ANALYSIS OF RAIN EROSION OF COATED MATERIALS George S. Springer Cheng-I. Yang Poul S. Larsen Approved for public release; distribution unlimited.

FOREWORD This report was prepared by the University of Michigan, D:.p tment of Mechanical Engineering, Ann Arbor, Michigan, under Air Fo: i Contract F33615-72-C-1563. It was initiated under Project No. 7340,'No.lietallic and Composite Materials, " Task No. 734007 "Coatings for Energy Utilization, Control and Protective Functions." The work was administ~ red under the direction of the Air Force Materials Laboratory, Air FI rce Systems Command, Wright-Patterson Air Force Base, Ohio, wit} 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 1972 through May 1973. The authors wish to thank 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 July 1973. This technical report has been reviewed and is approved. MERILL L. IINE S ~ Chief Ela stomers and Coatings Branch Nonmetallic Materials Division Air Force Materials Laboratory iv

ABSTRACT The behavior of coat-substrate systems subjected to repeated impingements of liquid droplets was investigated. The systems studied consisted of a thick homogeneous substrate covered by a single layer of homogeneous coating of arbitrary thickness. Based on the uniaxial stress wave model, the variations of the stresses with time were determined both in the coating and in the substrate. Employing the fatigue theorems established for the rain erosion of homogeneous materials, algebraic equations were derived which describe the incubation period, and the mass loss of the coating past the incubation period, in terms of the properties of the droplet, the coating and the substrate. The results were compared to available experimental data and good agreement was found between the present analytical results and the data. The differences between the uniaxial stress wave and the uniaxial strain wave models were also evaluated by calculating according to both models a) the stress at the coat-liquid interface, b) the stress that would occur in the substrate in the absence of the coating, and c) the stress in the coating after the first wave reflection from the substrate. v

TABLE OF CONTENTS Section Page I. Introduction 1 II. The Problem 3 III. Stress History of the Coating and the Substrate 9 IV. Incubation Period 20 V. Rate of Mass Removal 31 VI. Total Mass Loss 35 VII. Limits of Applicability of Model 37 VIII. Fatigue Failure of the Substrate 39 IX. Comparison Between the Results of the Uniaxial Stress and Strain Theories 40 X. Summary 47 Appendices I. Derivation of Equation 18 52 II. The Value of the Constant al for Homogeneous Materials 54 REFERENCES 57 vi

ILLUSTRATIONS Figure Page 1 Droplet Impingement on a Coat-Substrate System. Description of Problem............................... 4 2a Schematic of the Experimental Results................ 7 2b The Solution Model.................................... 7 3 Stress Wave Pattern in the Coating and in Substrate...... 10 4 The Variation of the Stress at the Coat-Substrate Interface............................................ 13 5 The Actual and Approximate Variation of the Stress at the Coat-Substrate Interface......................... 15 5 The Variation of the Number of Stress Wave Reflections in the Coating with y................................... 17 7 Force Distribution on the Surface of the Coating......... 21 8 The Variation of the Stress with Time at the Liquid Droplet-Coat Interface.......................... 23 9 Idealized ae-N Curve................................. 26 * -o 10 Incubation Period ni versus Se/. Solid Line: Model (Eq. 65). Symbols Defined in Table 1.................... 30 11 Rate of Erosion Versus the Inverse of the Incubation Period. Solid Line: Model (Eq. 75). Symbols Defined in Table I......................................... 34 12 Comparison of Present Model (Solid Line, Eq. 77b) with Experimental Results. Symbols Defined in Table 1........ 36 13 Rankine-Hugonoit Curve for a Homogeneous Solid........... 41 14 Impact of the Droplet on a Homogeneous Material.Calculation of the Stress at the Liquid-Solid Interface by (a) the Uniaxial Stress Wave Model and (b) the Uniaxial Strain Wave Model...................................... 43 15 Impact of a Droplet on a Substrate Covered with a Single Layer of Coating. Calculation of the Stress at the Liquid-Coating Interface ol, the Stress at the CoatingSubstrate Interface a2, and the Stress that Would Occur on the Surface of the Substrate in the Absence of Coating ao. (a) Uniaxial Stress Wave Model; (b) Uniaxial Strain Wave Model........................................ 45 vii

TABLES Table Page 1 Description of Data and Symbols Used in Figures 10, 11 and 12.......................... 50 viii

NOMENCLATURE a -a6 constants (dimensionless) A area (ft2) B, B2 constant related to wave velocity defined in Eq. (81) (dimensionless) b constant defined by Eq. (57) (dimensionless) b 1 constant in Eq. (54) (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.40 F force (lbf) h thickness of coat (ft) I rain intensity (ft/sec) k number of stress wave reflections in the coating required for the stress at coat-substrate interface to reach a value of 63.3 percent of Oa (dimensionless) kL total number of stress wave reflections in the coating during the impact period (dimensionless) k average number of stress wave reflections in the coating (dimensionless) m mass eroded per unit area (lbm/ft2) m* dimensionless mass loss defined by Eq. (76) 2 n number of drops impinging per unit area (number/ft2) n* number of drops impinging per site, see Eq.(61)(dimensionless) n* characteristic life (dimensionless) a N fatigue life (see Fig. 4) (dimensionless) ix

p probability defined by Eq. (27) (dimensionless) P stress (lbf/ft2) q drop density (number/ft3) r distance (ft) S parameter defined by Eq. (59) (lbf/ft2) S parameter defined by Eq. (60) (lbf/ft2) t time (sec) te time required for ke number of stress wave reflections to take place in the coating (sec) tL the duration of impact (sec) u particle velocity (ft/sec) v wave velocity defined by Eq. (81) V velocity of impact (ft/sec) Vt terminal velocity of a rain droplet (ft/sec) W weight loss due to erosion (lbf) 2 Z dynamic impedance (lbm/(ft -sec)) GREEK LETTERS a rate of mass loss (Ibm/impact) (see Fig. 2b) a ~~* dimensionless rate of mass loss (see Eq. 73) gB ~ Weibull slope in Eq. (67) (dimensionless) Y the ratio of kL to ke (y=kL/ke) v Poisson's ratio (dimensionless) p density (lbm/ft3) 9e ~ angle (radians) en stress (lbf/ft2) a stress amplitude (lbf/ft2) a x

a equivalent dynamic stress defined by Eq. (42) (lbf/ft2) e a mean stress (lbf/ft2) m a mean stress after kL number of stress wave reflections (lbf/ft2) aI endurance limit (lbf/ft ) a ultimate tensile strength (lbf/ft2) u'P parameter defined by Eqs. (13)-(14) SUBSCRIPTS c coating i end of incubation period f upper limit of validity of model k the number of stress wave reflections in the coating L liquid s solid sc coat-substrate interface Lc liquid-coat interface SUPERSCRIPTS B uniaxial strain wave model h coat -substrate interface o liquid-coat interface xi

