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Axial plastic waves in a rod
Pergamon
International Journal of Plasticity, Vol. 14, Nos. 1-3, pp. 25-42, 1998 © 1998ElsevierScienceLtd Printed in Great Britain. All rights reserved 0749-6419/98$19.00+ 0.00 PII: S 0 7 4 9 - 6 4 1 9 ( 9 7 ) 0 0 0 3 8 - 7
AXIAL PLASTIC WAVES IN A ROD T. W. Wright US. Army Research Laboratory, Aberdeen Proving Ground, MD 21005-5066, U.S.A.
(Received in final revised form 10 July 1997) Abstract--Equations are presented to describe nonlinear plastic wave propagation in a rod with the effects of radial inertia and radial shear included. Intrinsic constitutive equations are postulated for plastic deformation, and some of the principal aspects of wave propagation are described. In particular, a scaling argument is used to extract the nonlinear bar speed, which carries the principal pulse of energy, rather than the two characteristic speeds. © 1998 Elsevier Science Ltd. All rights reserved I. INTRODUCTION Equations o f motion, suitable for describing wave p r o p a g a t i o n in a circular cylindrical r o d with stress free sides, m a y be obtained f r o m the three dimensional equations o f a nonlinear c o n t i n u u m (e.g. A n t m a n , 1972) or they m a y be f o u n d directly by assuming dependence on an axial strain, radial strain, and the axial gradient o f radial strain. If the m o t i o n o f tlhe rod is a p p r o x i m a t e d by r = R[I -4- u(Z, t)], z = Z + w(Z, t)
(1)
where (r, z) are the current coordinates, (R, Z) are the material coordinates, and the functions (u(Z, t) and w(Z, t) are kinematic degrees o f freedom, then an a p p r o x i m a t i o n for the kinetic energy per unit length o f rod is K - - ½zraZp(w2 + ½a2t~2), where p is the reference mass density, a n d a is the reference radius o f the rod. In addition, if the material is elastic with stored energy per unit length o f W = zra2 W ( w t, u, u ~), where the dash means differentiation with respect to Z, then the corresponding E u l e r - L a g r a n g e equations m a y be written
S' = pf~, Q' - P = ~ pa 2 u"
(2)
where S - 9w ow In eqn (2) S m a y be interpreted as the average o f the i~w" p _ - - aw 0u ' and Q = ~-7. axial stress, P as the average o f the sum o f the radial and circumferencial stresses, and Q as the polar m o m e n t o f the average radial shear stress (e.g. Wright, 1982). The addition o f the second degree o f freedom, u(Z, t), has m a d e possible the inclusion o f radial shear and radial inertia in a consistent fashion. F o r the case o f a linear isotropic -
-
25
26
T.W. Wright
material equations equivalent to eqn (2) were first given by Mindlin and Herrmann (1950), but were superceded later by equations given by Mindlin and McNiven (1960) that included still a third degree of freedom at each cross section of the rod. This later version allowed cupping motions of the cross section as well as radial motions, so that the frequency dispersion curves were reasonably accurate up to the first cut off frequency, which might be either a purely radial or a purely axial motion of the whole rod depending on the elastic constants. Since the third degree of freedom, which would introduce significant additional complexity, has not been included here, the higher dispersion curves will not be accurately represented, but the lowest curve will show major effects due to the radial coupling. In the present paper the interest is in modifications to dominantly axial pulses, so the extra degree of freedom is not required. An intrinsic version of a rod theory that is kinematically equivalent to the case sketched above, but that includes plastic deformations will be developed. The constitutive equations and the complete system of equations that govern the motion are developed in the next two sections. Then a section devoted to a discussion of wave speeds in the bar and the associated asymptotic motions precedes a section that describes a typical initialboundary value problem to be treated. Next, in two sections devoted to elastic and plastic waves, respectively, elastic dispersion in the precursor is described, two new solutions for plastic wave propagation are presented, as well as the influence of the elastic dispersion on the subsequent development of the plastic wave. A few remarks about stability are made in the next section, followed by a final section of discussion and conclusions. II. CONSTITUTIVE EQUATIONS It will be assumed that the balance laws for axial and radial motion remain the same as eqn (2), but that the constitutive equations may be divided in the standard way into additive elastic and plastic parts. Thus, the strain increments may be written dw'=dwe
+ d w ' p,
du=due-~-dldp,
dut=du;--~-du;
(3)
where as before the dash indicates differentiation with respect to the axial coordinate, Z. (Strictly speaking, only the terms on the left hand side of eqn (3) are derivatives. The elastic components on the right hand side will be computed from stresses, as is usual in plasticity, and the plastic components are the remainders.) Clearly w', u, and u' have the interpretation of axial strain, radial strain, and the axial gradient of radial strain, respectively. For the moment u and u ' will be treated as independent variables. The elastic constitutive relations will be assumed to be the same as those given by Mindlin and Herrmann (1950). dw
'e =
dS dP - - -- v - - ~ , E
1 - u dP due - - - 2 E
dS v---~,
1 ~ a2du ' e
dQ
= --
ii~
(4)
where E,/z, and v are the elastic Young's modulus, shear modulus, and Poisson's ratio, respectively. The usual relationship among the three holds, E = 2/z(1 + v). To find reasonable constitutive expressions for the increments of plastic strain we first consider approximate representations for the deviatoric components of stress and strain in cylindrical coordinates. From the kinematics laid down in eqn (1) the linearized strain, E, and its deviator, e, have components
Axial plastic waves in a rod
tel=
i uo ,u,l 0
u
0
Ru' 0
,
[el=
w'
27
o
½(u-w') 0
0
½Ru'
oU, 1
(5)
]
(6)
-~(u-w')l
The average stress, Z, and its deviator, s, will be approximated by
LVIP f i l °,r l::/0
r
o
o ,
s
L,1:
1-½(s-½P) o
' ( s -°½ P ) o
•
o (s-½e)
In a fully three dimensional theory of isotropic plasticity, increments of plastic strain are proportional to stresses on the yield surface, dep = d)~. Z r, where dL is a scalar plastic multiplier. From eqns (5) and (6) it is clear that the diagonal components may be related in a consistent way by writing dwp - dup = dX. (S - ½P). Similarly if the off diagonal terms were required to hold pointwise, they would be related by d ( 1 R @ ) = d)~. r, but since only the polar moment of r appears as a generalized force, this latter relation, after multiplication by R and averaging over the cross section, yields ¼a2du'p = dX. Q. With plastic incompressibility also taken into account the complete flow rule becomes
a~ ; + 2a.. = o, aW'p- aU. = ax . ( s -
~ )
?,
~1 a2au p' = aX . Q .
(7)
The increment of plastic work, dWp = Sdw'p + Pdup + Qdu'p, again with plastic incompressibility taken into account becomes
aw~ =~2 ( S - 1 p5) d ( w ' p - Up) + Qdu'p
(8)
which is clearly consistent with eqn (7). The plastic work then becomes
dWp = ~ k2d~. where k
~_
[ ( S - i P1 )
(9)
2 +a--rQ 6 2 ]1/2
The function k = it(S, P, Q) will be taken to be the yield function. In a uniaxial tension or compression test k is just equal to the axial flow stress, S. Furthermore, in a uniaxial test the incremental plastic work is given by
SdS dWp = Sdep = H(S)
(10)
where H(S) is the slope of the plastic stress/strain curve, H(S) = dS/dEp. If S is replaced by k as the argument of H(.) in the general case when P and Q are nonzero, then'after equating eqns (9) and (10) the multiplier d~. is found to be
28
T . W . Wright
dX -
3 1 dk 2 H(k) k
(11)
and the plastic strain increments may be written as follows: dw'p .= -2dup = S H ½P'-dkk '
6 Q dk du'p = ~ "--'--H k "
(12)
Naturally the plastic strain increments are computed only if the stresses lie on the yield surface and dk > 0; that is, only if plastic loading occurs. IlL T H E C O M P L E T E S Y S T E M OF E Q U A T I O N S
So far the quantities u and u' have been treated as independent variables, but in the dynamical rod problem the second is the gradient of the first, and the quantity calculated for the second must be compatible with the first. Compatibility will be required only for the total strain, and not for the elastic and plastic parts separately. Thus if the identifications v=w,
e=w',
p=u,
q=u'
(13)
are made, the complete system of equations may be written as follows: k-v'=O,
q-p'=O,
S'-p~=O,
Q'
(14)
t~-p=O
1 2-p -~pa
P = 0
(15)
Eil:rc,Iil
(16)
where the 3 × 3 matrix of elastoplastic compliances, C p, is.symmetric and nonsingular. ok n(kl k where e and S here In symbolic form eqn (16) may also be written k = L S + a-~" stand for the integrated column vectors on the two sides of eqn (16), L stands for the matrix of elastic compliances, as deducible from eqn (4), and the second term, which gives the plastic strain rates, shows that plastic flow is normal to the generalized yield surface. With the convenient definitions o r - ( S - I p ) / k and fl=_ x/6Q/ak, so that 0/2 q-- B E = 1, the components of C p may be written as
[UP] =
v
a2
1-v
E
2H
2E
4g ~ a "H
1 ~2 I" ~ " ~
--
_,/-8 ~ 2a'H
"
2 + 6 ~21 ~
a~'-H I
•
(17)
Axial plastic waves in a rod
29
The determinant of eqn (17) is det ICp I= ~ {1 + 3~} > 0. Since the plastic modulus H is positive, the determinant is positive, and the matrix of compliances is invertible. For purposes of calculating wave speeds the inverted form, which is the matrix of plastic moduli, [MP], is simpler to work with. We have
Ii]= where the subscripts on
~: FMpEe :~qu M~Eu M~qqjMP1quqMI/~~tq~ M~uu
(18)
Mp indicate components, not derivatives, and
dee ICPl M~P~- a21E [ ( 1 - v)(1 + 3 H ) , 2 ( 1 - 2V)HOt2]
(19)
detICI'IM~u=detICPIM~-a22E[V(I + 3H)+(1-2V)H ~2]
(20)
detlCPlM~u--a2tzE[(1 + 3 ~ -det ICPl MPq =
-det
ICPl MPq~=
det
ICPl MPuq:
- (1-
det
ICPl MPqu=
det IC"l Mpq = ( 1 - 2v)(12k a +v)[1 + 3HO~2].
