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“Taylor Instability” of the Surface of an Elastic-Plastic Plate
Ill ' T a y l o r Instability" of the Surface of an Elastic-Plastic Plate Daniel C. Drucker University of Illinois at Urbana-Champaign, Urbana, Illinois Summary. Surface bumps or waves of initial amplitude h0 and wavelength λ grow in a ductile metal plate when it is subjected to a sufficiently large acceleration a in the direction of the inward normal from the surface to the plate. This "Taylor instability" is explained and explicit results are given that agree with the main features of the one set of experiments that have been reported. In this first approximation the metal of density p is supposed isotropic and perfectly plastic with yield strength σ 0 in uniaxial tension or compression that is twice the yield strength in shear. A threshold value is found for pa.h0 below which no (appreciable) growth will take place and above which the increase of amplitude h is very rapid with time t. For plates of thickness H appreciably greater than /, and p(xh0 above the threshold ( 1 + n/2)a0F, pah - (1 + n/2)a0F = [p
1
INTRODUCTION AND A SET OF EXPERIMENTAL RESULTS
The term "instability" arises very naturally in the consideration of waves at the interface between two fluids. When, for example, the top surface of water exposed to air in an ordinary glass or tank is perturbed, it will oscillate. The restoring forces are the gravitational pull and the surface tension. Viscous forces will damp the motion and eventually bring the water to rest. A downward acceleration of the system equal to the gravitational acceleration (free fall) nullifies the gravitational restoring force and leaves only the surface tension to stabilize the wave motion. A still larger acceleration directed 37
38
Daniel C. Drucker
downward causes all small amplitude waves of long wave length to grow exponentially in amplitude with time. Only waves of sufficiently short wave length can be stabilized by the surface tension. This break-up of the surface, or instability, at rather moderate magnitude of acceleration directed from the surface or interface into the denser fluid is a "Taylor instability". It has been described and analyzed in some detail following the early work of Rayleigh and Taylor [1-3]. Quite naturally, therefore, when a corresponding phenomenon was of concern at the surface of a solid plate or body subjected to very large accelerations directed inward from the surface (Fig. 1 ) the earlier analysis was carried over [4]. The expectation was that the important feature would be a wave length instability threshold governed by elastic restoring forces. />ah 0 A 0 /F
ï
f
f>
Fig. 1 A small bump adds a tensile force Q to the uniform pressure P.
The expected growth of "waves" machined or pressed in a solid surface was demonstrated by Barnes, Blewitt, McQueen, Meyer, and Venable in the Los Alamos Phermex facility. They obtained remarkably clear pictures of a free aluminum plate 2.54 mm thick with an initial sinusoidal surface pattern of 2/z0 = 0.203 mm and 5.08 mm wave length subjected to a shockless pressure P of the order of 100 kbars on the wavy surface [5 ]. In 7 or 8 microseconds the 2/z value grew to over 1 mm (Table 1). Table 1 Initial dimensions and observed growth from Tables 1 and II of Ref. [5] Shot no. 1 2 3 4 5 6 7
Material 1100-0 1100-0
ιιοο-υ
1100-0 6061-T6 304S.S. 304S.S.
H (mm)
λ (mm)
(mm)
2/i 0
t (^sec)
2Λ (mm)
2.54 2.54 2.54 2.54 2:54 1.90 1.90
5.08 2.54 5.08 5.08 5.08 5.08 5.08
0.203 0.102 0.203 0.203 0.203 0.203 0.203
8.00 8.01 6.41 7.32 8.09 5.54 6.50
1.515 0.165 0.880 1.168 LI 27 0.476 0.725
"Taylor Instability"
39
However, the concept of waves and the transition from a stable oscillatory mode to an unstable exponential growth is not the most helpful approach for solids. Periodicity in space is not relevant. Elastic restoring forces also have minor relevance at most. Visible deformations, far smaller than the spectacular growth of the surface "waves" seen in the aluminum plates, are plastic, not elastic. Therefore, whether or not they occur is determined by a plasticity analysis. As a first approximation it is reasonable to idealize a very ductile solid as isotropic and perfectly plastic with an appropriate yield strength σ0 about twice the yield strength in shear. Of course,
2
INITIAL SOLUTION AND SUBSEQUENT MOTION FOR A PERFECTLY PLASTIC SOLID
The (shockless) pressure P needed to give an acceleration a to a plate of thickness H and mass density p, Fig. 1, is P=p
(2.1)
A small bump of height h0 and plan area A0 on the otherwiseflatsurface of the accelerating plate adds a tensile force Q to the uniform pressure on that plane. M.T. V.5—C
40
Daniel C. Drucker
Q = poth0A0 for a mesa-like uniform thickness bump and is less for a rounded bump: Q = pxh0A0/F
(2.2)
with the geometric factor F > 1. In the two-dimensional case (Fig. 2) Q is a force per unit dimension perpendicular to the paper, A0 is replaced by λ/2 and F is unity for a rectangular bump, π/2 for a half-sine-wave.
