A nonlocal model for plug formation in plates

NONLOCAL MODEL FOR PLUG FORMATION IN PLATES? HILM DEMIRAYS Princeton University,

and A. CEMAL ERINGEN Princeton, NJ OWO, U.S.A.

Ab&act-By means of a nonlocal viscous fluid model, an investigation is carried out of the problem of penetration of a cylindrical projectile into a plate leading to a failure of the plate by a plug formation. The effect of impact is represented by a uniform initial velocity distribution over a circular region on the surface of the plate. The behavior of this plate material is assumed to be viscous and spatially nonlocal, and only the effects of vertical shearing stress are considered. The expression of stress, velocity and displacement are obtained and the calculated displacement profiles are compared with some existing experimental pre6les.

THESTUDYof

penetration and perforation problems has long been of interest to many sectors of industry which stimulated a large body of experimental research and empirical knowledge. Because of the complex mechanism involved and the limited knowledge of the behavior of materials under hi strain rates, or short-time loadings of high intensity, analytical investigations have focused on some simple problems. One of the basic difhculties in dealing with hypervelocity impacts and penetrations is the selection of a suitable form for the constitutive equations for the target material that are physically realistic and yet mathematically tractable. When a projectile strikes a plate at ordinance velocities (of order 3000fps) the type of failure, which might occur, depends on the kinetic energy of the projectile, the dynamic thermo-mechanical properties of the plate and projectile materials, and some geometric factors such as the ratio of plate thickness to diameter of projectile, angle of incidence, etc. To study such a complicated problem by mathematical models, it is generally assumed that the failure of the target plate occurs mainly by a single mechanism, all other effects being negligible. The assumed failure mechanism is usually taken as one of the types illustrated in Fig. 1. In this paper we analyse the plug formation mechanism by use of the nonlocal viscous flow theory. One of the main approaches to the high speed penetration problem is the so-called hydrodynamic model which employs the classical viscous Bow theory in various modified and simplified forms. The effect of plasticity is taken into account in some approximate way. In this spirit we have the works of Pytel and Davids [l] on the plug formation of plates, Thomson [2] on the penetration of cylindrical projectiles into visco-plastic plates, and Tate 131using a fluid model to assess the crater depth and the source of deceleration of the projectile. Comprehensive reviews of the literature on the field of dynamic penetrations into plates have been made by Goldsmith [4], Cristescu [5], and Johnson [6]. In all these works, the classical local theories have been utilized. As a result of this, stress singularities are predicted which are not physically

SCABBING

WCTILE HOLE ENLARGEMENT

DISHING OR PETAL FORMATK)N

PLUG FOtWATKIN

Ftg. 1. Failure mechanisms. tTbis work was supported by the Army Research Office at Durham. SOn leave from Marmara Research Institute, Division of Applied Mathematics,

P.K. 141, Kadikoy-Istanbul.

288

H.DEMIRAYandA.C.ERINGEN

realistic. In fact, according to classical theories, no matter how small the impact velocity of projectile and how thick the target plate is, because of the infinite stress concentration, the perforation (generally in plug form) will occur. This prediction is contrary to the experimental findings. The present paper is concerned with the analysis of plug formation in plates by a rigid cylindrical projectile. The model employed here assumes that the failure of the plate results from plug formation, and the effect of impact is represented by a uniform initial velocity distribution over a circular region of the plate surface. It is well-known that in hypervelocities the flow behavior is important. Fracture, on the other hand, requires the consideration of nonlocal (cohesive) forces. Thus, we are motivated to assume that the plate behaves like a nonlocal viscous fluid. For simplicity, we consider only the effect of vertical shearing stress. The expression of stress, velocity and displacement are obtained and the calculated displacement profiles are compared with some existing experiments. 2,PHYSICALMODELOFPLUG