(1 "
2v)oq~ aE--H
(22)
(23)
IV. WAVE SPEEDS
Equations (14), (15), and either (16) or (18) form a system of eight first order equations, which may be put in the symbolic form MUt + NU~ = BU, where U is an 8 x 1 column matrix, M, N, and B are 8 x 8 square matrices, and the subscripts t and Z indicate partial differentiation. The matrix M depends only on the generalized stresses and elastic constants, and the matrices N and B are sparse and contain only constants. The characteristic wave speeds of the system may be found by considering acceleration waves, that is, waves where U is continuous but may have jumps in its first derivatives. Since M depends only on the components of U, we can write
M[Ut] --1-N[Uz] = 0
(24)
where in this context, the square brackets indicate the jump of the quantity enclosed across the acceleration wave. But jumps in temporal and spatial derivatives must be consistent and satisfy [Ut]+ V[Uz]= 0 (25)
30
T.W. Wright
where V is the local speed of the acceleration wave. Equations (24) and (25) combined lead to
(M-1N)[Ut] = O.
(26)
Consequently the speeds of propagation must satisfy
(IN)
det M - ff
= 0
(27)
and [Ut] must be a right eigenvector of the matrix (M - I N ) . In the present case with the components of Ut given by [u, f, k, ,b, t~, P, Q, s], eqn (27) becomes 1 0 0 p 0 V -I 0 0 det 0 0 MPuu 0 g qu o o
0 0 0 0 1 0 0 lpa2 0 "V-1 Meus 0 o o
0 0 0 0 0 0 0 0 1 0 MPuq -1 o o
0 0 0 V-' 0 0
-1 o
0 V-I 0 0 0 0 0 -1
=0.
(28)
Equation (28) is expanded by a tedious but straight forward calculation to give a biquadratic equation for the speeds of propagation.
(RV2-Mpe)(2Ra2V2-MPqq) - (MPqe)2~O.
(29)
There are two forward and two backward waves possible. Equations (19) and (23), together with the expression for det ICp I, show that in the elastic limit when H ~ oo, the plastic moduli obey M~ ~ 1-2v 1-v "1+~' E MPq ---+la2lz, and MP~--* 0. In this limit the two factors in eqn (29) are uncoupled, and the wave speeds reduce to the bulk longitudinal and shear speeds for an isotropic, elastic material. Because of the limits the two branches will be referred to as the fast or longitudinal mode and the slow or shear mode. Of course, complete expressions for the roots of eqn (29) may easily be written down for computations. It is well known that although the characteristic speeds in an elastic bar are the same as those in bulk material, the main energy in a pulse propagates with the so-called elastic bar speed, Cb = E V ~ , which is intermediate to the two characteristic speeds. Since the bar speed is not itself characteristic, there are no discontinuities associated with it, but it can carry large, rapid variations that are dispersive in nature. If the material is only linear elastic, then application of the Laplace transform, followed by asymptotic analysis for long times, will bring out the bar speed and the dispersion of the pulse associated with it (e.g. Wright, 1982). But if the material is nonlinear elastic or, as in the present case, nonlinear elastic-plastic, then the problem becomes one of identifying the bar speed in terms of the elastic properties and the plastic flow.
Axial plastic waves in a rod
31
The scaling method used by Wright (1982) works here, as well as in the purely nonlinear elastic case, considered previously. First define a new set of independent variables ( = t~I/2z,
"c = 81/2t,
(30)
where 8 is a small parameter. This scaling implies that fixed values of (, r will correspond to long times and large distances as 8 tends to zero. Next define a second set of independent variables in terms of (, r: r/=0((,r), ~=8(. (31) In terms of these final variables derivatives with respect to Z, t are replaced as follows: 0
O--Z=
t~l/2//( " 0 --[-83/2 0
-~0
0
0
8~' Ot -- ~l/2rlr " ~-~"
(32)
The function 0((, r) has been designed to have properties similar to those of a characteristic variable. The procedure to be followed below is a modification of that first introduced by Varley and Cumberbatch (1966), who called it the method of relatively undistorted waves. Thus, with the definitions to = r/~ and x = rk, a speed, c, can be defined as the slope, of a curve along which ~ is constant. Thus,
to
c --
d~ dZ -. dr dt
--
K
(33)
This speed will be identified as the nonlinear bar speed. It is sometimes more convenient to use the slowness, s --- 1/c, and the incremental arrival time, l = 1/to. With independent variables ~, r/the compatibility condition, to¢= K~, rewritten for s, l becomes s~ = 8l~.
(34)
With these preliminaries out of the way and after writing ,b = / i and P = Q ' - ½pa2~ instead of (14.3) and (15.2), we now apply eqn (32) to the whole set of eqns (14), (15), and (18). The system becomes 0 0 0 0 0 g~uu
0 p s 0 0 0
MPeu 0
0 0 1 0 0 M~ue
1 0 0 0 0 0 0 0 s 1 0 MPq
M~ee 0
M~ q
0 0 0 1 0 -1 o-1 0
0 0 0 0 0 0 0
o
s 0 0 0 0 o --1
"
uo q v~ e~ i p, = q. P~ Q.
_ S~ _
8'/2 (tou.) 8lS~ 3lv~ 8'/2~(KQ,-½pa2top,)+83/2Q~ o ~lp~ 0 0 0
(35) At this stage it will be assumed a priori that each term on the right hand side of eqn (35) is at least 0(iil/2). The assumption must be justified later in each application. Symbolically eqn (35) may be written as
32
T.W. Wright A ( U ) U o = 31/2bl(U) + Sb2(U) + 33/2b3(U)
(36)
where A(U) is the 8 × 8 matrix on the left hand side o f e q n (35), U is the solution vector, and bi(U), i = 1, 2, 3, are the three column vectors at each order on the right hand side. A regular perturbation scheme for U with expansions for A ( U ) and hi(U), i = 1, 2, 3 U = U° + &I/2u1 + & V 2 + ....
(37)
a ( u ) = A ( u °) + A e ( U ° ) ( U - U °) + ....
(38)
0
bi(U ) = b i ( f O) -~- - . ~ b i ( U ° ) ( U - U O) -~- ....
(39)
leads to the following sequence of equations: = o
8 /2 . A(uO)u
+ Ae(V°)Ut
= a t ( V °)
(40)
~ 1 . . ........
The first of eqn (40) demands that the determinant of A (U °) vanishes and that U~ is a right proper vector, R, of A (U°). Now multiplying the second of eqn (40) by a left proper vector, L, of A ( U °) eliminates the first term and leads to the equation LAvU1R = Lbl. But since LA = 0 by definition of a left proper vector, we also have L A e = - L E A . When written in index form with a comma denoting differentiation with respect to components of U, we find that, ~ LiAij, kU~Rj = - ~ Li, kAijUIRj. Since A R = 0 by definition of a i,j,k
i,j,k
right proper vector, the sum is zero, and the only surviving term in the original equation gives a compatibility condition, L(U°)bl (U °) = 0. The same condition occurs at all higher orders, recursively, so Lb = 0 in general. When applied to eqn (35), the three conditions sketched out in the last paragraph, detA = 0, U~ = R, and Lb = 0, lead to an expression for the nonlinear bar speed, to simple ordinary differential equations for the components of the solution vector, and to some extra compatibility conditions. From detA = 0, we find only one algebraic condition, namely pc2 = M ~
M~u
(41)
After a certain amount of algebra, it can be shown that eqn (41) reduces to
pc 2
H + 3#fl 2
E
E + H + (1 - 2v)/zfl2"
Since 0 < 132 < 1 during plastic flow, the bar speed lies in the range
(42)
Axial plastic waves in a rod
H pc 2 < -- < 1 E+HE-
33 (43)
where the left hand side corresponds to the condition of no radial shear, Q = 0 or/32 = 0, and the right hand side corresponds to the condition S = ½P or/32 = 1. In the elastic limit as H ~ o~ the right hand side of eqn (42) reduces to 1 for every value of/32, so the speed reduces to Cb = ~ , as it should if it is to be identified as the bar speed. To interpret the left hand side of eqn (43), let us write cr = ~ (ee + ep) ="~ (~ + ep) for the uniaxial stressstrain curve;. Then differentiation with respect to ep leads to the relation 1 &r
H
E de
E+H
(44)
after recalling that the hardening modulus is defined by H = ~-~. The relation eqn (43) may now be rewritten as (45)
In other words the bounds of the plastic bar speed are determined below by using the tangent modulus of the uniaxial stress-strain curve and above by using the elastic Young's modulus. In particular, the plastic bar speed is determined from the tangent modulus only when there is no radial shear. In general the presence of radial shear will make waves travel faster. This will be the case whenever the axial strain is nonuniform. The second condition, U~ = R, with R proportional to the cofactors of the bottom row of A, leads to the ratios u0o : _ v p ::v~o.
c = %o : 1 = p ,o7 : O = % :oO = P ~ : O = Q ° 7 : M P q s ( l + v p ) = ~ : p c 2 .
(46)
In eqn (461, the plastic Poisson's ratio, which is the negative ratio of increments of radial strain to axial strain, - u J0% ,0 comes out to be
Vp = MP_/MP_
v + (/z/H)[(1 + v) - (1 - 2v)/32]
u,..u=/~1q--~¥(-1-2-7~]"
(47)
Again in the elastic limit this expression reduces to the elastic Poisson's ratio, v, as it should, and furthermore, in the incompressible limit where v---~ 1/2, we also have Vp ~ 1/2, for any value of H, as is to be expected. In fact, it is not hard to show that for all H,O < t l < o~, and all/3, 0 < 13 < 1, we have v < vp < 1/2. In the ratio involving Q0 in eqn (46), the relationship M~qe= - M ~ has also been used. Since eqn (46) is an autonomous system of ordinary differential equations, it may be rewritten with any one of the components o f U chosen as the new independent variable. With the Choice of e ° the equations become
34
T.W. Wright du o dv ° dp ° dq ° dp ° -0 d 7 - - Vp, de o - - c , de o - de o - de o dQ° = M~e(1 + vp), de o
aS o de o
(48)
= pc 2
In fact, since the right hand sides in eqn (48) only depend on the generalized stresses (pO, QO, SO), and p0 is at most a constant, only the last two equations need to be integrated simultaneously, and then u ° and v ° may be found by quadrature. Finally, with L proportional to the cofactors of the left hand column of A,
[L] = [sMPq,(1 + Vp), c, - p c 2, -Vp, -MPq,(1 + Vp), -vp, O, 1]
(49)
the complete compatibility condition at all orders is found to be l
2
+ 8[clS~ - pc21v~ - lp~MPqe(1 + vp)] -
-
(50)
83/2upQ~n.
