Plastic deformation behind strong shock waves

Mechanics of Materials 5 (1986) 13-28 North-Holland

13

PLASTIC DEFORMATION BEHIND STRONG SHOCK WAVES * J. W E E R T M A N ** Center for Materials Science, Los Alamos National Laboratory, Los Alamos, NM87545, U.S.A.

Received 5 September 1985

The dislocation produced plastic deformation that must occur behind the front of strong shock waves is analysed in this paper. (A strong shock has a driving stress that is large compared with the bulk elastic modulus.) Because the front of strong shock waves must be exceedingly narrow, there is a problem of how the effective shear stress, which is probably of the order of the theoretical shear strength at the immediate front itself, is relaxed in the region behind the shock front. Our analysis (Weertman and Follansbee, 1983)) of moderate strength shock waves is used for the relaxing region behind a strong shock wave front. It is concluded that a strong shock wave front has the following dislocation structure: The shock front, of atomic dimensions, consists of a Smith dislocation interface of dislocations that keep up with the front by moving at transonic or supersonic velocities. Immediately behind the Smith interface is a region of moving dislocations that do not keep up with the shock front. In this region normal dislocation motion and multiplication takes place. Within this region of normal plastic deformation the shock pressure rises by an amount that is quite small compared with the shock pressure itself.

1. Introduction

In a previous p a p e r we analysed the finite amplitude shock wave of moderate strength by use of o r d i n a r y dislocation dynamics ( W e e r t m a n a n d Follansbee, 1983). (By moderate is m e a n t that the pressure of the shock wave is small compared with the bulk modulus of the material.) The shock wave is treated in that analysis as a sum of infinitesimal amplitude elastic plastic waves. Each infinitesimal wave has a different propagation velocity with respect to the local crystal lattice. The finite amplitude shock wave has a steady-state profile and its velocity is a constant. The width of the moderate, finite amplitude shock was was found to decrease as the pressure driving the wave is increased. W h e n the pressure approaches the bulk modulus in magnitude, where the analysis breaks down, the extrapolated results are that the shock wave width is of " a t o m i c dimensions or smaller, the dislocation velocity would be greater than the sound velocity, and the dislocation density would be so large that more dislocations cross a unit area than there are atoms in that area". The conclusion reached is that " t h e s e extrapolated results are indications that ordinary dislocation mechanics breaks d o w n in strong shocks a n d Smith type dislocation interfaces between shocked a n d unshocked material must come into existence". Because the shock front of a strong shock wave is very n a r r o w it is not possible for plastic deformation (that can relax the stresses towards a hydrostatic condition) to occur in a ' n o r m a l ' way at the shock front. The Smith dislocation interface (Smith, 1958) produces plastic d e f o r m a t i o n at the shock front itself is an ' a b n o r m a l ' manner. But it is important to realize that the Smith interface, as is discussed in this paper, can not reduce the effective shear stress behind the shock front to a level below the theoretical shear strength of the material. Since plastic deformation can occur at such a stress level, additional plastic deformation occurs immediately behind the shock front. In moderate shock waves the resolved shear stress on any plane never exceeds the theoretical shear strength of the material. This is true whether or not plastic * This research has been performed under the auspices of the U.S. Department of Energy. **Permanent address: Department of Materials Science and Engineering, Northwestern University, Evanston, IL 60201, U.S.A. 0167-6636/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland)

14

J. Weertman / Plastic deformation

deformation occurs. In a strong shock the theoretical shear strength level would be exceeded if no plastic deformation were to occur. Therefore, plastic deformation must occur in strong shock waves. In our earlier paper, mentioned above, the analysis lead to the conclusion that the width of a moderate shock wave becomes smaller and smaller as the shock pressure increases. When the width of the shock wave is of an atomic dimension a Smith interface comes into existence. (The number of dislocations in a Smith interface is treated by Weertman, 1973 and 1981. See also, Fadeenko, 1978, and Mogilevsky, 1983.) But in that earlier paper no consideration was given to the possibility that behind a Smith interface there may exist a plastically deforming zone of finite extent. It is the purpose of the present paper to investigate the possible dislocation mechanics in a strong shock wave in the region immediately behind the shock front itself. Our main interest is in the region where a substantial amount of 'normal" plastic deformation may take place through dislocation movement and generation before any unloading of the shock wave begins. While reading the paper, one should appreciate that the plastic deformation which is analysed occurs only in the material immediately behind the front of a strong shock. The compressive stress o~ (in the direction of propagation) changes only by a small amount /to,: (13o.,./o.,-I<< !) compared with the magnitude of o.~, over a distance from the ffoiit to a position behind the front where plastic deformation essentially has stopped. Errors introduced into the analysis by the neglect o| lactors in the plastic zone such as the temperature rise produced by the increase of the compressive stress and by the plastic work are small and can be neglected to a first order. Although the strong shock wave phenomenon is considered here, the part of the strong shock wave which is analysed is essentially a plastic wave of moderate amplitude. This paper exploits the advantage that can be taken of this fact.

2. Theory Let the shock front propagate in the x direction. Let ox, Oy and o. be the three tensile/compressive stress components. Let c~, %, and c= be the corresponding strain components. Each strain component c, can be split into the subcomponents

(1)

Ci =Eei "[- Cpi "~-Cti

where ce, is the isothermal elastic strain, Cpi is the plastic strain and ct~ is the thermal expansion strain (ct, = a t ( T - To) where a t is the coefficient of linear thermal expansion, T is the temperature and To is a reference temperature.) The plastic strains must satisfy

(2)

Epx + Cpy "~"Ep: = O. The elastic strains must satisfy =-

fOP

de

(3)

and

%y=%:=

- fo P v ( P ) - - ~[~¢ex - dP.

