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
MECHANICAL STATES OF SOLIDS
John J. Gilman
Materials Science and Engineering, University of California at Los Angeles
Los Angeles, California 90095
Abstract. In the field of shock compression of solids, discussions are sometimes inconsistent because
descriptions of mechanical states are misused. For example, "elastic strains" and "plastic deformations" are
physically distinct entities, yet they are both called "strains" and are sometimes added together
inappropriately. This has led to the concept of a "plastic modulus"; to the assumption that plastic
deformation is a continuous process; and other errors. These are discussed. It is pointed out that the
intimate connections between electronic (chemical) and mechanical states must be taken into account at
shock fronts.
equation of state, and each specific state has a
corresponding set of state variables.
In passing from one state to another an elastic
material may lose, or gain, small amounts of energy
due to an elastic effects, which convert small amounts
of elastic energy into entropy and thus are not
recoverable. These losses may be associated with
specific mechanisms, and their amounts can be
calculated quantitatively (at least in principle).
In contrast to the well-behaved case of elastic
deformation, there is little about plastic deformation
that is not chaotic: starting with the distribution of
displacements that is associated with it.
For centuries it was thought that during plastic
deformation the distribution of plastic displacements
(the plastic displacement field) was microscopically
uniform. Until the acceptance of the atomic theory of
matter at the end of the 19th century, there was no
reason to think otherwise. Localization of the
deformation into shear bands was well-known, of
course, but it was not realized that this fragmentation
continued beyond microscopic dimensions for 3-4
orders of magnitude further; down to atomic
dimensions. Furthermore, the instrumentation that
recorded "stress-strain curves" drew pre-dominantly
smooth lines with little change in passing from the
elastic to the plastic regime (sometimes serrated
curves were observed, but they were not the norm).
INTRODUCTION
Three mechanical states of solids will be discussed
in this paper. They are: the elastic state; the plastic
state; and the electro-mechanical (or chemical) state.
Both elastic and plastic deformations produce
displacements within solid matter; i.e., changes of
shape. End of similarity.
Elastic deformation is usually temporary, well
ordered, and mostly affme. Plastic deformation is
usually permanent, chaotic, and anything but affme.
Since the work needed to cause elastic deformation is
stored in the deformed material in the form of
changes of electronic structure (strain energy), it can
be quickly recovered almost completely. Also, elastic
deformation can propagate rapidly from one position
to another. On the other hand, the work that causes
plastic deformation is dissipated as heat, as well as
configurationally entropy, so almost none of it can be
recovered, and the de-formation can propagate only
slowly.
An immediate conclusion is that elastic strains are
state variables that depend only on the values of other
state variables (+/- pressure, shear stresses,
temperature, electromagnetic fields, gravitational
fields, etc.). Whenever these are given a particular set
of values, the elastic strains (relative to a standard
state) acquire corresponding values. Thus the state
variables are connected in a definite way by an
36
It is now known that plastic deformation is
anything but uniform. It is heterogeneous all the way
down to the atomic level and some-what beyond. The
somewhat beyond refers to the fact that the quantum
mechanical amplitude functions are disturbed in a
non-uniform way at the kinks on dislocation lines.
The contrast with the homogeneous nature of elastic
deformation is striking. The latter is affine right
down to atomic dimensions.
To illustrate the discontinuity that occurs at the
"yield point" of an elastic plastic material consider
cyclic loading, and the corresponding "Q", or
vibrational quality of the deformation. This is
defined as the ratio of the total elastic work done on
the material per cycle divided by the amount of
energy lost per cycle. Qs up to about 107 are
observed for elastic deformations up to the yield
point. When yielding occurs, this drops immediately
to as little as 10. Thus a quite discontinuous change
occurs. Ideally the discontinuity can be even larger.
If a dislocation line is present, a plastic deformation
of the order of unity can occur in a volume of order,
b3 which is to be averaged over the specimen (say
one cm3). Here b is an atomic dimension, or about 2
x 10~8 cm., and b3 is about 10 ~ 23 cm3, so the change in
Q could be as much as 1023. In practice, this is much
too large to be observed, but it makes the point that in
principle plastic yielding is an exceedingly
discontinuous process.
differentiated by subscripts. But their physical bases
are fundamentally different. Therefore, their names
should be distinct.. Also, it is improper to add,
subtract, multiply, or divide them. In this paper, the
word "strain", and its corresponding symbol, 8 will be
restricted to the elastic case. For the plastic case, the
word "deformation", and its corresponding symbol, 5
will be used.
