THE UNIVERSITY OF MICHIGAN College of Engineering Department of Mechanical Engineering Cavitation and Multiphase Flow Laboratory Report No. 02643-2-PR Progress Report No. 2 A STATISTICALLY VERIFIED MODEL FOR CORRELATING VOLUME LOSS DUE TO CAVITATION OR LIQUID IMPINGEMENT (Submitted for presentation at ASTM Symposium on Char, acterization and Determination of Erosion Resistance) by: F G. Hammitt Y. C. Huang C. L. Kling T M.M Mitchell L. P. Solomon Financial Support Provided by: U. S. Naval Air Development Center Contract No. N62269-69-C-0285 This document is subject to special export controls and each transmittal to foreign governments or foreign nationals may be n.xade only with prior approval of Commanding Officer, Naval Air Development Center. April 1969

ABSTRACT Rocket sled data has been used to evaluate statistically best values for threshold velocity and velocity exponent as well as the coefficient n in the relation: MDPR = K(VsinG - V )a / sinn It is found that a reasonable assumption for most materials is that threshold velocity is zero, n=l, and that there is a rough correlation between K and a. The correlations with the data are relatively poor. However, within the basic precision attained, a single figure of merit for the material in terms of either K or a could be easily derived. A more basic relation for the evaluation of MDPR has been assumed in terms of the energy flux model suggested by Hoff et al (5). In this relation the material properties are described in terms of an energy per volume term describing material failure. It has been found that the material energy term is best described as proportional to ultimate resilience. No correlation with any efficiency term relating liquid and material properties alone has been found.

ii TABLE OF CONTENTS I. INTRODUCTION.1..........1 II. ROCKET SLED DATA CORRELATION.. 2 A. General Considerations...............2 B. Computer Correlation Results.. 4 III. GENERALIZED EROSION MODEL..... S A. General Considerations 6 B. Basic Equation Selected 7 C. Evaluation of Energy Parameter 8 1. General Remarks 8 2. Selection of Data for Evaluation.10 3. Best Fit Results Attained.. 11 IV. CONCLUSIONS................... 6 NOMENCLATURE...............19.... 19 BIBLIOGRAPHY.................. 20 TABLES.......................... 22 FIGURES.3 0 APPENDIX.......................33

I. INTRODUCTION One of the major objectives of much past and present erosion research, either cavitation or impingement, is to establish a mathematical model with fluid-flow, and material parameters as input data which would allow the engineering prediction of erosion rates for given, as yet untested, materials. A precise model of this sort has so far eluded investigators. This appears to be inevitable in view of the highly complex and varied natures of the erosion processes, even though produced by droplet impingement or cavitation, for example, alone. Nevertheless, it is desirable, using a large and diverse group of data, to attempt to determine optimum correlation relationships, and also to determine roughly what degree of precision can be expected from correlation models using easily measured standard engineering parameters as input data. The development of an optimum model and the examination of other possible models, particularly for droplet impingement in the range of interest of aircraft rain erosion, is the objective of the present paper. A fairly complex set of data, including both impingement and cavitation data has been used. This combination of data seems reasonable due to the presumed basic similarity of the erosion phenomena in impingement and cavitation. The model chosen for further investigation has been made dimensionally-consistent and as simple as possible, in hopes of obtaining a maximum generality and applicability for the results. This objective is also enhanced by the use of a diverse data set including items generated in different impact and cavitation type tests. It is expected thatadditional data items as they become available (or are reduced to the form here required) will be incorporated into the overall analysis, thus increasing its generality still further. Prior to the investigation of an overall erosion model, attempts were made to correlate data obtained on rain erosion materials in the

2 Holloman AFB rocket sled facility, using previously published semi-empirical relations between erosion rate, velocity, and angle of impact. The relatively poor fit achieved in this instance emphasized the necessity for a model more closely based on the details of the physical processes involved. For this reason it was decided to attempt a step-by-step development, relating as closely as possible to measurable data at all times, of a more basic model along the lines suggested by Hoff etal (1), relating the rate of erosion (MDPR = mean depth of penetration rate) to the kinetic eiergy impacting the target, the efficiency of energy transfer between drop and target (ft), and a material parameter (E) with dimensions of energy per unit target material volume. The equation adopted, explained in detail later, relates MDPR to the impinging kinetic evergy and the energy necessary to remove material: MDPR ( f)( V) - - - (1) e Our analysis to date, carried out within the framework of this equation and utilizing a data set which includes bo-Lh impingement and cavitation data, has concentrated on the optimum evaluation of the material parameter ~ in terms of mechanical material properties, and has also included the contribution to the energy transfer efficiency term z due to the material and fluid, rather than geometrical and flow, parameters. II. ROCKET SLED DATA CORRELATION A. General Considerations As a portion of the overall program aimed at the evaluation of potential rain erosion resistent materials, we have examined some of the data generated by the 1967 rocket sled tests at

3 Holloman AFB to determine the suitability of certain semi-empirical damage-predicting equations. The portion of the rocket sled datfe selected for this analysis comprised ten groups of materials including ceramics, plastics, and metals. They had been tested in the 6000 ft. rain field at Holloman AFB at Mach numbers ranging from 1.5 to 3.0, at various angles of impact ranging from 0 0 13. 5 to 90. The full details of this analysis have been reported previously (2). However, certain salient features will be repeated here for convenience. An earlier report by Tatnall, et al. (3), based upon an experimental fit of rocket sled data suggests the velocity appears in an exponential form: aV bV MDPR = Ce sinn + CZe — (2) where C1, C2, a and b are constants depending on material properties. Baker, et al. (4), proposed a relation based on their impact data, which includes the concept of a threshold velocity below which damage is essentially zero: MDPR = K(V sine -Vo) / sin) for Vsin e >V 0 0 for V sin e c V (3) More recently Hoff and Langbein (5) have suggested a modification of eq. (3) whereby the denominator is squared: MDPR = K(Vsine- V )a/ sine for V sin e > v = 0 for V sin e <V - - (4) Eq. (2) is simply a curve-fitting expression, not based on any physical model. Eq. (3) on the other hand assumed basically

4 that MDPR is proportional to the difference between the normal component of the impact velocity and some "threshold velocity", all raised to some power, a. A similar assumption has often been made in the cavitation literature,; 6, e.g.) where damage was assumed proportional to the 6th power of the flow velocity. In eq. (3), sinG has been added to the denominator to take some account of the damage due to shear from the high radial velocity after impact,which increases for oblique collisions. Actually, since in the rocket sled type test the specimen impacts a reduced number of raindrops if the impact is not normal, it might be argued that an additional sinG is required in the numerator, cancelling that in the denominator. This latter variation was not tried in the present analysis. Eq. (4) is identical to eq. (3) except that sin - appears in the denominator. This term can be derived logically from a model assuming energy:3ux on the target to be the predominent mechanism (5), if it is also assumed that the efficiency of energy transfer between impacting drop and target is a function of VsinQ only. However, it seems unlikely that this is strictly the case, so that eq. (3) and (4) remain semi-emp-3Sical in nature, and to be tested only in terms of a data fit. B. Computer Correlation Results The most comprehensive analysis of the rocket sled materials was made using Baker's proposed eq. (3). For each material a least mean square fit regression analysis was made to determine the best value of threshold velocity V, and of the amplitude constant K and the velocity exponent a. Full details are given elsewhere (2), but the important general results are included here. Table I shows the comparison between between predicted and measured MDPR for Pyroceram, along with the best values of V, K, and a. It is noted that these differ by factors in excess of 10 in many cases. This is typical for most of the results. It

