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Some scaling laws governing gun erosion
Wear, 39 (1976) 307 - 322 0 Elsevier Sequoia S.A., Lausanne
307 - Printed
in the Netherlands
SOME SCALING LAWS GOVERNING GUN EROSION
W. T. EBIHARA Research, Development and Engineering Directorate, Arsenal, Rock Island, Ill. 61201 (U.S.A.) A. A. HOCHREIN,
Februa’ry
Island
JR., and A. THIRUVENGADAM
Daedalean Associates, Incorporated, (U.S.A.) (Received
Rodman Laboratory;Rock
15110 Frederick
Road, Woodbine, Md. 21797
18, 1976)
Summary Gun barrel erosion is one of the major problems limiting the velocity, range and accuracy of the projectile. The various interacting mechanisms causing erosion are discussed. The importance of developing scaling laws governing gun tube erosion is emphasized. The approach adopted*in this paper for the development of scaling laws is inspired by similar attempts in cavitation and liquid impact erosion. The concepts of threshold erosion, erosion intensity, erosion strength and erosion parameter are extended, modified and applied to gun tube erosion. As a result, the outlook for developing scaling laws governing gun barrel erosion seems very promising. The nondimensional number called the,erosion parameter is given by the ratio between the output intensity of erosion representing the energy absorbed by the material and the input intensity of erosion contributed by the erosive forces. The output intensity of erosion is given by the product of the rate of depth of erosion and erosion resistance of the material, whereas the input intensity includes the Poisson’s ratio of the projectile, the maximum acceleration, the maximum velocity and the bore diameter. With the aid of a few justifiable assumptions, it has been shown that the rate of depth of erosion is directly proportional to the square of the bore diameter. Available experimental data support these results for velocity scale and size scale. Further experiments are required to test these predictions fully. The erosion state is highly dependent on the number of rounds fired; it has four stages including incubation, acceleration, deceleration and steady state periods very similar to other erosion phenomena. Previously developed erosion theory correlates well with the available gun tube erosion rates as a function of the number of rounds fired; this lends further support to the erosion model proposed in this paper. However, the interacting influences of thermal and chemical mechanisms must be carefully considered in any further developments.
308
Introduction The wear erosion in gun barrels is a detrimental material problem seriously limiting the projectile velocity, range and accuracy. The maximum muzzle velocity of 3000 ft s--l reached during the 1940s has not been substantially exceeded, mainly because of the extensive erosion problems. Several erosion-resistant materials have been developed, the most prominent being stellite. The advent of high temperature super alloys technology offers a great potential in improving the capability, performance and service life of gun barrels. The service life for different experimental barrels varies from 7000 to 35 000 rounds depending upon the material liner and plating combinations [ 11. Muzzle end erosion has been shown to be the dominant problem in some of the super alloy barrels in contrast to the traditional breech end erosion problem [2] . Moreover, it has become increasingly important to formulate analytical models incorporating the dynamic response of materials at elevated temperatures under the influence of erosive forces. Gun tube erosion is characterized by an enlargement of the bore as a result of material wear. Erosion is greatest at the breech end and decreases rapidly towards the muzzle. However, recent observations with super alloys show that the muzzle end erosion is the dominant factor. The erosion rate is generally represented in fractions of a thousandth of an inch (mil) per round. Although the erosion rate is generally a few mil per round, the amount of metal lost is sometimes impressive, particularly in large guns. For example, in a 12 in gun firing at 2600 ft s-l, nearly a pound of material may be removed from the bore surface during a single round [3]. In addition to the removal of material by mechanical abrasive effects, severe heating also contributes by melting a thin layer which is carried off by hot gases and the band [4] . This intense heating and cooling in the presence of burnt gases make the bore surfaces brittle, leading to cracks. The brittle cracking is intense at the breech end and decreases towards the muzzle end. Generally the crack pattern follows the machining direction. Rapid heating and cooling also leads to unfavorable metallurgical changes in the bore surface material [5,61. Gun tube material development programs date back to 1940 leading to the development of several desirable materials including high strength steels, stellite liners and chromium plating techniques [ 31. More recent investigations [7] have indicated superior materials such as stellite 21 and Timkin 16-26-6 as compared with the traditional gun steel. Super alloys and refractory materials are being tested for small arms rapid fire weapons and chromemolybdenum-vanadium steels with chrome plating have been used in gun tube applications. However, the material requirements for the gun tube are so complex that no ideal material has been found yet. Since some of the ideal requirements are conflicting, the final material selection is necessarily a compromise. Some of the materials exhibiting very high resistance to heat and erosion do not have adequate strength to resist the high pressures
309
and inertial forces. Cost and fabrication techniques are also important considerations. Moreover, recent developments in coating techniques offer great potential despite flaking and delamination problems associated with such coatings. The introduction of wear-reducing additives is a major step forward [8] . Some of these additives provide laminar cooling of the bore surface while others improve the thermal and chemical compatibility of the material thus increasing their fatigue endurance limits. Many creative design concepts have also been introduced for reducing-erosion. These include the proper design of the projectile bands and the introduction of new materials which minimize the engraving friction. However, the lack of a coordinate’d approach to design and material selection has led to the present situation wherein many successful approaches in one type of weapon system for a particular size and speed range are not directly applicable in a different application. This is primarily because there is no quantitative approach in estimating the rate of erosion in a given size of gun at a given speed for a specified material and projectile geometry. However, similar complex problems such as cavitation erosion are being approached from the point of view of developing scaling laws [9]. Engineering design procedures and calculations invoive problems too complex for exact analytical solutions. Such problems are generally investigated through a judicious combination of experiments and development of nondimensional parameters governing the processes investigated. This procedure is used extensively in the design of ships, airplanes, hydraulic pumps and turbines as well as in problems that deal with heat flow and with complex stresses [lo] . There are several scaling parameters known to engineers. Each of these nondimensional scaling parameters used in designs has physical significance in relating different processes. Although rational procedures such as dimensional analysis are available, it is still not possible to derive all the pertinent nondimensional parameters a priori and to predict the functional relations involving complex problems encountered in engineering designs. However, it is possible to determine the dominant parameters and their relations by careful experimentation, by a proper understanding of the important aspects of the complex phenomena involved and by deductive reasoning. Such an approach would greatly enhance the understanding of the dominant erosion mechanisms and the material property requirements, It would also lead to the correlation of experimental data with analytical models and field performance, thereby enabling the designers to use these results with confidence. With this objective in mind, the various mechanisms involved in the complex phenomenon of gun erosion are reviewed. A plausible erosion model is derived which makes use of the recent approaches utilized in cavitation erosion, droplet erosion and wear of materials. Available experimental data show that the land erosion increases with the number of rounds fired, reaches a maximum and then decreases tending to a stable rate. A recently developed theory of erosion is used to correlate these rates in a nondimensional form. The maximum erosion rate and the steady state
310
erosion rate depend upon several gun parameters, velocity and gun tube diameter.