Fig. 2
Two-dimensional bumps and waves.
The two-dimensional Prandtl solution for a rigid punch pressing against a half-space (Fig. 3) is directly applicable to the small isolated rectangular bump on a thick plate. A regularly spaced set of punches with the width of each punch equal to the separation λ/2 between punches (Fig. 4) provides the needed information for a regular rectangular "wave" pattern of wave length λ. The limit force per unit area that must be exceeded for large-scale plastic deformation to occur in the elastic-perfectly plastic body of yield strength σ 0 is (1 + π/2)σ0 for either the isolated punch or the regular array of punches. This two-dimensional result is a fair approximation for the three-dimensional problem of the limit force per unit of area for any punch with a convex area of contact on a half space, although the value of 3σ0 which is on the high side is closer [6-8].
Fig. 3
Prandtl solution.
"Taylor Instability"
41
Figures 2,3, and 4 demonstrate the equivalence or approximate equivalence of the downward accelerating body with an isolated bump or a wave to the punch problem with signs reversed. The alternating side-by-side tensile and compressive forces shown acting on the mean surface of Fig. 4, each QL/2 = | ( 1 + π/2)σ0λ/2, are equivalent to the equally spaced tensile forces QL = (1 + π/2)σ0λ/2 and provide a more generally useful picture for a wave. The obvious result that a rectangular wave of amplitude, h0, (Fig. 2), which is a set of bumps of height 2/z0, reaches limit load at just half the acceleration required for a single rectangular bump of height h0 carries over to the upperbound calculation for all wave shapes.
M-+-T-+-H-—
λ
— H
Fig. 4 Multiple punches.
The threshold value of pah0 or Ph0/H Eq. (2.1) is given by poLhl" = PhT0H/H = (1 + n/2)a0F
(2.3)
where F for a wave is one-half of the F for the corresponding isolated bump: one-half for the rectangular wave and π/4 for the sinusoidal. Below the threshold value the plastic deformation will be contained and no appreciable change in geometry will result. Above the threshold, the geometry change will be large. The continual increase in h from the initial h0 causes the driving forces to increase and the system to run away in an extremely short time. The initial value of dv/dt or d2h/dt2, the acceleration with respect to the
42
Daniel C. Drucker
surface of the moving plate, is proportional to the excess of the driving force Q over the required QL. Now the details of the assumed velocity pattern for plastic deformation and the volume of material participating play a far stronger role than in the calculation of the limit force QL. Therefore the result for d2h/dt2 is more approximate still. The equation of motion associated with the Prandtl velocity pattern follows directly from D'Alembert's principle. An upward virtual displacement Δ of the bump and the rigid triangular region below results in a displacement A^/2 everywhere in the two quarter-circle fan regions of shear and a downward displacement Δ in the rigid triangular side regions shown in Fig. 4 for the wave patterns (or a 45° inclined displacement of Δ/^/2 in the two rigid triangular sliding regions shown in Fig. 3 for the isolated bump). The acceleration of each point of each region is in the same ratio to d2h/dt2 as the virtual displacement is toA. d2^ Ιλλ 1 d2h 7i//L/2\ A A + 2p 224 2 dt2 4 \ 4
(Q-QL)A=p
le
„ d2h \ λλ A for the wave or 244 + 2p h λ pa F 2
1 +
1
ί/2/ι|"1/1/ΐΊ A
7ΐΛη_2 2 4_| π\
λ
=
2p2
d2hÀ2[l
for the isolated bump, π
+
^T|j Ï6
+
il
8j
2
. π\ ^ <χλ d h pa/I-|l+-)ff0F = p - ^ r
(2.4)
where
ß
16 (4 + n)F
(2.5)
If the wave or bump shape remains the same as in the case of a nondeforming rectangular wave or bump and the Prandtl velocity field, the solution to Eq. (2.4) is pah - |1 +-]a0F
= pah0
l+2l f f of cosh (t^/βφ)
(2.6)
provided of course that pah0 is above the threshold value. If the wave or bump is sinusoidal, the Prandtl velocity solution would indicate a changing shape
'Taylor Instability"
43
with F a function of h. Solutions which preserve wave shape appear to require slightly higher driving forces, but on the other hand bumps cannot carry (1 + π/2)σ0 at their free sides. In all cases, the general exponential nature of the early growth of bumps and waves persists in Eq. (2.4). The changes in F are moderate enough to make (2.6) a sufficiently useful result for all situations until the ratio of h to λ/2 becomes large enough to permit the extending bump to deform entirely on its own and produce a spike extending far beyond the displacement associated with the motion of the body of the solid. 3
COMPARISON OF EXPERIMENT AND PREDICTION
The threshold value of pah0 calculated from (2.3) is Ph0/H = (1 + n/2)a0F = 4 or 8kbars for σ0 = 2 or 4kbars and F = π/4 for sine waves. With P of 100 kbars as reported [5 ], the threshold value oihJH is hlH/H = 0.04 or 0.08.
