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The role of viscosity of solids in microparticle cratering
Acla Aslronautica Vol.5, pp. il29-1144 © PergamonPress Ltd., 1978. Printedin Great Britain
0094-5765/'7811101-1129~02.00]0
The role of viscosity of solids in microparticle crateringt M. R. S E I Z E W , $ A. L. K U H L § V. D. B L A N K E N S H I I ~
ASD
TRW Defense and Space Systems, One Space Park, Redondo Beach, CA 90278, U.S.A. (Received 30 March 1977; revised 2 August 1977)
Abstract--The importance of viscous effects on the impact cratering process has been investigated. The shock wave front region was characterized by the balance of inertia and viscous forces (Reynolds number ~ 1), and by a continuous distribution of all thermodynamic quantities. The width of the shock wave front region and its flow properties were then related to the impact particle size and velocity, and to the target material properties. Experimental data for aluminum demonstrated that cratering characteristics were changed when the impact particle size was the same order of magnitude as the shock wave front region. An analytical model was developed for cratering dominated by this "viscous" flow. Results from this model indicate a substantial decrease in crater radius and increase in crater surface temperature when compared with inviscid analysis. In addition, a new test method was identified to evaluate the viscosity of solids.
Introduction TsE IMPACTof particles on a high speed aerospace vehicle is a major area of concern to system designers. These particles can cause damage by penetrating and cratering thermal protection surfaces resulting in erosion of the heatshield and mass loss. These phenomena will also increase the benign aerodynamic heating while a vehicle is flying in the atmosphere, whether in a natural or an artificially produced particulate environment. The initial interaction of the impacting particle and target is an important phenomenon related to the partitioning of the kinetic energy of the particle and the subsequent increase in heating. In general, the analysis of the impact cratering process relies on sophisticated inviscid numerical methods, and on scaling empirical data to different impact conditions. However, at the present time there are no data available, either analytical or empirical, that can be used to evaluate the performance of the heatshield materials in erosive environments consisting of small particulates in the micron range. As pointed out by MacCormack (1969), application of the inviscid theory to this type of impact condition is not valid when the particle size is the same order of magnitude as the viscous region generated during impact. A more rigorous tThis work was performed under a TRW independent research and development program, Project No. 76004880. This paper was presented at the Sixth Colloquium on Gasdynamics of Explosions and Reactive Systems, Stockholm, Sweden, 22-26 August 1977. ~;Head of Energy Dynamics Section, Vulnerability and Hardness Laboratory. §Member of the Technical Staff, Vulnerability and Hardness Laboratory. IHead of the Reentry Systems Department. 1129
1130
M.R. Seizew, A. L. Kuhl and V. D. Blankenship
approach is needed which takes into account all physical process involved. The objective of this study was to determine how material viscosity affects the particle-heatshield interaction process as a function of the size and velocity of the impacting particle. First, a postulate relating the size and velocity of the impacting particle to the effects of the viscosity in the considered impact cratering process was formulated and rationalized by considering the properties of the shock-wave-front (SWF). The validity of this basic postulate was verified by using the available impact test data for aluminum. Then an approximate analytical model for this viscous interaction process was developed. Finally, the viscous flow model was used to correlate viscosity test data with analytical predictions. In addition, a new test method was identified to evaluate viscosity of solids. Analysis
A number of investigators have attempted to synthesize some aspects of the phenomenon associated with the effects of viscosity in a hypervelocity impact process and other shock-related problems. Direct analytical methods have been used by MacCormack (1969) and Zaidel (1967), while a number of investigators have performed experimental studies.t It is evident that the absence of the theoretical treatise for viscosity of the solids limits these studies. Nevertheless, the investigations were partially used in formulating and rationalizing the postulate for the viscous flow and the subsequent analysis.
Formulation of the viscous flow postulate Denardo et al. (1966, 1967) have reported some inconsistencies in a simple linear scaling rule for small particle, high velocity impact conditions. Penetration, radius of the crater formed, and transfer of momentum to the target decreased more rapidly than simple scaling would imply as the size of particle was reduced. This nonlinear effect was attributed to rate-dependent stress effects important at the initial stage of cratering, and materials strength effects which influence the last stage of crater formation. MacCormack (1969) attempted to correlate this experimentally observed deviation from a simple linear scaling rule by taking into account rate-dependentstress-viscosity effects. In that study, the Navier-Stokes equations were solved numerically with a finite-difference method which was second-order accurate in both time and space. Although the results were not conclusive, the method demonstrated increasing importance of viscosity as the size of the impacting particle diminished. Two important observations can now be noted from that study: (1) Viscosity is important when the strain-rate is high, which corresponds to the initial phase of cratering, and (2) The effects of viscosity become more predominant in a flow produced by an impact particle of small size.
al.
tSee Sakharov (1971).
et al.
(1965), Mineev and Savinov (1967), Mineev and Zaidel (1968), and Godunov
et
The role of viscosity o[ solids in microparticle cratering
1131
Let us now consider some important physical aspects of shock wave propagation in a compressible medium. On the one hand, the theoretical studies of Zeldovich and Raiser (1966), and Zaidel (1967) demonstrate that the shock front is a transition layer of finite thickness in which there occurs a change of all quantities that characterize the state of the medium. Furthermore, the studies show that without taking into account viscosity, one cannot produce a continuous distribution of thermodynamic properties in the shock front. On the other hand, experimental studies by Sakharov et al. (1965), Mineev and Savinov (1967), Mineev and Zaidel (1968), and Godunov et al. (1971), show perturbations of a plane shock wave are damped as the wave propagates due to the effects of viscosity. At this point, let us assume that the inertial forces and viscous forces in the SWF are in equilibrium, that is, that the Reynolds number based on the shock thickness (i.e. R~ = pvAh/vl) is of order one.t Thus, the thickness of the SWF can be deduced as •h =~ pv
(1)
where Ah = the width of the SWF, [Ah] = cm; 7/= the dynamic viscosity in the SWF region, [7] = gmlcm-sec; p = the density in the SWF region, [p] = gm/cm~; v = the material velocity in the SWF region, [v] = cmlsec. This is the characteristic dimension of the viscous flow region formed by the shock wave. As this relation shows, the transition layer thickness is dependent on viscosity, which is in turn a function of pressure and temperature of the medium. It is well known that when a infinite plate of a finite thickness impacts a semi-infinite half-space of the same material, the initial width of the flow region formed in the target is proportional to the thickness of the impacting plate. This proportionality is, of course, deduced from the time required for shock and rarefaction waves to traverse the thickness of the impacting plate. When a particle of a specified size impacts a semi-infinite target, the width of the flow region formed in the target is also identically proportional to the diameter, d, of the impacting particle. Thus, when the diameter of the impacting particle is large, the extent of the flow region in the target which has been engulfed by the shock is also large compared with the shock front thickness. Therefore, the flow in the region ecompassed by the shock wave can be considered inviscid. However, as d approaches Ah, the width of the flow region approaches the width of the SWF for a specified impact condition and the flow becomes predominantly viscous. This is consistent with the results of the numerical calculations performed by MacCormack (1969), which showed the increasing importance of viscosity as size of the impacting particle diminishes. Now we can formulate the hypothesis for the existence of the viscous flow in a hypervelocity impact process:
tlndeed, the above experiments show that viscosityis inverselyproportional to Reynoldsnumber.
1132
M.R. Seizew, A. L. Kuhl and V. D. Blankenship
The initial stage of a particleltarget iteration process is dominated by viscous effects when the impacting particle diameter is the same order of magnitude as the shock front thickness. Verification of the viscous flow postulate To examine the validity of the above postulate, two types of experimental data were required. First, data were required to evaluate the thickness of the SWF for materials whose viscosity is known as a function of pressure and temperature. A literature survey was made and reliable data were found for aluminum. Using relation (1), experimental viscosities from Mineev and Savinov (1967) and Mineev and Zaidel (1968), and the equation of state for 90% pure aluminum, the variation of the SWF thickness and temperature were established as a function of impact velocity as shown in Fig. 1. The bounds of the SWF, as shown in this figure, are a result of the experimental uncertainty in the viscosity data for aluminum. At this point, the available impact data for aluminum (Denardo et al. (1967) and Slattery (1966)) were grouped according to impact velocity, particle radius, and ratio of formed-crater radius to impacting-particle radius (Table 1). This data is displayed in Fig. 2. There the particle size effect (an effect strictly absent from 1600
I
I
I
l
3500
I /
MATERIAL:
/
ALUMINUM 90% PURE
/ /
1400 --
-- 3000 T
'u 1200
-
2500 ~_~ z
.~
o v
. 1000
2000
800
1500
i.-
v
~°°°I
~
400
2oo
-- 500
I
I
I
I
I
3
4
5
6
7
o 8
PARTICLE IMPACT VELOCITY, V e (km/sec) Fig. 1. Shock f r o n t thickness and temperature in aluminum as a function o f impacting
particle velocity.
