*Corresponding author.Fax:+65-6791-0676.
E-mail address:cqmli@ntu.edu.sg(Q.M.Li).
0734-743X/02/$-e front matter r2002Elvier Science Ltd.All rights rerved. PII:S0734-743X(02)00005-2
In general,deep penetration of non-deformable projectile depends on the impact velocity,geometry a
nd mass of the projectile,the material properties of the target and the frictional resistance at the projectile–target interface.Backman and Goldsmith [18],Zukas [19]and Corbett et al.[20]surveyed the rearch field of penetration mechanics and discusd the dependence of penetration mechanism on the factors.Young [21]carried out a number of experiments to determine the effects of various parameters,including projectile no shape,projectile mass and area and impact velocity,on the penetration depth.However,the functions of the listed parameters in a penetrating process are still not fully understood.For example,the vague definition of a no shape factor may cau uncertainties in the application of an empirical formula.Most empirical formulae on penetration depth are purely determined by the curve fitting of the test data and show unit-dependence.Meanwhile,it is still a challenge to evaluate the effect of dynamic sliding friction between projectile and target during penetration.
A dimensional analysis was conducted for the penetration of a non-deformable projectile into reinforced and plain concrete targets [22].It was found that only two dimensionless ,the impact function ðI Þand the geometry function of projectile ðN Þ;as combinations of dominant physical quantities,determine the final penetration depth in a mi-infinite concrete target hit by a non-deformable projectile.
In the prent paper,a general geometry function is introduced to define the geometrical characteristics of a projectile.Bad on the dimensional analysis and the dynamic cavity-expansion model,a non-dimensional formula of penetration depth is recommended for deep penetration by a non-deformable projectile.Comparison between predictions and penetration tests is performed for a range of no shapes,impact velocities and target mediums and shows good agreement when the penetration depth is larger than the projectile diameter and the projectile no length while projectile remains rigid without noticeable deformation and damage.
2.No geometry of a non-deformable projectile
散打比赛规则A non-deformable projectile having arbitrary no shape,as shown in Fig.1,impacts a target normally at velocity V 0and proceeds to penetrate the target medium at rigid-body velocity V :
X , sin )
Fig.1.Side outline of an arbitary no.
X.W.Chen,Q.M.Li /International Journal of Impact Engineering 27(2002)619–637
620
The dynamic cavity-expansion analysis yields the following relation between the normal compressive stress s n on the projectile no and the normal expansion velocity v[5–9]: s n¼AYþB r v2;ð1Þ
where Y and r are yielding stress and density of target material,respectively.A and B are dimensionless material constants.
The particle velocity at no–target interface caud by the rigid-body velocity V of a projectile is
v¼V cos y:ð2Þ
The tangential stress on the no is presumably determined by the frictional resistance on the interface
s t¼m m s n;ð3Þ
where m m is the sliding friction coefficient in impact.
The resulting axial resistant force on the projectile no can be integrated from the normal compressive stress and the tangential [9,22,23]).Thus,
F x¼p d2
4
ðAYN1þB r V2N2Þ;ð4Þ
where d is the calibre diameter of projectile shank.Here two dimensionless parameters relating to the no shape and friction are introduced:
N1¼1þ4m m a A
n
sin y d A
p d
;ð5aÞ
N2¼N nþ4m m a A
n
cos2y sin y d A浣溪沙贺铸
p d2
;ð5bÞ
in which
N n¼4a
A n
cos3y d A
p d
;ð5cÞ
where all the integrals are performed on the no surface A n:Specially,if m m¼0;then N1¼1and N2¼N n:N n is a type of‘‘no factor’’to reflect the geometrical characteristics of the projectile no,which plays an important role in geometry optimisation for a projectile.
If the no shape can be reprented by the no shape function y¼yðxÞfor an arbitrary no shape,as shown in Fig.1,then
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N1¼1þ8m m
d2
Z h
y d x;ð6aÞ
N2¼N nþ8m m
老生常谈什么意思d三建乡
Z h
yy02
五味俱全1þy
d x;ð6bÞ
X.W.Chen,Q.M.Li/International Journal of Impact Engineering27(2002)619–637621
N n ¼8d 2Z h 0yy 031þy 02d x ;ð6c Þ
天门冬的功效与作用where h is the height of no,as shown in Fig.1[23].
Basically,a non-deformable projectile is approximately characterid by its mass M ;shank diameter d and no shape function y ¼y ðx Þ:For complicated no shape,veral characteristic geometry dimensions are required to define the no shape function.For example,the diameter of the truncated area d 1in Fig.3is introduced for truncated-ogive projectile in addition to s and d :It is reaso
nable to assume that the length of shank has little influence on penetration.However,the projectile mass,reprented usually by calibre density M =d 3;is important for penetration.Figs.2–5show some common ,ogive,truncated-ogive,conical and blunt nos as well as their characteristic dimensions.A dimensionless no parameter is defined as
c ¼s d
;ð7Þfor various no shapes in Figs.2–5.For ogive projectile c is the caliber-radius-head (CRH).Correspondingly,the no shape parameters in Eqs.(6a)–(6c)for the common no shapes are calculated as follows:
Ogive no [10]in Fig.2:N 1¼1þ4m m c 2
p 2Àf 0 Àsin 2f 02
!;ð8a ÞN 2¼N n þm m c 2p 2Àf 0 À132sin 2f 0þsin 4f 04
!;ð8b ÞN n ¼13c À124c
2;0o N n p 12;ð8c Þf 0¼sin À11À12c ;where c X 12:ð8d
Þ
Conical no [10]in Fig.4:
N 1¼1þ2m m c ;N 2¼N n þ2m
m c
1þ4c 2;N n ¼1
1þ4c 2;0o N n p 1:
ð9a ;b ;c ÞBlunt no in Fig.5:
N 1¼1þ2m m c 2ð2f 0Àsin 2f 0Þ;ð10a
Þ
Fig.4.Conical no geometry.
X.W.Chen,Q.M.Li /International Journal of Impact Engineering 27(2002)619–637
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