Dimensionless correlation for sand erosion of families of polymers

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1 Dimensionless correlation for sand erosion of families of polymers Paul G. Cizmas and John C. Slattery 1 Texas A&M University College Station, TX , USA Abstract A single correlation for sand erosion of a family of polyurethanes is presented. By family is meant a group of chemically similar compounds. For the first time, an implication of Truesdell s [1] observations regarding the number of dimensional parameters in viscoelastic constitutive equations has been used together with the Pi theorem to prepare a single dimensionless correlation for a group of materials with similar but different forms of constitutive equations, different forms of stressdeformation behavior. To this point, such correlations have been for materials having the same form of constitutive equation, such as Hooke s law or Newton s law of viscosity. Key words: particle erosion, sand erosion, polyurethanes, protective coatings 1 Introduction Wear due to erosion by solid particle impact is a phenomenon that affects many practical applications, including gas turbine compressors, helicopter and propeller blades, pulverized coal combustion, powder transport, high speed vehicles and aircraft, canopies and radomes. To date, four types of coatings have been used to reduce erosion: metals, ceramics, polymers, and composites. Several experimental investigations have indicated that polymers or composites provide better erosion protection than metals or ceramics [2, 3, 4]. In 1 Departments of Aerospace Engineering, of Chemical Engineering, and of Mathematics Corresponding author. Department of Aerospace Engineering, Tel.: ; fax: ; cizmas@tamu.edu Preprint submitted to Elsevier Science 13 May 2006

2 addition, polymers and composite coatings are lighter, and they may be easier to install/replace. Compared to metals, however, the polymer and composites have more complex damage and failure mechanisms [5]. An exhaustive investigation of wear revealed a lack of predictive methods for both hard and rubbery coatings [6], although erosion resistance did correlate well with the rebound resilience [7]. Li and Hutchings [8] went on to show in a study of sand erosion of polyurethanes that, for similar values of rebound resilience, the erosion resistance decreased with increasing hardness, tensile strength and elastic modulus. In what follows, a single correlation for sand erosion of a family of polyurethanes is presented. By family is meant a group of chemically similar compounds. For the first time, an implication of Truesdell [1] observation regarding the number of dimensional parameters in viscoelastic constitutive equations has been used together with the Pi theorem to prepare a single dimensionless correlation for a group of materials with different forms of constitutive equations, different forms of stress-deformation behavior. To this point, such correlations have been for materials having the same form of constitutive equation, such as Hooke s law or Newton s law of viscosity. 2 Dimensionless correlation Truesdell [1] observed that, no matter how complex a constitutive equation for stress-deformation behavior might be, no matter how many dimensionless parameters it might involve, it would contain only three parameters having dimensions: a characteristic viscosity µ, the modulus of elasticity E, and a natural (relaxation, recovery or characteristic) time τ. For viscoelastic solids (solids having a fading memory of past deformations) such as those studied by Li and Hutchings [8], the constitutive equation is likely simpler, having only two parameters with dimensions: E and τ. To emphasize, the most important point here is that the forms of the constitutive equations for the polyurethanes studied by Li and Hutchings [8] are (almost certainly) all different. To our knowledge, this is the first time that a correlation for materials having different constitutive equations has been attempted. Typically, sand erosion experiments directly observe only a few variables, including: the mass of polymer lost M p, the mass M s of sand striking the surface, the nominal size of the sand particles D s, the speed v of the sand particles, their angle α of impact, density ρ of the polymer, and its thickness h p. The simplest interpretation of these experiments is that there are eight dimensional 2

3 parameters (M p, M s, E, τ, ρ, h p, D s, and v) and one dimensionless parameter (α) or M p = F (M s, E, τ, ρ, h p, D s, v, α) (1) In terms of the rate of erosion 2 R E M p M s (2) (1) requires R E = G (M s, E, τ, ρ, h p, D s, v, α) (3) For steady-state erosion, the dependence upon M s drops out [9, p. 173], and (3) becomes R E = H (E, τ, ρ, h p, D s, v, α) (4) The natural (relaxation, characteristic) time τ does not have a unique definition. It can be chosen on the basis of convenience for the materials and experiments being studied. Using the Pi theorem [10], we can now write (4) as ( R E = K E, τ, α, D ) s h p where E E ρv 2 is the dimensionless modulus of elasticity and τ τ h p E ρ (5) (6) (7) is the dimensionless natural time. Note that E/ρ is proportional to the speed of the longitudinal wave traveling in the polymer coating. 2.1 Dimensionless correlation of a family of polyurethanes To our knowledge, the only reliable sand erosion study currently available in the open literature for a family of polymers is that of Li and Hutchings [8], 2 If M p is a linear function of M s and if there is no initial incubation behavior, R E = M p /M s [9, pp ]. 3

