Hugoniots of Porous Materials at Non-Equilibrium
Conditions
A.D. Resnyansky
Weapons Systems Division
Defence Science and Technology Organisation
DSTO-TR-2137
ABSTRACT
This report is establishing the multi-phase modelling capability for evaluation of mitigants
against blast and fragmentation. The work analyses states of a highly porous material under
shock compression. A two-phase model accounting for the inter-phase heat transfer is suggested
and employed. Numerical analysis of wave structure in a porous copper is conducted and a
simplified method of Hugoniot analysis is developed, which is associated with specification of
equilibrium states within the shock transformation. Three types of inter-phase equilibrium within
the shock transformation are distinguished. This state classification results in different loci that
are seen as Hugoniots when certain thermodynamic parameters are in equilibrium; parts of the
loci form a composite Hugoniot curve. Corresponding composite Hugoniots demonstrate a good
correlation with available experimental data for porous metals.
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June 2008
APPROVED FOR PUBLIC RELEASE
Hugoniots of Porous Materials at Non-Equilibrium
Conditions
Executive Summary
The properties of porous materials are important for enhanced protection evaluation and
mitigation of blast and fragmentation due to warhead effects and when neutralising
Improvised Explosive Devices (IEDs). For example, porous mitigants can undergo the
hundreds-kilobar pressure loads due to explosive effects if a weapon is initiated in near
proximity to the mitigant and even higher loads in the case of impact by fragments driven
by the products of a high explosive charge. This report stresses that in such a regime the
heat transfer effects may play a surprisingly important role affecting the Hugoniot
behaviour.
Accurate description of the porous materials is more difficult than that of conventional
materials because the curves connecting material states in front of and behind the shock
front (the Hugoniot curves) are sensitive to non-equilibrium processes occurring in such
complex materials as heterogeneous explosives, composite and geological materials, and
various multi-phase mixtures. A Hugoniot analysis presumes complete equilibrium
behind a shock front, whereas actual states being achieved behind the head shock front are
out of equilibrium. At the same time, certain parameters may be in equilibrium, resulting
in ‘partial’ Hugoniots. In general, the shock transition in a porous material transforms the
material through several stages. Specifically, in the pressure-temperature diagram a
material state at the initial stage of loading (after the front of a compaction precursor) is
characterised by pressure and temperature inter-phase non-equilibrium; this state is
followed by a state under pressure equilibrium and temperature non-equilibrium; and,
finally, a state is achieved with inter-phase pressure and temperature equilibrium. The
present report outlines a mathematical model for inert two-phase materials, accounting for
the heat transfer effects, which is suitable for description of porous materials. Numerical
analysis reveals a multi-step wave structure in a porous copper specifying the pressuretemperature equilibrium configurations listed above. Using basic assumptions of the
model, a simplified approach is developed, which allows us to evaluate the partial
Hugoniots corresponding to the inter-phase equilibrium configurations for shock waves in
porous materials. The approach has enabled us to build up the partial Hugoniots, using a
simple procedure, and to evaluate the heat transfer contribution at different levels and
durations of shock load. The report has demonstrated that the heat transfer is an essential
factor of the Hugoniot description. This enabled us to analyse the Hugoniot abnormality
for highly porous metals resulting in a reasonable agreement of the analysis results with
available experiments.
Authors
A.D. Resnyansky
Weapons Systems Division
____________________
Anatoly Resnyansky obtained a MSc in Applied Mathematics and
Mechanics from Novosibirsk State University (Russia) in 1979. In
1979-1995 he worked in the Lavrentyev Institute of Hydrodynamics
(Russian Academy of Science) in the area of constitutive modelling
for problems of high-velocity impact. Anatoly obtained a PhD in
Physics and Mathematics from the Institute of Hydrodynamics in
1985. In 1996-1998 he worked in a private industry in Australia. He
joined the Weapons Effects Group of the Weapons Systems Division
(DSTO) in 1998. His current research interests include constitutive
modelling and material characterisation at the high strain rates,
ballistic testing and simulation, and theoretical and experimental
analysis of multi-phase flows. He has more than eighty papers
published internationally in this area.
________________________________________________
Contents
1. INTRODUCTION............................................................................................................... 1
2. CONSTITUTIVE MODEL WITH HEAT TRANSFER................................................. 2
3. WAVE STRUCTURE .......................................................................................................... 5
4. HUGONIOT RELATIONS.............................................................................................. 10
5. RESULTS OF HUGONIOT ANALYSIS AND EXPERIMENTS.............................. 17
6. CONCLUSIONS................................................................................................................ 20
7. REFERENCES .................................................................................................................... 21
DSTO-TR-2137
1. Introduction
Material characterisation of porous materials has a potential to extrapolate Equations Of State
(EOSs) of the solid constituent of powders into the high-temperature range of state
parameters. This is possible because the air trapped by pores is extremely hot during shock
compression. An accurate EOS is a key factor for establishing modelling capabilities. As such,
material characterisation has become a common task when developing the underground
nuclear test programs in the United States and the former Soviet Union accompanied by novel
use of experimental methods for Hugoniot analysis of a wide spectrum of condensed
materials. When realising these programs, the researchers have achieved the terabar level of
pressures; however, the most detailed results have been obtained in tests done under
laboratory conditions for a large variety of materials, covering the gigapascal range of
pressures [1]. The primary objective of the tests was to obtain Hugoniots that are the basic
data for derivation of EOSs.
Normally, when testing, experimentalists record the shock velocities in a tested and standard
material (the (D-D)-data). The Impedance Match (IM) technique [2] enables one to derive the
shock and particle velocities from the (D-D)-data when assuming equilibrium behind the
shock front. The majority of published data (e.g., see [1, 3]) have delivered the post-processed
Hugoniots in (P-ρ)- and (D-U)-coordinates, where P is pressure, ρ density, and U particle
velocity behind shock front propagating with velocity D. These test data have (additionally to
the IM-processing) been corrected in order to take into account various factors such as the
shock wave attenuation during its propagation in the experimental set-up, non-planarity of
impact, curvature of the shock front, etc. It was of great interest to researchers that at
moderate pressures of the order of several GPa the derived Hugoniots for highly porous
materials showed an anomalous behaviour resulting in a pressure increase with decrease of
density. In order to explain the abnormality, high temperature excitation mechanisms at the
atomic level (such as those manifested by variable specific heat [1]) have been employed in
EOSs when approximating the test data. Some papers (e.g., see [4]), however, questioned the
need to consider the high-temperature material response mechanisms when designing EOSs
in the pressure range not exceeding hundreds of GPa. This leads to an idea to look for an
explanation of the abnormality at the mesa-level, analysing the phase interaction of a porous
material.
One key assumption of the IM-technique is the jump approximation for shock wave in porous
material along with an assumption of the full equilibrium behind the shock front. The porous
materials, however, fail to demonstrate the full equilibrium behind the shock wave head front,
which is confirmed by available temporal records of stress or velocity (see, for example, [5-9]).
These experimental observations of the shock profiles in porous materials show that the shock
transformation occurs in several stages, resulting in the wave splitting, with only a part of the
thermodynamic parameter set being in equilibrium after each step of the complex wave
structure.
The present report outlines a two-phase constitutive model that is similar to the model used in
[9] and further developed towards the heat exchange effects in a subsequent Section. This
model employs conventional EOSs for gaseous and solid phases of a porous material, which
1
DSTO-TR-2137
have no abnormalities. Using porous copper as an example, a brief numerical analysis
confirms a complex wave structure in the material and suggests two transition zones during
the impact loading. The first transition seen as a compaction precursor wave of the whole
shock transformation is associated with equilibrating pressure. The second transition
occurring in the main compression part of the shock transformation results in temperature
equilibrium. The modelling results show that a state behind the head shock front might be
quite far from the inter-phase thermal equilibrium. Comparison of calculated Hugoniots with
experimental ones confirms a possibility of this non-equilibrium. The Hugoniot analysis of the
two-phase mixture for several porous metals demonstrates that within assumptions of either
Pressure Equilibrium (PE) or full Pressure-Temperature equilibrium (PTE) the partial
Hugoniots are monotonic and show no abnormalities. However, transition from the PE to PTE
regime may result in the monotonicity failure for the Hugoniot in the pressure-density
coordinates, which is treated in numerous literature references as the Hugoniot abnormality.