The role of viscosity o] solids in micropanicle cratering
5.5
j
I
J L
I
i
I
1133
I
"LARGE"PARTICLE~ DATA (R - 1500~ " ~ , , ~ /
leoo
•
) _~ 4.5
O 2Rp~:~h •
2Rp
1200
~3.S
~
o.ok/ / /i .
/
2
I
I
"
I
I
I
3 4 5 6 7 PARTICLEIMPACTVELOCITY, V° (kin/me)
8
Fig. 2. Viscous (i.e. particle size) effects on the ratio of crater radius to particle radius formed by the hypervelocity impact of aluminum spheres on a semi-infinite aluminum slab; large and small particle data converge when the shock front reaches maximum thickness.
an inviscid description of this flow process) is demonstrated dramatically. "Small" particles (Rp < 10/~) generate craters that are about half the size (i.e. half the non-dimensional crater radius) of craters formed by particles that are large compared to the shock thickness. Refer, for example, to the cratering data in Fig. 2 at an impact velocity of 7.5 km/sec. In addition, the crater results for "large" and "small" particles are similar in the region where the shock front is the thickest (particle velocities of 5.5 km/sec).t Outside this impact regime, "large" particles behave in a more or less inviscid manner, while "small" particles are always affected by viscosity. These observations, the convergence o f the i m p a c t data and their similar functional d e p e n d e n c e as S W F ( w h i c h i n c l u d e s m e l t region at 7.5 k m / s e c ) , are t a k e n to be a p r o o f o f the v i s c o u s flow postulate.
tit should also be noted that this is the impact resime where the characteristic dimension of "large" particles as well as "small" particles is less than the shock thickness, Ah.
Speed of Sound,
5.2 2.6 3.0
0.079 2.3 x IO-’ 2.1 x lo-’
0.04-0.03
7.0-7.5
tSee text for definition. Density, p0 = 2.8 gmlcc;
4.2 4.5
0.158 0.079 2.3 X lo-’
0.13-0.07
5.5-6.0
10’4
4.3 4.0
0.158 0.079
0.05-O. 13
5.0-5.5
a.3* 1.0 Extrapolated
0.08409
4.5-5.0
4.3 4.0 3.5 2.5 2.2 2.3
0.06-0.08
4.0-U
C z 5.0 X !u‘ cmisec.
722 Extrapoiated
IO%4
4.9 2 2.4 Extrapolated
2eo.5
0.317 0.158 0.079 4.1 x lo-’ 3.2 X lo-’ 2.4 x IO-’
0.02-0.045
0.635 3.317 0.158 0.079 0.158 0.079 3.5 x IO-’
3.04.0
3.85 3.80 3.0-3.7 3.6 3.84.0 4.0 2.0
Radius of particle R,-cm
Thickness of SWF Ah-cm
Impact velocity V,-kmlsec
Ratio of crater and particle radii
Yield,
50-200
50-200
50-200
50-200
50-200
50-200
r _ 9.4 Kb; Young
Ah > ZR, IO Ah C 2R, 5 Ah > 2R, 6.5 Ah < 2R, 6 Ah > 2R, 8 Ah < ZR, 6 Ah > 2R, I1
Ah i 2R, 4.0 Ah > 2R, 8 Ah < 2R, 4.0
Ah < 2R, 2.3 zt 0.3
Initial Rash-ligbttime s-see X 10m9 estimated
977- 1080
685-777
598-685
5 16-598
366-516
299-366
Initial impacr pressure P,-Kb
2700
Temperature T-OK experimental
data
Ratio,
2700
0.64-0.63
0.68-0.67
0.70-4I.68
0.724.70
0.74-0.72
0.78-0.74
%I
Constantt
D - 0.33: R,,
Temperature T-OK calculated
E = 689 Kb; Poisson
impact
Modulus.
of the aluminum-aluminum
Viscosity v-poise x IO’ calculated
1. Summary
Viscosity q-poise X l@ experimental or extrapolated
Table
-? 5.4 ,” 10 2 RD
I .52-I .49
1.56-1.52
1.67-1.56
*I
Constantt
The role of viscosity of solids in micropanicie crater/rag
1135
Analytical modeling o f v i s c o u s / l o w
An analytical model for this viscous impact process has been formulated as follows: (a) Crater formation in the target is accomplished by the radial expansion of a spherical cavity (see the treatise by Hopkins (1960) for a-general discussion of cavity dynamics in solids). (b) Impact velocity and particle size prescribe the driving conditions on the cavity wall. (c) The target medium is assumed to behave like viscous solid (i.e. stresses depend on strain rates) which locks up on compression. (d) The Cauchy equations of motion are then evaluated to determine the cavity expansion, in particular, the maximum crater radius. (e) Solution of this flow field defines the energy density distribution along the radius and also irreversible work done during cratering as well as the sudace temperature. Assuming spherically symmetrical flow in the target material, the Cauchy equation of motion in terms of the physical components of stress can be written: OcrR . 2 D( v ) OR * -R (err - ere) = p Dt '
(2)
where err = the stress in radial direction; t = the time; or, = stress in "hoop" direction; R = radial coordinate; p = density; v = mass velocity in radial direction, and D(v)
Or+
Dt
Ov
vT "
Next, using the definition of the strain field, one can easily derive the relation of strain rate and velocity. For the radial component, OU
0--R =
(3)
eR
and
O(eR)= 02u Ot
OtOR
Ov OR"
Likewise, for the "hoop" direction, U
(4)
and
R Ot
R"
In these relations, u is the deformation field, eR ffi strain in radial direction, and e, --strain in hoop direction.
1136
M.R. Seizew, A. L. Kuhl and V. D. Blankenship
At this point, an assumption concerning the constitutive relation of the target material is introduced, namely, that the dilitation of the target material is limited to a constant value. That is, the target material is assumed to be a rigid "locking solid" in compression.* Although this assumption is not physically realistic in a general sense, it does greatly reduce the complexity of the analysis. The assumption of a "locking" medium can be formulated in terms of strain components: • R + ~e + ~
= ~R + 2 ~ e ~ = ~L.
(5)
The functional relation of the velocity in terms of R and t can be derived by differentiating (5) with respect to time, and using relations (3) and (4). The resulting partial differential equation is ~_t(eR)+2~t(~e ) = a OR v+
2 ~v=
0.
(6)
Integration of (6) gives
f(t)
v = -~T,
(7)
where [ ( t ) is a continuous unknown function of time. Equation (7) will hold for all material states in the locking region. A further assumption concerning the constitutive law obeyed by the target material is now introduced. Using the above postulate, it is assumed that the differences in the principle stresses can be expressed in terms of the shear strength and the strain rate difference, or (trR-tr0)= Y
+ lair 7/~ ~
a~e]
c~t/
(8)
where Y represents the shear strength of material, and r/ is the constant of proportionality, the viscosity coefficient. From these relations, Cauchy's equation can now be formulated in terms of one principle stress component, ~rR, and in terms of f(t) and R as follows:
r/u) °'q--R-4-= PP[-~-'z
ao'R 4 2.2.2_~ .. f ( t ) OR
2~]
(9)
where
/ ( t ) = af(t) cgt pp = limiting "locking" denisty. tThis assumption, in a rigorous way, is related to the Hugoniot of the material with initial "critical" density, i.e. (I -PcR/P)= (21(2+ F)) where F is a Gruneisen coefficient. ~tln the case of spherical symmetry, note that e6 = e~ and cro= ~r~.
