Notes on Shock Compression, Wave propagation and Spall Strength W. G. Proud Institute of Shock Physics, Blackett Laboratory, Imperial College London, London, SW7 2AZ. Overview These notes give a brief overview of selected topics in the area of shock wave loading. They are brief, however, a list of additional reading is provided towards the end of the document. This document should be used in conjunction with that reading matter or as part of a lecture series. 1. Determining the Hugoniot (Compressive) of a unknown material by plate impact The Hugoniot of a material describes its shock compression properties, it is probably the most fundamental description of behaviour under shock loading. It is used to find the dynamic elastic limit, the Hugioniot Elastic Limit (HEL). A widely used technique to obtain the Hugioniot is by firing a well aligned plate of known properties against a plate of the material of undetermined shock properties. Often the target material is fitted with a backing plate, of known properties, with a gauge mounted in it. A flier of a well-defined material with a known Hugoniot which strikes the target at a velocity (V). One of the fundamental aspects of shock loading called impedance matching. If two surfaces remain in contact then the stress and particle velocity are the same on both sides of the impact interface. If this is not the case, then stress waves will be produced to make it so, alternatively the materials will separate and no longer be in contact. In the equations that are developed here, the following two simplifications are made. Firstly that the Hugoniots and Isentropes are straight lines and secondly, for the stress levels considered the Isentrope is the same as the Hugoniot for all materials. 1
Figure 1. Experimental Geometry Figure 2. Distance-time diagram Figure 3. Stress states from Hugoniot The distance-time diagram shows a region indicated by "1" which corresponds to stress state and region "2" which corresponds to ' state. In the following equations, Z f = impedance flier, Z b = impedance backing plate, Z? = impedance of test material In the straight line (elastic cases) Z = C B where = density and C B = bulk sound speed of that material. From figure 3 we can see that = Z? U, (1) = Z f (V-U), (2) ' = Z b U', (3) ' = Z? (2U-U'), (4) and from (1) and (4) - ' = Z f (U' - U). (5) The gauge in the block detects the level ' and U' is found from the Hugoniot of the backing material. The unknowns are Z?, and U. Equating (1) and (2) Z?U = Z f (V-U) Substituting for Z? in (4) and Z? = Z f (V/U - 1) ' = Z f (V/U - 1)(2U - U') 2
and 'U = Z f (V - U)(2U - U') Multiply brackets and collecting like terms in U we obtain: 2 Z f U 2 - (2 Z f V + Z f U' - ')U + U' Z f V = 0 Solve for the roots of a quadratic in U using the standard relationship U = (-b ± (b 2-4ac)) / 2a Where a = 2 Z f b = -(2 Z f V + Z f U' - ') c = U' Z f V From this U will be determined. Obtain stress,,related to the particle velocity using (2). Then obtain Z? by equation (1) 2. Shock Propagation across interfaces Many applications of shock waves involve interaction with boundaries and surfaces. The technique of impedance matching can be applied in these circumstance. When solving a problem involving the propagation of shock or release processes remember to keep firmly in mind the following points: 1. Put in your start conditions. Is the material moving or stationary? Does it have an initial stress? 2. Which material is the wave or fan propagating in? Waves do not appear at interfaces, they have to transmit through something to get to that interface, the only exception is the impact interface. 3. Draw an x-t diagram. Draw in the waves as lines, steeper slope = slower waves, shallow slope = faster wave. Which direction are the waves propagating? 4. Stress and particle-velocity are preserved across interfaces. However, if the interfaces separate (lect. 10, pg. 6) or spall occurs (lect. 11) this is no longer the case and two new interfaces are produced. The Hugoniot / Release for the material between these new interfaces is that of air or vacuum - which can be represented in the first instance by the x- axis through the origin. 5. Wave phase. In particle-velocity v. stress graphs a shallow Hugoniot corresponds to a low impedance material e.g. a polymer. A steep Hugoniot corresponds to a high impedance material e.g. a metal. As well as an amplitude, there may be a phase shift upon reflection at impedance-mismatched interfaces. 3