Although the right hand side is organized according to the powers of 8 that multiply the various terms, the terms themselves may have various orders of 8. Therefore, each case must be examined on its own merits in order to find the appropriate, lowest order compatibility condition. Now let us re-examine the right hand side of eqn (35) in light of the results so far. First note that the compatibility eqn (34) shows that either the slowness, s, is independent of ~/ to lowest order, or that the incremental arrival time,/, has a leading term that is 0 ( 8 - 1 ) . The first condition applies in an elastic wave, and then the wave is essentially steady. The elastic case will be examined later, but to see whether or not the steady case is possible for a fully plastic wave, s~ should be computed from eqn (42) and examined for feasibility. On the other hand, if the second condition applies, then oJ = 1-1 = 0 ( 8 ) , and since s = O(1), we must also have tc = 0 ( 8 ) , as well. Then to comply with the basic assumption that each term on the fight hand side of eqn (35) is at least 0(81/2), the leading terms of both S and v must be independent o f ~. That is, S = 80(/7) + 81/2S1(~, ~) --1-.....
1~ = v°(r]) + 81/2vl(T], ~) -~- .....
(51)
Equation (51) will be accepted for the moment and checked for consistency later. With this assumption, only the second and third rows on the fight hand side of eqn (35) are 0(81/2), all other rows are at least 0(83/2), and the inertia term in the fourth row is 0(83). The compatibility condition comes out to be S~-pev~
=0.
(52)
Equation (52) adds no new conditions to eqn (48) and so can be ignored in constructing the solution to lowest order.
Axial plastic waves in a rod
35
It should be noted that although ~ - # 0 so that radial shear contributes to the solution at lowest order, radial inertia does not contribute to the plastic wave in a simple wave. This last fact can be seen by observing that since o9 = 0(8) by assumption, and pn = 0(83/2) at the very least, the last term on the first line of eqn (50), which is the only contribution of radial inertia, is at least 0(83). This is in distinct contrast to the case of an elastic wave, as will be seen in the section after next. V. INITIAL-BOUNDARY VALUE P R O B L E M
It is now possible to consider the following initial-boundary value problem for a semiinfinite rod. Find the wave structure that exists in the rod at long times and large distances if the rod i,~ initially at rest with free sides, and if a known strain is impulsively applied to the end at Z = 0. To lowest order the solution may be constructed as follows: (a) At the origin Z = 0 the time of initiation of each level of strain is given simultaneously at t = 0. Thus if the speed of propagation (or the slowness) is known as a function of E, lines of constant strain may be drawn from the origin with slope equal to the speed. In coordinates ((, r) the equation for the family of straight lines may be written -
r
=
c(e)
=
1 s(e)
.
(53)
It will be assumed that the magnitude of c decreases as the magnitude of e increases so that the lines of constant strain form an opening fan at the origin. (b) Integrate the last two equations of eqn (48). These equations in full are
dS H + 3/zfl2 ~-e = E- E + H + (1 - 2v)/zfl2
(54)
dQ -6#2(1 + Vp). ( S - ½?)Q -~e = (H + 3tt)k 2 where /3 = x/-6Q/ak, k = _[(S _ ~p)l
2
+(v~a/ak)
2
(55)
1/2
]
, H(k) is a known function,
and Vp is given by eqn (47). With P = O the right hand sides of eqns (54) and (55) s
/
\
+ u_~.~}dS and Q ___0, are functions only of S and Q. One solution is e -- ey + ~ (1 \Sr
Sy and ey are the stress and strain at first yield. This is simply the KfirmfinTayl.~r solution (e.g. Kolsky, 1963). With e(S) known and the fan defined by where
= c= " .V~E+I~CS), the asymptotic solution is completely determined to lowest order, provided that H (S) decreases with increasing magnitude of S so that the fan opens out. Now look more closely at eqn (55). If S > 0, as in a tension wave, then eqn (55), plotted in the phase plane, gives a curve through the origin that passes from the 2rid to the 4th
36
T.W. Wright
quadrant. For a phase curve of this type every small disturbance in Q at the leading edge of the wave tends to evolve towards the origin as strain becomes increasingly positive in the developing wave, provided that k remains non-negative, of course. On the other hand, if S < 0, the phase curve for Q passes through the origin from the 3rd to the 1st quadrant. Again every disturbance in Q at the leading edge of the wave evolves toward the origin as strain now becomes increasingly negative in the developing wave. Thus, in either case the Kfirmfin-Taylor solution is stable for strains applied at the end of the bar, whether tensile or compressive, and can be expected to dominate eventually. If the laigher order rod theory only produced the K~irmfin-Taylor solution as its lowest order approximation, little would have been gained. However, if the disturbance in Q is large, the wave speeds are modified significantly, as was shown in the discussion preceding and following eqn (45), and this in turn can modify the structure of the wave for some distance behind its leading edge. Of course, the fastest part of the wave is elastic, and in the next section it is shown how substantial radial shear can be present in the elastic precursor if the loading on the end is applied rapidly enough. VI. ELASTIC WAVES
In an elastic wave all wave speeds are constant. The compatibility eqn (50) still applies, but now there is a substantial reordering of terms. To see how this comes about, label the lines of constant speed in an arbitrary way, r / = f ( r - (/Cb), where f is an arbitrary, but smoothly increasing function of its argument, ~0 = r - (/Cb, so that ~ increases at a fixed station in the rod as time increases. We have o~ = r/+ =f~o, and r = 17¢= -c~tf~, so that w/x = --Cb = -- Ex/E~. Equation (34) is satisfied identically at all orders, and s, to, r, and l are all O(1). Furthermore, for an elastic material the moduli M p = M~q, = 0, and M~qq = la2/z. Finally from eqn (35) it is clear that p, q, and Q all start at O(81/2), and P starts at 0(8). That is, p = 81/2p I + . . . ,
q=
81/2q1 + - - . ,
Q = 81/2Q 1 + . . . ,
P = 8P 2 + . . . .
(56)
At 0(81/:') the compatibility eqn (50) is satisfied identically, but at 0(8) the compatibility condition demands that 1 -v-~q0 (rQ',7--~1 pa2wp~) + cblS~-- pc2blv~= 0 .
(57)
F r o m the' basic eqns (35) it can be seen that
P" = O-~-(wu°)o'l
1 2 Q'n = ~a lzq,lv q,7l = _spl
(58)
and from the relationships in eqn (46) it can be seen that
= _pc
O,
uo0
p 0
(59)
Axial plastic waves in a rod
37
With eqns (58) and (59) put into eqn (57) and with the recognition that t°-aon= ~ ' then after some rearrangement of terms the compatibility equation becomes
v~o -- r3c3v~ ----O,
where
r3 =
8(1 + v) v2(1 -k- 2v)a 2 "
(60)
The super:script on the axial velocity, v, has been dropped for clarity in eqn (60). The development leading up to eqn (60) clearly shows that in an elastic wave radial inertia and radial shear both contribute substantially to the lowest order solution. It will be recognized that eqn (60) is a transport equation that describes the elastic dispersion at the leading edge of the wave. A similarity solution may be found by assuming that the axial velocity depends on the independent variables only in the combination X = ¢p~m for some constant, m. Then with v = V(x) we have v ~ = ~3m "7~r d3V and
v~ = m x U l '~av XX" With m = - 1 / 3 , the terms ~3m and ~-1 cancel from eqn (60), which now reduces to an ordinary differential equation d3V 1 33 d V dx 3 F~r cbX.~xx = O.
(61)
Equation (61) is a second order O.D.E. for dV/dx whose solutions may be written in terms of tile Airy functions Ai and Bi (Abramowitz and Stegun, 1965). To ensure that the solution rapidly tends to zero ahead of the wave, only the Ai function can be used. In the original wLriables, the solution may be written
rCb V¢ = C .£ZTV.~/3A t ( - ~ )
(62)
v = C I Ai(-z)dz
(63)
and
--00
where C is an arbitrary constant, and • = ~ (3~).,~ • Note that the scaling parameter 8, which occars implicitly in both numerator and denominator of ~, cancels out so that the solution does not actually depend on the exact choice of 8. This fact motivated the choice of scaling in the first place. The representation in eqn (63) is exactly the same as the one obtained in Wright (1982) for the problem of a step velocity applied to the boundary of a linear elastic rod. In that case a solution was found by LaPlace transform, followed by an asymptotic expansion. Note the familiar dispersion of the rapidly rising front according to the negative one third power of distance for elastic waves in a circular cylindrical rod, as first noted by Skalak (1957) for the full 3D theory of elasticity. VII. PLASTIC WAVES The corLstant C that sets the scale of the particle velocity in the elastic part of the wave cannot be specified precisely in the elastic-plastic problem because it is unknown when
38
T.W. Wright
yield first occurs other than to note that ker = S 2 + (6Q2/a2), where k r is the initial yield stress and P is taken to be zero. Referring back once again to eqns (56), (58), and (59), expressions for S and Q in the elastic wave are S = --pCbV
and
Q = --31/2 v/2a2 2c~ .v,.