(3.1)
At first it might seem remarkable that a single ratio had been selected for all but Shot 2 of the aluminum plates that is only 20 % below the h0/H of the stainless-steel plates (Table 1). In their paper, Barnes et al. [5] write of 3.3 kbars for σ0 which is intermediate in value between 2 and 4 kbars. Shot 2 with only half the initial amplitude h0 of the other shots is well below threshold and, as would be predicted, did not show large growth in 8 microseconds. The growth rate factor ^Jßu/λ = y/Ρβ/ρΗλ in (2.6) is 0.9 per microsecond for all the aluminum plates except Shot 2 so that the multiplier cosh (ί^/βα/λ ) in the rewritten form of (2.6) h-hT0H = (h0 - /z™)cosh(t^/βφ)
(3.2)
is over 100 at 6 microseconds and well over 1000 before 9 microseconds is reached. It is worth noting parenthetically that for Shot 2, with its wavelength half as large, the multiplier of 1000+ would require only 9/^/2 or 6.4 microseconds, well under the 8 of the test. For all other aluminum plates, an h0 that is a mere 5 % above the threshold value h'0H causes h to grow by a factor of 5 in 6 microseconds. Consequently it is necessary to have p
44
Daniel C. Drucker
y/Ï2 or not quite 1.5. At 6 microseconds, cosh (ty/ßoc/λ ) would be about 20 instead of the 100 for aluminum. An initial excess of 15 % for h0 compared with hlH would grow to 3/i0 and so be in the measurable range of the experiment. Of course, all these numbers are very crude and are not to be taken too literally. They do, however, agree with the main features of the results reported by Barnes et al [5 ] remarkably well for so idealized a description of very complex behavior. The calculation of h™ appears to be amply good to permit specification of the surfacefinishneeded to avoid growth of bumps or waves at any given P or a.
4
APPROXIMATION, IDEALIZATION, AND INTERPRETATION
Calculated threshold values pot.h0 = Ph0/H > (1 + n/2)a0F for thick plates and the growth rate factor y/βοϊ/λ appear reasonable for the data of Barnes et ai, but H of their plates was only λ or A/2. The limit stress ( 1 + π/2)σ0 is an upper bound which is an exact answer for H » λ and becomes more and more of an overestimate as λ approaches and exceeds H. Better estimates can be made for these relatively thinner plates. However, it is the short wave lengths with their potentially high growth rates that are of greatest interest. Even for a long wave λ = 2H, the threshold value for a half-sine-wave surface irregularity drops by less than one-third from (1 + π/2)σ0π/4 to about (π/2)σ0. The familiar cut-off and the attenuation of small, sub-millimeter, wave lengths that result from surface tension and viscosity in the Taylor instability offluidsunder moderate accelerations is missing here. A cut-off or threshold corresponding to surface tension would appear if the energy to form new surface were included in the analysis. However, it would not be significant until λ reached down to small multiples of atomic dimensions. At the enormous accelerations needed for Taylor instability of solids, the cut-off value of λ for fluids also would be down at atomic dimensions instead of submillimeters. The strain rates to which the metal is subjected are taken into account only to zeroth order in the analysis by the choice of σ0 as the averageflowstrength of the metal at the high strain rates encountered. Strain-rate distributions have been ignored. They certainly should be examined for very small λ because the average strain rates in the metal are of order (dh/dt)/À and the variations from point-to-point in a continuous velocity distribution will also be at least ofthat order. Temperature effects also need study. Only the initial stages of the increasingly unstable growth of surface bumps or waves have been modeled in the analysis presented. When the amplitude h
"Taylor Instability"
45
becomes comparable to A/2, local geometry takes over and the precise shape of the initial and growing bump assumes great importance. The tendency of the bump to become thin and for an initial sinusoidal wave to produce the familiar "spike and bubble" pattern is obvious. At the very beginning of the growth, some small but possibly significant increase in h can occur for h0 below the computed threshold (see Shot 2, Table 1). In part, some growth would be expected because the idealization of perfect plasticity ignores the plastic deformation prior to strain-hardening from the actual yield stress at the strain rates of interest to the yield strength σ0 applicable to large strains. In addition, the idealization ignores the much larger difference between the σ0 for the high strain rates of growth and the static yield strength. However, both of these idealizations are closer to reality here than in many other problems where they are employed successfully. A great part of the transition to σ0 is taken care of physically through the plastic compression at high strain rate that accompanies the smooth but enormous pressure and acceleration build-up prior to the growth stage. Also, the high pressure is maintained over the time interval of interest. It is not so surprising therefore that little difference is observed between the growth of a bump or a wave pattern in the originally very soft 1100-0 aluminum plate and its growth in the initially very much harder 6061-T6, both heated appreciably by the initial adiabatic compression. The idealization of perfect plasticity with σ0 appropriate to high strains and strain rates is a good first approximation to reality as well as a very helpful simplification.