SECTION I INTRODUCTION Components of high speed aircraft and missiles may experience heavy damage when subjected to repeated impingements of rain droplets. The damage to nonmetallic components, such as plastic radomes, may be particularly severe. To protect such surfaces from rain erosion, th -se surfaces are frequently covered with a thin layer of coating. Considerable research has been performed in the past to select the most suitable coating material, and to determine the behavior of various coat-substrate systems undergoing liquid impingement. The majority of the previous studies of rain erosion of coated materials have been experimental in nature, with the bulk of prior research concentrating on the measurement of an erosion parameter (e.g. weight loss) under specific conditions (References 1-6). These experimental studies provide information on the behavior of a given coat-substrate combination under a given condition, but fail to describe material behavior beyond the range of the experiments in which they were obtained. For the selection of the proper materials and for the design of the appropriate structures an analytical or semiempirical model would be needed, which would describe the response of coat-substrate systems in terms of the relevant parameters. These parameters should include the properties of the coating and the substrate, the thickness of the coating, and the impact velocity and size of the droplet. In recent years, Progress towards this goal has been made by Morris (Reference 7), Engel.ad Piekutowski (Reference 8) and by Conn and his coworkers (References 9-11), who analyzed the stress history in various coat-substrate systec4s. Al-1

though the results of these investigations further our understanding of the processes which contribute to the failure of the coating and the substrate, as yet they are not capable of correlating fully the existing data and generalizing the results obtained from a few experiments. The objective of this investigation is to develop a model which is consistent with experimental observation and which predicts quantitatively "erosion" of coated materials under previously untested conditions. In particular, the model proposed here is aimed at describing a) the "incubation period", i.e. the time elapsed before the mass loss of the coating becomes appreciable, arid b) the degradation of the coating past the incubation period, as manifested by its mass loss. The model is based on fatigue concepts (e.g. References 12, 13), and is along the lines developed previously for homogeneous (uncoated) materials (Reference 13). The success of this model in describing the damage of homogeneous materials warranted its extension to coated materials. -2

SECTION II THE PROBLEM The problem investigated is the following. Spherical liquid droplets impinge repeatedly upon a plane, semi-infinite material cm.i-sisting of a homogeneous substrate covered by a homogeneous coating (Feg. 1). The thickness of the coating is h. The substrate is taken to I semiinfinite normal to the plane of the surface (x direction in Fig. 1). The coating and the substrate are characterized by the following properties; density p, speed of sound C, modulus of elasticity E, Poissonts ratio v, ultimate tensile strength au and endurance limit aI. Parameters related to the coating and the substrate are denoted by c and s, respectively. Parameters related to the droplet are identified by the subscript L. A perfect bond is assumed between the coating and the substrate, i.e. at the interface (x=h) the stresses and the displacements are the same in the coating and the substrate. Furthermore, the stress wave propagating through the coating and the substsate are considered to be one dimensional, propagating normal to the surface (compression waves). Waves parallel to the surface (shear waves) are neglected. The diameter of the droplets d, the angle of incidence e, and the velocity of impact V are taken to be constant. The spatial distribution of the droplets is considered to be uniform. Accordingly, the number of droplets impinging on unit area in time t is (Reference 13) n = (Vcose)qt (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 -3

DROPLET Diameter: d Density: hL Speed of Sound: CL ~ I Y COATING h SUBSTRATE X COATING SUBSTRATE Density: P Ps Speed of Sound: Cc Cs Modulus of Elasticity: Ec Es Poisson Ratio: vC yv Ultimate Tensile Strength: (ou)c (ur)s Endurance Limit: (I)C I)S Thickness: h Semi - infi nite Fig. 1. Droplet Impingement on a Coat-Substrate System. Description of Problem.

called "intensity" (with units of length/time) which is related to q by the expression 6 I =q 6 -I (2) Equations (1) and (2) may be combined to yield 6 (V cose)I (3) 0n - ^ v.. t (3) Vtd3 The impingement rate is assumed to be sufficiently low so that all the effects produced by the impact of one droplet diminish before t'~e impact of the next droplet (References 13, 14), The pressure within the droplet varies both with position and with time. For simplicity, the pressure at the liquid-surface interface is taken to be constant, its value being given by the water hammer pressure (Reference 15) OLCL V cos p = (4) TL L o C C c Although more accurate representation of the pressure is possible (Reference 15) the accuracies afforded by the use of equation (4) will suffice in the present analysis. The duration of the pressure at the interface is approximated by 2d (5) tL CL The forces, created by the repeated droplet impacts, damage the iaterial as manifested by the formation of pits and cracks on the surface, and by weight loss of the coating material. Experimental evidence indicates that under a wide range of conditions the weight loss W varies with time t -5

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 ti 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 < ni (6a) m = a (n-n.) ni < n < nf (6b) Thus, the material loss m produced by a certain number of impacts n, can be calculated once the incubation period ni 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 if, the latter being the upper limit of validity of equation (6b). It is noted here that the above model is valid only if there is an incubation period. Problems in which even one impact results in appreciable damage will not be considered. -6

E ~ ~|~ 1~<w oData iJa,i Model -j 00 LL. I u Ilncuba- Incuba-l o) tion I o I n tion I -J Period |- PeriodI F I o Is o_ -- I| I n 03 tan-'a )~go I?l I tf U iBR TIME, t NUMBER OF IMPACTS PER UNIT AREA, n Fig. 2a. Schematic of the Experimental Results Fig. 2b. The Solution Model

In order to establish ni and a, the stress history in the coating must be known. Thus, first expressions are derived which describe, in suitable form, the variation of the stress with time in the coating and in the substrate. -8

SECTION III STRESS HISTORY OF THE COATING AND THE SUBSTRATE The variation of the stress with time may be evaluated by considering either uniaxial stress waves (References 10, 11) or uniax!ial strain waves (References 6, 7) propagating through the coating. As will be shown in Section VIII these two approaches yield similar resul s. The present calculations are based on the uniaxial stress wave moad-l. When a liquid droplet impinges upon the surface of the coating, a stress wave propagates through the coating (see Fig. 3). The magnitude of this initial stress wave, denoted by al, is identical to the hydrostatic pressure P, i.e., al e P (7) P is given by equation (4). At the coat-substrate interface a portion of the stress wave is transmitted into the substrate while a portion of it is reflected back into the coating. Thus, there is a "left" traveling wave in the coating of magnitude a2 (Fig. 3) o2 = al + a (8) In equation (8) ah represents the magnitude of the reflected wave which may be expressed as (Reference 8) Z -Z h s9 C (9) Ir 1 Z +Z s c -9

LIQUID COAT SUBSTRATE O<t< h/Cc cr h/C <t 2h/CC kh.-LIQUID -4- COATg+ — SUBSTRATEFig. 3. Stress Wave Pattern in the Coating and in Substrate. "-10-10

In the time interval t=C /2h the "left" traveling a2 wave reaches the coatC liquid interface and a new "right" traveling wave of magnitude a3 is generated at the x=0 surface a3 a + r0 (10) 3 2 r where a is the reflected wave from the surface of the coating (Reference 8) 2 L-Z a C r 2 ZL+Z L c In equations (9) and (11) Z is the impedance of the material Z a pC (12) Introducing the notation z -Z *sc - Z (13) L c the magnitudes of the "left" and "right" traveling waves become a^ ~P ~a c CT Cy P(13 + ~2 a 1 + ~ Wsc 1= SC +(15) a3 1 + s + C Lc) 04 ~ 1 + +' Lc cc) etc. -11