1137
The role of viscosity of solids in microparticle cratering
Equation (9) can be integrated with respect to R, giving cra - 2 Y In R +2./ R, = pp [f'(t) L-~-c- f- (~tj) ] + g ( t )
(10)
where g(t) denotes a new function of time. This new function, g(t), can be eliminated from relation (10) by considering the boundary conditions at the crater-cavity. Defining the crater-cavity radius by Ro and prescribing the pressure-time function at this depth, relation (10) becomes [f'(t) - P ( t ) - 2 Y In R~ + 2,/~(~ Ro = Po [2-~-/R(~t)] + g(t). (11) If we solve for g(t) in (11) and substitute back into relation (10), we have erR= - P ( t ) + 2 Y
In ~ + 2,pf(t)[~-~- ~--~]-~ f2(t)[~'~- ~ l ] + Op](t)[~_~_ 1 ] .
(12) Consideration of mass and momentum conservation at the shock front allows one to relate f(t) to the shock front parameters/~b, Rb, and Rb, the acceleration, velocity, and distance travelled by the shock front, respectively. From conservation of mass across the shock, we write (13)
po(Rb - v) ~ pORb
This relation will hold exactly for only a perfectly discontinuous single shock front moving into an undisturbed medium. When (7) is used in conjunction with (13), it follows that (14)
f ( t ) = RvRb2(1--~)
Conservation of momentum across the shock gives ~R~ b
~- [Rb - f(t)] f(t) L
(15)
where the minus sign is present because of the convention that stress is negative in compression. Equations (14) and (15) can be used in conjunction with relation (12) to derive the following relation: 1
Po/
1
2
Po/J
1
1
Po
,Op
\ - ~-pj ~ - ~ - ~-~bj + 2 Y In ~-~..
(16)
1138
M.R. Seizcw, A. L. Kuhl and V. D. Blankenship
To solve eqn (16) for Ro or Rb as a function of time, we need to prescribe P ( t ) and define the relationship between R~ and Rb. The relation between R, and Rh is defined by considering mass conservation in the deformed region, before and after deformation. For the material within radius Rb, the appropriate conservation relation is poRb = pp ( Rb 3 -- Ra3).
(17)
This relation is consistent with the assumption of a locking medium (i.e. p = pp = constant behind the shock front), and with the further assumption that the loss of ejected material f r o m the crater cavity is neglected. Equation (17) leads to the desired relationship between Ra and Rb, such as (18)
R~ 3 = R b 3 a
where
Differentiation of (18) with respect to time gives RoRo 2 = RbRbZ a = f ( t ).
(19)
This is consistent with (14), and with eqn (7) evaluated at R = R,. With the use of relations (18) and (19), eqn (16) can be put into the following form in terms o f / ~ , , Ro, and Ro : P ( t ) = A d ( R o R , ) + ( B - A)Ro2 + c d (ln R , ) + D
(20)
where A, B, C, and D are constants given by A = pp(l - ot ~n) B = ~ ( 3 - ot 4t3 - 2a v3)
(2])
C = 27(1 - a ) D = 2Y In (a-lJ3).
The constants C and D, proportional to the viscosity coefficient and dynamic yield stress, are several orders of magnitude larger than A and B for nearly all materials. Hence, it is a reasonable approximation to neglect the second term in the right hand side of (20) and write d
P ( t ) = ppro d ( / ~ o ) + 2~ro, ~ (In Ro)+ 2Y in rN
(22)
The role of viscosity of solids in microparticle cratering
1139
where go = ( 1 -
a l / 3 ) , ~o, = (1 - a ) , r N = a -'13.
If we assume that the pressure in the crater cavity is P = Poe-tl,, where ¢ is characteristic of the impact conditions, the material properties, and the configuration of the impacting particle, eqn (22) can be integrated with the result
R°
K~OpRoRo+ 2,/KoI In _-~--+ 2Yt In KN = -P0¢e -'/" + PO'. /(a,0
(23)
The radius of the cavity, Ro, can be estimated by integrating eqn (23) and by noting that dt = dRJRo and assuming R° = constant, to give 1
2
~Ra K~Op= - ~
[Ra ln ~o~,o-Ra] - Y ln gNtZ+ Po¢[f(e-tl" -1) + ¢].
(24)
In the derivation of eqns (23) and (24), the integrations were begun at time t = 0 and crater-cavity radius Re.0, which corresponds to the elastic deformation of the target surface caused by the impacting particle before the plastic deformation begins. Further, Re.0 is assumed to be negligible in comparison to Re and, therefore, appears only in the logarithmic terms in (23) and (24). If we use (23) to eliminate Re, an expression is found for Re, the crater cavity radius as a function of time: Re _ Re] f 2~/ra, LRo In ~ j RoK~o~ - Yt 2In KN ~Ra Kagp = Ra 2~lKo:,ln-~--+ 2tY In rN + Po¢(e -t/¢- 1) 1
Z
Ka,0
+ p o ¢ 2 ( e -'/" -
1) + Poet.
(25)
A similar expression can be obtained for Rb. Equation (25) is a complex algebraic expression for R, as a function of time. This expression has been evaluated numerically for a range of values of 7, the viscosity coefficient. For values of , / > 1000 poise, it has been demonstrated that the expression for Re(t) can be closely approximated by neglecting in eqn (23) the product RoRo in comparison with other terms, and writing
Poe
R° ~ R,,.oexp [ 2 - - ~ ( 1 - e - ' )
Yt In KN]
~?K"-~ J"
(26)
Differentiation of (26) with respect to time yields an expression for the rate of expansion of the crater-cavity boundary,
[Poe-`/" Y In ~N.]
Re ~ Re L 2~Ko~
~Ko, j"
(27)
1140
M.R. Seizew, A. L. Kuhl and V. D, Blankenship
At /1~a = O, the crater-cavity has reached its maximum dimension, and the so-called "flash time" is given by tF = - r
In [2Yln [ Po
]
(28)
Ku_l"
By knowing Ra and Ra as a function of time, the strain rate, hence, the energy density resulting from irreversible work, can be evaluated at a given point in the deformed region (i.e. the region between the maximum crater cavity radius and the radius of the shock front at which the stress has attenuated below the dynamic yield point). Here we assume that the Eulerian as well as the Lagrangian strain rates can be approximated by the corresponding rates of deformation or that, in general, 1 (aui+ Ouf~
(29)
where u and x are velocity and coordinates associated with i and j direction. The energy density can be formulated as an integral of the product of generalized stress and deformation rates, namely (30)
w(R) = J[2koxl
With a spherical coordinate system and the results of the previous analysis, the energy density can be expressed as H
0
0"0 - 3 ~ q ~ +
o
RaRo2 ~ o
0
0-0 0
o
o
0
0
--2~
0
RaRa2 -fir-
Y
0
0 dt
o'o
or
(31) In this expression, the symbol tR is the shock arrival time at the radius R, and tF is the "flash time" when all stresses in the target material have attenuated to a magnitude below the dynamic yield point• To integrate eqn (31) for a given radius R, a linear dependency of shock arrival time on radius is assumed and therefore R can be defined in terms of R. by considering deformed region Ro <-R <- Rb. Thus R = R~[I+ a--~-N N]
(32)
The roleof viscosityof solids in microparticlecratering
1141
Yln ] tR = - N ~ - l n [L2P0 KNj,
(33)
and
where N is a fraction of ( R b -- Ro) and 0 - N < 1. Substituting relations (26), (27), and (32) into eqn (31), we have
W(N) = I + ( A _ N6)
-(1+
6VlXo,~f:l,R (~---~°e-#S-YInKN) 2 dt
N2YN~3 K
ftlt(-~e-#5-Yln~cN)dt"
(34)
With the relations (28) and (33) taken for limits of integration in (34), the final relation for energy density as a function of N is
w(N)=M,+A_N]. [(2Yln 6~
~N {2Yl n
2
+PoYInKN[-~InKN _ {2Y \P0 In KN)N] + Y21n2xN In (2Y In KN)(N-- 1)} \ Po 2Y1"
[po [{2Yl n
- (l+a--~-N N) 3Ko, 1.2L\Po
N 2Yln KN) -- "~0 KN]
+ Y ln KN In f2YIn KN)(1-- N)}. \Po
(35)
The surface temperature for flow conditions when Ah ~ d will be predominantly dependent on the state of the matter caused by the viscous dissipation of the kinetic energy. The surface energy density associated with this "flash" temperature can be estimated from relation (35) when N = 0. Thus, the surface energy density is
6~.{~_.~2[l_(2YlnxN)]2+PoYlnxN[~lnxN_l]_Y21n2xN. \ Po