(4)

In (3) and (4) P = - ~(ox + Oy + o:) is the hydrostatic pressure, x ( P ) is the bulk modulus of the solid at pressure P, and v ( P ) is Poisson's ratio at pressure P. Under adiabatic conditions the terms ~ti can be dropped and adiabatic elastic constants can be substituted for the isothermal elastic constants. It is generally assumed that behind the shock front conditions are uniform in a transverse direction.

J. Weertman /Plastic deformation

15

However, a transverse homogeneous situation is not possible if the shock front is a Smith interface of dislocations which move with a velocity (measured with respect to the local lattice) in the transonic or supersonic range (Weertman, 1981; Weertman and Weertman, 1980). A transonic or supersonic dislocation generates planar sound waves of a width equal to the width of the dislocation itself. The stress amplitude within these sound waves can be of the order of the theoretical shear strength. Thus if a Smith interface exists a wildly fluctuating stress occurs behind a shock front. Only in regions where the amplitude of these dislocation produced sound waves have been reduced by dissipative processes can the material behind a shock front be in a state that is uniform in a transverse direction. No complete examination of requirements for the existence of a Smith dislocation shock front interface has yet been made.

2.1. Smith interface Consider Fig. 1. It shows a set of equally spaced dislocations that are moving at an angle ~ with respect to the shock front. In Fig. 2 are shown the two sets of sound waves emanating from one of the dislocations of Fig. 1 if the dislocation is moving supersonically. One set is a pair of shear waves and the other set is a pair of longitudinal sound waves. If the dislocation moves transonically, only one set of sound waves is generated by the dislocation. Th/s set is a pair of shear waves. The Smith interface may consist, as shown in Fig. 3, of two complimentary sets of the dislocations of Fig. 1 and two similar sets that are oriented perpendicular to the plane of Figs. 1 and 3. It previously was shown (Weertman 1973; 1981) that the number of dislocations n of one kind in the Smith interface is n = (1 -

(po/p)'/S)/b

(5)

sin 'k

where n is the number per unit length of shock front, b is the length of the Burgers vector of a dislocation,

~

'~'

"y,

LONGITUDINAL /~, WAVE

WAVE~.~ DIRECTION ,r

SHOCK

t

Fig. 1. One set of dislocations in a Smith interface. Fig. 2. Individual dislocation of Fig. 1. Dislocation is shown with a pair of shear waves and a pair of longitudinal waves emanating from it. Fig. 3. Two complementary sets of dislocations in a Smith interface.

J. Weertman / Plastic deformation

16

O is the density of the material at pressure P and P0 is the density at the reference pressure. Equation (5) was derived under the assumption that behind the shock front pure hydrostatic conditions prevail. If the stresses behind the shock front are not purely hydrostatic the number n of dislocations given by (5) must be modified. Weertman (1973) showed that if purely hydrostatic conditions prevail behind the shock front, the changes ~o, in the three stress components that Smith interface dislocations (which move at velocity V) produce behind the shock front are 8ax=O

(6a)

and

~%=6o:= {1 +v}

2 nbgc~ )

V2 [[tan(Ox-~)+r(Ox+~)]t-Bxsing~+aZcos~] -[tan(0s-ok) + tan(0s + 4~)][(a4/fl~)sin ~b+ aZcos 4~]].

In (6) the terms

a,/3s, and fix are

a z= 1 - VZ/2c~,

(6b)

given by

"?=ll-v% l

and =

v% l

(7)

where c~ is the shear wave velocity and cx is the longitudinal wave velocity behind the shock front. The angles O, and 0 x are equal to

O,=tan-'(cJV)

and

Ox=tan-'(cx/V ).

(8)

If the stress immediately behind the Smith interface is not purely hydrostatic the stress changes give by (6) must be modified. The stress changes given by (6) were found by taking a stress-displacement law on a slip plane that gave a stress which was equal to zero after passage of the dislocation. (The stress being considered is the shear stress arising from the presence of the dislocation.) If the hydrostatic situation is not attained then, of course, the shear stress on a slip plane cannot equal zero. If no 'ordinary' plastic deformation takes place and if the Smith interface dislocations do not come into existence, the shear stress at, and behind, the shock front would be larger than the theoretical shear strength. If no plastic deformation occurs, the shear stress at the shock front is indeed large enough to create dislocations in the bare lattice without the need of any dislocation sources. The Smith interface dislocations can.spontaneously come into existence. But once they exist at the shock front there is no obvious necessity for them to exist in sufficient number of reduce the triaxial stress situation to a purely hydrostatic condition. That is, the shear stress on planes that make neither a 0 ° nor a 90 ° angle with the shock front is not required to be equal to zero immediately behind a Smith dislocation interface. In fact, a simple argument reveals that the Smith interface dislocations cannot reduce the maximum shear stress to a level appreciably below the theoretical shear strength rtss. Suppose the number of Smith interface dislocations is just sufficient to bring the shear stress to the value equal to ~'ts~- Any further creation of Smith dislocations would reduce the shear stress to a level below rts~ and the shear stress is no longer sufficiently large to create dislocations without the presence of dislocation sources. Thus dislocations can no longer be created in the bare lattice. If the width of a strong shock front is of the order of atomic dimensions any dislocation source has a dimension very much larger than the width of the shock front. Hence the number of Smith dislocations can not be increased through the agency of 'normal' dislocation creation at ordinary dislocation sources. If Smith dislocations are removed from the shock front because

J. Weertman / Plastic deformation

17

they become stuck at obstacles in the lattice or because the number of jogs on them becomes too large or for whatever reason the maximum shear stress at the shock front on an oblique plane to the front must increase to a level above "rtss. Since the stress is large enough to create dislocations in the bare lattice new Smith dislocations will be created. Thus even if Smith dislocations are lost during propagation of the shock front it is to be expected that the number of dislocations in the Smith interface remains constant and this number is that required to bring the stresses immediately behind the shock front to the level at which the maximum shear stress that exists in the lattice is approximately equal to "rts~, the theoretical shear strength. If the condition is specified that immediately behind the shock front the maximum shear stress is equal to "rts~ the number n given by (5) must be changed to the value

n--

bsin,

p!

