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On normal impact of an infinite elastic-plastic thin-walled plate by a finite elastic rod
Nuclear Engineeringand Design 37 (1976) 225-230 © North-HollandPublishing Company
ON NORMAL IMPACT OF AN INFINITE ELASTIC-PLASTIC THIN-WALLED PLATE B Y A FINITE ELASTIC R O D * S.J. KOWALSKI, J.A. KO~,ODZIEJ and B. RANIECKI Polish Academy of Sciences, Institute of Fundamental Technological Research, 00-049 Warsaw,Poland
Received 13 December 1975 The problem of the transversecollisionof a finite circular cylinder (fiat-ended metal projectile) of length L and radius Ro, movingwith the initial velocity V0, with an immobilethin infinite plate, is considered. It is assumed that the plate is in a pure shear state. The one-dimensionalwavesboth in the plate (cylindricalelastic-plastic waves) and in cylinder (plane longitudinal elastic waves) are investigated.Graphs"of the peak dimensionless shear stress at the point of impact versus four non-dimensionalquantities representing the material and geometricalproperties of the plate and the projectile, are presented. These graphs allow the projectile speedjust sufficient to initiate ejection of the slug to be estimated.
1. I n t r o d u c t i o n
displacement of the cylinder, and (b) that the resultant forces acting on the plate and the projectile are equal. The aim of this paper is to determine the peak shear stress in the plate at the region of impact r = R 0 through the analysis of one-dimensional transverse cylindrical adiabatic elastic-Plastic waves in the plate. The numerical calculations show that the shear stress at r = R 0 undergoes jump discontinuity at the initial moment of the collision and is then an increasing function of time t until the first longitudinal elastic wave reflected from the free edge of the cylinder reaches the point of contact (z = g/2, fig. 1). The shear stress, therefore, attains its peak at time
The problem of transverse collision of a finite circular cylinder (flat ended metal projectile) of length L and radius R 0, moving with the initial velocity V0, with an immobile thin infmite plate of thickness g (fig. 1) is considered. According to the experimental evidence, at higher speeds (of the order of conventional bullet speed) such a projectile starts to penetrate the plate. The plug diameter usually is about the same size as that of the projectile, and white adiabatic shear line cracks parallel to the projectile are observed [1]. In our theoretical study of this type of impact we assume, therefore, that the shear stresses in the plate play a dominant role and the normal stresses are negligible. Such a simplified assumption is justified only when the speed of the projectile is sufficiently high and when the ratio of the mass of the plug to that of the projectile is less than unity. The problem is treated as a typical contact problem. Thus, at the point of impact we assume (a) that the transverse displacement of the plate equals the longitudinal *
(1)
t* = 2L/at e) ,
~ - ~"~
Paper SI/5 presented at the International Seminar on Extreme Load Conditions and Limit Analysis Procedures for Structural Reactor Safeguardsand Containment Structures (ELCALAP),Berlin, Germany, 8-11 September 1975.
prc~)e~tte /pLug
-W
Fig. 1. Projectile and plate geometry.
225
226
S.J.
Kowalski et aL / Impact o f a plate by an elastic rod
where at e)
=
!
(E/p l ) 1/2
__...L-=-- rp
hrar dl~ ni, n£
2. Basic equations 2.1. Equations o f m o t i o n f o r the plate
Assume that the thin-walled plate is in a pure shear state so that the only non-zero component of the stress tensor in a cylindrical coordinate system z-r-O (fig. 1) is Ozr = r, and the only non-zero component of the mean (along the thickness) displacement vector is u z = w. Assume that both r and w depend solely on two independent variables r and t. The equation of dynamic equilibrium of the plate then becomes 3r+ 1 (3) P20=~ r r, forr>Ro, t>O,
~intttmttcln of ,,~d ¢~bcatic ~ h~gar"
8~
Fig. 2. The adiabatic shear stress-shear strain curve.