2. Constitutive Model with Heat Transfer
A constitutive model for a porous material will be derived in the present report in a fashion
similar to [9]. In the model, relations between the phase and averaged variables for the
majority of thermodynamic parameters are identical to those from [9, 10]. Similarly to the
publications [9, 10], the gaseous (air) and solid phases are referred to by superscript indices 1
and 2, respectively. Phase velocities are in equilibrium; they are coincident with the particle
velocity of the mixture, coordinate components of which are uj. The volume and mass
concentrations of the first phase are θ and c with similar relations (see [9, 10]) for density ρ,
entropy s, pressure p, and internal energy e:
ρ = θ ρ(1) + (1 – θ ) ρ(2) ,
p = ρ2eρ = θ p(1) + (1 – θ ) p(2) ,
(1)
e=c
e(1)
+ (1 – c)
e(2)
,
s=c
s(1)
+ (1 – c)
s(2)
.
Here, formulas for pressure in mixture have been derived from the thermodynamic identity
applied to pressure in the phases. In contrast to the previous model, we introduce a new
definition of the entropy disequilibrium χ borrowed from report [11] which is dealing with
another two-phase material model:
χ = c s(1) – (1 – c) s(2) .
(2)
This definition enables us to link phase entropies and the averaged entropy (1) over the whole
range of thermal parameters. Following these definitions, we can express the independent
phase state variables via the averaged ones as follows:
ρ(1) = ρ c/θ , ρ(2) = ρ(1 – c) /(1 – θ ) , s(1) = ½(s + χ) /c , s(2) = ½ (s – χ) /(1 – c) .
(3)
Summarizing, the independent variables that fully describe the two-phase material are ρ, s, uj,
c, θ, and χ. Conservation laws and constitutive equations for mass and volume concentration
parameters have been obtained in [9] and result in the following system of equations:
2
DSTO-TR-2137
∂ρ ∂ρ u j
+
=0,
∂t
∂x j
∂ρ u i ∂ (ρ u i u j + pδ ij )
+
=0,
∂t
∂x j
∂ρ E ∂ (ρ u j E + pu j )
+
=0,
∂t
∂x j
∂ρ c ∂ρ cu j
+
= − ρϕ ,
∂t
∂x j
(4)
∂ρ θ ∂ρθ u j
+
= − ρψ .
∂t
∂x j
where E = e + uj·uj/2, and δij are components of the unit tensor. Here, in (4), φ and ψ are
constitutive functions that will be specified below. When the specific internal energy e is given
as a function of the independent thermodynamic parameters e = e(ρ, s, c, θ, χ), the
thermodynamic identity gives p = ρ2eρ and T = es.
We assume that the internal energy dependencies for each phase are given in the following
form with the natural independent variables:
e(1) = e(1)(ρ(1),s(1)) ,
e(2) = e(2)(ρ(2),s(2)) .
(5)
Similarly to the calculation of pressure in (1), the thermodynamic identities for the phases can
be used for calculation of averaged temperature by differentiation of the internal energy from
(1) with respect to entropy as follows:
(
T = e s = c e (1) + (1 − c ) e ( 2 )
)
s
( ) + (1 − c )T (s ( ) ) = (T
=cT (1) s (1)
( 2)
s
2
s
(1)
)
+ T (2 ) 2 .
(6)
In order to derive a constitutive equation for χ we remind (see [10]) that the conservation laws
for mixture could be obtained from the respective balance laws for the phases. Variables of
density and pressure for these laws are presented by partial densities ρ(1)p = ρ·c = θ·ρ(1) and ρ(2)p
= ρ·(1–c) = (1–θ)·ρ(2), and by partial pressures p(1)p = θ·p(1) and p(2)p = (1–θ)·p(2). The mass,
momentum, and energy balance laws for the phases have the same form as three first
equations of the system (4) in which density and pressure are replaced by corresponding
partial variables; the mass, momentum, and energy balance laws have the source (exchange)
terms that are denoted by m0, n0, and l0, respectively, for the first phase and by –m0, –n0, and –l0
for the second phase. It is obvious, from the constitutive equation for c in (4), that m0 = – ρφ.
We focus on the energy equations that take the following form after the reductions with use of
relevant mass and momentum balance laws
⎛ (1) u 2 ⎞
∂e (1)
(1) p ∂ u j
⎟,
= l 0 − un0 − m0 ⎜ e −
+p
ρ
⎜
2 ⎟
∂x j
∂t
⎝
⎠
2
⎛
⎞
∂u j
∂e (2 )
u
⎟.
= −l 0 + un0 + m0 ⎜ e (2 ) −
+ p (2 ) p
ρ (2 ) p
⎜
2 ⎟
∂x j
∂t
⎝
⎠
(1) p
(7)
3
DSTO-TR-2137
Using the thermodynamic identities for each phase and the mass and momentum balance
laws for the phases, from (5) and (7) we can obtain the following governing laws for phase
entropies:
(1) p (1)
∂ρ (1) p s (1) ∂ρ s u j l 0 − un0 − m0 μ ′ (1)
+
=
= R1 ,
∂t
∂x j
T (1)
(8)
(2 ) p (2 )
l − un0 − m0 μ ′ (2 )
∂ρ (2 ) p s (2 ) ∂ρ s u j
+
=− 0
= R2 .
∂t
∂x j
T (2 )
Here μ′ (i) = μ(i) – |u|2/2 and μ(i) = e(i) + p(i)/ρ(i) – s(i)T(i) = e(i) + p(i)p/ρ(i)p – s(i)T(i) are the Gibbs
energies. Summing up and subtracting the equations (8) and using (1) and (2), we can obtain
the entropy dissipation law
∂ρ s ∂ρ su j
+
= R1 + R2 + Δ ,
∂t
∂x j
(9)
and the following constitutive equation for parameter χ
∂ρ χ ∂ρ χ u j
+
= R1 − R2 = − ρω .
∂t
∂x j
(10)
Here the term Δ in the right hand side for the entropy law is added in order to compensate for
dissipation due to compaction. Thus, the whole system of equations includes three
conservation laws from (4), the constitutive equations for mass and volume concentrations c
and θ in (4) and the constitutive equation (10) for parameter χ. In order to verify
thermodynamic correctness of the model, we need to expand the energy conservation law.
Differentiating e = e(ρ, s, c, θ, χ) as a complex function in this law and using mass and
momentum conservation laws from (4) for elimination of convective terms, we have
ρ ec
dc
dθ
dχ
ds
+ ρ eθ
+ ρ eχ
+ ρ es
=0,
dt
dt
dt
dt
(11)
here d/dt = ∂/∂t + uj·∂/∂xj. The energy derivatives in (11) are obtained from (1), (3), and (5):
ec = μ(1) – μ(2) , eθ = – (p(1) – p(2))/ρ , eχ = (T(1) – T(2))/2 .
(12)
It is easy to check if the corollary (11) of the energy conservation law is consistent with the
choice of constitutive equations. We replace the particle derivatives in (11) by the right-hand
sides from the constitutive equation of (4), and by those from (9) and (10), so we have:
m0(μ(1) – μ(2)) + (p(1) – p(2))ψ + (R1 – R2)(T(1) – T(2))/2 +
(13)
+ (R1 + R2
4
)(T(1)
+
T(2))/2
+
∆(T(1)
+
T(2))/2
=0.
DSTO-TR-2137
Expanding denotations of R1 and R2 from (8), we can see that (13) is satisfied if we take
∆ = – (p(1) – p(2))·ψ/T .