5
EFFECT OF ACCOMPANYING LATERAL PLASTIC FLOW OF THE PLATE
A state of uniaxial strain with an enormous compressive stress on all planes parallel to the surface of the plate except those immediately adjacent to the traction-free back surface calls for almost equal lateral compressive stress on all perpendicular planes. If the perturbations due to the surface irregularities of waves are ignored, the direct compression varies linearly from the applied pressure P at the wavy surface to zero at the opposite free face. In the idealized perfectly plastic material, the lateral compression away from the perturbed region is σ0 less than the direct compression. Hydrostatic pressure is assumed to be without influence so that the state of stress everywhere away from the perturbations is equivalent simply to a uniform lateral tension σ0 on all planes perpendicular to the surface of the plate. This lateral tension σ0 is, of course, precisely what is needed to match the state of stress in the triangular regions on
46
Daniel C. Drucker
each side of the Prandtl solution. Plastic stretching of the entire plate away from the surface irregularities is a permissible alternative to uniaxial strain. Neither the threshold value of poch0 nor the local Prandtl solution is altered. If, instead, the idealized plate is plastically compressed laterally, the compressive lateral stress would vary from P + σ0 at the wavy surface to σ0 at the free face. The equivalent state of stress would be a uniform lateral compression σ0. With lateral compressive plastic flow, the threshold value of pa/z0 of the Prandtl solution or of any two- or three-dimensional perfectly plastic or workhardening solution would drop to zero. All bumps of any h0 and λ would grow at an initial rate given by y/ßcc/λ. Again, in reality, strain-rate effects would have considerable influence at small λ. Also at some multiple of atomic dimensions the surface energy would play a cut-off role.
6
CONCLUDING REMARKS
The main result is that there is an appreciable threshold amplitude hlH below which a small surface irregularity in a metal plate will not grow significantly under overall uniaxial strain conditions as in the tests of Barnes et al. A side comment is that there is no "wave" action. An isolated bump or a set of bumps or a regular wave pattern all behave in much the same way. When the plate thickness is not several times the lateral dimensions of the surface irregularity or when the irregularity is not of simple form and has a complex two-dimensional shape in the plane of the surface, pah™ = (1 + π/2)σ0 can still serve as a good guide to the polishing or buffing that is needed to avoid significant growth under the enormous accelerations imposed. Modern elastic-plastic computer codes can be employed when more accurate results are needed. In two-dimensional problems (plane strain or axial symmetry) visco-plasticity can be introduced and the continual changes in geometry followed in detail when warranted by the importance of more realistic solutions. It is interesting to speculate that had Barnes et al. run a complete set of computer solutions for different h0 and λ they might well have found the threshold cut-off in addition to their discovery that, contrary to the then prevailing opinion, the elastic modulus of the metal did not really matter. Finally, it is worth noting that the plate studies and tests should not be thought of as simulating the situation when there is the equivalent of simultaneous plastic lateral compression. The threshold value of poch0 then does sink to zero because the yield or flow strength of the metal is nullified.
"Taylor Instability"
47
Acknowledgment. It is a pleasure to acknowledge helpful discussions with Drs. J. R. Rice and J. J. Gilman during the course of the July 1977 Materials Research Council meeting sponsored by the Defense Advanced Research Projects Agency under Contract No. MDA 903-76C-0250 with the University of Michigan. 7
REFERENCES
1. Bellman, R. and Pennington, R. H., "Effects of Surface Tensions and Viscosity on Taylor Instability," Q. Appl. Math. 12 (1954) 151-162. 2. Axford, R. A., "Initial Value Problems of the Rayleigh-Taylor Instability Type", Los Alamos Scientific Laboratory Report LA-5378 (1974). 3. Menikoff, R., Mjolsness, R. C, Sharp, D. H., and Zemach, C, "The Unstable Normal Mode for Rayleigh-Taylor Instability in Viscous Fluids," Los Alamos Scientific Laboratory Report LAUR-77-1559 (1977). 4. Miles, J. W., "Taylor Instability of a Flat Plate," General Atomics Report GAMD-7335 (1960). 5. Barnes, J. F., Blewitt, P. J., McQueen, R. G., Meyer, K. A., and Venable, D., "Taylor Instability in Solids," J. Appl. Phys. 45 (1974) 727-732. 6. Shield, R. T. and Drucker, D. C, "The Application of Limit Analysis to Punch-indentation Problems," J. Appl. Mech. 20, Tïans. ASME 75 (1953) 453-460. 7. Drucker, D. C. and Chen, W. F., "On the Use of Simple Discontinuous Fields to Bound Limit Loads," Engineering Plasticity, J. Heyman and F. A. Leckie (eds.), Cambridge University Press (1968), pp. 129-145. 8. Chen, W. F., Limit Analysis and Soil Plasticity, Elsevier Scientific Publishing Company, Amsterdam (1975).
