p-p„=p„u,u^
2.7 Graphical representation of EOS data
T he shock EOS data for m aterials can be represented graphically in various equivalent ways; the three m ost useful plots are Shock V elocity- P article V elocity {U ^ ~ ^ p )’ P ressure- Particle V elocity { P - U p ) and Pressure- Specific volum e ( P - V ) graphs. A H ugoniot EOS is not the loading path but rather the locus o f points achieved via shock com pression from a given initial state (M elosh, 1989). The loading path is know n as the R ayleigh line and is a straight line connecting initial and com pressed states (figure 2.11). In principle, a reflected shock H ugoniot, starting from a point on the single shock H ugoniot, differs from the single shock H ugoniot starting from am bient pressure. H ow ever, reflected shock data are often used, w ith or w ithout correction, to approxim ate single shock H ugoniot points. U^ — U p plots directly show the
shock EOS data measured experim entally and for most materials the relationship is linear, unless a phase transform ation occurs, and can be fitted to (equation 2.4).
H u g o n i o t E la s tic • L i m i t , ela stic etastic Up
Figure 2.9, V - Up plot demonstrating the linear relationship between the shock wave velocity and the particle velocity of a material (right side of graph), modified after Melosh, 1989.
P — U ^ plots o f the shock EOS data dem onstrate the shock impedance o f a material, where those m aterials with high shock im pedances are represented by steeper curves, and those with lower shock impedances by shallow curves (figure 2.8). The shock impedance o f a material is a function o f density and shock wave velocity and is described by equation 2.5:
i = p U ^ equation 2.5
where;
i = shock impedance
p = density
U = shock wave velocity
As a general rule, denser m aterials have higher shock im pedances, but there are exceptions. Very ‘s tif f low-density materials like diam ond and sapphire have very high shock velocities, and som e soft dense materials like lead have very low shock velocities. Higher shock pressures and lower particle velocities will be produced in materials with higher shock im pedances com pared to those with lower shock impedances for the same projectile velocity
(figure 2.10).
Pressure in high shock impedance material
Pressure in low shock impedance material
Projectile velocity
velocity. The curve representing the projectile in this diagram is green (— ).
Figure 2.10, In P ~ U p plots materials with higher shock impedances (— ) have steeper curves than materials with lower shock impedances (__ ). Higher peak pressures and lower particle velocities are produced in materials with higher shock and particle velocities produced in materials with lower shock impedances, for the same projectile
The calculation o f the pressure produced by the impact o f a flyer plate on a target is achieved by sim ple graphical calculations in the P-Up plane, as illustrated in fig u re 2.10. At
impact, the pressure and particle velocity must be equal in both flyer plate and target and the Hugoniots o f flyer plate and target must be known. \n fig u re 2.10, the Hugoniots o f high- and low-im pedance targets are plotted normally. The m irror image o f the high-im pedance flyer plate is plotted with its origin at the flyer plate impact velocity. In an impact with a low- impedance target, the projectile velocity is slightly decreased and the particle velocity o f the target Jumps from zero to a very high value. In the case o f the impact with high impedance target, there is a greater decrease in projectile velocity and a higher pressure. In the case o f symmetrical impact i.e. when the flyer plate and target are com posed o f the same material, the particle velocity in target and flyer would be half the impact velocity.
The Hugoniot curve for the sam ple material can be represented on U,. - U p , P - V and
P - U ^ plots as well as other equivalent representations o f the EOS data, including P - U ^ ,
and, as long as the initial state o f the material (density etc.) is known, any o f the plots may be calculated from the others by rearranging the Hugoniot equations.
The EOS data o f a material plotted on a P - V graph maybe used to calculate the net internal energy increase (figure 2.11).
" t
0
Figure 2.11, The Hugoniot curve (---- ) on a F -T plot represents Hugoniot cur\'e
ayleigh line a loci o f final states achieved in a porous material from specific Release adiabat shock conditions. The release adiabat (--- ) represents a Recoverable energy continuous pathway followed by the material, and varies Irreversible energy depending on the shocked state the material reached. A line connecting the initial, pre-shocked state {Po, Vo) and the shocked state {P, V) is the Rayleigh line (— ) and represents the shock wave loading path. The total internal energy increase is represented by the area under the Rayleigh Line, and the irreversible energy or waste heat is represented by the area between the Rayleigh line and the release adiabat ( ) and the recoverable energy is the area under the release adiabat (0). This example represents a porous material that crushes irreversibly, which results in the release adiabat being steeper than the Hugoniot curve. In initally non-porous materials that undergo no phase transitions the release adiabat is shallower than (i.e. lies to the right of) the Hugoniot because o f thermal expansion.