-

(l+v)~

(9)

where O0 is the density for P - - - a ~ . Equation (6a) remains unchanged but (6b) becomes

~o,. = ~o. -- {1 + v }

v2

[[tan(O~ - , )

+ tan(O~ + ¢)][ - B ~ s i n , + . 2 c o s ¢]

-[tan(Os - , ) + tan(Os + * ) ] [ ( - 4 / f l ~ ) s i n , + .2cos ~]] + 2 ~',~s.

(10)

3. Plastic deformation behind the shock front The passage of the shock front, before the arrival of an unloading wave, leaves the solid in a stress state in which the maximum shear stress present is equal to ~'tss, the theoretical shear strength. In a metal, at least, further plastic deformation can occur. The additional plastic deformation must take the form of a plastic wave. Since the theoretical shear strength presumably is of the order of ~ to ~ of the shear modulus this plastic wave is one of moderate to moderately large amplitude. In other words, a postcursor, moderate multiple plastic wave should follow immediately behind a strong shock wave. (An analogue of the situation of a precursor elastic wave that preceeds a moderate or weak elastic plastic wave. However, the postcursor plastic wave should have the same velocity, not a different one, as the shock wave and should exist immediately behind the strong shock wave front. The strong shock wave and the plastic wave can be regarded as a single entity.) The velocity ap of the plastic postcursor wave is ( ap)la b = ( ap)mg + u.

(11)

In (11) u is the particle velocity of the material immediately behind the shock front and (Up)rag is the velocity of the plastic wave as measured in the moving coordinate system in which the atoms of the lattice are at rest. A necessary condition for the plastic postcursor to be a separate wave is U + (ap)mg q: a s

(12)

where a s is the shock wave velocity as measured in the laboratory system. But is it not possible to satisfy (12). If the plastic wave has a velocity greater than as, the plastic wave catches up to the shock wave and forms an even stronger shock wave. (As shown experimentally (McQueen et al., 1970; 1984) a wave traveling with the bulk velocity behind a shock wave will catch up to the shock wave. Once it catches up to

J. Weertman /Plastic deformation

i8

the shock wave it becomes part of the shock wave.) If the p!astic wave travels sl3wer :han the shock wave, as time increases it becomes further and further separated from the shock wave. In the region between the plastic wave and the shock wave the shear stress is equal to the theoretical stress. But no plastic deformation can take place at this stress level if the plastic wave is indeed separated from the shock wave. Clearly this is not possible. The analysis given in Weertman and Follansbee (1983) can be applied to the ?lastic wave with several modifications. In that analysis the initial value of the effective shear stress of ~he plastic wave is not a given value. The initial value of the shear stress is determined by dislocation raechanics. In the present situation the initial value of ,r is fixed by the condition that it is equal to the theoretical shear strength. Thus the dislocation mechanics behind the strong shock front must be rather different than the corresponding dislocation mechanics behind an elastic precursor of a moderate shock wave. Since the effective shear stress ~" is equal to the theoretical shear strength, the effective shear modulus for a further incremental increase in the shear strain is equal to zero. For an incremental decrease in the shear stress the shear modulus regains a finite value. Equations that must be satisfied for the plastic wave behind the strong shock front again are (Weertman and Follansbee, 1983) 6.~ + 2O.v= 3 ~ x ,

(13)

3acv2 6., - dy = 2p(i.,, - ~p.~ ) = --~3 (1 - "O)i.,,

(14)

and 3•vl ax =

(~x - - -

P3

Pl cpx

=--

1P3

(~x-

(15)

3vl ]

Equations (14) and (15) combine to give 6~ - 6y = {3v2(1 - T I ) / ( 3 V 1 -- 2~V2)} 6x.

(16)

In (13) through (16) the dot stands for time derivative, v~ = 1 - v, v2 = 1 - 2v, and v3 = 1 + v. The strain rate ( , is the total strain rate in the x direction and ~p.~ is the plastic c o mp o n e n t of the total strain rate in this direction. The stresses ox, % and a x are always compressive. Thus a~, %, ax ~ 0. The term 7/represents the ratio

= ~pdx/(~.

(17)

N o plastic deformation occurs behind the shock front when ~/= 0; when 7/= 1 the material behind the shock front behaves as a fluid with a shear modulus equal to zero. No te that when 0 < TI < 1, ¢iy> 6x (but lOyl < 16xl). When 7/= 0, 6y = 6x. When ~ is in the range of 1 < ~ < 3 v l / 2 v 2, 6v < 6x (but I~>,1> I~1). If the postcursor plastic wave could be regarded as being of infinitesimal amplitude its velocity in the coordinate system that moves at the particle velocity is a = { K ~ / v 3 }1/2(3v~ - 2nv2) 1/2.

