CP620, Shock Compression of Condensed Matter - 2001
edited by M. D. Furnish, N. N. Thadhani, and Y. Hone
© 2002 American Institute of Physics 0-7354-0068-7/02/$ 19.00
TRADITIONAL ANALYSIS OF NONLINEAR
WAVE PROPAGATION IN SOLIDS
Lee Davison
39 Canoncito Vista Road, Tijeras NM 87059 USA
Abstract Propagation of nonlinear plane waves of uniaxial strain is reviewed, emphasizing calculation of thermodynamic states produced by these waves. Both smooth waves and shocks are considered,
with emphasis on discussion of shocks as they occur in several settings.
CLASSICAL CONTINUUM MECHANICS
dx
dv
—— "PR —— = 0
3X
The analysis of this article is restricted to states
of uniaxial strain. Two coordinates are required to
describe these deformations. The first, X, called the
Lagrangean coordinate, gives the position of the
material particle when the body is in a reference
configuration. The second coordinate, x, called the
spatial or Eulerian coordinate, gives its current
position. Uniaxial strains can be described in terms
of a single independent coordinate and time in
either the Lagrangean or Eulerian representation,
= x(X,t)
or X =
dt
dtu
dx
dx
dt
d&
dx
—— —pPR — = 0
(2)
or the Eulerian form
dt a*
(1)
8/11
dx
. dx _
—— -P— -P^— -
respectively.
The particle velocity, x, is the time derivative
of the current particle position. The specific volume
ratio, V/VR (where the subscript R is used to
designate a value in the reference configuration) is
used to measure deformation.
Principles of conservation of mass, momentum,
and energy establish the governing equations of
classical continuum mechanics. When the uniaxial
fields are smooth, the conservation principles are
represented by partial differential equations, which
can be written in either the Lagrangean form,
dt
ox
(3)
dx
—
ox
In these equations p (= 1/v) is the density, t\\ is
the longitudinal stress component, and & is the
internal energy density.
A shock is a propagating surface at which the
fields are discontinuous. The Lagrangean and
Eulerian forms of the conditions the conservation
principles impose at a shock are
20
can also be derived from the jump conditions. This
equation represents a straight line, called the
Rayleigh line, connecting the initial and final states
of a shock transition in the t\\-v plane. The
Rankine-Hugoniot equation shows that the increase in internal energy density that results from a
shock transition is represented by the area under
this Rayleigh line.
When this equation is examined in the context
of a smooth steady wave, one sees that the Rayleigh
line is the t\\-v trajectory followed by a particle
as the wave passes.
The three shock jump equations and a boundary
condition are not sufficient to determine the five
variables defining a shock transition. Some information characterizing the differing responses of
specific materials is also needed. In the plane of any
two of the variables, the locus of states achievable
by shock transitions from the given initial state is
called an Hugoniot curve. These curves are usually
measured but they can also be calculated from theories of material response. The curve conveys the
minimum amount of information about the material
that is needed to solve the jump equations. Because
the five variables form ten distinct pairs, there are
ten equivalent representations of the Hugoniot
curve. Three of these curves are particularly important. The t\\ -v Hugoniot is a thermodynamic
relation, the t\\ -x Hugoniot is used for analysis of
shock interactions, and the Us-x Hugoniot frequently arises as the most direct representation of
shock measurements.
(4)
and
(5)
respectively. In these expressions [?] = ^+ -£~» *s
the jump experienced by a field as the shock passes.
The quantities f/s and MS are the Lagrangean and
Eulerian shock velocities, respectively. They stand
in the relation
PR
to one another and are the same only when the
shock is propagating into material at rest at its reference density.
The jump conditions presented for shocks also
apply to differences in the fields between any two
points in a steadily propagating smooth waveform.
In a steady wave, all of the fields are functions of a
single variable, either Z-X-Ct or z = x-ct.
When these fields are substituted into the PDEs, the
resulting equations are immediately integrable to
yield algebraic equations identical in form to the
jump conditions.