(14)
On the other side, we can formulate the entropy non-decrease condition, when replacing the
derivatives for parameters c, θ and χ by the right-hand sides from (4) and (10):
ds ϕ ec + ψ eθ + ω e χ
=
,
dt
T
which means that φec + ψeθ + ωeχ should be nonnegative. Similar to the choice of constitutive
functions in [9] this condition is easily satisfied with the following choice of functions φ, ψ,
and ω:
φ = ec·φ0 ,
ψ = eθ·ψ0 ,
ω = eχ·ω0 ,
(15)
where φ0, ψ0, and ω0 are arbitrary nonnegative functions. With this choice, the dissipation term
(14) transforms to
∆ = ρeθ·ψ/T = ρ(eθ)2·ψ0 /T ,
so the additive ∆ in (9) is non-negative too.
It is interesting to note that from (12) it follows that the derivatives ec, eθ, and eχ are still
associated with the Gibbs energy affinity, the pressure affinity and the temperature affinity,
respectively, similarly to the case of [9]. It is obvious that the system of equations of the model
is hyperbolic when normal condition PV < 0 is satisfied; here V=1/ρ is specific volume.
Conditions of convexity and regularity for mixture’s EOS and links to those for phases’ EOSs
are identical to ones for the model [9].
Thus, the system (4) along with the constitutive equation (10) generates a system describing
the behaviour of a two-phase inert porous material. The model is closed with the EOSs (5) for
each phase, where phase parameters are connected with the averaged parameters by relations
(1) and (3). The constitutive equations are complemented with selection of the constitutive
rates (the arbitrary nonnegative functions φ0, ψ0, and ω0) that will be specified in next Section.
3. Wave Structure
We choose the constitutive functions φ0, ψ0 in the same form that has been used for the
constitutive equations for sand and porous 2024-Aluminium [9]:
φ0 = 0 , ψ0 = θ(1–θ)·B0 .
5
DSTO-TR-2137
where B0 = a0A·[θ – θ0(p)] at θ > θ0(p) and B0 = 0, otherwise. The function θ0(p) is determined
from an equilibrium curve Peqv = Peqv(θ) taken from experiments (e.g., the gas gun test data [6]
for porous copper) and is close to that selected in [10]. Similarly to [10], we consider a porous
copper with porosity m = 4 (m = ρ0s/ ρ00), where ρ0s = 8.93 g/cm3 is density of the solid phase
and ρ00 is density of porous material.
EOSs for the two phases are taken in the same simple form as in [10]. Namely, EOS for the
gaseous phase was chosen in the ideal gas form
⎛
⎞ ⎛ ρ
e (ρ , s ) =
exp⎜ s ⎟ ⋅ ⎜
ρ 0 g (γ − 1) ⎝ cvg ⎠ ⎜⎝ ρ 0 g
(1)
S0
⎞
⎟
⎟
⎠
γ −1
(16)
,
and EOS for the solid phase was taken in the Mie-Gruneisen form:
2
α0
⎤
⎛ ρ
c02 ⎡⎛ ρ ⎞
p0
(2 )
⎜⎜
⎜
⎟
⎥
⎢
1
−
+
e (ρ , s ) =
c
T
⋅
−
vs
0
2
ρ
2α 0 ⎢⎜⎝ ρ 0 s ⎟⎠
⎥
⎝ ρ 0s
⎦
⎣
β
⎞ ⎡ ⎛ s ⎞ ⎤
⎟⎟ ⎢exp⎜
− 1 + e j0
c ⎟ ⎥
⎠ ⎣ ⎝ vs ⎠ ⎦
.
(17)
The material constants for the solid phase EOS in the case of copper were taken to be c0 =
3.94 km/s, ρ0s = 8.93 g/cm3, cvs = 0.45 J/(g·grad), α0 = 0.96, β = 1.99, p0 = 0.1 MPa, T0 = 293○K.
Specification of the material constants will be discussed in more detail in next Section. The
standard values were chosen for the air at the normal conditions with the exponent γ = 1.4, the
thermal capacity cvg = 0.7 J/(g·grad), and initial density ρ0g = 0.0012 g/cm3. The constant S0 is
calculated from the condition of adjustment of the phase pressures at the initial density to 0.1
MPa (1 atm), and ej0 is calculated such that the internal energies of the phases at normal
conditions are adjusted as well. It should be noted that the EOSs are deliberately
oversimplified for the pressure and temperature levels considered in the present report.
However, this consideration will be illustrated to be sufficient for description of the main
features of Hugoniot. Thus, more complex EOSs for the phases is the next level of
approximation whereas the taking into account of the two-phase nature of the material is the
most significant level.
In the present report, the choice of the rate ω for the constitutive equation for χ will be
deduced directly from the energy exchange law. Assuming the velocity equilibrium between
the phases (n0 = 0) and using (7), the heat exchange term l0 takes the following well-known
form:
l0 = – h·(T(1) – T(2)) ,
where h is the heat transfer coefficient. Then, from (10), it follows:
l0 ⎞
T (1) + T (2 )
⎛ l0
,
+
=
−
l
⎟
0
(1)
T (2 ) ⎠
T (1)T (2 )
⎝T
ρ ω = − (R1 − R2 ) = − ⎜
resulting in
6
DSTO-TR-2137
ω=
(
)(
)
h T (1) − T (2 ) T (1) + T (2 )
⋅
.
ρ
T (1)T (2 )
Using (12) and (15), we have:
ω0 =
2h T (1) + T (2 )
⋅
ρ T (1)T (2 )
(18)
that satisfies the necessary condition of non-negative ω0 provided that h is also non-negative.
We choose the heat transfer coefficient as simple as possible relying on a heat transfer model
[12] in the following general form:
h = AS·keff /d 2 .
(19)
Here, AS is a dimensionless factor that depends, according to [12], on porosity, the powder’s
surface area per unit of volume, parameters of convection such as the Reynolds number Re,
diffusivities characterised by the Prandtl number Pr, some thermo-physical relations between
matrix phase and carrier, etc. For simplicity, we fix this number to a constant value of AS = 400
which agrees with Re of the order of 100 and Pr of the order of 1, using one of the models [12].
The effective thermal conductivity keff is taken in the following symmetric form [13]:
⎛ θ 1−θ ⎞
⎟,
k eff = 1 ⎜⎜ +
k 2 ⎟⎠
⎝ k1
(20)
where k1 = 0.025 W/(m·K) is the thermal conductivity of air and k2 = 400 W/(m·K) is the
thermal conductivity of copper. The parameter d is a characteristic particle diameter of the
powder (grain size), which can be selected as d = 100 μm for a typical case. A certain freedom
dictated by interphase exchange mechanisms, while specifying the coefficients, should be
noted when the heat transfer coefficient in (11) is not changing as soon as AS/d2 is constant.
Thus, the same h can be obtained from (19) with d = 10 μm and a value of AS = 4 which is
correspondent to Re and Pr of the order of 1. Therefore, this phenomenological approach
somewhat integrates contributions of the micro-mechanisms into a value of AS.
It is traditionally believed that thermal equilibrium for a condensed material is achieved by
the heat transfer during a long time when compared with the shock compression timeframe.
Therefore, the heat transfer is usually ignored in considerations of the shock wave processes
for conventional condensed materials. However, in contrast to the conventional materials,
porous materials are the mixtures and they exhibit a complex behaviour. Below we will try to
estimate characteristic temperature equilibration time scales for a porous material seen as a
gas-solid mixture. It is accepted that the time scale of heat transfer in a homogeneous material
is characterized by τ = ρcvL2/k, where ρ is density, cv is the specific heat, L is a characteristic
distance between the points where the heat exchanges, and k is the thermal conductivity.
Considering the case of the copper powder with m = 4, we can evaluate the corresponding
time parameters for the solid phase (τs) and for the gaseous phase (τg). We take L = d, and for
7
DSTO-TR-2137
the solid phase cv = cvs, ρ = ρ0s, and k = k2, which results in τs ≈ 0.1 msec. Similarly, for the
gaseous phase with cv = cvg, ρ = ρ0g, and k = k1, the time scale estimate will give the
equilibration time of τg ≈ 0.33 msec. Therefore, the thermal equilibration does take a rather
long time of the order of hundreds of microseconds in corresponding mixture constituents.