Stress and Particle Velocity Process / direction Stress Particle Velocity Shock > Up Up Shock < Up Down Release < Down Up Release > Down Down For shocks the Rayleigh line is followed to a point on the Hugoniot of the material. For release the Isentrope is followed. At low stresses the Hugoniot and the Isentrope can be assumed to be the same. Transmission / Reflection - wave phase Wave type /Boundary type Transmission No phase change Reflection Possible phase change Shock wave Shock transmitted Shock Reflected / Shock wave Shock transmitted Release Reflected / Release fan Release transmitted Shock reflected / Release Fan / Release transmitted Release reflected The wave or fan is travelling though the first named material into the second named material Controlling Equations for transmission / reflection (lecture 2, pg.3) The wave is being transmitted into material 1 from material 2 Transmission factor T = 2Z 1 / (Z 1 + Z 2 ) Reflection factor R = (Z 1 - Z 2 ) / (Z 1 +Z 2 ) The transmitted factor is always +ve. The reflected factor may be positive or negative. For +ve factors the wave does not change phase (Z 1 >Z 2 ). For a -ve factor phase change occurs (Z 2 >Z 1 ). Consider a shock as a +ve pulse and release as a -ve pulse. For a negative reflection coefficient for a release wave, the situation is; -ve (reflection) x -ve (release) = +ve i.e. a shock wave is reflected The four following diagrams represent the four possible cases of interface-wave interaction. All propagation situations resolve into a succession of these interfaceproblems. In each case a Hugoniot diagram and a x-t diagram are drawn. 4
NOTE - If you incorrectly put a reflection as changing phase i.e. reflecting a shock instead of a release you will not be able to get the Hugoniots or Isentropes to overlap! This should alert you that something is wrong. Stress HUGONIOTS 1 2 Time X-T DIAGRAM Particle Velocity STATE 2 Release Shock STATE 1 Shock O state O state Distance Shock moving through a high impedance material and meeting a High / Low interface Shock moviing through high-impedance material and meeting a High / Low interface 5
HUGONIOTS Stress 2 1 Time X-T DIAGRAM Particle Velocity STATE 2 Shock Shock STATE 1 Shock O state O state Distance Shock moving through a low impedance material and meeting a Low / High interface Shock moviing through high-impedance material and meeting a Low / High interface 6
HUGONIOTS Stress (a) Both materials initially in state 1. (b) Release moves material to origin (state 2) (c) Reflected release in material and releases material (state 3) 1 2 3 Particle Velocity X-T DIAGRAM STATE 3 Time Release Release STATE 2 Release STATE 1 Distance Release moving through low-impedance material and meeting a Low / High interface Release moviing through low-impedance material and meeting a Low / High interface 7
HUGONIOTS Stress 1 3 (a) Both materials initially in state 1. (b) Release moves material to origin (state 2) (c) Reflection shock material and partially releases material (state 3) 2 Time X-T DIAGRAM Particle Velocity STATE 3 Shock Release STATE 2 Release STATE 1 Distance Release moviing through low-impedance material and meeting a High / Low interface Release moving through high-impedance material and meeting a High /Low interface 3. The Spall (Dynamic Tensile) Strength ( spall ) of Solids The spall strength is the dynamic tensile strength of a material at high strain rates. It is most commonly encountered in plate impact and explosive loading scenarios. The spall 8
strength of the material is taken from a stress pulse measured by a gauge, usually embedded in a back surface configuration, or from a VISAR trace. In either case the signal has the form shown below. The value taken for the spall strength ( spall ) is indicated on the diagram. Particle Velocity / Stress spall Time The following description shows how spall strength can be related to the Hugoniot of the material and explains why the so call "pull-back" signal can be used as the measure of this parameter. Wave Diagram for the Spall Process It is assumed that the Isentrope and the Hugoniot are straight lines. Secondly, at low stress levels they follow the same path. The x-t diagram for the spall experiment is shown below. The release fans from R and R' combine in one of three cases: (i) where the material is taken to peak tension but spall is not exceeded, (ii) spall occurs, (iii) the material has no spall strength Initial stage of Impact For the initial impact the stress in the material is state A, this would be recorded by a gauge in PMMA mounted on the rear of the sample as stress state B. 9
