High Dynamic Testing of Materials
Elastic Waves and Shock Waves
Dr.-Ing. Erhardt Lach
French-German Research Institute of Saint-Louis, ISL
[email protected]
1
On the myth of plastic waves
John J. Gilman has been advocating the view that „plastic waves“ cannot exist any
more than opacity waves or resistance waves can. He attributes the error to the
early pioneers such as Taylor, von Karman and Duwez. Since plastic deformation is
dissipative, such waves cannot exist. Mathematically, the wave equation for elastic
shear strain in one dimension
ρ (∂2u / ∂t2) = G ▼2 u
is differentiable on the right side and G is replaced by a „plastic modulus“. A plastic
modulus does not exist for plastic flow in terms of external variables. J. J. Gilman
argues that dislocations are the elementary carrier of plastic deformation and, so,
plastic deformation is a transport process. If this conjecture is right, there should be
a transport quantity that gives a more transparent concept of dynamic plastic
deformation. Y. Horie1) proved that the plastic deformation is characterized by a
diffusion equation rather than a wave equation.
1)
Y. Horie, On the myth of plastic waves, J. Phys. IV France 110 (2003), p. 3-7
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Types of Elastic Waves
Different types of elastic waves can propagate in solids, depending on how the
motion of particles of the solid is related to the direction of propagation of the
waves itself and on boundary conditions. A particle can be defined as a small
discrete portion of the solid. It is not an atom.
A) Longitudinal waves (in infinite or semi-infinite media, they are known as
dilatational waves)
B) Distortional (shear or transverse) waves
C) Surface (or Rayleigh) waves
D) Bending (or flexural) waves ( in bars or plates)
E) Interfacial (Stoneley) waves
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Bounded medium
Hammer impacting slender body
Unbounded medium
Hammer impacting semi-infinite body;
formation of longitudinal, shear and
Rayleigh waves. Interaction of
longitudinal wave with free-surface
wavelets that comprise head wave.
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A. Longitudinal waves: These waves correspond to the motion of the
particles back and forth along the direction of wave propagation. Particle
velocity UP is parallel to the wave velocity U. If the wave is compressive,
they have the same sense. If it is tensile, they have opposite senses.
E
Equation c0 =
medium (bar).
ρ
gives the velocity of the wave in a bounded
In an unbounded media (4), the longitudinal wave possesses a general
shape and is travelling at a velocity:
 λ + 2µ 

Vlong = 
ρ


1
2
It is also called dilatational wave. The velocity is often called bulk
sound speed.
From elasticity theory:
one assumes
υ
µ = E /[2(1 + υ )] and λ = υE /[(1 +ν )(1 − 2υ )]
= 0.3, then:
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1
2
Vlong

