Challenges in Understanding the Dynamic Behavior of Heterogeneous Materials
379
Fig. 7 Consider the stress
amplitude plotted over the
distance through a continuous
medium. After a sufficiently
large and rapid application of
load on a material, a shock
wave will develop which is a
traveling discontinuity in
thermodynamic state
variables. The state variables
behind the shock wave
depend on the conservation
relations and an equation of
state. (Adapted from [25])
2.2 Conservation Relations for a Shock Wave
The conservation of mass, momentum, and energy can be derived by considering
infinitesimal elements and matter/momentum/energy exchange therein, to formulate
the strong form of the equations. For the conservation of mass, influx and efflux of
matter in an infinitesimal volume lead to:
[ρ(x, t) + ∇ · (ρv)] dd = 0
∀dd.
(18)
This expression is valid for all possible integration volumes and can thus only be
true if the integrand is exactly zero, i.e.,
∂ρ
∂t
+ ∇ · (ρv) = 0,
(19)
or in indicial notation:
∂ρ
∂t
+
∂(ρv i )
∂x i
= 0,
(20)
where the density of the material ρ is a continuous function of space and time –
ρ = ρ(x, y, z, t) – and v is the spatial velocity of the medium. 2
In the case of a discontinuous change in state variables, the continuity equation (19) must take into account a moving discontinuity at velocity U s relative to
the disturbed material moving at a different material velocity U p which propagates
2 More details on the difference between material and spatial coordinates can be found in Malvern’s
excellent text [50] on continuum mechanics.
379
Fig. 7 Consider the stress
amplitude plotted over the
distance through a continuous
medium. After a sufficiently
large and rapid application of
load on a material, a shock
wave will develop which is a
traveling discontinuity in
thermodynamic state
variables. The state variables
behind the shock wave
depend on the conservation
relations and an equation of
state. (Adapted from [25])
2.2 Conservation Relations for a Shock Wave
The conservation of mass, momentum, and energy can be derived by considering
infinitesimal elements and matter/momentum/energy exchange therein, to formulate
the strong form of the equations. For the conservation of mass, influx and efflux of
matter in an infinitesimal volume lead to:
[ρ(x, t) + ∇ · (ρv)] dd = 0
∀dd.
(18)
This expression is valid for all possible integration volumes and can thus only be
true if the integrand is exactly zero, i.e.,
∂ρ
∂t
+ ∇ · (ρv) = 0,
(19)
or in indicial notation:
∂ρ
∂t
+
∂(ρv i )
∂x i
= 0,
(20)
where the density of the material ρ is a continuous function of space and time –
ρ = ρ(x, y, z, t) – and v is the spatial velocity of the medium. 2
In the case of a discontinuous change in state variables, the continuity equation (19) must take into account a moving discontinuity at velocity U s relative to
the disturbed material moving at a different material velocity U p which propagates
2 More details on the difference between material and spatial coordinates can be found in Malvern’s
excellent text [50] on continuum mechanics.