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An indirect method based on fretting tests to characterize the elastic properties of materials: Application to an epoxy resin RTM6 under variable temperature conditions
Wear 269 (2010) 632–637
Contents lists available at ScienceDirect
Wear journal homepage: www.elsevier.com/locate/wear
Short communication
An indirect method based on fretting tests to characterize the elastic properties of materials: Application to an epoxy resin RTM6 under variable temperature conditions S. Terekhina a,b , S. Fouvry a,∗ , M. Salvia a,∗∗ , I. Bulanov b a b
Laboratoire de Tribologie et Dynamique des Systèmes, UMR CNRS ECL ENISE ENSMSE 5513, École Centrale de Lyon, 69134 Ecully cedex, France Laboratory of Composite Materials, Bauman Moscow State Technical University, 5, 2nd Baumanskaya Str., 105005 Moscow, Russian Federation
a r t i c l e
i n f o
Article history: Received 30 October 2009 Received in revised form 31 May 2010 Accepted 16 June 2010 Available online 23 June 2010 Keywords: Fretting DMA Epoxy resin Hertzian contact
a b s t r a c t An original indirect method of elastic modulus measurement based on fretting tests is developed here. It consists in applying a normal force, then marking the contact area thanks to a tiny cyclic fretting loading. The elastic modulus of the epoxy resin is extrapolated from the post-mortem contact radius measurement using a Hertzian contact description. Applied to a RTM6 epoxy resin, the elastic properties are extracted from this partial slip fretting analysis and are compared to dynamic mechanical analysis data (DMA). Very good agreement was found between these fretting results and data given by DMA analysis, confirming the potential of this new contact method to extract the mechanical properties of materials. © 2010 Elsevier B.V. All rights reserved.
1. Introduction The industrial use of polymers as friction materials in both homogeneous and heterogeneous forms has widely increased over the past few years. As a result, the wear and friction properties of polymers are now extensively investigated. However, the wear analysis of polymers requires complete knowledge of their mechanical properties. One original approach which is currently being developed in the present research work consists in extracting these variables using a calibrated fretting test. Fretting is a small-amplitude oscillatory motion between contacting surfaces [1]. It may arise in any assembly of engineering components if vibration or cyclic stress is present, inducing localised wear or fatigue cracks. These forms of damage are encountered in all quasi-static loaded assemblies such as keys, cables, bearing races and shafts, orthopedic implants, cranes, turbine blade roots, electrical contacts [2], and also during mold fabrication (for example, by resin transfer molding (RTM) process [3]). The contact theory developed by Hertz for elastic conditions, smooth surfaces and the semi-infinite bulk hypothesis, allows the apparent area to be linked from the contact geometry, the elastic
∗ Corresponding author. Tel.: +33 4 72 18 65 62; fax: +33 4 78 43 33 83. ∗∗ Corresponding author. E-mail addresses: [email protected] (S. Fouvry), [email protected] (M. Salvia). 0043-1648/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.wear.2010.06.008
properties of the contacting bodies and the applied normal force. Hence, explicit expressions of the contact radius and pressure field distributions are provided for axisymmetrical sphere/plane contact or 2D cylinder/plane configuration. Alternatively, using a reverse approach, the elastic properties of the material can be extracted by measuring the contact radius induced by a given normal force [4]. This strategy is extensively applied for ‘transparent’ materials which allow direct access to the contact area [5]. However, for nontransparent materials direct determination of the elastic contact radius is not possible. One alternative strategy, extensively applied by the instrumented indentation test method, consists in analyzing the loading and unloading curves (i.e. evolution of the depth indentation versus the normal force) in order to extract the elastic properties [6]. However, this method requires the exact determination of the indentation depth, which is highly affected by external aspects like test apparatus stiffness or temperature dilation fluctuations. An alternative strategy which is developed here consists in applying tiny fretting cycles (i.e. tangential oscillatory displacements) in order to mark the contact obtained for a given normal force. Then, using post-mortem contact radius measurements, the elastic properties of the material can be extracted. This approach, which is not affected by test stiffness or other temperature fluctuations, is here applied to describe the temperature behavior of polymers. However, in the case of polymer material not only elastic but also viscous properties play important roles in contact deformation,
S. Terekhina et al. / Wear 269 (2010) 632–637
633
Table 1 Mechanical properties of ball-bearing steel (AISI 52100). E2 , GPa
2
Rm , MPa
RpO,2 , MPa
HRC
210
0.3
2200
2000
60–67
