Technique for the Determination of Dynamic-Tensile ...

Reprinted from JouRNAL OF APPLIED PHYSICS, Vol. 38, No. 8, 3271-3275, July 1967 Copyright 1967 by the American Institute of Physics Printed in U. S. A.

Technique for the Determination of Dynamic-Tensile-Strength Characteristics* B. R . BREED, CHARLES L. MADER, AND DOUGLAS VENABLE Los Alamos Scientific Laboratory, University of California, Los Alamos, New Mexico (Received 7 October 1966; in final form 15 March 1967) A technique for the examination of the dynamic-tensile-strength characteristics of materials is presented. The dependence of tensile strength upon stress gradient, hence stress rate, is demonstrated for AI, Cu, Ni, and Pb. The results were tested and found to be self-consistent in that once the dynamic-tensile-strength characteristics have been measured they can be incorporated into the numerical calculations, which then can be used to predict complex multilayer spall behavior even in layers of dissimilar materials.

INTRODUCTION That tensile strength of materials depends upon the rate at which stress is applied has been the subject of several earlier papers. The work reported here is relevant only to those few papers in which very high stress rates are considered. 1- 6 In all cases the dynamic tensile strength was determined by relating the observed thickness of a spalled layer to a hydrodynamic model. Sometimes the thickness was obtained by direct measurements on recovered specimens; sometimes it was inferred from the characteristics of the motion of the free surface of a specimen plate. Clearly, the validity of measurements on recovered samples is hinged, to some extent, upon the assumption that the observed spalled thickness is indeed the same as that which existed during the initial phase of spalling. Furthermore, the time-history of events is not obtainable by any of these methods if multiple spalling occurs. It is the intent of this paper to discuss a different, but very powerful technique for dynamically observing these various phenomena. A flash radiographic scheme was employed for all the experiments described here. This technique is much more than a mere addition to the repertory of schemes now employed, for it offers a method for establishing the self-consistency of the interpretation of results. Flash radiography provides a method by which, not only the thickness of the first spalled layer and its position with time can be measured, but of equal importance, the thickness and positions of subsequent layers are observed too. The fact that these additional data are available and are useful reduces the numbers of assumptions needed to provide quantitative information about the dependence of tensile strength upon stress rate and the mutually dependent stress gradient. The results obtained can be given a rather severe test *This work was performed under the auspices of the United States Atomic Energy Commission. 1 J. S. Rinehart and J. Pearson, Behavior of Metals Under Impulsive Loads (The American Society for Metals, Cleveland, 1954) . 2 R. G. McQueen and S. P. Marsh, J . Appl. Phys. 33, 654 (1962). 3 P. Whiteman, Atomic Weapons Research Establishment Report UNDEX 445 (1962). 4 D. V. Keller and J. G. Trulio, J. Appl. Phys. 34, 172 (1963). 1 G. Nahami, Les Ondes de Detonation (Centre National de Ia Recherche Scientifique, Paris, 1962), p. 451. 'I. C. Skidmore, Appl. Mater. Res. 4, 131 (1965).

in which it is demonstrated that characteristic spall curves, obtained from the experiments, are sufficient for predicting spalling behavior in more complex situations. This test and its results are discussed and representative data are presented on spalling in Al, Cu, Ni and Pb. HYDRODYNAMIC MODEL Tensile stresses can be produced under dynamic conditions in a slab of specimen material by the collision of a shock wave with a free surface. The material at the surface is accelerated outward by virtue of the particle motion in the shock and relief waves. If the material of the slab can sustain tensile stre~ses and if the pressure behind the shock wave decreases with distance, then a tension wave is developed as the relief wave propagates back into the slab. 7 In a homogeneous slab, whose equation of state is known, this phenomenon can be described by numerical solutions of the onedimensional hydrodynamic equations. In our experiments the numerical solutions were obtained using the SIN code, developed by Charles L. Mader of the Los Alamos Scientific Laboratory (LASL) . This code is rather thoroughly described elsewhere8 •9 and only a brief description will be given here. It is essentially a one-dimensional finite-difference computer program for solving the hydrodynamic equations in Lagrangian form. Up to 2000 finite-difference space zones were used for the calculations reported here. A complete solution is generated for a sequence of times separated by small time intervals. The program can be used to solve both reactive and nonreactive hydrodynamic problems. In the calculations reported here the problems of real viscosity and thermal conductivity have been neglected. The Grlineisen equation of state was used for states off the single-shock Hugoniot for larger than normal density states and off the zero pressure curve for subnormal density states. The single-shock Hugoniot equation of state was described by the linear relation U,=C+SUP between the shock velocity U. and particle velocity Up, with the coefficients obtained from B. Hopkinson, Trans. Roy. Soc. (London) 213A, 437 (1914). Charles L. Mader, Los Alamos Scientific Laboratory, Los Alamos, New Mexico, Report LA-2703 (1962). 'Charles L. Mader, Los Alamos Scientific Laboratory, Loa Alamos, New Mexico, Report LADC-.5795 (1963),

