Determination of Wear Rate Equation and Estimation of Residual Life of 155 mm Autofrettaged Gun Barrel

Wear of steel at high temperature is a complex phenomenon. Quantification of wear per event is an important aspect in assessing life of a component. Necessity of economics and light weighted components compels to study the system under common parameters and extrapolate the results. Temperatures/pressures inside a gun are impossible to be measured by direct methods. Mathematical modeling and comparison of results with end effect measurable parameters is only way to predict it’s accuracy. There is no one such quantitative demonstration which can predict accurate wear rate. Gun wear is unique to gun system and no one theory fits into predictions of gun wear. Gun wear is unique phenomena wrt any particular gun system and needs to be studied independently. Numerous studies have been carried to study steel wear below 600oC. Wear studies at higher temperature i.e. above 600oC is not enough to describe correct wear phenomena.


INTRODUCTION
Wear of steel at high temperature is a complex phenomenon.Quantification of wear per event is an important aspect in assessing life of a component.Necessity of economics and light weighted components compels to study the system under common parameters and extrapolate the results.Temperatures/pressures inside a gun are impossible to be measured by direct methods.Mathematical modeling and comparison of results with end effect measurable parameters is only way to predict it's accuracy.
Wear of steel at high temperature is a complex phenomenon.Quantification of wear per event is an important aspect in assessing life of a component.Necessity of economics and light weighted components compels to study the system under common parameters and extrapolate the results.Temperatures/pressures inside a gun are impossible to be measured by direct methods.Mathematical modeling and comparison of results with end effect measurable parameters is only way to predict it's accuracy.
Manufacture of gun tube is an expensive proposal.Requirement of safety is prime importance and thereby generally it is overdesigned.It costs dearly thereby restricts total available quantum.A tube designed for adequate strength will enhance overall availability.Presently existing tube life can be enhanced to reduce burden on manufacturing agencies/Exchequer.Assessing wear rate of steel is an important study for design engineer for design of optimum gun system.
There is no one such quantitative demonstration which can predict accurate wear rate.Gun wear is unique to gun system and no one theory fits into predictions of gun wear 1,2,3,4 .Gun wear is unique phenomena w.r.t.any particular gun system and needs to be studied independently.Numerous studies have been carried to study steel wear below 600ºC.Wear studies at higher temperature i.e. above 600 o C 1,2,5 is not enough to describe correct wear phenomena.
Gun barrel wear is a complex phenomenon and has been difficult to understand.A unique pattern is observed with each gun system.Early formulas derived by Jones 6 were modified by Shulyer 6 and Kent 7 stated that gun wear is independent of shot mass.However both deferred on relationship of wear with size.Shulyer 6 stated that wear is independent of size and Kent suggested wear is inversely proportional to 2.91 th power of bore.Thornhill 8 linked the wear with the heat transfer.Cuthbert and Parry 9 supported Thornhill 8 and stated that wear increases with flame temperature of gasses even when muzzle velocities were kept constant by varying charge masses .No significant wear was noticed below 600 o C .Between 600 o C to 1000 o C, wear increased with moderate rate and after 1000 o C, it increased rapidly.Kanhere 10 related volume removal rate of steel with KE of propellant gases and surface hardness.Criteria of Kanhere is not advocated by majority of authors.
Izak M Synmam 11 simulated gun wear assigning it to be caused by temperature and heat transfer alone as proposed by Thornhill 8 .Lawtan 12 stated that gas leakage through driving band causes excessive temperature rise and leads to high wear rates.He simulated and experimentally verified in close vessel experiments.However no experimental results on actual autofrettaged tube system were given.Peter 13 stated that plastic deformation in gun barrel above 750ºC plays a major role in wear gun in comparison to static or kinematics friction.
J. K. Kim B. D. Choi 14 carried out experimental study to predict approximate life gun 155mm (self propelled) gun.C98 tube was studied and results were extrapolated to approximate the new life.Study was carried out by firing few rounds with each tube and results were extrapolated.Autofrettaged tubes considered were in early stages of life, so no consideration was give to older and used tubes, so statement of Lawtan 12 could not be commented.Xiiaogang Huang 15 studied thermal behavior of 5.56 mm ceramic gun barrels.He stated that maximum peak temperature is achieved within 20% of total travel time and stays for less than 0.2% of time travelled.It is extremely difficult to verify results with desired accuracy.
In a study of barrel life assessment Munger 16 also attributed heat as a major player in barrel projectile interaction.Mechanical wear is insignificant as compared to wear due to heat.Yavon 17 developed a theoretical model to gun barrel wear and result were predicted against laboratory models of ballistic gun under controlled environment with known and verified projectiles.Regression formulas thus derived did not consider actual field conditions and were derived under ideal conditions with tubes in initial stages of life.Gun tube erosion is more rapid and prominent in enlarged guns due to higher leakage of gases in worn out guns 3 , 9,12,19,20 and variation in heat transfer rates 19 .
Xin 18 and Shang developed FE models for cold formed channels.He validated his models with results of Hans Cock 1998, and the developed virtual test results for range of cold formed channels.He suggested through simulation updating of coefficients in design formula for more reliability.He suggested that simulation is an efficient way of predictions without experimental validation over effective range.
Correct gun barrel wear and predictions of residual life is important information for any defence organization owing to fact of its importance in formulating defense potential indices, procurement plans and defense budgeting.Model is to be developed for each type of gun system as no universal answer is available.Model so developed should be more akin to operating variable of that system.From discussion above it appears that wear rate suggested by Schulyer, Kent and Thornhill may throw some light on wear patterns in autofrettaged tubes operating at very high temperature and pressure.
1.1 METHODOLOGY Shots were fired and measurement of wear was taken in 155mm tubes.Measurements of gun barrel wear were taken in guns with different life i.e. latest guns, mid life guns and worn out guns.
Result were plotted to ascertain surety levels 21 so R 2 analysis of data was carried out.Different equations were fitted in data series with help of Mathematica 7. R 2 (coefficient of determination) 21 for each equation were determined, selection was based on best fit as it gave satisfactory assurance level.
Mathematica 7 used for solving all equations.