5 is also noted that the best value for the threshold velocity in this particular case is zero, which is also fairly typical. Table II shows the effect of choice of V on the best values o for the exponent a and the amplitude constant. K. The effect on K of varying V between 0 and 2000 f/s is small but a varies over this range from 6.44 to 2.28. A plot of MDPR vs. V shows a small or zero MDPR for small velocities and then a rapidly increasing MDPR for larger velocities. Such a curve can be fit almost equally well by various combinations of V (including 0) and velocity exponents a, as the present calculations show. Since, strictly, it is unlikely that there will be zero damage for repeated impacts at any velocity, it may be permissible to avoid the concept of threshold velocity entirely. If it is used, it is obviously a function of number of impacts per second as well as velocity, and it may be necessary to define an arbitrarily small but finite limit for MDPR which will then define the threshold velocity. Fig. 1 shows two typical sketches for the relation between probable error and choice of threshold velocity for this data. For those materials exhibiting behavior of the type shown in sketch 1-a, the optimum choice for threshold velocity is zero. For other materials, as in sketch l-b, a definite optimum V appears. For some materials the best values of V, K and a were 0 computed from both eq. (3) and (4). Table III shows the comparison for an inorganic laminate, D-2 and a thermal plastic, I-2. While eq. (4) calls for an exponent 2 for the sing term, the effect of exponents ranging from 1.0 to 2.5 was examined (n=l corresponds to eq. (3)). It is noted that for these materials, the choice of n affects the best choice of threshold velocity (and of course a., which is not listed), but affects the minimum probable error

6 only slightly. From this data it appears that a choice of n=l, desirable for the sake of simplicity, would not significantly reduice the "goodness" of the correlation. The effect on probable error of assuming zero threshold velocity (also desirable for simplicity) is shown in the last column. It is noted that the additional error so introduced is not particularly large. For the best-fit values of V for the different materials as o analyzed under eq. (3), it has been noted that there is a rough correlation between K and a (Fig. 2). If a sufficiently precise correlation of this type existed, it might be possible to characterize a given material by a single figure of merit, which could be either K or a. III. GENERALIZED EROSION MODEL A. General Considerations The limited success achieved in correlating the rocket sled test data using eq. (3) or (4), leads to the general conclusion that a more basic mathematical model is required. However, in the present instance the lack of good correlation is partly due to the type of data used. It is not permissible to compare damage attained after a fixed exposure period for materials of widely differing resistances, since only a mean MDPR can then be computed for materials in very different portions of their MDP vs. time (or number of impacts) curve. It is thus necessary to use data wherein the total MDP vs. exposure curve is available so that only comparable portions of thi s curve will be compared. After further understanding and the verification of predicting equations has been achieved, it may then become possible to interpret data from a test such as that of the rocket sled in a more suitable manner. However, this data is not adequate for

7 the generation and verification of a basic model. For these reasons, data from various types of facilities, both impact and cavitation, have been compiled together and used for the remainder of the present investigation. B. Basic Equation Selected The best hope of achieving a relationship of the generality necessary to allow possible applicability over the broad range of rain erosion materials, lies in a relation which is directly related to a physical model of the erosion process, is dimensionally consistent, and is as simple as possible. While it will be possible often to achieve a better fit for a given data set with more complex mathematical expressions, the likelihood of fitting other data sets with the same relation is reduced. Following this line of reasoning, we have elected to use the basic energy flux model suggested by Hoff et al. (1). However, we have not carried this beyond the stage where verification from our available data was possible, and hence have not introduced some of the assumptions used in the Hoff paper (1). We assume simply that the product of the rate of volume loss per unit exposed area (MDPR) times the exposed area (A ) is proportional to the product of the impacting kinetic energy per unit projected area times the projected area. The constant of proportionality is the quotient of the efficiency of energy transfer between impacting drop and material damage processes (iv)) and a material parameter (E) describing the energy per unit material volume absorbed in the material in such a way as to cause damage. This relation is expressed by eq. (1).

8 To utilize this equation it is of course necessary to evaluate a and 4. Our analysis to the present has produced a simple best fit expression for 2, as will be explained later. We have not as yet, however, fully evaluated. For the moment it appears that the efficiency will be influenced by several factors, and may perhaps be considered as a product of several separate terms reflecting each of these mechanisms. Considering the details of the collision process between a liquid drop and a material surface, it seems likely that / will be a function of (a) material and liquid properties perhaps as reflected by the acoustic impedance ratio (7), (b) geometrical factors involved in the collision, i. e., shape of impacting drop, angle of impact, surface roughness, etc., and (c) velocity of impact which will affect the pressure applied to the surface and hence the degree of surface deformation and the departure from the concept of an elastic material. Since (a) material and liquid properties involves no other parameters of the collision, we have lumped its consideration into that of the energy parameter FL, assuming as a first approximation that this portion of the efficiency term /, may be some function of the acoustic impedance ratio. No attempt has yet been made to evaluate the remaining portions of C. Evaluation of Energy Parameter, 6 1. General Rema:rks It is desired to find a material mechanical property with units of e~~ergy per unit volume having the characteristic that for a given test (impingement or cavitation) with fixed test

9 parameters (velocity, fluid conditions, geometry, etc., the product of MDPR x a will be nearly constant as possible over a broad range of test materials. The material property must appear only to the first power, i.e., a polynomial expression of an energy term will not do, since this would destroy the dimensional consistency of eq. (1). To be of use in a predicting equation, the energy term must be measurable in a simple mechanical test, and hopefully already available in the literature for most standard materials. The only parameters meeting these conditions are those energy terms which can be computed from the standard stress-strain curve. Our own previous work (8, e.g.) as well as that of Hobbs (o), and Rao (10) (all, incidently for cavitation tests) and Heymann (11,12) for a combined data set suggests that the best single parameter correlation is to be 2 found with ultimate resilience = T.S. / 2E, i.e. the area under the elastic portion of a stress-strain curve if elastic strain were continued up to the full tensile strength (T.S.). Thiruvengadem (13, e.g.), on the other hand, has reported that the best fit is in terms of strain energy (area under the complete stress-strain curve of a material). This latter can be evaluated either as the "engineering strain energy" (SE), i.e., area under the conventional stress-strain curve where tensile strength is computed from the observed machine breaking load without consideration of reduced area) or "true strain energy" (TSE) where actual breaking stress is used. We have used approximations of both in this analysis. In the case of ultimate resilience, for simplicity we have used the observed breaking load only, since for many rain erosion materials, reduction of area data is not available. Also, our previous work with other materials indicated this to be preferable (8).