and/92 is the material density of the plate. The mean shear strain 3` is defined
(4)
3` = ~ w / ~ r
and is the sum of the elastic 3`e and plastic 3`P portions 3' = "lee + '7p -
(5)
The adiabatic curve ~- - 3`P is approximated by the straight line as shown in fig. 2. Since in the case considered r ~> 0, the constitutive relations are
(6)
3`e = (l/p) T,
, ,~P =
v = aW/at = ~v,
Lt n e ~ r
(2)
is the speed of elastic longitudinal waves in the bar, and E and Pl denote Young's modulus and the density of the cylinder, respectively. We construct in this paper homographs representing the dimensionless peak shear stress as a function of a minimum number of dimensionless quantities involving various material, geometric and kinematic constants. For this purpose the adiabatic shear stressshear strain curve is approximated by the linear hardening model. The universal nomographs presented here may be used to estimate the critical projectile speed at which the projectile starts to perforate the plate, provided that the critical shear stress just sufficient to initiate the so-called 'adiabatic shear' is known. The problem of this type for the rigid viscoplastic model of the plate material and for the rigid projectile was apparently first investigated by Bakhshian [2] and Kochetkov [3] (see also refs. [4] and [5]). The idealization adopted by those authors seems, however, to be more adequate for the phenomena occurring in the second stage of the perforation when the lozalized adiabatic shear is already advanced.
where
/'£-,,
if 7" --
r s --/2p')'P
=
0
and
~"> 0
Up O,
(7) i f 7" - 7"s - / 2 p T P < 0 ,
or if ~- - r s - pp3`P = 0
and ~"~< 0
where r s is the yield limit in pure shear, and/a and btp a r e the elastic shear modulus and the hardening modulus, respectively. Eqs. ( 3 ) - ( 7 ) constitute a complete set of the equations describing the motion of the plate. 2.2. S t r e s s - v e l o c i t y relation f o r an elastic cylinder
Assume that the motion of the cylinder is similar to that of the one-dimensional longitudinal motion of a bar, and denote by Ozz = o(z, t) and u z = u(z, t) (fig. 1) the stress field and longitudinal displacement field of the cylinder, respectively. Since o(g/2 + L, t) = a(z, 0) = 0 and t~(z, 0) = V0, the following relation between stress and particle velocity of the elastic
227
S.Z Kowalski et al. / Impact of a plate by an elastic rod
cylinder is valid (see, for example, ref. [ 1]), o(z, t) = 0
V 0 + a(z, t) + ~
(8)
Plate) for t < t* - (z - g/2)/a~ e). The above relation is used to express the boundary condition for the plate entirely in terms of v and v.
aO_aY+ly, 0F OF e-
2. 3. Initial and boundary conditions
Neglect the inertia of the plug and assume that the resulting force acting on the edge z = g/2 (fig. 1) of the cylinder is equal to the resulting shearing force driving the plug. Moreover, assume that the velocity ~(g/2, t) of the edge z = g/2 of the cylinder is equal to the mean velocity o(R 0, t) of the plug. Using eq. (8) we get the following boundary condition for the plate: Roate) p l .
~-
is the speed of the elastic transverse waves in the plate and r/2 is the coefficient of a linear hardening of the plate. Note that the product M?-* equals the ratio of the total mass of the projectile to the mass of the plug. The basic equations (3)-(7) when written in terms of dimensionless quantities ~-, ~ F andFtake the form
(o + V0) = r ,
M(~ + ~0) = ~-,
dr, O) = 0 .
(10)
2.4. Basic equations expressed in terms o f dimensionless quantities
We shall utilize the following dimensionless quantries f-
a ~e) t
r
F-
R0 '
;
R0
r* -a?)t*_ 2L ( R0
[
#
; T$ C . '
M
r/1 occurring in eq. (12) is defined as follows:
= '/ 1,
t
r/,
in the elastic and unloading regions, (14) in the elastic-plastic regions.
3. Remarks on the solution The set of partial differential equations (12) is hyperbolic. It can be replaced by the following differential relations aiongihe characteristic lines: dff - 1
d~-- 1 y d F = 0, r/1
along F - r/l/= const., (15)
F
d~+ 1 d e _ l ~ - d r 771 F
= O,
along F + r/1F = const. Rmgton oF laLcmLtc der°rmnLitan
~I/2
(I1)
RO [.PlE] 1/2
=¥
for F = 0 .