(18)

In our previous paper a finite amplitude plastic wave is regarded as a superposition of infinitesimal amplitude waves. Let the postcursor plastic wave be so regarded. The initial value of the plastic to total strain rate ratio ~ can be estimated through dislocation dynamics. In particular, let p be the dislocation density and f be the fraction of dislocations that are mobile at a given instant in time. Let the velocity v of a mobile dislocation be given by

v=b~/B

(19)

J. Weertmun / Plastic deformation

where B is the dislocation damping constant. Thus

~px is given

19

by

~p., = ( a / B ) fpb2"r = ( a / 2 B ) fpb2( o~ - o v)

(20)

where a is a geometric factor of order of magnitude of 0.1 to 1 and ~- = ½(ox - o,.) is the shear stress. (Note that r < 0). In (20) it is assmed that any interaction stress between dislocations is negligible compared with 'i'.

Combining (14), (17) and (20) gives

2a

!

[o.,-o:.1.

(21)

The initial value ~i is found by combining (12) and (18) 3vl 7h=2v2

v3

3vl

a.~sv3

~

2v2~12(a~- u ) 2 = ~ 2 + 2m,---~2(12o-12)"

(22)

where o.,~ is the value of ox at the shock front, 12 is the volume of a unit mass at the shock pressure, 120 is the volume at zero pressure, and x is the value of the compressibility modulus at the shock pressure. In (22) use has been made of the shock relationships (Rice et at., 1958)

a~= - [ 1 2 2 / ( 1 2 o - 12)] o,~s,

(23a)

u2= _ (12o _ 12)o.,,s'

(23b)

( a s - u )2 = _ [ 122/( 120 - 12)] o~.

(23c)

It should be noted in (22) that the term ( 3 v ] / 2 v 2) is of the order The second term on the right hand side of (22) is of the order of - 2 large compared with the value of K measured at zero pressure the since K becomes of the order of o~s at very high pressure and compared with one (if v remains constant in value). The value of 71 at an arbitrary value of ox behind the shock front Weertman and Follansbee (1983). It is

of 3. (If v = ~ it is exactly equal to 3.) when - Ox~/K << 1. When o~ becomes second term should decrease in value the ratio 12/(120- 12) becomes small can be obtained from equation (40) of (24)

n = h i - ( , , , - oxs)/,,*, where K* = 2 K V 2 / ( 2 V 3 --

For v = 1 and

3v]

(25)

+ 2~/iV2).

71i "- 3, K* --~ 1 I¢.

3.I. Constant dislocation density If (16) and (24) are combined the following equation is obtained (26) ÷= -

3vl _ 271v2 i/.

This equation integrates to (27) "r =

-'rts S -

l•,

(rl - rli) +

log

31,1 _ 2 r / i v 2

20

J. Weertman / Plastic deformation

and 3v3~* ( 2v2 (o,, - oxs)}. 8/y2 log 1 + ~*(3v: - 211iP2)

(28)

Equations (27) and (28) are almost the same as those found earlier (Weertman and Follansbee, 1983). At the rear of the plastic wave the shear stress • is reduced to a small value (iTI/rts~ << 1) if no dislocation multiplication occurs to harden the metal. Thus the final value ~/f of the plastic to total strain rate ratio is given by

2~ 2 ' i ] f "at-

{ 3vl - 2r/f v2 } log 3v I _ 27/iP2 "" ~/i

(29)

4"rtss 3r* "

The value of r/f from (24) must be larger than ~/i- Hence the following inequality must be satisfied

v3 ( 3vl -- 2r/iv2 ) 4'rtss > ~. 2/,2log 3v 1 _ 27/fv2 3~*

(30)

The shear stress "r must also satisfy (21). If (21) and (27) are combined:

[

3~,1 - 2-¢/v2

3v I -- 2T/ip 2

)/-1

,3.

The final value ~f is found from (31) by setting ~/= 0. Thus "//f=

"/~i - -

4"rtss 2 ~ 2 ( 3vl - 2~/iv2 ) 3x----g+ log 3v~ - 2//fv 2 '

(32a)

or T/f=(1/2v2) 3 v ~ - ( 3 v l - E ~ / i v E ) e x p -

v-T

~ / f - ~ i + 31¢*]

"

(32b)

Equation (32b) has the expectation that ~f ~ " 3 for v = ~-, ~ i " " 3 and ~'tss/X = 4 . The final value of ~'f= ½(o: - oy)f is, of course, zero because no work hardening occurs. In actuality "rf will equal the yield stress of the metal. The final value of (o,,)f is found from (24). It is ( Ox )f = Oxs -- ~*( ~ f -- ~i )

- o~

~

3v I - (3v ! - 2~iv2)ex p

2v2

4"qs~

(33)

In (24)-(33) the implicit assumption is made that in the stress range Ioxsl < Ioxl < I(g~)fl the bulk modulus s is a constant. If account is taken of the variation of ~ with pressure these equations are modified (see Weertman and Follansbee, 1983) by changing the definition of ~* from that given by (25) to the following one ~* =

21¢v2 2v 3 + ( q - - 1 ) ( 3 v , - 2~1iv2)

(34)

where q = t t K / ~ P = - ~ K / ~ O x . Typical values of q are oi the order of 4 to 6. The profile of o x versus distance from the the shock front is found from (24) and (31) and noting that in a coordinate system that moves with the shock front ~Ox/~t = ( u - ap)~Ox/~X.