Mechanisms
including
the muzzle
of gun barrel erosion
As in other erosion problems, the mechanisms of erosion of gun tubes include mechanical causes, high temperature effects and chemical interactions. As the gun is fired, the burning powder is converted into a whitehot gas under high pressure propelling the projectile along the bore. During this process, the projectile as well as the tube are subjected to short duration stress pulses of intense magnitude. The accelerating forces and the mechanical stresses reach maximum levels near the breech end. Such dynamic forces cause a swaying action of the projectile. This combined with unburnt propellent particles, reaction products and wear particles leads to severe mechanical erosion. At the same time, the mass of high pressure white-hot flame follows the projectile, heating the tube wall, sometimes melting and chemically altering, in addition to gouging and scouring, the barrel wall. The bore surface material is subjected to rapid heating and cooling which sometimes leads to a martensitic transformation of the barrel material. The propellent gas contains free radicals, ions and unstable molecules such as CO, COz, HzO, Hz, NH,, CH, and HsS, which may react with the gun tube surface material, a process greatly assisted by the high temperature of the propellent gas. Although the erosion rate is measured in mils (thousandths of an inch) per round, the amount of metal lost may be as much as a pound per round in large guns. Repeated firing causes substantial enlargement of the bore leading to loss of pressure, loss of muzzle velocity and firing inaccuracy. The complex nature of gun tube erosion mechanisms has defied many attempts in the past to construct quantitative analytical models to describe and predict erosion rates. Such attempts have been at least partially successful in predicting the service life of systems experiencing other erosion phenomena such as cavitation and liquid impact, by the development of scaling laws and model prototype correlations that lead to improved design techniques, material selection and service life predictions. It is the objective of this paper to extend those approaches in order to develop scaling parameters applicable to the gun tube erosion phenomenon.
Review of related erosion problems and approaches The approach used in this paper for the development of the scaling laws is inspired by similar experiences in cavitation erosion and in impact erosion such as rain and wet stream erosion. In all of these erosion problems, one of the important aspects is the prediction of the material response to a given set of input conditions such as the collapse of cavitation bubbles on or
311
Fig. 1. Definition sketch for the material response to erosive forces.
near the surface, the impingement of liquid drops or jets, and the impact of solid particles Ill]. In general, the impact energy of the input system (Fig. 1) will produce one of the following effects depending on the intensity of impact and on the number of repetitions: (1) there may not be any permanent deformation (threshoId); (2) the material may deform after a certain number of repetitive impacts (fatigue); (3) a percent deformation may develop at the onset of the first impact; (4) or the material may plastically flow as a result of high strain rates and temperatures. Based on these conditions one may arrive at two types of problem: the unders~ding of the threshold conditions wherein the impact stresses reach a limiting value just sufficient to initiate detectable erosion after several rounds and the.prediction of the amount of erosion if the erosive forces are above the threshold for the material. The objective of most investigators was to provide the designer with quantitative parameters for design. The threshold criterion for the case of liquid impact erosion is related to the fatigue endurance of the material and to the dynamic impact stress developed in the material: o,/plCiU1 = constant
(1)
where ue is the fatigue endurance of the material, p1 is the density of the liquid and C1 is the impact speed. The constant depends upon the material and the env~onment [12] . Similar results have been obtained for the case of cavitation erosion; the threshold intensity of cavitation erosion is related to the fatigue strength of the material [ 131. When the erosive forces exceed the threshold limit of the material, a certain volume of material is removed from the surface of the parent
312
Fig. 2. Master chart for cavitation erosion.
material as a result of the work done by the erosive forces. The energy absorbed by the volume of the material fractured is given by
E, = AVS,
(2)
where 3, is the erosion strength which represents the energy absorbing capacity of the material per unit volume under the action of the erosive forces, A V is the volume of material eroded and E, is the energy absorbed by the material eroded. The intensity of erosion is defined as the power absorbed by the material per unit area [ 143 and is given by I =-
AVS, AAt
=-AY Se At
(3)
313
Fig. 3. Effect of time on intensity of erosion.
where A is the area of erosion, At is the test duration and Ay is the mean depth of erosion (A V/A). This intensity parameter has the dimensions of power per unit area of erosion and physically represents the rate of energy absorption by the material. Figure 2 shows this intensity plotted against rate of depth of erosion for various materials ranging from soft lead to very highly resistant stellites. The range of intensities typical of practical machines varies from lo3 - lo4 in lb a-’ ine2. The screening tests .operate at intensity levels of the order of lo5 in lb a- ’ inB2 or 1 W mb2. The depth of erosion is generally in the range of a fraction of an inch per year. Chemical corrosion rates on steels are in the range lo- 3 to lo- 2 in a- ‘. Erosion rates of the order of 1 in a- ’ represent serious erosion which may warrant operational limitation or redesign [ 151. Although the intensity parameter is very useful as an engineering tool in design and research, the fact that the intensity of erosion is dependent on the exposure period complicates the analysis. The rate of erosion increases from negligible values, reaches a maximum and then decreases and levels off to a steady value. The erosion period may be divided into four periods [ 161; these are the incubation period, the acceleration period, the deceleration period and the steady state period (Fig. 3). An elementary theory of erosion has been formulated that makes use of some of the recent ideas such as the erosion’strength, intensity of erosion and fatigue failure probability [ 171. The following differential equation may be derived that makes use of these ideas: ti k15’2 _+--dt l/q2 k=
-2 S(AI,) 1’2
1 dq -= 7) dt
o
(4)
0
0
I 1
I
I
e
3 Refativo
I 4
I
I
I
I
J
5
6
7
8
9
E*poa.we
Ijms,
T
Fig. 4. Relative erosion rate as a function of relative exposure time for the seven materials.
where I is the intensity of erosion, k is a constant with dimensions of length, 17is the efficiency of erosion and t is the time of erosion. Equation (4) can be normalized with respect to the time at which the intensity is a maximum. The resulting normalized differential is dl iis/2 i d+i -+--__=o (5) ij dr d7 Mi2 where 7 = t/tl, 7 = I/I,,, , 77= q/q 1, h = (d7)/dT),= 1. Subscript 1 corresponds to the maximum intensity of erosion I,,,. The general solution of the normalized equation for erosion is given by
This theory predicts the relative rate of erosion (the ratio between the rate of erosion and the peak rate of erosion) and the relative exposure time (the ratio between the exposure time and the time corresponding to the peak erosion rate). A comparison between theory and experimental data for various materials exposed to cavitation erosion is shown in Fig. 4. This approach offers the possibility of using a weaker material at comparable intensities of erosion in a shorter time [IS] . Similar comparisons between theory and experimental results obtained from liquid impact erosion for several materials have been reported in ref. 12. A typical correlation for titanium 6 AL-4V (annealed) is shown in Fig. 5. The following development of scaling laws for gun tube erosion is greatly influenced by the experience derived from cavitation and liquid impact erosion models.
