A simple model for predicting energy dissipation of thin plates being perforated by “hard” missiles

Nuclear Engineering and Design 51 (1979) 157-161 © North-Holland Publishing Company

157

A SIMPLE MODEL FOR PREDICTING ENERGY DISSIPATION OF THIN PLATES BEING PERFORATED BY "HARD" MISSILES Ernst LIMBERGER Fachgruppe Tragfiihigkeit der Baukonstruktionen, Bundesanstalt 1"f~rMaterialpriifung (BAM}, D-1000 Berlin 45, Fed. Rep. Germany Received 22 May 1978

For the case of low velocity impact a simple model is derived for the determination of energy dissipation of thin plates being perforated by "hard" missiles. The predicted residual energy of the missile having passed through the target is compared with test results. The tests were carried out with plates made of wood-chips (a rather homogeneous and cheap materlal). For a projectile with large diameter relative to the thickness of the target it is shown that the energy absorption of the plate is essentially influenced by the fracture type.

1. Introduction Structures are often required to withstand the effects of missile impact. In this connection the question of energy absorption of the impacted target has to be solved. Considering for example a structure that consists of wails or shells in series where only the last one is not allowed to be perforated, one has to predict the exit energy of the missile after perforation of a single wall, i.e. after having passed through the wall. Assuming nondeformable missiles there are two main problems to be discussed. (1) If the diameter of the missile is small relative to the thickness of the target a large amount of the impact energy is needed for local damage and penetration of the missile into the target, depending upon the projectile shape. This problem only can be solved by using empirical formulae based on experimental results. (2) If the diameter of the fiat-ended missile is large relative to the thickness of the target, only a small amount of impact energy is needed for local indentation effects. If no perforation occurs almost all the kinetic energy will be converted into strain energy. This absorbed energy results in an overall target response that includes flexural deformation and deformation due to local shear effects.

This paper deals specifically with the second case postulating thin target wails and low impact velocities, neglecting "strain-rate" effects.

2. Failure modes of plates impacted by hard missiles For flat-ended "hard" missiles with a diameter small relative to the plate size and large relative to the plate thickness, one can classify two main failure modes. (a) For a "primary perforation" failure happens because of low punching shear capacity. This local failure mode should not be mixed up with penetration effects. (b) For a "secondary perforation" the missile passes through the target if the total strain energy capacity is exceeded. Between these two failure modes there is a great difference in the energy absorption of the plate. The type of failure mode that occurs depends upon the impact capacity, wave propagation conditions, and the relationship between the punching resistance and the failure in flexure capacity. The connection between the energy to obtain failure and the failure mode can be seen in a static experiment. In fig. 1 we show two load deflection curves with their failure

E. Limberger / Energy dissipation of thin plates perforated by "'hard" missiles

158

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(kN)

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Fig. 2. Simplified schematic for impact perforation analysis of plates, symbols used. (a) "Primary perforation" at the end of phase 1; (b) "secondary perforation" at the end of phase 2.

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Fig. 1. Connection of failure mode and work done by an external single static load (wood-chip plates, 13 mm, size 1.0 X 1.0 m, punching area 20 cm2). (a) Local failure; (b) overall failure.

modes obtained from tests carried out at wood-chip plates (single loaded with a punching area of 20 cm2). The total static work done until punching occurs differs by a factor of 2.7.

3. Determination of energy dissipation based on the model of an inelastic impact In this section the impact of a "hard" missile against a thin plate is studied using the model of an inelastic impact of short duration and assuming the structure to behave perfectly plastic. The effect of inertia will be taken into account by using an effective mass of the plate concentrated at the point of impact.

3.1. Model Fig. 2 illustrates schematically the impact effects during time assuming two ranges of impact time. For the first range the impact can be approximately assumed to be a "plastic" impact of short duration. At the end of the first impact range the missile mass ml and the effective inertial mass of the plate m 2 have the same velocity. In the case of "plastic impact" conservation of energy and momentum requires the energy loss ~ " associated with local plastic deformation of the plate (assuming a "hard" nondeformable missile). If AE exceeds the maximum local failure energy, AE m "primary perforation" occurs, see fig. 2a. For this case the velocities at the end of the first range are unequal. If AE is less than AE B the structure has to absorb the kinetic energy E 2 during the second impact range, assumed to be of long duration. The missile passes through the plate ("secondary perforation") if the total strain energy E2B is exceeded, see fig. 2b. As a simplification no rebound is assumed.