W.u~,~.=-~
x (2Yln
X] 2Y~"
[l_2YlnxN]+YlnKNln(2Yln
(36)
Comparison of analytical results with experimental data Comparisons have been made between the viscous flow model and experimental data, as available. No information was found on the energy density
1142
M.R. Seizew, A. L. Kuhl and V. D. Blankenship
distribution, however, data were located on material viscosity and surface temperature at the crater boundary for high velocity particle impacts in homogeneous aluminum.t Thus viscosity and surface flash temperature were evaluated analytically from relations (26) and (36), respectively. In these equations, the initial elastic radius of the crater, R,0, was determined from the Hertzian relation (eqn B-3), while the initial flash time was estimated from experimental impact data for tantalum and quartz targets (Friichtenicht, 1965). Calculated values of viscosity and surface flash temperature are compared with experimen~ tal values from Denardo et al. (1966, 1967) in Table 1. As can be seen from this table, calculated and experimental viscosities are in good agreement when the impacting particle diameter is less than or equal to the shock front thickness. At the lower impact velocities the experimental and calculated values of viscosities are in agreement, and they seem to be independent of the shock thickness and particle dimension. At the higher impact velocities, deviation between values of viscosities is noted and may be attributed to the phase change/melting of the aluminum. The temperature comparisons shown in Table 1 indicates a good agreement in calculated and experimental values. This observed temperature, 2700°K, and the calculated energy density shows that the surface of the aluminum target is vaporized when it is subjected to the impact of an aluminum particle travelling at 4km/sec. This is a very important aspect of the high velocity impact process which can now be properly evaluated with relation (36). It is also important to note (Appendix A) that the temperature behind shock front as calculated using an isentropic compression process and equation of state of aluminum is much lower--approximately equal to 1200°K for the considered impact velocity. Thus, the irreversibility of the process in the SWF region is verified. The limits of application of the viscous flow model developed here are difficult to estimate because of the limited data base. Assuming, however, that the experimental measurements are accurate at both high and low velocities, and that the assumptions concerning parameters of the models are valid, a definite limit of application would appear to exist within the velocity range studied herein. Conclusions The following conclusions are justified by the analysis of this study: (1) The flow region in the target generated during hypervelocity impact is related to the particle size and velocity. If the particle size is the same order of magnitude as the shock thickness (which depends on the impact conditions and material), viscous effects must be considered. (2) The analytical model developed here is valid for velocity range considered in this study for homogeneous metals. (3) Microparticle impact tests can be used to experimentally evaluate the viscosity of the solids at high pressures and temperatures. t(See Sakharov et al. (1965), Mineev and Savinov (1967), Godunov et al. (1971), Denardo et al. (1967), and Slattery (1966)). ~tExperimental values of temperature were obtained by using Pyrex glass particles ~ 50/~ diameter and aluminum targets.
The role of viscosity of solids in micropanicie cratering
1143
References Bitter, J. G. (1963) A study of erosion phenomena, part I,Wear, Vol. 6. Denardo, B. Pat and Mysmith, C. Robert (1966) Momentum transfer and cratering phenomena associated with the impact of aluminum spheres into thick aluminum targets at velocity to 24000 Ft/SCc. Agardograph 87, Vol. I. Gordon and Breach, New York. Denardo, B. Pat, Summers, James L. and Mysmith, C. Robert (1967) Projectile Size Elect on Hypervelocity Impact Crater in Aluminum. NASA TND-4067. Friichtenicht, J. R. (1965) Experiments on the Impact-Light-Flash at High Velocities. NASA Report 4158-6017-TU-000. Godunov, S. K. et al. (1971) Investigation of viscosity of metals in high-velocity collisions. Combustion, Explosion and Shock Waves Journal. Academy of Sciences USSR, No. I. Hopkins, H. G. (1960) Dynamic expansion of spherical cavities in metals. Progress in Solid Mechanics, (Edited by I. N. Sneddon and R. Hill), Vol. I, Chap. III. North Holland, Amsterdam. MacCormack, R. W. (1969) The effect of viscosity in hypervelocity impact cratering. AIAA Hypervelocity Impact Conference, paper No. 69-354. Mineev, V. N. and Savinov, E. V. (1967) Viscosity and melting point of aluminum, lead and sodium chloride subjected to shock compression. Soviet Physics--JETP 25(3). Mineev, V. N. and Savinov, E. V. (1968) Viscosity of water and mercury under shock loading. Soviet Physics~JETP 27(6). Rosen, F, D. and Scully, C. N. (1965) Impact flash investigations to 15.4" KM/Sec. Seventh Hyperveiocity Impact Symposium Proceeding. VoL VI. Sakharov, A. D., Zaidel, M. R., Mineev, V. N. and Oleinek, A. G. (1965) Experimental investigation of the stability of shock waves and the mechanical properties of substances at high pressure and temperature. Soviet Physics--Doklady 9(12). Seizew, M. R. and Danker, G. R. (1973) Irreversible Energy Density Distribution in Materials Subjected to a High Velocity Panicle Impact. Report No. 22168--6321-RU-00, TRW Systems Group, Redondo Beach, Calif. Slattery, J. C. (1966) Experimental Research on Hypervelocity Cratering by Microscopic Panicles. NASA Report 0346.-6001-RO-00. Zaidel, R. M. (1967) Development of perturbations in plane shock waves, Journal of Applied Mechanics and Technical Physics. Vol. 8, No. 4, Academy of Sciences, USSR. Zeldovich, Ya. B. and Raizer, Yu. P. (1966) Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena. Academic Press, New York. Appendix A
Temperature behind the shock due to isentropic compression If we neglect electronic excitation to pressure and temperature, which is a good assumption when pressures are less than 1 Mb, the total energy and pressure can be expressed as follows: E ffi Ex + ET
(At)
P = P~ + Pr
(A2)
where E~ is "cold" specific energy; Er is "thermal" specific energy; Px is "cold" compression pressure; Pr is "thermal" compression pressure. Relations of E. P, P~ and E are as follows: V0
P~(V)dV V-
C2
Pr --['Err V
(A3)
I,/o"
(AS)
1144
M.R. Seizew, A. L. Kuhl and V. D. Blankenship
1
E = ~(V o- V)P
(A6)
where F is the Gruneisen constant, and where C is speed of sound: ~ is adiabatic exponent; V is specific volume. Using these relations, one can solve for E r by
P (V o- V ) - f f ~ P ( V ) d V ET =
- f v° F dV
1 Jv
(A7)
V
or, after integrating,
Er
n In-1
1 - F ln-~ If we use the value of P and (V0] V) for impact condition of 4 km/sec for aluminum (see Table l) and F = 2.09 and n = 4, the specific thermal energy behind the shock (A8) was approximately 8.1 × 109 ergs/cm 3. The calculated temperature was found to be 1200*K as compared to the experimental value of 2700"K. Using relation (36) one also obtains a temperature of 2700"K, an exact correlation with the experiment.
Appendix
B
Evaluation of the "initial elastic" radius The "elastic depth" H~ is defined as follows:
H~=~_~4y12R [ 1 - g ' 2 + 1 -g22] 2
--g?/J
(BI)
where Yl is elastic limit of target; Rp is radius of impacting particle; E~ is Young's modulus of target; E 2 is Young's modulus of impacting particle; gl is Poisson ratio of target material; g~ is Poisson ratio of impacting particle. The "elastic radius" is defined as:
R,,o=X/(2RpHa).