especially in the glass transition region. Therefore, to separate the effect of the viscous and elastic response of the fretting contact behavior, a combined dynamic mechanical analysis (DMA) [7] is performed. 2. Experimental procedure 2.1. Materials The fretting tests and dynamic mechanical analysis were carried out using an epoxy resin HexFlow RTM6 (Hexcel, France), already degassed. This resin was specifically developed to fulfill the requirements of the aerospace and space industries in advanced resin transfer molding (RTM) processes. It is a premixed epoxy system of the tetraglycidyl-diamino-diphenyl-methane (TGDDM) type suitable for service temperatures from −60 up to 180 ◦ C. The standard curing cycle for RTM6 resin is 160 ◦ C for 75 min, followed by freestanding post-cure at 180 ◦ C for 120 min (ramp: 1 ◦ C/min). During the fretting tests the resin RTM6 was tested against ballbearing steel (AISI 52100). The mechanical properties of the steel are reported in Table 1. The surface roughness of the AISI 52100 steel ball is very low, around Ra ≈ 0.09 m. The RTM6 resin surface roughness is slightly higher, around Ra ≈ 0.5 m, but very small compared to the studied contact radius (aH > 1 mm). This indirectly justifies the smooth surface hypothesis imposed by the Hertzian theory. 2.2. Experimental techniques 2.2.1. Fretting tests To investigate the elastic properties of the RTM6 resin, a sphere/plane fretting test involving an electrodynamic shaker system has been applied (Fig. 1a). The normal force P is kept constant while varying tangential force (Q) and displacement (ı) are imposed. Thus, the fretting loop Q–ı, as shown in Fig. 1b, can be drawn to extract quantitative variables, including the dissipated energy (Ed ), (i.e. the area of the hysteresis), which is related to the friction work dissipated per fretting cycle. Note that for polymer materials it can be influenced by mechanical loss behavior induced by viscoelastic response. Other variables like the sliding amplitude (ı0 ) (i.e. the remaining displacement when Q = 0), the tangential force amplitude (Q*) and the displacement amplitude (ı*) can also be extracted. The sphere/plane contact consists of a 22.22 mm radius, in contact with a rectangular RTM6 specimen (4.1 mm × 20 mm × 25 mm) using a constant normal load (P = 75 N). The tests were conducted under the partial slip condition for which the fretting cycle displays a very closed elliptical form [8,9]. A constant small displacement amplitude of 4 m was imposed to protect the interface against cracking damage and avoid any wear damage. Applying the Hertzian formalism, the resin is considered in a first approximation as purely elastic. The normal indentation loading rate during the test was fixed at 1 Hz (i.e. 75 N/s), whereas the tangential displacement related to the fretting cycle was fixed at 10 Hz thus, to limit any fluctuations induced by the time-dependent behavior of the polymer. The tests were performed in a closed chamber with constant 40–50% relative humidity and variable temperature in the range of 25–220 ◦ C. The required high temperatures were reached using heating elements placed in the lower holder. The temperature was
Fig. 1. (a) Diagram of the experimental setup. (b) Fretting log of one fretting cycle.
measured by means of a thin thermocouple placed on the top and side surfaces of the RTM6 specimen. The rubbing AISI 52100 surface was heated by the lower specimen through thermal conductivity. In order to maintain the required temperature in the contact, thermo-insulating fabric, which limits heat dissipation and favors temperature homogenization, was positioned around the specimens. Prior to testing all specimens were cleaned with ethanol. The radius of the final fretting scar related to the contact radius was measured using optical microscopy. 2.2.2. Dynamic mechanical analysis Dynamic mechanical analysis (DMA) is a convenient and sensitive testing method for the rapid determination of the frequency thermo-mechanical properties of polymers and polymer-based materials. It consists in determining the time-dependent behavior of materials under dynamic periodic sinusoidal strain or stress. In this paper, dynamic mechanical analysis was performed on a (26 mm × 4.1 mm × 1.5 mm) rectangular specimen subjected to tension/compression loading using a DMA50 0.1dB Metravib test system. The tests were carried out under controlled strain in the linear domain of viscoelasticity of the material. From the stress and strain measurements [10] the complex modulus E* = E + jE , where E is the storage (elastic component) modulus, and E is the loss (viscous component) modulus, was found. The loss factor or damping tan ı = E /E where ı is the phase shift between stress and strain can therefore be extrapolated. A temperature ranging from −100 to 250 ◦ C was investigated by applying a heating rate of 1.5 ◦ C/min under nitrogen flow. The applied frequencies were in the same range as for the fretting tests, i.e. 1–10 Hz. Under these conditions the contact response can be assumed to be elastic. 2.2.3. Numerical modeling To verify the semi-infinite hypothesis imposed by the Hertzian model, the contact response of the studied finite thickness condition was modeled using the commercial finite element code
634
S. Terekhina et al. / Wear 269 (2010) 632–637
Fig. 2. Illustration of the axisymmetric sphere/plane model.