.~j
,
R 0 \EO 2 ]
0 < F< ?-*,
fore-= 1,
~=0,
a~e)rs'
II0//
v0---q)
ola
~-
:
~f%.gmnLtan o? ,'Loum~.i.c fR
'
dll
Far mo&i.an
~-0; 9-0
where r.1
a ? ) = (e/p2) 1/2
(12)
(13) ~=0;
171
o(r, O) = 0 ,
(F>I),
whereas the boundary condition, eq. (8), and initial conditions become
(9)
for r = R0, 0 < t < t*. It is assumed that the plate is initially immobile and unstressed, i.e.
a~_ 1 aY OF r/~ OF
T
Fig. 3. The wave picturein the t -r plane.
228
S.J. Kowalski et aL / Impact of a plate by an elastic rod
This set of differential relations together with the dynamic and kinematic compatibility conditions have been used to obtain the solution of the problem by means of the finite-difference technique with the step AY = 0.2. The solution in t - V plane is shown in fig. 3. The line Y - 1 = hepresents a strong discontinuity elastic wave. Behind this wave (~-= F + 1 - 0) the solution is ~= --if= (V) -1/2 .
(16)
The region D - O - F is the region of an elastic strain. Along the line V= 1 + r/~-+ 0, it is ~-= 1, and therefore the mixed problem for the set of eqs (12) (taking r/1 = 1) is determined in this region. The line V= 1 + r/trepresents a strong discontinuity plastic loading wave. Knowing the solution in the region D - 0 - F , the dimensionless stress ~-and the velocity 6behind this wave (F= 1 + r/F- 0) may be determined by using the dynamic compatibility condition and the relation, eq. (15)1 (taking r/1 = r/)" Thus, the nature of the mixed problem is again determined in the region O - B - C . This region is the region of an active (r > 0) elastic-plastic deformation. At time F = ?-* and V= 1 the shear stress decreases instantaneously. Therefore the line C - E represents a strong discontinuity unloading wave. Since our main purpose is to investigate the influence of various dimensionless parameters on the peak shear stress ~-(1,T* - 0) the solution is determined only in the region O - A - B - C - O (fig. 3). It is worthwhile to mention that the solution at point V= 1, t- = +0 may be presented explicitly
t-(l, +0) = 1 +
4. Discussion of results The changes in shear stress ~(1, t ) and in mechanical power PM at r = l PM = g(1,?-) V(1,?-)
as time ?- varies, are presented in figs 4 and 5, respectively, for various values of the dimensionless quantities V0, r/and 3/. It is seen from fig. 4 that the shear stress at f = 1 is an increasing function of time. This fact justifies our statement that the peak shear stress is reached at time ?- = T*. The dimensionless peak shear stress is denoted by
¢-p = ?(1,T* - o).
(20)
Since the basic equations (12) and the boundary condition eq. (1_3)1 involve only three dimensionless quantities, V0, M and r/, it follows that the peak shear stress is a function of four dimensionless quantities only (cf. eq. (11))
~-p = f(M, Po, r/, ?-*).
(21)
The mathematical properties of this function for fixed values of V0 may be seen from figs 6 - 8 . The characteristic feature of those diagrams is that there exist certain critical values o f M = M* above which the peak shear stress is 'almost independent' of M. These critical values are not significantly influenced by ?-* (in the range t-* < 15), i.e. M = M*(F"O, r/).
r/[M(V 0 - 1 ) - I]
(191}
(22)
r/+M
M(V"0 -- 1) - 1 0(1,+0) = - 1 +r/+M
(17)
Z(4'i)I
2~-M -M-I -?r/2=°'°1
on account ofeqs (13)1 and (15)2. Using eq. (17) 1 it can be verified easily that the plastic flow occurs at V= 1 provided that the condition
v0 > (l/M) + 1 is satisfied.
(18) Fig. 4. Shear stress versus time at the point of impact.
$.Y. Kowalski et ai. / Impact o[a plate by an elastic rod
229
~,I - 11'
I
I
/0 g.
"~' -( for
f
2.