(B2)
,nn [1-gl 2 . l-g22] R ~ ' ° = ~ - ~ Y I K " [ - - - ~ * E2 ]"
(B3)
Substituting He~ (1B) into (2B) we have:
0094-5765/'7811101-1129~02.00]0
The role of viscosity of solids in microparticle crateringt M. R. S E I Z E W , $ A. L. K U H L § V. D. B L A N K E N S H I I ~
ASD
TRW Defense and Space Systems, One Space Park, Redondo Beach, CA 90278, U.S.A. (Received 30 March 1977; revised 2 August 1977)
Abstract--The importance of viscous effects on the impact cratering process has been investigated. The shock wave front region was characterized by the balance of inertia and viscous forces (Reynolds number ~ 1), and by a continuous distribution of all thermodynamic quantities. The width of the shock wave front region and its flow properties were then related to the impact particle size and velocity, and to the target material properties. Experimental data for aluminum demonstrated that cratering characteristics were changed when the impact particle size was the same order of magnitude as the shock wave front region. An analytical model was developed for cratering dominated by this "viscous" flow. Results from this model indicate a substantial decrease in crater radius and increase in crater surface temperature when compared with inviscid analysis. In addition, a new test method was identified to evaluate the viscosity of solids.
Introduction TsE IMPACTof particles on a high speed aerospace vehicle is a major area of concern to system designers. These particles can cause damage by penetrating and cratering thermal protection surfaces resulting in erosion of the heatshield and mass loss. These phenomena will also increase the benign aerodynamic heating while a vehicle is flying in the atmosphere, whether in a natural or an artificially produced particulate environment. The initial interaction of the impacting particle and target is an important phenomenon related to the partitioning of the kinetic energy of the particle and the subsequent increase in heating. In general, the analysis of the impact cratering process relies on sophisticated inviscid numerical methods, and on scaling empirical data to different impact conditions. However, at the present time there are no data available, either analytical or empirical, that can be used to evaluate the performance of the heatshield materials in erosive environments consisting of small particulates in the micron range. As pointed out by MacCormack (1969), application of the inviscid theory to this type of impact condition is not valid when the particle size is the same order of magnitude as the viscous region generated during impact. A more rigorous tThis work was performed under a TRW independent research and development program, Project No. 76004880. This paper was presented at the Sixth Colloquium on Gasdynamics of Explosions and Reactive Systems, Stockholm, Sweden, 22-26 August 1977. ~;Head of Energy Dynamics Section, Vulnerability and Hardness Laboratory. §Member of the Technical Staff, Vulnerability and Hardness Laboratory. IHead of the Reentry Systems Department. 1129
1130
M.R. Seizew, A. L. Kuhl and V. D. Blankenship
approach is needed which takes into account all physical process involved. The objective of this study was to determine how material viscosity affects the particle-heatshield interaction process as a function of the size and velocity of the impacting particle. First, a postulate relating the size and velocity of the impacting particle to the effects of the viscosity in the considered impact cratering process was formulated and rationalized by considering the properties of the shock-wave-front (SWF). The validity of this basic postulate was verified by using the available impact test data for aluminum. Then an approximate analytical model for this viscous interaction process was developed. Finally, the viscous flow model was used to correlate viscosity test data with analytical predictions. In addition, a new test method was identified to evaluate viscosity of solids. Analysis
A number of investigators have attempted to synthesize some aspects of the phenomenon associated with the effects of viscosity in a hypervelocity impact process and other shock-related problems. Direct analytical methods have been used by MacCormack (1969) and Zaidel (1967), while a number of investigators have performed experimental studies.t It is evident that the absence of the theoretical treatise for viscosity of the solids limits these studies. Nevertheless, the investigations were partially used in formulating and rationalizing the postulate for the viscous flow and the subsequent analysis.
Formulation of the viscous flow postulate Denardo et al. (1966, 1967) have reported some inconsistencies in a simple linear scaling rule for small particle, high velocity impact conditions. Penetration, radius of the crater formed, and transfer of momentum to the target decreased more rapidly than simple scaling would imply as the size of particle was reduced. This nonlinear effect was attributed to rate-dependent stress effects important at the initial stage of cratering, and materials strength effects which influence the last stage of crater formation. MacCormack (1969) attempted to correlate this experimentally observed deviation from a simple linear scaling rule by taking into account rate-dependentstress-viscosity effects. In that study, the Navier-Stokes equations were solved numerically with a finite-difference method which was second-order accurate in both time and space. Although the results were not conclusive, the method demonstrated increasing importance of viscosity as the size of the impacting particle diminished. Two important observations can now be noted from that study: (1) Viscosity is important when the strain-rate is high, which corresponds to the initial phase of cratering, and (2) The effects of viscosity become more predominant in a flow produced by an impact particle of small size.
al.
tSee Sakharov (1971).
et al.
(1965), Mineev and Savinov (1967), Mineev and Zaidel (1968), and Godunov
et
The role of viscosity o[ solids in microparticle cratering
1131
Let us now consider some important physical aspects of shock wave propagation in a compressible medium. On the one hand, the theoretical studies of Zeldovich and Raiser (1966), and Zaidel (1967) demonstrate that the shock front is a transition layer of finite thickness in which there occurs a change of all quantities that characterize the state of the medium. Furthermore, the studies show that without taking into account viscosity, one cannot produce a continuous distribution of thermodynamic properties in the shock front. On the other hand, experimental studies by Sakharov et al. (1965), Mineev and Savinov (1967), Mineev and Zaidel (1968), and Godunov et al. (1971), show perturbations of a plane shock wave are damped as the wave propagates due to the effects of viscosity. At this point, let us assume that the inertial forces and viscous forces in the SWF are in equilibrium, that is, that the Reynolds number based on the shock thickness (i.e. R~ = pvAh/vl) is of order one.t Thus, the thickness of the SWF can be deduced as •h =~ pv
(1)
where Ah = the width of the SWF, [Ah] = cm; 7/= the dynamic viscosity in the SWF region, [7] = gmlcm-sec; p = the density in the SWF region, [p] = gm/cm~; v = the material velocity in the SWF region, [v] = cmlsec. This is the characteristic dimension of the viscous flow region formed by the shock wave. As this relation shows, the transition layer thickness is dependent on viscosity, which is in turn a function of pressure and temperature of the medium. It is well known that when a infinite plate of a finite thickness impacts a semi-infinite half-space of the same material, the initial width of the flow region formed in the target is proportional to the thickness of the impacting plate. This proportionality is, of course, deduced from the time required for shock and rarefaction waves to traverse the thickness of the impacting plate. When a particle of a specified size impacts a semi-infinite target, the width of the flow region formed in the target is also identically proportional to the diameter, d, of the impacting particle. Thus, when the diameter of the impacting particle is large, the extent of the flow region in the target which has been engulfed by the shock is also large compared with the shock front thickness. Therefore, the flow in the region ecompassed by the shock wave can be considered inviscid. However, as d approaches Ah, the width of the flow region approaches the width of the SWF for a specified impact condition and the flow becomes predominantly viscous. This is consistent with the results of the numerical calculations performed by MacCormack (1969), which showed the increasing importance of viscosity as size of the impacting particle diminishes. Now we can formulate the hypothesis for the existence of the viscous flow in a hypervelocity impact process:
tlndeed, the above experiments show that viscosityis inverselyproportional to Reynoldsnumber.
1132
M.R. Seizew, A. L. Kuhl and V. D. Blankenship
The initial stage of a particleltarget iteration process is dominated by viscous effects when the impacting particle diameter is the same order of magnitude as the shock front thickness. Verification of the viscous flow postulate To examine the validity of the above postulate, two types of experimental data were required. First, data were required to evaluate the thickness of the SWF for materials whose viscosity is known as a function of pressure and temperature. A literature survey was made and reliable data were found for aluminum. Using relation (1), experimental viscosities from Mineev and Savinov (1967) and Mineev and Zaidel (1968), and the equation of state for 90% pure aluminum, the variation of the SWF thickness and temperature were established as a function of impact velocity as shown in Fig. 1. The bounds of the SWF, as shown in this figure, are a result of the experimental uncertainty in the viscosity data for aluminum. At this point, the available impact data for aluminum (Denardo et al. (1967) and Slattery (1966)) were grouped according to impact velocity, particle radius, and ratio of formed-crater radius to impacting-particle radius (Table 1). This data is displayed in Fig. 2. There the particle size effect (an effect strictly absent from 1600
I
I
I
l
3500
I /
MATERIAL:
/
ALUMINUM 90% PURE
/ /
1400 --
-- 3000 T
'u 1200
-
2500 ~_~ z
.~
o v
. 1000
2000
800
1500
i.-
v
~°°°I
~
400
2oo
-- 500
I
I
I
I
I
3
4
5
6
7
o 8
PARTICLE IMPACT VELOCITY, V e (km/sec) Fig. 1. Shock f r o n t thickness and temperature in aluminum as a function o f impacting
particle velocity.