Equations (15) may readily be generalized to the following forms O2k 1+ SC k 1 - scLc - (*" c(16) 2k-l 2k k-l - OS - (* C*Ld (17) a1 a- c sc'Lc where k is an integer, k = 1, 2, 3,.... Note that the stress history in the coating depends on the relative magnitddes of ZL, Zc and Z. This is illustrated in Fig, 4, where the variation of the stress with time is shown for the four possible combinations of impedances. After a long period of time (i.e. after a large number of reflections, k-n) the stress at both on the surface of the coating (x=O) and at the coat-substrate interface (x=h) approaches the constant value 1 + c 1 + ZL/Z c (18) a a lim a2k =1- + Z/ (18) ~! k-~- ~2k 1-4s 4L l+Z/Z k-o SC Lsc L s a is the stress that would occur in the substrate if the droplet would impinge upon it directly in the absence of a coating (see Appendix I). It is evident from Fig. 4 that the coating reduces the stresses in the substrate only if the appropriate coating material (i.e. appropriate combination of ZL, Zc and Z ) is selected (Figs. 4c and 4d). For certain combinations of coating and substrate the mean stresses in the substrate are actually higher with the coating than without it (Figs. 4a and 4b). This result clearly indicates the importance of the proper selection of the material used as coating for a particular substrate. -12

C0'...0 0I 0 (c) ZL <ZC < Zs (b) Z_> Zc Zs TIME, t TIME, t Fig.A. The Variation of the Stress at the Coat-Substrate Interface. FiZ~... Th~e Variatio3n o~f the Stre~.~ at the Coat-Subs crate Interface,

Equations (16) and (17) describe the variation of the stress with time in the coating. For our further calculations it is convenient to replace the stepwise variation of the stress by a continuous function. To accomplish this, equation (16) is rewritten in the form a2k a=D a a k-o — k - - 0 -0 (, ) (19) o1 at a al scLc (9) Equation (19) is now approximated by the expression a2k a a a2 k- 1 1 1 1 e- (20) al l ~ 1 e By replacing equation (19) by equation (20) we replace, in effect, the stepwise stress function with an exponential curve, as illustrated in Fig. 5. In equation (20) k is the number of reflections required for the stress to reach 63.3 percent of a. To evaluate k we introduce the condition that the area under the actual (stepwise) and the exponential curves are to be the same. This condition requires that the following equality be satisfied - 0 0 a k-i 00 0 02 co 00 2 co co 2 k-i - -- (a-)'s (*cLc ]8 I f exp (- I]dk k=l 1 1 1 f a k1 (21) Evaluating the summation and the integral in equation (21) we obtain k 1 (22) e S-C* q sc Lc Substitution of equations (13) and (14)jnto equation (22) yields i + ZL/Z 1 + Z /Z k - *L S C S k -................. (23) e 2 + / (23) L S In the absence of coating Z =Z and k =I, which, as expected, shows that S c e there are no reflections in a semi-infinite material. -14

cT2 Approximate 0'l iActual TIME, t Fig. 5. The Actual and Approximate Variation of the Stress at the Coat-Substrate Interface.

The time required for k number of reflections to occur is (see Fig. 3) 2h t - k (24) eC c and the number of reflections during this time is C k - t c (25) e e 2h(25) Similarly, the number of reflections which occur during the duration of the impact tL (given by equation 5) is C C c c d k ^~~~-^ ti~~~ c, c d <(26) kL tL 2h CL h(26) L It is to be noted that k is independent of the thickness of the e coating (see equation 23), while kL depends on h. For thick coating (h/d -A) kL -+ 0 and for thin coating (h/d *0) kL - a. Thus, the ratio k (27) e may vary between zero and infinity. It is conveninet to bridge these two limits by the exponential curve k - ke [1 - exp(- -)] (28) e or k = k [1 - exp(-y)] (29) k represents the average number of reflections in the coating. The variation of k with Y is illustrated in Fig, 6. For thikk coating k becomes kh/d 0 (30) n/d 16-16

* ^r ke~~~~~^k itti k,~~~~~~k'I'Tke - - Isr kL rig. 6. The Variation of the Nurt~er of Stress Wave Refl~ections in the Coating with y.

For thin coating equation (29) reduces to kh/d. O ke (31) which is, by our definition, the maximum number of reflections which may occur in the coating. We may evaluate now the average values of the stresses at the coatliquid (x=O) and at the coat-substrate interfaces (x=h) during the period of impact tL. The average stress at x=O is kL -_ E a2k-l (32) L k=l and at x=h is kL -h 1 -a z ak (33) k=1 Substituting equations (16), (17) and (18) into equations (32) and (33) and utilizing the exponential approximation gives by equation (20), after some algebraic manipulation, we obtain -o 1+ s 1+ *IL a I sc 1 +_Lc 1- exp(-y) ] (34a) a, 1 - ^-sc sc 1 + (34a) 1 -sc4Lc sc ~ h + s 5 _1 ~ 1 +a SC ri- ri, l1-exp(-y) (3 ____[ ~ ~~ i-........(34b) CI 1 -'sc'Lc SC Lc Y If the coating is of the same material as the substrate c =0 and equation sc (34a) reduces to a~ = al P (35) a=1~ ~-18-18

The force exerted by the droplet on the surface of the coating also varies with time. The average force on the surface during the duration of one impact tL is F = a -d (36) The foregoing equations describe the stress history in the coating and in the substrate when the substrate is covered by a single layer of coating. The results could be generalized readily to include two or more layers of coatings. It is emphasized, however, that the express ions here developed are not restricted to thin coatings, but may be applied to coatings of arbitrary thicknesses. The thickness of the coating enters the results through the parameter Y. From equations (23), (26) and (27) we have Cc d 1 +2L/Z 1 + ZL/Zs Y C ( L 3) ( L —- - 5) (37) LC h c + Z/s 2 For a thick coating (h/d + ~)y becomes Y 0 (38) Yh/d - (38) For a thin coating (h/d + 0) y assumes the value Yh/d 3 0 (39) *h/d + 0 -19

SECTION IV INCUBATION PERIOD It has been recognized in the past that fatigue plays an important role in the erosion process (References 12, 14, 16-21), particularly in the "early" stages of the process, corresponding to the incubation period. Applying fatigue concepts to the problem of rain erosion, Springer and Baxi (Reference 13) recently established a semiempirical formula which describes the incubation period in a homogeneous material. Here, Springer and Baxi's analysis is extended to homogeneous materials covered by a single layer of coating. The analysis is based on the concept 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 22) f f N + + -- al (40) 1 2 q where fl, f2...f represent the number of cycles the specimen is subjected to specified overstress levels ael' a 2.oeq and N1, N2..N represent the life (in cycles) at these overstress levels, as given by the fatigue (e versus N) curve. al is a constant. Let us now consider a point B on the surface of the material as shown in Fig. 7. Each droplet impinging upon the surface creates 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 -20

0 0 0 0 0 0 0 0 o o O o I1~ mdr^ X^~~ l~~Impinging Droplets r J^ B Ir ^dr.^-^7Point Force o o o o o F 0 o o o O O O/0O O Ourface Fig. 7. Force Distribution on the Surface of the Coating.

droplet is (Reference 23) P (1-v ) F - ( —- (41) 2wrr where F is given by equation (36), Due to the propagation and reflection of the stress waves in the coating (as discussed in the previous section) the stress in the coating does not remain constant, but fluctuates, as illustrated in Fig, 8. Fatigue life Of the material is generally calculated using an "equivalent dynamic stress" (Reference 24) a a a a m (42) e o a a m where a is the ultimate tensile strength of the material. In the present case a may be separated into two parts ae a' + ". The first e e e e part, a' is due to oscillations about the mean a'= a with amplitude a' e m a The second part a" is due to "oscillation" about the mean a"=a/2, with a e m constant amplitude a"=a/2. Thus, a' is not a constant but varies with a a time. For simplicity, we assume that a' is a constant with a value equia valent to the maximum amplitude, i.e. a' 102 - a( (43) Equations (36) and (43) yield,CY 1Cy Itg2 (44) -22

i b C' O. a J cn 0al QCl) w) anm l \, TIME, t Fig. 6. The Variation of the Stress with Time at the Liquid Droplet-Coat Interface.