It is especially important to recognize, that
quantities derived from £ and 6 are also physically
distinct. In particular, (da/de), an elastic modulus, is
physically very different from (da/dS) which is not
(repeat not) a modulus. The latter relates to the
dissipation of stress, but not to its propagation. The
slope, (da/d6) may be legitimately taken to be a
strain-hardening coefficient, however. A formal
distinction is that (da/d8) may be either positive or
negative, whereas (da/de) must always be positive.
Negative values of (da/d8) are important be-cause
they produce a mode of plastic instability that results
in localized plastic shear bands. The negative values
are created whenever the rate of multiplication of
dislocation lines in-creases the deformation rate more
rapidly than is necessary to keep up with the
deformation rate that the external system is trying to
impose. The external system may be the loading
machine, the strained material around the tip of a
crack, etc.
ELASTIC AND PLASTIC DISPLACEMENTS
ELASTIC STATES AND PLASTIC
DEFORMATION
Physically, a displacement is the same geo-metric
thing whether it results from an elastic strain, or a
plastic deformation. Therefore, it should be the
variable of choice in dealing with elasto-plastic
problems. Since there are two kinds of displacement,
although they are physically identical entities, they
can be distinguished by means of subscripts and they
can appear together in equations. However, their
space and time derivatives have different physical
meanings, so
the derivatives cannot appear together in the same
equation. Thus most elasto-plastic problems should
be described by a set of at least two equations each of
which contains only one kind of derivative. Then the
solution of the corresponding problem requires the
simultaneous solution of the set (the elastic and the
plastic equations.
Small changes of the shapes of solid bodies can be
conveniently described in terms of displacement
gradients.
That is, in terms of a field of
displacements normalized by dividing them by a
corresponding field of local gauges; yielding a field
of displacement gradients. To provide a useful
description these fields must be continuous, and
compatible. Furthermore, in the elastic case, they
must be differentiable with respect to both time and
space. But, plastic displacement gradients do not
meet this last criterion because they are irreversible,
and they are microscopically discontinuous.
Unfortunately, for the historical reasons out-lined
above, the word "strain" has been used to describe
both elastic and plastic deformations; sometimes
37
An additional reason why the elastic and plastic
derivatives cannot be mixed is that the elastic
displacements are continuous, whereas the
microscopic plastic displacements are quantized in
units of the Burgers displacement, b. To some extent
these quantized displacements can be smoothed out
in macroscopic problems, but this does not eliminate
the other differences between the elastic and plastic
displacements; i.e., the fact that one is conservative,
while the other is non-conservative.
The second derivative of an elastic displacement
with respect to position is an elastic modulus (either
bulk or shear). But this is not the case for the second
derivative of the plastic displacement since the latter
is not conserved. Thus there is no "plastic modulus".
This is important in interpreting: elastic-plastic
impacts; plastic zones at crack tips; local yielding;
upper yield-points; etc.
Accordingly, deformation at a high rate and a low
temperature should be the same as de-formation at a
low rate and a high temperature. To a considerable
extent this is true, but the substitution of time for
temperature is not quantitative. Therefore, this
parameter is useful for making interpolations, but it is
not reliable for extrapolations.
DEBRIS PRODUCTION
The motion of dislocations through a solid (crystal,
glass, or composite) causes plastic deformation. This
was quantitatively verified in the 1950s by Oilman
and Johnston.3 One of the surprises of their direct
observations was that there is a stochastic component
of dislocation motion associated with the screw type.
Screw dislocations are not restricted to motion on
one glide plane. As a result they "double-cross-glide"
in an apparently random fashion; perhaps in response
to thermal fluctuations.4 In addition to multiplication,
this creates "debris" in the wakes of moving screw
dislocations. The debris consists of innumerable
dislocation dipoles having a variety of dipole
moments. It is this debris that causes deformation
hardening, and leads to fatigue cracks, and other
forms of degradation.
Dipole debris production occurs at a higher rate
when the applied stress, and the deformation rate, is
high, or the temperature is low. There is a long
standing myth that most (90-95%) of the work that is
done to produce plastic deformation is converted into
heat during the deformation. This is entirely at odds
with experimental measurements reviewed by Bever,
Holt, and Titchener.5 The partitioning of the work
into heat vs. debris depends on the conditions of
deformation. That is, on such factors as: temperature,
rate, prior deformation, initial perfection of the
specimen, and so on.