(35)

21

J. Weertman / Plastic deformation

In (35) the velocity ap is the velocity of the plastic wave in the laboratory system which is given by a p" = a + U,

(36)

where the velocity a, measured with respect to lattice atoms at rest, is given by (18). The combination of (18), (24), (31), (34), (35) and (36) gives

~ '~[_ (4,rtss/3k,)

- ~ + 7i

i

_2_~2log (3~',27v2)]-' 3v I _ 27iv 2

7d7

V/3~,l _

27P2

= _ 9K(afpb2/2B)( v3/k~)]/2x

(37)

where the distance x is measured from the shock front. Let T0 = 3v] = 2v 2 and let 87i and 87 be defined by (see also (22)) 87i=70--7i

=/Ioxs[V31( ~ )=(as_ [ 2Kv2 ] G 0 _ ~

~2( P3 ) u, ~2v2k ~

(38a)

and 87 = 7o - 7.

(38b)

(It should be noted that 0 < 87 < 87i, and, for extremely strong shocks, 0 < 87 < 87i << 1.) If (38) is inserted into (37): an'[ * - 87 + 8 7 , - ~ l o g ( ~ - ~ ) ]

,1 L7

- ' ( 7 ° - 87) d(87)

= 9t¢ ( a f p b 2 / B )(2v2P3/k9 )'/2 x

(39)

where 7* = 4"rtss/3k*Since 87 << 70, (39) integrates initially to 167o ]

, ~

(40)

If both sides of (40) are multipfied by the term (V~-~i + ~ - ~ ) / 8 7 i : ( 87i - 87 ) / 8 7 i

X/Xo

=

(41)

where

9ml*v2

~

)[ sO2 ],2osu,(2.o t (9o - 9 )

(42)

9Kp2J,~,afpb 2

where use has been made of (23c). For x ~ oo, (39) reduces to 8n -, 8 7 i e x p ( - 2v27*/v3).

(43)

Since 87i - 87 = 7 - 7i = -(ox - Oxs)/K* [see (24)], (41) and (43) also are equations that gives o~ - oxs as a function of x.

J. Weertman / Plastic deformation

22

1.o J

f

f

f

. P

f

1 ~- K*Xw

V3"0'

2 Kx o u 2 l a o - a )

8~

,..

8~7

1+ 0.045R 1"10

O" X O'XS

-,' "a

1

.

x--'~"-- 1 - exp ( - ~2V2 ~*-) Xw

0

"

0.09

I

X

1.0

XO

Fig. 4. Normalized plots partially schematic of ratio 8vl/Svli and ox/O~s versus x/xw for the case of constant dislocation density. Curves are calculated using (41), (42), (43) and (45) and the constants listed in Table 1.

F i g u r e 4 p r e s e n t s a s c h e m a t i c n o r m a l i z e d p l o t o f 8T//8~/i v e r s u s X / X w . H e r e x w, t h e w i d t h o f t h e p l a s t i c w a v e , is f o u n d b y n o t i n g t h a t a t x -- Xw, 8.~ h a s a v a l u e a p p r o x i m a t e l y e q u a l t o t h a t g i v e n b y (43). T h u s x ~ = Xo[ 1 - e x p ( -

2v2~/*/v 3)].

(44a)

F o r v = ~ a n d ~'t~/K = ~l , t h e t e r m 2v2Tl*/v 3 = 0 . 0 8 8 . T h u s t h i s t e r m is s m a l l a n d ( 4 4 a ) r e d u c e s t o

xw = (as - u )(4~o/9Kv3 )( B / a f P b 2 ).

(44b)

Table 1 Plastic wave width -- xw and initial dislocation velocity for case of constant dislocation density

a

P (m-2)

f

B (Pa s)

xw

Vi (km/s)

1012 1012 1014 1012 1012 1012

1 10 -4 10-4 1 10-4 1

5 x 10 -s 5 x 10 -s 5 X 10 -s 0.5 0.5 2 x 10-4

1012

10-4

2 X 10 - 4

1014 1012 1012

10 -4 1 10-4

2 X 10 -4 2 2

43/~m 43 cm 4.3 mm 43 cm 43/~m 170/~m 1.7 m 1.7 cm 1.7 m 170 Fm

42 42 42 0.0042 0.0042 10.5 10.5 10.5 0.00105 0.00105

I assuming s = 3Ko. where Ko is the value a Calculated using the following values for the constants appropriate for aluminum: z, = s; of K under zero pressure; So = 75 GPa; b = 0.28 nm; 7/0 = 3; 7" = 0.18; (as - u) = 5 km/s and a~ = 6 k m / s (from Marsh, 1980); R-1 = 2800 kg/m3; a = ~,1. a/y2 =1; l"~ss/~= ~~,• and q = 5.

J. Weertman /Plastic deformation

23

In Table 1 are listed values of Xw obtained using various values of the dislocation density p, fraction of mobile dislocations f, etc. in (44) The equation that relates oJo,,~ to initial values of X/Xo is

°x =1+ o~,

=1+

(:x)(,o)

~ I°x~l / ~0

~ Kx0

(45a)

2v2(~20-~ ) "

The final value of q~ is given by

,3° ) .

o~ = 1 + O,cs ~ ICXo ] 2V2(~o-- ~ )

(45b)

Figure 4 also shows a schematic plot of ax/O~s versus X/Xo. Table 1 also lists the initial velocity V~ of the dislocations immediately behind the shock front. (The velocity Vi is that of the ordinary dislocations, not the Smith interface dislocation, immediately behind the shock front.) The velocity Vi is calculated from the equation

V~ = "rtss/B --- Kb/30B.