10 2. Selection of Data for Evaluation In the interest of maximum applicability and generality of results we have elected to use as broad a group of data as possible in the evaluation of -//y, including some from cavitation tests and some from impingement tests. Howeve r, for incorporation in the analysis it is necessary that the stressstrain curves for the materials be accurately known to us, and that the damage data exist in such a fashion that the entire MDP vs. time curve is available so that a comparable portion of this curve can be used in all cases. Consistent with our previous practice (8), and that of Hobbs (9), we have selected the maximum MDPR as the characteristic value for the material. The largest single portion of the data we have used is that generated by our own vibratory cavitation facility in water (20 kHz, 2 mil nominal operating condition at 750 F). Other data has been incorporated into the analysis only when tests were available on at least one common material (i.e. identical material, from same bar stock, etc., if at all possible). In these cases, a ratio between maximum MDPR for the common material in the differing tests or facilities was established, and the additional materials tested in'the other facility (or test condition) normalized to the common material. in our vibratory facility. Thus values of the amplitude constants apply to this particular vibratory facility. In this manner it is possible to incorporate data from various types of tests since the efficiency factors involving test geometry and velocity are thus removed from consideration. Data from the following sources, in addition to our standard vibratory cavitation data has been used.

11 a. Impact tests by King of RAE (14) in Dornier rotating arm facility b. Impact tests by Electricite de France (15,16) on rotating wheel c. Venturi tests by Rao et al (10) d. Vibratory cavitation tests in our laboratory (17) using stationary specimen arrangement in close proximity to vibrating horn (same unit also used in standard set-up). As tests on additional common materials become available, it may be possible to include further data sets, hopefully including some from other impact facilities. The materials and their mechanical properties are listed in Table IV. Test data on Stellite 6-B was not included in the actual data fits since much previous data has indicated that its resistance is much greater than expected on the basis of its mechanical properties (a factor of about 10 in this case). Other exotic alloys for which we have data also were not included since these are very far removed in their properties from any rain erosion materials. 3. Best Fit Results Attained a. Predominant Mechanical Property Previous work here and elsewhere led us to the conclusion that the most likely form for the energy parameter ~_ would be a combination of ultimate resilience and strain energy so arranged that the resultant term would have the units of energy/ volume. To attain reasonable flexibility within this limitation, the following relation was postulated: a b tUR UR Cl(ESE)UR +c z (ESEsE 2

12 where C1, C2, a, and b are constants to be computed by a least square fit regression analysis of the data. Investigation of this relation showed that the best values for a and b were close to zero, so that the simpler relation of eq. (6) was indicated. An additive constant, C, was used since this improved the data fit. The physical interpretation of C is that of a threshold energy necessary to cause measurable damage, i.e., a concept analogous to that of threshold velocity. -= C + CUR + C SE (6) 1 2 Using the least mean square fit analysis with eq. (6), or the following special case versions of it: ~ = c + c UR (a) ~ =C + C 1SE -. (b) oC:1 CTSE (c) - (7) = C U1 R + C SE - - (d) = UR - -(e) it was found that the best correlation coefficient and the smallest percent standard error of estimate resulted from eq. (7-d), although in general (7-a) was in all cases nearly as good, indicating that ultimate resilience was the material parameter of major importance. This was further verified by the dominance of the second term over the third in eq. (7-d). The statistics of the correlation with either eq. (7-b) or (7-c) were relatively very poor with TSE worse than SE. Hence SE is used in eq. (7-d). This data is summarized in Tables V and VIII. While the correlation with eq. (7-d) is better than that with (7-a), it is only slightly so. Hence for the present data set it is permissible to use eq. (7-a) in preference to (7-d) in the

13 interests of increased simplicity, so that the only mechanical property involved in the correlation becomes ultimate resilience, which is much more easily measured for materials such as those used for rain erosion than is engineering strain energy. Since the best value of C in eq. (7-a) is relatively very small, o it isjustified to use the form of (7-e) where this threshold energy term is neglected. The standard error of estimate has been computed in such a way that it is always approximately proportional to to give equal weight to both weak and strong materials in the correlation, and allow the reasonably accurate prediction of MDPR for materials of low ~. The applicable relations are shown in the Appendix. b. Determination of Efficiency Factor, b As previously discussed, it has been assumed that one factor of the overall energy-transfer efficiency term in the basic eq. (5), i.e. Y7A), may be represented to a first approximation as a function of the acoustic impedance ratio (AI) between liquid and material (AI = PLC PSCS ) A consideration of the "water-hammer equation" for materials of finite elasticity, usually assumed to give a reasonable approximation of the pressure applied to the material surface under droplet impact (18, e.g.), indicated the importance of AI in determining this pressure, and in fact suggests a functional form of AI,

14 f(AI), which might be tried. ~p -LcV AI + 1 (8) so f(AI) = AI + 1 Here f(AI) is taken as a direct factor in the relation describing the pressure generated at the point of impact. Since pressure has units of energy per volume, the consideration of pressure is dimensionally consistent with the general model assumed. Another possible form of f(AI) is the "transmission coefficient.' giving the ratio of absorbed to reflected energy for the case of a shock wave impinging upon a solid surface in a continuous medium (which is not identical to the present case). (AI + 1)2 Then f(AI) = (AI+ ) (9) 4AI The best fit correlations have been investigated for both forms. It was assumed that: n + + 13 - = f(AI), where n= -1, -2,- (10) Table VI summarizes the results. It appears that there is no substantial improvement in the correlation to be attained by the use of f(AI) in any of these forms. This is surprising in the light of Heymann's result (11, 12) that the fit with UR was improved by using 2, 2 2 UR x E, since E E c for the metals used. As also suggested by Heymann's discussion (12) it seems necessary that /. differ substantially between materials, since the ratio between

15 the extreme material erosion resistances is orders of magnitude greater than that between the corresponding material energy properties. Nevertheless, in light of the present results 4k9 has been assumed unity, and omitted from subsequent relations. c. Non-Linear Parameter Fits i. Polynomial Energy Parameter Fit Our postulated basic equation (1) requires a first power energy term for dimensional consistency. In order to verify that the assumption of such a linear relationship with energy is reasonable, polynomial data fits of the type E=- C + C1(UR) + C (UR) + C (UR) -- (11) o 1 2 3 were investigated. An earlier incomplete data set was used, but it is felt that the values shown in Table VII are typical. As expected there was some improvement over the linear fit, but it was not great, Table VIII indicating that the linear relationship is physically reasonable and suitable for the present purpose where the maintenance of dimensional consistency is important. 2 ii. Fit with UR x E Heymann's correlation (11, 12) was improved by using 2 UR x E rather than E. However, this statistical fit for our present data is not as good as that with UR alone, and of course is dimensionally inconsistent with the assumed model (Table VII). d. Recommended Relation Based on the foregoing, the following relations are recommended for common metals and alloys at this time. As additional data is incorporated, it is anticipated that the best values of the constants may change slightly, but it is believed that such a change will be small, and that the form of the equation will remain unchanged.