PLASTIC EQUATION OF STATE
If a general equation of state existed, it would be
possible to approach any given point in a stress, a,
deformation, 5, temperature, T, space along an
arbitrary path, and the state of the material at the
point would always be the same. In other words the
material would obey an equation analogous to the
state equation for gases (A = constant):
PV = AT
0)
This has been extensively tested by comparing its
partial derivatives at various plastic states (P, V, T
points). It is obeyed as long as a material is elastic,
but as soon as the yield point is reached, it fails to
give consistent values. Equations of this type were
extensively tested in the 1930-1950-time period.
Some of the results are discussed by Tietz and Dorn1.
Another type of state equation is one in which the
rate of a reaction depends only on time and
temperature if the stress, or the deformation, is held
constant. This is true for simple chemical reactions
in dilute systems. If it is also true for mechanical
systems, the plastic behavior should depend only on a
combination of time and temperature known as the
Zener-Hollomon parameter2:
rate ~ t [exp (- Q/kT)]
HEAT PRODUCTION
A dislocation moving through a nearly perfect pure
metal produces very little heat. This is clear from the
small amount of viscous damping that it experiences
as indicated by internal friction measurements, and
by direct velocity measurements. The viscosity
values are of the order of 10"4 Poise. Since the
(2)
38
dislocation motion is of the "stick-slip" type, the
maximum viscous losses occur during the fast slip
events when the dislocation velocities reach about
v
max = vJn where vs is the shear wave velocity » 3.2 x
105 cm/sec. Then the maximum deformation rate at
the core of a dislocation is about (d6/dt)max = vmax /b »
10 5 /2.5x 10'8 = 4x 1012 cm/sec; and the maximum
drag stress is 4 x 1 0 8 d/cm2. From this, U = the strain
energy per atom = (i2/G)b3 » 1.8 x 10 ~18 erg. The
maximum temperature rise, T can be found by
equating this to the thermal energy per atom = kT,
where k = Boltzmann's const. = 1 .38 x 10 ~16 erg/deg.
The result is T » 0.013 °K; a small rise indeed.
Even when the viscosity is much larger, say 10" 3
Poise, the temperature rise is a small fraction of a
degree K.
Since heat production by individual dislocations is
small, and the total heat production is large, it is
obvious that this part of the plastic deformation
process is poorly understood for the case of pure
metals. For alloys containing hard particles, the heat
production may well be located within the hard
particles where there are covalent bonds to be broken
irreversibly. In general, the Fluctuation-Dissipation
Theorem of thermodynamics can be invoked6, but
this says nothing about the source of the fluctuations,
so it is not very satisfying. The apparent drag caused
by fluctuating internal stresses has been analyzed by
Chen, Oilman and Head7, but again this is rather
formal theory.
that that the r.h.s. of Equation (3) is differentiable,
but it is not since the displacement due to dislocations
is quantized. He also assumed that plastic stressdeformation curves are continuous and therefore
differentiable. In reality, however, they consist of a
series of steps because the deformation is dissipative.
That is, when the yield stress is reached a small
quantity of deformation occurs, absorbing some
elastic strain energy, and thereby reducing the local
stress slightly. The deformation then stops, waiting
for the stress to rise back to the yield stress. This
repeats if there is no deformation hardening, and the
material is said to be elastic-perfectly plastic. The
deformation is discontinuous in time. Obviously,
there is no propagation of the deformation, and the
"plastic modulus" is zero.
If there is deformation hardening, the pattern is
similar, except that the stress-deformation curve takes
the shape of a rising staircase. There is still no
propagation of the deformation (L_ders band
propagation is another matter).
FLUCTIATION-DISSIPATION THEOREM
A fundamental theorem of statistical thermodynamics is the Fluctuation-Dissipation Theo-rem.6
An important manifestation of it is the EinsteinStokes Equation relating diffusivity and viscosity.
This theorem is not intuitively obvious because it is a
subtle consequence of the asymmetry of time if it is
assumed that the conservation laws for momentum
and energy hold.12
Consider a large particle, of mass M, moving
through a sea of small particles, of mass m (in one
dimension for simplicity). The large particle will
experience both "front-end" and "back-end"
collisions with the small particles. Front-end
collisions will be more frequent than back-end ones
when the large particle moves at constant velocity, v.