FORMATION

When a cylindrical projectile strikes a very hard plate, it is often observed that the projectile pushes a portion of the plate material forward. This separated piece of material, the so-called “plug”, is approximately in cylindrical shape (cf., for example, Zenner and Peterson [7] and Zenner and Holloman [8]). Mathematical analyses applicable to the study of plug formation have been proposed by Awerbuch and Bodner [91,Awerbuch [lo], Goldsmith and Finnegan [l l] and Woodward and de Morton [12]. In these works the balance of total momentum has been utilized, and only the effect of vertical shearing stress is taken into account. For obvious reasons, the energy of plastic deformation and elastic vibrations of the projectile are usually neglected in analyzing the impact problems of the type being discussed here. In this way, the problem is reduced to that of a rigid projectile of given geometry striking a nonrigid plate. The assumption of rigid projectile is certainly valid for high speed projectiles striking thin plates (see Taylor [ 131). In high-speed penetration, the process of penetration is so fast, i.e. the perforation time is so short, that the effect of strain rate is very important. As a first approximation, one may neglect the elastic part of the stress tensor and consider only the viscous part accounting for the high strain rates. Moreover, because of the nearly cylindrical shape of the plugging material and the symmetry of the problem, it can be assumed that the vertical shearing stress is the only stress which need be considered. Since the yielding continues all through the perforation, we also assume that the shearing stress does not change through the thickness of the plate. Experimental study by Pytel and Davids[l] shows that this assumption is not quite valid, e.g. the vertical displacement changes through the thickness of the plate. Nevertheless, for thin plates this assumption holds. The main role of the projectile is in the setting of a cylindrical portion of the target plate into a uniform motion. Finally, in failure, the cohesive forces between material points near the sheared surface play an important role because a new free-surface is created. This consideration suggests the use of a nonlocal viscous fluid model for the plate material. Although, it may seem that these assumptions are somewhat oversimplifying of the plugging mechanism, the results turned out to be in reasonable agreement with experiments. 3.FORMULATlONOFTHEPROBLEM

In nonlocal continuum mechanics the equation of the balance of momentum is identical to the Cauchy’s equation t&l.& + Plfi - 61)= 0

(3.1)

where tkl are the COttIpOnentS of the Stress tensor in rectangular coordinate xk, fr are that of body force density, nf are the components of the velocity field, and p is the mass density. A comma indicates partial differentiation with respect to Cartesian coordinates and a superposed dot the material time rate, e.g.

(3.2) As usual, repeated indices indicate the summation over the range 1-3.

A nonlocalviscous model for plug formationin plates

289

The nonlocal con~uum theories of elastic solids and viscous fluids have been formulated by E~~en~l~l6] in a series of papers and some appIications of the theories, such as the crack tip problem of elasticity and the surface tension problem of viscous fluids, have been worked out in several of his works. Experimental studies show that [17], in high-speed penetrations and perforations, the strain rate is so high that the elastic part of the stress tensor is negligible as compared to the viscous part. Thus, following Eringen[lS. 161, we write for constitutive equations for the stress tensor at xk tkl =

-P&k,

+

IV

d((X’-

Xl)(n

dbk,

+

2JLd;,) dU’

(3.3)

where p is the pressure and du is the deformation rate tensor defined by (3.4) A and JL are the viscosity coefficients and a’ is a coefficient representing the effects of distant points xi to the stress at the pointxk. Introducing (3.3) into (3.1) and converting some volume integrals into surface integrals and neglecting the body force, we obtain

8s

I

a’(lx’-x~)u~~,~du’-p at = ,fl’@’ - xl)~~n~du’ f

V

(3.5)

where u; = Ad&&,+ 2pd;,.

(3.6)

Here S is the material surface of the body and n; is the unit normal directed outward at the surface of the body. In the present problem the plate is considered very thin and, therefore, the field quantities are independent of the x3 = z coordinate in the thickness direction. Thus we may assume that a’ is only a function of xf - x1 and xi-x2, in Cartesian coordinate. Integration of (3.5) across the thickness (from z’ = - ItO/2to z’ = + hd2) gives

I

$(lx - x1)&k dx’ dy’ - p 2 =

G’(jx’- xj)cr&n;ds’

(3.7)

A

where we wrote x = xl, y = y1 and &‘s

hd2

I -hd2

a’ dz’.