1
INTRODUCTION AND A SET OF EXPERIMENTAL RESULTS
The term "instability" arises very naturally in the consideration of waves at the interface between two fluids. When, for example, the top surface of water exposed to air in an ordinary glass or tank is perturbed, it will oscillate. The restoring forces are the gravitational pull and the surface tension. Viscous forces will damp the motion and eventually bring the water to rest. A downward acceleration of the system equal to the gravitational acceleration (free fall) nullifies the gravitational restoring force and leaves only the surface tension to stabilize the wave motion. A still larger acceleration directed 37
38
Daniel C. Drucker
downward causes all small amplitude waves of long wave length to grow exponentially in amplitude with time. Only waves of sufficiently short wave length can be stabilized by the surface tension. This break-up of the surface, or instability, at rather moderate magnitude of acceleration directed from the surface or interface into the denser fluid is a "Taylor instability". It has been described and analyzed in some detail following the early work of Rayleigh and Taylor [1-3]. Quite naturally, therefore, when a corresponding phenomenon was of concern at the surface of a solid plate or body subjected to very large accelerations directed inward from the surface (Fig. 1 ) the earlier analysis was carried over [4]. The expectation was that the important feature would be a wave length instability threshold governed by elastic restoring forces. />ah 0 A 0 /F
ï
f
f>
Fig. 1 A small bump adds a tensile force Q to the uniform pressure P.
The expected growth of "waves" machined or pressed in a solid surface was demonstrated by Barnes, Blewitt, McQueen, Meyer, and Venable in the Los Alamos Phermex facility. They obtained remarkably clear pictures of a free aluminum plate 2.54 mm thick with an initial sinusoidal surface pattern of 2/z0 = 0.203 mm and 5.08 mm wave length subjected to a shockless pressure P of the order of 100 kbars on the wavy surface [5 ]. In 7 or 8 microseconds the 2/z value grew to over 1 mm (Table 1). Table 1 Initial dimensions and observed growth from Tables 1 and II of Ref. [5] Shot no. 1 2 3 4 5 6 7
Material 1100-0 1100-0
ιιοο-υ
1100-0 6061-T6 304S.S. 304S.S.
H (mm)
λ (mm)
(mm)
2/i 0
t (^sec)
2Λ (mm)
2.54 2.54 2.54 2.54 2:54 1.90 1.90
5.08 2.54 5.08 5.08 5.08 5.08 5.08
0.203 0.102 0.203 0.203 0.203 0.203 0.203
8.00 8.01 6.41 7.32 8.09 5.54 6.50
1.515 0.165 0.880 1.168 LI 27 0.476 0.725
"Taylor Instability"
39
However, the concept of waves and the transition from a stable oscillatory mode to an unstable exponential growth is not the most helpful approach for solids. Periodicity in space is not relevant. Elastic restoring forces also have minor relevance at most. Visible deformations, far smaller than the spectacular growth of the surface "waves" seen in the aluminum plates, are plastic, not elastic. Therefore, whether or not they occur is determined by a plasticity analysis. As a first approximation it is reasonable to idealize a very ductile solid as isotropic and perfectly plastic with an appropriate yield strength σ0 about twice the yield strength in shear. Of course,
2
INITIAL SOLUTION AND SUBSEQUENT MOTION FOR A PERFECTLY PLASTIC SOLID
The (shockless) pressure P needed to give an acceleration a to a plate of thickness H and mass density p, Fig. 1, is P=p
(2.1)
A small bump of height h0 and plan area A0 on the otherwiseflatsurface of the accelerating plate adds a tensile force Q to the uniform pressure on that plane. M.T. V.5—C
40
Daniel C. Drucker
Q = poth0A0 for a mesa-like uniform thickness bump and is less for a rounded bump: Q = pxh0A0/F
(2.2)
with the geometric factor F > 1. In the two-dimensional case (Fig. 2) Q is a force per unit dimension perpendicular to the paper, A0 is replaced by λ/2 and F is unity for a rectangular bump, π/2 for a half-sine-wave.
Fig. 2
Two-dimensional bumps and waves.