(64)
The ratio fl/a = v/-6Q/aS is independent of the constant C and indicates how much contribution the radial shearing makes to initial yielding. In terms of the elastic solution the ratio is given by
fl [3 /2 var~ z/2 . . . . . cx
E (3~) 1/3
Ai(-~)
~ 0.4(Z/a) -x/3
(65)
f Ai(-z)dz
where • is given following eqn (63). The approximate value holds for a wide range of Poisson's ratios because the Airy function and its integral are nearly equal near the wave front (small ~) and because the remainder of the coefficient is only a weak function of v for a wide range of values, at least 1/6 < v < 1/2. Since ot2 +/32 = 1, an estimate for the relative amounts of axial stress and radial shearing at first yield is
~/6(Z/a) 1/3 I°tY 1~ [1 + 6(Z/a)2~3] 1/2'
1 IflY I~ [1 + 6(Z/a)2~3] 1/2"
(66)
The net result is that radial shear accounts for a substantial fraction of the yield stress at first yield until the wave has travelled a substantial distance, Z/a >> 1. Consequently, not only is Q not zero as an initial condition for eqn (55), which describes the evolution of Q in the fully plastic wave, but if eqn (66) were to be taken literally, at the moment of initiation when the front is at Z = 0, all yielding is due only to radial shear, i.e. fl = 1. Furthermore, if, fl = 1, then ot = 0, so the initial rate of decay of Q would also be zero. Of course, eqn (66) should not be taken too literally since the whole theory developed so far is intended to be asymptotic for "long" times and "large" distances, which must be interpreted as Z/a >> 1 and cbt/a >> 1. Nevertheless, some initial condition must be given for eqns (54) and (55), and the choice S = 0, Q = 4- akr/v/-6 at e = 0 is one possible and consistent choice. (The plus or minus sign on the initial value of Q applies to tension and compression respectively at the origin for a wave moving to the right.) As a simple example, consider an incompressible material that has no work hardening. Then v = 1/2, so that E = 3/2. Also, H _-- 0, the yield function is constant, k = kr, and eqns (54) and (55) simplify to
de = 3lzfl2 = E 1 dO_ dE which have the immediate solutions
E. SQ k2
(67) (68)
Axial plastic wavesin a rod S = krtanh(Ee/kr),
v--
Q = ~(akr/v/-6)sech(Ee/kr)
4k, y- ~ t.a n
-1(
sinh kEg~, y] u = - e l 2
39 (69)
(70)
where 0 < e < oo in eqns (69) and (70). Since the wave speed for this case is given by pc 2 = 3/zfl2, according to eqn (42), the curves of constant e make a fan centred on the origin and are distributed according to c = Z/t = x/~
sech(Ee/kr).
(71)
The solution laid down by eqns (69), (70), and (71) is given parametrically in e and has an entirely reasonable interpretation. The fastest wave travels with the elastic bar speed, and ahead of the first arrival all fields are quiescent. At impact and at first arrival (if elastic dispersion is ignored) the material jumps to full yield, but since there can be no jump in the axial stress, which would imply a jump in radial strain as well, and hence a discontinuity in the deformed radius of the bar, there is a jump in the radial shear stress sufficient to cause yield instead. Then as the wave develops, there is a transfer from radial shearing to axial loading, and the wave continues to evolve until the maximum applied strain is reached. It is interesting to note that since 0 < tan -1 x < rr/2, no matter what the applied strain, the magnitude of the maximum axial velocity is limited to [ v I< Jrky/2v/-PE. Behind the fan of the loading wave further deformation ceases. As the total strain builds up to the static yield strain, er = k r / E , and then exceeds it, the wave speed and Q both begin to drop rapidly, and the axial stress tends toward the full yield stress. Note that even though the axial stress approaches the static yield stress, but never reaches it, in fact the material is yielding throughout the whole wave. When elastic dispersion is included, the solution is somewhat modified. The elastic wave is described by eqns (63) and (64). Now suppose the axial stress and strain in the elastic portion of ~thewave at the moment of first yield are So and e0 = SolE. Integration of eqns (67) and (68) now gives the plastic wave as
[
S = k ~ : t a n h t a n h -1
-t
E-So] ~rr
'
Q=
+a~/6 sech[tanh_,S0 E k-S0] k k--Yr+ r J
(72)
where So/kr is estimated from the first of eqn (66) and l e0 [
dy . . de
.
1 . I+R
xy x 2 + y 2"
(74)
40
T . W . Wright
With the change of variables x = rsinqJ,y = rcosqJ these equations become
dr de
R --. I+R dq~ --
de
sin qJ
(75)
cos ko -
(76)
r
It is no longer possible to integrate the equations completely in closed form, but one integration is possible and the other reduces to a quadrature.
/cosq,0"~&
r = ro ~ , ~ )
(77)
qJ
I( ,+2R r0(cos qJ0)i~R COS k o ) - ~ d ~
(78)
: E -- E0.
koo
Since S / E = r sin ~o and v ~ Q / a E = +r cos qJ, the solution is now given parametrically in qJ. If elastic dispersion is ignored, qJ0 --- 0 and the parameter evolves toward Jr/2 in a tensile wave or -zr/2 in a compressive wave. If elastic dispersion is included, it may be accounted for in the same manner as before. Since pc 2 : -l~' - iR+fl2 - ~ in this example and
r = k/E, the solution is distributed parametrically over the wave fan Z --=c=
Vf•[ 1
sin 2 qJ'l
u2
/
(79)
VIII. STABILITY CONSIDERATIONS
Although the solutions in the last section are perfectly well defined as the strain becomes arbitrarily large, they may no longer make physical sense if the strain is substantially more than the yield strain and k//~ • i -R~ IS • less than the slow characteristic speed. In the first of the two examples where there is no work hardening it is clear that the wave speed tends toward zero at large strain, so that all the deformation piles up near the origin. In the second example, clearly if R is small enough, the same thing can happen. This remark brings up a new issue. The wave speed actually can become so slow that it is less than the slower of the two characteristic speeds. Thus, revisiting eqn (29) for the characteristic speeds, we can write down the solution of the quadratic as I~
pV~2' = ~ (M~, + a2/2]
p
2
p 2
- a 2 / 2 ] -- a--~-J
(80)
Axial plastic wavesin a rod
41
where the subscripts 1 and 2, as well as the plus and minus signs, refer to the fast and slow characteristic speeds, respectively. Whereas it is easy to see that 0I:21 is always greater than pc 2 (subtract eqn (80) with the plus sign from eqn (41) and note that the result is always negative), it can happen that pc2 is less than p V 2 for some values of I z / H and u 2. For example, in the simpler problem above, the fast speed works out to be infinite, but the slow speed is given by p V 2 =/xot 2. Since the bar speed was found to be given by pc 2 = 3/zf12., these two speeds are equal when f12 = 1/4, ot2 = 3/4, or when akr Q - 2~#6'
S = ~-kr.
(81)
From the second of eqn (81) and the first of eqn (69), it can be seen that this condition is reached when the strain is given by e ~ 1.32er or Iv I~ 1.05kr/v/p-E where er is the strain at static yield. It has been shown by Wu (1961) that there is a stable hierarchy of waves (in Whitham's (1974) terminology) in the linear problem only if 1:2 < c < Vl, but it is unknown to what extent the same condition applies in the nonlinear case. If a similar condition applies, then the straight forward asymptotic solution is unlikely to hold for the complete problem. It may be speculated that in a compressive wave mush- rooming would be predicted to occur at the loaded end of the bar when c becomes too small for the bar wave to keep ahead of the slow characteristic speed, and that similarly necking may be predicted in a tensile wave. Further exploration of these aspects of the problem will be deferred to another day. IX. DISCUSSION AND CONCLUSIONS A theory that includes the effects of radial shear and radial inertia has been proposed and partialtly developed for propagation of axial waves in a circular cylindrical bar. Although based on the limited kinematical foundation provided by the Mindlin and Herrimann (1951) theory, there is ample scope for a representation of predominantly axial motions. The theory shows that the Khrm~n-Taylor solution should hold eventually at large distances from the impact end of a bar, where radial shear has decayed essentially to zero, but since elastic dispersion is relatively slow (only a one third power in eqns (62) and (63)) the distance may be several tens of radii from the impact end before the K~trmfinTaylor solution becomes a good approximation. Alternatively, if the axial loading at the impact end is slow enough so that the wave fan is distributed over the time axis, rather than issuing completely from the origin, radial shear is always small, and the K~irm~inTaylor solution will be valid throughout. An impact problem constructed within the context of the complete theory would require additional consideration of the leading and trailing characteristic waves so as to develop the asymptotic transitions into the main bar wave. However, the main structure of the bar wave seems perfectly clear as laid out in the exposition above. The experimental implications of the theory have yet to be explored. However, it would seem perfectly possible to observe a wave fan whose rays are straight and whose slopes (wave speeds) depend only on the level of axial strain, but which represents a solution with substantial levels of radial shear, rather than the K~irm~m-Taylor limit of vanishing radial shear. Then integration of the 7aS/ = Pc2, where the right hand side is determined from the observed slopes, could result in a calculated S - e curve that is higher than the
42
T.W. Wright
quasi-static curve at the same axial strain because of eqn (45). In fact, that appears to be exactly what has been observed by Hauver (1978), who measured strains and arrival times in a series of experiments on instrumented penetrators, which were made from a precipitation hardened, and nearly rate insensitive, tool steel. His stress-strain curve, as calculated from measured wave speeds at constant strains and the relationship dg/dE = pc 2, lies close to the quasi-static curve at first yield, but diverges from it with increasing axial strain. By about 15% strain, where the experiment ended, the calculated curve lies at least 30% above the quasi-static curve, far higher than could be explained by rate dependence alone. The explanation seems to be not that the dynamic constitutive response is higher than the quasi-static response, but that radial shear also does plastic work and contributes substantially to work hardening. The reader should be reminded explicitly that the material has been assumed to be rate independent throughout this work. If there were rate dependence as well as inertial and shear effects, the present analysis would not apply without some modification, and Hauvet's experiment would become even more difficult to analyze and understand. Acknowledgements--This paper is an extension of work that was presented at the Ninth U. S. National Congress of Applied Mechanics (but never published) in a session on Viscoelastic and Plastic Waves, organized by the late Professor James F. Bell. The work was inspired by Professor Bell's superb experimental observations of plastic wave propagation that he conducted over many years.