The role of viscosity o] solids in micropanicle cratering
5.5
j
I
J L
I
i
I
1133
I
"LARGE"PARTICLE~ DATA (R - 1500~ " ~ , , ~ /
leoo
•
) _~ 4.5
O 2Rp~:~h •
2Rp
1200
~3.S
~
o.ok/ / /i .
/
2
I
I
"
I
I
I
3 4 5 6 7 PARTICLEIMPACTVELOCITY, V° (kin/me)
8
Fig. 2. Viscous (i.e. particle size) effects on the ratio of crater radius to particle radius formed by the hypervelocity impact of aluminum spheres on a semi-infinite aluminum slab; large and small particle data converge when the shock front reaches maximum thickness.
an inviscid description of this flow process) is demonstrated dramatically. "Small" particles (Rp < 10/~) generate craters that are about half the size (i.e. half the non-dimensional crater radius) of craters formed by particles that are large compared to the shock thickness. Refer, for example, to the cratering data in Fig. 2 at an impact velocity of 7.5 km/sec. In addition, the crater results for "large" and "small" particles are similar in the region where the shock front is the thickest (particle velocities of 5.5 km/sec).t Outside this impact regime, "large" particles behave in a more or less inviscid manner, while "small" particles are always affected by viscosity. These observations, the convergence o f the i m p a c t data and their similar functional d e p e n d e n c e as S W F ( w h i c h i n c l u d e s m e l t region at 7.5 k m / s e c ) , are t a k e n to be a p r o o f o f the v i s c o u s flow postulate.
tit should also be noted that this is the impact resime where the characteristic dimension of "large" particles as well as "small" particles is less than the shock thickness, Ah.
Speed of Sound,
5.2 2.6 3.0
0.079 2.3 x IO-’ 2.1 x lo-’
0.04-0.03
7.0-7.5
tSee text for definition. Density, p0 = 2.8 gmlcc;
4.2 4.5
0.158 0.079 2.3 X lo-’
0.13-0.07
5.5-6.0
10’4
4.3 4.0
0.158 0.079
0.05-O. 13
5.0-5.5
a.3* 1.0 Extrapolated
0.08409
4.5-5.0
4.3 4.0 3.5 2.5 2.2 2.3
0.06-0.08
4.0-U
C z 5.0 X !u‘ cmisec.
722 Extrapoiated
IO%4
4.9 2 2.4 Extrapolated
2eo.5
0.317 0.158 0.079 4.1 x lo-’ 3.2 X lo-’ 2.4 x IO-’
0.02-0.045
0.635 3.317 0.158 0.079 0.158 0.079 3.5 x IO-’
3.04.0
3.85 3.80 3.0-3.7 3.6 3.84.0 4.0 2.0
Radius of particle R,-cm
Thickness of SWF Ah-cm
Impact velocity V,-kmlsec
Ratio of crater and particle radii
Yield,
50-200
50-200
50-200
50-200
50-200
50-200
r _ 9.4 Kb; Young
Ah > ZR, IO Ah C 2R, 5 Ah > 2R, 6.5 Ah < 2R, 6 Ah > 2R, 8 Ah < ZR, 6 Ah > 2R, I1
Ah i 2R, 4.0 Ah > 2R, 8 Ah < 2R, 4.0
Ah < 2R, 2.3 zt 0.3
Initial Rash-ligbttime s-see X 10m9 estimated
977- 1080
685-777
598-685
5 16-598
366-516
299-366
Initial impacr pressure P,-Kb
2700
Temperature T-OK experimental
data
Ratio,
2700
0.64-0.63
0.68-0.67
0.70-4I.68
0.724.70
0.74-0.72
0.78-0.74
%I
Constantt
D - 0.33: R,,
Temperature T-OK calculated
E = 689 Kb; Poisson
impact
Modulus.
of the aluminum-aluminum
Viscosity v-poise x IO’ calculated
1. Summary
Viscosity q-poise X l@ experimental or extrapolated
Table
-? 5.4 ,” 10 2 RD
I .52-I .49
1.56-1.52
1.67-1.56
*I
Constantt
The role of viscosity of solids in micropanicie crater/rag
1135
Analytical modeling o f v i s c o u s / l o w
An analytical model for this viscous impact process has been formulated as follows: (a) Crater formation in the target is accomplished by the radial expansion of a spherical cavity (see the treatise by Hopkins (1960) for a-general discussion of cavity dynamics in solids). (b) Impact velocity and particle size prescribe the driving conditions on the cavity wall. (c) The target medium is assumed to behave like viscous solid (i.e. stresses depend on strain rates) which locks up on compression. (d) The Cauchy equations of motion are then evaluated to determine the cavity expansion, in particular, the maximum crater radius. (e) Solution of this flow field defines the energy density distribution along the radius and also irreversible work done during cratering as well as the sudace temperature. Assuming spherically symmetrical flow in the target material, the Cauchy equation of motion in terms of the physical components of stress can be written: OcrR . 2 D( v ) OR * -R (err - ere) = p Dt '
(2)
where err = the stress in radial direction; t = the time; or, = stress in "hoop" direction; R = radial coordinate; p = density; v = mass velocity in radial direction, and D(v)
Or+
Dt
Ov
vT "
Next, using the definition of the strain field, one can easily derive the relation of strain rate and velocity. For the radial component, OU
0--R =
(3)
eR
and
O(eR)= 02u Ot
OtOR
Ov OR"
Likewise, for the "hoop" direction, U
(4)
and
R Ot
R"
In these relations, u is the deformation field, eR ffi strain in radial direction, and e, --strain in hoop direction.
1136
M.R. Seizew, A. L. Kuhl and V. D. Blankenship
At this point, an assumption concerning the constitutive relation of the target material is introduced, namely, that the dilitation of the target material is limited to a constant value. That is, the target material is assumed to be a rigid "locking solid" in compression.* Although this assumption is not physically realistic in a general sense, it does greatly reduce the complexity of the analysis. The assumption of a "locking" medium can be formulated in terms of strain components: • R + ~e + ~
= ~R + 2 ~ e ~ = ~L.
(5)
The functional relation of the velocity in terms of R and t can be derived by differentiating (5) with respect to time, and using relations (3) and (4). The resulting partial differential equation is ~_t(eR)+2~t(~e ) = a OR v+
2 ~v=
0.
(6)
Integration of (6) gives
f(t)
v = -~T,
(7)
where [ ( t ) is a continuous unknown function of time. Equation (7) will hold for all material states in the locking region. A further assumption concerning the constitutive law obeyed by the target material is now introduced. Using the above postulate, it is assumed that the differences in the principle stresses can be expressed in terms of the shear strength and the strain rate difference, or (trR-tr0)= Y
+ lair 7/~ ~
a~e]
c~t/
(8)
where Y represents the shear strength of material, and r/ is the constant of proportionality, the viscosity coefficient. From these relations, Cauchy's equation can now be formulated in terms of one principle stress component, ~rR, and in terms of f(t) and R as follows:
r/u) °'q--R-4-= PP[-~-'z
ao'R 4 2.2.2_~ .. f ( t ) OR
2~]
(9)
where
/ ( t ) = af(t) cgt pp = limiting "locking" denisty. tThis assumption, in a rigorous way, is related to the Hugoniot of the material with initial "critical" density, i.e. (I -PcR/P)= (21(2+ F)) where F is a Gruneisen coefficient. ~tln the case of spherical symmetry, note that e6 = e~ and cro= ~r~.
