plate deflection decreases with rotatory inertia, but in the absence of perforation, the plate deflection increases with rotatory inertia. Acknowledgement--The author wishes to express his thanks to Prof. Wang Ren for his support and help in connection with the work presented.

REFERENCES 1. M. E, BACKMANand W. GOLDSMITH,The mechanics of penetration of projectiles into targets. Int. 2. Engno Sci. 16, 1 (1978). 2. C. A. CALDER,J. M. KELLY and W. GOLDSmTH, Projective impact on an infinitive viscoplastic plate. Int. 2. Solids Struct. 7, 1143 (1971). 3. J. M. KELLY and T. WIERZBICKI,Motion of a circular viscoplastic plate subject to projectile impact. Z. Angew. Math. Mech. 18, 235 (1967). 4. J. M. KELLY and T. R. WXLSHAW,A theoretical and experimental study of projectile impact on clamped circular plates. Proc. R. Soc. Lond. A306, 435 (1968). 5. A. l%r~L and N. DAVIDS,A viscous model of plug formation in plates. J. Franklin Inst, 276, 349 (1958). 6. N. JON~, Recent progress in the dynamic plastic behavior of structures. Shock Vib. Dio., Part 1:10 (9), 21-33 (1978); Part 2:10 (10), 13-19 (1978). 7. N. JON~ and J. GoM~s DE OLIVmRA,The influence of rotatory inertia and transverse shear on the dynamic plastic behavior of beams. ASME £. appl. Mech. 46, 303-310 (1979). 8. N. JONESand J. Go~ss D~ OLIWlRA, Dynamic plastic response of circular plates with transverse shear and rotatory inertia. ASME J. appl. Mech. 47, 27-34 (1980). 9. JIN QUANLIN,An analytical solution of dynamic response for the rigid-perfectly plastic Timosheako beam (in Chinese). Acta Mechanica Sinica 15, 504 (1984). 10. HUANG ZHUPlNG, Discontinuities in dynamics of rigid-perfectly plastic continua (in Chinese). Acta Meehanica Sinica 15, 500 (1983). 11. YIAN6 GtJITONGand XONG ZHOHUA,Plastic Dynamics, pp. 164-166 (in Chinese). Qinghua University Press (1984).