Elimination of the shock and particle velocities
from the jump conditions yields the RankineHugoniot equation,
COMPRESSIBLE FLUIDS
An Hugoniot curve falls far short of a complete
description of material response, so it is necessary
to consider a more comprehensive characterization.
The theory of inviscid compressible fluids is widely
used for modeling the response of solids to strong
shocks. This is justified because the ability of materials to withstand shear is limited by the onset of
flow or fracture although they can withstand any
applied pressure. When the limiting shear stress is
considered negligible compared to the longitudinal
stress, the latter is effectively a pressure and the
material behaves as a fluid.
(7)
relating the thermodynamic variables. This equation
is the vehicle for introducing the effect of a shock
transition into a thermodynamic analysis.
The Lagrangean shock speed relation,
vR w
(8)
21
Thermodynamics can be combined to yield the differential equation
The thermodynamic response of a fluid is captured by an equation of state such as the internal
energy density function
(9)
(15)
[A- A- |
H
0( )(A) 2pR
In this expression A = I-(V/VR) is the Lagrangean
compression and T| is the entropy density.
Thermodynamic analysis leads to the usual
pressure and temperature equations
for the entropy density, r|(H)(A), at points on the
Hugoniot curve. In this equation, 9(H)(A) is the
temperature on the Hugoniot.
Before this equation can be integrated, it is necessary to determine 0( H )(A). Evaluation of the
temperature equation of state, Eq. 11, on the
Hugoniot, differentiation, and substitution from
Eq. 15 gives
(10)
and
(ii)
9(A,ri) =
p(H)(A)-/r
A-A-
We also have the First Law of Thermodynamics,
1
1-A
(12)
: ———
PR
(16)
The derivative
2pR
^1-A d/?(A,s)
PR
d&
where Griineisen's parameter and the specific heat
appear through identification of their defining derivatives. This equation is a first order ordinary
differential equation that can be integrated numerically once expressions for Griineisen's parameter
and the specific heat are adopted. In the case that
CO) is constant (e.g., 3M), the equation is linear
and can be reduced to quadrature. After the temperature has been determined, Eq. 15 can be integrated to give the entropy density on the Hugoniot.
Equation 15 can be applied to consideration of
the important issue of existence of shocks. The
jump equations, Hugoniot curve, and boundary
condition yield a solution for a shock that may exist, but they imply nothing about the question of
existence itself. The entropy production principle
holds that a solution that results in a decrease in
entropy is inadmissible and that the predicted shock
cannot actually exist. Equation 15 shows that entropy increases upon passage of a shock when the
Hugoniot is concave upward but decreases when it
is concave downward. This means that decompression shocks cannot exist in the former case and
(13)
defines an important thermodynamic property
called Griineisen's parameter. Lattice dynamical
models indicate that y is a function of compression
alone, and this fact accounts for much of its usefulness.
Since y depends on A alone, integration of the
defining equation, with A regarded as a parameter,
leads to the result
p=
—A
[8 - S(H)(A)] .
A-A-
(14)
The Hugoniot pressure and internal energy appear
in this equation because the Hugoniot curve was
used to evaluate the undetermined function of A
that results from the integration. This equation is
called a Mie-Griineisen equation of state. It plays a
central role in analysis of the thermodynamic aspects of shock compression.
Now we consider temperature and entropy. The
Rankine-Hugoniot equation and the First Law of
22
compression shocks cannot exist in the latter.
Smooth solutions exist in exactly those cases where
shocks are inadmissible.
It is interesting that there is an increase in entropy when a shock passes through material. The
shock transition is irreversible, and the entropy increase is a consequence of the discontinuous motion
and occurs despite the fact that the material is perfectly elastic.
Analyses similar to those just discussed (often
invoking the Mie-Griineisen equation) can be used
to establish the relationship between other thermodynamic curves. Isentropes and isotherms can be
calculated from Hugoniots and vice versa. We
know that an Hugoniot curve depends on the given
initial state, but a given Hugoniot can be transformed to one for a different initial thermodynamic
state. Sometimes this is done to account for
changed initial temperature. In many cases, interest
in a shock propagating into material that has already been compressed by a shock necessitates determining a Hugoniot that is centered on the state
prevailing behind the leading shock. An additional
possibility is re-centering an Hugoniot to a lower
initial density. This produces an Hugoniot for a
strongly shocked porous material.