Here A is the surface area of the plate and I’ is its boundary curve. Since the plate is of infinite extent and a’ dies out as exp (- Kr’), r’ = v/xl2 + yri, when ~‘+QJ, the right-hand side of (3.7) vanishes. Thus, we have the final form of the balance equations

I &(1x’- xl)a~,,~

dx’ dy’ - p 9 = 0.

A

(3.8)

In this particular problem the only impo~ant stress component is t,, and the remaining stress components are assumed to vanish. For the displacement field, we assume that only the z-component is different from zero and it is a function of (x’, y’, t). Denoting the downward velocity component by 11(x,y, t), for the non-zero stress components we have

t y* =

I

A

(3.9) &‘f[x’ - Xl)@zdx’dy’.

290

H. D~MIRAY and A. C. ERINGEN

Introducing (3.9) into eqn (3.8) we obtain

j-_-/_-j’@’ - xI>p ($ +2)

dx’ dy ’ - p 2 = 0

(3.10)

This integro-differential equation is to be solved under the following initial conditions tr(r,O)= V.

for

r=v/X2+R

(3.11) u(f,O)=O

for

r>R

where R is the radius of the circle over which the initial velocity is uniform and V, is the impact velocity of the projectile, Fig. 2. 4. THE SOLUTION

To obtain the solution of this problem, it is convenient to employ the Fourier transform technique. Taking the double Fourier transform of (3.10) and utilizing the convolution theorem we obtain 2?7&(5,n)j.&(Z2 + ?$)fi + p $ = 0

(4.1)

where E and 6 are defined by h(r) exp W

+ vu)1dxdy,

1=(4.2)

1 mm WI v)= G _ _oDW exp ii(Zx + rly)ldxdy.

IJ

Noting that G and u are only functions of r, the Fourier transform defined by (4.2) is also equivalent to the zero order Hankel transform of ai and u, i.e.

(4.3) a([, q) = a(k) = Ii(

0

dr

where Jo(kr) is the first kind of Bessel function of order zero. Thus, eqn (4.1) becomes [2mi(k)pk’]a(k)+p~=O.

Fig. 2. Plate showing initial conditions.

(4.4)

A nonlocal viscous model for plug formation in plates

291

The solution of this equation is given by 5(k, t) = A(k) ewBtL)’

(4.5)

where A(k) is an unknown coefficient and /3(k) is defined by (4.6) Applying the inverse Hankel transform to eqn (4.5) we have U(T,t) =

(4.7)

The initial conditions (3.11) then takes the form OD

I

0

kA(k)Jo(kr) dk = V&(R - r)

(4.8)

where H(R - r) is the Heaviside unit step function. It is interesting to note that this integral equation is independent of coefficient /3(k) that characterizes the nonlocal effects. However, as time progresses the stronger effect of nonlocality comes into play, eqn (4.7). Noting that the function A(k) is the zero order Hankel transform of Vfi(R - T), from (4.8), we find A(k) =

=i’oH(R - r)rJo(kr) dr = (V,,R/k)J,(kR).

(4.9)

Then from (4.7) the velocity field may be expressed as u(r, t) = VJZ

mJI(kR)Jo(kr)

I0

e-@(‘)’ dk.

(4.10)

The non-zero components of the stress tensor in Cartesian coordinates are expressed by,eqn (3.9). tXT=

&'(r', 8';r,e)p %os al

e’d8’r’

(4.11)

sin 8’ d&r’ dr’.

(4.12)

m *a

tyr=

If

0 0

Noting the relation between I,, tXZand fyr tn = tXZcos 8 + ty, sin e

(4.13)

we obtain t, =

DI 2n o o w,e';

II

r,ew

’aul arJcos(et- e)de’dt'

.

(4.14)

Substituting (4.10) into (4.14) we get the explicit expression of the shear stress as tn = - V&p 1

k.l,(kR) emB(lr)’dkj-f**

0 0

ayr),8’;r,e)J,(kr’)f)r’

cos (et- e)

de’dr’.