R 2 (Pearson product moment correlation coefficient) is
Wear rates were also determinded based on barrel wear theories and comparison graphs drawn:-Schulyer's Criteria 3 for wear is, C is charge mass, m is projetile mass, l is length of projectile and d is diameter of projectile.Value of constant is unique for each system.Maximum permissible wear taken, so as to avoid mechanical failure, is 51.87 mm 19,24,25 Value of P Max is calculated as per Internal ballistics relationships 22 Thornhill's Criteria 1 is where T cr is melting temperature of steel.Value of T max is calculated as per Internal ballistics relationships 22 Best fit equation for Measured Wear is; Wear ϭ 6.54554 ϫ 10 Ϫ14 EFC 4 Ϫ 5.01507 ϫ 10 Ϫ10 EFC 3 ϩ 1.19058ϫ10 Ϫ06 EFC 2 ϩ 4.90864 ϫ 10 Ϫ03 EFC ϩ 7.27686 ϫ 10 Ϫ01 Relationship of Charge mass with EFC, for the given system is EFCϭ Ϫ0.397415ϩ0.37056ϫϪ0.080205 x 2 ϩ0.00610115 x 3 x ϭ Propellant charge weight When propellant weight and EFC equations are corelated with wear equation the solution gives four roots, one positive, one negative, two imaginery values.Collecting all positive values and plotiing best fit for the same with respect to EFC is given in fiugre 1 below Plot for estimated wear as per Shulyer, Kent and Thornhill are plottted against measured wear and are given in figure 02.

RESULTS AND DISCUSSION
Kents' and Thornhill wear pattern are almost parallel.If maximum wear is taken as equal to measured wear graph, then they are almost overlapping.Schulyer's criteria can not be applied due to very high deviations from measurements.
Thornhiil's criteria is almost a strainght line.It may be due to the fact that maximum rounds fired by this gun system are charge 6 and less then 1.5% are higher charges.Multiplying thornhills's constant by a fixed value gives a overlapping graph to measured wear graph.Thornhill's criteria with this amedned wear rate constant is best solution for this gun system.
Kents' criteria agrees with measured wear pattern for initial stages of progress of wear.It's deviation there after is higher over mesured wear pattern.Multiplying kent's criteria graph data with an additional constant makes it almost parallel to measured wear graph with an offset.
Wearϭ5.83073ϫ10 Ϫ3 EFC ϩ 6.16663ϫ10 Ϫ1 , R 2 ϭ 97.4851% (Linear Relationship) is best fit equation with respect to EFC for 155mm gun tube with assurance level of 97.48 for the given set of gun tubes.
Simple relationship for Resudual life of each tube; derived for Balance Life in EFC is