1137
The role of viscosity of solids in microparticle cratering
Equation (9) can be integrated with respect to R, giving cra - 2 Y In R +2./ R, = pp [f'(t) L-~-c- f- (~tj) ] + g ( t )
(10)
where g(t) denotes a new function of time. This new function, g(t), can be eliminated from relation (10) by considering the boundary conditions at the crater-cavity. Defining the crater-cavity radius by Ro and prescribing the pressure-time function at this depth, relation (10) becomes [f'(t) - P ( t ) - 2 Y In R~ + 2,/~(~ Ro = Po [2-~-/R(~t)] + g(t). (11) If we solve for g(t) in (11) and substitute back into relation (10), we have erR= - P ( t ) + 2 Y
In ~ + 2,pf(t)[~-~- ~--~]-~ f2(t)[~'~- ~ l ] + Op](t)[~_~_ 1 ] .
(12) Consideration of mass and momentum conservation at the shock front allows one to relate f(t) to the shock front parameters/~b, Rb, and Rb, the acceleration, velocity, and distance travelled by the shock front, respectively. From conservation of mass across the shock, we write (13)
po(Rb - v) ~ pORb
This relation will hold exactly for only a perfectly discontinuous single shock front moving into an undisturbed medium. When (7) is used in conjunction with (13), it follows that (14)
f ( t ) = RvRb2(1--~)
Conservation of momentum across the shock gives ~R~ b
~- [Rb - f(t)] f(t) L
(15)
where the minus sign is present because of the convention that stress is negative in compression. Equations (14) and (15) can be used in conjunction with relation (12) to derive the following relation: 1
Po/
1
2
Po/J
1
1
Po
,Op
\ - ~-pj ~ - ~ - ~-~bj + 2 Y In ~-~..
(16)
1138
M.R. Seizcw, A. L. Kuhl and V. D. Blankenship
To solve eqn (16) for Ro or Rb as a function of time, we need to prescribe P ( t ) and define the relationship between R~ and Rb. The relation between R, and Rh is defined by considering mass conservation in the deformed region, before and after deformation. For the material within radius Rb, the appropriate conservation relation is poRb = pp ( Rb 3 -- Ra3).
(17)
This relation is consistent with the assumption of a locking medium (i.e. p = pp = constant behind the shock front), and with the further assumption that the loss of ejected material f r o m the crater cavity is neglected. Equation (17) leads to the desired relationship between Ra and Rb, such as (18)
R~ 3 = R b 3 a
where
Differentiation of (18) with respect to time gives RoRo 2 = RbRbZ a = f ( t ).
(19)
This is consistent with (14), and with eqn (7) evaluated at R = R,. With the use of relations (18) and (19), eqn (16) can be put into the following form in terms o f / ~ , , Ro, and Ro : P ( t ) = A d ( R o R , ) + ( B - A)Ro2 + c d (ln R , ) + D
(20)
where A, B, C, and D are constants given by A = pp(l - ot ~n) B = ~ ( 3 - ot 4t3 - 2a v3)
(2])
C = 27(1 - a ) D = 2Y In (a-lJ3).
The constants C and D, proportional to the viscosity coefficient and dynamic yield stress, are several orders of magnitude larger than A and B for nearly all materials. Hence, it is a reasonable approximation to neglect the second term in the right hand side of (20) and write d
P ( t ) = ppro d ( / ~ o ) + 2~ro, ~ (In Ro)+ 2Y in rN
(22)
The role of viscosity of solids in microparticle cratering
1139
where go = ( 1 -
a l / 3 ) , ~o, = (1 - a ) , r N = a -'13.
If we assume that the pressure in the crater cavity is P = Poe-tl,, where ¢ is characteristic of the impact conditions, the material properties, and the configuration of the impacting particle, eqn (22) can be integrated with the result
R°
K~OpRoRo+ 2,/KoI In _-~--+ 2Yt In KN = -P0¢e -'/" + PO'. /(a,0
(23)
The radius of the cavity, Ro, can be estimated by integrating eqn (23) and by noting that dt = dRJRo and assuming R° = constant, to give 1
2
~Ra K~Op= - ~
[Ra ln ~o~,o-Ra] - Y ln gNtZ+ Po¢[f(e-tl" -1) + ¢].
(24)
In the derivation of eqns (23) and (24), the integrations were begun at time t = 0 and crater-cavity radius Re.0, which corresponds to the elastic deformation of the target surface caused by the impacting particle before the plastic deformation begins. Further, Re.0 is assumed to be negligible in comparison to Re and, therefore, appears only in the logarithmic terms in (23) and (24). If we use (23) to eliminate Re, an expression is found for Re, the crater cavity radius as a function of time: Re _ Re] f 2~/ra, LRo In ~ j RoK~o~ - Yt 2In KN ~Ra Kagp = Ra 2~lKo:,ln-~--+ 2tY In rN + Po¢(e -t/¢- 1) 1
Z
Ka,0
+ p o ¢ 2 ( e -'/" -
1) + Poet.
(25)
A similar expression can be obtained for Rb. Equation (25) is a complex algebraic expression for R, as a function of time. This expression has been evaluated numerically for a range of values of 7, the viscosity coefficient. For values of , / > 1000 poise, it has been demonstrated that the expression for Re(t) can be closely approximated by neglecting in eqn (23) the product RoRo in comparison with other terms, and writing
Poe
R° ~ R,,.oexp [ 2 - - ~ ( 1 - e - ' )
Yt In KN]
~?K"-~ J"
(26)
Differentiation of (26) with respect to time yields an expression for the rate of expansion of the crater-cavity boundary,
[Poe-`/" Y In ~N.]
Re ~ Re L 2~Ko~
~Ko, j"
(27)
1140
M.R. Seizew, A. L. Kuhl and V. D, Blankenship
At /1~a = O, the crater-cavity has reached its maximum dimension, and the so-called "flash time" is given by tF = - r
In [2Yln [ Po
]
(28)
Ku_l"
By knowing Ra and Ra as a function of time, the strain rate, hence, the energy density resulting from irreversible work, can be evaluated at a given point in the deformed region (i.e. the region between the maximum crater cavity radius and the radius of the shock front at which the stress has attenuated below the dynamic yield point). Here we assume that the Eulerian as well as the Lagrangian strain rates can be approximated by the corresponding rates of deformation or that, in general, 1 (aui+ Ouf~
(29)
where u and x are velocity and coordinates associated with i and j direction. The energy density can be formulated as an integral of the product of generalized stress and deformation rates, namely (30)
w(R) = J[2koxl
With a spherical coordinate system and the results of the previous analysis, the energy density can be expressed as H
0
0"0 - 3 ~ q ~ +
o
RaRo2 ~ o
0
0-0 0
o
o
0
0
--2~
0
RaRa2 -fir-
Y
0
0 dt
o'o
or
(31) In this expression, the symbol tR is the shock arrival time at the radius R, and tF is the "flash time" when all stresses in the target material have attenuated to a magnitude below the dynamic yield point• To integrate eqn (31) for a given radius R, a linear dependency of shock arrival time on radius is assumed and therefore R can be defined in terms of R. by considering deformed region Ro <-R <- Rb. Thus R = R~[I+ a--~-N N]
(32)
The roleof viscosityof solids in microparticlecratering
1141
Yln ] tR = - N ~ - l n [L2P0 KNj,
(33)
and
where N is a fraction of ( R b -- Ro) and 0 - N < 1. Substituting relations (26), (27), and (32) into eqn (31), we have
W(N) = I + ( A _ N6)
-(1+
6VlXo,~f:l,R (~---~°e-#S-YInKN) 2 dt
N2YN~3 K
ftlt(-~e-#5-Yln~cN)dt"
(34)
With the relations (28) and (33) taken for limits of integration in (34), the final relation for energy density as a function of N is
w(N)=M,+A_N]. [(2Yln 6~
~N {2Yl n
2
+PoYInKN[-~InKN _ {2Y \P0 In KN)N] + Y21n2xN In (2Y In KN)(N-- 1)} \ Po 2Y1"
[po [{2Yl n
- (l+a--~-N N) 3Ko, 1.2L\Po
N 2Yln KN) -- "~0 KN]
+ Y ln KN In f2YIn KN)(1-- N)}. \Po
(35)
The surface temperature for flow conditions when Ah ~ d will be predominantly dependent on the state of the matter caused by the viscous dissipation of the kinetic energy. The surface energy density associated with this "flash" temperature can be estimated from relation (35) when N = 0. Thus, the surface energy density is
6~.{~_.~2[l_(2YlnxN)]2+PoYlnxN[~lnxN_l]_Y21n2xN. \ Po
W.u~,~.=-~
x (2Yln
X] 2Y~"
[l_2YlnxN]+YlnKNln(2Yln
(36)