ABAQUS [11]. The 2D axisymmetric, elastic, standard model is applied to simulate the indentation test. Details of the sphere/plane model and mesh configuration are described in Fig. 2. The mesh is composed of linear axisymmetric triangular (CAX3) elements. The triangular elements are considered to decrease the analysis cost. The sphere has a radius of 22.22 mm. It is fully constrained except in the vertical direction. The plane, measuring 120 mm × 4 mm in the first case, and 120 mm × 10 mm in the second, is fully constrained. Taking advantage of symmetry, half of the considered sample was modeled, using the defined boundary conditions (Fig. 2). A 75 N normal force was applied and the corresponding pressure field distribution was extracted using a similar procedure to that described in [12,13]. 3. Results and discussion 3.1. Dynamic mechanical analysis In order to analyse the viscoelastic behavior of RTM6 resin, dynamic mechanical analysis (DMA) was performed. Fig. 3 shows the temperature dependence of the storage modulus Er and the loss factor tan ı at 1 and 10 Hz. The drops in the Er evolution and
the peaks of the tan ı value characterize the physical transitions of the polymer. Usually, the transition temperatures are taken at the maximum turndown rate of the Er modulus or at the maximum tan ı peak value. The loss factor spectra exhibit three different molecular relaxations, as reported by other authors in epoxy systems [14]. The lower temperature peak or relaxation ˇ, observed around −60 ◦ C for 10 Hz, is related to the motion of small units of macromolecular chains (hydroxyether groups and diphenylpropane units). This relaxation mode is characterized by high values of the Er modulus. A small drop of the Er value can be noticed between 50 and 100 ◦ C, which is related to the ω relaxation (Tω = 95 ◦ C at 10 Hz). This relaxation may be induced by the presence of water in the polymer and structural relaxations. The main temperature damping of the RTM6, the so-called ␣-transition, occurs at 225 ◦ C for f = 10 Hz loading frequency. It is associated to the glass transition involving long distance molecular motions. This relaxation is characterized by a very sharp drop of the Er storage modulus (about two decades). Moreover, the tan ı evolution shows that above 200 ◦ C the viscous response of the polymer became significant (Fig. 3). This suggests that the Hertzian analysis will be limited above this threshold temperature, so that the polymer could not be considered in a first approximation, as purely elastic. However, the values of the elastic modulus Er extracted from the DMA analysis for the different temperatures investigated will be applied and transposed to the fretting test investigation. 3.2. Identification of mechanical properties through a fretting investigation The present contact analysis involves the following three steps: 1. Determining the contact radius by coupling an indentation followed by tiny fretting loadings to mark the contact area and extract the contact radius (a) through post-mortem optical observation of the fretting scar. 2. Applying a reverse identification approach based on the Hertzian formalism, thus extracting the elastic properties of the polymer from the experimental value of the contact radius (a) (point 1). 3. Comparing the elastic properties extracted from the contact investigation and the value defined from the DMA analysis. According to the elastic Hertzian theory, the contact radius of a sphere/plane interface is expressed by: a=
3PR∗ 1/3 4Ec
(1)
where P is the normal force; R* is the equivalent radius given by: 1 1 1 = + R∗ R1 R2
(2)
where R1 is the plane radius, here infinite, and R2 is the sphere radius. Thus, R* ∼ R2 = R. From Eq. (1) the equivalent elastic modulus of the contact Ec is deduced: Ec =
3PR 4a3
(3)
The Hertzian formalism also permits this variable to be expressed by: 1 − 12 1 − 22 1 = + Ec E1 E2 Fig. 3. Evolution of the elastic modulus Er and the loss factor (frequencies of 1 and 10 Hz) for various temperatures.
(4)
where E1 and E2 are the storage modulus of the epoxy plane and the elastic modulus of the steel sphere respectively, and 1 , 2 are
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635
Fig. 4. Evolution of the contact width for different thicknesses of the resin RTM6.
the Poisson coefficients of the plane (RTM6) and the sphere (52100) respectively (Table 1). Due to industrial considerations, the RTM6 resin specimens are restricted to a thickness of 4 mm. This thickness is relatively small compared to the Hertzian semi-infinite assumption which can invalidate the current analysis. Indeed, it has been shown that the Hertzian formalism can only be applied if the contact radius remains smaller then the specimen thickness (i.e. e/a > 10). To quantify the error induced by the thin 4 mm thickness plates, a FEM analysis comparing the pressure field distributions obtained for e = 4 and e = 10 mm was carried out. Assuming Hertzian formalism, it corresponds to an e/a ratio equal to 5 and 12.5 respectively (a(Hertz) = 0.80 mm for P = 75 N). Fig. 4 compares the contact pressure profiles versus the contact radius for the two mentioned thicknesses. The pressure profiles are quasi superimposed. The finite element analysis leads to a similar contact radius a = 0.79 mm for a 75 N normal load. Comparing this FEM value and the analytical Hertzian approach, a negligible difference of less than 1.25% can be seen (see Fig. 5 and Table 2). Thus, these results confirm that the Hertzian theory can be adopted to characterize 4 mm thickness specimen experimental results. Substituting the equivalent elastic contact modulus Ec of Eq. (4) into Eq. (3), the storage modulus of the plane E1 (i.e. RTM6) is then derived by: E1 =
(1 − 12 ) 4a3 /3P · R − (1 − 22 )/E2
(5)
where P, R, 2 and E2 are known and the contact radius (a) is measured from the post-mortem observation of fretting scars. The Poisson coefficient 1 varies with temperature. To simplify our analysis, it will be taken to be constant at an average value of 0.35 [15]. As mentioned previously, a major difficulty is measure the contact radius (a). This can be indirectly obtained by analyzing the loading and unloading curves of instrumented indentation tests [16–19] or through the in situ observation of the contact area for transparent materials [5]. Theses two methods are, however, limited when high temperatures are applied and non-transparent materials are investigated. The originality of the present method
Fig. 5. Determination of Hertzian contact radius: (a) Hertz description; (b) experiment by using optical microscope (plane-sphere contact, RTM6-AISI 52100, P = 75 N, T = 90 ◦ C, ı* = ±4 m).