The P- V graph dem onstrates that the specific volum e (V) decreases (i.e. density increases) with increasing peak pressure. The release adiabat o f a material is also represented in the P - V plot, and unlike the Hugoniot curve, the curve it describes is a continuous path. For shocked materials that have not undergone a phase transform ation the non-porous Hugoniot curve is a very good approxim ation o f the release adiabat. For m aterials that have transform ed to another phase the release adiabat changes due to the difference in volume between the different phases, this can be seen \n fig u re 2.12 (Ahrens, 1968) in which the release ad i a bats for silica shocked to various peak pressures were measured experim entally.
O 20
SPECIFIC VOLUME,
Figure 2.12, An example o f a P - V plot including measured released adiabats for a fine grained polycrystalline quartz (Arkansas Novaculite) and single crystal quartz, reproduced from Ahrens, 1968.
The R ayleigh line links the initial, p re-sh ock co n d itio n s ( / q and Vq) to the sh ock ed
state o f the m aterial ( P and V ) and the area under the R ayleigh line is equal to the total internal en ergy increase. A portion o f this en ergy is d issip ated as w a ste heat, h en ce sh ock co m p ressio n is irreversible. T h e area b etw een the R ayleigh line and the release adiabat represents an ap proxim ation o f this w aste heat or net internal en ergy increase.
2.8 C alculating shock conditions
For the ex p erim en ts in this study w e calcu lated the sh ock co n d itio n s in our sa m p les and represented them grap h ically. T h e sa m p les o f porous quartz w ere held in con tain ers c o m p o sed o f various m aterials during the sh ock e v e n t (fu ll d escrip tion o f exp erim en t d esig n in C h a p te r i ) . O ne o f the ex p erim en ts w a s m atched im p ed an ce, the rest w ere m ism atch ed im pedan ce exp erim en ts. T h e term m atched im p ed an ce in dicates that the con tainer m aterial and the sam p le
m aterial have the sam e sh ock im p ed an ce, i.e. they share the sam e cu rve on a P - J J ^ graph
(figure 2.13).
P Figure 2.13, The term matched impedance experiments refers to the
match in shock impedances between the container material (— ) and the sample material (— ), represented by the container material and the sample material having the same or very similar curves on a
P - U ^ graph.
T h is m ean s that sh ock co m p re ssio n w ill produce the sam e pressures and particle v e lo c itie s in both m aterials via a sin g le , direct sh ock w a v e. In e s s e n c e , the sh o ck w a v e 62
travelling through the container and sample behaves as if it is travelling though a single uninterrupted material (figure 2.14). In this case the sample is described as having a direct loading path, i.e. the path o f the shock wave through the sam ple that loaded it to the peak shock pressure was not reflected, but direct and the sample reached the peak shock pressure by loading directly from its am bient conditions. This is illustrated graphically in fig u re 2.15 using a plot o f pressure (P) against time (t).
.Flyer plate Figure 2.14, Sketch o f cross section through the flyer plate, .Flyer plate- sample container and sample illustrating a direct loading path,
interface Conditions at point A are described graphically in fig u r e 2.15. Sample The sample is instantaneously loaded from its ambient conditions Container to the peak shock pressure induced at the impact interface by the ~ arrival o f the first shock wave. The dashed lines indicate that the flyer and sample assembly would be much wider in a real experiment to prevent reflections from the edges from affecting the sample.
F
t
Figure 2.15, P-t plot demonstrating the change in conditions at point A (figure 2.14) in a sample shocked via a direct loading path. The pressure directly increases to the peak pressure induced at the impact interface from the ambient pre-shock conditions o f the sample.
M ism atched-im pedance experim ents are those in which the container material has a different shock impedance to the sample material (figure 2.16).
Figure 2.16, Mismatched impedance experiments are those in which the container material (___) and the sample material (— ) have different shock impedances, represented here by their different curves on a P - U p plot. Generally in mismatched experiments the container
material has a greater shock impedance than the sample material.