The equivalent dynamic stresses corresponding to the two modes of stress oscillations just described may thus be written as a IJ IS au a' =, ISCI U (45) U, (a/2) a e= (46) e a -a/ u 2 The number of cycles for which the material at point B is subjected to a given stress between ae and cae+de is equal to the number of impacts on a dr wide annulus located at r (Fig. 7). During the incubation period the total number 6f impacts on the annulus is f, - ni2wrdr (47) For each single impact the number of stress oscillations in the coating is k (equation 29). The total number of stress oscillations during f impact is, therefore, kfi. Accordingly, Miner's rule becomes f k k f. it Is= a1 (48) Ni T 1 where N' is the fatigue life for overstress levels at a' and Ni is the where Ni e Ni is the fatigue life for overstress levels at a"e. Since r varies continuously from zero to infinity, equations (47) and (48) may be written as 7 n 2rr k ni 2rr / N dr + dr a (49) 0 0 The first term on the left hand side represents the stress oscillation about a "a/2 and the second term the oscillation about a ma. From m m -24

equation (41) rdr is 1 F(1-2v ) rdr 2- 2 da (50) 2o da is determined by differentiating equations (45) and (46) do [ C]-1 do' (51) [ (Ouc) -a 2(o )2 2 V(a Ud -1 doa [- ] do" (52) [2(oac) -o]2 Substitution of equations (50-52) into equation (49) results in 1 2m F (1-2v ) a - | sc (1-2vc) 2- ~I (4a2) do i -- ~- k2dn, t s 2 2i (4o2) ( i 4r " 2 fJ e- N* ---- e do' e - j -e. da" (53) N' e N e u a U The lower and upper limits of the integrals have been changed to the ultimate tensile strength au and the endurance limit aI, respectively. In order to perform the integration the fatigue life N must be known as a function of the stress ae. For most materials the fatigue curve between a and a may be approximated by (Fig. 9) N -b (54) N = blae (54) where bl and b are constants. Equation (54) must satisfy the conditions N 1 for oe a (55a) b2 N - 10 for a = aI (55b) -25

~ I I I b (f) (F) ^^s. I c- Curv II V)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~I O1"~Ib2 LIFE CYCLES, Log N Fig. 9. Idealized ae-N Curve

b2 In equation (55b), 10 corresponds to the "knee" in the fatigue curve (Fig. 9). Equations (54) and (55) yield N = (o/a e) (56) u ce b2 au b = (ba (57) log10 (A —) c "IC Substituting equations (56) and (36) into equation (53) and integrating we obtain b-1 b-1 2 ~ -a I ba C 2 (s I +k) = aI 7 d ni0 - ( b - ( + 2 *sc k) 58 4 i C (b1) C (58) 4(b-l)ouc Introducing the definitions 4a (b-1) 4(a u) (b-1) St (59) b-1 1-2v a- c (1-2v)[1-( ) ] C UC S ~- 1 | j(60) e + 2A 1ksc nd2 n*, n (61) i i 4 (61) equation (58) becomes S In a a e (62) i 1-o a The parameter S characterizes the "strength" of the material. Thus, the number of impacts needed to initiate damage is propositional to the -o ratio of the "strength" of the material S to the stress a prcduced by -27

the impinging droplets. Such a dependence of n on S and -~ is reasone able, since the length of the incubation period is expected to increase -o with increasing S and with decreasing ao, However, in view of the fact that equation (62) is based on the fatigue properties of materials in pure torsion and bending, one cannot expect a linear relationship to hold beeween ni and S /a. In order to extend the range of applicability of equation (62), while retaining its major feature (iamely the functional dependence of ni on S /o~) we write e S a2 a2 * e S 1 ni a1 ( al- 1 + 2T (63) 0 - 0 I Sol+ 2X where both al and a2 are as yet undetermined constants. For a homogeneous material (in the absence of coating) the incubation period is (Reference 13) a2 *iH - al ( (64) iH Both P and ao denote an average stress at the surface. Note, that ni -ad niH differ only by the factor l/(l+2k|#scl). This factor represents the damping effect of the coating. A homogeneous material may be viewed as either a material with very thick coating (h/d + a, k - O0, equation 30), or one in which the coating and the substrate are made of the same material ( s =0, equation 13). It is evident that for either one of these conditions equation (63) reduces to equation (64), provided that the constants a1 and a2 have the appropriate values. To ensure that in the limits (k+ 0 and/or 8sc + 0) equations (63) and (64) become equal we adopt here the same values for -28

al and a2 as were derived by Springer and Baxi (Reference 13) for homo* -6 geneous materials. Using the values a =7.1xlO and a2=5.7 we obtain * -6 S, 5,7 ni 7. lxlO (-6) (65) Equation (65) gives the incubation period of a single layer of chating of arbitrary thickness. The validity of the model must now be evaluated by comparing this result to experimental data. The comparison iL presented in Fig. 10. In this figure all the data are included for which both ni and the relevant material properties (o, aIo b2, v, E,p for both the coating and the substrate were available. As can be se n, there is excellent correlation between the model and the data, lending 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 n >1 or, according toequation(65),when S /ao > 8. Thus, an e incubation period exists if * n.i > 1 S /a~ > 8 (66) e When S /a~ is equal to or less than 8 damage will occur even upon one impact per site. This is most likely to occur at high impact velocities in which case a~ is high (since ao~ P a V). The value for the constant al was given in Reference 13 as 3. x10-4. This value was obtained by using the stress a instead of ae in calculating the fatigue life. When a is replaced by ae al becomes 7.1x10-6 (see Appendix II). -29

104 103 AA nA 0'2 - ^ Present Model A I0 J10 ~wDotoa: Lapp et al 1955 or A Lapp et al 1956 Lapp et al 1958 8 I o Schmitt et oal 1967 * Schmitt 1970 10 102 103 Se/ao Fig. 10. Incubation Period ni versus Se/o-~. Solid Line: Model (Eq. 65). Symbols Defined in Table 1. -30

SECTION V RATE OF MASS REMOVAL The mass removal rate of coat-substrate systems can be cPalculated in a manner analogously to the mass removal rate of homogeneo- materials. The analysis relevant to homogeneous materials is given in Reference 13. Parts of this analysis will be repeated here for the sake of i- pleteness, and to enable the reader to follow the discussion withos. the need of constant referral to the earlier reference. Beyond the incubation period, erosion of the surface of t' -e 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". Fo specimens in torsion and bending tests the probability that failure will occur between minimum life ni and any arbitrary longer life n may be estimated from the Weibull distribution (Reference 25) n-n. p = 1- exp[- (- ) ] (67) na where na is the characteristic life corresponding to the 63.2 percent failure point and 8 is a constant (Weibull slope). For (n-n i) ia <1 a equation (67) may be approximated by n-nh p n' ( i) (68) -31

The probability p can also be taken as the number of specimens that fail between ni and n. If the material undergoing erosion due to liquid impingements is considered to be made up of many small "parts", then the amount of material eroded (mass loss) is proportional to p, i.e. 8 8 n-n i n*-n* m' a3(-a' a3 *) (69) a a p is the density of the material being eroded. In equation (69) m was nondimensionalized with respect to pd in order to render the proportionality constant a3 dimensionless. Equation (6b) is now rewritten in dimensionless form m p d3 (n*-n) (70) Pd d3 d3A Li wTP <r/4 Equations (69) and (70) give 8-1 (n*-n*t) - a3 (71) rpd3/4 (n*") According to equation (71) 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 ni<n<nf. This requirement can be met by setting B=1. Such a value for 8 is not unreasonable under high frequency loading (Reference 21). The characteristic life na is related to the minimum life ni. This relationship may be expressed suitably as a5 n - a n (72) -32

