CP620, Shock Compression of Condensed Matter - 2001
edited by M. D. Furnish, N. N. Thadhani, and Y. Horie
© 2002 American Institute of Physics 0-7354-0068-7/02/$ 19.00
NONLOCAL THEORY OF MACRO-MESO-LEVEL ENERGY
EXCHANGE IN THE SHOCK COMPRESSED MATTER
Tatyana A. Khantouleva
Department of Physical Mechanics, Faculty of Mathematics and Mechanics, S.Petersburg State University,
S.Petersburg, 198904, Bibliotechnaya, 2, RUSSIA
Abstract. Series of experimental results of the shock loading of materials [1-2] had demonstrated that
before the heat dissipation the energy exchange between macroscopic and mesoscopic scale levels took
place. It had been found out that the macro-meso-energy exchange during stress relaxation determines
dynamical properties of materials. A new self-consistent non-local approach developed by the author on
the base of non-equilibrium statistical mechanics and resonance theory [3-4] presents integral relaxation
model as non-local generalization of the Maxwell model for a medium with mesoscopic internal
structures. In scope of this theory the problem on the shock wave propagation in semi-space had been
formulated as a nonlinear operator set with the branching solutions determining the time evolution of a
spectrum of mesoscopic scales. The relationships between the experimentally measured mesoscopic
characteristics (velocity dispersion and wave amplitude loss) allow predicting the conditions of
structure transitions and extremal strength properties of materials under dynamical loading.
resonance systems presents integral relaxation
model for a medium with changing mesoscopic
structures.
INTRODUCTION
Multiscale and multistage energy exchange
follows high-rate straining of materials and excites
fluctuations at the mesoscopic scale level. The
collective effects provoke new internal structure
formation at the mesoscopic scale level. Rotational
and translational mesoscopic structures had been
found out in series of experimental studies on the
shock loading of materials [1-2].
At present it is clear that the relaxation in a
shock-compressed solid cannot be described in the
framework of the traditional elastic-plastic theory.
Unlike quasi-static straining the most important
feature of the dynamical deformation of solids is the
emergence of space-time correlation among
elementary carriers of deformation.
A new self-consistent non-local approach
developed by the author [3-4] on the base of nonequilibrium statistical mechanics and theory of
NONLOCAL GENERALIZATION OF THE
MAXWELL MEDIUM MODEL
The well-known Maxwell model of a medium
determines a deviator stress component S. On the
initial condition S(t=0)=0 the model can be written
in the integral form
dx'
263
u
j!
For the simple exponent integral kernel the
model (1) with the structure effects neglected is
reduced to the Maxwell model of a medium without
internal structures.
Additional nonlinear integral relationships
derived from boundary conditions determine an
evolution of a discrete spectrum of the internal
structure scales and introduce a feedback into the
system, make the model completed and selfconsistent.
So, dynamical behavior of a medium is,
characterized by some functionals (integrals) of
macroscopic fields depending not only on medium
properties but also on the history of the dynamical
deformation in the whole medium-filled volume
including boundaries. For the steady straining or
for structureless media the parameters become
constant values corresponding to the usual medium
characteristics like the shear modulus and viscosity
oo
= Jf
dx'
r ft , r
—exp[
— (x'-x-
Here u denotes macroscopic velocity component in
the x-direction, TJ is a shear viscosity, S is a yield
limit.
The non-local model (1) includes three internal
parameters a , y , € :
1) £ is a typical correlation length ;
2) (1 +a ) is a relative effective viscosity of a
medium with mesoscopic internal structure;
3) y is a polarization parameter making the
shear stress tensor asymmetrical.
In the case of the frozen relaxation as S-»QQ the
model results the Hook's law for the stress deviator
in an elastic solid.
S,-
dx
3
SHOCK WAVE PROPAGATION
Here elastic shear modulus G is connected with the
shear viscosity: r| =2Gtr and the relaxation time in
the limit tr =s/C is determined by the relaxation
length s and sound velocity C
In the opposite case after the mesoscopic
structure relaxation has already completed as s-»0
the newtonian liquid is resulted in the limit.
The mass and momentum balance equations are
written in Lagrangian coordinates in the onedimensional case where axis x is the shock wave
propagation direction
p 2 dt
dx
=;
(2)
(3)
In the absence of dissipation the last term can be
neglected and over the threshold Sj a flow of ideal
fluid takes place. It is the model of ideal plasticity.
Sj is resulted from the internal structure relaxation
already completed.
In the resonance case s determines a typical size
of the medium internal structure element The other
parameter y is closely connected with a kinematic
type of straining (rotation or translation).
The deviator S and spherical part P of the stress
tensor should be divided into two parts: S=S*+S m,
P=Pe+Pm. The first items S*, Pe correspond to the
cold shear and cold compression respectively
presenting the elastic properties of a medium:
S°=-Ge, Pe=(p-pQ)C2. The second ones S mt Pm
assume being connected with the mesoscopic
effects. Herewith, S m can be determined by the
264
non-local relaxation model (1) and Pm should be
determined further.