16 R = C + cUR - _ _ (12-a) o 1 = C1 UR -. —--- (12 - b) (12-a) (12-by where C = 0.463 o C,= 1.999 2. 330 Coefficient of Correlation = 0.808 0.808 Standard Error of Estimate = 1. 981 2. 007 Since the improvement due to the inclusion of C is small, the 0o form (12-b) is recommended. Table VIII lists the full data set used along with measured and predicted values of E (which is equivalent to MDPR for data normalized in the fashion here used), according to eq. (12) and the coefficient of correlation and standard error of estimate, computed as shown in the Appendix. The predicted and measured values are tabulated for both eq. (12-a) and (12-b) along with the deviations for each material. Fig. 3. presents the same information graphically for the recommended eq. (12-b) where the "conical " standard error of estimate band is shown. The amplitude constants apply to the UM vibratory facility only. Constants for other facilities are found by multiplying the given constants by the ratio between maximum damage rates in the other facility and the UM facility. IV. CONCLUSIONS It is postulated that the most likely form for an equation relating material, liquid, and test parameters with impingement or cavitation erosion rates with good hope for general applicability, is one which is based on a clear physical model with dimensional consistency. For the evaluation of impingement erosion rates, consistent with the previous suggestion of Hoff, et al (1) the equation A V MDPR = ('"cY..4.)( P )( eff ) - --- (l-a) A 2 e

17 has been chosen. A statistical evaluation of ~, which must have units of energy per volume, has shown the best fit with a comprehensive data set including both impingement and cavitation data, in the form: F= C1 UR -..... (12-b) Neither higher power terms in UR or terms in SE improved the statistics of the fit substantially, and the fit in terms only of SE was relatively very poor. It is thus concluded that for the large group of metals here used the best linear energy per volume mechanical property correlation for volume loss rate under droplet impingement or cavitation attack is the expression eq. (10-b) in ultimate resilience alone. Rocket sled rain erosion data has been statistically evaluated to find best values for threshold velocity and velocity exponent, as well as the coefficient n in the expression (4,5): MDPR = K(VsinQ -V ) / sinne - -(4-a) It was found that the statistical fit is relatively insensitive to n so that n=l is a suitable value. It was also found that for many materials, the statistical fit is also insensitive to the choice of a threshold velocity V, so that only slight reduction 0

18 in the "goodness" of the fit occurs for most materials if it is assumed that V = 0. However, the best fit values for K o and a are sensitive to the choice of V and n. It was found o that there is a rough correlation between best fit K and a, with K decreasing approximately linearly with increasing a. Thus it might be possible to characterize a material by a single figure of merit in terms of eq. (4-a), if a best fit relation between K and a is determined, so that either may be eliminated in terms of the other in eq. (4-a). However, the statistical precision of the correlations is relatively very poor for the rocket sled data. Acknowledgments - Financial support for this investigation has been derived primarily from the United States Naval Air Development Center, Johnsville, Penna, under Contract N62269-69-C-0285. In addition the cavitation testing of some of the materials was financed by the Worthington Corporation, Harrison, N. J. The authors wish also to acknowledge the work of Messrs. Dale Kemppainen and Edward Timm of this laboratory for the supervision and data reduction of test data.

19 NOMENC LATURE MDPR - Mean depth of penetration rate (= volume loss rate/ exposed area). K = Amplitude constant eq. (3) a = Velocity exponent eq. (2) V = Impact velocity V = Threshold velocity o = Angle between tangent to surface and dirrection of impact = Efficiency of energy transfer between impacting drop or jet and surface - - eq. (5) = Removal energy (= energy/volume to remove given volume from surface). A - Projected target area in flight direction - - eq. (5) p A = Exposed target area - - eq. (5) eff = Effective liquid density, mass of liquid per unit volume of gas-liquid mixture - - eq. (5) C, C1, C2, C = Constants, eq. (6), (7), (9), (10) Ap = Pressure differential due to liquid drop impacting surface - - eq. (8) = density c = Sonic velocity, or velocity of propagation of shock wave. AI = Acoustic impedance ratio between impacting liquid and target material = UR = Ultimate resilience SE = Engineering strain energy TSE = True strain energy

20 B IB LIOGRAPH Y 1. G. Hoff, G. Langbein, and H. Rieger, "Material Destruction Due to Liquid Impact", ASTM STP 408, 42-69, 1966. 2. T. M. Mitchell and FE. G. Hammitt,'Preliminary Analyses Applied to a Portion of Holloman AFB Rocket Sled Data on Rain Erosion Materials", ORA Report No. 01077-4-T, University of Michigan, Oct. 1968. 3. G. Tatnall, K. Foulke, and G. Schmitt, Jr., "Joint Air Force-Navy Supersonic Rain Erosion Evaluation of Dielectric and Other Materials", Report No. NADCAE-6708, 1967. 4. W. C. Baker, K. H. Jolliffe, and D. Pierson, "The Resistance of Materials to Impact Erosion Damage", A Discussion on Deformation of Solids by the Impact of Liquids, Phil Trans. Roy. Soc., A, No. 1110, Vol. 260, 193-203, July 1966. 5 G. Hoff, G. Langbein, and H. Rieger, "Investigation of the Angle-Time Dependence of Rain Erosion", Progress Report No. 62269-7-002050, Dornier System GmbH, March 1968. 6. R. T. Knapp, "Recent Investigations of Ca'vitation and Cavitation Damage," Trans. ASME, v. 77, 1045-1054, 1955. 7. F. G. Hammitt, etal, "Initial Phase of Damage of Tes, Specimens in a Cavitating Venturi as Affected by Fluid and Material Properties and Degree of Cavitation", Trans. ASME, J. Basic Engr., D, 87, 453-464, 1965. 8. R. Garcia and F. G. Hammitt, "Cavitation Damage and Correlations with Materials and Fluid Properties", Trans. ASME, J. Basic Engr., D, 89, 4, 753-763, 1967.

21 9. J. M. Hobbs, "Experience With a 20 -Kc Cavitation Erosion Test", ASTM, STP 408, 159-185, 1967. 10. B. C. S. Rao, N. S. L. Rao, K. Seetharamiah, "Cavitation Erosion Studies With Venturi and Rotating Disc Equipment in Water", to be presented ASME Annual Meeting 1969 and published Trans. ASME, J. Basic Engr. 1IL F. J. Heymann, "Erosion by Cavitation, Liquid Impingement, and Solid Impingement", Engineering Report E-1460, Westinghouse Electric Corporation, March 15, 1968. 12. F. J. Heymann, "Toward Quantitative Prediction of Liquid Impact Erosion Damage", ASTM Symposium on Characterization and Determination of Erosion Resistance", Atlantic City, N.J., June 1969. 13. A. Thiruvengadam, "A Unified Theory of Cavitation Dama ge", Trans. ASME, J. Basic Engr., D, 85, 3, 365-376, 1963. 14. R. B. King, letter to F. G. Hammitt, June 13, 1968. 15. C. Chao, F. G. Hammitt, C. L. Kling, D. O. Rogers, "ASTM Round-Robin Test With Vibratory Cavitation and Liquid Impact Facilities of 6061-T 6511 Aluminum Alloy, 316 Stainless Steel, Commercially Pure Nickel", to be presented ASTM Symposium on Characteristization and Determination of Erosion Resistance, and published ASTM STP 16. R. Canavelis, "Comparison of the Resistance of Different Materials with a Jet Impact Test Rig", HC/061-230-9, Electricite de France, Chaton, France, Nov. 1967. 17. D. J. Kemppainen and F. G. Hammitt, "Effects of External Load on Cavitation Damage", to be presented IAHR Symposium, Kyoto, Japan, Sept. 1969 and published IAHR Proc. 18. J. H.'Brunton, "The Physics of Impact and Deformation: Single-Impact", Phil. Trans. Roy. Soc., A, v. 260, No. 1110, pp. 79-85, July 1966. 19. Murray R. Spiegel, Theory and Problems of Statistics, Schaum Publishing Co., New York, 1961

22 TABLE I Comparison of Actual and Predicted MDPR's for Material A-i, Pyroceram, using MDPR = (K(VsinO-V ))a/senO as a Predicting Equation with a = 6. 27, V = 0, K = 5. 34x10. (Standard Deviation of Predicting Equation = 4~88y/s. ) V (f/s) 0 () (MDPR)pr iced (MDPR ) 1580 30.9 7.9 1580 30.9 0 1580 45 5.5 10.5 1580 45 5.5 10.5 1580 60 16.1 0 1580 60 16.1 5.3 2197 30 6.8 0 2197 30 6.8 0 2197 45 43. 7 0 2197 45 43. 7 3. 6 2197 60 127.3 7.3 2197 60 127.3 80.6 2594 30 9.6 0 2594 30 9. 6 0 2594 45 124.1 0 2594 45 124.1 4.3 2594 60 361.4 3,849. 2594 60 361.4 2, 240. 290 5 30 40.6 0 2905 30 40.6 14.5 2905 45 252.4 2, 189. 2905 45 252.4 179. 2905 60 735.3 4,465.

23 TAB LE II Effect of Vo on Values of K and a for Material C-2, an Epoxy Laminate. V (f/s) a K (x105 0 6.44 25.7 200 6.36 27.9 400 6.24 29.7 600 6. 08 32.3 800 5.87 34.8 1000 5. 59 36. 7 1200 5.22 39.6 1400 4.73 43.4 1600 4.09 41.3 1800 3. 28 40.1 2000 2.28 28.4

24 TABLE III Results of Evaluation of Equation MDPR sinn 0 = LK(V-Vo)sinI a for Various Values of n. Material D-2. n Threshold Minimum Probable Velocity (f/s) Probable Error for Error (ri/s) Vo = 0 (1/s) 1. 0 1100 82 146 1. 5 1000 88 143 2. 0 900 95 141 2.5 800 101 140 Material I-2. 1.0 350 7.3 7.7 1. 5 200 7. 3 7.4 2.0 100 7.2 7.3 2.5 0 7.2 7.2

T ABLEiY Mechanical Properties of Materials in Data Set YS TS ~ EL HARD MDPR OR SE BS1433 COPPER 0.300E 05 0.360E 05 0.180E 08 0.180E O0 0.900E 02 0.647E 01 0.360E 02 0.648E 04 1.000 1.000 STAINLESS STEEL 316 0.310E 05 0o813E 05 0.260E 08 0.690E O0 0.748E 02 0.301E 00 0.127E 03 0.561E 05 3.531 8.657 NICKLE 270 O.800E 06 0.488E 05 0.277E 08 0.610E 00 0.249E 02 0.128E 01 0.430E 02 0.298E 05 1.194 4.594 AL 6061 0.407E 05 0.475E 05 0.910E 07 0,220E O0 0.600E 02 0.436E 01 0.124E 03 0.104E 05 3~444 1.613 STAINLESS STEEL 306 0,647E 05.0,945E 05 0,290E 08 0,638E 00 0,237E 03 0,330E 00 0,154E 03 0,603E 05 4,277 9,304 BRONZE #1 0,24.3E 05 0,452E 05 0,128E 08 0,230E 00 0,189E 03 0,189E 01 0,798E 02 0,104E 05 2,217 J,604 BRONZE #2 0,790E 05 0,112E 06 0,147E 08 0o205E O0 0,304E 03 0,163E 00. 0,426E 03 0,229E 05 11,834 3,537 BRONZE #3 0.880E 05 0,119E 06 0,172E 08 0,150E 00 0,225E 03 0,220E 00 0,411E 03 0,178E 05 11,410 2.752 BRONZE #4 0.190E 05 0,282E 05 0,121E 08 0,600E-01 0,152E 03' 0,176E 01 0,329E 02 0,169E 04 0,913 0.261 BRONZE #5 0,105E 05 0,189E 05 0,558E 07 0,130E O0 0,974E 02 0,330E 01 0,320E 02 0,246E 04 0,889 0,379 BRONZE #6 0,162E 05 0,193E 05 0,711E 07 0,300E-01 0,152E 03 0,257E 01 0,262E 02 0,579E 03 0,728 0,089 STAINLESS STEEL #1 0.115E 06 0,157E 06 0,263E 08 0,220E 00 0,290E 03 0,252E 00 0,470E 03 0,346E 05 13,050 5,337 STAINLESS STEEL #2 0.186E 06 0.188E 06 0,257E 08 0,750E-01 0,418E 03 0,270E O0 0,691E 03 0o141E 05 19,189 2,179 STAINLESS STEEL #3 0.104E 06 0.126E 06 0,251E 08 0,195E O0 0,264E03 0,43'0E 00 0,319E 03 0,247E 05 8,865 3,807 COPPER 0,282E 05 0,333E 05 O. 160E 08 0,543E O0 0,968E 02 0,671E 01 0,347E 02 0,181E 05 0,963 2,790 BRASS(65-35! 0,489E 05 0,605E 05 0,157E 08 0,393E 00 0o146E 03 0,170E 01 0,117E 03 0,238E 05 3,238 3,669 MILD S'IEEL 1020 0.897E 05 0.965E 05 0,300E 08 0,259E O0 0,227E 03 0,808E O0 0,155E 03 0,250E 05' 4,311 3,857 STAINLESS STEEL 304 0.410E 05 0.994E 05 0,290E 08 0.168E 00 0,315E 03 0,332E 00 0,170E 03 0,167E 05 4,732 2.577 A ST~ B144iSAE660) — 0.175E 05 0.225E 05 O.140E 08 0.173E 00 0.174E 03 0.147E O1 0.181E 02 0.389E 04 0.502 0.601 MAGNESIUM 0,241E 05 0,392E 05 0,650E 07 0,255E O0 0,885E 02 0,434E01 0,118E 03 0,100E 05 3,283 1,543 AL-U~i~NOM —~-003-O 0,680E 04 O.15gE 05 0,900E 07 0,541E O0 0,512E-02 0,304E 02 0,140E 02 Oo8bOE 04 0,390 1,327 COPPER O.300E 05 0.360E 05 O. 180E 08 0,180E O0 0,900E 02 0,647E 01 0,360E 02 0,648E 06 1,000 1,000 CR-130 STEEL 0.290E 05 0.780E 05 0.2qOE 08 0,280E O0 0.255E 03 0,465E 01 0,105E 03 0,218E 05 2,914 3,370 ALLOY 0.450E 05 0.560E 05 O. IOOE 08 0,100E O0 0,114E 03 0,802E 01 0,157E 03 0,560E 06 6,356 0,864 [LUMiNOM 0.150E 05 O.160E 05 0.900E 07 0.500E-01 0.270E 02 0.255E 02 0.142E 02 0.800E 03 0.395[ 0.123 OPPER 0.142E 05 0.3IOE 05 O. 170E 08 0.500E 00 0.600E02 0.82~E 01 0.283E 02 0.155E 05 0.785/ 2.392 [PHi]SPHOR BRONZE 0.394E 05 0,4[6E 05 0.150E 08 O, 110E O0 0,950E 02 0,440E01 0,577E 02 0,458E 06 1,602 0,706 Ln tBRASS 0.157E 05~ 0.260E 05 O.160E 08 0.530E O0 O,150E 03 0,200E01 0,211E 02 0,138E 05 0,587 2,127 IMILD STEEL 0.484E 05 0.650E 05 0,280E 08 0,600E-01 0,950E 02 0,236E 01 0,754E 02 0,390E 04 2,096 0.602 STAINLESS STEEL 0.354E 05 0.930E 05 0,280E 08 0,570E O0 0.170E 03 0,653E O0 0,154E 03'0,530E 05 4,290 8,18! ~TA[NLESS STEEL 316 0.310E 05 0.813E 05 0.260E 08 0.690E O0 0.748E 02 0.713E 00 0.127E 03 0.561E 05 3.53[ 8.657 NICKLE 270 0,800E 04 0,488E 05 0,277E 08 0,610E O0 0,249E 02 0,126E 01 0'430E 02 0,298E 05 1,194 4,594 AL 6061 0.407E 05 0.475E 05 O.910E 07 0,220E O0 0.600E 02 0,436E 01 0,124E 03 0,104E 05 3,444 {.,613 STELLITE 6-B 0.710E 05 0.138E 06 0.304E 08 0.210E 00 0.322E 03 0.180E-01 0.313E 03 0.290E 05 8. 728 4. 475 TOOL STEEL #1 0.540E 05 0.110E06 0.275E 08 0.175E-01 0.235E 03 0.730E -01 0.220E 03 0.193E 04 6.111 0.298 YS = Yield Strength (psi) MDPR - Maximum Mean Depth of Penetration Rate (mils/hr) ~ (All ~~lues are corrected to U. M. vibratory facility) T5 = Tensile Strength (psi) - Ultirr-.~te v.~o;1;.... - m ~r i,-,o;~ = Elastic. Modulus (psi)...... EL ~j~ = Strain Energy to ~ aziure = l~ x ~ (psi) = Elongation (%) NUR = Ultimate Resilience normalized to BS 1433 Copper HARD = Brine11 Hardness NSE = Strain Energy normalized to BS 1433 Copper

TABLE V Summary of Statistical Correlation Data for Eq. 7. C; = -1. (173 C. 0. 914 C.=2.8 75 C. 6.487 C, =1. 633 e&=i. 735' C 1. 897 c= 1824 C = 0.445 C =0.889 C =1.139 2 2,2 2 -2 MATERIAL &NORMALIZED EPSILON CC2U R C1+C2*SE Cl+C2*TSE Cl*UR.C2*SE OC~J+2S _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _S O U R C E S BS1433 COPPER 1.000 2.811 4.699 6.931 2.522 1.102) STAINLESS STEEL 316 21.482 7.611 18.664 12.474 13.464 14.218 NICKLE 270 5.044 3.179 11.254 14.510 6. 035 5.534 AL 6061. 1.4892 7.446 5.817 7.103 7.057 6.040 STAINLESS STEEL 304 19.594 9.026 1.845 13.565 15.258 16.250 BRONZE #1 3.421 5.1.19.801 6.924 5.046 3.902 UM Vibratory BRONZE #2 39.669 23.359.326 7.612 22.468 22.789 Cavitation Faciiy BRONZE #3 29.391 22.556.895 7.278 21.079 21.161 BRONZE #4 3.674 2.645 3526553172019 BRONZE #5 1.959 2.601 3`6 567 6.581178022 ____________6__ 2.5016 7.0956 9.567 7.4551 8.550 80.4026U8irtr STAINLESS TEEL 8.052 95.091 92.691 7.4571 10.470 10.9510CaitonFcly STALINLESSTE 303- 03.913 1.6540 5.296 6.7643.7 3 0 COPPINESSSER 15.000 172.81 4.699 6.931 27.522 17.109AEDri46ottn CR-10, STE0.931 6.7441902 7.654 7.74.55 3e712 mFciy BALS ALLOY5 0.R01706 9.17.517 6.7405 7.880 6.7269Vbrtr ALMINUMT 12 80025 1.663 3.9100 6.523 10.4755 -0.946avttinF STAOPP SSTE L3401.4785 2.403 7.238 7.301 3.4098 2.315 wn-ith Stat~inn PHONSPHOR BROZE1.470 3.953 4.1639 6.7879 3.244 5.682'RASUINM3033.228 2.027 6.754 67632 2.849 1.669J MILDE STEEL 2.739 4.8863973 6.924 3.9522 2.549 A-Drn STAINLES STEEL 316164 9.06976128.6 12.474 13.464 7142187 ARotain WheFacilt NICLE 270 5.111 3.179 11.254 14.50 6.803 6.769i ALOPH6061 NZ 1.482 7.446 5.817613 78.057 6.04012 CORRLATON3OEFICINT2.8087 0.746 0.3236 0.8549 0.566 STAINDARE EROSO ESTIMATE 295914 87.8745 9.744 14.280 514912

27 TABLE VI Acoustic Impedance Correction f(AI) AI +1 (AI +1) 4AI n Correlation Standard Error Correlation Standard Error Coefficient of Estimate Coefficient of Estimate 0 0. 808 2. 007 0. 808 2. 007 1 0. 807 2. 005 0. 807 2. 101 2 0.807 2. 003 0. 782 2. 324 3 0. 806 2. 001 0. 743 2. 668 -1 0.808 2. 009 0. 781 2. 070 -2 0. 808 2.011 0. 721 2. 431 -3 0.809 2. 014 0. 582 3. 745

28 TABLE VII Equations Using Non-linear Parameters Eauation Correlation Standard Error Coefficient of Estimate = 2.330 UR 0. 808 2. 007 2 = -2.681 + 3.343 UR -0.087 UR 0.870 5. 616* = 0.266 UR + 0.412 UR - 0.019UR 0.919 4.459* _ =3. 685 UR x E2 0. 678 5. 714 = 1. 147 + 1.444 UR x E 0. 678 4. 271 i These values are more comparable to the results of Table V

Table VIII Recommended Correlating Equations C = 0.463 d ______ -,GC ~4. 9992. 330 MATERIAL & NORMALIZED EPSILON CUR*UR -— c~_C? U - +-C *UR -UR + c2, BS1433 COPPER 1.000 2.462 1.981 2.330 2.007 STAINLESS STEEL 316 21.482 7.520 6.993 8.225 7.086 U'M Vibratory Cavitation Facility NICKLE 270 5.044 2.850 2.365 2.782 2.396 AL 6061 1.482 7.346 6.820 8.022 6.911 STAINLESS STEEL 304 19.594 9.011 8.471 9.964 8.584 BRONZE #1 3.421 4.894 4.391 5.164 4.449 BRONZE #2 39.669 24.115 23.438 27.568 23.750 BRONZE #3 29.391 23.269 22.600 26.582 22.900 BRONZE #4 3.674 2.288 1.808 2.127 1.832 BRONZE #5 1.959 2.240 1.761 2.071 1.784 BRONZE #6 2.516 1.918 1.442 1.696 1.461 STAINLESS STEEL #1 25.659 26.547 25.847 30.402 26.191 STAINLESS STEEL #2 23.948 38.817 38.007 44.704 38.513 STAINLESS STEEL #3 15.037 18.182 17.559 20.653 17.792 COPPER 0.963 2.387 1.906 2.242 1.932 BRASS (65-35 ) 3.801 6.935 6.413 7.543 6.499 UM Vibratory Cavitation Facility MILD STEEL 1020 8.002 9.080 8.539 10.043 8.653 STAINLESS STEEL 304 19.476 9.921 9.372 11.024 9.497 Stationary Specimen ASTM B144(SAE660) 4.384 1.467 0.995 1.170 1.008 MAGNESIUM 1.490 7.026 6.503 7.649 6.590 - ALUMINUM 3003-0 0.213 1.243 0.773 0.909 0.783 N COPPER 1.000 2.462 1.981 2.330 2.007 CR-130 STEEL 1.391 6.287 5.771 6.788 5.848 RAEDnornier Rotafting Arm F'cility AL ALLOY 0.806 9.169 8.627 10.147 8.742 ALUMINUM 0.254 1.253 0.782 0.920 0.793. COPPER 0.785 2.032 1.555 1.829 1.576 PHOSPHOR BRONZE 1.470 3.666 3.174 3.733 3.216 Ventuiri Facility BRAASS 3.228 1.636 1.162 1.367 1.178 MILD STEEL 2.739 4.652 4.151 4.882 4.206 STAINLESS STEEL 9.902 9.038 8.497 9.994 8.6109 STAINLESS STEEL 316 9.069 7.520 6.993 8.225 7.086 NICKLE 270 5.111 2.850 2.365 2.782 2.396 Rotating Wheel Impact Facility AL 6061 1.482 7.346 6.820 8.022 6.911ting Wheel Impact Facility CORRELATION COEFFICIENT 808 0808 PERCEAGE STANDARD ERRR F ESTIMATE0 808 0808 PERCEN.YAGE STANDARD ERROR OF ESTIMATE 1.911200

30 Probable Error 0 2500 0 2500 V V 0 0 (a) (b) Figure 1. Typical Curves for Probable Error as a Function of Threshold Velocity.

31 80- K AS A FUNCTION OF oC FOR VARIOUS ROCKET SLED RESULTS 70 E 60 50 D E K E E E (x05) E 40 E SC 30 -i c 20 - 10 LETTER-SYMBOLS REPRESENT THE GROUP OF MATERIALS FROM WHICH A GIVEN DATA POINT COMES 0 2 3 4 5 6 7 8 c> Figure 2

32 Tool Steel 13.6 hr Stellite 6B 1/MDPR = 13.6 hr/mil 6 1/MDPR = 55.5 hr/mil 7 / o Linear Standard Error of Estimate 5 / Best Fit Line 4 0 0 I 0 UM Vibratory Cavitation Facility O UM Vibratory Cavitation Facility f3 r r / |with Stationary Specimen /)/ 0 RAE - Dornier Rotating Arm Facility ) Venturi Facility X Rotating Wheel Impact Facility / 200 400 500 600 ~0 U200 400 500 600 UR (psi) Figure 3.

33 V. APPENDIX A. Correlation Coefficient r The correlation coefficient is defined as usual (19) as N _XY - X[Y - (1-A) r = 2 N~Y -2 Y) N - _ 2 between two variables X and Y, if a linear relationship is as sumed. B. Standard error of estimate, s If we let Y represent the value of Y for a est given value of X as estimated from a least square regression line of Y on X Y = a + bX - - - - - - - - -(2-A) the scatter about the regression line is measure by the so-called standard error of estimate of Y on X namely s (Y Y- -(3-A) y,x N which is derived from the following equation Y = a + bX + s = Y + s (4-A) Y, X est- Y, X if we assume the scatter is independent of X. However, it was found, from preli minary plotting of Y vs. X in our data, that the scatter is nearly linearly dependent on X so that Y = a+ bX - sX - (5-A) was used. The constants a, b, and standard error of estimate were determined by least square regression analysis according to Y a + =-(-+ b) s (6-A) =

34 TABLE I-A Least Square Fit Program FORTRAN IV G COMPILER MAIN 05-14-69 23:20.17 PAGE 0001 C C INVESTIGATION OF EROSION FAILURE ENERGY 0001 DIMENSION GROUP(60) 0002 DIMENSION NAME(200,5),DEN(290 ),WL(200) DPR(200),Z(200),AI(200) 0003 DIMENSION YS(200),TS(200),Y(200),EL(200),RDA(200),A(200) 0004 DIMENSION URO(200),SEO(200),TSEO(200)tUR(200),SE(200),TSE(200 0005 DIMENSION CON(200) 0006 DIMENSION EA(200)a EB(200), EC(200), ED(200)vEE(200) 0007 DIMENSION RA(200), RB(200), RC(200), RD(200), RE(200) 0008 DIMENSION CG(10), SG(10) 0009 DIMENSION H(100) 0010 DIMENSION EF(100) EG(100) RF(100)tRG(100) 0011 COMMON N 0012 COMMON /S4/ CONUR,SEtTSE 0013 COMMON /S5/ GROUP,NAME C C INPUT 0014 DENW=1.0 0015 YW=0.28E6 0016 1 READ (5, 7) N, (GROUP(K)t K=1l15) 0017 DO 20 I=1,N 0018 READ (5,10) (NAME(I,J), J=l,5), DPR(I), DEN(I) 0019 READ (5,12) YS(I), TS(I), Y(I), EL(I), RDA(I) 0020 RFAD (5,4) H(I) 0021 A( I)=1.0-RDA(I) 0022 CON( I )=l. 0023 20 CONTINUE 0024 7 FORMAT (15, 5X, 15A4) 0025 10 FORMAT (5A4, 2F10.3) 0026 12 FORMAT (3E15.3, 2F10.3) 0027 4 FORMAT (F1O.1) C C ANALYSIS & CALCULATIONS 0028 SEl = TS(1)*EL(1) 0029 TSE1 =0.5*((YS(1)+TS(1)/A(I))*EL(l-TS(1)*YS(1)/(A(1)*Y(l))) 0030 UR1=0.5*TS ( 1**2.0/Y( ) 0031 DO 11 I=1,N 0032 SEO(I) = TS(I)*EL(I) 0033 SEMI) = SEO(I)/SEI A 0034 TSEO( I )=0.5*((YS( I +TS(I)/A( I )*EL( I )-TS(I)*YS(I)/(A(I )*Y(I))) 0035 TSE(I)=TSEO(I)/TSE1 0036 URO(I-)=O.5*TS(I)**2.0/Y(I) 0037 UR( I )=URO(I)/UR1 0038 11 CONTINUE 0039 Z I=SORT( DENW*YW/ ( DEN( ) *Y ( 1 ) ) 0040 AIl=l.O+Z1 0041 DO 30 I=1,N 0042 Z(I)=SQRT(DENW*YW/(DEN(I)*Y(I))) 0043 AI ( I )=.O+Z(I) 0044 AllI)=AI(I)/All 0045 EA( I )=DPR( 1)/2PR(I) 0046 EB(I)=EA(I)*A (I) 0047 ECI)=EA( I)*AI( I)**2.0 0048 ED( I )=EA(I)*iI ( I )**3.0 0049 EEI)=EA(I)/AI( I )