The two-dimensional Prandtl solution for a rigid punch pressing against a half-space (Fig. 3) is directly applicable to the small isolated rectangular bump on a thick plate. A regularly spaced set of punches with the width of each punch equal to the separation λ/2 between punches (Fig. 4) provides the needed information for a regular rectangular "wave" pattern of wave length λ. The limit force per unit area that must be exceeded for large-scale plastic deformation to occur in the elastic-perfectly plastic body of yield strength σ 0 is (1 + π/2)σ0 for either the isolated punch or the regular array of punches. This two-dimensional result is a fair approximation for the three-dimensional problem of the limit force per unit of area for any punch with a convex area of contact on a half space, although the value of 3σ0 which is on the high side is closer [6-8].
Fig. 3
Prandtl solution.
"Taylor Instability"
41
Figures 2,3, and 4 demonstrate the equivalence or approximate equivalence of the downward accelerating body with an isolated bump or a wave to the punch problem with signs reversed. The alternating side-by-side tensile and compressive forces shown acting on the mean surface of Fig. 4, each QL/2 = | ( 1 + π/2)σ0λ/2, are equivalent to the equally spaced tensile forces QL = (1 + π/2)σ0λ/2 and provide a more generally useful picture for a wave. The obvious result that a rectangular wave of amplitude, h0, (Fig. 2), which is a set of bumps of height 2/z0, reaches limit load at just half the acceleration required for a single rectangular bump of height h0 carries over to the upperbound calculation for all wave shapes.
M-+-T-+-H-—
λ
— H
Fig. 4 Multiple punches.
The threshold value of pah0 or Ph0/H Eq. (2.1) is given by poLhl" = PhT0H/H = (1 + n/2)a0F
(2.3)
where F for a wave is one-half of the F for the corresponding isolated bump: one-half for the rectangular wave and π/4 for the sinusoidal. Below the threshold value the plastic deformation will be contained and no appreciable change in geometry will result. Above the threshold, the geometry change will be large. The continual increase in h from the initial h0 causes the driving forces to increase and the system to run away in an extremely short time. The initial value of dv/dt or d2h/dt2, the acceleration with respect to the
42
Daniel C. Drucker
surface of the moving plate, is proportional to the excess of the driving force Q over the required QL. Now the details of the assumed velocity pattern for plastic deformation and the volume of material participating play a far stronger role than in the calculation of the limit force QL. Therefore the result for d2h/dt2 is more approximate still. The equation of motion associated with the Prandtl velocity pattern follows directly from D'Alembert's principle. An upward virtual displacement Δ of the bump and the rigid triangular region below results in a displacement A^/2 everywhere in the two quarter-circle fan regions of shear and a downward displacement Δ in the rigid triangular side regions shown in Fig. 4 for the wave patterns (or a 45° inclined displacement of Δ/^/2 in the two rigid triangular sliding regions shown in Fig. 3 for the isolated bump). The acceleration of each point of each region is in the same ratio to d2h/dt2 as the virtual displacement is toA. d2^ Ιλλ 1 d2h 7i//L/2\ A A + 2p 224 2 dt2 4 \ 4
(Q-QL)A=p
le
„ d2h \ λλ A for the wave or 244 + 2p h λ pa F 2
1 +
1
ί/2/ι|"1/1/ΐΊ A
7ΐΛη_2 2 4_| π\
λ
=
2p2
d2hÀ2[l
for the isolated bump, π
+
^T|j Ï6
+
il
8j
2
. π\ ^ <χλ d h pa/I-|l+-)ff0F = p - ^ r
(2.4)
where
ß
16 (4 + n)F
(2.5)
If the wave or bump shape remains the same as in the case of a nondeforming rectangular wave or bump and the Prandtl velocity field, the solution to Eq. (2.4) is pah - |1 +-]a0F
= pah0
l+2l f f of cosh (t^/βφ)
(2.6)
provided of course that pah0 is above the threshold value. If the wave or bump is sinusoidal, the Prandtl velocity solution would indicate a changing shape
'Taylor Instability"
43
with F a function of h. Solutions which preserve wave shape appear to require slightly higher driving forces, but on the other hand bumps cannot carry (1 + π/2)σ0 at their free sides. In all cases, the general exponential nature of the early growth of bumps and waves persists in Eq. (2.4). The changes in F are moderate enough to make (2.6) a sufficiently useful result for all situations until the ratio of h to λ/2 becomes large enough to permit the extending bump to deform entirely on its own and produce a spike extending far beyond the displacement associated with the motion of the body of the solid. 3
COMPARISON OF EXPERIMENT AND PREDICTION
The threshold value of pah0 calculated from (2.3) is Ph0/H = (1 + n/2)a0F = 4 or 8kbars for σ0 = 2 or 4kbars and F = π/4 for sine waves. With P of 100 kbars as reported [5 ], the threshold value oihJH is hlH/H = 0.04 or 0.08.