316 Titanium
bAL -f Y hwmakd)
‘<;;+jf 0
I
0
1
I 2.
I 3
I t=4t&A.5
I
I 0
I
I
I
I
7
8
9
10
Fig. 5. Comparison of experimental results with the erosion theory for titanium 6AL-4V.
Scaling law development Intensity of gun tube erosion The problem of gun barrel erosion can be divided into two aspects. First, a deeper understanding of the factors governing the threshold intensity of erosion for candidate materials is needed. The threshold intensity depends upon the stresses on the tube wall, the acceleration and velocity of the projectile, the temperature, the chemical compatibility of the material with the propellent gas and the fatigue strength of the material. It would be very useful to formulate a definition of intensity of erosion for gun barrel erosion and relate the threshold values of this intensity with material characteristics. This necessity leads us to the second aspect, namely the definition of intensity of erosion. As shown in Fig. 6, the output intensity experienced by the material is defined as the rate of energy absorbed per unit area eroded and is given by AVS, 11 = AAt =-AY St? At Ay AN =--s, AN At where I1 is the output intensity of erosion, AV is the volume of material eroded, A is the area of erosion, Ay is the mean depth of erosion, At is the
(6)
316
I
EQO
I 10,000
wocr by WJ) Fig. 6. Output intensity of erosion. Fig. 7, Dependence of erosion on muzzle velocity. Source of data was the Gun Erosion Handbook, 1946.
exposure period, AN is the number of rounds fired during the exposure period, Ay/AN is the mean depth of erosion per round, AN/At is the number of rounds per unit time and S, is the erosion strength of the material. The output intensity of erosion as seen by the gun tube material is equal to the product of the mean depth of erosion per round, the number of rounds fired in a given period of time and the erosion strength of the material as determined from a screening test. Similarly, the input density of erosion may be derived from the rate of work input to the tube wall material from the expansion of the projectile as it is accelerated along the gun tube. In order to derive an expression for the input intensity, it is assumed that the projectile expands and produces a compressive stress on the tube wall. Exerting this stress, it also moves along the tube rubbing against the wall. The wear produced by the contact stress is given by the product of the contact stress and the local velocity of the projectile 1191: I, = P, v,
(71
where Ia is the input intensity of erosion, P, is the contact stress and V, is the velocity of the projectile. However, P, a vFof&
(81
where v is the Poisson’s ratio of the projectile, F. is the force accelerating the projectile and A*’ is the cross-sectional area of the projectile. Furthermore
where i& = ~~~~j==ti~~ + If3 Mpowder ; Atprojectile is the mass of the projectile, 1Mpowderis the mass of powder and dV/dt = a,,, , the maximum acceleration of the projectile. The cross-sectional area of the projectile is given by A0 aDb
W)
where Do is the bore diameter of the gun tube. Substituting eqns. (8), (9) and (10) into eqn. (7) gives 42 a-
L&&V, 0;
dV
-
dt
(11)
Moreover, since the maximum acceleration of the projectile is proportional to the second power of the maximum velocity of the projectile, and the contact velocity V, is also proportional to the maximum-velocity V, of the projectile, eqn. (11) may be written as VM, V&
I, a _
G?
(12)
While the relation (12) represents the power input from the firing of the bullet to the material surface through the erosion process, eqn. (6) gives the power absorbed by the surface in material erosion. The ratio between the output intensity and input intensity is given by (13) This ratio is a nondimensional number which can be used as an essential scaling parameter in the analysis of gun tube erosion data as well as in the design of gun tubes. With this motivation, eqn. (13) was tested on available gun tube erosion data [3]. According to the relation (13), the erosion rate varies as the third power of the velocity. This relation has been compared with experimental data in Fig. 7. The data points are the average erosion rates as a function of muzzle velocity for more than 100 tested guns at 400 and 800 rounds. The line through the data points at 800 rounds is the third power law. The data plotted on a logarithmic scale indicate the validity of the erosion model and the scaling law suggested in eqn. (13). Similarly, the dependence of the rate of erosion on the bore diameter of the gun may be evaluated from eqn. (13) following eqn. (12): (14) and as adesign rule [3] Me aD$
(15) Furthermore, the acceleration of the projectile must increase with the bore diameter in order to compensate for the increased frictional forces. Assuming
318
12or. I10 -
6140
90 -
El
&Iw4030. to -
31
I-#s.I
I9
10 -
fPS 0
100-
80 -
to *oo
a300 to L9oe FPJ
88 0
H
0
CF
Fig. 8. Dependence of erosion intensity on caliber.
Fig. 9. Erosion rate at origin of rifling as a function of the number of rounds for a 4 - ‘7 in gun.
that the maximum accelerationis directly proportional to the bore diameter, the relation (14) becomes I, a DgD,/D$ I2 =Dg
(16)
for constant valuesof ttand V,. This is the case demonstratedby the data presentedin Fig. 8. An approximately lineardependence of erosion rate on
319
the square of the bore diameter for guns fired at three nominal velocities is shown in Fig. 8. Thus there seems to be evidence that the relation (13) represents both the speed scale and the size scale effects of gun tube erosion. The next aspect investigated related to the dependence of the rate of erosion on the number of rounds fired. As with cavitation erosion and liquid impact erosion, the gun tube land erosion rates are also small (incubation) at the beginning; they increase (acceleration), reach a maximum value and then decrease to a more or less steady value as shown in Fig. 9. This gives further credence to the erosion model proposed since the relation shown in Fig. 9 is in full agreement with such a model. The result that the decrease in erosion rates is caused by the formation of grooves in the lands is explained by the fact that there are no decreasing erosion rates in the grooves because the erosive forces are already attenuated in the grooves. For this reason, the erosion rates in the grooves are constant (Fig. 9). Because of this similarity between the gun tube erosion data and the previously developed erosion, the analytical approach for gun tube erosion data was extended. The rate of erosion for gun tube lands follows the corresponding four stages of deterioration. As shown in Fig. 9, the relation between the rate of erosion and the number of rounds fired includes the incubation, acceleration, deceleration and steady state periods. If it is assumed that this relation is a result of the mechanisms associated with the energy transmission, absorption and failure of the material which are very similar to the other processes, then eqn. (4) is also applicable to gun tube erosion. The general solution of the normalized erosion equation is given by eqn. (6). The erosion rate and the corresponding number of rounds are normalized with respect to the peak erosion rate and the corresponding number of rounds. Figure 10 shows a plot of relative erosion rate as a function of the relative number of rounds normalized with respect to peak erosion rate conditions. The data source includes refs. 3 and 4 for various test-fired guns as well as fleet data. The solid curve represents the general solution of eqn. (6), corresponding to a Weibull shape parameter of 3.5. The general agreement between theory and experimental data is encouraging and an understanding of the nonlinear behavior of gun tube erosion evolves. In addition, design calculations for gun tube life can be made on a more quantitative basis. However, it must be emphasized at this point that the complexity of the processes involved in gun tube erosion requires further extensive investigations to take into account the thermal and chemical effects. At higher temperatures, the strain rate effects become significant. What has been presented is only a first step towards the development of realistic scaling laws that can be useful in understanding the processes involved as well as in the design applications. Further experiments are required to test carefully the simplifying assumptions made during the development of quantitative models. The use of stellite liners and chromium plating is another approach to the prevention of gun tube erosion. With the use of such liners and tough coatings, the analysis of the effectiveness of protective methods is becoming
Fig. 10, Comparison of experimental results with the gun tube erosion theory.