This mismatch in shock im pedance results in the shock wave being m odified at the interface between the container material and the sample material because o f the requirem ent that
pressure and particle velocity m ust both be continuous across the interface betw een tw o m aterials. I f the m aterials have different H ugoniots, i.e. different pressure vs. particle velocity relationships, an adjustm ent m ust take place. W hen the shock w ave reaches the interface betw een high-im pedance and low im pedance m aterials, a shock is transm itted into the low- im pedance m aterial and a rarefaction is transm itted back into the high-im pedance m aterial resulting in a decrease o f pressure. W hen the shock arrives at an interface betw een low- im pedance and high-im pedance m aterials, a shock is transm itted into the high-im pedance m aterial and a reflection is transm itted back into the low -im pedance m aterial, resulting in an increase o f pressure. In the usual situation o f a low -im pedance sam ple in a high-im pedance container, the shock w ave is reflected back and forth through the sam ple, w ith the pressure produced in the sam ple increasing w ith each reflection until the peak shock pressure, i.e. the pressure at the projectile- container interface, is reached (figure 2.21). T hese experim ents are described as having a reflected loading path because the peak pressure is reached in the sam ple via a num ber o f reflections o f the shock w ave. M ism atched im pedance experim ents are also referred to as reverberation technique experim ents because o f the ‘bouncing’ back and forth o f the shock w ave through the sam ple. R eflected loading path can be described as indirect as they do not load the sam ple directly from am bient conditions to the peak shock pressure, but the sam ple is shocked via a num ber o f interm ediate states (figure 2.17).
M ism atched im pedance experim ents are the m ost com m only used in shock recovery experim ents on m inerals and rocks. There are tw o reasons fo r their prevalence. Firstly, until recently the m axim um velocities o f accelerated flat plates w ere about ~7 km /sec. By using high im pedance container m aterials, for exam ple stainless steel or copper, w ith a high im pedance projectile m aterial, the peak pressures produced are m uch higher in both the container and, after the reflections o f the shock w ave, in the sam ple. T hus the range o f available peak pressures is increased. T he second reason is that it allow s an easier sam ple recovery. Stainless steel and copper, the m ost com m only used container m aterials are quite ductile and instead o f shattering or m elting on im pact, they deform but rem ain coherent. T hus w hen they are used as a container m aterial the com plete container m ay be rem oved from the target cham ber post-shock and the sam ple rem oved.
Figure 2.17, P-t plot showing the pressure conditions induced in the sample in figures 2.23 and 2.21. The reflected loading path
induces intermediate pressure conditions (Pi, P 2 and P 3 ) in the
sample, before reaching the peak shock pressure P.,. The release path is not shown in this sketch because it can be very complex and depends on the details of flyer properties and thickness, container properties and thickness, compressed sample properties and thickness, where properties include release adiabats.
The term matched impedance experim ents should not be confused with the term ‘impedance m atching experim ents’, which has been used by some authors to refer to an experimental technique in which a ‘long’ sam ple is loaded in a m ismatched, high impedance container (Langenhorst, 1993; Deutsch, pers. com., 2001). ‘Im pedance m atching’ experim ents are designed so that the first shock wave produced in the sam ple is released by the rarefaction wave produced in the flyer plate before the first reflection occurs, so the final pressure produced in the sample is lower than the pressure at the projectile- container interface (figure 2.18).
Flyer plate Sample material Container F igure 2.18, Sketch o f a cross-section through the flyer plate ( ), sample ( ) and container ( ) in an “impedance matching” experiment. The container material has a mismatched shock impedance with the sample. The sample is long enough to ensure that the release wave (— ) reachs the shock front (— ) before it reaches the downstream sample-container interface, so that the pressure is released before the first reflect can occur. The sketch is simplified and does not illustrate the compression o f the flyer plate and target and the acceleration o f the impact interface that would occur.
2.9 G raphical Calculation of Peak Shock Pressure
The peak shock pressure is the maximum pressure produced in a material by a shock wave. Only the material that the shock wave passes through before the release waves reach the advancing shock wave will experience the peak shock pressure. It is a function o f the velocity o f the projectile and the shock impedance o f the m aterials either side o f the impact surface.
Prior to the developm ent o f modern com puters and appropriate wave propagation codes, simple graphical methods were used to make shock wave calculations. Pressure vs. particle velocity Hugoniots were plotted on transparent media and m anipulated like a slide rule to determine im pedance match conditions. (DeCarli, pers. com., 1999). Today, there are a variety o f one-dim ensional wave propagation codes that can be used to determ ine im pedance match conditions. N evertheless, many shock physicists continue to use the graphical m ethods because they are convenient and ju st as accurate as the code calculations. Accuracy is limited by the accuracy o f the experim ental Hugoniot data rather than by errors in the graphical methods them selves, which have been improved by adaptation to the spreadsheet.
Since Hugoniot data for most materials o f interest are available in the form o f shock velocity vs. particle velocity relationships, the spreadsheet is a useful tool for transform ing the Hugoniot into its various forms (figure 2.19).
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The fill comm and is used to enter the particle velocities in column A, beginning with zero and increasing by increments o f 0.01 km/s up to the maximum value o f interest. The shock velocity is calculated in column B, using the fit to Hugoniot data;
Pressure is calculated in column C, using the relationship:
^ - P o ^ s ^ p
(V o- y) may be calculated from the relationship: