1951J L. E. MALVERN 405
non-steady layer would equal the steady layer in times given approximately by
I ~ for velocity,
Qx
t = jj for temperature.
Thus to a reasonable approximation it can be said that by the time a point on a suddenly
accelerated plate moves 5 times its distance from the leading edge, its boundary layers
will have become steady state ones.
PLASTIC WAVE PROPAGATION IN A BAR OF MATERIAL
EXHIBITING A STRAIN RATE EFFECT1
By L. E. MALVERN (Carnegie Institute of Technology)
1. Introduction. The propagation of a transient wave of plastic deformation due to
longitudinal impact on a bar has been treated by Donnell,2 and White and Griffis,3 by
a non-linear superposition method. The partial differential equations governing the
wave propagation were derived independently by Taylor4 and von Karman5 under the
assumption of a relation between stress and strain independent of strain rate. Constant
velocity tension impact tests at the California Institute of Technology6,7 gave fair
agreement with the theory. Some systematic discrepancies were, however, observed.
In the tension impact tests the maximum residual strain was smaller than predicted
by the theory, and the observed force-time variation at the fixed end during impact
showed that the stress there was greater than the theory predicted. It has been sug-
gested6 that these discrepancies were due to the use in the theory of an invariant relation
between stress and strain independent of strain rate. At the high strain rates involved
in deformation under impact a considerable deviation from the static stress-strain
relation may be expected. The present work extends the theory to apply to materials in
which the stress is a function of the instantaneous plastic strain and strain rate.
deceived June 5, 1950. The results presented here were obtained in the course of research conducted
at Brown University under Contract N7onr-358 sponsored jointly by the Office of Naval Research and the
Bureau of Ships. This paper is part of a thesis submitted in partial fulfillment of the requirements for
the degree of Doctor of Philosophy at Brown University, October, 1949.
2L. H. Donnell, Longitudinal wave transmission and impact, A.S.M.E., Trans. 52 (1), APM 153-167
(1930).
3M. P. White and L. Griffis, The permanent strain in a uniform bar due to longitudinal impact, J. Appl.
Mech., A.S.M.E., Trans. 69, A-337-A-343 (1947).
4G. I. Taylor, Propagation of earth waves from an explosion, British Official Report R.C. 70 (1940).
6Th. v. Karman, On the propagation of plastic deformation in solids, N.D.R.C. Report No. A-29
(O.S.R.D. No. 365) (1942).
6P. E. Duwez, D. S. Wood, D. S. Clark, and J. V. Charyk, The effect of stopped impact and reflection
on the propagation of plastic strain in tension, N.D.R.C. Report No. A-108, (O.S.R.D. No. 98S) (1942).
'P. E. Duwez and D. S. Clark, An experimental study of the propagation of plastic deformation under
conditions of longitudinal impact, A.S.T.M., Proc. 47, 502-532 (1947).
406 NOTES [Vol. VIII, No. 4
2. Proposed flow law for tension and compression. It is assumed that in longi-
tudinal impact on a cylindrical, or prismatic bar a relation of the form
<r = *(«", e"') (1)
exists between the values of the nominal tensile stress a (longitudinal force per unit of
initial cross-sectional area), plastic strain e" (permanent change in length per unit
initial length), and the plastic strain rate t". Since 4> is in general an increasing function
of t" this determines t" as a function of a and This relation may be expressed as
Eoe" = g(<r, «),
where the factor E0 is Youngs modulus and e is the total strain. Elastic deformation is
assumed to be independent of strain rate. Thus, if «' denotes the elastic strain,
E0e' = cr". (2)
The relation between total strain, strain rate, and stress is then
E0( = <7* + g{a, «). (3)
The static stress-strain relation a — f(e) is interpreted as a succession of equilibrium
states so that plastic flow occurs only when the plasticity condition
* > /(«) (4)
is satisfied. Otherwise the elastic law (2) applies instead of the plastic flow law (3).
The elastic law also applies until the initial yield strain e„ is reached on the first loading.
The plasticity condition (4) as stated applies to tensile impact (<r and e positive) but
the same form of law may be used in compressive impact if compressive stress and strain
are reckoned positive.
There is some evidence that the right-hand member of (1) should have the form of
J(e) plus a term depending logarithmically on the plastic strain rate.8 If this is the case
g(a, e) will depend exponentially on a — /(«), the excess of the instantaneous stress
over the static stress at the same strain. Sokolovsky9 has treated wave propagation
in a material without work-hardening using a law of the form (3) in which g(a — <r„)
was a function only of the excess of the stress over the initial yield stress .
3. Equations governing the wave propagation. The propagation of the wave of
plastic deformation is governed by the following system of three partial differential
equations, in which x denotes the initial distance of a cross section from the impact
end, v the particle velocity (assumed constant over each cross section, and p the initial
density.
da dv n
dx ~ P dt - °-
de dv_
dt dx
= o, (5)
_ de d<x , .
EaJt~
8See, for example, H. Deutler, Experimentelle Untersuchung ueber die Abhaengigkeit der Zugspannun-
gen von der Verformungsgeschwindigkeit, Phys. Z. 33, 247-259 (1932).
9V. V. Sokolovsky, The propagation of elastic-viscous-plastic waves in bars, Prikl. Mat. i Mek. 12,
261-280 (1948).
1951J L. E. MALVERN 407
The first equation is the longitudinal equation of motion. The second equation is a
consequence of the fact that v = du/dt and e = du/dx, where u(x, t) is the displacement
at time t of the cross section which was initially at distance x from the impact end of
the bar, and the third equation is the law (3).