7 A
c /
t
3L
4-
3-
,?-
+
J -
Fig. 11. Stress wave interactions in layered materials. Fig. 12. Influence of impedance variation on tensile stress coefficient.
important. Such an analysis would be useful to the designer in order to reduce cladding separation. The analysis includes the understanding of the failure modes by evaluation of the stress wave interactions in layered materials. The failure modes include erosion, chipping due to scabbing and
321
tensile failure of the coatings at their interface. Figure 11 shows the cross section of a gun tube substrate material which has a protective plating or a liner. The stresses in the cross-sectional view schematically indicate how the reflected and absorbed initial compressive stresses become tensile forces causing rapid deterioration of the protective coating or liner. Figure 12 shows typical results of a stress wave analysis relating the magnitude of tensile stresses developed in the coating system with the relative impedance of the coating and substrate [ 201. By properly matching the impedance ratio of the protective coatings, the tensile stresses can be reduced by an order of magnitude.
Concluding
remarks
As a result of these studies, the outlook for developing scaling laws governing gun barrel erosion appears encouraging. The erosion parameter is defined as the ratio between the output intensity of erosion representing the energy absorbed by the material and the input intensity of erosion arising out of the erosive forces caused by the dynamic factors. Specifically, the input intensity of erosion includes the rate of depth of erosion and the erosion strength of the material in that environment, whereas the output intensity of erosion includes the Poisson’s ratio of the projectile, the maximum acceleration, the maximum velocity and the bore diameter. With the aid of a few justifiable assumptions, it can be shown that the erosion rate increases as the cube of the velocity of the projectile and decreases as the square of the bore diameter. Available experimental data support these theoretical predictions but further experiments are required to verify these results and to confirm the erosion scaling parameter. An important variable in these analyses is the number of rounds fired. The erosion rate is highly dependent on the number of rounds fired; it has four stages as is the case with similar erosion phenomena such as cavitation and liquid impact. These periods include incubation, acceleration, deceleration and steady state. Previously developed erosion theory correlates equally well with the available gun tube erosion data. This correlation further supports the erosion model proposed. However, the interacting influences of thermal and chemical mechanisms must be carefully considered in any further developments. Such an effort would hopefully lead to a better understanding of the relative influences of these interacting mechanisms and to the development of better materials, protective systems and erosion-free designs. References 1 W. T. Ebibara, Wear and erosion characteristics of a cast cobalt base alloy, Tech. Rep. SWERR-TR-73-2, Research Directorate, Weapons Laboratory, WECOM, Jan., 1973. 2 W. T. Ebihara, Erosion in 7.62 mm machine gun barrels, Tech. Rep. RE-70-196, Science and Technology Laboratory, U.S. Army Weapons Command, December, 1970.
322 3 J. S. Burlew (ed.), Hypervelocity guns and the control of gun erosion, Summary Tech. Rep. of NRDC, Dirt. 1, Vol. 1, Office of Scientific Research and Development, Washington, D.C., 1946. 4 E. L. Bannister, Thermal theory for the erosion of guns by propellent gases, Tech. Note 1740, U.S. Army, Aberdeen Research and Development Center, Ballistic Research Laboratories, Aberdeen Proving Ground, Maryland, September, 1970. 5 W. T. Ebihara, Structural stability of a cast Co-Cr-Mo alloy during impulsive thermalmechanical loading, Tech. Rep. RE-70-197, Science and Technology Laboratory, U.S. Army Weapons Command, December 1970. 6 D. J. Taylor and J. Morris, Gun erosion and methods of control, Royal Armament Research and Development Establishment, Fort Halstead, Seven Oaks, Kent, England. 7 R. P. O’Shea, G. S. Allison, J. D. DiBenedetto and K. R. Iyer, Improved materials and manufacturing methods for gun barrels, Tech. Rep. SWERR-TR-72-54, Weapons Laboratory, U.S. Army Weapons Command, August, 1972. 8 Wear reduction additives, U.S. Patent 3, 148, 620, (Sept., 1964), to D. E. Jacobsen and S. Y. Ek. 9 A. Thiruvengadam, Scaling laws for cavitation erosion, Proc. Symp. on High Speed Flow of Water, International Union of Theoretical and Applied Mechanics, Leningrad, U.S.S.R., June 22 - 26, 1971. 10 A. Thiruvengadam, The concept of erosion strength, Erosion by Cavitation or Impingement, Am. Sot. Test. Mater., Spec. Tech. Publ. 408, 1967. 11 P. Bridgman, Dimensional Analysis, Yale University Press, New Haven, Conn., 1931. 12 A. Thiruvengadam, S. L. Rudy and M. Gunasekaran, Experimental and analytical investigations on liquid impact erosion, Am. Sot. Test. Mater., Spec. Tech. Publ. 474 (1970) 249 - 287. 13 A. Thiruvengadam, Intensity of cavitation erosion encountered in field installations, Symp. on Cavitation in Fluid Machinery, Winter Annual Meeting, Chicago, Ill., Nov. 1965. 14 A. Thiruvengadam, A comparative evaluation of cavitation damage test devices, ASME Symp. on Cavitation Research Facilities and Techniques, 1964, pp. 157 - 164. 15 A. Thiruvengadam, Cavitation erosion, Appl. Mech. Rev., March (1971) 245 - 253. 16 F. J. Heymam, On the time dependence of the rate of erosion due to impingement or cavitation, Am. Sot. Test. Mat., Spec. Tech. Publ. 408 (1967) 70 - 110. 17 A. Thiruvengadam, Theory of erosion, Proc. 2nd Meersburg Conf. on Rain Erosion, Royal Aircraft Establishment, Farnborough, England, 1967. 18 A. Thiruvengadam, On the selection of modeling materials to scale long term erosion behavior of prototype systems, Proc. 3rd Int. Conf. on Rain Erosion and Allied Phenomena, Royal Aircraft Establishment, Farnborough, England, 1970. 19 E. Rabinowicz, Friction and Wear of Materials, Wiley-Interscience, New York, 1965. 20 F. Envent, unpublished results obtained at Daedalean Associates, Incorporated, 1973.