(3.1)
At first it might seem remarkable that a single ratio had been selected for all but Shot 2 of the aluminum plates that is only 20 % below the h0/H of the stainless-steel plates (Table 1). In their paper, Barnes et al. [5] write of 3.3 kbars for σ0 which is intermediate in value between 2 and 4 kbars. Shot 2 with only half the initial amplitude h0 of the other shots is well below threshold and, as would be predicted, did not show large growth in 8 microseconds. The growth rate factor ^Jßu/λ = y/Ρβ/ρΗλ in (2.6) is 0.9 per microsecond for all the aluminum plates except Shot 2 so that the multiplier cosh (ί^/βα/λ ) in the rewritten form of (2.6) h-hT0H = (h0 - /z™)cosh(t^/βφ)
(3.2)
is over 100 at 6 microseconds and well over 1000 before 9 microseconds is reached. It is worth noting parenthetically that for Shot 2, with its wavelength half as large, the multiplier of 1000+ would require only 9/^/2 or 6.4 microseconds, well under the 8 of the test. For all other aluminum plates, an h0 that is a mere 5 % above the threshold value h'0H causes h to grow by a factor of 5 in 6 microseconds. Consequently it is necessary to have p
44
Daniel C. Drucker
y/Ï2 or not quite 1.5. At 6 microseconds, cosh (ty/ßoc/λ ) would be about 20 instead of the 100 for aluminum. An initial excess of 15 % for h0 compared with hlH would grow to 3/i0 and so be in the measurable range of the experiment. Of course, all these numbers are very crude and are not to be taken too literally. They do, however, agree with the main features of the results reported by Barnes et al [5 ] remarkably well for so idealized a description of very complex behavior. The calculation of h™ appears to be amply good to permit specification of the surfacefinishneeded to avoid growth of bumps or waves at any given P or a.
4
APPROXIMATION, IDEALIZATION, AND INTERPRETATION
Calculated threshold values pot.h0 = Ph0/H > (1 + n/2)a0F for thick plates and the growth rate factor y/βοϊ/λ appear reasonable for the data of Barnes et ai, but H of their plates was only λ or A/2. The limit stress ( 1 + π/2)σ0 is an upper bound which is an exact answer for H » λ and becomes more and more of an overestimate as λ approaches and exceeds H. Better estimates can be made for these relatively thinner plates. However, it is the short wave lengths with their potentially high growth rates that are of greatest interest. Even for a long wave λ = 2H, the threshold value for a half-sine-wave surface irregularity drops by less than one-third from (1 + π/2)σ0π/4 to about (π/2)σ0. The familiar cut-off and the attenuation of small, sub-millimeter, wave lengths that result from surface tension and viscosity in the Taylor instability offluidsunder moderate accelerations is missing here. A cut-off or threshold corresponding to surface tension would appear if the energy to form new surface were included in the analysis. However, it would not be significant until λ reached down to small multiples of atomic dimensions. At the enormous accelerations needed for Taylor instability of solids, the cut-off value of λ for fluids also would be down at atomic dimensions instead of submillimeters. The strain rates to which the metal is subjected are taken into account only to zeroth order in the analysis by the choice of σ0 as the averageflowstrength of the metal at the high strain rates encountered. Strain-rate distributions have been ignored. They certainly should be examined for very small λ because the average strain rates in the metal are of order (dh/dt)/À and the variations from point-to-point in a continuous velocity distribution will also be at least ofthat order. Temperature effects also need study. Only the initial stages of the increasingly unstable growth of surface bumps or waves have been modeled in the analysis presented. When the amplitude h
"Taylor Instability"
45
becomes comparable to A/2, local geometry takes over and the precise shape of the initial and growing bump assumes great importance. The tendency of the bump to become thin and for an initial sinusoidal wave to produce the familiar "spike and bubble" pattern is obvious. At the very beginning of the growth, some small but possibly significant increase in h can occur for h0 below the computed threshold (see Shot 2, Table 1). In part, some growth would be expected because the idealization of perfect plasticity ignores the plastic deformation prior to strain-hardening from the actual yield stress at the strain rates of interest to the yield strength σ0 applicable to large strains. In addition, the idealization ignores the much larger difference between the σ0 for the high strain rates of growth and the static yield strength. However, both of these idealizations are closer to reality here than in many other problems where they are employed successfully. A great part of the transition to σ0 is taken care of physically through the plastic compression at high strain rate that accompanies the smooth but enormous pressure and acceleration build-up prior to the growth stage. Also, the high pressure is maintained over the time interval of interest. It is not so surprising therefore that little difference is observed between the growth of a bump or a wave pattern in the originally very soft 1100-0 aluminum plate and its growth in the initially very much harder 6061-T6, both heated appreciably by the initial adiabatic compression. The idealization of perfect plasticity with σ0 appropriate to high strains and strain rates is a good first approximation to reality as well as a very helpful simplification.