3. 2. Determination o f the energy dissipation In the case of "plastic impact", conservation of momentum and energy result in

E. Limberger /Energy dissipation of thin plates perforated by "hard" missiles mlVla = mlV~ + m2V~, mx

m x V,l: + ?

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(1) V~:

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(2)

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/Z m /

a)

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If ZkE is less than ZkEB, the mass m l and the mass m2 have the same velocity at the end of the first impact range. Their kinetic energy is called E 2. F r o m eqs. (1) and (2), using the mass ratio M = mx/m2, we find

1

E2

= E1 M + 1 '

(3)

M Ea M + 1

(4)

=

If the impact energy E 1 is such that the critical value z2xEB or E2B is not exceeded, the following equations for the limiting curve o f E 1 with two branches are obtained:

Eli = Z~'B(1 E l l l = E2B

+M'),

M+I M

(7)

(8)

and is given by 2M+l M EA = ZkEB M + 1 +E2BM--+ 1 " eq. (9) is shown in fig. 3a b y curve D. If the impact energy E 1 exceeds a magnitude o f

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b)

t

(6)

see fig. 3a, curve C. For the case o f "secondary perforation" the enlargement of E 1 will at first cause an increase o f energy absorption E A up to a limiting value determined by reaching the local failure limit. This limiting value E A is related to the impact energy

El = ZXEB(1 + M)

m2

(5)

With the assumption made in section 3 . 1 , the limit values calculated by eq. (5) or (6) are the limit for the energy absorption E A for the case that no perforation occurs, see fig. 3a, curves A and B. If the impact energy E , increases, perforation will occur. Which perforation type occurs depends upon the magnitude of E l , M, ZkEB and E2B. For the case o f "primary perforation" we will obtain a limit value for the energy absorption E A with V~ = V~: 2M+l EA = z~EB +M- - - - T '

i

(9)

EI

f Fig. 3. Model for impact perforation analysis. (a) Energy absorption E A as a function of mass ratio M (shown for the ratio E2B/Z~?B = 4): A = limiting curve ofE A for "local" failure; B = limiting curve ofE A for total failure; C = E A for "primary perforation"; and D = maximum E A for "secondary perforation". (b) Energy absorption E A versus impact energy E 1 for the mass ratio M = Mc. (e) Energy absorption E A versus impact energy E l for the mass ratio M = Md.

E 1 defined by eq. (8), there will be a change of failure mode: the energy absorption E A will drop to a magnitude given by eq. (7). A further increase o f impact energy will cause a further decrease o f E A. Starting with eqs: (1) and (2) E A is obtained by

m2 V.~2,

EA:ZXEB+-T

(10)

E. Limberger /Energy dissipation of thin plates perforated by "hard" missiles

160

(11) white E 1/> AEa(M + 1). I f E 1 > > AE B the limit value of E A is AE a. For two selected ratios o f m l / m 2 we obtain the curves for energy absorption E A versus impact energy E 1 shown in fig. 3b and 3c.

can be described by the model, see fig. 3a, curves A and B. Finally, in fig. 5 the energy absorption E a is shown as a function of the impact energy E 1 for two sizes and fixed mass ratios. The energy absorption E A given by fig. 5a is to be compared with fig. 3b, and E A given by fig. 5b is to be compared with fig. 3c. Fig. 5a shows the effects that describe "prim'ary perforation". Fig. 5b shows the effects that describe "secondary perforation" which changes to "primary perforation" for greater impact velocities.