307 - Printed
in the Netherlands
SOME SCALING LAWS GOVERNING GUN EROSION
W. T. EBIHARA Research, Development and Engineering Directorate, Arsenal, Rock Island, Ill. 61201 (U.S.A.) A. A. HOCHREIN,
Februa’ry
Island
JR., and A. THIRUVENGADAM
Daedalean Associates, Incorporated, (U.S.A.) (Received
Rodman Laboratory;Rock
15110 Frederick
Road, Woodbine, Md. 21797
18, 1976)
Summary Gun barrel erosion is one of the major problems limiting the velocity, range and accuracy of the projectile. The various interacting mechanisms causing erosion are discussed. The importance of developing scaling laws governing gun tube erosion is emphasized. The approach adopted*in this paper for the development of scaling laws is inspired by similar attempts in cavitation and liquid impact erosion. The concepts of threshold erosion, erosion intensity, erosion strength and erosion parameter are extended, modified and applied to gun tube erosion. As a result, the outlook for developing scaling laws governing gun barrel erosion seems very promising. The nondimensional number called the,erosion parameter is given by the ratio between the output intensity of erosion representing the energy absorbed by the material and the input intensity of erosion contributed by the erosive forces. The output intensity of erosion is given by the product of the rate of depth of erosion and erosion resistance of the material, whereas the input intensity includes the Poisson’s ratio of the projectile, the maximum acceleration, the maximum velocity and the bore diameter. With the aid of a few justifiable assumptions, it has been shown that the rate of depth of erosion is directly proportional to the square of the bore diameter. Available experimental data support these results for velocity scale and size scale. Further experiments are required to test these predictions fully. The erosion state is highly dependent on the number of rounds fired; it has four stages including incubation, acceleration, deceleration and steady state periods very similar to other erosion phenomena. Previously developed erosion theory correlates well with the available gun tube erosion rates as a function of the number of rounds fired; this lends further support to the erosion model proposed in this paper. However, the interacting influences of thermal and chemical mechanisms must be carefully considered in any further developments.
308
Introduction The wear erosion in gun barrels is a detrimental material problem seriously limiting the projectile velocity, range and accuracy. The maximum muzzle velocity of 3000 ft s--l reached during the 1940s has not been substantially exceeded, mainly because of the extensive erosion problems. Several erosion-resistant materials have been developed, the most prominent being stellite. The advent of high temperature super alloys technology offers a great potential in improving the capability, performance and service life of gun barrels. The service life for different experimental barrels varies from 7000 to 35 000 rounds depending upon the material liner and plating combinations [ 11. Muzzle end erosion has been shown to be the dominant problem in some of the super alloy barrels in contrast to the traditional breech end erosion problem [2] . Moreover, it has become increasingly important to formulate analytical models incorporating the dynamic response of materials at elevated temperatures under the influence of erosive forces. Gun tube erosion is characterized by an enlargement of the bore as a result of material wear. Erosion is greatest at the breech end and decreases rapidly towards the muzzle. However, recent observations with super alloys show that the muzzle end erosion is the dominant factor. The erosion rate is generally represented in fractions of a thousandth of an inch (mil) per round. Although the erosion rate is generally a few mil per round, the amount of metal lost is sometimes impressive, particularly in large guns. For example, in a 12 in gun firing at 2600 ft s-l, nearly a pound of material may be removed from the bore surface during a single round [3]. In addition to the removal of material by mechanical abrasive effects, severe heating also contributes by melting a thin layer which is carried off by hot gases and the band [4] . This intense heating and cooling in the presence of burnt gases make the bore surfaces brittle, leading to cracks. The brittle cracking is intense at the breech end and decreases towards the muzzle end. Generally the crack pattern follows the machining direction. Rapid heating and cooling also leads to unfavorable metallurgical changes in the bore surface material [5,61. Gun tube material development programs date back to 1940 leading to the development of several desirable materials including high strength steels, stellite liners and chromium plating techniques [ 31. More recent investigations [7] have indicated superior materials such as stellite 21 and Timkin 16-26-6 as compared with the traditional gun steel. Super alloys and refractory materials are being tested for small arms rapid fire weapons and chromemolybdenum-vanadium steels with chrome plating have been used in gun tube applications. However, the material requirements for the gun tube are so complex that no ideal material has been found yet. Since some of the ideal requirements are conflicting, the final material selection is necessarily a compromise. Some of the materials exhibiting very high resistance to heat and erosion do not have adequate strength to resist the high pressures
309
and inertial forces. Cost and fabrication techniques are also important considerations. Moreover, recent developments in coating techniques offer great potential despite flaking and delamination problems associated with such coatings. The introduction of wear-reducing additives is a major step forward [8] . Some of these additives provide laminar cooling of the bore surface while others improve the thermal and chemical compatibility of the material thus increasing their fatigue endurance limits. Many creative design concepts have also been introduced for reducing-erosion. These include the proper design of the projectile bands and the introduction of new materials which minimize the engraving friction. However, the lack of a coordinate’d approach to design and material selection has led to the present situation wherein many successful approaches in one type of weapon system for a particular size and speed range are not directly applicable in a different application. This is primarily because there is no quantitative approach in estimating the rate of erosion in a given size of gun at a given speed for a specified material and projectile geometry. However, similar complex problems such as cavitation erosion are being approached from the point of view of developing scaling laws [9]. Engineering design procedures and calculations invoive problems too complex for exact analytical solutions. Such problems are generally investigated through a judicious combination of experiments and development of nondimensional parameters governing the processes investigated. This procedure is used extensively in the design of ships, airplanes, hydraulic pumps and turbines as well as in problems that deal with heat flow and with complex stresses [lo] . There are several scaling parameters known to engineers. Each of these nondimensional scaling parameters used in designs has physical significance in relating different processes. Although rational procedures such as dimensional analysis are available, it is still not possible to derive all the pertinent nondimensional parameters a priori and to predict the functional relations involving complex problems encountered in engineering designs. However, it is possible to determine the dominant parameters and their relations by careful experimentation, by a proper understanding of the important aspects of the complex phenomena involved and by deductive reasoning. Such an approach would greatly enhance the understanding of the dominant erosion mechanisms and the material property requirements, It would also lead to the correlation of experimental data with analytical models and field performance, thereby enabling the designers to use these results with confidence. With this objective in mind, the various mechanisms involved in the complex phenomenon of gun erosion are reviewed. A plausible erosion model is derived which makes use of the recent approaches utilized in cavitation erosion, droplet erosion and wear of materials. Available experimental data show that the land erosion increases with the number of rounds fired, reaches a maximum and then decreases tending to a stable rate. A recently developed theory of erosion is used to correlate these rates in a nondimensional form. The maximum erosion rate and the steady state
310
erosion rate depend upon several gun parameters, velocity and gun tube diameter.