However, heat is a mass related characteristic; therefore, in contrast to the case of isolated
homogeneous components, in the case of a porous material we are dealing with the mixture of
the components, which has a very small mass concentration of the gaseous component. Thus,
whereas a typical powder grain heats up slowly (during hundreds of microseconds), the
cooling down of the gaseous phase may occur relatively quickly. Addressing to (8) and
roughly evaluating internal energies as those proportional to the product of heat capacity and
temperature, we can estimate the characteristic time of thermal equilibrating in mixture τm to
be inversely proportional to h[1/(c·cvg)+1/((1–c)·cvs)]/ρ. Estimating the mass concentration
from (3) with the present mechanical and thermo-physical constants for the porous (m = 4)
copper, we have c ≈ 0.0004. From the evaluation (19-20) of the heat transfer coefficient as h ≈
ASk1/(θd 2) ≈ 1.3·109 W/(m3grad), we have τm ≈ ρc·cvg/h ≈ 0.3 μsec. Thus, even this rough
estimate for the mixture shows a significant deviation from the timescale estimates for the
individual constituents. Therefore, in order to avoid loss of the wave structure details when
compressing a powder with shock wave, we should take the heat exchange into account while
calculating the wave profiles. It should also be noted that the radiative cooling down of the
gaseous phase may occur even quicker and according to various estimates it could be as quick
as thousands of degrees per microsecond [14].
The resulting kinetic (18-20) for χ, which employs the above stated thermo-physical constants,
will be referred to as a Basic Heat Transfer (BHT) kinetic. We will also consider two other
kinetics that are a Small Heat Transfer (SHT) kinetic obtained from (19) with the factor of AS =
0, and a Large Heat Transfer (LHT) kinetic resulted from (19) with AS >> 1 (instantaneous
thermal equilibrium forcing in T(1) = T(2)). With these choices, we aim to compare the results for
the SHT, LHT and BHT kinetics in order to evaluate a contribution of the heat transfer effects
to the shock compression behaviour.
Figure 1
Calculated density profiles corresponding to the shock compression by a hard plate with the
following impact velocities: (a) U0 = 0.3 km/s; (b) U0 = 1 km/s
The constitutive relations obtained above have been used for calculation of propagation of the
plane shock wave in a sample made from porous copper (m = 4). The shock wave is a result of
impact by a hard plate (punch) that is compacting a 1 cm-thickness sample with a constant
8
DSTO-TR-2137
velocity U0 from the impact interface x = 0. The interface x = 1 is free of stresses. The density
profiles (distribution of density along the sample’s thickness) are shown in Figs. 1 and 2 at
time intervals marked by numbers (the time measure in μsec) at every solid profile that was
calculated with the BHT kinetic. The thermal contribution is assessed by comparison of these
profiles with calculations made with the SHT and LHT kinetics. The SHT and LHT
calculations are shown with dashed and point-and-dash profiles, respectively, that are taken
at the last time interval reached for every given U0 (except for the profiles in Fig. 2(b) where
the corresponding profiles are taken at t = 1 μsec).
For the case of Fig. 1(a), the wave non-equilibrium is essentially caused by interaction of the
compaction precursor reflected from the free surface with the main compression wave.
Profiles calculated with the LHT kinetic are very close to the BHT kinetic profiles because the
thermal ‘relaxation’ occurs almost fully for this range of loads, so we could not distinguish
corresponding profiles in Fig. 1(a) (i.e., the point-and-dash profile is coincident with
corresponding solid one). The averaged temperature is moderate for such a small load;
however, the thermal contribution is already considerable. The loading duration in the
present case is significant; therefore, after density has approached a value corresponding to
the pressure equilibrium (observed from the dashed profile for the SHT calculation), a further
density increase takes place when the phase temperatures equilibrate. The wave splitting is
not noticeable due to an integrated long-time character of the load.
For a moderately stronger wave (U0 = 1 km/s) illustrated by profiles in Fig. 1(b), the wave
structure changes significantly. To stress the non-equilibrium effect, the characteristic points
of the profile at t = 6 μsec in Figs. 1(b) are indicated with the arrows labelled by letters ‘L’ and
‘F’ (points at the main compression front and at the loading surface, respectively). As in the
previous calculation, the SHT (dashed) profile is characterised by a subdued density when
pressure is equal between the phases, but the temperature equilibrium is not achieved. The
shock wave (BHT calculation) exhibits a clear three-step structure where the steps are: a lowamplitude fast-going precursor (that is hardly visible in the density profiles, but it is very
noticeable in stress and velocity profiles as similar profiles in [10] show); main PE wave; and a
delayed PTE wave. As in the previous case, when temperature is equilibrating, pressure and
density increase in the PTE wave due to a relatively moderate heating. Thus, the state ‘F’
corresponds to the state behind the PE wave and in front of the PTE wave; and the state ‘L’
corresponds to the state behind the PTE wave if the process is long enough to equilibrate the
phase temperatures. With further load increase, temperature is elevating (along with an
increase of thermal non-equilibrium) and density is gradually decreasing down to a value
corresponding to the pressure and temperature equilibrated (PTE) state in accordance with
the well-known predictions of the anomalous behaviour for a porous medium [1, 15].
Due to the gradual density decrease with the wave strength, there is a load (an alignment
point), at which the density behind the PE wave is aligned with the density behind the PTE
wave. This behaviour is observed in Fig. 2(a) for loading at U0 = 3 km/s. In this case, it is seen
that the shock velocities for all three calculations with the BHT, LHT, and SHT kinetics are
close to each other. It is also seen from comparison with the previous calculation that, with an
increase of the wave strength, the distance lag between the PE and PTE waves increases due
to a higher velocity of the PE wave and, obviously, due to a slower change of the PTE wave
velocity within the current timeframe. This distance lag, which is the heat transfer zone
9
DSTO-TR-2137
resulting in the density and pressure relaxation, is occurring, however, during a shorter
absolute time.
Figure 2
The density profiles corresponding to the shock compression by a hard plate with the
following impact velocities: (a) U0 = 3 km/s; (b) U0 = 5 km/s
The heat transfer effects manifest themselves much more clearly in calculations of propagation
of a very strong shock wave (U0 = 5 km/s) shown in Fig. 2(b), which complete the analysis.
For this case, it is seen from the BHT calculation that state after the PE wave evolves in a
kinetic manner accompanied by density and pressure relaxation. The density after the PE
wave is actually tending down to the value corresponding to the LHT calculation (the dotand-dash profile). In contrast, calculations for a smaller load (below U0 = 3 km/s) result in
states characterised by a monotonic increase of pressure and density after PE and PTE waves
successively, due to a relatively low incoming thermal energy.
Thus, the numerical results and the comparison of the results for different heat transfer
kinetics show that the time of the shock wave transformation, which depends on the load
level, is large enough for the loads lower than the alignment pressure in order to equilibrate
temperature while reaching the prescribed pressure behind the shock wave. On the other side,
this time is too small at a higher loading pressure, resulting in temperature non-equilibrium in
the head wave followed by the kinetic relaxation zone. In next Section we will trace down the
thermodynamic path of material states within the shock wave by analysing partial Hugoniots.
4. Hugoniot Relations
We derive partial Hugoniots, using the Rankine-Hugoniot relations from the mass,
momentum and energy conservation laws balanced across shock jump [2]:
ρ (D − U ) = ρ 0 (D − U 0 ) ,
p − p 0 = ρ 0 (D − U 0 )(U − U 0 ) ,
e − e0 = 12 ( p + p 0 )(1 ρ 0 − 1 ρ ) ,
(21)
where, using an EOS in the caloric form e = e(ρ,p), the Hugoniot curve pH = pH(ρ) can be
obtained from (21) and all parameters behind the shock front found. For a conventional
material, the equation of state within the Mie-Gruneisen EOS assumptions can be found by an
10
DSTO-TR-2137
inverse procedure, using a known Hugoniot. In (21), p, ρ, U, e are pressure, density, velocity
and internal energy behind the shock wave, and p0, ρ0, U0, e0 are the same variables in front of
the shock wave.