The system (5) is a hyperbolic system of quasi-linear partial differential equations
which may be integrated numerically by the method of characteristics under appropriate
boundary conditions. The characteristics in the x,£-plane are the three families of straight
lines defined by the characteristic differential equations
dx = 0, dx — c0 dt = 0, dx + c0 dt - 0, (6)
where
Co = (£?o/p)1/2
is the constant speed of propagation of longitudinal elastic waves in the bar.
The following three equations hold respectively along the three characteristics
defined by (6).
E0 dt — da = g{a, e) dt
da — pc0 dv = — g(a, e) dt (7)
da + pc0 dv = — g(a, e) dt
The equations (6) may be integrated immediately to give the fixed straight character-
istics of the plastic region of the £,<-plane, but the equations (7) will in general require
0
Fig. 1 Characteristics in the x,J-plane.
step-by-step numerical integration. For this purpose the differentials of (7) are replaced
by finite differences, and the value of g{a, e) in each equation is taken as an average of
the values along the appropriate segment of the characteristic.
4. Boundary conditions for continued impact on a semi-infinite bar. Consider a
continued tensile impact on a semi-infinite bar which is initially at rest. The extension
of the theory to finite bars and finite durations of impact is not difficult, although the
408 NOTES [Vol. VIII, No. 4
numerical integration becomes more involved. The z-axis is chosen so that at the instant
of impact, the impact end is at the origin and the bar lies along the positive x ax^. Suppose
that at time t = 0, the end of the rod is instantaneously set in motion with velocity
v0 such that [ v01 > c0e„ after which the end velocity is maintained constant at the value
v0 . For a tensile impact the velocity is negative.
Immediately upon impact a shock wave of elastic deformation begins to travel
along the bar at the speed c0 . This leading wave front is represented in Fig. 1 by the
line x = c0t. The shock wave conditions which hold across an elastic shock wave traveling
in the positive direction are
Act = — pc0Av,
Av = —C0 Ae, (8)
A <7 = pc20Ae = E0Ae,
where A a, Ae, and Av are the jumps in stress, strain, and velocity, respectively, as the
shock wave passes. The first condition results from equating impulse to change of
momentum for the traversing of an element of the bar by the shock wave. The second
condition is a consequence of the continuity of displacement across the shock and the
third condition follows from the first two.
Since a = e = v = 0 in the undisturbed region ahead of the shock wave, the jump
conditions (8) yield
<r = pelt = — pc0v on x = c0t (9)
just after the shock wave passes. The line x — c0t is also a characteristic of (5) along
which the second equation of (7) holds. This equation may be integrated after elimi-
nating e and v by use of (9) to yield
/'
V a n
dT 1 t, (10)
q{t, t/pc0) 2
where <r0 = — pc0v0 is the stress at x = 0, t = 0. Equations (8) and (10) thus determine
a, e, and v along x = c0t . With this information and the boundary condition v = v0
on x = 0, the numerical integration of the equations (7) may be performed to determine
<r, e and v throughout the plastic region. The values so obtained should be checked
with the plasticity condition (4) at each point to make sure that the point is in the
plastic region.
5. Impact of finite duration: unloading. If the impact is of duration t0 , after which
the boundary condition is a — 0 on x = 0, the solution may be constructed as in the
preceding section up to the characteristic of the family dx — c0 dt = 0 passing through
the point (0, t0), i.e. up to the line MN of Fig. 2. In the unloading region Eq. (2) replaces
Eq. (3) so that g(a, e) = 0 for unloading. Equations (6, 7) then yield
<r — E0e = const. on x = const.,
ct — pc0v — const. on x — c0t = const., (11)
u + pc0v = const. on x + c0t = const.
1951] L. E. MALVERN 409
The various constants are determined by matching solutions at the elastic-plastic un-
loading boundary in the x,i-plane.
N
0
Fig. 2 Unloading shock wave MN after impact of duration to on a semi-infinite bar.
A sudden reduction of the impact end stress to zero causes the initial unloading wave
to be a shock wave. Since the shock wave is elastic the jump conditions (8) apply and
the wave travels at the speed c0 , i.e. along MN, Fig. 2. The shock intensity decreases
as the wave progresses and may decrease to zero. If the shock wave travels as far as
Q, Fig. 2, the solution is determined in the triangle MQR by Eqs. (11), the boundary
condition a = 0 on x = 0, and the jump conditions (8) together with the known values
of tr, e, and v along MN before the shock wave passes.
If the shock wave is absorbed, say at Q, the elastic-plastic boundary becomes a
boundary of continuous transition from the plastic to the elastic state, and does not in
general continue along the line MN. Along a continuous unloading boundary the static
relation a = /(e) holds. This condition and the characteristic conditions in the elastic
and plastic regions suffice to determine the position of the unloading boundary, although
a trial and error procedure is usually needed to find the boundary points. A continuous
transition from plastic to elastic behavior may begin even before the impact ends. The
unloading shock wave then travels through an elastic region until it overtakes the
continuous unloading boundary.
6. Solutions of the equations. Numerical integration of the system (5) has been
carried out for an idealized form of the law (3) in which g(a, e) = k[<r — /(e)] where fc
is a multiplicative constant, with the function /(e) chosen in a simple form which ap-
proximates the static stress-strain curve of a hardened aluminum. Even with the idealized
law used the solutions indicate that a law of the type (3) can account for the discrepancies
observed in the stress-time variation at the fixed end of impact specimens. This type
of law does not, however, account for the discrepancy observed in the maximum residual
strain.