(4.15)

So far we have not placed any restriction on the form of the influence function $(1x’-xl). In this work we assume that ai’ is of the following form ~‘(Ix’- xl>= ($)

exp I- [(x’ - x)* + (y’ - #l/K}.

(4.16)

H. DEMIRAY and A. C. ERINGEN

292

Here one should note that as K -+ 0 the influence function oi’(Ix’-xl) approaches to Dirac delta function in two dimensional space. Applying the double Fourier transforms to both sides of eqn (4.16) we get a(k) = $ exp [ - Kff

+ $)/4] = & exp f- K/?/4).

(4.17)

Thus, from (4.6) we have p(k) = 5

exp (- Kk*/4).

(4.18)

The influence function &‘(r’, 8’; r, e), in cylindrical coordinate, takes the form oi’(r’, 8’; r, e) = --& exp {- [r’2+ r* - 2rr’ cos (e’ - e)l/K).

(4.19)

Introducing the following dimensionIess quantities 5 = r/R, Pt ?=P

C’= f/R,

u=ulVo,

a,=2

c = K/R’, -Rt I.Lvo ’

p = kR

Y(P) = Pz exp ( - &4)

(4.20)

X’W,8’; 6, et= f exp {- (f* + 5”- 25’4 cos (et - e)lk) the velocity and the shear stress take the following form (4.21)

a&, 7) = /omplr(p) exp [ - YWTIdpjO~jo2*ht’~et; 5, eWJ&$‘) cos (0’ - 0) de’ df’.

(4.22)

Here we first carry out the integration related to trigonome~ic functions, i.e. 1s

rzr

I0

I

/*s

xf cos (et - e) de’ = 2 expI - (5” + S2MJ exp ((2rgr) cos (et- e)] cos (et - e) de’. 0

Since the function cos a exp (4 cos a) is periodic in above integral is independent of 8. Thus we have

a

with the period 27r, the result of the

exp [(21’&) cos eq cos 8’de’.

(4.23)

However, this integral is related to the modified Bessel function of order one, Watson[l8], pp. 181, as

I

II

exp (2 c0s.e) cos e de = 3711(f).

0

Utilizing this relation in (4.23), we have (4.24) Introducing (4.24) into (4.22) we obtain the shearing stress. Thus

A nonlocai viscous model for

or, after pe~orming the intention,

plug

formation

in plates

293

see Watson~l71, p. 395, we have (4.26)

This integral will have to be calculated by a numerical procedure. Before doing this, it is of interest to investigate a special case. The case of e = 0 (Local Theory). In this limiting case, from equations (4.21) and (4.26) we obtain

or 1 a, =-exp 2

c

- !$

3

1,(g/27).

(4.28)

This is exactly the same as that given by Pytel and Davids[l]. Thus the classical result is obtained as a special case of the nonlocal approach developed here. It is interesting to note that the shear stress cr, given by (4.28) vanishes everywhere, except g = 1, as T+O. Thus, the classical theory gives a stress dist~bution that is initially infinite at 5 = 1 and zero otherwise. However, if one sets T = 0 in eqn (4.26) we have

or

&(g, 0) = 3I,($) exp (-q).

(4.29)

From (4.29) it is clear that a, is finite for all vaiues of e+ 0 and g. From physical considerations such a result seems rather reasonable. In~oducing the non-dimensional displacement 6 as

from (4.21) we have p=

“‘bang’[1- exp ( - ye)]

These integrals cannot be expressed an~ytic~ly; utilized. 5. EVALUATION

OF DISPLACEMENT

dp.

(4.30)

a numerical integration will have to be

AND STRESS FIELDS

The stress and displacement fields have been calculated as functions of the reduced time 7, reduced radial coordinate g and Q= 0.04. This value of t is selected to match the theoretical displacement (at 5 = 0, for 7 = 0) to that found by experiment conducted by Pytel and Davids[l]. For all other points of all curves (Figs. 3-6) this particular value of L was used. Figure 3 is a plot of the nondimensional shearing stress u, vs reduced radial distance for several values of reduced time 7, as calculated from eqns (4.26). The classical local theory gives infinite stress singularity for e&1,0), while the present solution gives finite value for the

294

H.