- - g,~ JMqO
8
f~ S( 4[
7 $
$ 4
1o
3
--
¢-I"3 ~-I*'7.5 3"~"12
i
}~--'- 3f~} for M,g ~.-
2,~JTJ
!
2
Fig. 5. Mechanical power versus time at the point of impact. ,;
10 'g 8
s
4 3 2
'1
8
!
i
I
[ close' to the straight lines. Thus, the function eq. (21) is 'approximately' of the form
I
~p =fl(rl,/-*) + f2(r/, t * )
~
"-'-" -"-"---
""/'1
,, m ~ . • 4.7q ,.,71
fo'r M.IO - i
J~
'
~S21
I
S
four M-~O
l;
7
M
Fig. 6. Peak shear stress versus M for Vo = 20.
V0
8
. "-~'- 7.4S 1
Cr
S
f
3
_L '-o.~ f~r M-to
0
I
:~
6
Fig. 7. Peak shear stress versus M for Vo = 40.
The function eq. (22) is shown in fig. 9 for ,/2 = 0.01 and r/2 = 0.001. The 'asymptotic' values of ~'p corresponding to high values ofM(M ~, M*) are shown in figs 6 - 8 and are indicated by arrows. In fig. 10 ~p versus V0 is plotted for various values o f / * , and f o r M > M * . It is seen that the curves are 'very
(23)
f o r M > M * (V o, n) (fig. 9). It is worthwhile to mention that when M-~ o. the solution of our problem approaches the solution of a problem with the step-velocity type of the boundary condition i~l, 7) = VO" Therefore, the condition M > M* may be regarded as the criterion for the validity of the latter idealization.
t*P-3 2- [*- 7,s
6
M
Fig. 8. Peak shear stress versus M for F'o = 60.
- E' -3 2- ~*'7,~ 3" ~ * ' f 2
7 G
7
Fig. 9. Critical value o f Mversus the initial velocity o f a projectile.
230
S.J. Kowalski et al. / Impact o f a plate by an elastic' rod
The answer is obtained as lollows. The above data give
it
t f*~
12 "
VO ~ 4 0 •
r~2 ~ 0.001 '
r-cr ~ 2 . 5 , '
.
4
•
'
and from fig. 7 we find
,
M= 0.6. The plate thickness corresponding to the value M = 0.6 is g=4cm. ~b 2b ~o ~o to
6o
Tb
Fig. I0. Peak shear stress versus F'o for supercritical values of M.
References
5. Simple example Assume that the plate is made of mild steel and has the temperature 400°C. For mild steel at this temperature and a strain rate o f the order 10 - 2 sec, it is ¢cr ~" 2600 kg/cm 2 [7], ~'s ~ 1050 kg/cm 2, /1 ~ 6.5 X 105 kg/cm 2,/a D ~ 700 kg/cm 2 and P2 = 7.8 X lO - 6 kg sec 2 c m - 4 . Suppose that the cylinder moving with the initial velocity V0 = 190 m/sec is also made o f the mild steel but has the temperature 20°C, so that P l = P2, E = 2 X l06 kg/cm 2. Let the cylinder have the following dimensions: R 0 = 2.8 c m ; L = 30 cm. Find the critical plate thickness at which the localized shearing is imminent.
[ 1 ] W. Johnson, Impact Strength of Materials, Arnold, London (1972) ch. 10. [2] A. Bakhshian, On the visco-plastic flow during impact of a plate by a cylinder, Prikl. Math. Mekh. 1 (XII) (1948) (in Russian). [3] A.M. Kochetkov, Approximate solutions of some problems of non-steady plastic flow, Prikl. Math. Mekh. 5 (XIV) (1950) (in Russian). [4] X.A. Rakhmatulin and Yu.A. Demianov, Strenght of Materials at Impact, Moscow (1961) (in Russian). [5] N. Cristescu, Dynamic Plasticity, North-Holland, Amsterdam (1967). [6] R.F. Recht, Catastrophic thermoplastic shear, J. Appl. Mech., Trans. ASME, Ser. E 31 (2) (1964). [7] Z. Mr6z and B. Raniecki, On the uniqueness problem in coupled thermoplasticity, to appear in Int. J. Eng. Sci. 14 (1976).