Mechanisms
including
the muzzle
of gun barrel erosion
As in other erosion problems, the mechanisms of erosion of gun tubes include mechanical causes, high temperature effects and chemical interactions. As the gun is fired, the burning powder is converted into a whitehot gas under high pressure propelling the projectile along the bore. During this process, the projectile as well as the tube are subjected to short duration stress pulses of intense magnitude. The accelerating forces and the mechanical stresses reach maximum levels near the breech end. Such dynamic forces cause a swaying action of the projectile. This combined with unburnt propellent particles, reaction products and wear particles leads to severe mechanical erosion. At the same time, the mass of high pressure white-hot flame follows the projectile, heating the tube wall, sometimes melting and chemically altering, in addition to gouging and scouring, the barrel wall. The bore surface material is subjected to rapid heating and cooling which sometimes leads to a martensitic transformation of the barrel material. The propellent gas contains free radicals, ions and unstable molecules such as CO, COz, HzO, Hz, NH,, CH, and HsS, which may react with the gun tube surface material, a process greatly assisted by the high temperature of the propellent gas. Although the erosion rate is measured in mils (thousandths of an inch) per round, the amount of metal lost may be as much as a pound per round in large guns. Repeated firing causes substantial enlargement of the bore leading to loss of pressure, loss of muzzle velocity and firing inaccuracy. The complex nature of gun tube erosion mechanisms has defied many attempts in the past to construct quantitative analytical models to describe and predict erosion rates. Such attempts have been at least partially successful in predicting the service life of systems experiencing other erosion phenomena such as cavitation and liquid impact, by the development of scaling laws and model prototype correlations that lead to improved design techniques, material selection and service life predictions. It is the objective of this paper to extend those approaches in order to develop scaling parameters applicable to the gun tube erosion phenomenon.
Review of related erosion problems and approaches The approach used in this paper for the development of the scaling laws is inspired by similar experiences in cavitation erosion and in impact erosion such as rain and wet stream erosion. In all of these erosion problems, one of the important aspects is the prediction of the material response to a given set of input conditions such as the collapse of cavitation bubbles on or
311
Fig. 1. Definition sketch for the material response to erosive forces.
near the surface, the impingement of liquid drops or jets, and the impact of solid particles Ill]. In general, the impact energy of the input system (Fig. 1) will produce one of the following effects depending on the intensity of impact and on the number of repetitions: (1) there may not be any permanent deformation (threshoId); (2) the material may deform after a certain number of repetitive impacts (fatigue); (3) a percent deformation may develop at the onset of the first impact; (4) or the material may plastically flow as a result of high strain rates and temperatures. Based on these conditions one may arrive at two types of problem: the unders~ding of the threshold conditions wherein the impact stresses reach a limiting value just sufficient to initiate detectable erosion after several rounds and the.prediction of the amount of erosion if the erosive forces are above the threshold for the material. The objective of most investigators was to provide the designer with quantitative parameters for design. The threshold criterion for the case of liquid impact erosion is related to the fatigue endurance of the material and to the dynamic impact stress developed in the material: o,/plCiU1 = constant
(1)
where ue is the fatigue endurance of the material, p1 is the density of the liquid and C1 is the impact speed. The constant depends upon the material and the env~onment [12] . Similar results have been obtained for the case of cavitation erosion; the threshold intensity of cavitation erosion is related to the fatigue strength of the material [ 131. When the erosive forces exceed the threshold limit of the material, a certain volume of material is removed from the surface of the parent
312
Fig. 2. Master chart for cavitation erosion.
material as a result of the work done by the erosive forces. The energy absorbed by the volume of the material fractured is given by
E, = AVS,
(2)
where 3, is the erosion strength which represents the energy absorbing capacity of the material per unit volume under the action of the erosive forces, A V is the volume of material eroded and E, is the energy absorbed by the material eroded. The intensity of erosion is defined as the power absorbed by the material per unit area [ 143 and is given by I =-
AVS, AAt
=-AY Se At
(3)
313
Fig. 3. Effect of time on intensity of erosion.
where A is the area of erosion, At is the test duration and Ay is the mean depth of erosion (A V/A). This intensity parameter has the dimensions of power per unit area of erosion and physically represents the rate of energy absorption by the material. Figure 2 shows this intensity plotted against rate of depth of erosion for various materials ranging from soft lead to very highly resistant stellites. The range of intensities typical of practical machines varies from lo3 - lo4 in lb a-’ ine2. The screening tests .operate at intensity levels of the order of lo5 in lb a- ’ inB2 or 1 W mb2. The depth of erosion is generally in the range of a fraction of an inch per year. Chemical corrosion rates on steels are in the range lo- 3 to lo- 2 in a- ‘. Erosion rates of the order of 1 in a- ’ represent serious erosion which may warrant operational limitation or redesign [ 151. Although the intensity parameter is very useful as an engineering tool in design and research, the fact that the intensity of erosion is dependent on the exposure period complicates the analysis. The rate of erosion increases from negligible values, reaches a maximum and then decreases and levels off to a steady value. The erosion period may be divided into four periods [ 161; these are the incubation period, the acceleration period, the deceleration period and the steady state period (Fig. 3). An elementary theory of erosion has been formulated that makes use of some of the recent ideas such as the erosion’strength, intensity of erosion and fatigue failure probability [ 171. The following differential equation may be derived that makes use of these ideas: ti k15’2 _+--dt l/q2 k=
-2 S(AI,) 1’2
1 dq -= 7) dt
o
(4)
0
0
I 1
I
I
e
3 Refativo
I 4
I
I
I
I
J
5
6
7
8
9
E*poa.we
Ijms,
T
Fig. 4. Relative erosion rate as a function of relative exposure time for the seven materials.
where I is the intensity of erosion, k is a constant with dimensions of length, 17is the efficiency of erosion and t is the time of erosion. Equation (4) can be normalized with respect to the time at which the intensity is a maximum. The resulting normalized differential is dl iis/2 i d+i -+--__=o (5) ij dr d7 Mi2 where 7 = t/tl, 7 = I/I,,, , 77= q/q 1, h = (d7)/dT),= 1. Subscript 1 corresponds to the maximum intensity of erosion I,,,. The general solution of the normalized equation for erosion is given by
This theory predicts the relative rate of erosion (the ratio between the rate of erosion and the peak rate of erosion) and the relative exposure time (the ratio between the exposure time and the time corresponding to the peak erosion rate). A comparison between theory and experimental data for various materials exposed to cavitation erosion is shown in Fig. 4. This approach offers the possibility of using a weaker material at comparable intensities of erosion in a shorter time [IS] . Similar comparisons between theory and experimental results obtained from liquid impact erosion for several materials have been reported in ref. 12. A typical correlation for titanium 6 AL-4V (annealed) is shown in Fig. 5. The following development of scaling laws for gun tube erosion is greatly influenced by the experience derived from cavitation and liquid impact erosion models.