DEMIRAY and A.

C.

ERINCEN

Fig. 3. Variations of nondimensional stress vs I.

shearing stress at the same location. The value of this nondimensional stress at this point is 2.623 for this particular value of c. Recalling the definition of the dimensionless stress, i.e. a, = - Rt,& Vo), eqn (4.20), for given values of R and CL,it is possible to assign a critical value for the impact velocity V. at which the plug formation initiates. Denoting the critical value of the stress initiating the plug formation by t, = t, we find for the critical velocity

v; =

(

2.623

R

>

c1 L

(5.1)

Since the classical theory gives infinite stress at g = 1, it precludes the definition of such a critical impact velocity. The nonlocal approach, however, makes it possible to obtain such a critical value, as given by eqn (5.1). Examination of Fig. 3 shows that in the beginning the stress is concentrated at g = 1, but, as time goes by, this peak of the stress becomes smooth and moves in outward direction. This latter observation is a result of waves propagating in the radial direction. Figure 4 is a plot of u, vs the reduced time T for several values of g. Again, this shows that large initial stresses resulting from impact disturbances die out very fast as time progresses. It is interesting to note that, because of the interference of ingoing and outgoing stress waves in the region g c 1, the magnitude of stress decreases very fast in time. However, this is not the case for the region 3 > 1. Here we also note that the stress predicted by the present formulation is small as compared to the result in the classical theory. Figure 5 displays the nondimensional velocity distribution with radial distance for several values of time parameter 7. As can be seen from the figure, because of the viscous effects in the body, the velocity of the central part of the plate decreases rapidly as time T increases.

A nonlocalviscous model for plug formationin plates

Fig. 4. Noadimensianalshear stress vs time.

lo

I

0

7=0

20

L 7=l

IO

_

0

2

792

i!

Fig. 5. Nondimensional

s

velocity vs &

296

H.

DEMIRAY and A. C. ERINGEN

--

THEORETICAL - -

EXPERIMENTAL

Fig. 6. Nondimensional displacement vs (; theoretical and experimental.

The variation of displacement D with radial distance l for various values of time is shown in Fig. 6. It can be noted that the displacement attains its maximum value along the axis l= 0 and asymptotically approaches to zero as l-, QQ for all values of fixed time. From eqn (4.30), as 7 + m, we have +(5, a) =

I0

w~-2J0 VW) exp (ep ‘/4) dp.

(5.2)

This integral does not have any finite value. Physically this means that once the plate is set into motion, the motion continues indefinitely. Since we have neglected the elastic part of the stress tensor such a result is to be expected. 6. NUMERICAL

RESULTS

In this section, we shall discuss the implications of a numerical result obtained from previous analytical treatment. From Bakhsiyan[l9], the appropriate material constants for a steel plate are p = 7.31 X 10e4lb-sec/in4.

(6.1)

p = 5.69 lb-seclin’. From equations (4.20) we find the following expressions for various physical quantities in terms of their nondimensional counterparts for the impact of a 0.30 cal. projectile striking a steel plate at a velocity of 3000 fps (see Pytel and Davids [ 11) t = @ 7 = 2 88 x 10m67set CL * _,,2=E$m=

1.37x 106a, psi

r = Rf = O.lSYin.