REFERENCES
Abramowitz, M. and Stegun, I. A. (1965) Handbook of Mathematical Functions. Dover, New York. Antman, S. S. (1972) The Theory of Rods. Handbuch der Physik, Vol. Via/2. Springer-Verlag, New York. Hauver, G. E. (1978) Penetration with instrumented rods. Int. J. Engng Sei. 16, 871-877. Kolsky, H. (1963) Stress Waves in Solids. Dover, New York. Mindlin, R. D. and McNiven, H. D. (1960) Axially symmetric waves in elastic rods. J. Appl. Mech. 27, 145-151. Mindlin, R. D. and Herrmann, G. (1950) A one-dimensional theory of compressional waves in an elastic rod. Proe. 1st U.S. Nat. Cong. Appl. Mech., pp. 187-191. Skalak, R. (1957) Longitudinal impact of a semi-infinite circular elastic bar. J. Appl. Mech. 24, 59~:~4. Varley, E. and Cumberbatch, E. (1966) Non-linear high frequency sound waves. J. Inst. Maths. Applies. 2, 133-143. Whitham, G. B. (1974) Linear and Nonlinear Waves. Wiley-Interscience, New York. Wright, T. W. (1982) Nonlinear waves in rods. Proc. IUTAM Syrup. on Finite Elasticity, eds D. E. Carlson and R. T. Shield. Martinus Nijhoff, The Hague. Wu, T. T. (1961) A note on the stability condition for certain wave propagation problems. Comm. Pure and Appl. Math. 14, 745-747.
International Journal of Plasticity, Vol. 14, Nos. 1-3, pp. 25-42, 1998 © 1998ElsevierScienceLtd Printed in Great Britain. All rights reserved 0749-6419/98$19.00+ 0.00 PII: S 0 7 4 9 - 6 4 1 9 ( 9 7 ) 0 0 0 3 8 - 7
AXIAL PLASTIC WAVES IN A ROD T. W. Wright US. Army Research Laboratory, Aberdeen Proving Ground, MD 21005-5066, U.S.A.
(Received in final revised form 10 July 1997) Abstract--Equations are presented to describe nonlinear plastic wave propagation in a rod with the effects of radial inertia and radial shear included. Intrinsic constitutive equations are postulated for plastic deformation, and some of the principal aspects of wave propagation are described. In particular, a scaling argument is used to extract the nonlinear bar speed, which carries the principal pulse of energy, rather than the two characteristic speeds. © 1998 Elsevier Science Ltd. All rights reserved I. INTRODUCTION Equations o f motion, suitable for describing wave p r o p a g a t i o n in a circular cylindrical r o d with stress free sides, m a y be obtained f r o m the three dimensional equations o f a nonlinear c o n t i n u u m (e.g. A n t m a n , 1972) or they m a y be f o u n d directly by assuming dependence on an axial strain, radial strain, and the axial gradient o f radial strain. If the m o t i o n o f tlhe rod is a p p r o x i m a t e d by r = R[I -4- u(Z, t)], z = Z + w(Z, t)
(1)
where (r, z) are the current coordinates, (R, Z) are the material coordinates, and the functions (u(Z, t) and w(Z, t) are kinematic degrees o f freedom, then an a p p r o x i m a t i o n for the kinetic energy per unit length o f rod is K - - ½zraZp(w2 + ½a2t~2), where p is the reference mass density, a n d a is the reference radius o f the rod. In addition, if the material is elastic with stored energy per unit length o f W = zra2 W ( w t, u, u ~), where the dash means differentiation with respect to Z, then the corresponding E u l e r - L a g r a n g e equations m a y be written
S' = pf~, Q' - P = ~ pa 2 u"
(2)
where S - 9w ow In eqn (2) S m a y be interpreted as the average o f the i~w" p _ - - aw 0u ' and Q = ~-7. axial stress, P as the average o f the sum o f the radial and circumferencial stresses, and Q as the polar m o m e n t o f the average radial shear stress (e.g. Wright, 1982). The addition o f the second degree o f freedom, u(Z, t), has m a d e possible the inclusion o f radial shear and radial inertia in a consistent fashion. F o r the case o f a linear isotropic -
-
25
26
T.W. Wright
material equations equivalent to eqn (2) were first given by Mindlin and Herrmann (1950), but were superceded later by equations given by Mindlin and McNiven (1960) that included still a third degree of freedom at each cross section of the rod. This later version allowed cupping motions of the cross section as well as radial motions, so that the frequency dispersion curves were reasonably accurate up to the first cut off frequency, which might be either a purely radial or a purely axial motion of the whole rod depending on the elastic constants. Since the third degree of freedom, which would introduce significant additional complexity, has not been included here, the higher dispersion curves will not be accurately represented, but the lowest curve will show major effects due to the radial coupling. In the present paper the interest is in modifications to dominantly axial pulses, so the extra degree of freedom is not required. An intrinsic version of a rod theory that is kinematically equivalent to the case sketched above, but that includes plastic deformations will be developed. The constitutive equations and the complete system of equations that govern the motion are developed in the next two sections. Then a section devoted to a discussion of wave speeds in the bar and the associated asymptotic motions precedes a section that describes a typical initialboundary value problem to be treated. Next, in two sections devoted to elastic and plastic waves, respectively, elastic dispersion in the precursor is described, two new solutions for plastic wave propagation are presented, as well as the influence of the elastic dispersion on the subsequent development of the plastic wave. A few remarks about stability are made in the next section, followed by a final section of discussion and conclusions. II. CONSTITUTIVE EQUATIONS It will be assumed that the balance laws for axial and radial motion remain the same as eqn (2), but that the constitutive equations may be divided in the standard way into additive elastic and plastic parts. Thus, the strain increments may be written dw'=dwe
+ d w ' p,
du=due-~-dldp,
dut=du;--~-du;
(3)
where as before the dash indicates differentiation with respect to the axial coordinate, Z. (Strictly speaking, only the terms on the left hand side of eqn (3) are derivatives. The elastic components on the right hand side will be computed from stresses, as is usual in plasticity, and the plastic components are the remainders.) Clearly w', u, and u' have the interpretation of axial strain, radial strain, and the axial gradient of radial strain, respectively. For the moment u and u ' will be treated as independent variables. The elastic constitutive relations will be assumed to be the same as those given by Mindlin and Herrmann (1950). dw
'e =
dS dP - - -- v - - ~ , E
1 - u dP due - - - 2 E
dS v---~,
1 ~ a2du ' e
dQ
= --
ii~
(4)
where E,/z, and v are the elastic Young's modulus, shear modulus, and Poisson's ratio, respectively. The usual relationship among the three holds, E = 2/z(1 + v). To find reasonable constitutive expressions for the increments of plastic strain we first consider approximate representations for the deviatoric components of stress and strain in cylindrical coordinates. From the kinematics laid down in eqn (1) the linearized strain, E, and its deviator, e, have components
Axial plastic waves in a rod
tel=
i uo ,u,l 0
u
0
Ru' 0
,
[el=
w'
27
o
½(u-w') 0
0
½Ru'
oU, 1
(5)
]
(6)
-~(u-w')l
The average stress, Z, and its deviator, s, will be approximated by
LVIP f i l °,r l::/0
r
o
o ,
s
L,1:
1-½(s-½P) o
' ( s -°½ P ) o
•
o (s-½e)
In a fully three dimensional theory of isotropic plasticity, increments of plastic strain are proportional to stresses on the yield surface, dep = d)~. Z r, where dL is a scalar plastic multiplier. From eqns (5) and (6) it is clear that the diagonal components may be related in a consistent way by writing dwp - dup = dX. (S - ½P). Similarly if the off diagonal terms were required to hold pointwise, they would be related by d ( 1 R @ ) = d)~. r, but since only the polar moment of r appears as a generalized force, this latter relation, after multiplication by R and averaging over the cross section, yields ¼a2du'p = dX. Q. With plastic incompressibility also taken into account the complete flow rule becomes
a~ ; + 2a.. = o, aW'p- aU. = ax . ( s -
~ )
?,
~1 a2au p' = aX . Q .
(7)
The increment of plastic work, dWp = Sdw'p + Pdup + Qdu'p, again with plastic incompressibility taken into account becomes
aw~ =~2 ( S - 1 p5) d ( w ' p - Up) + Qdu'p
(8)
which is clearly consistent with eqn (7). The plastic work then becomes
dWp = ~ k2d~. where k
~_
[ ( S - i P1 )
(9)
2 +a--rQ 6 2 ]1/2
The function k = it(S, P, Q) will be taken to be the yield function. In a uniaxial tension or compression test k is just equal to the axial flow stress, S. Furthermore, in a uniaxial test the incremental plastic work is given by
SdS dWp = Sdep = H(S)
(10)
where H(S) is the slope of the plastic stress/strain curve, H(S) = dS/dEp. If S is replaced by k as the argument of H(.) in the general case when P and Q are nonzero, then'after equating eqns (9) and (10) the multiplier d~. is found to be
28
T . W . Wright
dX -
3 1 dk 2 H(k) k
(11)
and the plastic strain increments may be written as follows: dw'p .= -2dup = S H ½P'-dkk '
6 Q dk du'p = ~ "--'--H k "
(12)
Naturally the plastic strain increments are computed only if the stresses lie on the yield surface and dk > 0; that is, only if plastic loading occurs. IlL T H E C O M P L E T E S Y S T E M OF E Q U A T I O N S
So far the quantities u and u' have been treated as independent variables, but in the dynamical rod problem the second is the gradient of the first, and the quantity calculated for the second must be compatible with the first. Compatibility will be required only for the total strain, and not for the elastic and plastic parts separately. Thus if the identifications v=w,
e=w',
p=u,
q=u'
(13)
are made, the complete system of equations may be written as follows: k-v'=O,
q-p'=O,
S'-p~=O,
Q'
(14)
t~-p=O
1 2-p -~pa
P = 0
(15)
Eil:rc,Iil
(16)
where the 3 × 3 matrix of elastoplastic compliances, C p, is.symmetric and nonsingular. ok n(kl k where e and S here In symbolic form eqn (16) may also be written k = L S + a-~" stand for the integrated column vectors on the two sides of eqn (16), L stands for the matrix of elastic compliances, as deducible from eqn (4), and the second term, which gives the plastic strain rates, shows that plastic flow is normal to the generalized yield surface. With the convenient definitions o r - ( S - I p ) / k and fl=_ x/6Q/ak, so that 0/2 q-- B E = 1, the components of C p may be written as
[UP] =
v
a2
1-v
E
2H
2E
4g ~ a "H
1 ~2 I" ~ " ~
--
_,/-8 ~ 2a'H
"
2 + 6 ~21 ~
a~'-H I
•
(17)
Axial plastic waves in a rod
29
The determinant of eqn (17) is det ICp I= ~ {1 + 3~} > 0. Since the plastic modulus H is positive, the determinant is positive, and the matrix of compliances is invertible. For purposes of calculating wave speeds the inverted form, which is the matrix of plastic moduli, [MP], is simpler to work with. We have
Ii]= where the subscripts on
~: FMpEe :~qu M~Eu M~qqjMP1quqMI/~~tq~ M~uu
(18)
Mp indicate components, not derivatives, and
dee ICPl M~P~- a21E [ ( 1 - v)(1 + 3 H ) , 2 ( 1 - 2V)HOt2]
(19)
detICI'IM~u=detICPIM~-a22E[V(I + 3H)+(1-2V)H ~2]
(20)
detlCPlM~u--a2tzE[(1 + 3 ~ -det ICPl MPq =
-det
ICPl MPq~=
det
ICPl MPuq:
- (1-
det
ICPl MPqu=
det IC"l Mpq = ( 1 - 2v)(12k a +v)[1 + 3HO~2].