Fig. 6. Temperature dependence of Hertzian contact radius (P = 75 N): () experimental data (post-mortem examination of the fretting scar); (—) and (– – –) theoretical prediction (Eq. (1)) computed using the mechanical properties given by the DMA analysis at 1 and 10 Hz respectively.
is that: first, the normal force P is applied, then a small number of fretting cycles less than 1000 are applied to mark the contact area without cracking or damaging the interface (Fig. 5). In order to maintain a partial slip contact and constrain the sliding zone to the tiny borders of the contact area, very small tangential force amplitudes were applied: less than ±4 m. After the test, the contact radius can easily be measured and the elastic properties (E1 ) extrapolated. This methodology is currently applied, fixing the normal force at 75 N, but varying the temperature from 25 to 220 ◦ C. Table 2 compiles the measured contact radius using this indirect fretting test method and the extrapolated E1 values defined from Eq. (5). Fig. 6 shows the evolution of the measured contact radius (a) versus the temperature. The evolution of the contact radius is smooth, quasi linear, until 200 ◦ C. After this threshold temperature
Table 2 Experimental determination of the contact radius a and extrapolation of the storage modulus E 1 assuming 1 = 0.35 in the range of temperature [25–220 ◦ C], P = 75 N. T, ◦ C
25
90
150
175
200
210
215
220
a, mm E1 (RTM6), GPa (Eq. (5))
0.8 2.2
0.85 1.8
0.9 1.52
0.91 1.47
0.98 1.17
1.15 0.72
1.4 0.4
2.1 0.12
636
S. Terekhina et al. / Wear 269 (2010) 632–637
Fig. 7. Evolution of the elastic modulus defined from the contact analysis (i.e. postmortem examination of fretting scars) and the corresponding value defined from DMA analysis (studied temperature range: 25–220 ◦ C): () 1 Hz; (♦) 10 Hz. (—) Theoretical curve related to a perfect correlation.
a sharp increase can be observed. This explains that the contact area is dependent on the viscoelastic response of the material, i.e. on tan (ı) and Er as illustrated by the DMA analysis. Above 200 ◦ C the storage modulus of the RTM6 resin is sharply reduced, which promotes a sharp increase of the contact radius (Fig. 3). These experimental results are compared to the prediction given by the Hertzian expression (Eq. (1)) imputing the storage modulus evolution, obtained by the DMA experiments at 1 and 10 Hz. Excellent agreement is observed. All the experimental contact radius values are bracketed by the model considering the DMA elastic values analysis established at 1 and 10 Hz. Besides, the interval error remains very small: less than 5% below 200 ◦ C and around 20% for the worst estimation at 220 ◦ C. This suggests that the temperature contact radius variation can be predicted by coupling the Hertzian formalism and accurate mechanical analysis by DMA. This conclusion is confirmed in Fig. 7 where Young’s modulus defined from the DMA analysis (Er ), is plotted versus Young’s modulus extrapolated from the contact analysis (E1 ), for the different studied temperatures. A very good correlation is observed, particularly for the lowest temperatures which correspond to the highest modulus values. This confirms the reliability of this methodology as long as the Hertzian formalism can be considered. Nevertheless, above Tth ≈ 200 ◦ C a sharp increase of the viscous response of the material (Fig. 3) induces a decay of the Hertzian estimation of the elastic properties compared to the reference DMA values. This is confirmed by a larger discrepancy observed for the smallest elastic modulus values (Fig. 7). However, for industrial applications the resin RTM6 is never worked in the range of glass transition temperatures and systematically applied at temperatures lower than 200 ◦ C. Therefore, it appears that the given indirect Hertzian contact analysis can be considered as a useful and practical method to extrapolate the elastic properties of the material under ambient but also high temperature conditions. 4. Summary and conclusion A major objective of this work was to propose a new alternative methodology, applied to epoxy RTM6 resin, to measure the elastic modulus as a function of temperature, using an indirect sphere/plane indentation methodology. This consists in applying a given normal force followed by few tiny partial slip fretting cycles to mark the contact area. The post-mortem contact radius