Then the mass and momentum balance
equations (2)-(3) can be reduced to a wave-type
equation
dt2
•-a.
dx2
The balance equation for the internal energy E=
Em can be written as follows
cvi - Jf cfadx\
0
-1
dtdx
MULTISTAGE AND MULTISCALE
ENERGY EXCHANGE
(6)
(4)
The
experimentally
measured
value
characterizing mesoscopic fluctuations is the
velocity dispersion D. The spherical part of the
stress tensor related to the mesoscopic effects Pm
(fluctuative pressure) and a part of the internal
energy Em should be connected with the velocity
dispersion via equations of state
Here a=Cp/pQ is the Lagrangian velocity of the
wave propagation and ai2=a2+(4/3)Gp0~1 is the
longitudinal sound velocity in Lagrangian
coordinates. The source in the right-hand part in
Eqn. (16) consists of the two competing parts: nonlinear term making the wave front steeper and the
term connected with the mesoscopic relaxation
effects leading to the more sloping front.
At the initial stage t«tRi (Typical rise-time tR=
dx
Pm=MpQD,
Em=cmD+E™, cm=\
dE
dD
(7)
is defined by the maximal strain-
Here M is a thermodynamic characteristics of the
mesofluctuations in a given material and cm is an
energy capacity of the mesoscopic fluctuations. The
full energy at the mesoscopic scale level Em consists
of two parts: kinetic energy of the mesoscopic
fluctuations cmD and potential energy Ems stored in
the mesoscopic structures.
At the initial stage for t«tr when mesoscopic
fluctuations are frozen cm-^oo,. Eqn. (6) splits into
the two independent parts, and the last one results:
D=D(t=0).
At the last dissipation stage for t»tr the
mesoscopic fluctuations have already relaxed
cm—>0, Eqn. (6) is reduces to the usual heat balance
equation
max
rate) mesoscopic degrees of freedom are frozen S m
,Pm =0, Eqn. (4) governs simple wave propagation
in elastic solids at the velocity a\ when tr~ tR or in
ideal liquids at the velocity a when t»tr.
At the last stage for t»tR), In the absence of
dissipation Eqn. (4) governs adiabatic Hugonio
waves propagation.
At the intermediate stage when the mesoscopic
relaxation isn't over inside the load front Eqn. (4)
results the wave amplitude loss at the plateau of the
compressive pulse which can be measured
experimentally
(5)
f =o.
The time 9 is counted inside the wave front
265
At the intermediate stage when
mesoscopic
fluctuations are gene rating Eqn. (6) governs the
velocity dispersion D:
DYNAMICAL EQUILIBRIUM
The energy balance at the mesoscopic level (9)
can be rewritten in the integral form and estimated
using Eqn. (5) as follows
(12)
Here p0R= S n'. The macro-meso-energy exchange
is reversible if SE^/01=0. If dE™/d t*0, structure
transitions occur and the macro-meso-energy
exchange becomes irreversible.
For materials with small cm when 8Ems> Ot the
structure transitions dominantly occur. Formaterials with large cm when SEms < 0,
destructive processes would prevail.
In the dynamical equilibrium SEms=0, Eqn. (12)
results that structure scales at meso-1, meso-2 and
macro-scale levels are approximately equal:
ENTROPY PRODUCTION
The entropy production at the mesoscopic scale
level is determined only by the irreversible energy
flux dEms/@t=J"1 conditioned by structure transitions
(cmD)l*(crnD)2*AuSU
(13)
DISCUSSION
= jmXm =
dt
~.
dx
(10)
Yu Mescheryakov [1-2] had experimentally found
out the condition.(13) as criterion of the maximal
spall strength.
This remarkable accordance between the
theoretical result obtained and the experimental data
allows conclusion that dynamical properties of
materials are really defined by stages of the macromeso-energy exchange.
The thermodynamic force at the mesoscopic scale
level leading to the structure transitions arises due
to the growing fluctuations X"= (^-Pm) —.
dx
In case where new degrees of freedom and
structure scales arise at the mesoscopic scale level
dEms/01 > 0, the entropy production lowers the full
entropy of a system: am< 0. In case where the
elastic potential energy contributes to the
mesoscopic fluctuations dEms/d t < 0, the entropy
production increases am> 0 provoking destruction
at the mesoscopic scale level and initiating the more
large-scale shear structures. The case where there is
no structure transitions dE™/d t =0, am= 0,
corresponds to the dynamical equilibrium.
It can be easily shown that the rate of the
entropy production in the dynamical equilibrium
has a minimum.
d2Ems
(11)
dt
dXn
dt2
REFERENCES
266
1.
Mescheryakov Yu. L, Divakov A.K. DYMATJ.l,
271-287(1994).
2.
Mescheryakov Yu. I., Divakov A.K., Zhigacheva
N.I. Mater. Phys. Mech. 3. 63-100 (2001).
3.
Khantuleva T.A. Mescheryakov Yu.I. Intern. J.
Solids and Structures 36. 3105-3129 (1999).
4.
Khantuleva T.A. Mater. Phys. Mech. 2. 51-62
(2000).