ON NORMAL IMPACT OF AN INFINITE ELASTIC-PLASTIC THIN-WALLED PLATE B Y A FINITE ELASTIC R O D * S.J. KOWALSKI, J.A. KO~,ODZIEJ and B. RANIECKI Polish Academy of Sciences, Institute of Fundamental Technological Research, 00-049 Warsaw,Poland
Received 13 December 1975 The problem of the transversecollisionof a finite circular cylinder (fiat-ended metal projectile) of length L and radius Ro, movingwith the initial velocity V0, with an immobilethin infinite plate, is considered. It is assumed that the plate is in a pure shear state. The one-dimensionalwavesboth in the plate (cylindricalelastic-plastic waves) and in cylinder (plane longitudinal elastic waves) are investigated.Graphs"of the peak dimensionless shear stress at the point of impact versus four non-dimensionalquantities representing the material and geometricalproperties of the plate and the projectile, are presented. These graphs allow the projectile speedjust sufficient to initiate ejection of the slug to be estimated.
1. I n t r o d u c t i o n
displacement of the cylinder, and (b) that the resultant forces acting on the plate and the projectile are equal. The aim of this paper is to determine the peak shear stress in the plate at the region of impact r = R 0 through the analysis of one-dimensional transverse cylindrical adiabatic elastic-Plastic waves in the plate. The numerical calculations show that the shear stress at r = R 0 undergoes jump discontinuity at the initial moment of the collision and is then an increasing function of time t until the first longitudinal elastic wave reflected from the free edge of the cylinder reaches the point of contact (z = g/2, fig. 1). The shear stress, therefore, attains its peak at time
The problem of transverse collision of a finite circular cylinder (flat ended metal projectile) of length L and radius R 0, moving with the initial velocity V0, with an immobile thin infmite plate of thickness g (fig. 1) is considered. According to the experimental evidence, at higher speeds (of the order of conventional bullet speed) such a projectile starts to penetrate the plate. The plug diameter usually is about the same size as that of the projectile, and white adiabatic shear line cracks parallel to the projectile are observed [1]. In our theoretical study of this type of impact we assume, therefore, that the shear stresses in the plate play a dominant role and the normal stresses are negligible. Such a simplified assumption is justified only when the speed of the projectile is sufficiently high and when the ratio of the mass of the plug to that of the projectile is less than unity. The problem is treated as a typical contact problem. Thus, at the point of impact we assume (a) that the transverse displacement of the plate equals the longitudinal *
(1)
t* = 2L/at e) ,
~ - ~"~
Paper SI/5 presented at the International Seminar on Extreme Load Conditions and Limit Analysis Procedures for Structural Reactor Safeguardsand Containment Structures (ELCALAP),Berlin, Germany, 8-11 September 1975.
prc~)e~tte /pLug
-W
Fig. 1. Projectile and plate geometry.
225
226
S.J.
Kowalski et aL / Impact o f a plate by an elastic rod
where at e)
=
!
(E/p l ) 1/2
__...L-=-- rp
hrar dl~ ni, n£
2. Basic equations 2.1. Equations o f m o t i o n f o r the plate
Assume that the thin-walled plate is in a pure shear state so that the only non-zero component of the stress tensor in a cylindrical coordinate system z-r-O (fig. 1) is Ozr = r, and the only non-zero component of the mean (along the thickness) displacement vector is u z = w. Assume that both r and w depend solely on two independent variables r and t. The equation of dynamic equilibrium of the plate then becomes 3r+ 1 (3) P20=~ r r, forr>Ro, t>O,
~intttmttcln of ,,~d ¢~bcatic ~ h~gar"
8~
Fig. 2. The adiabatic shear stress-shear strain curve.