Thermodynamic equations for elastic solids are
developed exactly as for fluids. A simplifying factor is that the limited shear strength restricts the
theory to rather modest strains. Expansion of the
internal energy density function to include cubic
terms in the strain usually provides an adequate
model.
dX
dX
dt
-+- = 0
dt
(17)
when written in Lagrangean form. The function
C 2 (A) = VRd/?( H )(A)/dA is the Lagrangean isentropic soundspeed.
Consider the situation depicted in the X-t plot of
Fig. 1. Because of the instantaneous encounter of
the shock with the boundary, the reflected wave
emerges from a single point on the X-t diagram. In
this case, the fields are functions of the single variable Z = -Xlt. This is called a centered simple
wave. When fields dependent on this variable are
substituted into the PDEs, we find that we must
have
C(A) = +Z = -X/t.
(18)
Substitution of this result into either of the field
equations produces an equation that can be integrated immediately to give the particle velocity
r
(19)
C(A>/A'.
From these two equations, waveforms and other
features of the solution are easily calculated.
The analysis shows that the particle velocity behind the decompression fan is approximately twice
the particle velocity behind the shock. Since the
decompression isentrope lies above the Hugoniot,
this factor will actually be greater than two. In the
particular example in the figure, it is 2.03. When a
Hugoniot curve is measured, it is often assumed
that the measured velocity of the stress-free surface
ANALYSIS OF SMOOTH WAVES
Shock Reflection at a Stress-Free Boundary
When a compression shock encounters a stressfree boundary, it is reflected back into the material
as a decompression wave taking the material to zero
pressure. This decompression wave must be
smooth, so it is a solution of the partial differential
equations. The decompression process is isentropic
and the energy conservation equation is automatically satisfied. The two remaining PDEs are
- 2 0 2 4 6 8 10 12 14 16 18 20
Compression, %
-1.0
FIGURE 1. Hugoniot curve, decompression isentrope, and X-t
plot for a decompression wave produced by reflection of a 50
GPa shock at the stress-free surface of a copper slab.
23
is twice that of the particles behind the shock (the
free-surface velocity approximation). This overestimates the true particle velocity slightly; in this
case by 1.5%. If this small error in particle velocity
and the wave spreading can be neglected, the problem can be solved as a shock interaction although
the decompression shock cannot really exist.
FIGURE 3. The x-t diagram and waveforms of Fig. 2 extended
in time. Dotted lines show the triple valued part of the solution.
Smooth Compression Wave Propagation and
Shock Formation
able from a physical standpoint because one cannot
have three different densities at the same time and
place.
The solution to this dilemma lies in insertion of
a shock into the flow so that the new solution will
be single valued. One inserts the discontinuity into
the multivalued solution so that mass is conserved.
The product of reference density times the distance
from the initial position of the boundary to the
shock must equal the integral of the density from
the current position of the boundary to the shock.
Notice that, in this example, the shock forms at the
low-pressure edge of the waveform and increases in
strength until it reaches the full wave amplitude.
This example shows how a shock can arise naturally, from what began as a smooth wave.
Let us consider a smooth compression wave that
is introduced into the material by motion of the
boundary, as shown in the x-t diagram of Fig. 2.
As with the decompression wave just discussed,
pressure wavelets propagate at constant speed and
the solution in the wave region is a function of only
a single variable. This solution is a simple wave,
but not a centered simple wave because it is introduced gradually as the boundary moves rather than
emerging at an instant. The solution process is more
difficult than before, but still tractable. We know
that when a normal material is compressed, the
soundspeed increases. This means that the highpressure part of the waveform propagates faster
than the lower-pressure part and the wave becomes
increasingly steep, as shown in the right panel of
Fig. 2.