where a, and a5 are constants. Introducing the dimensionless lass loss sate * c a pc-3/4 (73) ipa /4 equations (71-73), together with the assumption 8=1 yield a* a (74) a6 (ni) The a given by equation (74) applies to both homogeneous materials and to coat-substrate systems. For homogeneous materials the:lues of a3 and a6 were determined by Springer and Baxi (Reference 13) and were found to be a3-0.023 and a6-0.7. Similarly as for the incubation period, we adopt the same values of these constants for the present problem of homogeneous substrates covered by a single layer of coating, i.e. k 1 (75) a - 0.023 - (75) (nP In the case of k-0 and/or Sc - the incubation period n* reduc-s to ni (see Section IV). Consequently, under these conditions, a*(given by equation 75) becomes the same as given by Springer and Baxi's formula for homogeneous materials. The validity of the foregoing model was assessed by comparing a calculated by equation (75) to available experimental data. This comparison, given in Fig. 11, shows very good agreement between the calculated and measured a values. This lends further confidence to the ur del. -33

~~~~~~~~~~~~~%100 s-2 o AA l0 A A A ILA A A A &\ 10 A A AP Present Model -A1 A -; ~10 —.y A *t Dato: * Lopp etol 1955 A Lapp etal 1958 ~ Schmitt etol 1967 o Schmitt 1970 id-4 I I t io'4 -3 -2 Io 10 10 10 10 10 I/ n Fig. 11. Rate of Erosion Versus the Inverse of the Incubation Period. Solid Line: Model (Eq. 75). Symbols Defined in Table 1. -34

SECTION VI TOTAL MASS LOSS The total mass loss was given by equation (6b) as m = a(n-ni) (6b) Introducing the dimensionless parameter m m m = In (76) pd c equations (6b), (70) and (73) yield m = a* (n -n*) (77a) or m = n -n (77b) According to equation (77b) it should be possible to correlate all erosion data on a m*/a* versus (n -n*) plot. Therefore, we have included all the existing data on such a plot (Fig. 12). In this figure the theoretical result given by our model (equation 77a) is also indicated. The agreement between the model and the data is quite good, particularly in view of the large errors inherent in many of the measurements. -35

IO4 103 _,, Ha~ s Present Model 102 - a& a AbA A y ja A * \t ^ & Data: o10 _ * Lappet al 1955 A Lapp et al 1956 A Loppet al 1958 o0^~*h./r~ o Schmitt etal 1967 * Schmitt 1970 Schmitt 1971 10 10I I2 103 IO4 (n* n*i) Fig. 12. Coparison of Present Model (Solid Line, Eq. 77b) with Experimental Results. Symbols Defined in Table 1. -36

SECTION VII LIMITS OF APPLICABILITY OF MODEL The results presented in Sections II-VI are valid when (a) there is a finite incubation period, and (b) the mass loss varies lL& aily either with time t or with the number of impacts n. The first 7f this condition is met when the following inequality is satisfied (se; equation 66) ni > 1 (66a) According to equation (65) this condition may also be expressed as S /o > 8 (66b) e o Equations (66a) or (66b) provide the lower limit of the applicability of the model. The upper limit beyond which the present model cannot be applied is determined by the second condition given above, namely that the mass loss must vary linearly with t or n. An estimate of tlis limit was made by observing that up to about n=3n. the data obtained at various values of n did not show any systematic deviation from the model. Thus, the results are valid as long as the number of impacts is less than three times the incubation period, i.e. n < 3ni (78a) or in dimensionless form n < 3ni (78b) -37

Using equation (65) we obtain the following expression for the upper limit -6 Se 57 n <21.3 x 10 (_) (78c) -o Note that the two limits expressed by equations (66) and (78) do not impose any constraints on either the material or the impact velocity. Thus, the results are valid for any material and for any velocity, provided that the experimental conditions fall within the range specified by equations (66) and (78). -38

SECTION VIII FATIGUE FAILURE OF THE SUBSTRATE The foregoing analysis was based on the assumption that t&o coating fails before the substrate. Under some conditions, however, i - substrate may fail before the coating. The analyses presented in Sections IV, V and VI can be applied readily to such a situation. To c iculate the behavior and failure of the substrate only minor modificat.ions need be made in the previous results. The average stress at the surface of the -o coating a (equation 34a) must be replaced by the average stre&s at the -h coat-substrate interface a (equation 34b). Consequently, equation (62) must be written as Se n =a aI (79) Furthermore, in calculating Se (equation 59) the parameters (auc), (Ic) and c must be replaced by the properties of the substrate (aus), (oIs) nd v. All other results remain unaltered. -39

SECTION IX COMPARISON BETWEEN THE RESULTS OF THE UNIAXIAL STRESS AND STRAIN THEORIES It was discussed in Section III that the stresses in the coating may be evaluated by assuming either uniaxial (one dimensional) stress waves or uniaxial (one-dimensional) strain waves propagating through the material. The uniaxial stress wave model was applied to the problem by Conn et al (References 10, 11) and by Engel and Piekutowski (Reference 8). The uniaxial strain model was employed by Morris (Reference 7). There has been considerable speculation in the literature (References 16, 26, 27) as to which approach yields more accurate results. Here, we examine briefly the differences in the uniaxial stress and strain models. These differences can best be illustrated using a graphical solution method (Reference 7). First let us consider the impact of a droplet on a homogeneous (uncoated) material. Upon impact one dimensional stress waves propagate into the solid and the liquid with velocities v and vL, respectively. The stress at any point behind the wave front in either the solid or in the liquid is given by a = pvu (80) where u is the particle velocity at the point and p is the density of the material. The wave velocity v is specified by the relationship v = C + B1u+ B2U (81) C is the velocity of the sound in the material. B1 and B2 are constants. The a versus u curve, shown in Fig. 13, is called the Rankine-Hugoniot -40

T = pVU b V) 7J LUJ U) / 41- (~~~n I-, 7- uj. pC PARTICLE VELOCITY, U Fig, 13. Rankine-Kugonoit Curve for a Homogeneous Solid.

curve. Note that the slope of the curve at the origin (u~O) is the dynamic impedance pC. The slope of the curve at u>0 is always larger than u0. It is assumed now that at the liquid-solid interface the displacements of the liquid and solid surfaces are equal (perfect contact). Then, if the particle velocity at the interface is denoted by uo, then the stresses at the interface are (see equation 80) a= p v u (solid) (82) aL = PLVL(V-uo) (liquid) (83) Since a aoL the intersection of the a versus u and aL versus u curves a u ~ ~s o L o (i.e. the Rankine-Hugoniot curves for the solid and the liquid) yield the stress a. and the particle velocity u1 at the interface (Fig. 14). In the uniaxial stress model proposed by Conn (References 10, 11) v is taken to be constant and equal to C, i.e. BB 2=0 for both the liquid and the solid. In the uniaxial strain wave model described by Morris (Reference 7) v is a function of u (equation 81). The uniaxial o stress model yields a lower stress at the interface than the uniaxial strain model, as indicated in Fig. 14. In this figure, and in the subsequent discussions the superscript B implies stresses evaluated by the uniaxial stress model (B1 and B2 are not zero). The foregoing procedure can be extended to a substrate covered by a thin layer of coating (Reference 7). The Rankine-Hugoniot relationships -42

(O) (b) ^o'sP.sUo CopUasVsUo b b w croo U)I - I < ^ /PLCLUo c2 P PVLUo U) ui V ui V PARTICLE VELOCITY, U. PUTIC, /L V OCITYFu Fig. 14. Impact of the Droplet on a Homogeneous Maierial. Calculation of the Stress at the LiquidSolid Interface by (a) the Uniaxial Stress Wave Model and (b) the Uniaxial Strain Wave Model.