316 Titanium
bAL -f Y hwmakd)
‘<;;+jf 0
I
0
1
I 2.
I 3
I t=4t&A.5
I
I 0
I
I
I
I
7
8
9
10
Fig. 5. Comparison of experimental results with the erosion theory for titanium 6AL-4V.
Scaling law development Intensity of gun tube erosion The problem of gun barrel erosion can be divided into two aspects. First, a deeper understanding of the factors governing the threshold intensity of erosion for candidate materials is needed. The threshold intensity depends upon the stresses on the tube wall, the acceleration and velocity of the projectile, the temperature, the chemical compatibility of the material with the propellent gas and the fatigue strength of the material. It would be very useful to formulate a definition of intensity of erosion for gun barrel erosion and relate the threshold values of this intensity with material characteristics. This necessity leads us to the second aspect, namely the definition of intensity of erosion. As shown in Fig. 6, the output intensity experienced by the material is defined as the rate of energy absorbed per unit area eroded and is given by AVS, 11 = AAt =-AY St? At Ay AN =--s, AN At where I1 is the output intensity of erosion, AV is the volume of material eroded, A is the area of erosion, Ay is the mean depth of erosion, At is the
(6)
316
I
EQO
I 10,000
wocr by WJ) Fig. 6. Output intensity of erosion. Fig. 7, Dependence of erosion on muzzle velocity. Source of data was the Gun Erosion Handbook, 1946.
exposure period, AN is the number of rounds fired during the exposure period, Ay/AN is the mean depth of erosion per round, AN/At is the number of rounds per unit time and S, is the erosion strength of the material. The output intensity of erosion as seen by the gun tube material is equal to the product of the mean depth of erosion per round, the number of rounds fired in a given period of time and the erosion strength of the material as determined from a screening test. Similarly, the input density of erosion may be derived from the rate of work input to the tube wall material from the expansion of the projectile as it is accelerated along the gun tube. In order to derive an expression for the input intensity, it is assumed that the projectile expands and produces a compressive stress on the tube wall. Exerting this stress, it also moves along the tube rubbing against the wall. The wear produced by the contact stress is given by the product of the contact stress and the local velocity of the projectile 1191: I, = P, v,
(71
where Ia is the input intensity of erosion, P, is the contact stress and V, is the velocity of the projectile. However, P, a vFof&
(81
where v is the Poisson’s ratio of the projectile, F. is the force accelerating the projectile and A*’ is the cross-sectional area of the projectile. Furthermore
where i& = ~~~~j==ti~~ + If3 Mpowder ; Atprojectile is the mass of the projectile, 1Mpowderis the mass of powder and dV/dt = a,,, , the maximum acceleration of the projectile. The cross-sectional area of the projectile is given by A0 aDb
W)
where Do is the bore diameter of the gun tube. Substituting eqns. (8), (9) and (10) into eqn. (7) gives 42 a-
L&&V, 0;
dV
-
dt
(11)
Moreover, since the maximum acceleration of the projectile is proportional to the second power of the maximum velocity of the projectile, and the contact velocity V, is also proportional to the maximum-velocity V, of the projectile, eqn. (11) may be written as VM, V&
I, a _
G?
(12)
While the relation (12) represents the power input from the firing of the bullet to the material surface through the erosion process, eqn. (6) gives the power absorbed by the surface in material erosion. The ratio between the output intensity and input intensity is given by (13) This ratio is a nondimensional number which can be used as an essential scaling parameter in the analysis of gun tube erosion data as well as in the design of gun tubes. With this motivation, eqn. (13) was tested on available gun tube erosion data [3]. According to the relation (13), the erosion rate varies as the third power of the velocity. This relation has been compared with experimental data in Fig. 7. The data points are the average erosion rates as a function of muzzle velocity for more than 100 tested guns at 400 and 800 rounds. The line through the data points at 800 rounds is the third power law. The data plotted on a logarithmic scale indicate the validity of the erosion model and the scaling law suggested in eqn. (13). Similarly, the dependence of the rate of erosion on the bore diameter of the gun may be evaluated from eqn. (13) following eqn. (12): (14) and as adesign rule [3] Me aD$
(15) Furthermore, the acceleration of the projectile must increase with the bore diameter in order to compensate for the increased frictional forces. Assuming
318
12or. I10 -
6140
90 -
El
&Iw4030. to -
31
I-#s.I
I9
10 -
fPS 0
100-
80 -
to *oo
a300 to L9oe FPJ
88 0
H
0
CF
Fig. 8. Dependence of erosion intensity on caliber.
Fig. 9. Erosion rate at origin of rifling as a function of the number of rounds for a 4 - ‘7 in gun.
that the maximum accelerationis directly proportional to the bore diameter, the relation (14) becomes I, a DgD,/D$ I2 =Dg
(16)
for constant valuesof ttand V,. This is the case demonstratedby the data presentedin Fig. 8. An approximately lineardependence of erosion rate on
319
the square of the bore diameter for guns fired at three nominal velocities is shown in Fig. 8. Thus there seems to be evidence that the relation (13) represents both the speed scale and the size scale effects of gun tube erosion. The next aspect investigated related to the dependence of the rate of erosion on the number of rounds fired. As with cavitation erosion and liquid impact erosion, the gun tube land erosion rates are also small (incubation) at the beginning; they increase (acceleration), reach a maximum value and then decrease to a more or less steady value as shown in Fig. 9. This gives further credence to the erosion model proposed since the relation shown in Fig. 9 is in full agreement with such a model. The result that the decrease in erosion rates is caused by the formation of grooves in the lands is explained by the fact that there are no decreasing erosion rates in the grooves because the erosive forces are already attenuated in the grooves. For this reason, the erosion rates in the grooves are constant (Fig. 9). Because of this similarity between the gun tube erosion data and the previously developed erosion, the analytical approach for gun tube erosion data was extended. The rate of erosion for gun tube lands follows the corresponding four stages of deterioration. As shown in Fig. 9, the relation between the rate of erosion and the number of rounds fired includes the incubation, acceleration, deceleration and steady state periods. If it is assumed that this relation is a result of the mechanisms associated with the energy transmission, absorption and failure of the material which are very similar to the other processes, then eqn. (4) is also applicable to gun tube erosion. The general solution of the normalized erosion equation is given by eqn. (6). The erosion rate and the corresponding number of rounds are normalized with respect to the peak erosion rate and the corresponding number of rounds. Figure 10 shows a plot of relative erosion rate as a function of the relative number of rounds normalized with respect to peak erosion rate conditions. The data source includes refs. 3 and 4 for various test-fired guns as well as fleet data. The solid curve represents the general solution of eqn. (6), corresponding to a Weibull shape parameter of 3.5. The general agreement between theory and experimental data is encouraging and an understanding of the nonlinear behavior of gun tube erosion evolves. In addition, design calculations for gun tube life can be made on a more quantitative basis. However, it must be emphasized at this point that the complexity of the processes involved in gun tube erosion requires further extensive investigations to take into account the thermal and chemical effects. At higher temperatures, the strain rate effects become significant. What has been presented is only a first step towards the development of realistic scaling laws that can be useful in understanding the processes involved as well as in the design applications. Further experiments are required to test carefully the simplifying assumptions made during the development of quantitative models. The use of stellite liners and chromium plating is another approach to the prevention of gun tube erosion. With the use of such liners and tough coatings, the analysis of the effectiveness of protective methods is becoming
Fig. 10, Comparison of experimental results with the gun tube erosion theory.