and/92 is the material density of the plate. The mean shear strain 3` is defined
(4)
3` = ~ w / ~ r
and is the sum of the elastic 3`e and plastic 3`P portions 3' = "lee + '7p -
(5)
The adiabatic curve ~- - 3`P is approximated by the straight line as shown in fig. 2. Since in the case considered r ~> 0, the constitutive relations are
(6)
3`e = (l/p) T,
, ,~P =
v = aW/at = ~v,
Lt n e ~ r
(2)
is the speed of elastic longitudinal waves in the bar, and E and Pl denote Young's modulus and the density of the cylinder, respectively. We construct in this paper homographs representing the dimensionless peak shear stress as a function of a minimum number of dimensionless quantities involving various material, geometric and kinematic constants. For this purpose the adiabatic shear stressshear strain curve is approximated by the linear hardening model. The universal nomographs presented here may be used to estimate the critical projectile speed at which the projectile starts to perforate the plate, provided that the critical shear stress just sufficient to initiate the so-called 'adiabatic shear' is known. The problem of this type for the rigid viscoplastic model of the plate material and for the rigid projectile was apparently first investigated by Bakhshian [2] and Kochetkov [3] (see also refs. [4] and [5]). The idealization adopted by those authors seems, however, to be more adequate for the phenomena occurring in the second stage of the perforation when the lozalized adiabatic shear is already advanced.
where
/'£-,,
if 7" --
r s --/2p')'P
=
0
and
~"> 0
Up O,
(7) i f 7" - 7"s - / 2 p T P < 0 ,
or if ~- - r s - pp3`P = 0
and ~"~< 0
where r s is the yield limit in pure shear, and/a and btp a r e the elastic shear modulus and the hardening modulus, respectively. Eqs. ( 3 ) - ( 7 ) constitute a complete set of the equations describing the motion of the plate. 2.2. S t r e s s - v e l o c i t y relation f o r an elastic cylinder
Assume that the motion of the cylinder is similar to that of the one-dimensional longitudinal motion of a bar, and denote by Ozz = o(z, t) and u z = u(z, t) (fig. 1) the stress field and longitudinal displacement field of the cylinder, respectively. Since o(g/2 + L, t) = a(z, 0) = 0 and t~(z, 0) = V0, the following relation between stress and particle velocity of the elastic
227
S.Z Kowalski et al. / Impact of a plate by an elastic rod
cylinder is valid (see, for example, ref. [ 1]), o(z, t) = 0
V 0 + a(z, t) + ~
(8)
Plate) for t < t* - (z - g/2)/a~ e). The above relation is used to express the boundary condition for the plate entirely in terms of v and v.
aO_aY+ly, 0F OF e-
2. 3. Initial and boundary conditions
Neglect the inertia of the plug and assume that the resulting force acting on the edge z = g/2 (fig. 1) of the cylinder is equal to the resulting shearing force driving the plug. Moreover, assume that the velocity ~(g/2, t) of the edge z = g/2 of the cylinder is equal to the mean velocity o(R 0, t) of the plug. Using eq. (8) we get the following boundary condition for the plate: Roate) p l .
~-
is the speed of the elastic transverse waves in the plate and r/2 is the coefficient of a linear hardening of the plate. Note that the product M?-* equals the ratio of the total mass of the projectile to the mass of the plug. The basic equations (3)-(7) when written in terms of dimensionless quantities ~-, ~ F andFtake the form
(o + V0) = r ,
M(~ + ~0) = ~-,
dr, O) = 0 .
(10)
2.4. Basic equations expressed in terms o f dimensionless quantities
We shall utilize the following dimensionless quantries f-
a ~e) t
r
F-
R0 '
;
R0
r* -a?)t*_ 2L ( R0
[
#
; T$ C . '
M
r/1 occurring in eq. (12) is defined as follows:
= '/ 1,
t
r/,
in the elastic and unloading regions, (14) in the elastic-plastic regions.
3. Remarks on the solution The set of partial differential equations (12) is hyperbolic. It can be replaced by the following differential relations aiongihe characteristic lines: dff - 1
d~-- 1 y d F = 0, r/1
along F - r/l/= const., (15)
F
d~+ 1 d e _ l ~ - d r 771 F
= O,
along F + r/1F = const. Rmgton oF laLcmLtc der°rmnLitan
~I/2
(I1)
RO [.PlE] 1/2
=¥
for F = 0 .