35 FORTRAN IV G COMPILER MAIN 05-14-69 23:20.17 PAGE 0002 0050 EF(I)=EA(I)/AI(II**2.0 0051 EG(I)=EA(I)/AI(I)**3.0 0052 30 CONTINUE C DATA 0053 WRITE (6,28) (GROUP(K), K=1,15) 0054 28 FORMAT (1H1,15A4) 0055 WRITE (6,25) 0056 25 FORMAT (1HOT28,'YS',T40,'TS',T52,'Y',T64,tELtT76,'HARD', 1 T88,'MDPRTl0OO,'UR',T112,'SE') 0057 WRITE (6,24) ((NAME(ItJ),J=1S5),YS(I),TS(I),Y(I),EL(I[),H(I}), 1 DPR( II),URO(I),SEO(I),UR(I),SE(I), I=1,N) 0058 24 FORMAT (5A4, 8E12.3,2F8.3) C 0059 CALL CAL(EA, RA, CG(l), SG(1) ) 0060 CALL CAL(EB, RB, CG(2), SG(2) ) 0061 CALL CAL(EC, RC, CG(3), SG(3) ) 0062 CALL CAL(ED. RDt CG(4), SG(4 ) 0063 CALL CAL(EE, RE, CG(5), SG(5) ) 0064 CALL CAL(EFvRFCG(6),SG(6)) 0065 CALL CAL(EGRGCG(7),SG(7)) C 0066 GO TO 1 0067 END

36 FORTRAN IV G COMPILER CAL 04-24-69 15:55.10 PAGE 0001 0001 SUBROUTINE CAL(EAG,OCC,OSDJ 0002 DIMENSION URA(200)tSEA(200),TSA(200)tUSA(200),AUS(200),EA(200) 0003 DIMENSION GROUP(60),NAME(200,5),CON(200) 0004 DIMENSION UR(200),SE(200),TSE(200) 0005 DIMENSION EU(100), PUI 100) 0006 DIMENSION G(100) P(100),S(100) 00C7 DIMENSION H( 100JT(100) 0008 DIMENSION COC(10), SOD(10) 0009 DIME NSION CO(100) 0010 COMMCN N /S1/ CCSDEV /S2/ C1,C2 /S3/ AO,B,C 0011 COMMCN /S4/ CON,URSE,TSE 0012 COMMON /S5/ GROUPNAME C -C REGRESSICN ANALYSIS 0013 WRITE (6,80) 0014 80 FORMAT (1H1) C 00 15 SEU= O. 0016 DO 40 I=1,N 0017 EU(I )=EA( I)/UR( I) 0018 PU(I )=.C/UR(I) 0019 SEU=SEU+EU(I) 0020 40 CONTINUE 0021 C=SEU/N 0022 CALL REG(EU, PU, CON) 0023 DO 41 I=1,N 0024 G(I)=C1+C2*UR(I} 0025 P(I)=G(I )/UR(I) 0026 H( I)=C*UR I) 0027 CO(I)=C 0028 41 CONTINUE C C CORRELATICN ANALYSIS 0029 CALL COR(EA, GCOC(1),SOD(1)) 0030 OCC=COC( 1 ) 0031 CALL COR(EU,P,COC(2),SOD(2)) 00 32 OSD= SOD ( 2 ) 0033 DO 42 I=1,N 0034 S(I)=OSD*UR(I) 0035 42 CONTINUE 0036 CALL COR(EA,HCOC(3) SOD (3)) 0037 CALL COR(EU,CO,COC(4) SO D(4)) 0038 DO 43 I=I,N 0039 T(I)=SOD4 )*UR( I ) 0040 43 CONTINUE C C SOLUTION 0041 WRITE (6,8) (GROUP(K), K=L,15) 0042 8 FORMAT I ///.10X,15A4 0043 WRITE (6,21) 0044 21 FORMAT ( //,' MATERIAL & NORMALIZED', T30,'EPSILON', 1 T47,' A+EUR',T62,'+-D*UR',T77,'C*UR',T92,'+-D*UR',/) 0045 WRITE (6,22) ((NAME(I,J),J=1,5),EA( I ),G(I),S(I),H(I),T(,I)=1,N) 0046 22 FORMAT ( 5A4,5F15.3) 0047 WRITE (6,23) COC(1),COC(3),SOD(2),SOD(4)

37 FORTRAN IV G COMPILER CAL 04-24-69 15:55.110 PAGE 0002 0048 23 FORMAT (/,16X,2F30.3,T1'CORRELATION COEFFICIENT',/,16X,2F30.3, i T,'PERCENTAGE STANDARD ERROR OF ESTIMATE') OC49 RETURN 0050 END

38 FORTRAN IV G COMPILER MAIN 04-24-69 15:55.15 PAGE 0001 0001 SUBROUTINE REGIEtRS) 0002 DIMENSION E(200), R(200), S(200) 0003 COMMON N /S2/ CL1C2 C REGRESSION ANALYSIS 00C4 T1 = O. 0005 T2 = 0. 0006 T3 " 0. 0007' T4 = O. 0008 T5 = 0. 0009 T6 = 0. 0010 DO 102 I = 1, N 0011 TI=Tl+R(I *Ri ) 0012 TZ=T2+R( I )*S (I I 0013 T3=T2 0014 T4=T4+S (I)*S ( I) 0015 T5=T5+E( I)*R(I ) 0016 T6=T6+E( I )*S (I 0017 102 CONTINUE 0018 D=.TI*T4-T2*T3 0019 D1=T5*T4-T6*T3 002Q D2=T 1*T6-T2*T5 0021 5 CI=D1/D 0022 4 C2=D2/D 0023 WRITE (6,1) Cl1 C2 0024 1 FORMAT (2F15.3) 0025 RETURN 0026 END

39 FORTRAN IV G COMPILER MAIN 04-24-"69 15:55.16 PAGE 0001 C 0001 SUBROUTINE COR(XYtCCSDEV) 0002 DIMENSION X(200), Y(200), DEV(200) 0003 COMMON N C CORRELATICN ANALYSIS 0004 XN = 0. 00C5 YN = 0. 0006 XY = 0. 0007 XX = 0. 0008 yy = 0. 0009 VAR=O. 0010 DO 6 I=1,N 0011 XN = XN+XlI) 0012 YN = YN+YII) 0013 XY = XY+X(I)*Y(I) 0014 XX = XX+X(I)*X(I) 0015 YY = YY+Y(I)*YV(I) 0016 DEV( I')=X I)-Y( I ) 0017 VAR=VAR+CEV( I)*DEV( I) 0018 6 CONTINUE 0019 XYM= XN*YN/N 00 20 XXM= XN*X N/N 0021 YYM=YN*Y N/N 0022 V1=XY-XYPI 0023 V2=XX-XXt 0024 V3=YY-YY M 0025 WRITE (6,8} V1, V2, V3 0026 8 FORMAT ( 3E-15.3) 0027 V4=V2*V3 C028 IF (V4) 2,2, 1 0029 1 CONTINUE 0030 CD=( Vl*V 1) /V2*V3) 0031 CC=S QRT(.C C) 0032 GO TO 3 0033 2 CONTINUE 0034 CC =. 0035 3 CONTINUE 0036 FN=N-2. 0037 SDEV=SQRT(VAR/FN) 0038 WRITE (6,7) CCZ SDEV 0039 7 FORMAT (2F12.3) 0040 RETURN 0041 END