For all cases, the release isentrope from the rear of the flyer must pass through the state B. The release isentropes must cross at the different states implied by the above cases. Case (i) Case (ii) Case (iii) This means that the stress pulse (up to C) must look as follows Case (i) Case (ii) Case (iii) The Reload Signal To complete the process we must consider what happens to the part of the release trapped between the spall plane and the rear of the target. This pulse is essentially reflected at the spall plane and changes phase to become a compressive signal as the spall plane is a free surface. This "trapped" part of the wave will become a release following reflection at the rear surface of the target, giving rise to an oscillating shape gradually decreasing magnitude. In case (i) there is no spall plane as nothing is trapped. The signal ramps to zero in t 1 -t 3 In case (ii) a free surface opens up on which the stress state is zero. Thus the state on the spall plane must lie on the line BX at the = 0 axis, i.e. the point Y in the diagram below. 10
This gives the reloading signal. Further oscillation of the trapped wave between the PMMA interface and the ( = 0) spall plane gives a series of subsequent reload signals which may or may not open up further spall planes depending on whether spall is exceeded. For the case (iii) where s = 0 (no spall strength), the lines CX and DY are at the same position and are collinear. D is the only possible final state. This results in pull-back signal. This may appear suprising since most of the release wave is trapped and reflects from the spall plane. However it is a reflected wave of the height of the spall strength i.e. zero. It must be remembered that the release is to state D, not to zero stress as in the case (i) where the spall strength was not exceeded. The full diagram of the spall experiment can now be drawn. 11
The measured height of the pull back signal is D - C. Which can be converted to an inmaterial value of P - Q. Now consider the parallelogram PYQX. Drop a perpendicular to R and S such that the angles PRQ annd YSX are right angles. Now PY is parallel to QX and PQ parallel to XY. Since SX and PR are also parallel, the angle QPR and SXY are the same. The angles PQR and XYS are related in the same manner. Overall this means that PR = SX. So the magnitude PR = spall The reload signal in particle velocity measured by VISAR on a free rear surface would correspond to the difference PR. If a gauge is used on the rear surface, the signal PR has to be converted into that recorded on the gauge. Now PR corresponds to the state C - D on our converted mean stress. Overall using a stress gauge or a VISAR system we can measure the spall strength. The above arguments all use straight lines and that the Isentrope = Hugoniot, for large stresses these assumptions are not followed but the arguments are the same. 12
Shock and Thermodynamic Properties of Selected Materials For use with shock equation of state: U s = c 0 + Su p Material 0 (g cm -3 ) c 0 (mm µs -1 ) S C p (J g -1 K -1 ) Ag 10.49 3.23 1.60 0.24 2.5 Au 19.24 3.06 1.57 0.13 3.1 Be 1.85 8.00 1.12 0.18 1.2 Bi 9.84 1.83 1.47 0.12 1.1 Co 1.55 3.60 0.95 0.66 1.1 Cr 7.12 5.17 1.47 0.45 1.5 Cs 1.83 1.05 1.04 0.24 1.5 Cu 8.93 3.94 1.49 0.40 2.0 Fe 7.85 3.57 1.92 0.45 1.8 Hg 13.54 1.49 2.05 0.14 3.0 K 0.86 1.97 1.18 0.76 1.4 Li 0.53 4.65 1.13 3.41 0.9 Mg 1.74 4.49 1.24 1.02 1.6 Mo 10.21 5.12 1.23 0.25 1.7 Na 0.97 2.58 1.24 1.23 1.3 Nb 8.59 4.44 1.21 0.27 1.7 Ni 8.87 4.60 1.44 0.44 2.0 Pb 11.53 2.05 1.46 0.13 2.8 Pd 11.99 3.95 1.59 0.24 2.5 Pt 21.42 3.60 1.54 0.13 2.9 Rb 1.53 1.13 1.27 0.36 1.9 Sn 7.29 2.61 1.49 0.22 2.3 Ta 16.65 3.41 1.20 0.14 1.8 U 18.95 2.49 2.20 0.12 2.1 W 19.22 4.03 1.24 0.13 1.8 Zn 7.14 3.01 1.58 0.39 2.1 KCl 1.99 2.15 1.54 0.68 1.3 LiF 2.64 5.15 1.35 1.59 2.0 NaCl 2.16 3.53 1.34 0.87 1.6 Al-2024 2.79 5.33 1.34 0.89 2.0 Al-6061 2.70 5.35 1.34 0.89 2.0 SS-304 7.90 4.57 1.49 0.44 2.2 Brass 8.45 3.73 1.43 0.38 2.0 Water 1.00 1.65 1.92 4.19 0.1 Teflon 2.15 1.84 1.71 1.02 0.6 PMMA 1.19 2.60 1.52 1.20 1.0 PE 0.92 2.90 1.48 2.30 1.6 PS 1.04 2.75 1.32 1.20 1.2 13