.~j
,
R 0 \EO 2 ]
0 < F< ?-*,
fore-= 1,
~=0,
a~e)rs'
II0//
v0---q)
ola
~-
:
~f%.gmnLtan o? ,'Loum~.i.c fR
'
dll
Far mo&i.an
~-0; 9-0
where r.1
a ? ) = (e/p2) 1/2
(12)
(13) ~=0;
171
o(r, O) = 0 ,
(F>I),
whereas the boundary condition, eq. (8), and initial conditions become
(9)
for r = R0, 0 < t < t*. It is assumed that the plate is initially immobile and unstressed, i.e.
a~_ 1 aY OF r/~ OF
T
Fig. 3. The wave picturein the t -r plane.
228
S.J. Kowalski et aL / Impact of a plate by an elastic rod
This set of differential relations together with the dynamic and kinematic compatibility conditions have been used to obtain the solution of the problem by means of the finite-difference technique with the step AY = 0.2. The solution in t - V plane is shown in fig. 3. The line Y - 1 = hepresents a strong discontinuity elastic wave. Behind this wave (~-= F + 1 - 0) the solution is ~= --if= (V) -1/2 .
(16)
The region D - O - F is the region of an elastic strain. Along the line V= 1 + r/~-+ 0, it is ~-= 1, and therefore the mixed problem for the set of eqs (12) (taking r/1 = 1) is determined in this region. The line V= 1 + r/trepresents a strong discontinuity plastic loading wave. Knowing the solution in the region D - 0 - F , the dimensionless stress ~-and the velocity 6behind this wave (F= 1 + r/F- 0) may be determined by using the dynamic compatibility condition and the relation, eq. (15)1 (taking r/1 = r/)" Thus, the nature of the mixed problem is again determined in the region O - B - C . This region is the region of an active (r > 0) elastic-plastic deformation. At time F = ?-* and V= 1 the shear stress decreases instantaneously. Therefore the line C - E represents a strong discontinuity unloading wave. Since our main purpose is to investigate the influence of various dimensionless parameters on the peak shear stress ~-(1,T* - 0) the solution is determined only in the region O - A - B - C - O (fig. 3). It is worthwhile to mention that the solution at point V= 1, t- = +0 may be presented explicitly
t-(l, +0) = 1 +
4. Discussion of results The changes in shear stress ~(1, t ) and in mechanical power PM at r = l PM = g(1,?-) V(1,?-)
as time ?- varies, are presented in figs 4 and 5, respectively, for various values of the dimensionless quantities V0, r/and 3/. It is seen from fig. 4 that the shear stress at f = 1 is an increasing function of time. This fact justifies our statement that the peak shear stress is reached at time ?- = T*. The dimensionless peak shear stress is denoted by
¢-p = ?(1,T* - o).
(20)
Since the basic equations (12) and the boundary condition eq. (1_3)1 involve only three dimensionless quantities, V0, M and r/, it follows that the peak shear stress is a function of four dimensionless quantities only (cf. eq. (11))
~-p = f(M, Po, r/, ?-*).
(21)
The mathematical properties of this function for fixed values of V0 may be seen from figs 6 - 8 . The characteristic feature of those diagrams is that there exist certain critical values o f M = M* above which the peak shear stress is 'almost independent' of M. These critical values are not significantly influenced by ?-* (in the range t-* < 15), i.e. M = M*(F"O, r/).
r/[M(V 0 - 1 ) - I]
(191}
(22)
r/+M
M(V"0 -- 1) - 1 0(1,+0) = - 1 +r/+M
(17)
Z(4'i)I
2~-M -M-I -?r/2=°'°1
on account ofeqs (13)1 and (15)2. Using eq. (17) 1 it can be verified easily that the plastic flow occurs at V= 1 provided that the condition
v0 > (l/M) + 1 is satisfied.
(18) Fig. 4. Shear stress versus time at the point of impact.
$.Y. Kowalski et ai. / Impact o[a plate by an elastic rod
229
~,I - 11'
I
I
/0 g.
"~' -( for
f
2.