(1 "
2v)oq~ aE--H
(22)
(23)
IV. WAVE SPEEDS
Equations (14), (15), and either (16) or (18) form a system of eight first order equations, which may be put in the symbolic form MUt + NU~ = BU, where U is an 8 x 1 column matrix, M, N, and B are 8 x 8 square matrices, and the subscripts t and Z indicate partial differentiation. The matrix M depends only on the generalized stresses and elastic constants, and the matrices N and B are sparse and contain only constants. The characteristic wave speeds of the system may be found by considering acceleration waves, that is, waves where U is continuous but may have jumps in its first derivatives. Since M depends only on the components of U, we can write
M[Ut] --1-N[Uz] = 0
(24)
where in this context, the square brackets indicate the jump of the quantity enclosed across the acceleration wave. But jumps in temporal and spatial derivatives must be consistent and satisfy [Ut]+ V[Uz]= 0 (25)
30
T.W. Wright
where V is the local speed of the acceleration wave. Equations (24) and (25) combined lead to
(M-1N)[Ut] = O.
(26)
Consequently the speeds of propagation must satisfy
(IN)
det M - ff
= 0
(27)
and [Ut] must be a right eigenvector of the matrix (M - I N ) . In the present case with the components of Ut given by [u, f, k, ,b, t~, P, Q, s], eqn (27) becomes 1 0 0 p 0 V -I 0 0 det 0 0 MPuu 0 g qu o o
0 0 0 0 1 0 0 lpa2 0 "V-1 Meus 0 o o
0 0 0 0 0 0 0 0 1 0 MPuq -1 o o
0 0 0 V-' 0 0
-1 o
0 V-I 0 0 0 0 0 -1
=0.
(28)
Equation (28) is expanded by a tedious but straight forward calculation to give a biquadratic equation for the speeds of propagation.
(RV2-Mpe)(2Ra2V2-MPqq) - (MPqe)2~O.
(29)
There are two forward and two backward waves possible. Equations (19) and (23), together with the expression for det ICp I, show that in the elastic limit when H ~ oo, the plastic moduli obey M~ ~ 1-2v 1-v "1+~' E MPq ---+la2lz, and MP~--* 0. In this limit the two factors in eqn (29) are uncoupled, and the wave speeds reduce to the bulk longitudinal and shear speeds for an isotropic, elastic material. Because of the limits the two branches will be referred to as the fast or longitudinal mode and the slow or shear mode. Of course, complete expressions for the roots of eqn (29) may easily be written down for computations. It is well known that although the characteristic speeds in an elastic bar are the same as those in bulk material, the main energy in a pulse propagates with the so-called elastic bar speed, Cb = E V ~ , which is intermediate to the two characteristic speeds. Since the bar speed is not itself characteristic, there are no discontinuities associated with it, but it can carry large, rapid variations that are dispersive in nature. If the material is only linear elastic, then application of the Laplace transform, followed by asymptotic analysis for long times, will bring out the bar speed and the dispersion of the pulse associated with it (e.g. Wright, 1982). But if the material is nonlinear elastic or, as in the present case, nonlinear elastic-plastic, then the problem becomes one of identifying the bar speed in terms of the elastic properties and the plastic flow.
Axial plastic waves in a rod
31
The scaling method used by Wright (1982) works here, as well as in the purely nonlinear elastic case, considered previously. First define a new set of independent variables ( = t~I/2z,
"c = 81/2t,
(30)
where 8 is a small parameter. This scaling implies that fixed values of (, r will correspond to long times and large distances as 8 tends to zero. Next define a second set of independent variables in terms of (, r: r/=0((,r), ~=8(. (31) In terms of these final variables derivatives with respect to Z, t are replaced as follows: 0
O--Z=
t~l/2//( " 0 --[-83/2 0
-~0
0
0
8~' Ot -- ~l/2rlr " ~-~"
(32)
The function 0((, r) has been designed to have properties similar to those of a characteristic variable. The procedure to be followed below is a modification of that first introduced by Varley and Cumberbatch (1966), who called it the method of relatively undistorted waves. Thus, with the definitions to = r/~ and x = rk, a speed, c, can be defined as the slope, of a curve along which ~ is constant. Thus,
to
c --
d~ dZ -. dr dt
--
K
(33)
This speed will be identified as the nonlinear bar speed. It is sometimes more convenient to use the slowness, s --- 1/c, and the incremental arrival time, l = 1/to. With independent variables ~, r/the compatibility condition, to¢= K~, rewritten for s, l becomes s~ = 8l~.
(34)
With these preliminaries out of the way and after writing ,b = / i and P = Q ' - ½pa2~ instead of (14.3) and (15.2), we now apply eqn (32) to the whole set of eqns (14), (15), and (18). The system becomes 0 0 0 0 0 g~uu
0 p s 0 0 0
MPeu 0
0 0 1 0 0 M~ue
1 0 0 0 0 0 0 0 s 1 0 MPq
M~ee 0
M~ q
0 0 0 1 0 -1 o-1 0
0 0 0 0 0 0 0
o
s 0 0 0 0 o --1
"
uo q v~ e~ i p, = q. P~ Q.
_ S~ _
8'/2 (tou.) 8lS~ 3lv~ 8'/2~(KQ,-½pa2top,)+83/2Q~ o ~lp~ 0 0 0
(35) At this stage it will be assumed a priori that each term on the right hand side of eqn (35) is at least 0(iil/2). The assumption must be justified later in each application. Symbolically eqn (35) may be written as
32
T.W. Wright A ( U ) U o = 31/2bl(U) + Sb2(U) + 33/2b3(U)
(36)
where A(U) is the 8 × 8 matrix on the left hand side o f e q n (35), U is the solution vector, and bi(U), i = 1, 2, 3, are the three column vectors at each order on the right hand side. A regular perturbation scheme for U with expansions for A ( U ) and hi(U), i = 1, 2, 3 U = U° + &I/2u1 + & V 2 + ....
(37)
a ( u ) = A ( u °) + A e ( U ° ) ( U - U °) + ....
(38)
0
bi(U ) = b i ( f O) -~- - . ~ b i ( U ° ) ( U - U O) -~- ....
(39)
leads to the following sequence of equations: = o
8 /2 . A(uO)u
+ Ae(V°)Ut
= a t ( V °)
(40)
~ 1 . . ........
The first of eqn (40) demands that the determinant of A (U °) vanishes and that U~ is a right proper vector, R, of A (U°). Now multiplying the second of eqn (40) by a left proper vector, L, of A ( U °) eliminates the first term and leads to the equation LAvU1R = Lbl. But since LA = 0 by definition of a left proper vector, we also have L A e = - L E A . When written in index form with a comma denoting differentiation with respect to components of U, we find that, ~ LiAij, kU~Rj = - ~ Li, kAijUIRj. Since A R = 0 by definition of a i,j,k
i,j,k
right proper vector, the sum is zero, and the only surviving term in the original equation gives a compatibility condition, L(U°)bl (U °) = 0. The same condition occurs at all higher orders, recursively, so Lb = 0 in general. When applied to eqn (35), the three conditions sketched out in the last paragraph, detA = 0, U~ = R, and Lb = 0, lead to an expression for the nonlinear bar speed, to simple ordinary differential equations for the components of the solution vector, and to some extra compatibility conditions. From detA = 0, we find only one algebraic condition, namely pc2 = M ~
M~u
(41)
After a certain amount of algebra, it can be shown that eqn (41) reduces to
pc 2
H + 3#fl 2
E
E + H + (1 - 2v)/zfl2"
Since 0 < 132 < 1 during plastic flow, the bar speed lies in the range
(42)
Axial plastic waves in a rod
H pc 2 < -- < 1 E+HE-
33 (43)
where the left hand side corresponds to the condition of no radial shear, Q = 0 or/32 = 0, and the right hand side corresponds to the condition S = ½P or/32 = 1. In the elastic limit as H ~ o~ the right hand side of eqn (42) reduces to 1 for every value of/32, so the speed reduces to Cb = ~ , as it should if it is to be identified as the bar speed. To interpret the left hand side of eqn (43), let us write cr = ~ (ee + ep) ="~ (~ + ep) for the uniaxial stressstrain curve;. Then differentiation with respect to ep leads to the relation 1 &r
H
E de
E+H
(44)
after recalling that the hardening modulus is defined by H = ~-~. The relation eqn (43) may now be rewritten as (45)
In other words the bounds of the plastic bar speed are determined below by using the tangent modulus of the uniaxial stress-strain curve and above by using the elastic Young's modulus. In particular, the plastic bar speed is determined from the tangent modulus only when there is no radial shear. In general the presence of radial shear will make waves travel faster. This will be the case whenever the axial strain is nonuniform. The second condition, U~ = R, with R proportional to the cofactors of the bottom row of A, leads to the ratios u0o : _ v p ::v~o.
c = %o : 1 = p ,o7 : O = % :oO = P ~ : O = Q ° 7 : M P q s ( l + v p ) = ~ : p c 2 .
