This is an important topic that we did not discuss earlier, and which involves some important principles.

Transcription

1 Lecture 8 1. Precursor Decay This is an important topic that we did not discuss earlier, and which involves some important principles. Elastic precursors often are observed to decay with target thickness, until a final value is achieved. For example, the figure below shows data for LiF. This phenomenon can make measurement of HEL tricky one must be sure the target is thick enough. FIG. 1. Stress-time profiles in lithium fluoride. Material III-b, after Asay et al. 4 What happens is that on first impact the shock is wholly elastic. It follows Ds /De = K + 4 G. The total strain is 3 e = e e + e P = DV /V where e e is the elastic part and the e P is the plastic part. The stress strain path is

2 s Peak shock stress Ds HEL Slope K + 4/3 G Ds is an overstress. At impact the strain rate is infinite. The behavior of elastic plastic materials is probably more correctly considered in terms of strain rate being a function of stress. Think of it like this: e

3 Plastic strain rate Slope = 1/n Y At low strain rates, the deviator is equal to Y, which is the quasi-static yield stress. When the deviator exceeds Y, it causes the material to flow with a strain rate equal to S/n, where n is the effective viscosity. When the strain rate is extremely high, then the flow stress is very high, and the material behaves as if it is very strong; e.g. it is elastic. The stress strain path follows the elastic modulus. As the shock propagates, the strain rate drops to a steady wave value, e ss. Grady has shown that e ss is proportional to the fourth power of the shock stress. This strain rate corresponds to some flow stress, Y ss, which corresponds to the HEL, S Y ss = (1-2n)/(1-n) s HEL, Where n is now Poisson s ratio. In practice Y ss is usually pretty close to Y static.

4 This kind of stress strain-rate relationship is called a Maxwell model. It has a mechanical analog: s The spring has a constant E and the damper has a viscosity h. The response of the system is s = eg + h de/dt If the load is applied instantaneously, all the strain is first carried by the elastic element. As deformation occurs in the viscous element, the load is relaxed. If a fixed strain rate is applied, D, then the stress increases to the limiting value s = hd{1-exp(-gt/h)] h/g is the relaxation time. 2. Phase Changes The most general form of a Hugoniot is probably this one:

5 R1 R2 PC P R3 E The waves are elastic precursor, plastic wave, phase change, elastic release in the high pressure phase, release shock due to phase reversal, and release in the low pressure phase. Phase changes are an important application in shock physics. They involved in material synthesis, for example diamonds and cubic BN from hexagonal BN. About a decade ago there was a flurry of activity about high pressure superconductors and there is still a lot of interest in high pressure hydrogen. Lets introduce a new notation, which I noticed in Gathers book, which I think is a little better than the standard notation. In this notation we will use D i to represent the velocity of a shock wrt the material into which the shock propagates. So D i = U i u i-1. Thus, a double shock is described thusly:

6 P P 2,u 2 u 1 D 2 +u 1 D 1 So D 1 is 1/V 0 times the slope of the Rayleigh line. These shocks are only stable if D 2 + u 1 < D 1. Using the usual relationships, viz. x D 1 = sqrt((p 1 -P 0 )/(V 0 -V 1 )), D 2 = sqrt[(p 2 -P 1 )/(V 1 -V 2 )], u 1 = sqrt[(p 1 -P 0 )(V 0 -V 1 )] Gives the formula we have seen before (P 2 -P 1 )/(V 1 -V 2 ) >(P 1 -P 0 )/(V 0 -V 1 ) for the shocks to merge, and otherwise the shocks spread apart. So this has the graphical interpretation:

7 P 2 1 If the slote of the line 12 is steeper than 01, a single shock forms. If the slope of 01 is steeper than 12, then there are two shocks. Hugoniots of the unstable kind are caused by shock waves. This causes a break in the D,u relationship: 0 V D 1 2 C0 u From the Hugoniot equations for internal energy, it is straightforward to show that the difference in internal energy between states 1 and 2 is H 2 H 1 = (1/2)(P 2 -P 1 )(V 1 +V 2 )

8 There is a rather complicated derivation that relates this to the latent heat of the phase transition (found in either Gathers or Duvall and Graham). Phases changes occur when the free energy of the high pressure phase is less than the free energy of the low pressure phase. Consider the Gibb s free energy of phase i G = E i TS i +PV i So the situation is: G 1 2 P Where the surface cross, the phase change occurs. First order phase changes like this are described by the Clausius-Clapeyron equation DP/dT = (S 2 -S 1 )/(V 2 -V 1 ) = DS/DV (which comes from V = G/ P and S = - G/ T) If the surfaces are tangent, it is a second order phase transition which is harder to detect. There is a mixed phase region in PVT space where the phases coexist. If 2 is the high pressure phase, usually DV<0. DS<0 implies dp/dt>0; DS>0 implies dp/dt<0. Note also DS = L/T where L is the latent heat of transformation.