for the coating and the liquid a = p v u (curve 1) (84) C C C 0 aL = PLVL(V-uo) (curve 2) (85) are drawn on a c versus u0 plot, The intercept of these curves yields the stress cland the particle velocity u1 at surface of the coating (x0o). Equation (84) is based on the properties of the undisturbed coating. The Rankine Hugonoit relationship for the coating behind the stress wave is (a c-1) - Pcvc(U -uo) (curve 3) (86) Finally, for the substrate we have Ca pv u (curve 4) (87) S S S 0 Curves (3) and (4) are also drawn on the a versus u plot. The inter0 cepts of curves (3) and (4) and (2) and (4) give a2 and a., respectively. Construction of a typical a versus u0 plot is illustrated in Fig. 15. Figure 15a shows the results for the uniaxial stress theory (vL=CL, v =C, v -C ) for the condition c c s L PLCL > PcCc < c s s (88) For the uniaxial strain model the wave velocities vL, vc and vs are not constants. However, if the condition PvL > Pcv < P v (89) PLVL c P c c s s-44

(o) (b) (Curve 4) (Curve 4) (Curve 3) b oao fC(Curve 3). b (Curve 2) (Curve 2) w: / 2 ) I- - <n 0., / Cr I ) (Curve I2 > } / c 1)(Curve I) C') O'Iu" V u, V PARTICLE VELOCITY, u, PARTICLE VELOCITY, u Fig. 15. Impact of a Droplet on a Substrate Covered with a Single Layer of Coating. Calculation of the Stress at the Liquid-Coating Interface al, the Stress at the Coating-Substrate Interface r2, and the Stress that Would Occur on the Surface of the Substrate in the Absence of Coating c~. (a) Uniaxial Stress Wave Model; (b) Uniaxial Strain Wave Model.

is satisfied for each value of u0 then the Rankine-Hugonoit curves are as shown in Fig. 15b. Thus, as long as the condition in equation (89) is B B satisfied a2 < a. This is in agreement with the result of the uniaxial 2 M stress wave model. If the condition expressed by equation (89) is not B B satisfied for all values of u then a2 may be larger than a. Whether O 2 00 B B a2 is larger or smaller than aB, depends on the relative magnitudes of B1 and B2 for the liquid, the coating and the substrate. The conditions under which this might occur cannot be specified at present time, because values for B1 and B2 are unavailable for most materials. Plots similar to those presented in Fig. 15 could also be drawn for materials with different relative impedances (i.e. PLvL< cvc < p v, PLvL < Pcvc > Psvs, pLvL > pcvc > pvs;see Fig. 4). However, the conclusions presented in the foregoing would not be altered. It is noted here that curves (3) and (1) in Fig. 15 are symmetric with respect to a = ol, regardless of the values of B1 and B2. This symmetry was not satisfied by the Rankine-Hugonoit plot presented in Reference 7. -46

SECTION X SUMMARY The following formulae may be used to estimate the incubatfion time and the mass loss of the coat material of coat-substrate sy.te i subjected to repeated impingement of liquid droplets. a) Incubation Period ~* -6 S, 5,7 ni = 7.1x10 [ e] (90) o -o or 5.7 -6 S 57 i 9.05x10 e (no. of impact) (9) "i 2 L.0J v unit area d~ or 5.7 where 4c (b-1) S u e (1-2vc)[1 + 2 k isc (9 -~ PLGLVcose 1 + 1 + l-exp(-) (94) a PLC 14, el-x(~(i (94) L, L 1 - *scc L sc 1+ S, Y 1 _P C, C c c and PC-P C PLCL-P C * _ s s c c yE = L L c c sc P C +P C' Lc P C +P C ss cc L L c c eC p (95) CL h [1 scLc sc Lc L -47

b) Rate of Mass Removal 4 *8 o a - 92 [-] (96) e or 4 -o 3 0 mass loss a n 70.6pCd [] ( (97) e impact -o S and a are defined in equations (93) and (94). c) Total Mass Loss * * * * m - a (n -ni) (98) or mass loss. m a(n-ni) m s lo (99) Equations (91), (97) and (99) yield the mass loss per unit area in time t -o -6 S 5.7 m = 70.6 pcd3 [ ] { (q t Vcos) - 9 10 (100) e d -O S and a are defined as in equations (93) and (94). The foregoing results are subject only to the following two constraints. a) Incubation time must be greater than zero (ti>O), a requirement satisfied by the condition S e > 7.96 (101) -o -O -48

b) Total time elapsed must be less than three times the incubation period, i.e. t < 3ti n < ni (102) ni < 3n* i i or 5.7 3 (Vcos)It 2.13x-5 e] (103) 2 3 2.3x 0 vtd S and o are defined in equations (93) and (94). -49

* TABLE 1. Description of Data and Syinbols Used in Figures 10, 11 and 12 Symbol Investigator Coating Substrate Coating Velocity Intensity Drop Size Thickness ft/sec (in/hr) mm ___________ _________________ __________________________________ * Lapp et al Neoprene Polyester 8.9, 10 731 1 1.9 1955 Aluminum 15, 20 Steel 20 Aluminum 5 877 Teflon Aluminum 5, 10 731 A Lapp et al Neoprene Polyester 4-5,7,10,20 877 1 1.9 1956 Aluminum 15-17 877?P~~~ ~Polyethylene Polyester 9-15 731 Teflon Epoxy 2, 10 Lapp et al Polyurethane Epoxy 15-30 731 1 1.9 1958 Aluminum 8-11 Polyester 10 Teflon Epoxy 10 Aluminum 62.5 O Schmitt et al Aluminum A1P04 20 1596 2.5 1.9 1967 PBI 20, 40 2360 Polymide 30 3169 Epoxy 30 Gaco Epoxy 20 Polyurethane 26,30 Teflon 35 Nickel 10

TABLE 1 (continued) Symbol Investigator Coating Substrate Coating Velocity Intensity Drop Size Thickness ft/sec in/hr nn mils * | Schmitt Alumina Polymide 10 1596 2.5 1.9 1970 2360 Epoxy 40 2751 3169 Gaco Epoxy 10 1596 l ___l______________ ___3169 Urethane 10,15,20,30 1596 _______________ 2360 Polyethylene Epoxy 30 2751 Cn>y | |Nickel Polymide 12 cj Schmitt Urethane Aluminum 15 731 I 1.9 19 71 Neoprene Epoxy 22 Aluminum Material properties ued in obtaining Figs. 10-12 are from References (6) (9)-( Material properties used in obtaining Figs. 10-12 are from References (6), (9)-(10)

APPENDIX I DERIVATION OF EQUATION (18) After k number of wave reflections the stress at the coat-substrate interface is (equation 16) 1+4 sc k 2k ~ l s- [1-(scL (A.l. 1) sc Lc After a large number of reflections (kw-a) the stress approaches the limit ao< - lim a2k (A.1.2) k-en Noting that Z9-Z ZL-Z S C L c1 (A.13) *scLc ( +Z z+z 1 A.1.3) s c L c we obtain lim ( s Lc) 0 (A.1.4) k-*w sc Lc Equations (A.1.1), (A.1.2) and (A.1.4) give a aC, 14? 2k 1scSC -- = lim k sc (A.1.5) 1 k-w ~1 1'sc Lc Using the notations (13) and (14) of Section III, equation (A.1.5) may be written as 1 +ZL/Z ZLVcose/(1 + ZL/Z ) L S L L C -52- 1 1+ Z/Z ZLVc /(+ ZL ) -52

We now observe that the denominator of equation (A.1.6) is equal to the stress at the surface of the coating [P(al, see equations (4) and (7)]. Thus, a is Z Vcose a - L (A. 1. 7) ^'TT1+ z/Z This is the stress that would be produced on the surface of thi substrate if the droplet would impinge upon it directly (see eqx^a-on 4). -53

APPENDIX II THE VALUE OF THE CONSTANT al FOR HOMOGENEOUS MATERIALS Springer and Baxi (Reference 13) calculated the incubation period from Miner's rule -+ -+.. + N a_ (A.2.1) N1 +2 k basing Ni on the stress a (equation 10 of Reference 13) a F(-2v) (A.2.2) 2rr Introducing (see equation 11 of Reference 13) f - ni27rrdr (A.2.3) and (see equation 16 of Reference 13) N = b ab (A.2.4) Springer and Baxi obtained ni 2wrdr -b = a (A.2.5) 0 b1a Equation (A.2.2) and (A.2.3) yield rdr 1 F(1-2) d (A.2.6) 2w 2o2 Substitution of equation (A.2.6) into equation (A.2.5) gives _ffd2 2 nI Int i (1-2v)/212] bj -4<( —) do (A.2. 7) blo Mu1 -54

Evaluating the integral Springer and Baxi obtained wd2 S G4 n = a1 (A.2.8) where 2a (b-1) 2o (b-l) S a b 1 -2v (A.2.9) b 1-2v (1-2V)[1-(o-) ] u and a constant a2 was introduced in Springer and Baxi's work d2 a2 ni.4. (- afd) (A.2.10) Comparing equation (A.2.10) with data, Springer and Baxi deduced the -4 values of a1 3 3.7x10 and a2 - 5.7, i.e. d2 4 S 5.7 ird2 -4S 4 ni = 3.7x10() (A. 2. 11) We compute now the above results basing the fatigue stress N on the equivalent dynamic stress a a a m a m (A.2.12) e a — o n m a a Since a =2 and a, equation (A.2.12) yields a 2 m 2 ao Ca m.u (A.2.13) e 2a -a u The replacement of a by a in equations (A.2.2), (A.2.4) gives e -d2 1 4 o (b-l) -rd —ni a1 -- (A.2.14) 4 i 1 p b-1 (l-2v) [l-(-5) ] -55

Introducing the notation 4a(b- 4 (b-)) 4 ) S __u ___ - UU (A.2.15) e b-1 (1-2v) (A.2 (1-2v) [l(-() ] u we obtain wd2 Se 4 f"X a p (A12. 16) Comparison of equations (A.2.9) and (A.2.16) shows that S - 2S (A.2.17) Accordingly equation (A.2.11) becomes 5.7 5.7 fd2 -4 Se -6 Se I 3. 7xlO (? - 7. lxlO (-) (A.2. 18) Thus in terms of Se the incubation period is 5.7 wd2 -6 Se 4 ni - 7. lxl0 (p) (A. 2.19) -56

REFERENCES [1] Lapp, R.R., Stutzman, R.H., Wahl, N.E., "A Study of the Rain Erosion of Plastic and Metals," WADC Technical Report 53-185, Part 2. Wright-Patterson Air Force Base, Dayton, Ohio. May 1955 [2] Lapp, R.R., Stutzman, R.H., ah, N.E,"Summary Report the Rain Erosion of Aircraft Material," WADC Technical Report 5S;.85, Part 3. Wright-Patterson Air Force Base, Dayton, Ohio. Sept. 1956. [3] Lapp, R.R., Thorpe, D.H., Stutzman, R.H., Wahl, N.E.,.Tihe Study of Erosion of Aircraft Materials at High Speed in Rain,' WADC Technical Report 53-185, Part 4. Wright-Patterson Air Fc-ce Base, Dayton, Ohio. May 1958. [4] Schmitt, G.F., Tatnall, G.J., Foulke, K.W., "Joint Air Force-Navy Supersonic Rain Erosion Evaluations of Materials," APFIM TR-67-164. Air Force Materials Laboratory, Wright-Patterson Air Forc- Base, Dayton, Ohio. 1967. [5] Schmitt, G.F., "Research for Improved Subsonic and Supersonic Rain Erosion Resistant Material," AFML-TR-67-211, Air Force Materials Laboratory, Wright-Patterson Air Force Base, Dayton, Ohio. January 1968. [6] Schmitt, G.F., "Erosion Behavior of Polymeric Coatings and Composites at Subsonic Velocities," Proceedings of the Third International Conference on Rain Erosion and Associated Phenomena," (Eidted by A.A. Fyall and R.B. King). Royal Aircraft Establishment, England. pp. 107-128. August, 1970. [7] Morris, J.W., "Supersonic Rain and Sand Eroiion Research" AFMLTR-69-287, Part II. "Mechanic Investigation of Rain Erosion," Air Force Materials Laboratory, Wright-Patterson Air Force Base, Dayton, Ohio. September 1969. [8] Engel, O.G., A.J. Piekutowski,':Investigation of Composite-Coating Systems for Rain-Erosion Protection," Prepared under contract N00019-71-C-0108 for Naval Air Systems Command, Departments of the Navy, at the University of Dayton, Research Institute, Dayton, Ohio. [9] Conn, A.F., "Research of Dynamic Response and Adhesive Failures of Rain Erosion Resistant Coating," Technical Report 811-1 Hydronautics, Laurel, MD. January 1969. [10] Conn, A.F., "Prediction of Rain Erosion Resistance from M.asurements of Dynamic Properties," Technical Paper, Hydronautics, La rel, MD. April 1970. [11] Conn, A.F., Rudy, S.L., "Further Research on Predicting t e Rain Erosion Resistance of Material," Technical Report 7107-1, Hydronautics, Laurel, MD. May 1972. -57

[12] Mok, C.H., "A Cumulative Damage Concept in Rain Erosion Studies," AIAA Journal, Vol. 7, pp. 751-753, 1969. [13] Springer, G.S., Baxi, C.B., "A Model for Rain Erosion of Homogeneous Material," Technical Report AFML-TR-724106, Air Force Materials Laboratory, Wright-Patterson Air Force Base, Dayton, Ohio. June 1972. [14] Heymann, F.J., "Erosion by Cavitation, Liquid Impingement and Solid Impingement: A Review," Engineering Report E-1460, Westinghouse Electric Corporation, Lester, Pennsylvania, March 1968. [15] Heymann, F.J., "On the Shock Wave Velocity and Impact Pressure in High Speed Liquid-Solid Impact," Trans. ASME, J. of Basic Engineering, Vol. 90, pp. 400-405, 1968. [16] Brunton, J.H., "Liquid Impact and Material Removal Phenomena," Technical Memorandum No. 33-354, Jet Propulsion Laboratory, California, Institute of Technology, Pasadena, California, June 1964. [17] Heymann, F.J., "A Survey of Clues to the Relation Between Erosion Rate and Impingement Conditions," Proceedings of the Second Meersburg Conference on Rain Erosion and Allied Phenomena, (Edited by A.A. Fyall and R.B. King), Royal Aircraft Establishment, Fawnborough, England, pp. 683-J60. August 1967. [18] Leith, W.C., Thompson, A.L., "Some Corrosion Effects in Accelerated Cavitation Damage," Trans. ASME, J. of Basic Engineering, Vol. 82D, pp. 795-807, 1960. [19] Mathieson, R., Hobbs, J.M., "Cavitation Erosion: Comparative Tests,"' Engineering, Vol. 189, pp. 136-137, 1960. [20] Ripken, J.F., "A Testing Rig for Studying Impingement and Cavitation Damage," in Erosion by Cavitation on Impingement, ASTM STP 408, American Society for Testing and Materials, pp. 3-11, 1967. [21] Thiruvengadam, A., Rudy, S.L., and Gunasekaran, M., "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. [22] Miner, M.A., "Cumulative Damage in Fatigue," J. Appl. Mech., Vol. 12, pp. A159-A164, 1945. [23] Timoshenko, S., Theory of Elasticity, McGraw-Hill Book Co., New York, 1934. [24] Juvinall, R.C., Stress, Strain and Strength, McGraw-Hill Book Co., New York, 1967. [25] Weibull, W., Fatigue Testing and Analysis of Results, Pergamon Press, New York, 1961. -58

[26] Conn, A.F., "Discussion on'Erosion Behavior of Polymeric Coating and Composites at Subsonic Velocity' by G.F. Schmitt," Proceedings of the Third International Conference on Rain Erosion and Associated Phenomena, (Edited by A.A. Fyall and R.B. King) Royal Aircraft Establishment, England, pp. 135-138. August 1970. [27] Morris, J.W., "Discussion on'Erosion Behavior of Polymre ic Coatings and Composites at Subsonic Velocity' by G.F. Schmit tT Proceedings of the Third International Conference on Rain Erosion and Associated Phenomena, (Edited by A.A. Fyall and R.B. King), Royal Aircraft Establishment, England. pp. 139-144. -59

UNCLAS SI FIED Sr.,,rty Cl^trttflj..rrn DOCUMtIT CONTROt DATA. R & D ('.fer.lty rletfllrlf lon of ttl*, horth of ah.ftmr t end IntO. ind nnol^tnn metuf he entered whe, thFe ovrall report I. clnaeliled *. QRIOINATINO ACTIVITY (Cotporrte a"lhotp) 2.. REPORT SECURITY CLASSIFICATION The University of Michigan UNCLASSIFIED Mechanical Engineering Department Ab. oRouP Ann Arbor, Michigan 48104 3. REPORT TITLE "Analysis of Rain Erosion of Coated Materials" 4. otEscRPTlve NOt' (Typ.t of teport and Inclielve date*) Technical Report, June 1972-June 1973 S. Au THOs S.) frlpae name, middle Inintial, tat name) George S. Springer Cheng-I. Yang Poul S. Larsen a. REPORT OATsE 7a. TOTAL NO. OF PAGES lb. NO. OF REFS September 1973 59 27 GI. CONNTRACT OR GRANT NO. ea. ORIGINATOR'S REPORT NUMBERIS) F33615-72-C-1563 b. PROJECT NO. c. Task No. 734007 eb. OTHER REPORT NO(S) (Any other numbers Ihfa may be aselgned thli report) --- ~d.^~~~~~~ AFML-TR-73-22 7 o1. O.ISTRIUTION STATEMENT Approved for public release; distribution unlimited. I. SUPPLIEMEiTAtR NOTES I. SPONSORING MILITARY ACTIVITY Air Force Materials Laboratory Wright-Patterson Air Force Base Ohio 45433 t1. ABSTRAC T The behavior of coat-substrate systems subjected to repeated impingements of liquid droplets was investigated. The systems studied consisted of a thick homogeneous substrate covered by a single layer of homogeneous coating of arbitrary thickness. Based on the uniaxial stress wave model, the variations of the stresses with time were determined both in the coating and in the substrate. Employing the fatigue theorems established for the rain erosion of homogeneous materials, algebraic equations were derived which describe the incubation period, and the mass loss of the coating past the incubation period, in terms of the properties of the droplet, the coating and the substrate. The results were compared to available experimental data and good agreement was found between the present analytical results and the data. The differences between the uniaxial stress wave and the uniaxial strain wave models were also evaluated by calculating according to both models a) the stress at the coat-liquid interface, b) the stress that would occur in the substrate in the absence of the coating, and c) the stress in the coating after the first wave reflection from the substrate. DD,o,,.1473 UNCLASSIFIED Security Classsifcation

UNCLASSIFIED Security Classi(ication KEY WO RD1 INK A LINK 8 LINK C OL____ WT ROLE WT- ROLEC - WT Rain Erosion Erosion Mechanism Fatigue Model Incubation Period UNCLEAS S IFIED Stcurity Cl4uawifcut&on n

UNIVERSITY OF MICHIGAN 3 9015 03526 6892