3271

7

8

3272

BREED,

MADER,

p

Po'+--+-~>------ x T -- ---

x.

xt

AND

VENABLE

When spalling occurs, the thickness of the spalled layers can be measured by flash radiographic techniques,11·12 and the thickness can be expressed in terms of g/ cm2. Then, with this information, the stress by which this layer separated from the parent material can be determined by means of the numerical description of the hydrodynamics. It is further assumed that separation is initiated at a tension T, located behind the front surface at a critical position x •. At the instant of separation, the spalled layer thickness is x1 - x.; however, the density in this layer is not necessarily constant. The spalled layer thickness, in g/ cm 2, is

FIG. 1. Typical distributions of pressure and density in the rarefaction wave.

p

M=

experimental data. 10 The Grlineisen 'Y was computed from the approximate relationship 'Y = 2 S -1. The equation of state is described in more detail in Refs. 8 and 9. Whiteman3 has suggested that the tensile stress at spalling is linearly related to the square root of the stress rate. For computational convenience we chose to relate it to the stress gradient

( 1) For purposes of comparison Whiteman's data were converted to this form. Once a relation of the form of Eq. ( 1) is determined experimentally, it can be inserted into the computer code and subsequent spall phenomena can be predicted, providing a method for determining the self-consistency of the procedure. EXPERIMENTAL METHOD At least two different schemes can be employed to produce the kind of hydrodynamic conditions necessary for these experiments. A very successful method uses the flying plate technique. 2·3 In our experiments it was the nonuniform shock arising from the impact of a detonation wave that was used. Typical pressure and mass density distributions for a one-dimensional case are illustrated in Fig. 1. The experiments reported here are limited to singlephase systems. A few experiments were carried out on iron but numerical interpretation of the results is difficult since an accurate numerical description of the equation of state of iron is not presently available and a phase change results in a tension wave with large pressure gradients. Direct radiographic observation of the double plastic wave structure in iron has indicated that the region between the two plastic waves may not be in thermodynamic equilibrium, and the SIN code is not yet able ta account for such a situation. 1oM. Hf Rice,· J. M . Walsh, and R. G. McQueen, in Solid State Physics, · F. Seitz and D. Turnbull, Eds, (Acl\demic Press · Jnc., New York1 19~8) Vol. 6. '

'

'

·:,..

'

!

Xf

(2)

pdx.

"• After the layer has had time to reach an equilibrium condition at a uniform constant density po, the layer thickness is still M but now (3)

M=poo,

where o is the radiographically observed thickness of the spalled layer, and po is taken to be essentially the value of the density of the unshocked material. By working backward through the hydrodynamic equation, ocan be related to x., hence the value ofT corresponding to this critical value of x. can be determined. The stress gradient was calculated using the relation aPjax=

I (T-Po)/(xJ-Xs) I'

(4)

where Po, T, Xf> and x. are as indicated in Fig. 1. The radiographic technique used for this work has been described rather thoroughly elsewhereP These particular experiments, which require very simple geometry, are represented schematically in Fig. 2. To depict the nature of the raw data, a typical radiograph of spalling in copper is shown in Fig. 3. For purposes of setting up an experiment, a suitable time for radiography is estimated by assuming that the first spalled layer moves out at about the free surface velocity. The actual time of observation, to be matched to the hydrodynamic model, is measured to within 0.01 J.LSec by means of electrical contractors located at the back surface of the specimen plate and an electronic radiation

PLANE WAVE GEN:tl_ERATOR PROTECTIVE NOSE H.E

FIG. 2. Schematic diagram of the experimental setup.

TARGET RADIOGRAPH SPECIMEN

iBEAM~J·

FILM PLANE

L

304cm 11

D

70cm

Douglas Venable, Phys. Today 17', 19 (1964). 12 Douglas Venable, The Fourth Symposium on Detonation (The United States Naval Ordnance Laboratory, White Oak, ~ilver Spring1 Mar{' land, 1965) ,

DYNAMIC

TENSILE STRENGTH

3273

CHARACTERISTICS

Frc. 3. Spalling in a copper slab.

detector. The time interval which is measured is the sum of the transit time of the shock wave through the specimen and the period during which the front surface has moved. Magnification of the radiographic image, which depends upon the experimental arrangement, is always accounted for in the data analyses. The x-ray film is protected from blast and shrapnel by a hollow conical cassette as discussed in Ref. 11. Various thicknesses of composition B-3 (60% RDX and 40% TNT by weight) are used to provide different shock-wave profiles, thus different stress rates. A position-time history of the interface between the plane wave generator and the explosive charge was measured radiographically and its velocity was accounted for in the calculations, although the calculations were found to be insensitive to the rear boundary used. The effects of lateral rarefaction waves in the explosive reaction products were made insignificant by simply using high explosive charges with sufficiently large cross sectional dimensions. Flash radiographs, which displayed the rarefaction-wave configurations for various geometries facilitated the choice of a suitable cross section for the charge. The usual charge cross section was 10 em X 10 em.

measurements is the finite resolution of the radiographs. Controlled experiments were run using thin specimen layers of known thickness. The radiographic thickness was in agreement with these thicknesses to within 0.1 mm. Also a few of the spalled layers were recovered and their thicknesses determined directly. These results also indicated a possible 0.1-mm error. The layers were found to be nonunifmm in thickness in some instances, which also contributes to the possible errors indicated in the table. The equation-of-state parameters used in the calculations are presented in Table II. The a, Cv, and p0 are the linear coefficient of thermal expansion, the specific heat, and the normal density, respectively. Interpretations were made by means of the hydrodynamic model. In all cases reported in Table I, the initial temperatures were ambient, about 20°C. Experimental data, expressed in terms of tension vs stress rate as indicated in Eq. (1), are displayed in Figs. 4 and 5 for aluminum, copper, nickel, and lead. The error bars indicated on the experimental points reflect the possible errors in observed spall-layer thick-

RESULTS The raw experimental data consist of the radiographic measurements of the thickness of the first spalled layer in a specimen, the initial thickness of the specimen, and the thickness of the high-explosive driver. These data are recorded in Tab!e I. The accuracy with which the layer thickness was measured is: indicated in the table. The most important limitation on the accuracy of these TABLE I. Observed spall-layer thicknesses (mm).

.12

COPPER

T

ALUMINUM

f

D AND 0 ARE WHITEMAN'S DATA 0

Ul

~ .10

0

m -

1-

l.LI

g-.30

1--

..J

u

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~ .02

>~40 o~LJ~3-L-L~s~~~9-L~J.I2~

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DISTANCE - em

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1.0

1.2

"p)l/2- ( MBARS/CM) ..1.t (l>X

1.4

13 Results of Recent R esearch Projects at the Los Alamos Scientific Laboratory (Los Alamos, New Mexico, 1965) pp. 159- 184. 14 Charles L. Mader, Los Alamos Scientific Laboratory Los Alamos, New Mexico, Report LADC-7692 (October 1965) .'

DYNAMIC TENSILE STRENGTH CHARACTERISTICS

curve for negative stresses, one can easily show that the P(p) relationship predicted by SIN compares well with the P (p) predicted by a Murnaghan equation of state using Fowles' constants15 which, in turn, were obtained from manipulations of published data. 10 •14 This merely indicates that simpler equations of state could also provide useful results. 16

G. R . Fowles,

J. Appl.

Phys. 31, 655 (1960).

3275

ACKNOWLEDGMENTS The authors wish to express their appreciation for the stimulating and useful discussions with Dr. John W. Taylor and Dr. Ian C. Skidmore concerning these matters. They also thank F. M. Jackson, R. K. London, B. F. Poe, G. W. Rodenz, and R. W. Taylor for their contributions to the planning and execution of these experiments.