BACKGROUND READING MATERIALS. Akhavan J, The Chemistry of Explosives, 3 rd Edition, (Royal Society of Chemistry 2011) Altgilbers, L. L., M. D. J. Brown, et al. Magnetocumulative Generators. (Berlin, Springer, 2000) L.V. Altshuler, R.F. Trunin, V.D. Urlin, V.E. Fortov, A.I. Funtikov Development of dynamic high-pressure techniques in Russia Physics Uspekhi 42 (1999) 261-280 Antoun, T., L. Seaman, et al. Spall Fracture. (Berlin, Springer, 2003) Asay J R and Shahinpoor M, High-Pressure Shock Compression of Solids, (Springer Verlag, New York, 1992) Bailey A and Murray S G, Explosives, Propellants and Pyrodynamics, (Brassey's, UK 1989) Batsanov A A, Effects of Explosion on Materials: Modification and Synthesis under High- Pressure Shock Compression (Springer Verlag) Bhandari, S. "Engineering Rock Blasting Operations", (A.A. Balkema, Rotterdam, The Netherlands 1997) Borovikov, V.A. and Vanyagin, I.F., "Modelling the Effects of Blasting on Rock Breakage", (A.A. Balkema, Rotterdam, The Netherlands, 1995) Cherét R, Detonation of Condensed Explosives (Springer Verlag) Cooper, P.W., Explosives Engineering, (Wiley-VCH, 1996) Davison L, Grady D and Shahinpoor M (eds.), High-Pressure Shock Compression of Solids II (Springer Verlag) A.N. Dremin Toward Detonation Theory (Springer-Verlag, Berlin, 1999) D.S. Drumheller Introduction to Wave Propagation in Nonlinear Fluids and Solids (Cambridge University Press, 1998) Davison L and Shahinpoor M (eds.), High-Pressure Shock Compression of Solids III (Springer Verlag 1997) Davison L, Grady D and Shahinpoor M (eds.), High-Pressure Shock Compression of Solids IV (Springer Verlag) Dodd, B. and Bai, Y, Adiabatic Shear Bands: Frontiers and Advances, 2 nd Ed. (Elsevier Insights, 2012) Drumheller, D S, Introduction to Wave Propagation in Nonlinear Fluids and Solids (Cambridge University Press, Cambridge, 1998) Fordham S, High Explosives and Propellants (New York, Pergamon Press 1980) Grady, D. E., Fragmentation of Rings and Shells: The Legacy of N.F. Mott. (Berlin, Springer, 2006) 14
Graham R, Solids Under High-Pressure Shock Compression (Springer Verlag) Hustrulid, W. "Blasting Principles for Open Pit Mining", (A.A. Balkema, Rotterdam, The Netherlands, 1999) J.N. Johnson, R. Chéret Shock waves in solids: An evolutionary perspective Shock Waves 9 (1999) 193-200 Johnson J N and Cherét (eds.), Classic Papers in Shock Compression Science (Springer Verlag 1998) Kanel, G. I., S. V. Razorenov, et al. Shock-Wave Phenomena and the Properties of Condensed Matter. (Berlin, Springer, 2004) J.G. Kirkwood Shock and Detonation Waves (Gordon & Breach, New York, 1967) Melosh H J, Impact Cratering: a geologic process (Oxford University Press 1989) Meyers, M A, Dynamic Behavior of Materials (New York, 1994) Nesterenko, V. F. (2001). Dynamics of Heterogeneous Materials. (Berlin, Springer, 2001) Petrosyan, M.I."Rock Breakage by Blasting"(A.A. Balkema, Rotterdam, The Netherlands, 1994) Porter, D., Group Interaction Modelling of Polymer Properties, (Dekker, New York, 1995) Ray S F, High Speed Photography and Photonics, (Focal Press, Oxford, 1997) Roger G, Gathers, Selected Topics in Shock Wave Physics and Equation of State Modelling (World Scientific, London, 1994) Rosenber, Z and Dekel, Terminal Ballistics, (Springer, 2012) Suceska M, Test Methods for Explosives (Springer Verlag, 1995) Trunin R F, Shock Compression of Condensed Materials (Cambridge University Press 1998) Wilkins, M. L. Computer Simulation of Dynamic Phenomena. (Springer, Berlin, 1999) Zukas J, High Velocity Impact Dynamics (John Wiley & Sons 1990) Zukas J and Walters W P (eds.), Explosive Effects and Applications (Sringer Verlag New York 1997) CONFERENCE PROCEEDINGS There are a number of relevant conferences, however the following are amongst the best known - (a) Shock Compression of Condensed Matter publisher- American Physical Society (AIP) (b) Detonation Symposium - publisher - Office of Naval Research (ONR) (c) DYMAT publisher - Journal de Physique IV (pre-2009) and European Physics Conferences (2009 onwards) (d) Novel Trends in Research in Energetic Materials - publisher - University of Parduvice, Czech Republic 15