(46)

The velocity given by (46) is the maximum dislocation velocity. The velocity decreases the greater is the distance from the shock front. In Table 1 the value of the damping constant B = 5 x 10 -s k g / m 3 is that found experimentally at room temperature far dislocations moving at velocities very small compared with the shear wave velocity. Experimentally it has been demonstrated (see Weertman and Weertman, 1980) that when the dislocation velocity becomes greater than about one third of the shear wave velocity that the effective value of B increases. In Table 1 vi is also calculated with a value of B that is a factor of four larger. The widths of the plastic waves given in Table 1, for the values of f << 1, and B of the order of those found experimentally, are so large as to appear to be physically unrealistic. This result might cast doubt on the assumption that the damping constant B determines the dislocation velocity. Recent work of Follansbee et al. (1985) and Follansbee and Gray (1985) on the upturn of the work hardening in rapidly deformed metals (for strain rates greater than 104 s-l), cannot be explained by assuming that the dislocation motion is controlled through a damping mechanism of strength of order 10 -4 Pa s. (The upturn was previously explained assuming that at the greater strain rates the damping mechanism controls the dislocation motion. This assumption, in turn, required that f << 1.) In Table 1 are listed 'pseudo' B values that are a factor 104 greater than the experimental values. These artificially high values of B, as can be seen in Table 1, do lead to a reduction of Xw. The results of Table 1 actually are physically satisfactory only for the case in which all the dislocations move ( f = 1) and B has its experimental value. For this case the dislocation width has reasonable values. However, the assumption that no dislocation multiplication occurs is unrealistic. In the next section this assumption is not made.

3.2. Dislocation density not constant Next consider the case in which the dislocation density does not remain constant during deformation. However, assume that the effect of work hardening can be ignored because the shear stress during most of the course of deformation is very much larger than the instantaneous value of the yield stress. (Work hardening does occur because the dislocation density increases. However, the velocity of dislocations is not changed appreciably because the 'applied' shear stress is so much larger than any back stress produced in the work hardening.) The rate of change/5 of the dislocation density is proportional to the expression f p V / A where V again

24

J. Weertman / Plasticdeformation

is the average dislocation velocity of the mobile dislocation and A is the average distance a dislocation must move before a new dislocation is created. Thus A is the average dimension of a F r a n k - R e a d source. Since A is of the order of 1/X/-ff the rate of change of the dislocation density is given by

i) = "YfPa/2v= _ .yfp3/2 ( q,, _ oy ) b / 2 B.

(47)

In (47) the term ~, is a constant of order of magnitude of one. Combining (14), (17), (20) and (47) gives (48)

¢rff = - ~ 9,~,2]~ l _ , , ] , a f l J ( O x - O y ) " If i / = 0, a condition which is justified below, (48) integrates to VrO = ~ i -

/)3

~i

9-~v2)(~)(5){2"rtss

+ (ox-Oy)}-

(49)

The final value of the dislocation density is

N=N_(

~_~ijt--~]

t

,~ + 2~'f}.

(50)

In (50), 2"rf = (o x - O.v)f where the subscript f stands for the final value of these quantities. (An upper limit on 0f is found by setting "rf = 0. For Pi/0f << 1, "rtsJ~: = ~ , "t/a = 1, ~i -- 3, b = 0.28 nm, and v = 3, the upper limit is #f < 2.5 × 1016 m/m3.) The final value of the quantity (q,: - %) is equal to zero if the assumption is made that no back stress % affects the dislocation velocity. However, this assumption is a realistic approximate one only when the back stress is small compared with the applied stress. When the back stress and. the applied stress are of comparable magnitude plastic deformation ceases. The back stress is of order

Tb "-- -a*Kbyt-O

(51)

where the constant a* is of the order of 0.1 to 0.5. On setting % = ~'f and combining (50) and (51):

['l'biJC(2"rtssP31( 1]i

)]/[

(2P31(1~---~_ I(~-~)]

(52)

In (52) %i is the initial value of the back stress. Equation (52) predicts that ~-¢ is within an order of magnitude of the value of - ~t~s- (Note that 2v3/9~, 2 --- -~, ~i/(1 - ~i) -- - 3, and the term V a * / a should be within an order of magnitude of a value 0.1 to 1.) Thus the work hardening in very strong shock wave might be expected to reach levels approaching the theoretical strength of the metal provided that the shock wave runs a sufficiently large distance. If (14), (17), (20) and (49) are combined, we have ( ( i , - 6 y ) = c , ( a x - oy)[1 - c : ( a x - Oy)]:

(53)

where (54a) and

(

l.i ,(v)

(54b)

J. Weertman /Plastic deformation

25

The integration of (53) gives

2"r,.~.,

1 + Czl°.,-- 0,.I exp (1 + ~ 2 1)~ : ~.]~1 . . + ~CCzrst~'. .

= e x p ( c , t ) = exp(,a,!C'X]

(55) where x is the distance to the shock front and t is the time elapsed since the passage of the shock front past a specific point in the solid. For a low initial dislocation density, which is assumed to be the case hereafter, (54a) and (54b) reduce to

c , "- -

-~¢ ] -~r2

2Ba ](

(56a)

Tli

and c2 = 1/2-q.,.,.

(56b)

From (55) the width Xw of the plastic wave is of the order of - a J c l .

Thus

Equation (55) gives the profile of the deviatonc stress component ( 0 ~ - 0,') versus x. A more useful profile that might possibly be compared in the future with experimental results is one of o.~ versus x. An equation that relates O~ with x is obtained by using (16) and the result that ~ changes by only a very small amount. From (16), we get 25T/i

I0.,-I =10~1 + 3(hi- 1)

(2~,.,. - Ia,. - a,. l)

(58)

where $~i is given by (38a) and, for very strong shocks, T/~= 3. Since rt.,.Jx = .~1 and, in very strong shocks, $ ~ << 1, the stress difference -Io,-I- Iox~l<< K. Hence the plastic deformation behind the shock front does occur, as asserted in the Section 1, under conditions of the "moderate' amplitude plastic wave. 1 Combining (55) and (58), and assuming that Oi is so small that, in (54b), c2 = ~'t~.,, gives

(("

=exp(c,t)=exp(

l° x( '°,--°,-s'-]

(59)

c'xi \asJ

where

2v2k~'l ]

"r* = 3(T/i _ 1)

(60)

When X/Xw << 1 and (o~ - O~s)/O~., << 1, (59) reduces to

oJox., = 1 + ( ~*/lox, I)(8X/Xw) '/'-.

(61a)

When X/Xw >> 1, (59) becomes ox/¢s = 1 +

( ,*/10~s l) {1

-

½exp[- (X + xw)/Xw] }.

(61b)

J. Weertman / Plastic deformation

26

o"X

1.0

-

O'x

r*

"1

(1-

1__)

T,

O.xs - 1 + ~xxsl -

2e

o-x O-xs

o 0

I

I

1.0

2.0

3.0

x

XW Fig. 5. Normalized plots of Io~I/K versus x / b given by (59). In the calculation of the curve the values of constants given in Table 1 . . . . . . . -! dig; U~-aLI.

In Fig. 5 is plotted the wave profile found from (59). Table 2 lists values of the plastic wave widths found from (57). In the equations above use has been made of the relationship i / = 0. This relationship can be justified as follows: From (22), ~i = 3vl/2v2 = 3 (for v = ~) when ~ / ( 9 0 - 17) << 1 for very strong shocks. From (24) rl = ~i - (o~ - o~s)/K*. But for very strong shocks, Iox - Oxsl/K* << 1. Thus 7/= *li3. 3. Dislocation density and work hardening not constant

The logical next case to treat is the one in which dislocation multiplication occurs and the back stress Tb from work hardening causes the dislocation velocity to be reduced. Because the back stress is effective in reducing the dislocation velocity only when % becomes of the same order of magnitude as ( o , , - o y ) the analysis of the previous section is applicable over most of the plastic wave. Only near the end of the plastic deformation would account have to be taken of %. The analysis of the last section should give a reasonable approximation for the case where explicit account is taken of the effect of % on the dislocation velocity. For this reason an analysis of the case when % reduces the dislocation veio,-ity is not developed here.

Table 2 Plastic wave width - xw for the case of increasing dislocation density

B/f

Xw

0.5 2

18 lgm 72/~m

a

Calculated from (57) with use of the constants given in Table 1.

a

J. Weertman /Plastic deformation

27

3. 4. Width of plastic wave it is not possible to make an accurate quantitative prediction of how the width of the plastic wave varies with the shock pressure o,.~. However, a qualitative prediction can be made. From (57) the width x , of the plastic wave is proportional to ( a J s ) ( B / f ) . (Note in (57) that the term Tt.,JK is roughly a constant. If the pressure is increased, thus increasing ~, the theoretical shear strength Tt.,~ is increased a corresponding amount.) The term (a,/K) at very high pressure must vary approximately as 1 / ~ since x must be roughly proportional to Io,d at very high pressure and a,, from (23a), is proportional to ~ . The uncertainty in a prediction of xw versus o~ resides in the term ( B / f ) . It is not known what is the value the fraction f of mobile dislocations. If f is large ( f = 1) then from (46) (see also Table 1) the dislocation velocity is very much smaller than the sound velocity. If f is small ( f < < 1) then the dislocation velocity must approach the sound velocity. (In Table 1 the initial dislocation velocity Vi found from (46) is greater than the sound velocity. This result is obtained assuming that the damping constant B is independent of the dislocation velocity. Actually, as the sound velocity is approached by a moving dislocation the damping constant B increases orders of magnitude in value (Weertman, 1973). The damping constant B, other than at temperatures close toabsolute zero which never exist in a strong shock wave, is proportional to the expression kTb/12*c, where k is Bolzmann's constant, T is the temperature, b is the Burgers vector, 12" is the atomic volume, and c is the sound velocity (Hirth and Lothe, 1982). Since 12" is proportional to 12 and 12 is only a weak function of 0~ in very strong shocks and since c is proportional to vr~-~, the term B is roughly proportional to k T / ~ . The temperature T in strong shocks is roughly proportional to Io~l. (If s is roughly proportional to Io~.,I the work done in compressing a solid is roughly proportional to Io~l and hence the temperature rise is also roughly proportional to Io~1.) Hence the width of a plastic wave, in a first order approximation, is independent of the shock pressure.