- - g,~ JMqO
8
f~ S( 4[
7 $
$ 4
1o
3
--
¢-I"3 ~-I*'7.5 3"~"12
i
}~--'- 3f~} for M,g ~.-
2,~JTJ
!
2
Fig. 5. Mechanical power versus time at the point of impact. ,;
10 'g 8
s
4 3 2
'1
8
!
i
I
[ close' to the straight lines. Thus, the function eq. (21) is 'approximately' of the form
I
~p =fl(rl,/-*) + f2(r/, t * )
~
"-'-" -"-"---
""/'1
,, m ~ . • 4.7q ,.,71
fo'r M.IO - i
J~
'
~S21
I
S
four M-~O
l;
7
M
Fig. 6. Peak shear stress versus M for Vo = 20.
V0
8
. "-~'- 7.4S 1
Cr
S
f
3
_L '-o.~ f~r M-to
0
I
:~
6
Fig. 7. Peak shear stress versus M for Vo = 40.
The function eq. (22) is shown in fig. 9 for ,/2 = 0.01 and r/2 = 0.001. The 'asymptotic' values of ~'p corresponding to high values ofM(M ~, M*) are shown in figs 6 - 8 and are indicated by arrows. In fig. 10 ~p versus V0 is plotted for various values o f / * , and f o r M > M * . It is seen that the curves are 'very
(23)
f o r M > M * (V o, n) (fig. 9). It is worthwhile to mention that when M-~ o. the solution of our problem approaches the solution of a problem with the step-velocity type of the boundary condition i~l, 7) = VO" Therefore, the condition M > M* may be regarded as the criterion for the validity of the latter idealization.
t*P-3 2- [*- 7,s
6
M
Fig. 8. Peak shear stress versus M for F'o = 60.
- E' -3 2- ~*'7,~ 3" ~ * ' f 2
7 G
7
Fig. 9. Critical value o f Mversus the initial velocity o f a projectile.
230
S.J. Kowalski et al. / Impact o f a plate by an elastic' rod
The answer is obtained as lollows. The above data give
it
t f*~
12 "
VO ~ 4 0 •
r~2 ~ 0.001 '
r-cr ~ 2 . 5 , '
.
4
•
'
and from fig. 7 we find
,
M= 0.6. The plate thickness corresponding to the value M = 0.6 is g=4cm. ~b 2b ~o ~o to
6o
Tb
Fig. I0. Peak shear stress versus F'o for supercritical values of M.
References
5. Simple example Assume that the plate is made of mild steel and has the temperature 400°C. For mild steel at this temperature and a strain rate o f the order 10 - 2 sec, it is ¢cr ~" 2600 kg/cm 2 [7], ~'s ~ 1050 kg/cm 2, /1 ~ 6.5 X 105 kg/cm 2,/a D ~ 700 kg/cm 2 and P2 = 7.8 X lO - 6 kg sec 2 c m - 4 . Suppose that the cylinder moving with the initial velocity V0 = 190 m/sec is also made o f the mild steel but has the temperature 20°C, so that P l = P2, E = 2 X l06 kg/cm 2. Let the cylinder have the following dimensions: R 0 = 2.8 c m ; L = 30 cm. Find the critical plate thickness at which the localized shearing is imminent.
[ 1 ] W. Johnson, Impact Strength of Materials, Arnold, London (1972) ch. 10. [2] A. Bakhshian, On the visco-plastic flow during impact of a plate by a cylinder, Prikl. Math. Mekh. 1 (XII) (1948) (in Russian). [3] A.M. Kochetkov, Approximate solutions of some problems of non-steady plastic flow, Prikl. Math. Mekh. 5 (XIV) (1950) (in Russian). [4] X.A. Rakhmatulin and Yu.A. Demianov, Strenght of Materials at Impact, Moscow (1961) (in Russian). [5] N. Cristescu, Dynamic Plasticity, North-Holland, Amsterdam (1967). [6] R.F. Recht, Catastrophic thermoplastic shear, J. Appl. Mech., Trans. ASME, Ser. E 31 (2) (1964). [7] Z. Mr6z and B. Raniecki, On the uniqueness problem in coupled thermoplasticity, to appear in Int. J. Eng. Sci. 14 (1976).