Comparison of analytical results with experimental data Comparisons have been made between the viscous flow model and experimental data, as available. No information was found on the energy density
1142
M.R. Seizew, A. L. Kuhl and V. D. Blankenship
distribution, however, data were located on material viscosity and surface temperature at the crater boundary for high velocity particle impacts in homogeneous aluminum.t Thus viscosity and surface flash temperature were evaluated analytically from relations (26) and (36), respectively. In these equations, the initial elastic radius of the crater, R,0, was determined from the Hertzian relation (eqn B-3), while the initial flash time was estimated from experimental impact data for tantalum and quartz targets (Friichtenicht, 1965). Calculated values of viscosity and surface flash temperature are compared with experimen~ tal values from Denardo et al. (1966, 1967) in Table 1. As can be seen from this table, calculated and experimental viscosities are in good agreement when the impacting particle diameter is less than or equal to the shock front thickness. At the lower impact velocities the experimental and calculated values of viscosities are in agreement, and they seem to be independent of the shock thickness and particle dimension. At the higher impact velocities, deviation between values of viscosities is noted and may be attributed to the phase change/melting of the aluminum. The temperature comparisons shown in Table 1 indicates a good agreement in calculated and experimental values. This observed temperature, 2700°K, and the calculated energy density shows that the surface of the aluminum target is vaporized when it is subjected to the impact of an aluminum particle travelling at 4km/sec. This is a very important aspect of the high velocity impact process which can now be properly evaluated with relation (36). It is also important to note (Appendix A) that the temperature behind shock front as calculated using an isentropic compression process and equation of state of aluminum is much lower--approximately equal to 1200°K for the considered impact velocity. Thus, the irreversibility of the process in the SWF region is verified. The limits of application of the viscous flow model developed here are difficult to estimate because of the limited data base. Assuming, however, that the experimental measurements are accurate at both high and low velocities, and that the assumptions concerning parameters of the models are valid, a definite limit of application would appear to exist within the velocity range studied herein. Conclusions The following conclusions are justified by the analysis of this study: (1) The flow region in the target generated during hypervelocity impact is related to the particle size and velocity. If the particle size is the same order of magnitude as the shock thickness (which depends on the impact conditions and material), viscous effects must be considered. (2) The analytical model developed here is valid for velocity range considered in this study for homogeneous metals. (3) Microparticle impact tests can be used to experimentally evaluate the viscosity of the solids at high pressures and temperatures. t(See Sakharov et al. (1965), Mineev and Savinov (1967), Godunov et al. (1971), Denardo et al. (1967), and Slattery (1966)). ~tExperimental values of temperature were obtained by using Pyrex glass particles ~ 50/~ diameter and aluminum targets.
The role of viscosity of solids in micropanicie cratering
1143
References Bitter, J. G. (1963) A study of erosion phenomena, part I,Wear, Vol. 6. Denardo, B. Pat and Mysmith, C. Robert (1966) Momentum transfer and cratering phenomena associated with the impact of aluminum spheres into thick aluminum targets at velocity to 24000 Ft/SCc. Agardograph 87, Vol. I. Gordon and Breach, New York. Denardo, B. Pat, Summers, James L. and Mysmith, C. Robert (1967) Projectile Size Elect on Hypervelocity Impact Crater in Aluminum. NASA TND-4067. Friichtenicht, J. R. (1965) Experiments on the Impact-Light-Flash at High Velocities. NASA Report 4158-6017-TU-000. Godunov, S. K. et al. (1971) Investigation of viscosity of metals in high-velocity collisions. Combustion, Explosion and Shock Waves Journal. Academy of Sciences USSR, No. I. Hopkins, H. G. (1960) Dynamic expansion of spherical cavities in metals. Progress in Solid Mechanics, (Edited by I. N. Sneddon and R. Hill), Vol. I, Chap. III. North Holland, Amsterdam. MacCormack, R. W. (1969) The effect of viscosity in hypervelocity impact cratering. AIAA Hypervelocity Impact Conference, paper No. 69-354. Mineev, V. N. and Savinov, E. V. (1967) Viscosity and melting point of aluminum, lead and sodium chloride subjected to shock compression. Soviet Physics--JETP 25(3). Mineev, V. N. and Savinov, E. V. (1968) Viscosity of water and mercury under shock loading. Soviet Physics~JETP 27(6). Rosen, F, D. and Scully, C. N. (1965) Impact flash investigations to 15.4" KM/Sec. Seventh Hyperveiocity Impact Symposium Proceeding. VoL VI. Sakharov, A. D., Zaidel, M. R., Mineev, V. N. and Oleinek, A. G. (1965) Experimental investigation of the stability of shock waves and the mechanical properties of substances at high pressure and temperature. Soviet Physics--Doklady 9(12). Seizew, M. R. and Danker, G. R. (1973) Irreversible Energy Density Distribution in Materials Subjected to a High Velocity Panicle Impact. Report No. 22168--6321-RU-00, TRW Systems Group, Redondo Beach, Calif. Slattery, J. C. (1966) Experimental Research on Hypervelocity Cratering by Microscopic Panicles. NASA Report 0346.-6001-RO-00. Zaidel, R. M. (1967) Development of perturbations in plane shock waves, Journal of Applied Mechanics and Technical Physics. Vol. 8, No. 4, Academy of Sciences, USSR. Zeldovich, Ya. B. and Raizer, Yu. P. (1966) Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena. Academic Press, New York. Appendix A
Temperature behind the shock due to isentropic compression If we neglect electronic excitation to pressure and temperature, which is a good assumption when pressures are less than 1 Mb, the total energy and pressure can be expressed as follows: E ffi Ex + ET
(At)
P = P~ + Pr
(A2)
where E~ is "cold" specific energy; Er is "thermal" specific energy; Px is "cold" compression pressure; Pr is "thermal" compression pressure. Relations of E. P, P~ and E are as follows: V0
P~(V)dV V-
C2
Pr --['Err V
(A3)
I,/o"
(AS)
1144
M.R. Seizew, A. L. Kuhl and V. D. Blankenship
1
E = ~(V o- V)P
(A6)
where F is the Gruneisen constant, and where C is speed of sound: ~ is adiabatic exponent; V is specific volume. Using these relations, one can solve for E r by
P (V o- V ) - f f ~ P ( V ) d V ET =
- f v° F dV
1 Jv
(A7)
V
or, after integrating,
Er
n In-1
1 - F ln-~ If we use the value of P and (V0] V) for impact condition of 4 km/sec for aluminum (see Table l) and F = 2.09 and n = 4, the specific thermal energy behind the shock (A8) was approximately 8.1 × 109 ergs/cm 3. The calculated temperature was found to be 1200*K as compared to the experimental value of 2700"K. Using relation (36) one also obtains a temperature of 2700"K, an exact correlation with the experiment.
Appendix
B
Evaluation of the "initial elastic" radius The "elastic depth" H~ is defined as follows:
H~=~_~4y12R [ 1 - g ' 2 + 1 -g22] 2
--g?/J
(BI)
where Yl is elastic limit of target; Rp is radius of impacting particle; E~ is Young's modulus of target; E 2 is Young's modulus of impacting particle; gl is Poisson ratio of target material; g~ is Poisson ratio of impacting particle. The "elastic radius" is defined as:
R,,o=X/(2RpHa).
(B2)
,nn [1-gl 2 . l-g22] R ~ ' ° = ~ - ~ Y I K " [ - - - ~ * E2 ]"
(B3)
Substituting He~ (1B) into (2B) we have: