AFML-TR- 72-106 A MODEL FOR RAIN EROSION OF HOMOGENEOUS MATERIALS GEORGE S. SPRINGER CHANDRAKANT B. BAXI This document has been approved for public release and sale; its distribution is unlimited.

AFML-TR-72-106 FOREWORD This report was prepared by the University of Michigan, Department of Mechanical Engineering, Ann Arbor, Michigan, under Air Force Contract F33615-71-C-1572. It was initiated under Project No. 7340 "Nonmetallic and Composite Materials", Task No. 734007 "Coatings for Energy Utilization, Control and Protective Functions". The work was administered under the direction of the Air Force Materials Laboratory, Air Force Systems Command, Wright-Patterson Air Force Base, Ohio, with George F. Schmitt, Jr. of the Elastomers and Coatings Branch, Nonmetallic Materials Division, acting as project engineer. This report covers the work carried out during the period from June 1971 through May 1972. The authors wish to thank Professor P. S. Larsen for his many helpful suggestions. The authors are also grateful to Mr. G. F. Schmitt, Jr. for his valuable comments and for providing many of the references and data used in this investigation. This report was submitted by the authors in May 1972. This technical report has been reviewed and is approved. WARREN P. NSON, Chief Elastomers and Coatings Branch Nonmetallic Materials Division Air Force Materials Laboratory iii

AFML-TR- 72-106 ABSTRACT The behavior of homogeneous materials subjected to repeated impingements of liquid droplets is investigated. Based on fatigue theorems, a model is presented for describing both the incubation period ni (i.e., the time elapsed before the mass loss of the material becomes appreciable), and the mass loss past the incubation period m. The parameters are established which govern the length of the incubation period and the subsequent mass loss rate, and simple algebraic expressions are developed relating ni and m to the properties of the impinging droplets and the material. The limits of applicability of the model are also established. The results obtained are compared to available experimental data. Reasonable agreement is found between the present results and the data, indicating that the model developed can be used to estimate the incubation period and the mass loss of the material. iv

TABLE OF CONTENTS SECTION PAGE I INTRODUCTION 1 II THE PROBLEM3 III INCUBATION PERIOD 8 IV RATE OF MASS REMOVAL 16 V TOTAL MASS LOSS21 VI UPPER LIMIT OF APPLICABILITY OF MODEL 23 VII NO INCUBATION PERIOD 24 VIII SUMMARY 28 REFERENCES 31 APPENDIX LITERATURE SURVEY 40 v

AFML-TR- 72-106 ILLUSTRATIONS FIGURE PAGE 1. Droplet Impingement on a Homogeneous Material. Description of the Problem. 4 2a. Schematic of the Experimental Results. 7 2b. The Solution Model. 7 3. Force Distribution on the Surface. 9 4. Idealized a-N Curve. 11 5. Incubation Period Versus S/P. Symbols for Data Defined in Table I. Solid Line: Best Fit -to Data. 14 6. Rate of Erosion Versus the Inverse of the Incubation Period. Symbols for Data Defined in Table I. Solid Line: Best Fit to Data. 19 7. Comparison of Present Model (Solid Line, Eq. 38) with Experimental Results. Symbols for Data Defined in Table I. 22 8. Limits of Region I. 25 9. Schematic of Experimental Results in Region II, and Description of Model. 26 TABLES TABLE PAGE I. Description of Data and Symbols Used in Figures 5, 6 and 7. 37 II. Classification of the Literature. 45 III. Summary of Experimental Data on the Erosion of Homogeneous Materials Subjected to Liquid Impingement. 46 vi

AFML-TR- 72-106 NOMENCLATURE+ al-a1O constants (dimensionless) A area (ft2) b constant defined by Eq. (19), (dimensionless) bl constant in Eq. (16), (dimensionless) b2 knee in the fatigue curve (see Fig. 4) C speed of sound (ft/sec) d diameter of the droplet (ft) E modulus of elasticity (lbf/ft2) f number of stress cycles, (see Eq. 9) F force (lbf) I rain intensity (ft/sec) m mass eroded per unit area (lbm/ft2) m" dimensionless mass loss defined by Eq. (40) n number of drops impinging per unit area (number/ft2) n* number of drops impinging per site, (see Eq. (22), dimensionless) n characteristic life (dimensionless) a N fatigue life (see Fig. 4), (dimensionless) p probability defined by Eq. (27), (dimensionless) P stress (Ibf/ft2) q drop density (number/ft3) r distance (ft) In this list the units are indicated in the English Engineering System (ft, sec, Ibm and lbf). However, any consistant set of units may be used in the equations. vii

AFML-TR- 72-106 S parameter defined by Eq. (21), (lbf/ft2) t time (sec) V velocity of impact (ft/sec) Vt terminal velocity of a rain droplet (ft/sec) W weight loss due to erosion (lbf) GREEK LETTERS a rate of mass loss (Ibm/impact) (see Fig. 2b) a* dimensionless rate of mass loss (see Eq. 33) a Weibull slope in Eq. (27), (dimensionless) v Poisson's ratio (dimensionless) p density (Ibm/ft3) 9 angle (radians) a stress (lbf/ft2) -a endurance limit (lbf/ft2) a ultimate tensile strength (lbf/ft2) SUBSCRIPTS II quantities associated with Region II i end of incubation period f upper limit of validity of model L liquid s solid viii

AFML-TR-72- 106 SECTION I INTRODUCTION Aircraft components such as radomes, leading edge surfaces, helicopter blades, and various structural members may experience heavy damage when subjected to repeated impingements of rain droplets. Liquid droplets may also cause significant damage to steam turbine blades. Owing to the severity of the problem numerous investigations have been concerned with the effects of liquid impingement on surfaces. Excellent reviews summarizing previous experimental and analytical work have been given, among others, by Eisenberg (Reference 13), Engel (Reference 15), Heymann (References 32 and 33), Heymann and Arcella (Reference 35) and Wahl (Reference 65). Many of the recent studies dealing with the problem are also described in the Proceedings of the First, Second and Third Rain Erosion Conferences (References 23-25). The majority of previous studies on the subject of rain erosion have been experimental in nature, with the bulk of prior research concentrating on the measurement of an erosion parameter (e.g. total weight loss) of a sample subjected to specific conditions. Such empirical studies provide information on the behavior of a given material under a given condition, but fail to describe material behavior beyond the range of the experiments in which they were obtained. Recently, attempts have been made to derive analytical or semi-empirical formulae which describe the dependence of material damage on selected operating variables, such as impact velocity and droplet size (References 29, 32, 33, 35, 53 and 58). Although these studies shed light on some of the mechanisms which contribute to material damage, a satisfactory method has not yet been devised which is capable of correlating the existing data and generalizing the -1

AFML-TR-72- 06 results obtained from a few experiments. The objective of this investigation is to develop a model which is consistent with experimental observations and which predicts quantitatively "erosion" of materials under previously untested conditions. It is evident, that such a model is needed for the selection of the proper materials and for the design of appropriate structures and components subject to severe liquid impingemen t. The model proposed here is aimed at describing a) the "incubation period", i.e. the time elapsed before the mass loss of the material becomes appreciable, and b) the degradation of the material past the incubation period as manifested by its mass loss. The existence of an incubation period suggests that the damage produced in the material is the consequence of cumulative fatigue damage produced by the repeated impacts of the droplets. Therefore, the present model is based on fatigue concepts. The role of fatigue in rain erosion has been recognized in the past (References 9, 48, 49, 57 and 63), and attempts have been made to describe the material damage in terms of fatigue parameters (References 32, 33 and 51). However, expressions for the incubation period and for the subsequent material loss have not yet been derived. -2

AFML-TR- 72-106 SECTION II THE PROBLEM The problem investigated is the following. Spherical liquid droplets of constant diameter d impinge repeatedly upon a semi-infinite, homogeneous material (Fig. 1). The angle of incidence of the droplets 0, and the velocity of impact V are taken to be constant. The spatial distribution of the droplets is considered to be uniform. Thus, the number of droplets impinging on unit area in time t is n = (V cos0)q t (1) where q is the number of droplets per unit volume. Rain, falling with constant terminal velocity Vt, is usually characterized by a parameter I called "intensity", which is related to q by the expression 6 I q 6 - (2) t 7r V d3 In Eq. (2) I has the units of length/time. Values for Vt may be obtained from charts (Reference 27), or may be calculated from the empirical relationship (Reference 1) -6d V = 965 - 1030 e (cm/sec) (3) t where d is in cm. Equations (1) and (2) may be combined to yield 6 (V cos0)I I( n = - t(4) Vtd Any consistent set of units may be used in Eq. (2). Customarily, I is expressed in in/hr, Vt in ft/sec and d in mm. For these units q - 1250 I (number/cu ft) Vtd -3

DROPLETP Diameter: d Density: PL Speed of Sound:CL MATERIAL Density: Ps Speed of Sound: Cs Modulus of Elasticity: Es Poisson Ratio: v Ultimate Tensile Strength: au Endurance Limit: aFigure 1. Droplet Impingement on a Homogeneous Material. Description of the Problem.

AFML-TR- 72-106 The impingement rate is assumed to be sufficiently low so that all the effects produced by the impact of one droplet diminish before the impact of the next droplet. This assumption is justified since, in practice, the time between subsequent impingements at a point is of the order -2 of 10 sec, while the stresses produced in the material become negligible -6 in about 10 sec (Reference 33). The pressure within the droplet varies both with position and with time. For simplicity, the pressure will be taken to be constant, its value being given by the water hammer pressure (Reference 34) PLCLVcosO p = L(5) PLC 1+ L P C s s where p is the density, and C the speed of sound. The subscripts L and s refer to the liquid and the solid, respectively. In terms of the modulus of elasticity E Cs = /Ps (6) Although more accurate representation of the pressure is possible (Reference 34), the accuracies afforded by the use of Eq. (5) will suffice in the present analysis. Thus, the force imparted to the surface by each droplet is F = P d (7) The forces, created by the repeated droplet impacts, damage the material as manifested by the formation of pits and cracks on the surface, and by weight loss of the material. Experimental evidence indicates that under -5

AFML-TR-72- 106 a wide range of conditions the weight loss W varies with time t as shown, schematically, in Fig. 2a. For some period of time, referred to as incubation period, the weight loss is insignificant. Between the end of the incubation period t. and a time denoted by tf the weight loss varies nearly linearly with time. After tf the relationship between W and t becomes more complex. Here, we will be concerned only with the behavior of the material up to time tf. In most practical situations the usefulness of the material does not extend beyond tf. It is advantageous to replace the total weight loss of the sample by the mass loss per unit area m, and the time by the number of droplets impinging upon unit area n. In terms of the parameters m and n, schematic representation of the data is given in Fig. 2b. It is now assumed that the data can be approximated by two straight lines as shown in Fig. 2b, i.e. m = 0, 0 < n. (8a) 1 m = a (n-ni), ni < n nf (8b) Thus, the material loss m produced by a certain number of impacts n, can be calculated once the incubation period n. and the rate of subsequent mass loss (as characterized by the slope a) are known. Therefore, the problem at hand is to determine the parameters ni, a, and nf, the latter being the upper limit of validity of Eq. (8b). It is noted here that the above model is valid only if there is an incubation period. It has been observed experimentally that under some conditions even one impact will result in appreciable damage, a situation in which ti 0. Problems of this type will be discussed in Section VII. -6

E 3$Z~ wQ1 ~< 0 oData L | -MModel <~ I Z I (I~) I 100~0 D LL I o0 0 i!o0~ io ~~~ o I o 0 I w I o() Incuba-l L Incuba-I tin I tion I tion'Periodl I U Period I ~, m o I o I 1 10 I an I (9 I o | I 0 1 | cn tan-,a w i~O Il/ i ti tf ni nf TIME, t NUMBER OF IMPACTS PER UNIT AREA, n Figure 2a. Schematic of the Experi- Figure 2b. The Solution Model. mental Results.

AFML-TR- 72-106 SECTION III INCUBATION PERIOD n It has been recognized that fatigue plays an important role in the erosion process (References 9, 32, 33, 48, 49, 51, 57 and 63), particularly in the "early" stages of the process, corresponding to the incubation period. Therefore, it is expected that fatigue theorems established for the torsion and bending of bars might be applied, at least qualitatively, to materials subjected to repeated liquid impingement. The failures of bars undergoing repeated torsion or bending have been found to follow Miner's rule (Reference 50), which is expressed by the following equation f! f2 fk N1 + $6N a1 (9) 1 2 k where fl, f2... fk represent the number of cycles the specimen is subjected to specified overstress levels al, o2a...ok and N1, N2...Nk represent the life (in cycles) at these overstress levels, as given by the fatigue (a vs N) curve. a1 is a constant which in torsion and bending tests generally varies between 0.6 and 2.2. Let us now consider one element at point B on the surface of the material as shown in Fig. 3. Each droplet impinging on the surface will create a stress at point B. Assuming that the force created by the droplet at its point of impact is a "point force", the stress at point B due to any one droplet is (Reference 64) F(1-2v ) a(r) = (10) 2rrr where v is the Poisson ratio. Every droplet which falls at a radius r -8

*~Q no B, -d~, Impinging Droplets 0 0 0o *-^-^'-,^^^/ Point Force F= P 4P Figure 3. Force Distribu tion on the Surface. F i g u r e^. 3 F D i t i u io _h Su r f a c e Figure 3. Force Distribution on the Surface.

AFML-TR- 72-106 produces the same stress at point B. Therefore, the number of cycles for which the material at point B is subjected to a given stress between a and a + da is equal to the number of impacts on a dr wide annulus located at r (Fig. 3). During the incubation period the total number of impacts on the annulus is f(r) = ni27rrdr (11) Therefore, fl, f2'f in Eq. (9) are replaced by f(rl), f(r2)...f(rk),i.e., f(r1) f(r2) f(rk) N + N2 + = a1 (12) Since r varies continuously from zero to infinity, Eqs. (11-12) mav be written as 00, n. 2r rdr nJ. rd = a1 (13) 0 Using Eq. (10), rdr can be expressed in terms of o F(1-2 v ) rdr = - do (14)'r 2a2 2 Equations (13) and (14), together with Eq. (7) yield.d2 )2 5I n[P7- (1-2v)/2] -J If^ -- -— s do = a (15) a N u The lower and upper limits of the integrals have been changed to the ultimate tensile strength a and the endurance limit a, respectively. U In order to perform the integration the fatigue life N must be known as a function of o. For most materials the fatigue curve between a and a may be approximated by (Fig. 4) N = blO (16) -10

Cru b1N )b.J c fU) LFE- CYCLES Log N )Figure 4 Idealized Curve. aI fl. I b2 LIFE CYCLES, Log N Figure 4, Idealized a-N Curve.

AFML-TR- 72-106 where b and bl are constants. Equation (16) must satisfy the conditions N = 1 for a= a (17a) u N = 10b2 for a= a (17b) b2 In Eq. (17b),10 corresponds to the "knee" in the fatigue curve (Fig. 4). Equations (16) and (17) yield b N = (a) (18) b2 b - (19) a log10 u(~ I Substituting Eq. (18) into Eq. (15) and integrating we obtain b-1 b-L ~2 alo -aI - n. P(1-2vs) b = a (20) 4" ~ s 2(b-l)o 1 u Introducing the definitions 2a (b-1) 2a (b-1) 3 = — u ab-i u (21) cy b-1 (1-2v) [-() ] -2vs s CT u c* _7 d2 ni ni 4. (22) equation (20) becomes * S ni = a1 P (23) Defining by "site" the surface area equal to the cross sectional area of one droplet, the number of sites per area A is A/(7rd2/4). Since n. is the number of impacts per unit area, ni is the total number of impacts per "site". The minimum value of ni is unity. -12

AFML-TR- 72-106 The parameter S characterizes the "strength" of the material. Thus, the number of impacts per sight needed to initiate damage is proportional to the ratio of the "strength" of the material S to the stress P produced * by the impinging droplets. Such a dependence of n. on S and P is reasonable, since the length of the incubation period is expected to increase with increasing S and with decreasing P. However, in view of the fact that Eq. (23) is based on the fatigue properties of materials in pure torsion and bending, one cannot expect a linear relationship to hold between n. and S/P. In order to extend the range of applicability of Eq. (23), while retaining its major feature (namely the functional de* pendence of n. on S/P) we write * Sa2 ni al(P (24) where both a and a2 are as yet undetermined constants. Since the form of the above functional relationship between n. and S/P is arbitrary, its validity must be evaluated by plotting experimentally observed values of n, versus S/P. The relationship will prove to be correct if on a log-log scale such a plot results in a straight line. The equation of this line would provide the constants a1 and a2. * The n. and S/P values deduced from all the available experimental data are shown in Fig. 5. The symbols used in this figure and the corresponding experimental conditions are identified in Table I. It must be pointed out that only those data could be included in Fig. 5 for which not only n. but also the material properties (a, ao, b2, vs, Es, ps) were available. It is seen from Fig. 5 that all the data can be correlated by a straight line. The equation of this line, obtained by -13

*e' a 03 3T I3sO:auyT PoI~S'I aTqv, uT puTJlaa veIa o0J sToqmirS *d/S snsa.A poTaec uo3BqnouI *S ajnSA d/S v01 z O1 01 I e 01 -01 0 80 01 _L ~01 l~~ ~~~~ He901

AFML-TR- 72-106 a least square fit of the data, is * -4 5.7 n. - 3.7xlO (S/P) (25) The excellent correlation in Fig. 5 lends support to the validity of the model. As was discussed in Section II, the present model is valid only when the incubation time is greater than zero. This condition is met when ni>l or, according to Eq. 25, when S/P>4.0. Thus, an incubation period exists if * ni>l (26) S/P>4.0 The foregoing analysis (Eq. 25) can be applied only when the above conditions are satisfied. When S/P is equal to or less than 4.0 damage will occur even upon one impact per site (Section VII). This is most likely to occur at high impact velocities in which case P is high (since Po-V) and S/P is low. -15

AFML-TR- 72-106 SECTION IV RATE OF MASS REMOVAL Beyond the incubation period, erosion of the surface of the material (as expressed in terms of mass loss) proceeds at a nearly constant rate as shown in Fig. 2b. In order to calculate this erosion rate,an analogy is drawn again between the behavior of the material upon which liquid droplets impinge, and the behavior of specimens subjected to torsion or bending fatigue tests. Experimental observations show that in the latter case the specimens do not all fail at once at some "minimum life", but their failure is scattered around a "characteristic life". For specimens in torsion and bending tests the probability that failure will occur between minimum life n. and any arbitrary longer life n may be estimated from the Weibull distribution (Reference 67) n-n. B p = 1 - exp [-( -- ) ] (2 7) na where na is the characteristic life corresponding to the 63.2 percent failure point and B is a constant (Weibull slope). For (n-ni)/na<< 1 Eq. (27) may be approximated by n-n. $ p ( 1) (28) a The probability p can also be taken as the number of specimens that fail between n. and n. If the material undergoing erosion due to liquid impineements is considered to be made up of many small "parts", then the amount of material eroded (mass loss) is proportional to p, i.e. n-n. n -n. d a3 (n_) = a3 (,) (29) s a na -16

AFML-TR-72- 106 In Eq. (29) m was nondimensionalized with respect to psdin order to render the proportionality constant a3 dimensionless. Equation (8b) is now rewritten in dimensionless form m a *-* p a= (n -ni) (30) t pd/4 Equating Eqs. (29) and (30) we obtain * * 8 n -n. 3 = a3 (- ) (31) = Psd /4 na According to Eq. (31) the mass loss rate a depends on the total number of impacts n. However, our model postulates a constant mass loss rate (i.e. a is independent of n, see Fig. 2b), at least when n< n<nf This requirement can be met by setting 6 = 1. Such a value for 6 is not unreasonable under high frequency loading (Reference 63). The characteristic life na is related to the minimum life ni. This relationship may be expressed suitably as *a5 n = a4ni (32) where a4 and a5 are constants. Introducing the dimensionless mass loss rate a*= (33) Psd /4 equations (31-33), together with the assumption B =1 yield * 1 a= a3 1 (34) 3 * a6 (ni) The new constant a6 was introduced in place of Sa5. -17

AFML-TR- 72- 106 The validity of Eq. (34) can be assessed (and the constants a3 and a6 can be determined) by plotting experimentally obtained values of a versus (1/ni). According to Eq. (34), on a log-log scale such a plot should result in a straight line. All the available experimental data are presented in such a manner in Fig. 6. As can be seen the data follows reasonably closely a straight line of the equation * 1 ~0.7 a =0.023 (* ) (35) ni indicating that the arguments leading to Eq. (35) were reasonable. The * somewhat larger scatter of this data as compared to the data on the n. vs S/P curve is due to the facts that a)n. can be estimated more accurately from the available data than the slope of a, and b) all available data have a rather wide margin of error, but the value of ni is less sensitive to these errors than the value of a. * * * Instead of plotting a versus 1/n. as was done in Fig. 6, a could have been correlated directly with the parameter S/P by using the relationship between n. and S/P given in Eq. (25). However, if a were plotted versus S/P then in this figure only those data could be included for which the material properties needed for calculating S were known. By plotting a against 1/n the use of these properties could be avoided; an advantage because for many of the data only n. were known but not the 1 material properties. -18

I0-I a* f 0 <e 10"'0 10-2__ #/niw Figure 6. Rate of Erosion Versus the Inverse of the Incubation eie B F 10-4L e 0. " 10-4 - vve>^ 4 11319 10 106 10 103 10-2 10' *n.

AFML-TR-72-106 The relationship between the time rate of mass loss (am/at) and the impact velocity V can now be established. For a rain of constant diameter and intensity impinging upon a given material, a may be expressed as 3 3 ps P * r dpd,0.7 4 a = --- = [0.023( —) ] P - V (36) 4 4 ni Noting, that 3n/at = Vq cose (see Eq. 1), we may write am= am an = aVqcoseaVV5 (37) 3t an at This result is consistent with the experimental observation that the time rate of mass loss varies approximately with the 5th power of the impact velocity. -20

AFML-TR- 72- 106 SECTION V TOTAL MASS LOSS In Figures 5 and 6 only those data could be included for which the variation of the mass (or weight loss) with time was known. There are numerous data available where such complete information is unavailable, but where the mass loss is given at one instant of time. A comparison of the present model with the latter "single point" results would be desirable, since agreement with such "single" data points would lend confidence to the validity of the model. To facilitate such a comparison Eq. (8b) is rewritten in dimensionless form m = a (n*-n.) (38) or m * * -= n - n. (39) a* n where the dimensionless mass loss is defined as = - (40) -Bed According to our model, expressed by Eq. (39), all data should correlate on a m*/a* versus (n*-ni) plot. Such correlation is presented in Fig. 7. This figure includes all existing data known to us in which droplets impinge continuously on a homogeneous material (see Table I). The results of experiments in which a jet impinges upon the surface (References 5, 6, 36 and 37) were not included in this figure. As can be seen, the agreement between data and the theoretical line given by Eq. (39) is quite reasonable, particularly in view of the large errors inherent in most of the experimental data. -21

107 107, _ - - -= Bl Presn Md 104 - af m ao0O O 0 E. 10*- ~ ) t-22. — e. Present Model 102 ^s- Al ~ -v 102o e: (10~ 10 103 104 105 10 10 10 (, (n-ni) Figure 7. Comparison of Present Model (Solid Line, Eq. 38) with Experimental Results. Symbols for Data Defined in Table I. -22

AFML-TR- 72-106 SECTION VI UPPER LIMIT OF APPLICABILITY OF MODEL As was stated in Section II, the model proposed is valid only as long as the mass loss varies linearly either with time t or with the number of impacts n. Consequently, the upper limit of t ( or n) must be established beyond which the present model cannot be applied. An estimate of this limit nf was made by observing that for many of the data given in Figs. 5, 6, and 7 the number of impacts n was as high as 3ni. Up to this point the data obtained at higher values of n did not show any systematic deviation from the data obtained at lower n values. It was concluded, therefore, that the model is valid at least up to nf=3ni, i.e. nf <3ni (41) or, in dimensionless form nf < 3ni (42) -23

AFML-TR- 72-106 SECTION VII NO INCUBATION PERIOD The model discussed thus far can be applied only if there is an incubation period or, as shown previously, if * n. > 1 1 S/P >4 (26) The domain in which the above conditions are satisfied will be referred to as Region I, as illustrated in Fig. 8. The domain corresponding to zero incubation period will be referred to as Region II. One would wish also to extend the model developed for Region I to Region II. To accomplish this, data of the form shown in Fig. 9 would be needed, i.e. the mass loss would have to be measured as a function of time (or number of impacts) for different materials of known properties. Unfortunately, to date such data have not yet been obtained. Therefore, one can only hypothesize that the data in Region II have the general trend shown in Fig. 9. If this trend is correct then, similarly to Region I, the data may be approximated by two straight lines (Fig. 9). Accordingly, the mass loss may be written as m mII + aII (n - 1) (43) * * where mii is the mass loss caused by one impact per sight, and aII is the rate of mass loss beyond n = 1. Intuitively, one might argue that * * both mII and aII should be directly proportional to the force of the impact (i.e. the pressure P), and inversely proportional to the "strength" -24

E Region II 0 ^^ Region I ]'; S/P >4 I NUMBER OF IMPACTS PER SIGHT, n' Figure 8. Limits of Region I. -25

o Data M odel,* Region II 0 R o Fr Se c Et Region I E S/P< 4 () and Description of Model. -26U) NUMBER OF IMPACTS PER SIGHT, n" Figure 9. Schematic of Experimental Results in Region II, and Description of Model. -26

AFML-TR- 72-106 * * of the material S. Thus, miIand aII are expected to be of the form a8 mi = a ( II a7 ((44) a10 aII a () The validity of Eqs. (43)-(44), and the constants a7-a10 could be assessed only by comparing these expressions to data, similarly as was done for the model in Region I. However, such comparisons will have to await the measurements of the appropriate data. -27

AFML-TR- 72-106 SECTION VIII SUMMARY In this report a model has been developed which can be used to estimate -the incubation time and the mass loss of homogeneous materials subjected to repeated impingements of liquid droplets. The following results were obtained a) Incubation Period 5.7 PLC 2au(b-l) ( + p- ) k ~-4 P Cs n., 3.7x10 -------- qno, of im actsC n (1-2vS)(PLCLVcose) J site or 5.7 LCL -4 2u(b-)(+P C no. of impac) (46) 4. 71xlO S: (46) n. = 471x1 ss unit area d2 (1-2vs) ( LCLVcose) ( or 5.7 OLCL -4 2a (b-l) (1 + -L) 4.71x104 s Cs (time) (47) ti= 2P qVcos d2 (1-2 v ) ( PCLVcos0) -28

AFML-TR- 72- 106 b) Rate of Mass Loss 4 * [ (l-2 v ) CLVcoLs9 a = 5.75 -S)LCL O (48) 2oU(b-l)(l + L) s Cs or 4 3 (l-2v ) PLCLVcos ] mass loss a= 4.51p3 1 ) CLco impact (49) PLCL impact 2a (b-l)(l + P Cs c) Total Mass Loss * * * * m = a (n - ni) (50) or ~M ~~~a ~(n-n,~ (mass loss) (51) m a(n-ni) i'unit area' Equations (46), (49) and (51) yield the mass loss per unit area in time t 3 (1-2v ) PLCLVcos6 m = 4.51 p d P LL CO j4 (q t Vcose) s PLCL 2aL(b-l)(l + p ) L -a- PsC s 2a (b-l)(l + LL) 4.71x10 4 u s s (52) d2 (l-2vs)(PCLVcos- ) -29

AFML-TR- 72-106 The foregoing results are subject only to the following two constraints a) incubation time must be greater than zero (ti>0), a requirement satisfied by the condition PLC 2o (b-l)(1 + p C) S s s -- > 4.0 (53) (1-2v) ( PCLVcos ) b) total time elapsed must be less than three times the incubation period, i.e. t < 3 t n < 3 n (54) n < 3 n. or 5.7 PLL 2a (b-l)(1 +- ) 3 (VcosO)It -3 2a(b-l)(I +PsC (55) ~2 VdX< 11x103 Vtd (1-2vs)( CLVcosO) A tentative model has also been suggested for those problems where constraint (a) (Eq. 53) is not satisfied (see Section VII). Owing to lack of data the validity of this model could not yet be assessed. -30

AFML-TR- 72- 106 REFERENCES 1. D. Atlas, R.C. Srivastava and R.S. Sekhon, Dpler Radar Characteristics of Precipitation at Vertical Incidence, Tech. Report No. 22, Laboratory for Atmospheric Probing, Department of Geophysical Sciences, University of Chicago and Electrical Engineering Department, Illinois Institute of Technology, Chicago, Illinois, May 1971. 2. D.W.C. Baker, K.M. Jolliffe and D. Pearson, "The Resistance of Materials to Impact Erosion Damage," Philosophical Transactions, Roy. Soc., Vol. A260, pp. 193-203, 1966. 3. D.W.C. Baker, D.E. Elliott, D.C. Jones and D. Pearson, "The Erosion of Steam Turbine Blade and Shield Materials," roceedings of Second Meersbur Conference on Rain Erosion and Allied Phenomena, (edited by A.A. Fyall and R.B. King), Royal Aircraft Establishment, Farnborough, England, pp. 449-515, August 1967. 4. J.L. Beal, R.R. Lapp and N.E. Wahl, A Study of the Rain Erosion of Plastics and Metals, WADC Technical Report 52-20, Wright-Patterson Air Force Base, Dayton, Ohio, September 1952. 5. D.J. Beckwith and J.B. Marriott, Erosion Damage in a Cobalt-Chromium5. D.J. Beckwith and J. B. Marriott,......_ Tungsten Alloy, Paper No. 126, Central Metallurgical Laboratories, English Electric, Whetstone, Liecester, U.K., June 1966. 6. D.J. Beckwith and J.B. Marriott, "Factors Affecting Erosion in 12% Chromium Steel," Proceedings of the Second Meersburg Conference On Rain Erosion and Allied Phenomena, (edited by A.A. Fyall and R.B. King), Royal Aircraft Establishment, Farnborough, England, pp. 761-784, August 1967. 7. F.P. Bowden and J.H. Brunton, "Deformation of Solids by Liquid mpact at Supersonic Speeds," Proc. Roy. Soc., Vol. A263, pp. 433450, 1961. 8. F.P. Bowden and J.E. Fields, "The Brittle Fracture of Solids by Liquid Impact', Proc. Roy. Soc., Vol. A282, pp. 331-352, 1964. 9. J.H. Brunton, Liquid Impact and Material Removal Phenomena, Technical Memorandum No. 33-354, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, June 1967. 10. J.H. Brunton, "Erosion by Liquid Shock," Proceedings of the Second Meersburg Conference on Rain Erosion andllied Phenomena, (edited by A.A. Fyall and R.B. King), Royal Aircraft Establishment, Farnborough, England, pp. 535-560, August 1967. 11. Characterization and Determination of Erosion Resistance, ASTM STP 474, American Society for Testing and Materials, 1970. -31