measurement combined with the Hertzian formalism then allows extrapolating of the elastic properties of the contacting material. To validate this original approach, the elastic values obtained have been compared to reference DMA data. A rather good correlation has been observed. The relative error compared to the reference DMA values remains lower than 5% below 200 ◦ C. However, above 200 ◦ C a decay of the Hertzian estimation is observed due to a significant viscous response of the polymer. DMA appears unquestionably as the reference method to extract mechanical properties of polymer materials, as the tension/compression test is for metals. However, these methods require a large quantity of material to make the specimens, which is quite restrictive when a very small quantity of material is available or if a load analysis is required. Hence, alternative indentation methods have been developed to overcome such limitations. The most extensively applied strategy is to perform instrumented indentation and to analyze both loading and unloading curves [6]. This given alternative method is quite similar to the conventional instrumented indentation. However, it displays two main advantages. 1. Using indentation, the contact area is determined through the estimated normal displacement within the contact which is highly affected by indentation apparatus stiffness. The proposed methodology allows a direct estimation of the elastic contact area, thanks to the surface marking induced by fretting displacement. 2. This contact methodology can easily be adopted for high temperatures and very high temperatures because it is not affected by any thermal dilation effects, unlike the conventional instrumented test systems where the depth indentation displacement is highly affected by any thermal gradient. This alternative approach, however, displays some limitations. Like for the instrumented test system, it can be affected by surface roughness effects [20,21]. This implies a systematic polishing of the surface. Besides, unlike the instrumented test method, this approach is currently not adopted to quantify the plastic and viscosity responses of materials. However, it could be interesting to combine the two methods – the instrumented indentation and the given indirect fretting marking approach – to take advantage of each strategy. Acknowledgements The authors are grateful to Hexcel Composites (France), especially to C. Dauphin, for supplying materials. References [1] J. Teng, K. Sato, In situ observations of fretting wear behaviour in PMMA/steel model, Mater. Des. 25 (2004) 471–478. [2] M. Varenberg, G. Halperin, I. Etsion, Different aspects of the role of wear debris in fretting wear, Wear 252 (2002) 902–910. [3] K. Sanjay, Composites Manufacturing: Materials, Product, and Process Engineering, CRC Press, USA, 2002, pp. 158–175. [4] K.-L. Johnson, Contact Mechanics, Cambridge University Press, 2003, p. 454. [5] J.F. Lamethe, P. Sergot, A. Chateauminois, B.J. Briscoe, Contact fatigue behaviour of glassy polymers with improved toughness under fretting wear conditions, Wear 255 (2003) 758–765. [6] W.C. Oliver, G.M. Pharr, An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation measurements, J. Mater. Res. 7 (6) (1992) 1564–1583. [7] J.D. Ferry, Viscoelastic Properties of Polymers, 3rd ed., Wiley, John & Sons, Incorporated, 1980, 668 pp. [8] O. Vingsbo, S. Soderberg, On fretting maps, Wear 126 (1998) 131–147.
S. Terekhina et al. / Wear 269 (2010) 632–637 [9] L. Vincent, Y. Berthier, M. Godet, Testing methods in fretting fatigue: a critical appraisal, ASTM STP 1159 (1992) 23–32. [10] S. Etienne, J.Y. Cavaillé, J. Perez, M. Salvia, Automatic system for analysis of micromechanical properties, Rev. Sci. Instrum. 53 (1982) 1261–1266. [11] ABAQUS Software, User’s Manual, Version 6.6, Karlsson and Sorensen Inc., Hibbitt, 2006. [12] C. Paulin, S. Fouvry, C. Meunier, Finite element modeling of fretting wear surface evolution: application to a Ti–6Al–4V contact, Wear (264) (2008) 26–36. [13] C.T. Tsai, Sh. Mall, Elasto-plastic finite element analysis of fretting stresses in pre-stressed strip in contact with cylindrical pad, Finite Elem. Anal. Des. (36) (2000) 171–187. [14] D. Colombini, J.J. Martinez-Vega, G. Merle, Influence of hydrothermal ageing and thermal treatments on the viscoelastic behaviour of DGEBA–MCDEA epoxy resin, Polym. Bull. (48) (2002) 75–82. [15] R. Younes, Approches et Démarches pour l’Optimisation en Mécanique non linéaire, Rapport d’activités scientifiques, 2007.
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[16] C.C. White, M.R. Vanlandingham, P.L. Drzai, N.-K. Chang, S.-H. Chang, Viscoelastic Characterization of Polymers Using Instrumented Indentation. II. Dynamic Testing. Published online in Wiley InterScience (www.interscience.wiley.com), doi:10.1002/polb.20455. [17] M. Zhao, N. Ogasawara, N. Chiba, X. Chen, A new approach to measure the elastic–plastic properties of bulk materials using spherical indentation, Acta Mater. 54 (2006) 23–32. [18] L.M. Farrissey, P.E. McHugh, Determination of elastic and plastic material properties using indentation: development of method and application to a thin surface coating, Mater. Sci. Eng. A 399 (2005) 254–266. [19] H. Lu, B. Wang, J. Ma, G. Huang, H. Viswanathan, Measurement of creep compliance of solid polymers by nanoindentation, Mech. Time-Dependent Mater. 7 (2003) 189–207. [20] J.A. Greenwood, J.H. Tripp, The elastic contact of rough spheres, ASME J. Appl. Mech. 34 (1967) 153–160. [21] M. Ciavarella, The generalized Cattaneo partial slip plane contact problem. I – Theory, Int. J. Solids Struct. 35 (18) (1998) 2349–2362.
Contents lists available at ScienceDirect
Wear journal homepage: www.elsevier.com/locate/wear
Short communication
An indirect method based on fretting tests to characterize the elastic properties of materials: Application to an epoxy resin RTM6 under variable temperature conditions S. Terekhina a,b , S. Fouvry a,∗ , M. Salvia a,∗∗ , I. Bulanov b a b
Laboratoire de Tribologie et Dynamique des Systèmes, UMR CNRS ECL ENISE ENSMSE 5513, École Centrale de Lyon, 69134 Ecully cedex, France Laboratory of Composite Materials, Bauman Moscow State Technical University, 5, 2nd Baumanskaya Str., 105005 Moscow, Russian Federation
a r t i c l e
i n f o
Article history: Received 30 October 2009 Received in revised form 31 May 2010 Accepted 16 June 2010 Available online 23 June 2010 Keywords: Fretting DMA Epoxy resin Hertzian contact
a b s t r a c t An original indirect method of elastic modulus measurement based on fretting tests is developed here. It consists in applying a normal force, then marking the contact area thanks to a tiny cyclic fretting loading. The elastic modulus of the epoxy resin is extrapolated from the post-mortem contact radius measurement using a Hertzian contact description. Applied to a RTM6 epoxy resin, the elastic properties are extracted from this partial slip fretting analysis and are compared to dynamic mechanical analysis data (DMA). Very good agreement was found between these fretting results and data given by DMA analysis, confirming the potential of this new contact method to extract the mechanical properties of materials. © 2010 Elsevier B.V. All rights reserved.
1. Introduction The industrial use of polymers as friction materials in both homogeneous and heterogeneous forms has widely increased over the past few years. As a result, the wear and friction properties of polymers are now extensively investigated. However, the wear analysis of polymers requires complete knowledge of their mechanical properties. One original approach which is currently being developed in the present research work consists in extracting these variables using a calibrated fretting test. Fretting is a small-amplitude oscillatory motion between contacting surfaces [1]. It may arise in any assembly of engineering components if vibration or cyclic stress is present, inducing localised wear or fatigue cracks. These forms of damage are encountered in all quasi-static loaded assemblies such as keys, cables, bearing races and shafts, orthopedic implants, cranes, turbine blade roots, electrical contacts [2], and also during mold fabrication (for example, by resin transfer molding (RTM) process [3]). The contact theory developed by Hertz for elastic conditions, smooth surfaces and the semi-infinite bulk hypothesis, allows the apparent area to be linked from the contact geometry, the elastic
∗ Corresponding author. Tel.: +33 4 72 18 65 62; fax: +33 4 78 43 33 83. ∗∗ Corresponding author. E-mail addresses: [email protected] (S. Fouvry), [email protected] (M. Salvia). 0043-1648/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.wear.2010.06.008
properties of the contacting bodies and the applied normal force. Hence, explicit expressions of the contact radius and pressure field distributions are provided for axisymmetrical sphere/plane contact or 2D cylinder/plane configuration. Alternatively, using a reverse approach, the elastic properties of the material can be extracted by measuring the contact radius induced by a given normal force [4]. This strategy is extensively applied for ‘transparent’ materials which allow direct access to the contact area [5]. However, for nontransparent materials direct determination of the elastic contact radius is not possible. One alternative strategy, extensively applied by the instrumented indentation test method, consists in analyzing the loading and unloading curves (i.e. evolution of the depth indentation versus the normal force) in order to extract the elastic properties [6]. However, this method requires the exact determination of the indentation depth, which is highly affected by external aspects like test apparatus stiffness or temperature dilation fluctuations. An alternative strategy which is developed here consists in applying tiny fretting cycles (i.e. tangential oscillatory displacements) in order to mark the contact obtained for a given normal force. Then, using post-mortem contact radius measurements, the elastic properties of the material can be extracted. This approach, which is not affected by test stiffness or other temperature fluctuations, is here applied to describe the temperature behavior of polymers. However, in the case of polymer material not only elastic but also viscous properties play important roles in contact deformation,
S. Terekhina et al. / Wear 269 (2010) 632–637
633
Table 1 Mechanical properties of ball-bearing steel (AISI 52100). E2 , GPa
2
Rm , MPa
RpO,2 , MPa
HRC
210
0.3
2200
2000
60–67
especially in the glass transition region. Therefore, to separate the effect of the viscous and elastic response of the fretting contact behavior, a combined dynamic mechanical analysis (DMA) [7] is performed. 2. Experimental procedure 2.1. Materials The fretting tests and dynamic mechanical analysis were carried out using an epoxy resin HexFlow RTM6 (Hexcel, France), already degassed. This resin was specifically developed to fulfill the requirements of the aerospace and space industries in advanced resin transfer molding (RTM) processes. It is a premixed epoxy system of the tetraglycidyl-diamino-diphenyl-methane (TGDDM) type suitable for service temperatures from −60 up to 180 ◦ C. The standard curing cycle for RTM6 resin is 160 ◦ C for 75 min, followed by freestanding post-cure at 180 ◦ C for 120 min (ramp: 1 ◦ C/min). During the fretting tests the resin RTM6 was tested against ballbearing steel (AISI 52100). The mechanical properties of the steel are reported in Table 1. The surface roughness of the AISI 52100 steel ball is very low, around Ra ≈ 0.09 m. The RTM6 resin surface roughness is slightly higher, around Ra ≈ 0.5 m, but very small compared to the studied contact radius (aH > 1 mm). This indirectly justifies the smooth surface hypothesis imposed by the Hertzian theory. 2.2. Experimental techniques 2.2.1. Fretting tests To investigate the elastic properties of the RTM6 resin, a sphere/plane fretting test involving an electrodynamic shaker system has been applied (Fig. 1a). The normal force P is kept constant while varying tangential force (Q) and displacement (ı) are imposed. Thus, the fretting loop Q–ı, as shown in Fig. 1b, can be drawn to extract quantitative variables, including the dissipated energy (Ed ), (i.e. the area of the hysteresis), which is related to the friction work dissipated per fretting cycle. Note that for polymer materials it can be influenced by mechanical loss behavior induced by viscoelastic response. Other variables like the sliding amplitude (ı0 ) (i.e. the remaining displacement when Q = 0), the tangential force amplitude (Q*) and the displacement amplitude (ı*) can also be extracted. The sphere/plane contact consists of a 22.22 mm radius, in contact with a rectangular RTM6 specimen (4.1 mm × 20 mm × 25 mm) using a constant normal load (P = 75 N). The tests were conducted under the partial slip condition for which the fretting cycle displays a very closed elliptical form [8,9]. A constant small displacement amplitude of 4 m was imposed to protect the interface against cracking damage and avoid any wear damage. Applying the Hertzian formalism, the resin is considered in a first approximation as purely elastic. The normal indentation loading rate during the test was fixed at 1 Hz (i.e. 75 N/s), whereas the tangential displacement related to the fretting cycle was fixed at 10 Hz thus, to limit any fluctuations induced by the time-dependent behavior of the polymer. The tests were performed in a closed chamber with constant 40–50% relative humidity and variable temperature in the range of 25–220 ◦ C. The required high temperatures were reached using heating elements placed in the lower holder. The temperature was
Fig. 1. (a) Diagram of the experimental setup. (b) Fretting log of one fretting cycle.
measured by means of a thin thermocouple placed on the top and side surfaces of the RTM6 specimen. The rubbing AISI 52100 surface was heated by the lower specimen through thermal conductivity. In order to maintain the required temperature in the contact, thermo-insulating fabric, which limits heat dissipation and favors temperature homogenization, was positioned around the specimens. Prior to testing all specimens were cleaned with ethanol. The radius of the final fretting scar related to the contact radius was measured using optical microscopy. 2.2.2. Dynamic mechanical analysis Dynamic mechanical analysis (DMA) is a convenient and sensitive testing method for the rapid determination of the frequency thermo-mechanical properties of polymers and polymer-based materials. It consists in determining the time-dependent behavior of materials under dynamic periodic sinusoidal strain or stress. In this paper, dynamic mechanical analysis was performed on a (26 mm × 4.1 mm × 1.5 mm) rectangular specimen subjected to tension/compression loading using a DMA50 0.1dB Metravib test system. The tests were carried out under controlled strain in the linear domain of viscoelasticity of the material. From the stress and strain measurements [10] the complex modulus E* = E + jE , where E is the storage (elastic component) modulus, and E is the loss (viscous component) modulus, was found. The loss factor or damping tan ı = E /E where ı is the phase shift between stress and strain can therefore be extrapolated. A temperature ranging from −100 to 250 ◦ C was investigated by applying a heating rate of 1.5 ◦ C/min under nitrogen flow. The applied frequencies were in the same range as for the fretting tests, i.e. 1–10 Hz. Under these conditions the contact response can be assumed to be elastic. 2.2.3. Numerical modeling To verify the semi-infinite hypothesis imposed by the Hertzian model, the contact response of the studied finite thickness condition was modeled using the commercial finite element code
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Fig. 2. Illustration of the axisymmetric sphere/plane model.