5
EFFECT OF ACCOMPANYING LATERAL PLASTIC FLOW OF THE PLATE
A state of uniaxial strain with an enormous compressive stress on all planes parallel to the surface of the plate except those immediately adjacent to the traction-free back surface calls for almost equal lateral compressive stress on all perpendicular planes. If the perturbations due to the surface irregularities of waves are ignored, the direct compression varies linearly from the applied pressure P at the wavy surface to zero at the opposite free face. In the idealized perfectly plastic material, the lateral compression away from the perturbed region is σ0 less than the direct compression. Hydrostatic pressure is assumed to be without influence so that the state of stress everywhere away from the perturbations is equivalent simply to a uniform lateral tension σ0 on all planes perpendicular to the surface of the plate. This lateral tension σ0 is, of course, precisely what is needed to match the state of stress in the triangular regions on
46
Daniel C. Drucker
each side of the Prandtl solution. Plastic stretching of the entire plate away from the surface irregularities is a permissible alternative to uniaxial strain. Neither the threshold value of poch0 nor the local Prandtl solution is altered. If, instead, the idealized plate is plastically compressed laterally, the compressive lateral stress would vary from P + σ0 at the wavy surface to σ0 at the free face. The equivalent state of stress would be a uniform lateral compression σ0. With lateral compressive plastic flow, the threshold value of pa/z0 of the Prandtl solution or of any two- or three-dimensional perfectly plastic or workhardening solution would drop to zero. All bumps of any h0 and λ would grow at an initial rate given by y/ßcc/λ. Again, in reality, strain-rate effects would have considerable influence at small λ. Also at some multiple of atomic dimensions the surface energy would play a cut-off role.
6
CONCLUDING REMARKS
The main result is that there is an appreciable threshold amplitude hlH below which a small surface irregularity in a metal plate will not grow significantly under overall uniaxial strain conditions as in the tests of Barnes et al. A side comment is that there is no "wave" action. An isolated bump or a set of bumps or a regular wave pattern all behave in much the same way. When the plate thickness is not several times the lateral dimensions of the surface irregularity or when the irregularity is not of simple form and has a complex two-dimensional shape in the plane of the surface, pah™ = (1 + π/2)σ0 can still serve as a good guide to the polishing or buffing that is needed to avoid significant growth under the enormous accelerations imposed. Modern elastic-plastic computer codes can be employed when more accurate results are needed. In two-dimensional problems (plane strain or axial symmetry) visco-plasticity can be introduced and the continual changes in geometry followed in detail when warranted by the importance of more realistic solutions. It is interesting to speculate that had Barnes et al. run a complete set of computer solutions for different h0 and λ they might well have found the threshold cut-off in addition to their discovery that, contrary to the then prevailing opinion, the elastic modulus of the metal did not really matter. Finally, it is worth noting that the plate studies and tests should not be thought of as simulating the situation when there is the equivalent of simultaneous plastic lateral compression. The threshold value of poch0 then does sink to zero because the yield or flow strength of the metal is nullified.
"Taylor Instability"
47
Acknowledgment. It is a pleasure to acknowledge helpful discussions with Drs. J. R. Rice and J. J. Gilman during the course of the July 1977 Materials Research Council meeting sponsored by the Defense Advanced Research Projects Agency under Contract No. MDA 903-76C-0250 with the University of Michigan. 7
REFERENCES
1. Bellman, R. and Pennington, R. H., "Effects of Surface Tensions and Viscosity on Taylor Instability," Q. Appl. Math. 12 (1954) 151-162. 2. Axford, R. A., "Initial Value Problems of the Rayleigh-Taylor Instability Type", Los Alamos Scientific Laboratory Report LA-5378 (1974). 3. Menikoff, R., Mjolsness, R. C, Sharp, D. H., and Zemach, C, "The Unstable Normal Mode for Rayleigh-Taylor Instability in Viscous Fluids," Los Alamos Scientific Laboratory Report LAUR-77-1559 (1977). 4. Miles, J. W., "Taylor Instability of a Flat Plate," General Atomics Report GAMD-7335 (1960). 5. Barnes, J. F., Blewitt, P. J., McQueen, R. G., Meyer, K. A., and Venable, D., "Taylor Instability in Solids," J. Appl. Phys. 45 (1974) 727-732. 6. Shield, R. T. and Drucker, D. C, "The Application of Limit Analysis to Punch-indentation Problems," J. Appl. Mech. 20, Tïans. ASME 75 (1953) 453-460. 7. Drucker, D. C. and Chen, W. F., "On the Use of Simple Discontinuous Fields to Bound Limit Loads," Engineering Plasticity, J. Heyman and F. A. Leckie (eds.), Cambridge University Press (1968), pp. 129-145. 8. Chen, W. F., Limit Analysis and Soil Plasticity, Elsevier Scientific Publishing Company, Amsterdam (1975).