AFML-TR-72 - 106 12. S, DeCorso,'"Ero'sidon. Tests of Stems Turbin.e Blade MBade1ateri al a ProceedinSs of the American Society rTesting d at.ils Vol. 64, pp. 782-796, 1964. 13. P. Eisenberg, "Cavitation and Impact Erosion-Concepts, Correlations, Controversies, in Characterization and Determination of Erosion Resistance, ASTM STP 474, American Society for Testing and Materials, pp. 3-28, 1970. 14. 0. Engel, Mechanism of Rain Er on-Iact Pressu in Solid Liquid Sphere Collisions, WADC Technical Report 53-192, Part 1, Wright-Patterson Air Force Base, Dayton, Ohio, July 1953. 15. 0. Engel, Mechanism of Rain Erosion-A Critical Review of Erosion by_Water Droj Im Jt_ WADC Technical Report 53-192, Part 2, WrightPatterson Air Force Base, Dayton, Ohio, August 1953. 16. 0. Engel, Mechanism of Rain Erosion, Dimensiona al alysis of Rain Erosion Damage, WADC Technical Report 53-192, Part 6, WrightPatterson Air Force Base, Dayton, Ohio, July 1955, 17. 0. Engel, "Pits in Metal Caused by Collision with Liquid Drops and Soft Metal Spheres," Journal of Research of the National Bureau of Standards, Vol. 62, pp. 229-246, 1959. 18. 0. Engel, "Note on Particle Velocity in Collisions Between Liquid Drops and Solids," Journal of Research of the National Bureau of Standards, Vol. 64, pp. 497-498, 1960. 19. 0. Engel, An Investation of Ve r- h-Spied.-Drn M e Erosion of 1100 Aluminum, AFML-TR-71-104, Air Force Materials Laboratory, Wright-Patterson Air Force Base, Dayton, Ohio, May 1971. 20. 0. Engel and J.A. Almo, A Model for Multiple-D roson of Brittle Solids, Contract NASW-1481, Nuclear Systems Program Space Division, General Electric Company, Cincinnati, Ohio, May 1971. 21. Erosion by Cavitaion and ement, ASTM STP 408, American Society for Testing and Materials, 1967. 22. J.E, Field, "The Importance of Surface Topography on Erosion Damage," Proceedins of the Second Meersbur Conference on Rain Erosion and Allied Phenomena, (edited by A.A. Fyall and R.B. King), Royal Aircraft Establishment, Farnborough, England, pp. 593-603, August 1967. 23. A.A. Fyall and R.B. King, editors, Proceedings of the Rain Erosion Conference held at Meersburg, West Germany, May 1965. Translated proceedings obtainable from: Royal Aircraft Establishment, Farnborough, England. -32

AFML-TR- 72-106 24. A.A. Fyall and R.B. King, editors, Proceedings of the Second Meersburg Conference on Rain Erosion and Allied Phenomena, August 1967. Translated Proceedings Available from: Royal Aircraft Establishment, Farnborough, England. 25. A.A. Fyall, editor, Proceedings of the Third Conference on Rain Erosion and Allied Phenomena, Royal Aircraft Establishment, Farnborough, England, August 1970. 26. A.A. Fyall, R.B. King and R.N.C. Strain, A Gravimetric Assessment of the Erosion Resistance of Various Materials, Report No. Chem 513, Royal Aircraft Establishment, Farnborough, England, 1957. 27. A.A. Fyall, "Meteorological Parameters Relevant to the Phenomenon of Rain Erosion," Proceedings of the Meersburg Conference on Rain Erosion and Allied Phenomena, (edited by A.A. Fyall and R.B. King), Royal Aircraft Establishment, Farnborough, England, pp. 3042, May 1965. 28. A.A. Fyall, "Single Impact Studies with Liquids and Solids," Proceedings of the Second Meersburg Conference on Rain Erosion and Allied Phenomena, (edited by A.A. Fyall and R.B. King), Royal Aircraft Establishment, Farnborough, England, pp. 563-591, August 1967. 29. F.G. Hammitt, Y.C. Huang, C.L. Kling, T.M. Mitchell, Jr. and L.P. Solomon, "A Statistically Vertical Model for Correlating Volume Loss Due to Cavitation or Liquid Impingement," in Characterization and Determination of Erosion Resistance, ASTM STP 474, American Society for Testing and Materials, pp. 288-322, 1970. 30. W. Herbert, "Influence of Heat Treatment of Steels Upon Rain Erosion Resistivity," Proceedings of the Rain Erosion Conference held at Meersburg, (edited by A.A. Fyall and R.B. King), Royal Aircraft Establishment, Farnborough, England, pp. 114-119, May 1965. 31. W. Herbert, "Behavior of Iron and Hardened Steels Towards Drop Impact," Proceedings of the Second Meersburg Conference on Rain Erosion and Allied Phenomena, (edited by A.A. Fyall and R.B. King), Royal Aircraft Establishment, Farnborough, England, pp. 359-388, August 1967. 32. F.J. Heymann, "A Survey of Clues to the Relation Between Erosion Rate and Impingement Conditions," Proceedins of the Secondeersburg Conference on Rain Erosion and Allied Phenomena, (edited by A.A. Fyall and R.B. King), Royal Aircraft Establishment, Farnborough, England, pp. 683-760, August 1967. 33. F. Heymann, Erosion by Cavitation, Liquid Impingement and Solid Impingement: A Review, Engineering Report E-1460, Westinghouse Electric Corporation, Lester, Pennsylvania, March 1968. -33

AFML-TR-72- 106 34. F. Heymann, "On the Shock Wave Velocity and Impact Pressure in HighSpeed Liquid-Solid Impact," Trans. ASME, J of Basic Enneerin, Vol. 90, pp. 400-405, 1968. 35. F. Heymann and F. Arcella, Analytical Investigation of Turbine Erosion Phenomena, WANL-PR-(DD)-014, Westinghouse Astronuclear Laboratory, Westinghouse Electric Corp., Lester, Pennsylvania, November 1966. 36. J.M. Hobbs, Factors Affecting Damage Caused by Liquid Imact, NEL Report No. 262, National Engineering Laboratory, East Kilbridge, Glasgow, England, December 1966. 37. J.M. Hobbs and W.C. Brunton, Comparative Erosion Tests on Ferrous Materials. Part I Drop Impact Tests, NEL Report No. 205, National Engineering Laboratory, East Kilbridge, Glasgow, England, November 1965. 38. G. Hoff, "Rain Erosion of Glass and Ceramics," Proceedings of the Rain Erosion Conference held at Meersbur, (edited by A.A. Fyall and R.B. King), Royal Aircraft Establishment, Farborough, England, pp. 90-94, May 1965, 39. G. Hoff, Rain Erosion Investigations of Different Materials with the Rotating Arm, Final Report to Contract #62269-7-002050, Dornier System, Friedrichshafen, West Germany, July 1968. 40. G. Hoff, W. Herbert and H. Rieger, "Rain and Sand Erosion, Phenomena of Material Destruction Caused by Repeated Loads," in Characterization of Erosion Resistance, ASTM STP 474, American Society for Testing and Materials, pp. 353-382, 1970. 41. G. Hoff and G. Langbein, "Resistance of Materials Towards Various Types of Mchanial Stress," Proceedings of the Second Meersburg Conference on Rain Erosion and Allied Phenomena, (edited by A.Ao Fyall and R.B. King), Royal Aircraft Establishment, Farnborough, England, pp. 655-681, August 1967. 42. Y. Huang, Numerical Studies of Unsteady Two-Dimensional Liquid Impact Phenomena, Ph.D. Thesis, Dept. of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan, July 1971. 43. R.C. Juvinall, Stress, Strain and Strength, McGraw-Hill Book Co., New York, 1967. 44. R.B. King, "Rain Erosion Testing at Supersonic Speeds Using RocketPropelled Vehicles," Proceedings of the Rain Erosion Conference Propelled Vehicles, C.on.ference held at Meersburg, (edited by A.A. Fyall and R.B. King), Royal Aircraft Establishment, Farnborough, England, pp. 49-57, May 1965. 45. R.B. King, "Multiple Impact Rain Erosion Studies at Velocities up to 450 m/s (M=l.3)," Proceedings of the Second Meersburg Conference on Rain Erosion and Allied Phenomena, (edited by A.A. Fyall and -34

AFML-TR- 72-106 R.B. King), Royal Aircraft Establishment, Farnborough, England, pp. 201-213, August 1967. 46. G. Langbein, "Rain Erosion of Polymers," Proceedings of the Rain Erosion Conference held at Meersburg, (edited by A.A. Fyall and R.B. King), Royal Aircraft Establishment, Farnborough, England, pp. 81-94, May 1965. 47. R.R. Lapp, R.H. Stutzman and N.E. Wahl, A Study of the Rain Erosion of Plastics and Metals, WADC Technical Report 53-185-Part 2, Wright-Patterson Air Force Base, Dayton, Ohio, November 1954. 48. W.C. Leith and A.L. Thompson, "Some Corrosion Effects in Accelerated Cavitation Damage," Trans. ASME, J. of Basic Engineering, Vol. 82D, pp. 795-807, 1960. 49. R. Mathieson and J.M. Hobbs, "Cavitation Erosion: Comparative Tests," Engineering, Vol. 189, pp. 136-137, 1960. 50. M.A. Miner, "Cumulative Damage in Fatigue," J. Appl. Mech., Vol. 12, pp. A159-A164, 1945. 51. C.H. Mok, "A Cumulative Damage Concept in Rain Erosion Studies," AIAA Journal, Vol. 7, pp. 751-753, 1969. 52. J.W. Morris, Jr., Supersonic Rain and Sand Erosion Research, Part II. Mechanistic Investigation of Rain Erosion AFML-TR-69-287, Part II, Air Force Materials Laboratory, Wright-Patterson Air Force Base, Dayton, Ohio, September 1969. 53. J.W. Morris, Jr. and N.E. Wahl, Supersonic Rain and Sand Erosion Research: Erosion Characteristics of Aerospace Materials, AFML-TR70-265, Air Force Materials Laboratory, Wright-Patterson Air Force Base, Dayton, Ohio, November 1970. 54. J.W. Morris, Jr., W.F. Adler, and N.E. Wahl, Supersonic Rain and Sand Erosion Research: Characterization and Development of Erosion Resistant Materials, AFML-TR-72-85, Air Force Materials Laboratory, WrightPatterson Air Force Base, Dayton, Ohio, May 1972. 55. G.P. Peterson, S.A. Maralo, Rain Erosion Flight Test Program, WADC Technical Report 58-454, Wright-Patterson Air Force Base, Dayton, Ohio, May 1959. 56. H. Rieger, "The Damage to Metals on High Speed Impact with Water Drops," roceedings of the Rain Erosion Conference held at Meersburg, (edited by A.A. Fyall and R.B. King), Royal Aircraft Establishment, Farnborough, England, pp. 107-113, May 1965. 57. J.F. Ripken, "A Testing Rig for Studying Impingement and Cavitation Damage," in Erosionby Cavitation or Impingement, ASTM STP 408, American Society for Testing and Materials, pp. 3-11, 1967. -35

AFML-TR- 72- 106 58. G.F. Schmitt, Jr., G.J. Tatnall and K.W. Foulke, Joint Air ForceNavy Supersonic Rain Erosion Evaluations of Materials, AFML-TR-67164, Air Force Materials Laboratory, Wright-Patterson Air Force Base, Dayton, Ohio, December 1967. 59. G.F. Schmitt, Jr. and A.H. Krabill, Velocity-Erosion Rate Relationships of Materials in Rain at Supersonic Speeds, AFML-TR-70-44, Air Force Materials Laboratory, Wright-Patterson Air Force Base, Dayton, Ohio, October 1970. 60. G.F. Schmitt, Jr., Material Parameters that Govern the Rain Erosion Behavior of Polymerc Coatings and Composites at Subsonic Velocities, AFML-TR-71-197, Air Force Materials Laboratory, Wright-Patterson Air Force Base, Dayton, Ohio, December 1971. 61. G.J. Tatnall, K.W. Foulke and G.F. Schmitt, Jr., "The U.S. Air ForceNavy Supersonic Rain Erosion Program for 1966," Proceedings bf he Second Meersburg Conference on Rain Erosion and Allied Phenomena, (edited by A.A. Fyall and R.B. King), Royal Aircraft Establishment, Farnborough, England, pp. 143-173, August 1967. 62. A. Thiruvengadam and S.L. Rudy, Experimental and Analytical Investigations Multiple Liquid Imact Erosion, Contract No. NASA CR1288, Hydronautics Inc., Laurel, Maryland, March 1969. 63. A. Thiruvengadam, S.L. Rudy, and M. Gunasekaran, "Experimental and Analytical Investigations on Liquid Impact Erosion," in Characterization and Determination of Erosion Resistance, ASTM STP 474, American Society for Testing and Materials, pp. 249-287, 1970. 64. S. Timoshenko, Theory of Elasticity, McGraw-Hill Book Co., New York, 1934. 65. N.E. Wahl, Investigation of the Phenomena of Rain Erosion at Subsonic and Supersonic Speeds, AFML-TR-65-330, Air Force Materials Laboratory, Wright-Patterson Air Force Base, Dayton, Ohio, October 1965. 66. N.E. Wahl and R.R. Lapp, Rain Erosion of Aircraft Materials, WADC Technical Report 55-308, Wright-Patterson Air Force Base, Dayton, Ohio, August 1955. 67. W. Weibull, Fatigue Testing and Analysis of Results, Pergamon Press, New York, 1961. -36

AFML-TR- 72-106 TABLE I DESCRIPTION OF DATA AND SYMBOLS USED IN FIGURES 5, 6 and 7. VelocityDiameter Symbol Material elocit of Drop Author (ft/sec _(mm) Perspex 730 1.9 Fyall et al (1957) \r~~a~ 1 ~1180 1.9 Schmitt et al (__1970) |s ^_ i __._____ _ |1395 2.0 King (1965) | V Alkathane 2 585-730 1.9 Fyall et al __________________________.(1957) V Q.S.-4 730 1.9 Fyall et al Polyethylene (1957) V |Polyphenylene 535-3720 1.9 Schmitt et al Oxide (1970)'WJ7 Cast Urethane 730 1.8 Morris et al.........____........____ _____... (1972) Polypropylene 980-1470 1.2 King (1967) Teflon 1180 1.9 Schmitt (1970) O Aluminum Alloy, 730 1.9 Fyall et al D.T.D. 423B (1957) 1100-0 1120 1.8 Morris & Wahl Aluminum (1970) 1145-H19 1120-2240 1.8 Morris & Wahl Aluminum (1970) 2024-T6 1120-2240 1.8 Morris & Wahl Aluminum (1970) 5052-0 1120 1.8 Morris & Wahl u6061-T6 1120-2240 1.8 Morris & Wahl Aluminum ( 1970) -37

AFML-TR- 72-106 Velocity Diame ter Symbol | Material (f/sc f Dro Author (ft/sec) of Drop (mm) 7075-T6 1120-2240 1.8 Morris & Wahl Aluminum (1970) 1120 Morris et al (1972);?) Aluminum 820-980 1.2 King (1967) (Pure) ~ Aluminum 1650-1420 1.2 Rieger (1965) (Z) |Aluminum 1340 1.2 Hoff et al Alloys (1969) Magnesium Alloy 730 1.9 Fyall et al D.T.D. 259 (1957) Copper Alloy 585-730 1.9 Fyall et al B.S. 1433 (1957) Copper 1120 1.8 Morris & Wahl (Electrolytic) (1970) 3 Nickel 1000 0.866 Engel et al (1971) 1120 1.8 Morris et al (1972) C C oCobalt-Chromium 1020 0.66 Baker et al Alloys (1966) OE3 Iron 1000 0.866 Engel et al (1971) Itsa Steels 1020 0.66 Baker et al (1967) __3 __ [| 1455 0.64 Herbert (1965) 6~3 1120 1.8 Morris et al (1972) -38

AFML-TR- 72-106 Diameter Velocity of Drop Author ymbol Material (ft/sec Drop.......___.________._(m m ) Titanium Alloys 1020 0.66 Baker et al,~.(_______________ (1967) 1120 1.8 Morris et al (1972) I~^)\~ ~1340 1.2 Hoff et al (1965) Tantalum 1000 0.866 Engel et al (1971) Udimet 700 1000 0.866 Engel et al 4\jI7(1971) | Magnesia Ceramic 1340 1.2 Hoff (1965) C | Zirconia 1340 1.2 Hoff (1965) Alumina Ceramic 1340 1.2 Hoff (1965) Spinell 1340 1.2 Hoff (1965) Glass 1340 1.2 Hoff (1965) -39

AFML-TR- 72- 106 APPENDIX: LITERATURE SURVEY In this section a brief summary is given of the existing analytical and experimental investigations on the response of homogeneous materials subjected to the repeated impingement of liquid droplets. For convenience, the available information is divided into groups as shown in Table A-I. The present summary will follow the pattern of this table. It is emphasized that only homogeneous materials are included here. Composite and laminated materials are not discussed. A-I. REVIEWS One of the earliest reviews dealing with the subject of rain erosion of materials was given by Engel in 1953 (Reference 15). A considerable amount of work has been done in the years following this review. Excellent summaries of the later work are given by Eisenberg (Reference 13), Heymann (References 32,33), Heymann and Arcella (Reference 35) and Wahl (Reference 65). It is noted that many of the results of the latest investigations were reported at the First, Second and Third rain erosion conferences (References 23-25). Comprehensive articles dealing with the subject may also be found in two special publications of the American Society for Testing and Materials (References 11 and 21). A-II. EXPERIMENTAL INVESTIGATIONS By far the largest number of investigations utilized experimental techniques to determine the behavior of materials subjected to liquid impingement. A comprehensive summary of the materials tested and the ranges of parameters covered in past experiments are given in Table A-I I. -40

AFML-TR- 72-106 As can be seen from Tables A-I and A-II the experiments can be categorized as 1) single impact studies, 2) in-flight tests, 3) whirling arm tests, 4) rocket sled tests, and 5) jet experiments. A-II-l. Single Impact Studies The response of materials subjected to the impact of a single droplet was studied by Engel (Reference 17), Fyall (Reference 28) and Bowden and Brunton (Reference 7). It is difficult to apply these results to problems where multiple impingement occurs, the latter being the situation which is encountered commonly in practice. A-II-2. In-Flight Tests In-flight tests, in which the speciman is mounted on or forms part of an airplane flying through rainstorms are expensive and provide only limited control of the experimental conditions. For these reasons, only a few such tests have been conducted most notably by the United States Air Force (Reference 47) and by Cornell Aeronautical Laboratory (Reference 55). A-II-3. Whirling Arm Tests In the whirling arm experiments the model is placed on an arm rotating in a simulated fain fall. This technique has been employed successfully to study behavior of materials both in the United States and abroad. In the United States, whirling arm experiments have been performed by Schmitt (Reference 60), Morris and Wahl (Reference 53) and Engel and Almo (Reference 20); in England by Baker and his coworkers (References 2 and 3), by King (References 44 and 45) and by Fyall, King and Strain (Reference 26). In Germany such experiments have been per-41

AFML-TR- 72-106 formed at Dornier Systems (References 31,32,38,39,40,41,46,56). The whirling arm technique is advantageous in that it provides reasonably good control over the experimental parameters of interest. However, the impact velocities that can be achieved are limited. The maximum Mach number attained thus far with this type of apparatus is 2.2. A-II-4. Rocket Sled Tests Impact velocities of up to 4292 ft/sec (Mach Number.i4.0) may be achieved in rocket sled tests. In this type of experiment the specimens are fastened on a sled propelled by a rocket and moving through an artificial rain field. Rocket sled tests for homogeneous materials have been conducted at Holloman Air Force Base, New Mexico, by Schmitt and his coworkers (References 58,59,60,61), and by Engel (Reference 19). A-I1-5. Jet Experiments Continuous jets of liquids have also been used to simulate the impingement of droplets on surfaces. This is generally achieved by mounting the test specimen on a wheel rotating with its axis parallel to the jet. Such experiments have been performed by Beckwith and Marriott (References 5 and 6), Hobbs (Reference 36) and Hobbs and Brunton (Reference 37). A-III. ANALYTICAL INVESTIGATIONS The damage incurred by materials subjected to liquid impingement depends upon the force imparted by the droplet to the material and on the material properties. The simplest expression for the pressure at the liquid-solid interface is obtained by assuming that the pressure is equal to the water hammer pressure caused by the impingement of an infinitely long liquid -42

AFML-TR- 72- 106 cylinder on a rigid solid. Formulae for the pressure, including the elastic behavior of materials,were developed by Engel (References 14 and 15) and by Heymann (Reference 34). The dependence of the pressure on time and position was considered by Huang (Reference 42). Experimental evidence indicates that, in addition to the normal stress (i.e. the pressure discussed above), shear stresses are also present between the liquid and the solid. An estimate of the magnitude of these stresses was made by Beal, Lapp and Wahl (Reference 4), who showed that, in most cases, the magnitude of the shear stress is negligible. The foregoing studies provide only the stresses (normal and tangential) at the surface. However, the prime interest is not in these stresses but in the behavior of materials, as expressed in terms of some convenient "erosion" parameter, such as the mass loss, incubation time etc. It has been observed that under many conditions the mass loss is related to the fatigue properties of the materials. Based on this information Heymann (Reference 32) and Thiruvengadam, Rudy and Ganasekaran (Reference 63)established qualitative relationships between the mass loss rate and time. Using fatigue concepts Mok (Reference 51) derived an integral equation which, under specific conditions, can be solved numerically to relate the incubation time to the maximum erosion rate. Owing to the large number of parameters a complete analytical solution to the problem has not yet been achieved. Attempts have been made to establish empirical and semi-empirical relationships between the various parameters describing the problem. One of the first of these attempts was made by Engel (Reference 16) who used dimensional analysis to find the dimensionless groups which govern the depth of pit formation. -43

AFML-TR- 72-106 Parametric studies, aimed at correlating data, were also done by practically every investigator who performed experiments (see Table A-II). One of the most comprehensive correlations of existing data was given by Heymann (Reference 33). -44

TABLE II CLASSIFICATION OF THE LITERATURE Li Liquid Impact Erosion. Reviews Experiments Analysis,imE E,.i Single I. Flight Whirling Rocket Impact Arm Sled

H TABLE III SUMMARY OF EXPERIMENTAL DATA ON EROSION OF HOMOGENEOUS MATERIALS SUBJECTED TO LIQUID IMPINGEMENT Investigator Material Drop or Jet Impact Vel. Intensity Parameter Remarks Dia., (mm) (ft/sec) (Inch/hr) Studied Baker et al Cobalt- 0.66 1020 Not Mass Loss Results given in (1966) chromium Given terms of mass loss alloys vs. mass of water impacted Baker et al Cobalt- 0.66 1020 Not Mass Loss Results given in (1967) chromium Given terms of mass loss alloys, vs. mass of water turbine blade impacted steels, high strength steels, titanium alloys, cemented tungsten, titanium carbides Beckwith Chromium 0.398 1030 (Jet) Mass Loss and Marriot tungsten (1966) alloy

Investigator Material Drop or Jet Impact Vel. Intensity Parameter Remarks Dia., mm (ft/sec) (Inch/hr) Studied Beckwith Austenitized 0.398 1030-1400 (Jet) Mass Loss and Marriot steel (1967) Bowden and Polymethyl- 1.5 to 3940 (Jet) Diameter of High speed photoBrunton methacrylat, 6.0 ring crack graphs used. Con(1961) natural eluded that for rubber, velocities above polyvinyl 1650 ft/sec. a chloride, liquid mass (water) plate glass, behaves in a com*>~~~ ~ stainless pressible manner steel Bowden and Glasses, 3.0 3940 (Jet) Development of Field hard poly- fracture is (1964) mers followed by high speed photography Brunton Perspex, 3.0 2620 (Jet) Pit depth (1967) duralumin and photographs