(1 − υ )E  =  1.346 E 
=

 ρ 


 (1 + υ )(1 − 2υ )ρ 
1
2
The longitudinal wave is faster in an unbounded (Vlong) or infinite medium
1.2 than in a bounded (c0) or finite medium.
B. Distortional (or shear) waves: If the motion of the particles conveying the
wave are perpendicular to the direction of the wave’s propagation, it is a
distortional wave.
C. Surface waves: They are analogous to waves on the surface of water.
This type of wave is restricted to the region adjacent to the surface.
Particle velocity UP decreases very rapidly.
D. Bending waves: These waves involve propagation of flexure in a onedimensional (bar) or two-dimensional configuration.
E. Interfacial waves: If two semi-infinite media with different properties are
in contact, special waves form at their interface.
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Incident wave of a SHPB
σ = ρ c0 VP / 2
t = 2LP / c0
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Shock Waves
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Definition of Shock Waves
uS2
Compression impuls
a) weak shock wave
b) strong shock wave
Pulse duration
A weak shock wave is
characterized by a
decreasing particle
velocity. US2 is smaller
than US1
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Definition of Shock Waves
Release
wave
shock front
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Shock front
Pressure P
Volume V = 1/ρ
Particle velocity
UP
Specific intrinsic
energy E
Behind the
shock front
In front of the
shock front
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Hugoniot equation:
U = V0 ((P – P0)/(V0 – V))1/2
u = ((P – P0) (V0 – V))1/2
U = shock velocity
u = particle velocity
P = pressure
V = specific volume
LACH E., NAHME H., Die Eigenschaften stickstofflegierter austenitischer Stähle unter hoher dynamischer Druckbelastung ,
Werkstoffwoche 1998, München, BRD, 12. - 15. 10. 1998
NAHME H., LACH E., Determination of the Mechanical Behavior of Nitrogen Alloyed Steel (P900) at Strain Rates 10-3 s-1 <
de/dt < 2*106 s-1, Congress on Shock Compression of Condensed Matter, Snowbird/UT, USA, June 29 - July 30 1999
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Realistic shock profile1)
1)Grady
D. E., Dynamic Material Properties of Armor Ceramic , Report No.
SAND 91-0147.4C-704, Sandia National Laboratory, 1991
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U
specimen
flyer plate
spall plate
time
SHOCK WAVE
HARDENING OF
MATERIALS
release wave
release
wave
shock wave
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Confinement
Detonator
Flyer plate
Explosive
Specimen
Cover plate
Momentum trap
Spallation plate
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Propagation of Shock Waves in Materials
the release wave is faster than the shock wave
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strain:
ε = 2/3 ln V1/V0
ε = 4/3 ln V1/V0
shock wave + release wave
shock wave pressure 100 kbar:
compression: V1 = 0.95 V0
ε = -3.4%
ε = 34x106 s-1
ε = n b v
n = density of dislocations b = Burgers vector
v = velocity of dislocations
assumption:
n = 108 /cm2
v < c0
max.: ε = 7.5 105 s-1
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Model of a super sonic edge dislocation
glide direction
glide plane
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Progress of shock front
according to
homogeneous dislocation
nucleation model
Direction of
the front
Meyers M. A., Dynamic Behavior of Materials, John Wiley &
Sons, ISBN-13: 978-0471582625
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Shock wave hardening of polycrystalline
materials
Shock wave pressure
Shock wave duration
Material
Strain rate
Temperature
Micro structure
Low stacking fault
energy
High stacking fault
energy
Point defects
Dislocation
Mechanical twinning
Dislocation cells
Phase transformation
Martensite
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Temperature increase
shock temperature
retained
temperature
Development of
shock and
retained
temperature with
increasing shock
wave pressure
Shock wave pressure, kBar
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Phases of Iron
α-Fe =
room temperature
γ-Fe =
high temperature
ε-Fe =
high pressure
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Vickers hardness HV30
Hadfield steel
C steel
austenitic steel
Shock wave pressure, kBar
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Schematic presentation of
shock-induced substructures
in FCC metals as a function
of shock pressure and SFE1).
1)
Murr L. E., in: Shock Waves
in Condensed Matter, eds.:
Schmidt S. C. and Holmes N.
C., Elsevier, Amsterdam,
1988, p. 315
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TRUE STRESS
3000
MPa
2500
P900
5
4
2000
3
1500
2
1:
2:
3:
4:
5:
1
1000
500
solution annealed
shock wave hardened, 345 HV30
shock wave hardened, 456 HV30
rolled at 400 °C , 479 HV30
cold rolled, 461 HV30
0
0,0
0,1
0,2
0,3
0,4
TRUE STRAIN
0,5
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Development of micro structure of
P900 according to shock wave
pressure.
a) Solution annealed,
Shock wave pressure b) 100 kBar,
c) 300 kBar and d) 500 kBar.
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Recrystallization of a shock wave
hardened specimen of P900 due to
heating by machining. Shock wave
pressure was 300 kBar.
Micro structure near the edge of a
specimen due to shock wave
hardening. The micrograph is showing
crack opened recrystallization twins.
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Effect of shock wave pressure
P900N
P900
500 kBar, 3 µs
100 kBar
3 µs
8 µm
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Effect of shock wave duration
PANACEA
100 kBar, 0.5 µs
100 kBar, 4 µs
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