4. Discussion

The analysis of this paper considered the plastic deformation that appears likely must occur behind the front of strong shock waves. We found previously (Weertman and Follansbee, 1983) that a weak to moderate strength plastic wave front becomes narrower and narrower as the shock pressure is increased. In the limit of strong shocks the width becomes of atomic dimension. In the plastic waves that we considered, the plastic deformation commences once the elastic limit is reached. At relatively weak pressure the plastic wave is preceded by an elastic precursor. If the plastic wave width is of atomic dimensions there is "not enough room' for dislocation sources to become activated within the plastic wave front. Thus the deformation within the shock front must be entirely elastic, provided that the theoretical strength of the solid is not exceeded. If it is exceeded, an extremely narrow front can come into existence through the agency of a Smith dislocation interface. In either situation, behind the shock front large deviatoric stresses exist. Until an unloading wave reaches the shock front, these large deviatoric stress can cause plastic deformation (at a rate ranging from 'leisurely' to very rapid) if they exceed the plastic yield stress. The deviatoric stresses in very strong shock waves will equal the theoretical yield strength of the material immediately behind the shock front. The plastic deformation in the region behind the shock front itself will form a traveling plastic wave. The plastic wave width cannot be of atomic dimensions; if it were, the wave would become part of the narrow shock front and (for strong shocks) the theoretical strength of the material is exceeded. However, the wave must not travel at a slower velocity than the shock wave; if it did, the wave eventually would travel in material that is subjected for relatively long periods of time to a very high deviatoric stress and yet did not deform plastically. This is not possible. Since the plastic wave behind the shock front has a finite width, the type of analysis that we presented

J. Weertman / Plastic deformation

28

earlier for moderate strength plastic waves need only be modified slightly to describe it. This modification is carried out in this paper. It is found that the width of the plastic wave is relatively insensitive to the value of the shock pressure of a very strong shock wave. It has been shown (Mikkola and Wright, 1983; Wright and Mikkola, 1985) that, if a shock wave is unloaded very quickly, the final dislocation density is a function of the time of existence of the shock wave for shorter time periods. For longer periods the dislocation density remains constant and no longer depends on time. Their results are qualitatively consistent with our picture of the plastic deformation processes behind strong shock fronts when dislocation multiplication takes place. It is to be noted that the equations and plots of o.~ versus x behind the shock front given in this paper are easily converted into approximate equations and plots of ( u - us) versus x where us is the particle velocity at the shock front. Since (o x -o.~s)/oxs << 1, from (15) and (17) o f W e e r t m a n a n d F o l l a n s b e e (1983): u - us -- [ ~ J ( a s

- u)](o.~ - o.~s) ~ ( ~ J a ~ ) ( o

x - o x s ) where 1~ is the value of 1~ at the s h o c k front.

Acknowledgement I wish to thank Paul Follansbee for his helpful comments and suggestions.

References Fadeenko, Yu.l. (178), "Mechanisms for plastic relaxation of a solid in a shock wave", J. Appi. Phys. Tech. Phys. 19, 123. Follansbee, P.S. and G.T. Gray (1985), "'Threshold stress measurements in shock-deformed copper", submitted to proceedings of 1985 American Physical Society Topical Conference on Sock Waves in Condensed Matter, Spokane, Washington, July 22-26, 1985. Follansbee, P.S., U.F. Kocks and G. Regazzoni (1985), "The mechanical threshold of dynamically deformed copper and Nitronic 40", submitted to the International Conference on Mechanical and Physical Behavior of Materials under Dynamic Loading, (DYMAT 85), Association pour la promotion des etudes du comportement dynamique des materiaux et applications, Paris, September 2-5, 1985. Hirth, J.P. and J. Lothe (1982), Theory of Dislocations, Wiley, New York, 2nd ed. Marsh, S.P. (1980), LASL Shock Hugoniot Data, University of California Press, Berkeley. McQueen, R.G., J.N. Fritz and C.E. Morris (1984), "The velocity of sound behind strong shock waves in 2024 AL", in: J.R. Asay, R.A. Graham and G.K. Straub, eds., Shock Waves in Condensed Matter--1983, North-Holland, Amsterdam, 95. McQueen, R.G., S.P. Marsh, J.W. Taylor, J.N. Fritz and W.J. Carter (1976), "The equation of state of solids from shock wave studies", in: R. Kinslow, ed., High-Velocity Impact Phenomena, Academic Press, New York, 293. Mikkola, D.E. and R.N. Wright (9183), "Dislocation generation and its relation to the dynamic plastic response of

shock loaded metals", in: J.R. Assay, R.A. Graham and G.K. Straub, eds., Shock Waves in Condensed Matter--1983, North-Holland, Amsterdam, 415. Mogilevsky, M.A. (1983), "Mechanisms of deformation under shock loading", Physics Reports 9.7, 357. Rice, M.H., R.C. McQueen and J.M. Walsh (1958), "Compression of solids by strong shocks", in: F. Seitz and D. Turnbull, eds., Solid State Physics, VoL 6, Academic Press, New York, 1. Smith, C.S. (1958), "Metallographic studies of metals after explosive shock", Trans. Metall. Soc. AIME 212, 574. Weertman, J. (1973), "Dislocations mechanics at high strain rates", in: R.W. Rhode, B.M. Butcher, J.R. Holland and C.H. Karnes, eds, Metallurgical Effects at High Strain Rates, Plenum Press, New York, 319. Weertman, J. (1981), "Moving dislocations ion a shock front", in: M.A. Meyers and L.E. Murr, eds., Shock Waves and High-Strain Rate Phenomena in Metals, Plenum Press, New York, 469. Weertman, J. and P.S. Follansbee (1983), "Finite amplitude plastic shock wave treated with summed infinitesimal amplitude elastic plastic waves through dislocation dynamics", Mechanics of Materials 2, 265. Weertman, J. and J.R. Weertman (1980), "Moving dislocations", in: F.R.N. Nabarro, ed., Dislocations in Solids, Vol. 3, North-Holland, Amsterdam, Ch. 8, 1. Wright, R.N. and D.E. Mikkola (9185), "High strain rate deformation of Mo and Mo-33Re by shock loading: Part I. Substructive development", Met. Trans, 16A, 881.