~> Investigator Material Drop or Jet Impact Vel. Intensity Parameter Remarks Dia., mm (ft/sec) (Inch/hr) Studied DeCorso Titanium 0.127 to 4600 (Jet) Maximum Different fluids: (1964) alloys, 1.52 Depth water, trichlorochromium ethylene, mercury, steel, transformer oil, s tellite, salt water, alcohol, -'lll copper ether and benzene Engel 1110-0 1 to 2 695-2445 Single Pit Depth Theoretical rela(1959) aluminum, Impact tion for pit depth 2024-0 is derived and aluminum, compared with ex00 copper, perimental results lead, steel, soft iron, and zinc Engel 1110- 1.9 1635 to 2.5 tMicroscopic Tests conducted on (19 71) aluminum 4202 examination rocket sled Engel and Zinc, iron 0.866 1000 Not Mass Loss Almo tantalum, mentioned (1971) nickel, and udimet 700 Field Aluminum, 2 to 3 2115-2460 (Jet) Photographic Conclusion: Change (1967) brass, per- in surface profile spex and in excess of 1000A0, glass can be significant in acting as sites for erosion damage

Investigator Material Drop or Jet Impact Vel. Intensity Parameter Remarks Dia., mm (ft/sec) (Inch/hr) Studied Fyall, et al Perspex, 1.9 981-1640 1 to 3 Mass Loss (1957) p olymers, copper, a luminum, m agnesium alloys, s tainless steel, b rass Fyall Perspex, 2-3 1000 Single Area of High speed photo@ (1967) Impact cracking graphy, photomicrography and profilimetry used Herbert Ferritic 1.2 1455 xl105 Mass Loss (1965) and auste- gr/cm3 nitic chrome steel, n monic 90 Herbert Low carbon 0.52 to 1340 3x10-6 to Mass Loss (1967) steels and 2.1 1.2x10-5 high alloy gr/cm3 steels with different heat treatments Hobbs Brass, 1.6 33-660 (Jet) Mass Loss (1966) aluminum, steel and nimonic

Investigator Material Drop or Jet Impact Vel. Intensity Parameter Remarks Dia.,mm (ft/sec) (Inch/hr) Studied Hobbs and Steel and 1.6 660 Jet Mass Loss Brunton cast iron (1965) Hoff Glass and 1.2 1340 lx10 Mass Loss (1965) ceramics gr/cm3 Hoff Glass, 1.2 220-466 1.2x10-6 to MDPR Sin2O 1) MDPR -Mean (1968) plexiglas 1.5x10-6 depth of penetragr/cm3 tion rate. 2) Review of Van other data and attempted correlation between anglee, velocity V and MDPR Hoff,et al Aluminum 1.2 1340 1.2x10-5 Erosion Effect of pressure (1965) and titanium gr/cm3 depth and temperature alloys also studied Hoff and Glass, mag- 0.5 to 981-1310 1.2x10-5 Mass eroded Energy approach Langbein nesium oxide, 2.1 gr/cm3 after 12 min. considered. Re(1967) aluminum exposure sistance to erosion oxide, Plexi- = (energy of liquid)' glass, aluminum (volume of ercded pure iron material) King Perspex 2.0 1395 1 Mass Loss Testing by use of (1965) rocket propelled vehicles is discussed

Investigator Material Drop or Jet Impact Vel. Intensity Parameter Remarks Dia., mm (ft/sec) (inch/hr) Studied r King Perspex, 1.2 820-1475 1 to 8 Mass Loss (1967) polypropylene, pure aluminum, aluminum alloy, copper alloy and stainless steel Langbein Plastic 1.2 1345 2.5x106 Depth of erosion (1965) gr/cm and mass loss Morris Aluminum 1.8 730-1120 1 Micrographs (1969) and photographs Morris and Aluminum 1.8 1120-2240 1 Mass Loss Wahl alloys, ti(1970) tanium, copper, coatings, laminated plastics and ceramics Morris Ceramic 1.8 730-1120 1 Mass Loss et al coatings, (1972) polymeric coatings, ductile metals (aluminum, nickel, steel and titanium) and solid elastometers

Investigator Material Drop or Jet Impact Vel Intensity Parameter Remarks Dia.,mm (ft/sec) (Inch/hr) Studied -5 l Rieger Aluminum, 1.2 650-1420 lxlO Mass Loss Also results of (1965) berilium gr/cm3 increase in hardcopper ness and depth of erosion after a fixed time interval given Schmitt Isotropic 1.9 1614 to 2.5 Mass loss a) Only one point et al ceramics, 3048 and MDP available after (1967) sandwich total time of ceramics, (MDP = test. plastic mean depth b) Properties of 1 l. ~~~~~~~~~~ ~ ~ ~~~~mean depth,~ ~ laminates,of penetra- some materials electroplated ti not specified. plastics, ( lo inorganic (mass loss) laminates, (density x area) ) ceramic and elastomeric coated laminates, glasses, thermo-plastics, sandwich plastics, metals

Investigator Material Drop or Jet Impact Vel. Intensity Parameter Remarks Dia., mm (ft/sec) (Inch/hr) Studied Schmitt and Isotropic 1.9 1614 to 2.25 to Mass Loss Rocket sled tests H Krabill ceramics, 3040 2.50 and MDP P (1970) sandwich ceramics, Plastic laminates, ceramic coated laminates, glasses, andwich plastics, bulk plas~.@^ ~ ~tics, metals metal coated laminates, cast resins and fibre composites. Schmitt Bulk and re- 1.8 735 to 1 Mass Loss (1971) inforced 880 plastics Tatnall Furnace epoxy 1.9 1110-2240 2.5 Mass Loss Rocket sled tests et al laminate, (1967) plexiglas, teflon, fused silica Thiruvengadam 316 Stainless 0.78 133-522 Jet Threshold Fatigue endurance et al steel, 1110-0 impact velo- limit also deter(1968,1969) aluminum, city at 10 mined nickel, cycles t itanium

UNCLASS IFIED__ Secut rity Classification_ DOCUMhENT CONTROL DATA - R & D (Security classification of title, body of abstract and indexing annotation nm st be entered wher the coverao!! r,! port is clcssififti4j. ORi G I N A T I N G ACTIVITY (Corpora te athor) i2a. R E PO R T SECUR TY CLASSI FI CA ON The University of Michigan UNCLASSIFIED ___ Mechanical Engineering Department 2b. GROUP Ann Arbor, Michigan 48104 3. REPORT TITLE "A Model for Rain Erosion of Homogeneous Materials" 4. DESCRIPTIVE NOTES (Type of report and irclusive dates) Technical Report, June 1971-May 1972 5. AUTHOR(S) (First name, middle initial, last name) George S. Springer Chandrakant B. Baxi 6. REPORT DATE 7a. TOTAL NO. OF PAGES 7b. NO. OF REFS May 1972 __53_ 67 8a. CONTRACT OR GRANT NO. 9a. ORIGINATOR'S R EPO T NUMBER(S) F33615-71-C-1572 AF ML-TR- 72-106 b. PROJEC T NO. 7340 __ c. 9b. OTHER REPORT NO(S) (Any other numbers that yr?.v be assigred this report) Task No. 734007s report) d. 10. DISTRIBUTION STATEMENT This document has been approved for public release and sale; its distribution is unlimited. I1. SUPPLEMENTARY NOTES 12. SPONSORING MIL!TARY ACTIVITY Air Force Materials Laboratory Wright-Patterson Air Force Base Ohio 45433 13. ABSTRACT The behavior of homogeneous materials subjected to repeated impingements of liquid droplets is investigated. Based on fatigue theorems, a model is presented for describing both the incubation period ni ( i.e. the time elapsed before the mass loss of the material becomes appreciable), and the mass loss past the incubation period m. The parameters are established which govern the length of the incubation period and the subsequent mass loss rate, and simple algebraic expressions are developed relating ni and m to the properties of the impinging droplets and the material. The limits of applicability of the model are also established. The results obtained are compared to available experimental data. Reasonable agreement is found between the present results and the data, indicating that the model developed can be used to estimate the incubation period and the mass loss of the material. vD 1 NcV 651473 UNCLASSIFIED Security ClassiticaIlo n

UNCLASSIFIED Security Classification 14. LINK A 4 LINK B. LINw C KEY WORDS ROLE WT ROLE WT ROLE WT Rain Erosion Erosion Mechanism Fatigue Model Incubation Period U NCLASSIFIED Security Classification

UNIVERSITY OF MICHIGAN III905 0326 69111111 3 9015 03526 6959