Notice that the wavelet trajectories shown in the
x-t diagram must eventually intersect. Figure 3
illustrates the result of extending the solution to
later times.
The intersecting characteristic curves lead to a
triple-valued solution. This result clearly unaccept-
Steady Waveforms (Structured Shocks)
As an example of steady wave analysis, we consider a material described by a third-order elastic
relation to which a linear viscous stress is added.
For compression waves, this stress relation is
-A). (20)
In a steady wave, the Lagrangean form of the
solution is a function of the single variable
Z = X - Us t . When the field variables vanish far
ahead of the wave and approach their equilibrium
values far behind it, the hi - A path, which is the
Rayleigh line, is given by
T
(21)
6 *, mm
FIGURE 2. Eulerian x—t diagram and density waveforms for a
smooth compression wave introduced into copper by boundary
motion
In a steady wave, A = - Us dk/dZ . When this
expression is substituted into the stress relation, and
that result substituted into the Rayleigh line equa24
taken together with the jump conditions, determines
the state behind the shock to within one variable.
This is the same as for a nonreactive shock. The
difference is that a detonation wave is sustained by
chemical energy rather than by forces imposed on
the boundary, so the issue we face is how to use our
knowledge of the chemical energy release to replace the missing boundary condition.
Since the detonation products form a gas, the
polytropic gas theory provides a convenient equation of state that takes the form
FIGURE 4. Steady waveforms and a schematic illustration of the
compression process occurring in a smooth steady wave.
tion, we obtain an ordinary differential equation
that can be integrated immediately to give
=f —
J A + / 2 (A+- A)(1-A)A
S(/7,V) =
(22)
_pv_
r-i -q- r-i
(23)
The positive parameter q is the internal energy that
is liberated by the chemical reaction.
When this equation of state is substituted into
the Rankine-Hugoniot equation one obtains the
Hugoniot curve for the detonation products. This
Hugoniot curve is centered on the reference state of
the unreacted explosive, but it does not pass
through this state. At the reference volume, the
pressure on the Hugoniot exceeds the center point
value by the amount PR (r -1) q .
where the characteristic time T is given by
T = 2v/(3Cii+Cm).
A graph of this result is shown in Fig. 4. Also
shown is a schematic stress-compression plot illustrating the elastic part of the stress and the
Rayleigh line, which is the path followed during
passage of the wave. The viscous part of the stress
is the difference between the two. The steady waveform arises as a standoff in the competition between
the tendency of the nonlinear elastic behavior to
produce wave steepening and that of the viscous
behavior to produce dispersion. The compression
rate in the wave adjusts itself so the viscous stress is
just the required amount. When the wave amplitude
is large, a large viscous stress, and correspondingly
high deformation rate, is required. As the viscosity
coefficient decreases, the waveform becomes increasingly steep, tending to a shock in the limit.
To obtain a solution to the detonation problem,
we consider the Hugoniot and Rayleigh line shown
in the/7-v plot of Fig. 5. The state behind the detonation shock must lie on both the Hugoniot and the
Rayleigh line corresponding to the detonation velocity. The Rayleigh line that intersects the
Hugoniot at one point, called the Chapman-Jouguet
point, provides a unique solution to the problem.
The point of intersection that determines the CJ
state can be calculated and, once this is done, the
remaining detonation parameters are obtainable
from the jump conditions. In essence, this analysis
produces a value of the shock velocity (detonation
velocity) that replaces the boundary condition that
would be imposed in the nonreactive case.
THE CHAPMAN- JOUGUET DETONATION
A detonation wave is a shock followed by a
chemical reaction zone and an unsteady decompression wave. In the most idealized view of a detonation, each particle of the explosive undergoes an
instantaneous transition from its initial form to reaction products as the shock passes. The shock at
the front of a detonation wave obeys the same jump
conditions that describe nonreactive shocks. The
material is described by an Hugoniot curve that,
FIGURE 5. Reaction product Hugoniot and Rayleigh line for a
Chapman-Jouguet detonation.
25