(46)
In eqn (461, the plastic Poisson's ratio, which is the negative ratio of increments of radial strain to axial strain, - u J0% ,0 comes out to be
Vp = MP_/MP_
v + (/z/H)[(1 + v) - (1 - 2v)/32]
u,..u=/~1q--~¥(-1-2-7~]"
(47)
Again in the elastic limit this expression reduces to the elastic Poisson's ratio, v, as it should, and furthermore, in the incompressible limit where v---~ 1/2, we also have Vp ~ 1/2, for any value of H, as is to be expected. In fact, it is not hard to show that for all H,O < t l < o~, and all/3, 0 < 13 < 1, we have v < vp < 1/2. In the ratio involving Q0 in eqn (46), the relationship M~qe= - M ~ has also been used. Since eqn (46) is an autonomous system of ordinary differential equations, it may be rewritten with any one of the components o f U chosen as the new independent variable. With the Choice of e ° the equations become
34
T.W. Wright du o dv ° dp ° dq ° dp ° -0 d 7 - - Vp, de o - - c , de o - de o - de o dQ° = M~e(1 + vp), de o
aS o de o
(48)
= pc 2
In fact, since the right hand sides in eqn (48) only depend on the generalized stresses (pO, QO, SO), and p0 is at most a constant, only the last two equations need to be integrated simultaneously, and then u ° and v ° may be found by quadrature. Finally, with L proportional to the cofactors of the left hand column of A,
[L] = [sMPq,(1 + Vp), c, - p c 2, -Vp, -MPq,(1 + Vp), -vp, O, 1]
(49)
the complete compatibility condition at all orders is found to be l
2
+ 8[clS~ - pc21v~ - lp~MPqe(1 + vp)] -
-
(50)
83/2upQ~n.
Although the right hand side is organized according to the powers of 8 that multiply the various terms, the terms themselves may have various orders of 8. Therefore, each case must be examined on its own merits in order to find the appropriate, lowest order compatibility condition. Now let us re-examine the right hand side of eqn (35) in light of the results so far. First note that the compatibility eqn (34) shows that either the slowness, s, is independent of ~/ to lowest order, or that the incremental arrival time,/, has a leading term that is 0 ( 8 - 1 ) . The first condition applies in an elastic wave, and then the wave is essentially steady. The elastic case will be examined later, but to see whether or not the steady case is possible for a fully plastic wave, s~ should be computed from eqn (42) and examined for feasibility. On the other hand, if the second condition applies, then oJ = 1-1 = 0 ( 8 ) , and since s = O(1), we must also have tc = 0 ( 8 ) , as well. Then to comply with the basic assumption that each term on the fight hand side of eqn (35) is at least 0(81/2), the leading terms of both S and v must be independent o f ~. That is, S = 80(/7) + 81/2S1(~, ~) --1-.....
1~ = v°(r]) + 81/2vl(T], ~) -~- .....
(51)
Equation (51) will be accepted for the moment and checked for consistency later. With this assumption, only the second and third rows on the fight hand side of eqn (35) are 0(81/2), all other rows are at least 0(83/2), and the inertia term in the fourth row is 0(83). The compatibility condition comes out to be S~-pev~
=0.
(52)
Equation (52) adds no new conditions to eqn (48) and so can be ignored in constructing the solution to lowest order.
Axial plastic waves in a rod
35
It should be noted that although ~ - # 0 so that radial shear contributes to the solution at lowest order, radial inertia does not contribute to the plastic wave in a simple wave. This last fact can be seen by observing that since o9 = 0(8) by assumption, and pn = 0(83/2) at the very least, the last term on the first line of eqn (50), which is the only contribution of radial inertia, is at least 0(83). This is in distinct contrast to the case of an elastic wave, as will be seen in the section after next. V. INITIAL-BOUNDARY VALUE P R O B L E M
It is now possible to consider the following initial-boundary value problem for a semiinfinite rod. Find the wave structure that exists in the rod at long times and large distances if the rod i,~ initially at rest with free sides, and if a known strain is impulsively applied to the end at Z = 0. To lowest order the solution may be constructed as follows: (a) At the origin Z = 0 the time of initiation of each level of strain is given simultaneously at t = 0. Thus if the speed of propagation (or the slowness) is known as a function of E, lines of constant strain may be drawn from the origin with slope equal to the speed. In coordinates ((, r) the equation for the family of straight lines may be written -
r
=
c(e)
=
1 s(e)
.
(53)
It will be assumed that the magnitude of c decreases as the magnitude of e increases so that the lines of constant strain form an opening fan at the origin. (b) Integrate the last two equations of eqn (48). These equations in full are
dS H + 3/zfl2 ~-e = E- E + H + (1 - 2v)/zfl2
(54)
dQ -6#2(1 + Vp). ( S - ½?)Q -~e = (H + 3tt)k 2 where /3 = x/-6Q/ak, k = _[(S _ ~p)l
2
+(v~a/ak)
2
(55)
1/2
]
, H(k) is a known function,
and Vp is given by eqn (47). With P = O the right hand sides of eqns (54) and (55) s
/
\
+ u_~.~}dS and Q ___0, are functions only of S and Q. One solution is e -- ey + ~ (1 \Sr
Sy and ey are the stress and strain at first yield. This is simply the KfirmfinTayl.~r solution (e.g. Kolsky, 1963). With e(S) known and the fan defined by where
= c= " .V~E+I~CS), the asymptotic solution is completely determined to lowest order, provided that H (S) decreases with increasing magnitude of S so that the fan opens out. Now look more closely at eqn (55). If S > 0, as in a tension wave, then eqn (55), plotted in the phase plane, gives a curve through the origin that passes from the 2rid to the 4th
36
T.W. Wright
quadrant. For a phase curve of this type every small disturbance in Q at the leading edge of the wave tends to evolve towards the origin as strain becomes increasingly positive in the developing wave, provided that k remains non-negative, of course. On the other hand, if S < 0, the phase curve for Q passes through the origin from the 3rd to the 1st quadrant. Again every disturbance in Q at the leading edge of the wave evolves toward the origin as strain now becomes increasingly negative in the developing wave. Thus, in either case the Kfirmfin-Taylor solution is stable for strains applied at the end of the bar, whether tensile or compressive, and can be expected to dominate eventually. If the laigher order rod theory only produced the K~irmfin-Taylor solution as its lowest order approximation, little would have been gained. However, if the disturbance in Q is large, the wave speeds are modified significantly, as was shown in the discussion preceding and following eqn (45), and this in turn can modify the structure of the wave for some distance behind its leading edge. Of course, the fastest part of the wave is elastic, and in the next section it is shown how substantial radial shear can be present in the elastic precursor if the loading on the end is applied rapidly enough. VI. ELASTIC WAVES
In an elastic wave all wave speeds are constant. The compatibility eqn (50) still applies, but now there is a substantial reordering of terms. To see how this comes about, label the lines of constant speed in an arbitrary way, r / = f ( r - (/Cb), where f is an arbitrary, but smoothly increasing function of its argument, ~0 = r - (/Cb, so that ~ increases at a fixed station in the rod as time increases. We have o~ = r/+ =f~o, and r = 17¢= -c~tf~, so that w/x = --Cb = -- Ex/E~. Equation (34) is satisfied identically at all orders, and s, to, r, and l are all O(1). Furthermore, for an elastic material the moduli M p = M~q, = 0, and M~qq = la2/z. Finally from eqn (35) it is clear that p, q, and Q all start at O(81/2), and P starts at 0(8). That is, p = 81/2p I + . . . ,
q=
81/2q1 + - - . ,
Q = 81/2Q 1 + . . . ,
P = 8P 2 + . . . .
(56)
At 0(81/:') the compatibility eqn (50) is satisfied identically, but at 0(8) the compatibility condition demands that 1 -v-~q0 (rQ',7--~1 pa2wp~) + cblS~-- pc2blv~= 0 .
(57)
F r o m the' basic eqns (35) it can be seen that
P" = O-~-(wu°)o'l
1 2 Q'n = ~a lzq,lv q,7l = _spl
(58)
and from the relationships in eqn (46) it can be seen that
= _pc
O,
uo0
p 0
(59)
Axial plastic waves in a rod
37
With eqns (58) and (59) put into eqn (57) and with the recognition that t°-aon= ~ ' then after some rearrangement of terms the compatibility equation becomes
v~o -- r3c3v~ ----O,
where
r3 =
8(1 + v) v2(1 -k- 2v)a 2 "
(60)
The super:script on the axial velocity, v, has been dropped for clarity in eqn (60). The development leading up to eqn (60) clearly shows that in an elastic wave radial inertia and radial shear both contribute substantially to the lowest order solution. It will be recognized that eqn (60) is a transport equation that describes the elastic dispersion at the leading edge of the wave. A similarity solution may be found by assuming that the axial velocity depends on the independent variables only in the combination X = ¢p~m for some constant, m. Then with v = V(x) we have v ~ = ~3m "7~r d3V and
v~ = m x U l '~av XX" With m = - 1 / 3 , the terms ~3m and ~-1 cancel from eqn (60), which now reduces to an ordinary differential equation d3V 1 33 d V dx 3 F~r cbX.~xx = O.
(61)
Equation (61) is a second order O.D.E. for dV/dx whose solutions may be written in terms of tile Airy functions Ai and Bi (Abramowitz and Stegun, 1965). To ensure that the solution rapidly tends to zero ahead of the wave, only the Ai function can be used. In the original wLriables, the solution may be written
rCb V¢ = C .£ZTV.~/3A t ( - ~ )
(62)
v = C I Ai(-z)dz
(63)
and
--00
where C is an arbitrary constant, and • = ~ (3~).,~ • Note that the scaling parameter 8, which occars implicitly in both numerator and denominator of ~, cancels out so that the solution does not actually depend on the exact choice of 8. This fact motivated the choice of scaling in the first place. The representation in eqn (63) is exactly the same as the one obtained in Wright (1982) for the problem of a step velocity applied to the boundary of a linear elastic rod. In that case a solution was found by LaPlace transform, followed by an asymptotic expansion. Note the familiar dispersion of the rapidly rising front according to the negative one third power of distance for elastic waves in a circular cylindrical rod, as first noted by Skalak (1957) for the full 3D theory of elasticity. VII. PLASTIC WAVES The corLstant C that sets the scale of the particle velocity in the elastic part of the wave cannot be specified precisely in the elastic-plastic problem because it is unknown when
38
T.W. Wright
yield first occurs other than to note that ker = S 2 + (6Q2/a2), where k r is the initial yield stress and P is taken to be zero. Referring back once again to eqns (56), (58), and (59), expressions for S and Q in the elastic wave are S = --pCbV
and
Q = --31/2 v/2a2 2c~ .v,.
