Challenges in Understanding the Dynamic Behavior of Heterogeneous Materials
381
where the differential δ denotes path dependence, i.e., the total differential of
internal energy E, path independent by definition of a total differential (Pfaffian
form), can be composed of path-dependent changes in heat Q and work W . The
work done by the passage of the shock wave is composed of the force term (P A)
and displacement (Uδt). Using intensive quantities, each term becomes [22, 54]:
= KE 2 − KE 1 =
1
2
[ρA(U s − U p )δt]U
2
p −
1
2
[ρ 0 A(U s − U 0 )δt]U
2
0 , (27)
P U p =
1
2
ρ(U s − U p )U
2
p + Eρ(U s − U p ) − E 0 ρ 0 U s .
(28)
The common form of the conservation of energy in a shock wave is obtained by
modifying this general form noting that the conservation of mass must also apply:
P U p =
1
2
ρ 0 U s U
2
p + ρ 0 U s (E − E 0 ).
(29)
Rearranging the equation by dividing through by ρ 0 U s to isolate the energy term
gives:
E − E 0 =
P U p
ρ 0 U s
−
1
2
ρ 0
U s U 2
p
ρ 0 U s
.
(30)
Invoking the conservation of momentum, Eq. (24) for the U p term gives, after
rearranging (cf. Gonzales [25] for the full derivation):
E − E 0 =
P (P − P 0 )
(P − P 0 )
· (V 0 − V ) −
1
2
(P − P 0 ) 2
P − P 0
· (V 0 − V )
(31)
∴ E − E 0 =
1
2
(P + P 0 )(V 0 − V ),
(32)
which is the commonly used form of the conservation of energy. Equation (32) is
integral to this work, as it is the partition of energy between the compaction of
the loose powder and the shock compression that leads to the complex behavior of
powder mixtures (i.e., the long rise times and dispersed wave fronts) under shock
compression. The conservation equations are completely general, and there are
five variables [54]: pressure P , material (particle) velocity U p , shock velocity U s ,
specific volume V , and energy E. Thus, a fourth equation is necessary for closure.
This equation is in the form of a thermodynamic equation of state (EOS) which
relates state variables, usually U s − U p or some other equation.
The equations presented in this section are completely general and satisfied if a
strictly one-dimensional uniaxial strain loading configuration is maintained during
shock compression. This will allow the bulk compressibility of the material to
stiffen the shock response and increase the pressure to yield the discontinuous
381
where the differential δ denotes path dependence, i.e., the total differential of
internal energy E, path independent by definition of a total differential (Pfaffian
form), can be composed of path-dependent changes in heat Q and work W . The
work done by the passage of the shock wave is composed of the force term (P A)
and displacement (Uδt). Using intensive quantities, each term becomes [22, 54]:
= KE 2 − KE 1 =
1
2
[ρA(U s − U p )δt]U
2
p −
1
2
[ρ 0 A(U s − U 0 )δt]U
2
0 , (27)
P U p =
1
2
ρ(U s − U p )U
2
p + Eρ(U s − U p ) − E 0 ρ 0 U s .
(28)
The common form of the conservation of energy in a shock wave is obtained by
modifying this general form noting that the conservation of mass must also apply:
P U p =
1
2
ρ 0 U s U
2
p + ρ 0 U s (E − E 0 ).
(29)
Rearranging the equation by dividing through by ρ 0 U s to isolate the energy term
gives:
E − E 0 =
P U p
ρ 0 U s
−
1
2
ρ 0
U s U 2
p
ρ 0 U s
.
(30)
Invoking the conservation of momentum, Eq. (24) for the U p term gives, after
rearranging (cf. Gonzales [25] for the full derivation):
E − E 0 =
P (P − P 0 )
(P − P 0 )
· (V 0 − V ) −
1
2
(P − P 0 ) 2
P − P 0
· (V 0 − V )
(31)
∴ E − E 0 =
1
2
(P + P 0 )(V 0 − V ),
(32)
which is the commonly used form of the conservation of energy. Equation (32) is
integral to this work, as it is the partition of energy between the compaction of
the loose powder and the shock compression that leads to the complex behavior of
powder mixtures (i.e., the long rise times and dispersed wave fronts) under shock
compression. The conservation equations are completely general, and there are
five variables [54]: pressure P , material (particle) velocity U p , shock velocity U s ,
specific volume V , and energy E. Thus, a fourth equation is necessary for closure.
This equation is in the form of a thermodynamic equation of state (EOS) which
relates state variables, usually U s − U p or some other equation.
The equations presented in this section are completely general and satisfied if a
strictly one-dimensional uniaxial strain loading configuration is maintained during
shock compression. This will allow the bulk compressibility of the material to
stiffen the shock response and increase the pressure to yield the discontinuous