(6.2)

By examining the Fig. 3 we see that the peaks in the shear stress occur for small values of impact time. The maximum value of stress is 2.623, and it occurs at the initial contact (7 = 0) of

A nonfocai viscous model for plug formation in plates

297

the projectile with the plate (see Fig. 3). Thus the maximum initial value of the shear stress is 3,~,~psi. This large shear stress must be compared with the cohesive stress holding the atoms of the plate together rather than the engineering yield stress. However, for T = 2(t = 5.76 X 10” set) the maximum stress is 48,000 psi, which is 82 times smaller than its initial value. This value of peak stress is less than what classical local theory predicts. Also, from Fig. 6 it can be seen that for T = l(t = 2.88 x 10m6set) the maximum displacement is approximately 0.05 in., which is almost the same as that found in the classical theory. Thus the viscous effects cause high stress and small displacements in the first few microseconds after impact. As the time increases the maximum stress decreases rapidly and the displacement continues to increase at a slower rate. CONCLUSION

In this paper we have attempted to describe the behavior of a plate subject to impact of a cylindrical projectile that may lead to formation of plug-type of failure. For the material response, a nonlocal viscous fluid model is assumed (linear theory) and all stress components, except the vertical shearing stress, are neglected. Assuming that the plate is sufficiently thin, the variation of field quantities with the thickness of plate is disregarded. The effect of projectile impact is characterized by a uniform initial velocity distribution over a circular region. With this model, the disturbance created by the projectile does not propagate, i.e. it is felt everywhere instantaneously. In contrast to the result of classical analysis, the present solution does not give infinite initial stress on the periphery of the phtg; it rather gives a large but finite stress distribution. But this stress decreases very rapidly with increasing time. For instance, for a typical case, the stress at t = 0 (3,~,~ psi) is 82 times larger than the stress calculated at t = 5.76 x 10d see (~,~ psi). The interestingdint here is that, although there are no stress waves prop~ting in the medium, the peaks of the shear stress move outward with increasing time. REFERENCES [l] A. PYTEL and N. DAVIDS, 1. Frank Institute, p. 394 (1963). [2] R. G. THOMSON, Analysis of Hypemelocity Perforation of a Visco-Plastic Solid including the Targrl Material Yield Strengrh, NASA, TR R-211 (1%5). [3] A. TATE, 1. Mech. Phys. Solids 15,387 (1967). [4] W. GOLDSMITH, Impact. Arnold, London (1960). [S] N. CRISTESCU, Proc. 2nd Symp. on Nasal Structural hie-chanics (Edited by E. H. Lee and P. S. Symonds). Pergamon Press, New York (t!XiO). [6] W. JOHNSON, Impact Stmagth of laterals. Arnold, London (1970). 17] C. ZENNER and R. II. PETERSON, mechanism of Armor Pen~ra~ion. Wa~~own Arsenal Laboratory, Second Partial Report, No. WAL 7101492,(May 1943). ]8] C. ZENNER and J. H. HOLLO~AN, mechanism of A~o~Pe~e~~~io~. Watertown Arseral Laboratory, First Partial Report, No. WAL 710/454,September 1942. 191I. AWERBUCH and S. R. BODNER, Int. I. Soiids S&UC. 10,671 (1974). [IO] I. AWERBUCH, Israeli 1. of Technology 8. 375 (1970). [II] W. GOLDSMITH and S. A. FINNEGAN, Int. 1. Mech. Sci. 13,843 (1971). [12] R. L. WOODWARDand M. E. DeMORTON, Int. J. hfech. Sci. IS, 1IY(1976). [13] G. I. TAYLOR, Pmt. Roy. Sot. London A, 194,289 (1948). [14] A. C., ERINGEN, Lett. Engng Sci. 2, 145(1974). [IS] A. C. ERINGEN, Continuum Physics (Edited by A. C. Eringen), Vol. 4, Part III. Academic Press, New York (1976). I161 A. C. ERINGEN, Int. 1. Engng Sci. 10,561 (1972). [17] R. F. FRECHET and T. W. IPSON, Ballistic Penetration Dynamics, f. Appi. Me& ASME, Series E, 384 (1963). (IS] G. N. WATSON, A Treatise on the Theory of Beset Functions, 2nd Edn. Macmillan, New York (1945). [19] F. A. BAKHISYAN, Frihf, Mat. M&h. I4 (in Russian) (cited by C~stescu[5]~1950).

(Received 10 Juiy 1977)