4. Experimental results Impact tests carried out on plates of different materials are described in [1]. The test results are in good agreement with the effects described in section 3 regarding the limitation of the model. For example, we show in fig. 4 the "perforation resistance" of wood-chip plates of two different sizes versus the impacting mass m 1. The "perforation resistance" is defined by the average impact energy E o which just forces perforation. For the large plate size (curve E) and low mass ratio M, the increase of impact mass rn 1 will cause an increase Of ED. For small plates (curve F) and greater mass ratio M, the increase of impact mass m I will cause a decrease o f E o. This behaviour also

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Fig. 4. Results of impact tests, Perforation resistance E D as a function of impact mass m 1 and plate size (wood-chip plates 13 mm, punching area 20 cm2): E = size 1.0 m X 1.0 m ; F = size 0.25 m X 0.25 m.

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Fig. 5. Results of impact tests. Energy absorption E A as a function of impact energy E l and plate size (wood-chip plates 13 ram, impact massm z = 1 kg): (a) size 1.0 m × 1.0 m, punching area 20 cm2; (b) size 0.25 m X 0.25 m, punching area 10 era2. O,test without perforation; × test with perforation.

-~

E. Limberger / Energy dissipation of thin plates perforated by "hard" missiles 5. Conclusions The model described in section 3 shows, for the case of a hard missile impact, that different energy absorptions can be obtained depending upon the fracture mechanisms caused. It was assumed that the energy absorption will mainly be influenced by two failure modes: the "primary perforation" corresponds to local shear failure and the "secondary perforation" corresponds to overall flexure. To obtain those failure types the two failure energies, AE B and E2B, were assumed to be necessary. The applicability was reduced to thin plates and a large diameter of the punching area relative to the plate thickness and flat-ended missiles. Therefore a third fracture mechanism, corresponding to indentation and penetration problems, could be neglected. For an application of the model presented here the fracture energies AE B and E2B and the effective target mass during impact should be determined. The total failure energy E2B can be conservatively evaluated by a yield line analysis using an estimation of the ultimate rotational capacity of the yield hinges [2,3]. A simplified technique to determine the effective target mass is contained in refs. [3] and [4]. Methods to estimate the local failure energy &E B are not well established. This local collapse energy could be obtained by performing a nonlinear analysis of a clamped circular plate section, the diameter of which depends upon the diameter of the impacting mass and the plate thickness. The analysis has to determine the external work done under quasi-static conditions until failure

(2) For a high value of the mass ratio M impact will at first cause "secondary perforation". If the velocity exceeds the first perforation point, the absorbed energy will increase. Further increases in velocity will cause a change of failure mode into "primary perforation" with a further decrease of the absorbed energy. (3) For structures consisting of wails or shells in series it follows that the perforation resistance of the whole construction might be by a multiple less than the sum of the single perforation resistances.

Nomenclature E1

AE E2

EA AEB

E2B ml m2 M

Via

vi

v~'

occurs.

Another method is to estimate, as far as possible, an approximate value of E2B and AE B based on external work done in static experiments. Based on the simple model, as shown above, the following conclusions for the energy dissipation during impact up to final perforation can be made. (1) The energy absorbed depends upon the local failure energy AEB,the total failure energy EZB, the ratio mass of the missile to the effective mass of the target, and the impact energy. For a low value of the mass ratio M impact will always cause "primary perforation". The maximum adsorbed energy results at the limit impact energy when perforation first occurs. With increasing velocity the absorbed energy will decrease.

161

ED

= total kinetic energy of missile just prior to impact = energy loss during inelastic impact = kinetic energy of missile and effective mass of target = energy absorbed by the target = maximum local failure energy = maximum total failure energy = mass of missile = effective mass of target = ratio o f m 1 to m 2 = the missile's initial velocity = velocity of rn I at the end of the first impact range = velocity of m 2 at the end of the first impact range = velocity of m 1 at the end of the second impact range = velocity of rn 2 at the end of the second impact range = "perforation resistance", defined by the average impact energy just forcing perforation.

References [1] E. Limberger, BAM-Berichte no. 35, Berlin (1976). [2] W. Struck, E. Limberger and H. Eiffel 2nd SMIRT Conf., Berlin (1973) Paper J3/4. [3] R.P. Kennedy, ELCALAP Seminar, Berlin (1975) Paper Sl/1.

[4] C.H. Norris et al., Structured Design for Dynamic Loads (McGraw-HiU,New York, 1959) p. 132.