(64)
The ratio fl/a = v/-6Q/aS is independent of the constant C and indicates how much contribution the radial shearing makes to initial yielding. In terms of the elastic solution the ratio is given by
fl [3 /2 var~ z/2 . . . . . cx
E (3~) 1/3
Ai(-~)
~ 0.4(Z/a) -x/3
(65)
f Ai(-z)dz
where • is given following eqn (63). The approximate value holds for a wide range of Poisson's ratios because the Airy function and its integral are nearly equal near the wave front (small ~) and because the remainder of the coefficient is only a weak function of v for a wide range of values, at least 1/6 < v < 1/2. Since ot2 +/32 = 1, an estimate for the relative amounts of axial stress and radial shearing at first yield is
~/6(Z/a) 1/3 I°tY 1~ [1 + 6(Z/a)2~3] 1/2'
1 IflY I~ [1 + 6(Z/a)2~3] 1/2"
(66)
The net result is that radial shear accounts for a substantial fraction of the yield stress at first yield until the wave has travelled a substantial distance, Z/a >> 1. Consequently, not only is Q not zero as an initial condition for eqn (55), which describes the evolution of Q in the fully plastic wave, but if eqn (66) were to be taken literally, at the moment of initiation when the front is at Z = 0, all yielding is due only to radial shear, i.e. fl = 1. Furthermore, if, fl = 1, then ot = 0, so the initial rate of decay of Q would also be zero. Of course, eqn (66) should not be taken too literally since the whole theory developed so far is intended to be asymptotic for "long" times and "large" distances, which must be interpreted as Z/a >> 1 and cbt/a >> 1. Nevertheless, some initial condition must be given for eqns (54) and (55), and the choice S = 0, Q = 4- akr/v/-6 at e = 0 is one possible and consistent choice. (The plus or minus sign on the initial value of Q applies to tension and compression respectively at the origin for a wave moving to the right.) As a simple example, consider an incompressible material that has no work hardening. Then v = 1/2, so that E = 3/2. Also, H _-- 0, the yield function is constant, k = kr, and eqns (54) and (55) simplify to
de = 3lzfl2 = E 1 dO_ dE which have the immediate solutions
E. SQ k2
(67) (68)
Axial plastic wavesin a rod S = krtanh(Ee/kr),
v--
Q = ~(akr/v/-6)sech(Ee/kr)
4k, y- ~ t.a n
-1(
sinh kEg~, y] u = - e l 2
39 (69)
(70)
where 0 < e < oo in eqns (69) and (70). Since the wave speed for this case is given by pc 2 = 3/zfl2, according to eqn (42), the curves of constant e make a fan centred on the origin and are distributed according to c = Z/t = x/~
sech(Ee/kr).
(71)
The solution laid down by eqns (69), (70), and (71) is given parametrically in e and has an entirely reasonable interpretation. The fastest wave travels with the elastic bar speed, and ahead of the first arrival all fields are quiescent. At impact and at first arrival (if elastic dispersion is ignored) the material jumps to full yield, but since there can be no jump in the axial stress, which would imply a jump in radial strain as well, and hence a discontinuity in the deformed radius of the bar, there is a jump in the radial shear stress sufficient to cause yield instead. Then as the wave develops, there is a transfer from radial shearing to axial loading, and the wave continues to evolve until the maximum applied strain is reached. It is interesting to note that since 0 < tan -1 x < rr/2, no matter what the applied strain, the magnitude of the maximum axial velocity is limited to [ v I< Jrky/2v/-PE. Behind the fan of the loading wave further deformation ceases. As the total strain builds up to the static yield strain, er = k r / E , and then exceeds it, the wave speed and Q both begin to drop rapidly, and the axial stress tends toward the full yield stress. Note that even though the axial stress approaches the static yield stress, but never reaches it, in fact the material is yielding throughout the whole wave. When elastic dispersion is included, the solution is somewhat modified. The elastic wave is described by eqns (63) and (64). Now suppose the axial stress and strain in the elastic portion of ~thewave at the moment of first yield are So and e0 = SolE. Integration of eqns (67) and (68) now gives the plastic wave as
[
S = k ~ : t a n h t a n h -1
-t
E-So] ~rr
'
Q=
+a~/6 sech[tanh_,S0 E k-S0] k k--Yr+ r J
(72)
where So/kr is estimated from the first of eqn (66) and l e0 [
dy . . de
.
1 . I+R
xy x 2 + y 2"
(74)
40
T . W . Wright
With the change of variables x = rsinqJ,y = rcosqJ these equations become
dr de
R --. I+R dq~ --
de
sin qJ
(75)
cos ko -
(76)
r
It is no longer possible to integrate the equations completely in closed form, but one integration is possible and the other reduces to a quadrature.
/cosq,0"~&
r = ro ~ , ~ )
(77)
qJ
I( ,+2R r0(cos qJ0)i~R COS k o ) - ~ d ~
(78)
: E -- E0.
koo
Since S / E = r sin ~o and v ~ Q / a E = +r cos qJ, the solution is now given parametrically in qJ. If elastic dispersion is ignored, qJ0 --- 0 and the parameter evolves toward Jr/2 in a tensile wave or -zr/2 in a compressive wave. If elastic dispersion is included, it may be accounted for in the same manner as before. Since pc 2 : -l~' - iR+fl2 - ~ in this example and
r = k/E, the solution is distributed parametrically over the wave fan Z --=c=
Vf•[ 1
sin 2 qJ'l
u2
/
(79)
VIII. STABILITY CONSIDERATIONS
Although the solutions in the last section are perfectly well defined as the strain becomes arbitrarily large, they may no longer make physical sense if the strain is substantially more than the yield strain and k//~ • i -R~ IS • less than the slow characteristic speed. In the first of the two examples where there is no work hardening it is clear that the wave speed tends toward zero at large strain, so that all the deformation piles up near the origin. In the second example, clearly if R is small enough, the same thing can happen. This remark brings up a new issue. The wave speed actually can become so slow that it is less than the slower of the two characteristic speeds. Thus, revisiting eqn (29) for the characteristic speeds, we can write down the solution of the quadratic as I~
pV~2' = ~ (M~, + a2/2]
p
2
p 2
- a 2 / 2 ] -- a--~-J
(80)
Axial plastic wavesin a rod
41
where the subscripts 1 and 2, as well as the plus and minus signs, refer to the fast and slow characteristic speeds, respectively. Whereas it is easy to see that 0I:21 is always greater than pc 2 (subtract eqn (80) with the plus sign from eqn (41) and note that the result is always negative), it can happen that pc2 is less than p V 2 for some values of I z / H and u 2. For example, in the simpler problem above, the fast speed works out to be infinite, but the slow speed is given by p V 2 =/xot 2. Since the bar speed was found to be given by pc 2 = 3/zf12., these two speeds are equal when f12 = 1/4, ot2 = 3/4, or when akr Q - 2~#6'
S = ~-kr.
(81)
From the second of eqn (81) and the first of eqn (69), it can be seen that this condition is reached when the strain is given by e ~ 1.32er or Iv I~ 1.05kr/v/p-E where er is the strain at static yield. It has been shown by Wu (1961) that there is a stable hierarchy of waves (in Whitham's (1974) terminology) in the linear problem only if 1:2 < c < Vl, but it is unknown to what extent the same condition applies in the nonlinear case. If a similar condition applies, then the straight forward asymptotic solution is unlikely to hold for the complete problem. It may be speculated that in a compressive wave mush- rooming would be predicted to occur at the loaded end of the bar when c becomes too small for the bar wave to keep ahead of the slow characteristic speed, and that similarly necking may be predicted in a tensile wave. Further exploration of these aspects of the problem will be deferred to another day. IX. DISCUSSION AND CONCLUSIONS A theory that includes the effects of radial shear and radial inertia has been proposed and partialtly developed for propagation of axial waves in a circular cylindrical bar. Although based on the limited kinematical foundation provided by the Mindlin and Herrimann (1951) theory, there is ample scope for a representation of predominantly axial motions. The theory shows that the Khrm~n-Taylor solution should hold eventually at large distances from the impact end of a bar, where radial shear has decayed essentially to zero, but since elastic dispersion is relatively slow (only a one third power in eqns (62) and (63)) the distance may be several tens of radii from the impact end before the K~trmfinTaylor solution becomes a good approximation. Alternatively, if the axial loading at the impact end is slow enough so that the wave fan is distributed over the time axis, rather than issuing completely from the origin, radial shear is always small, and the K~irm~inTaylor solution will be valid throughout. An impact problem constructed within the context of the complete theory would require additional consideration of the leading and trailing characteristic waves so as to develop the asymptotic transitions into the main bar wave. However, the main structure of the bar wave seems perfectly clear as laid out in the exposition above. The experimental implications of the theory have yet to be explored. However, it would seem perfectly possible to observe a wave fan whose rays are straight and whose slopes (wave speeds) depend only on the level of axial strain, but which represents a solution with substantial levels of radial shear, rather than the K~irm~m-Taylor limit of vanishing radial shear. Then integration of the 7aS/ = Pc2, where the right hand side is determined from the observed slopes, could result in a calculated S - e curve that is higher than the
42
T.W. Wright
quasi-static curve at the same axial strain because of eqn (45). In fact, that appears to be exactly what has been observed by Hauver (1978), who measured strains and arrival times in a series of experiments on instrumented penetrators, which were made from a precipitation hardened, and nearly rate insensitive, tool steel. His stress-strain curve, as calculated from measured wave speeds at constant strains and the relationship dg/dE = pc 2, lies close to the quasi-static curve at first yield, but diverges from it with increasing axial strain. By about 15% strain, where the experiment ended, the calculated curve lies at least 30% above the quasi-static curve, far higher than could be explained by rate dependence alone. The explanation seems to be not that the dynamic constitutive response is higher than the quasi-static response, but that radial shear also does plastic work and contributes substantially to work hardening. The reader should be reminded explicitly that the material has been assumed to be rate independent throughout this work. If there were rate dependence as well as inertial and shear effects, the present analysis would not apply without some modification, and Hauvet's experiment would become even more difficult to analyze and understand. Acknowledgements--This paper is an extension of work that was presented at the Ninth U. S. National Congress of Applied Mechanics (but never published) in a session on Viscoelastic and Plastic Waves, organized by the late Professor James F. Bell. The work was inspired by Professor Bell's superb experimental observations of plastic wave propagation that he conducted over many years.
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