Let r| be the friction, or viscosity coefficient, and (p
be a small random force, so the equation of motion
(with both steady and random forces) is:
NON-PROPAGATION OF PLASTIC
DEFORMATION
The wave equation for elastic shear strain (one
dimension) is:
where p = density, u = displacement, and G = shear
modulus. Partially by analogy, Taylor8, Rakmatulin9,
as well as von Karman and Duwez10 developed a
similar equation for "plastic waves" with G replaced
by a "plastic modulus". However, since plastic
deformation is dissipative, such waves cannot exist as
pointed out by Gilman11. Taylor's analysis came first;
the others perpetuated his mistake. He assumed-ed
F = M (dv/dt) = - nv + (p
(4)
Denoting TC as the very short dwell-time of each
collision, a standard thermodynamic deduction yields
a connection between the average thermal energy
39
compression they may plastically deform, or they
may undergo a phase transformation. According to
conventional wisdom, the behavior is driven by
pressure. In fact, shear strains are the important ones.
For chemical reactions, this was discovered toward
the end of the 19th century by Carey Lea.14 More
recently it was deduced from the nature of phase
changes15 where it is bond bending, rather than bondcompression that leads to the phase transitions in
semiconductors.
Allotropic phase changes are the most simple of all
chemical reactions because only the bond angles need
to change during such reactions. Isomerizations are
the next in complexity; then decompositions, and so
on. The evidence is that all of these are facilitated by
shear strains as might be expected since they all
involve changes of shape; whereas changes of size
are a secondary factor. The conventional idea that
pressure induces chemical reactions, disagrees with
both macro- and micro-observations. The macroobservations of Lea14, Bridgman16, and many others,
indicate that shear deformation is most important.
Micro-observations show that bond angles change
while bond lengths don't15.
The reason for the large effect of shear on reactions
is that shear profoundly affects the electronic
structures of solids and molecules.17 The effects are
much larger than the effects of hydrostatic pressure,
and are often of the opposite sign.
In molecules, shear (bond-bending) may be
considered to be an inverse Jahn-Teller effect
(closing the LUMO-HOMO gap).17
In solids, shear extends one direction while
compressing a perpendicular direction. This reduces
the minimum band gap.18
Since the band-gap, and the LUMO-HOMO gap,
are measures of stability, reducing them leads to
structural
changes,
decompositions,
and
metallization.
kT/2, and the mean square2 fluctuating energy,
4rjic<F 2>, relating the fluctuating force and the
viscosity:
= (T c /2kT)<F 2 >
(5)
Thus fluctuating forces yield an apparent drag on the
motion of the larger particles as a result of the
asymmetry of the collisions between the small
randomly moving particles and the large steadily
moving ones.
Rearranging Equation (5):
T = (i c /2kn)<F 2 >
(6)
This makes it apparent that if the large particle is
driven through the viscous sea of randomly moving
small particles, the temperature will rise. The nonintuitive feature of this is the inverse dependence on
the viscosity. The strong dependence of the
temperature on the magnitude of the fluctuating
forces might be expected.
It may be helpful to note that this is related to the
ancient rowboat puzzle in which there are two boats
with equally strong rowers. They start at the same
place on the bank of a river flowing with a velocity,
v. One goes across the river whose width is, w, and
comes back to the starting point. The other goes
downstream a distance, w, and then comes back.
Which one gets back first? One of the rowers
represents motion through a constant background,
and the other through a fluctuating background.
In the case of dislocations, the most important
fluctuations are those of shear strains, Ay. Zero-point
vibrations of the shear type are always present and
play an important role in determining the specific
heat. The average square fluctuation is given by:13
<(Ay)2> = k T / G V 0
(7)
where G = shear modulus, and V0 = initial volume. A
dislocation being driven through these fluctuations by
an applied stress causes the temperature to rise.
SHEAR AT SHOCK FRONTS
There is a sharp change in the state of uniaxial
strain at a shock front. This is sometimes taken to be
a change of pressure, but uniaxial strain consists of
both shear and dilatation. In pure metals, the shear
strains are quickly dissipated by plastic deformation,
but they always accompany the jump in deformation.
INSTABILIITES OF MECHANICAL STATES
When solids are subjected to uniaxial
compression (a combination of shear strain and
negative dilatation), at critical values of the
40
Electronic changes are very much faster than
acoustic ones, so they always result from the shear
strains and are present within a shock front. They
vary in magnitude, of course, with the size of the
jump in the uniaxial strain; and the front's thickness.
In an exothermic substance, the change in
electronic structure induced by a large shear strain
initiates the chemical reaction. The released energy
perpetuates the reaction.
7.
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Hollomon, J. H. and Jaffe L. D., Ferrous
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