4 Table 1 Erosion rates R E at 30 impact angle obtained using the two largest values of the mass of sand M s and the corresponding mass loss M p read from Fig. 6 of [8] Monothane M s M p R E [g] [mg] A e A e A e A e A e A e A e who used eight different grades of a polyester-based, one-component, castable polyurethane. These are Monothane grades A20 to A90 produced by Chemical Innovations Limited (CIL). Our objective is to form a dimensionless correlation of these data using (5). In their tests, Li and Hutchings [8] did not vary h p = 7 mm, D s = 120 µm, or v = 50 m/s. While they did investigate two values of the impact angle, we were able to use data only for α = 30 ; sticking was a serious problem for α = 90. The densities were not given for each polymer, but all had similar values: 1,190 to 1,220 kg/m 3. We chose to use 1,200 kg/m 3 for all of the polymers in our computations. For these reasons, (5) reduces to R E = K (E, τ ) (8) Table 1 shows the erosion rates R E obtained using the two largest values of the mass of sand M s and the corresponding mass loss M p read from Fig. 6 of [8]. (The values corresponding to A50 could not be read from the figure.) Two sets of data were available for E. The first set was measured by Li and Hutchings [8], while the second set was provided by the manufacturer [11]. As shown in Table 2, there are significant differences between them, although the trends are similar. There could be several reasons for the differences in the values of E. One possible explanation is that the cross sectional area was measured differently 4

5 [12]. Another explanation is that E was reported at different conditions. In the Li and Hutchings experiments [8], it was measured at half the extension to failure, while the manufacturer measured it at 300% extension. There is also the possibility of the presence of defects, such as edge nicks and bubbles, in the cast material [12]. It is also possible that the formulations of the polyurethanes changed since 1990, although we were told that they had not changed. A change in the formulations could also explain the differences in the values of rebound resilience shown in Table 3. Note that, although the rebound resilience is relatively easy to measure, the two sets of results differ in average by a factor of Li and Hutchings measured the rebound resilience by dropping a steel ball 6.35 mm in diameter from a height of 500 mm [8]. The procedure for measuring the rebound resilience used by the manufacturer [11] is not known by the authors. Table 2 Elasticity modulus of Monothanes Units Source Monothane A20 A30 A40 A50 A60 A70 A80 A90 MPa [11] MPa [8] Although we believe that the rebound resilience shown in Table 3 is dependent upon the natural time τ, we are unable to suggest a definition for τ in terms of it. If τ = τ = 0, these polyurethanes would be classified as perfectly elastic solids, described perhaps by Hooke s law, and (8) would further reduces to R E = K (E ) (9) While these polyurethanes are not perfectly elastic solids (their rebound resilience is not 100% in Table 3), it is likely that their characteristic relaxation time (recovery time) τ is very small, perhaps less than 10 6 s [13, Fig. 6]. In this case τ < 10 2, suggesting that (9) might still apply. This appears to be confirmed by Fig. 1 below. Table 3 Rebound resilience of Monothanes Units Source Monothane A20 A30 A40 A50 A60 A70 A80 A90 [%] [11] [%] [8] Fig. 1 shows the application of (9) to the data of Li and Hutchings [8] for the Monothane A20 to A90 family of polyurethane at 30 impact angle. The points corresponding to Monothane A30 and A40 almost overlap, for both sets of E. 5