9 Usually the change in the Hugoniot causes a second shock. In the mixed phase region, the Duff-Minshall formula relates the slope of the Hugoniot in the mixed phase region to the dp/dt slope of the phase boundary:

10 dv/dp) H = -bv A + 2aV A (dt/dp) + C p1 /T A (dt/dp) 2 There are various cases concerning shocking from one phase to another. First consider phase changes for which dp/dt>0. This is normal, where B is solid and A is liquid, and if we increase the pressure we make it harder to melt. E.g. the high pressure phase is also the high temperature phase. The temperature along the phase boundary my increase more rapidly along the Hugoniot than the phase boundary, or it might not, giving rise to these situations: {Remember dp/dt = -L/T(V A -V B )} P B A B A T The right fig. shows the case where dp/dt on the Hugoniot in the mixed phase region is steeper than on the phase line. You can shock from the solid to the melt. The left side is the reverse case. One can shock from the low density phase to the high density phase. In all cases, assuming the Hugoniot crosses the phase line. Consider dp/dt <0. This is like anomalous melting (e.g. water) or many crystallographic changes. If there is a phase change, the Hugoniot will cross the boundary.

11 P A B T Generally phase transitions take some time to happen, so the shock will overdrive the phase transition resuling in a decay. Graphically it looks like this: P V About the effects of shear stress. Although these are clearly important, they are not usually accounted for. Nevertheless, the mechanism of phase changes is generally assumed to be heterogeneous yielding. That is flow behind the plastic shock takes place on shear planes which get hot.

12 There are generally two types of phase changes: reconstructive and nonreconstructive. Reconstructive phase changes require atomic diffusion. For example, the change to high pressure quartz. These are usually relatively slow. Non-reconstructive require only a slight shift in atom locations. The best known such change is martensitic. These are very fast, they often take place promptly under shock loading, and static pressure and shock data often agree. The best review of this subject, although it is somewhat dated, as a paper by Duvall and Graham that appeared in Reviews of Modern Physics in Melting Melting is a first order phase change. Usually V L > V S, DS > 0, dp/dt > 0. It is hard to detect melting from the Hugoniot curve, usually. High pressure melting is usually described using the Simon equation: P/P 0 = B[(T/T 0 ) c 1] 1. Table of melt observations. Next we will examine some of the most important shock-induced phase transitions. 4. Iron Iron is in many ways the most important phase transition. Historically it was quite important, having been discovered by Bancroft et al in It was observed in shock waves before it was found statically, and traces of the phase transition were then discovered in meteorites. Plus, melting of iron, or at least sulfur is a major study in geophysics, since the core-inner core boundary is generally believed to be a melt transition. The high pressure phase is e. The phase transition is martensitic. 2. Slide of phase space of iron Plus latest thermo quantities for Chen and Ahrens.

13 3. Slide showing slight effect of melting on Hugoniot and large effect on sound speed. 5. Bismuth Bismuth is an important material because it is a high pressure standard and because the transition from Bi I to Bi II is an archetypical phase transition followed by several other materials. It also melts. 4. Slide of phase space of bismuth. 6. Silicates Most oxides and silicates have shock-induced phase transitions. Exceptions are MgO and Al 2 O 3. Consider the examples of CaO, which has a martensitic transformation and Fe2O3, which is electronic. Both are non-reconstructive transitions. 5. Slide comparing Hugoniots of Fe2O3 and CaO. Even complex silicates like olivine have reconstructive type transitions. Olivine is the most important constituent of the mantle. The phase transition is at 40 GPa. Olivine displays shock metamorphosis shocked material is recognized by glassy inclusions and kink bands. 6. Slide of fosterite. 6. Quartz SiO 2 is one of the most studied materials. In 1961 Stishov discovered a high pressure phase, named stishovite. Soon this phase was also found in shock experiments, by Wackerle and by McQueen and Marsh. Silica changes from 4-fold to 6-fold coordination. The high pressure phase is 62% denser than crystalline quartz and 95% denser than fused quartz. Static the transition starts at about 9 GPa, but it takes about 13 GPa in shock experiments. It must be overdriven. Glass also changes coordination number to a denser phase. Stishovite as a rutile structure. There is also a high temperature phase, coesite.

14 7. Slide of Hugoniot of quartz as determined by Wackerle. More recently the phase diagram has been expanded to include a further transition to 8-fold coordination, and finally melting. 8. Slide of recent paper of quartz phase diagram. Glasses display similar features. They densify. At, at least for soda lime glass, the high density glass is recovered. 9. Shock and release paths in soda lime glass. 7 Germania GeO 2 is like quartz, only more so. It has a bigger volume change. Like silica, it also has unstable shocks at low pressure. 10. Slide of shock properties of GeO2. 8. Titanium EOS models and tables prepared by the National labs include phase transition information. For example, I just today received a report on new EOS data for titanium. Models are constructed from Hugoniot data, static data, the Birch-Murnaghan equation, and a model for the Gruneisen parameter g = g0/h +.5(1-1/h)2, where h = r/r0 11. Slide of computed Hugoniots, including porous titanium.