This Section describes how the partial Hugoniots are constructed. First, we analyse a
hypothetic Non-Equilibrium (NE) case when phase pressures and temperatures in a mixture
are out of equilibrium. From (21) it follows:
e − e0 =
1
2
( p + p0 ) (1 ρ 0 − 1 ρ ) , (U − U 0 )2 = ( p − p0 )(1 ρ 0 − 1 ρ ) .
(22)
To start the Hugoniot analysis in case of mixture we first consider Hugoniots for each phase.
We can derive the Hugoniots for the constituents of a porous metal, using EOSs e(1)(ρ(1), s(1))
and e(2)(ρ(2), s(2)) from (16) and (17). For the present derivation, we can rewrite the EOSs in the
caloric form:
(1)
e (1) ρ (1), p (1) = p
(
)
ρ (1) (γ − 1)
(2 )
, e (2 ) ρ (2 ), p (2 ) = ec (δ ) + p
(
)
ρ (2 )β
,
(2 )
2
c2
p (β + 1)
c2
+ e j 0 − 0 ⋅ δ α0 δ α0 − 1 , δ = ρ
ec (δ ) = 0 2 ⋅ δ α 0 − 1 − 0
ρ 0s ,
ρ 0 s δβ
α0β
2α 0
(
)
(
)
(23)
or in the following equivalent form
(
)
(
)
e (1) ρ (1), T (1) = c vg T (1) , e (2 ) ρ (2 ), T (2 ) = e y (δ ) + cvs T (2 ) ,
e y (δ ) =
(
(24)
)
2
c 02
p
⋅ δ α 0 − 1 − c vs T0δ β − 0 + e j 0 .
2
ρ 0sδ
2α 0
Here, we used the thermodynamic identity in order to calculate pressure and temperature
from EOSs with the natural variables for internal energy. When pressure is found, the last
system can also be completed by expressions p(1) = p(1)(ρ(1), T (1)) and p(2) = p(2)(ρ(2), T (2)):
(
)
(
p (1) ρ (1), T (1) = ρ (1) (γ − 1) cvg T (1) ,
e p (δ ) =
p0
ρ 0 s δβ
+
c02
α0β
(
)
(
)
p (2 ) ρ (2 ), T (2 ) = ρ (2 )β e p (δ ) + cvs T (2 ) ,
)
⋅ δ α 0 δ α 0 − 1 − c vs T0δ β .
(25)
Referring, when necessary, to the assumptions of the present two-phase model, we further
consider three partial Hugoniots that are loci connecting the state in front of the shock wave
and the states within a shock wave, which have been emphasised in previous modelling
Section. These three Hugoniots form the following cases: i) case when the phase pressures and
temperatures are not in equilibrium (NE partial Hugoniot); ii) case when the pressures are in
equilibrium and the temperatures are not in equilibrium (PE partial Hugoniot); and iii) case
when the pressures and temperatures are in equilibrium (PTE Hugoniot). For the present
analysis we assume that the phase mass concentration c is constant.
11
DSTO-TR-2137
When addressing the first case, we can deduce the individual Hugoniots from (23) and from
the equation (22), which is formulated for each phase as follows:
(
)
p (1) = p H(1) ρ (1) ; ρ 0(1) , p0(1) =
p
(2 )
(2 )
(
(2 )
(2 )
(2 )
= p H ρ ; ρ 0 , p0
)
2e0(1) + p0(1) (v10 − v1 )
,
κ 1v1 − v10
2e0(2 ) − ec (v 2 ) + p 0(2 ) (v 20 − v 2 )
=
,
κ 2 v 2 − v 20
(26)
here ν1 = 1/ ρ(1), ν2 = 1/ ρ(2), κ1 = 1 + 2/(γ – 1), κ2 = 1 + 2/β. The variables with zero subscript
index denote initial values of corresponding variables, which correspond to the state in front
of the wave. Assuming the velocity equilibrium and pre-selecting Δ2=(U – U0)2, we can derive
two closing relations from the second equations of (22) for each of the phases with respect to
four variables p(1), p(2), ρ(1), and ρ(2):
( p ( ) − p ( ) )(v
1
1
0
10
− v1 ) = Δ 2 ,
( p ( ) − p ( ) )(v
2
2
0
20
− v2 ) = Δ 2 .
(27)
Thus, non-equilibrium phase pressures, temperatures, and specific volumes can be obtained
for a given Δ2 from (26) and (27). From (3) and (6) we can obtain the averaged density and
pressure as follows:
1 ρ = c ρ (1) + (1 − c ) ρ (2 ) ,
p = ρ ⋅ (cp (1) ρ (1) + (1 − c ) p (2 ) ρ (2 ) ) .
(28)
To illustrate the derivation, corresponding NE Hugoniots p = pH(ρ; ρ0(1), p0(1), ρ0(2), p0(2)) are
drawn by dashed curves in Fig. 3 and marked with Roman numbers i, ii, iii, and iv for copper
powders with porosities m = 1, 2, 3, and 4, respectively.
The pressure equilibrium case still involves individual Hugoniots (26) with an additional
condition for densities at the final state:
(
)
(
)
p H(1) ρ (1) ; ρ 0(1) , p 0(1) = p H(2 ) ρ (2 ) ; ρ 0(2 ) , p 0(2 ) .
(29)
Using a brief notation for the right-hand side of (29) as pH(2)(ν2), the equation (29) allows us to
express ν1 as a function of ν2:
2e0(1) + p0(1) v10 + p H(2 ) (v 2 )v10
v1 (v 2 ) =
.
p 0(1) + κ 1 p H(2 ) (v 2 )
(30)
A closing equation for ν2 (and, thus, for ρ(2) = 1/ν2 and δ = ρ(2)/ρ0s) follows from the second
equation of (22), where pressure and density are taken from (28) and (30) and Δ2 is given:
12
DSTO-TR-2137
⎡ 2e0(1) + p 0(1) (v10 − v1 (v 2 ))
2e0(2 ) − ec (v 2 ) + p 0(2 ) (v 20 − v 2 )
(
)
c
⋅
+
−
c
⋅
−
1
⎢
κ 1v1 (v 2 ) − v10
κ 2 v 2 − v 20
⎣
⎤
p 0 (cv1 (v 2 ) + (1 − c )v 2 )⎥ ⋅ [v 0 − cv1 (v 2 ) − (1 − c )v 2 ] = [cv1 (v 2 ) + (1 − c ) v 2 ] ⋅ Δ 2 .
⎦
(31)
Resolving equation (31) with respect to ν2 at a given Δ2, we can find densities from (31) and
(30) and pressures from (26) followed by calculation of averaged pressure and density from
(28). Corresponding partial PE Hugoniots in Fig. 3 are marked by numbers 1, 2, 3, and 4 for
the same porosities, as in the previous NE case.
Figure 3
Schematic of the Hugoniots at various assumptions of equilibrium and at various porosities
(the porosity number correspond to the curve number). Curves i-iv, 1-4, and I-VI
correspond to partial NE, PE, and PTE Hugoniots, respectively.
The last case of full pressure and temperature equilibrium is considered when taking T = T(1) =
T(2) and p = p(1) = p(2). Using the EOSs in the form (25) we can calculate phase density ν1 from
the first EOS of (25) via p and T that are in equilibrium for both phases and temperature T via
p and δ (i.e., ν2) from the second EOS of (25) as follows:
v1 =
(γ − 1)cvg T
p
, T=
p
ρ 0 s δβcvs
−
e p (δ )
cvs
.
(32)
Thus, the volume ν1 and the averaged density (with the use of (28)) can be expressed via δ and
p.
Using (24) and (1), we can express the internal energy e for the mixture via δ and T:
e = c v T + (1 − c ) e y (δ ) , c v = c ⋅ c vg + (1 − c ) ⋅ c vs .
(33)
13
DSTO-TR-2137
Thus, with the help of (32), the averaged internal energy e can be expressed from (33) via δ and
p as well. Substituting averaged volume ν = c ν1 + (1 – c) ν2 from (28), expressed through δ and
p, into the following second equation of (22): (p – p0)(ν0 – ν) = Δ2, we have a quadratic equation
with respect to pressure. After solving this equation, we can obtain p = p(δ). The parameter Δ2
is involved in each of the functions dependent on δ. Thus, the averaged volume can be found
as ν(δ) as well, and calculating backward the second equation of (32), we can determine T =
T(δ). The last step is calculation of δ at a given Δ2. This is performed with use of the first
equation of system (22) in the following form:
c v T (δ ) + (1 − c ) e y (δ ) − e0 =
1
( p(δ ) + p0 ) (v0 − v(δ )) .
2
Calculating this equation with respect to δ allows us to find pressure and density. Results of
the solution are drawn in Fig. 3 representing the PTE Hugoniots; they are marked by capital
Roman numbers I-VI for porosities 1 to 6, respectively.
It is seen from Fig. 3 that at the porosity m > 2, the ‘abnormal’ behaviour of the PTE Hugoniot
for the two-phase mixture of porous copper is obtained (as noted in [15]). Depending on the
state of equilibrium reached behind the shock front, the composite Hugoniot describing the
shock states of a material may transfer from one equilibrium branch to another; an example of
possible composite Hugoniot is shown by a bold curve for m = 2.
Using the partial Hugoniots, we can now analyse the shock wave behaviour observed in
calculations of the previous Section. For the case of porosity m = 4 we take the corresponding
Hugoniot curves from Fig. 3 and gather them into a detailed diagram drawn in Fig. 4.
In order to stress the kinetic character of the curves we attribute a characteristic time τ (which
is needed to achieve the relevant equilibrium states) to each of the partial Hugoniots.
According to this attribution, the NE Hugoniot (dashed curve ‘iv’) is associated with a small
time τ which is not enough to put the phase pressures and temperatures in equilibrium (the
curve is marked with τ = 0 in Fig. 4); the PTE Hugoniot ‘IV’ is associated with a time τ that is
large enough in order for the phase pressures and temperatures to achieve equilibrium (the
curve is marked with τ = ∞ in Fig. 4); and the PE Hugoniot ‘4’ is associated with an
intermediate time τp which is enough to reach pressure equilibrium but not enough to reach
temperature equilibrium (the curve is marked with τ = τp in Fig. 4). Concentrating on the postcompaction shock compression, it is well-known that the material states in shock wave follow
the Rayleigh line that is determined by the following equation obtained from the jump
conditions (22):
D = v0 ⋅
p − p0
1
1
, v0 =
, v= ,
v0 − v
ρ0
ρ
(34)
here the particle velocity in front of the wave is neglected. It should be noted that the origin of
the Rayleigh line (point ‘O’ in Fig. 4) is actually located at a state behind the compaction
precursor, which is fairly close to the point p = 0, V = 1/ρ00 marked as ‘O’ in Fig. 4, but does
14
DSTO-TR-2137
not coincide with it. Location of this point is governed by the equilibrium (p,θ)-curve used for
design of the kinetic ψ in (4). Details of calculation of this point can be found in [9].
Figure 4
Schematic for analysis of the shock wave transformation, using the partial Hugoniots of
porous copper at m = 4
Turning to the Rayleigh line (34), for a specified load we can draw a single line with the
corresponding shock velocity: p – p0 = (ρ0D)2(v0 – v). For instance, for a load similar to that of
the calculation shown in Fig. 1(b), the Rayleigh line RB is drawn in Fig. 4. It is seen that point IB
in Fig. 4 corresponds to the state ‘F’ in Fig. 1(b) and point FB in Fig. 4 corresponds to the state
‘L’ in Fig. 1(b). Because the intermediate point IB lying on the PE partial Hugoniot (this
Hugoniot originates at point ‘P’) is approached earlier in the thermodynamic space (after the
T1-transition from τ = 0 to τ = τp), we observe the first step of the wave, behind which the
phase pressures are in equilibrium. The final state FB is approached next (after the T2transition from τ = τp to τ = ∞); at this state the phase temperatures are in equilibrium as well
because the point FB is on the PTE Hugoniot (that originates at point ‘S’). Referring to
calculations in Fig. 1(b) with SHT and LHT kinetics, we have to note that states calculated
with the SHT kinetic roughly relate to the states of PE partial Hugoniot and those of the LHT
kinetic relate to the states of PTE Hugoniot. At approximately the same load level (in fact, the
load level in Fig. 1(b) is managed by U = U0, alternatively to a prescribed loading pressure) the
behind front state of the SHT calculation corresponds to point I’B in Fig. 4, which results in the
Rayleigh line rB. It is interesting to note that the shock velocity is faster in this case, which is
seen from the dashed profile in Fig. 1(b), because slope of rB (and corresponding shock
velocity) is larger than that of RB. Then, it is not surprising that the LHT calculation gives the
same velocity (the point-and-dash profile) as the BHT calculation does because the Rayleigh
line for the LHT case is coincident with the RB line.
In contrast, when increasing the load substantially (for instance, for a load similar to that of
the calculation shown in Fig. 2(b)), the Rayleigh line RA first approaches point FA (after the T1transition from τ = 0 to τ = τp). After the T2-transition from τ = τp to τ = ∞, the FA-state relaxes
to a state characterised by point IA lying on the PTE Hugoniot. Referring to the profiles in Fig.
15
DSTO-TR-2137
2(b), the behind-front state is characterised by a higher pressure and density corresponding to
the point FA that is on partial PE Hugoniot. This state is followed by a relaxed IA-state with
pressure and temperature in equilibrium. Let us compare shock velocities, referring to the
comparison of the BHT profile with the SHT and LHT profiles at approximately the same load
level. In the BHT case slope of RA-line is coincident with the Rayleigh line for the SHT
calculation (the dashed profile in Fig. 2(b)), Therefore, in contrast to the previous case, the
shock velocities for the BHT and SHT calculations are close to each other in the case of the
strong load. On the other side, the LHT calculation (the dash-and-point profile in Fig. 2(b)) has
the behind front state corresponding to point I’A on the PTE Hugoniot. In this case, the shock
velocity for the LHT profile is determined by slope of rA-line, which is larger than the shock
velocity calculated from RA-line.
The alignment point ‘A’ in Fig. 4 presents an obvious case because for the corresponding load
(the relevant profiles are drawn in Fig. 2(a)) the corresponding Rayleigh lines are identical for
the partial PE and PTE Hugoniots. In this case the T2-transition degenerates to a
transformation around a single state characterised by the point ‘A’ in (p, ρ)-coordinates.
Correspondingly, the shock velocities for BHT, SHT, and LHT profiles are all close to each
other.
Thus, the observed shock velocity behaviour from the Hugoniot diagram in Fig. 4 is in
agreement with the behaviour of profiles in Figs. 1-2.
It is interesting to note that in the case of full equilibrium the PTE Hugoniots are strictly
monotonic. In considering the Hugoniots, the powder constituents behave ‘normally’ at
compression according to (26), because the denominators in (26) never happen to be negative
for an individual phase. A negative denominator appearing in the traditional Hugoniot
analyses of powders [15] is a consequence of accepting the Mie-Gruneisen EOS for a twophase mixture with a single Gruneisen coefficient.
Concluding the present Section, we outline building up EOSs for metal constituents in order
to apply the EOSs to the Hugoniot analyses of various metallic powders. In order to do so, we
can use data [2] on shock compression of metals. These data provide Gruneisen coefficients
and other thermo-mechanical constants, in particular, cp, c0, and ρ0. However, an important
parameter in (17), responsible for degree of ‘cold’ compression, α0 is not known (for copper it
has been found earlier and reported for EOS (17) as α0 = 0.96). In order to complete the
derivation, we outline below an algorithm for calculation of α0 and list all the necessary EOS
parameters for a number of metals.
We apply the Hugoniot conditions (21) with EOS to the case of solid material, neglecting
pressure and velocity in front of the wave. In addition, we consider pressures significantly
exceeding the atmospheric pressure p0 and, therefore, we neglect p0 in EOS (23) for the present
derivation as well. This results in the following Hugoniot relation for a metal:
ec (δ ) +
16
p
ρδ
=
1
2
p ⎛⎜ 1 − 1 ⎞⎟ .
ρ⎠
⎝ ρ0
DSTO-TR-2137
Specifying ec(δ) from (23), we can pre-give pressure p and density ρ from an experimental set
of Hugoniot (p, ρ)-pairs from [2] and fit α0 (that enters ec(δ)) to the test data, provided that the
Gruneisen coefficient β, the bulk sound velocity c0 and initial density ρ0 (ρ0s in (23)) are taken
from [2].
Table 1
The EOS data for a number of metals
Metal
Co
Cr
Cu
Fe
Mg
Mo
Ni
Pb
Ta
W
ρ0, g/cm3
8.82
7.18
8.93
7.84
1.735
10.2
8.87
11.34
16.38
19.17
c0, km/s
4.75
5.22
3.9395
4.6336
4.45
5.42
4.65
1.936
3.4
4.0
cv, J/(g·ºC)
0.4145
0.272
0.45
0.45
1.0467
0.256
0.44
0.126
0.1424
0.13
α0
0.6
0.9
0.96
0.635
0.8
0.35
0.84
0.91
0.54
0.6
β
1.99
1.5
1.99
1.6723
1.46
1.58
1.91
2.775
1.689
1.54
Conducting this procedure for a number of metals we summarise the thermo-mechanical
constants for the metal phase EOS (17) in Table 1, where the Gruneisen coefficient, the bulk
sound velocity and thermal capacity are taken from [2]. These constants are used for
calculation of partial Hugoniots for various metal powders at different porosities, similarly to
the calculations for copper powder, results of which are drawn in Fig. 3.
5. Results of Hugoniot Analysis and Experiments
The present Section describes results of Hugoniot analysis for two metal powders.
We select the copper and lead powders for the consideration, because experimental data for
these porous metals provide the points, pressure at which essentially exceeds the alignment
pressure (a result of intersection of PE and PTE Hugoniots), mentioned earlier. For many
other porous metals, data for which are available in literature (e.g., see [1]), the experimental
data above the alignment pressure are scarce and either the relevant points are just isolated
occasions or these points are obtained for the powders with porosities close to 1; the latter
does not allow one to separate the data between the partial PE and PTE Hugoniots with
confidence. Copper powders have been extensively studied in a number of publications for a
fairly wide range of pressures including pressures above the 20-40 GPa (e.g., see [1, 3, 16-18]).
This 20-40 GPa pressure range covers a value of the alignment pressure calculated normally
for the porous copper as an intersection of the PE and PTE Hugoniots. On the other hand, the
alignment pressure is quite low for the lead porous metals; the lead powders have been
studied in detail in [1, 17].
Results of the Hugoniot analysis along with the test data for porous copper and lead are
shown in Figs. 5-9. The marking of curves in figures of the present Section by letters NE, PE,
17
DSTO-TR-2137
and PTE refers to NE (Non-Equilibrium), PE (Pressure Equilibrium), and PTE (PressureTemperature Equilibrium) partial Hugoniots, respectively. An interval ER drawn at every plot
indicates the 5%-density variation for corresponding material and gives an idea of possible
data scatter due to experimental errors.
Figure 5
Partial Hugoniots for porous copper; (a) – m = 1.412; (b) – m = 2.0; (c) – m = 2.5
Fig. 5 describes data for porous copper at moderate porosities (for m from 1.4 up to 2.5).
Crosses are experimental points: the test data m = 1.412 in Fig. 5(a) are from [16, 18]; the test
data m = 2.0 in Fig. 5(b) from [16-19], and the test data m = 2.5 in Fig. 5(c) from [17]. It should
be noted that the data from [16, 18] are not in full agreement with the porosity indicated for
each diagram; they have a slight porosity variation. For example, an obviously deviating point
at p ≈ 20GPa in Fig. 5(b) (the porosity m = 2.0 means ρ00 = 4.465 g/cm3), which was reported in
[16], has actually been obtained for a porous copper with ρ00 = 4.533 g/cm3. Such the
variations may cause some of the deviations occurring in the test data distribution.
Figure 6
18
Partial Hugoniots for porous copper; (a) – m = 3.0; (b) – m = 3.5; (c) – m = 4.0
DSTO-TR-2137
Similarly, results are outlined in Fig. 6 for relatively high porosities of copper powders (from
m = 3.0 up to m = 4.0). The test data m = 3.0 in Fig. 6(a) are taken from [16-18, 20-22], the test
data m = 3.5 in Fig. 6(b) are from [17], and the data m = 4 in Fig. 6(c) from [16, 17, 22-23].
Detailed review of the test data available in literature for copper powders with porosity m = 4
can also be found in [10].
Figure 7
Partial Hugoniots for porous copper; (a) – m = 5.45; (b) – m = 7.2
The Hugoniot results along with the experimental data for very high porosities of copper
powders (for m = 5.45 and m =7.2) are shown in Fig. 7. These test data have been obtained in
[17].
Figure 8
Partial Hugoniots for porous copper; m = 10.034
19
DSTO-TR-2137
Finalising the Hugoniot analysis for the porous copper, results for an extreme porosity of m =
10.034 are drawn in the diagram of Fig. 8. The test data for this porosity are taken from paper
[24].
The overall trend in Figs. 5-8 for the distribution of the porous copper test data around partial
Hugoniots is in agreement with the analysis of previous Section. The data at pressures below
20-40GPa (the alignment pressure level) are clearly separated from the data above
corresponding alignment points. Thus, the transition from PTE to PE Hugoniot is likely to
take place.
Figure 9
Partial Hugoniots for porous lead; (a) – m = 1.35; (b) – m = 1.67; (c) – m = 2.41
Similarly, analysing porous lead, the Hugoniot results and corresponding test data are
summarised in Fig. 9. The porosity values might look fairly moderate (from m = 1.35 up to m =
2.41). However, due to a high molar weight of the material the present porosities are of a wide
range, so the porosity m = 2.41 is considered to be high for this material. The experimental
data for the lead powders are taken from paper [17]. Due to very high molar weight the
anomalous branch is achieved within this porosity range and the alignment pressure is very
low (lower than 10 GPa). As a result, data points in the low-pressure region (below the
alignment points) is very hard to attribute to a specific partial Hugoniot. Nevertheless,
concentration of the points around the partial PE Hugoniot is obvious. It is interesting to note
that due to the material specific thermo-mechanical characteristics the material exhibits fairly
unusual behaviour of the PTE Hugoniots. Overall, results in Fig. 9 demonstrate a quite high
likelihood of the PTE-PE transition of the material states within the shock wave
transformation for this powder as well.
6. Conclusions
The report has demonstrated that the description of the behaviour of metal powders with
partial Hugoniots could substantiate the anomalous behaviour of the porous metals at high
20
DSTO-TR-2137
porosities. In doing so, this behaviour is a consequence of the non-equilibrium transitions
within the shock wave. Stability of this transition structure within a shock wave should be
specifically confirmed for each porous material of interest. This wave structure and associated
transition configuration may depend on the material structure influencing equilibrium of
various thermodynamic phase parameters at the transition stages. The Non-Equilibrium,
Pressure-Equilibrium and Pressure-Temperature-Equilibrium transition structure was clearly
observed for porous copper with the help of numerical simulation, using the present twophase material model. Corresponding Hugoniot analysis and comparison with available
experimental data have confirmed this transition structure. At the same time, the comparisons
and analyses have emphasised that the full equilibrium (that must be achieved for the classic
Hugoniot) is unlikely to be achieved for a shock wave recorded in experiments as the time of
arrival information. A trend for this transition structure to take place for porous lead has also
been indicated with the Hugoniot analysis and comparison with experiments. The importance
of the transition structure within the shock wave transformation results in an importance of
more complete shock behaviour information such as the velocity or stress profiles recorded by
continuous-record gauges; for discussion on this issue address to [10, 25].
7. References
1. Trunin R.F., Shock Compression of Condensed Materials, Cambridge University Press,
Cambridge, UK, 1998.
2. Walsh J.M., Rice M.H., McQueen R.G., and Yarger F.L., Shock-Wave Compressions of
Twenty-Seven Metals. Equations of State of Metals, Phys. Rev., 1957, v. 108, n. 2, pp.
196-216.
3. Trunin R.F., Shock compressibility of condensed materials in strong shock waves
generated by underground nuclear explosions, Physics-Uspekhi, 1994, v. 37, n. 11, pp.
1123-1145.
4. Nellis W.J., Moriarty J.A., Mitchell A.C., Ross M., Dandrea R.G., Ashcroft N.W.,
Holmes N.C., and Gathers G.R., Metals Physics at Ultrahigh Pressure: Aluminum,
Copper, and Lead as Prototypes, Phys. Rev. Lett., 1988, v. 60, n. 14, pp. 1414-1417.
5. Boade R.R., Compression of Porous Copper by Shock Waves, J. Appl. Physics, 1968, v.
39, n. 12, pp. 5693-5702.
6. Boade R.R., Principle Hugoniot, Second-Shock Hugoniot, and Release Behavior, of
Pressed Copper Powder, J. Appl. Physics, 1970, v. 41, n. 11, pp. 4542-4551.
7. Linde R.K., Seaman L., and Schmidt D.N., Shock response of porous copper, iron,
tungsten, and polyurethane, J. Appl. Physics, 1972, v. 43, n. 8, pp. 3367-3375.
8. Butcher B.M., Carroll M.M., and Holt A.C., Shock-wave compaction of porous
aluminum, J. Appl. Phys., v. 45, n. 9, 1974, pp. 3864-3875.
9. Resnyansky A.D. and Bourne N.K., Shock-wave compression of a porous material, J
Appl. Physics, 2004, v. 95, n. 4, pp. 1760-1769.
10. Resnyansky A.D., Constitutive Modelling of the Shock Behaviour of a Highly Porous
Material, DSTO Report DSTO-TR-2026, Edinburgh, Australia, 2007.
11. Resnyansky A.D., A Thermodynamically Complete Model for One-Dimensional TwoPhase Flows With Heat Exchange, DSTO Report DSTO-TR-1862, Edinburgh, Australia,
2006.
21
DSTO-TR-2137
12. Alazmi B. and Vafai K., Analysis of Variants Within the Porous Media Transport
Models, J. Heat Transfer, v. 122, 2000, pp. 303-326.
13. Rees D. A. S. and Pop I., Local thermal nonequilibrium in porous medium convection,
in Transport Phenomena in Porous Media, Vol. III, edited by D. B. Ingham and I. Pop
(Elsevier, Oxford, 2005), pp. 147-173.
14. Dijken D. K. and De Hosson J. Th. M., Thermodynamic model of the compaction of
powder materials by shock waves, J Appl. Physics, 1994, v. 75, n. 1, pp. 203-209.
15. Linde R.K. and Schmidt D.N., Shock Propagation in Nonreactive Porous Solids, J.
Appl. Physics, 1966, v. 37, n. 8, pp. 3259-3271.
16. Marsh S.P. (Ed.), LASL Shock Hugoniot Data, University of California Press, Berkley
and Los Angeles, California, 1980.
17. Trunin R.F., Simakov G.V., Sutulov Yu.N., Medvedev A.B., Rogozkin B.D., and
Fedorov Yu.E., Compressibility of porous metals in shock waves, Soviet Physics JETP,
1989, v. 69, n. 3, pp. 580-588.
18. van Thiel M., Shaner J., and Salinas E. (Eds.), Compendium of Shock Wave Data,
Lawrence Livermore Laboratory Report UCRL-50108, University of California,
Livermore, Vol. 1, 1977.
19. Al'tshuler L. V., Bushman A. V., Zhernokletov M. V., Zubarev V. N., Leont'ev A. A.,
and Fortov V. E., Unload isentropes and equation of state of metals at high energy
densities, Soviet Physics JETP, 1980, v. 51, n. 2, pp. 373-383.
20. Alekseev Yu. L., Ratnikov V. P. and Rybakov A. P., Shock adiabats of porous metals, J.
Appl. Mech. and Techn. Phys., 1971, v. 12, n. 2, pp. 257-262.
21. Bakanova A. A., Dudoladov I. P., and Sutulov Yu. N., Shock compressibility of porous
tungsten, molybdenum, copper, and aluminum in the low pressure domain, J. Appl.
Mech. and Techn. Phys., 1974, v. 15, n. 2, pp. 241-245.
22. Trunin R.F., Medvedev A.B., Funtikov A.I., Podurets M.A., Simakov G.V., and
Sevast’yanov A.G., Shock compression of porous iron, copper, and tungsten, and their
equation of state in the terapascal pressure range, Soviet Physics JETP, 1989, v. 68, n. 2,
pp. 356-361.
23. Kormer S.B., Funtikov A.I., Urlin V.D., and Kolesnikova A.N., Dynamic compression
of porous metals and the equation of state with variable specific heat at high
temperatures, Soviet Physics JETP, 1962, v. 15, n. 3, pp. 477-488.
24. Gryaznov V. K., Fortov V. E., Zhernokletov M. V., Simakov G. V., Trunin R. F., Trusov
L. I., and Iosilevski I. L., Shock compression and thermodynamics of highly nonideal
metallic plasma, Soviet Physics JETP, 1998, v. 87, n. 4, pp. 678-690.
25. Resnyansky A.D., Constitutive Modelling of Shock Compression of a Porous Copper,
In: Shock Compression of Condensed Matter - 2007, edited by M. Elert, M.D. Furnish,
R. Chau, N. Holmes, and J. Nguyen, (AIP, New York, 2007), CP955, 2007, pp. 93-96.
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2. TITLE
Hugoniots of Porous Materials at Non-Equilibrium Conditions
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Title
Abstract
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4. AUTHOR(S)
5. CORPORATE AUTHOR
A.D. Resnyansky
DSTO Defence Science and Technology Organisation
PO Box 1500
Edinburgh South Australia 5111 Australia
6a. DSTO NUMBER
6b. AR NUMBER
6c. TYPE OF REPORT
7. DOCUMENT DATE
DSTO-TR-2137
AR-014-201
Technical Report
June 2008
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9. TASK NUMBER
10. TASK SPONSOR
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12. NO. OF REFERENCES
2008/1051263/1
LRR 07/249
DSTO
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13. URL on the World Wide Web
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http://www.dsto.defence.gov.au/corporate/reports/DSTOTR-2137.pdf
Chief, Weapons Systems Division
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17. CITATION IN OTHER DOCUMENTS
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18. DSTO RESEARCH LIBRARY THESAURUS http://web-vic.dsto.defence.gov.au/workareas/library/resources/dsto_thesaurus.shtml
Two-phase modelling; shock physics; porous materials
19. ABSTRACT
This report is establishing the multi-phase modelling capability for evaluation of mitigants against blast and fragmentation. The work
analyses states of a highly porous material under shock compression. A two-phase model accounting for the inter-phase heat transfer is
suggested and employed. Numerical analysis of wave structure in a porous copper is conducted and a simplified method of Hugoniot
analysis is developed, which is associated with specification of equilibrium states within the shock transformation. Three types of inter-phase
equilibrium within the shock transformation are distinguished. This state classification results in different loci that are seen as Hugoniots
when certain thermodynamic parameters are in equilibrium; parts of the loci form a composite Hugoniot curve. Corresponding composite
Hugoniots demonstrate a good correlation with available experimental data for porous metals.
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