6 deg impact angle Experimental data with CIL values for E Linear regression Experimental data with Li & Hutchings values for E Linear regression Fig. 1. Dimensionless representation of erosion at 30 impact angle. Note that each triangle and circle symbol represent a different polymer. The data were fitted using standard linear regression by least squares. The Pearson correlation coefficient [14] for the data using the values of E provided by the manufacturer CIL values was 0.975; for the values proved by Li and Hutchings [8], it was Note that data correlation improves as the absolute values of the Pearson correlation coefficient increases. 3 Conclusions There are two conclusions to be drawn from this work. (1) We have constructed successfully a single correlation for sand erosion of the family of polyurethanes studied by Li and Hutchings [8]. It indicates desirable trends of properties, helping in the design of new coatings. The limitation of this correlation is that we do not know how broadly it applies. While our expectation is that this particular correlation applies only to polyurethanes, without further testing, we are uncertain as to what portion of the very extensive class of polyurethanes it includes. (2) For the first time, an implication of Truesdell s [1] observations regarding 6

7 the number of dimensional parameters in viscoelastic constitutive equations has been used together with the Pi theorem to prepare a single dimensionless correlation for a group of materials with similar but different forms of constitutive equations, different forms of stress-deformation behavior. To this point, such correlations have been for materials having the same form of constitutive equation, such as Hooke s law or Newton s law of viscosity. (3) Our assumption in using Eq. (9) that τ should not be included appears to be confirmed by the results in Fig. 1. Acknowledgments We would like to thank Mr. David Stone of the Army Aviation & Missile Research Development Engineering Center for his continuous support during the project. We would also like to thank Mr. Edwin Rosenzweig of the Navy Air Systems Command for many fruitful discussions. We are grateful to Prof. Ian Hutchings of the University of Cambridge for providing enlarged figures of the erosion rates for the Monothanes. Last but not the least, we would like to thank Prof. Vikram Kinra of Texas A&M University for sharing with us his observation on the longitudinal wave velocity term of the dimensionless time. Nomenclature Roman h p - thickness of polymer coating D s - diameter of sand particle E - modulus of elasticity R E - rate of erosion F, G, H, K - functionals M p - mass loss M s - mass of sand v - speed of sand particle 7

8 Greek α - impact angle ρ - mass density of polymer τ - natural (relaxation, characteristic)time τ - dimensionless time References [1] C. Truesdell, The natural time of a viscoelastic fluid: Its significance and measurement, The Physics of Fluids 7 (8) (1964) [2] V. K. Agarwal, D. Mills, J. S. Mason, A comparison of the erosive wear of steel and rubber bends in pneumatic conveying system pipelines, in: J. E. Field, N. S. Corney (Eds.), Proceedings 6th International Conference on Erosion by Liquid and Solid Impact, Cavendish Laboratory, Cambridge, UK, [3] R. J. Nuttall, The selection of abrasion resistant lining materials, Bulk solids handling 5 (5) (1985) [4] H. J. Ephithite, Rubber lining - the soft option against abrasion, Bulk solids handling 5 (5) (1985) [5] M.-N. Barkoula, J. Karger-Kocsis, Processes and influencing parameters on the solid particle erosion of polymers and their composites, Journal of Materials Science 37 (2002) [6] H. C. Meng, K. C. Ludema, Wear models and predictive equations: their form and content, Wear (1995) [7] I. M. Hutchings, D. W. T. Deuchar, A. H. Muhr, Erosion of unfilled elastomers by solid particle impact, Journal of Materials Science 22 (1987) [8] J. Li, I. M. Hutchings, Resistance of cast polyurethane elastomers to solid particle erosion, Wear 135 (1990) [9] I. M. Hutchings, Tribology: Friction and Wear of Engineering Materials, CRC Press, Inc., Boca Raton, Florida, [10] L. Brand, The Pi theorem of dimensional analysis, Archive for Rational Mechanics and Analysis 1 (34) (1957) [11] Properties furnished by personal communication (2005). [12] I. M. Hutchings, Private communication (2005). [13] A. K. Rizos, G. Fytas, R. J. Ma, C. H. Wang, V. Abetz, G. C. Meyer, Local molecular motion in polyurethane-poly(methyl methacrylate) interpenetrating polymer networks, Macromolecules 26 (1993) [14] R. L. Ott, An Introduction to Statistical Methods and Data Analysis, 4th Edition, Duxbury, Belmont, California,

Chapter 7. Highlights:

Chapter 7. Highlights: Chapter 7 Highlights: 1. Understand the basic concepts of engineering stress and strain, yield strength, tensile strength, Young's(elastic) modulus, ductility, toughness, resilience, true stress and true