7 A
c /
t
3L
4-
3-
,?-
+
J -
Fig. 11. Stress wave interactions in layered materials. Fig. 12. Influence of impedance variation on tensile stress coefficient.
important. Such an analysis would be useful to the designer in order to reduce cladding separation. The analysis includes the understanding of the failure modes by evaluation of the stress wave interactions in layered materials. The failure modes include erosion, chipping due to scabbing and
321
tensile failure of the coatings at their interface. Figure 11 shows the cross section of a gun tube substrate material which has a protective plating or a liner. The stresses in the cross-sectional view schematically indicate how the reflected and absorbed initial compressive stresses become tensile forces causing rapid deterioration of the protective coating or liner. Figure 12 shows typical results of a stress wave analysis relating the magnitude of tensile stresses developed in the coating system with the relative impedance of the coating and substrate [ 201. By properly matching the impedance ratio of the protective coatings, the tensile stresses can be reduced by an order of magnitude.
Concluding
remarks
As a result of these studies, the outlook for developing scaling laws governing gun barrel erosion appears encouraging. The erosion parameter is defined as the ratio between the output intensity of erosion representing the energy absorbed by the material and the input intensity of erosion arising out of the erosive forces caused by the dynamic factors. Specifically, the input intensity of erosion includes the rate of depth of erosion and the erosion strength of the material in that environment, whereas the output intensity of erosion includes the Poisson’s ratio of the projectile, the maximum acceleration, the maximum velocity and the bore diameter. With the aid of a few justifiable assumptions, it can be shown that the erosion rate increases as the cube of the velocity of the projectile and decreases as the square of the bore diameter. Available experimental data support these theoretical predictions but further experiments are required to verify these results and to confirm the erosion scaling parameter. An important variable in these analyses is the number of rounds fired. The erosion rate is highly dependent on the number of rounds fired; it has four stages as is the case with similar erosion phenomena such as cavitation and liquid impact. These periods include incubation, acceleration, deceleration and steady state. Previously developed erosion theory correlates equally well with the available gun tube erosion data. This correlation further supports the erosion model proposed. However, the interacting influences of thermal and chemical mechanisms must be carefully considered in any further developments. Such an effort would hopefully lead to a better understanding of the relative influences of these interacting mechanisms and to the development of better materials, protective systems and erosion-free designs. References 1 W. T. Ebibara, Wear and erosion characteristics of a cast cobalt base alloy, Tech. Rep. SWERR-TR-73-2, Research Directorate, Weapons Laboratory, WECOM, Jan., 1973. 2 W. T. Ebihara, Erosion in 7.62 mm machine gun barrels, Tech. Rep. RE-70-196, Science and Technology Laboratory, U.S. Army Weapons Command, December, 1970.
322 3 J. S. Burlew (ed.), Hypervelocity guns and the control of gun erosion, Summary Tech. Rep. of NRDC, Dirt. 1, Vol. 1, Office of Scientific Research and Development, Washington, D.C., 1946. 4 E. L. Bannister, Thermal theory for the erosion of guns by propellent gases, Tech. Note 1740, U.S. Army, Aberdeen Research and Development Center, Ballistic Research Laboratories, Aberdeen Proving Ground, Maryland, September, 1970. 5 W. T. Ebihara, Structural stability of a cast Co-Cr-Mo alloy during impulsive thermalmechanical loading, Tech. Rep. RE-70-197, Science and Technology Laboratory, U.S. Army Weapons Command, December 1970. 6 D. J. Taylor and J. Morris, Gun erosion and methods of control, Royal Armament Research and Development Establishment, Fort Halstead, Seven Oaks, Kent, England. 7 R. P. O’Shea, G. S. Allison, J. D. DiBenedetto and K. R. Iyer, Improved materials and manufacturing methods for gun barrels, Tech. Rep. SWERR-TR-72-54, Weapons Laboratory, U.S. Army Weapons Command, August, 1972. 8 Wear reduction additives, U.S. Patent 3, 148, 620, (Sept., 1964), to D. E. Jacobsen and S. Y. Ek. 9 A. Thiruvengadam, Scaling laws for cavitation erosion, Proc. Symp. on High Speed Flow of Water, International Union of Theoretical and Applied Mechanics, Leningrad, U.S.S.R., June 22 - 26, 1971. 10 A. Thiruvengadam, The concept of erosion strength, Erosion by Cavitation or Impingement, Am. Sot. Test. Mater., Spec. Tech. Publ. 408, 1967. 11 P. Bridgman, Dimensional Analysis, Yale University Press, New Haven, Conn., 1931. 12 A. Thiruvengadam, S. L. Rudy and M. Gunasekaran, Experimental and analytical investigations on liquid impact erosion, Am. Sot. Test. Mater., Spec. Tech. Publ. 474 (1970) 249 - 287. 13 A. Thiruvengadam, Intensity of cavitation erosion encountered in field installations, Symp. on Cavitation in Fluid Machinery, Winter Annual Meeting, Chicago, Ill., Nov. 1965. 14 A. Thiruvengadam, A comparative evaluation of cavitation damage test devices, ASME Symp. on Cavitation Research Facilities and Techniques, 1964, pp. 157 - 164. 15 A. Thiruvengadam, Cavitation erosion, Appl. Mech. Rev., March (1971) 245 - 253. 16 F. J. Heymam, On the time dependence of the rate of erosion due to impingement or cavitation, Am. Sot. Test. Mat., Spec. Tech. Publ. 408 (1967) 70 - 110. 17 A. Thiruvengadam, Theory of erosion, Proc. 2nd Meersburg Conf. on Rain Erosion, Royal Aircraft Establishment, Farnborough, England, 1967. 18 A. Thiruvengadam, On the selection of modeling materials to scale long term erosion behavior of prototype systems, Proc. 3rd Int. Conf. on Rain Erosion and Allied Phenomena, Royal Aircraft Establishment, Farnborough, England, 1970. 19 E. Rabinowicz, Friction and Wear of Materials, Wiley-Interscience, New York, 1965. 20 F. Envent, unpublished results obtained at Daedalean Associates, Incorporated, 1973.