ABAQUS [11]. The 2D axisymmetric, elastic, standard model is applied to simulate the indentation test. Details of the sphere/plane model and mesh configuration are described in Fig. 2. The mesh is composed of linear axisymmetric triangular (CAX3) elements. The triangular elements are considered to decrease the analysis cost. The sphere has a radius of 22.22 mm. It is fully constrained except in the vertical direction. The plane, measuring 120 mm × 4 mm in the first case, and 120 mm × 10 mm in the second, is fully constrained. Taking advantage of symmetry, half of the considered sample was modeled, using the defined boundary conditions (Fig. 2). A 75 N normal force was applied and the corresponding pressure field distribution was extracted using a similar procedure to that described in [12,13]. 3. Results and discussion 3.1. Dynamic mechanical analysis In order to analyse the viscoelastic behavior of RTM6 resin, dynamic mechanical analysis (DMA) was performed. Fig. 3 shows the temperature dependence of the storage modulus Er and the loss factor tan ı at 1 and 10 Hz. The drops in the Er evolution and
the peaks of the tan ı value characterize the physical transitions of the polymer. Usually, the transition temperatures are taken at the maximum turndown rate of the Er modulus or at the maximum tan ı peak value. The loss factor spectra exhibit three different molecular relaxations, as reported by other authors in epoxy systems [14]. The lower temperature peak or relaxation ˇ, observed around −60 ◦ C for 10 Hz, is related to the motion of small units of macromolecular chains (hydroxyether groups and diphenylpropane units). This relaxation mode is characterized by high values of the Er modulus. A small drop of the Er value can be noticed between 50 and 100 ◦ C, which is related to the ω relaxation (Tω = 95 ◦ C at 10 Hz). This relaxation may be induced by the presence of water in the polymer and structural relaxations. The main temperature damping of the RTM6, the so-called ␣-transition, occurs at 225 ◦ C for f = 10 Hz loading frequency. It is associated to the glass transition involving long distance molecular motions. This relaxation is characterized by a very sharp drop of the Er storage modulus (about two decades). Moreover, the tan ı evolution shows that above 200 ◦ C the viscous response of the polymer became significant (Fig. 3). This suggests that the Hertzian analysis will be limited above this threshold temperature, so that the polymer could not be considered in a first approximation, as purely elastic. However, the values of the elastic modulus Er extracted from the DMA analysis for the different temperatures investigated will be applied and transposed to the fretting test investigation. 3.2. Identification of mechanical properties through a fretting investigation The present contact analysis involves the following three steps: 1. Determining the contact radius by coupling an indentation followed by tiny fretting loadings to mark the contact area and extract the contact radius (a) through post-mortem optical observation of the fretting scar. 2. Applying a reverse identification approach based on the Hertzian formalism, thus extracting the elastic properties of the polymer from the experimental value of the contact radius (a) (point 1). 3. Comparing the elastic properties extracted from the contact investigation and the value defined from the DMA analysis. According to the elastic Hertzian theory, the contact radius of a sphere/plane interface is expressed by: a=
3PR∗ 1/3 4Ec
(1)
where P is the normal force; R* is the equivalent radius given by: 1 1 1 = + R∗ R1 R2
(2)
where R1 is the plane radius, here infinite, and R2 is the sphere radius. Thus, R* ∼ R2 = R. From Eq. (1) the equivalent elastic modulus of the contact Ec is deduced: Ec =
3PR 4a3
(3)
The Hertzian formalism also permits this variable to be expressed by: 1 − 12 1 − 22 1 = + Ec E1 E2 Fig. 3. Evolution of the elastic modulus Er and the loss factor (frequencies of 1 and 10 Hz) for various temperatures.
(4)
where E1 and E2 are the storage modulus of the epoxy plane and the elastic modulus of the steel sphere respectively, and 1 , 2 are
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Fig. 4. Evolution of the contact width for different thicknesses of the resin RTM6.
the Poisson coefficients of the plane (RTM6) and the sphere (52100) respectively (Table 1). Due to industrial considerations, the RTM6 resin specimens are restricted to a thickness of 4 mm. This thickness is relatively small compared to the Hertzian semi-infinite assumption which can invalidate the current analysis. Indeed, it has been shown that the Hertzian formalism can only be applied if the contact radius remains smaller then the specimen thickness (i.e. e/a > 10). To quantify the error induced by the thin 4 mm thickness plates, a FEM analysis comparing the pressure field distributions obtained for e = 4 and e = 10 mm was carried out. Assuming Hertzian formalism, it corresponds to an e/a ratio equal to 5 and 12.5 respectively (a(Hertz) = 0.80 mm for P = 75 N). Fig. 4 compares the contact pressure profiles versus the contact radius for the two mentioned thicknesses. The pressure profiles are quasi superimposed. The finite element analysis leads to a similar contact radius a = 0.79 mm for a 75 N normal load. Comparing this FEM value and the analytical Hertzian approach, a negligible difference of less than 1.25% can be seen (see Fig. 5 and Table 2). Thus, these results confirm that the Hertzian theory can be adopted to characterize 4 mm thickness specimen experimental results. Substituting the equivalent elastic contact modulus Ec of Eq. (4) into Eq. (3), the storage modulus of the plane E1 (i.e. RTM6) is then derived by: E1 =
(1 − 12 ) 4a3 /3P · R − (1 − 22 )/E2
(5)
where P, R, 2 and E2 are known and the contact radius (a) is measured from the post-mortem observation of fretting scars. The Poisson coefficient 1 varies with temperature. To simplify our analysis, it will be taken to be constant at an average value of 0.35 [15]. As mentioned previously, a major difficulty is measure the contact radius (a). This can be indirectly obtained by analyzing the loading and unloading curves of instrumented indentation tests [16–19] or through the in situ observation of the contact area for transparent materials [5]. Theses two methods are, however, limited when high temperatures are applied and non-transparent materials are investigated. The originality of the present method
Fig. 5. Determination of Hertzian contact radius: (a) Hertz description; (b) experiment by using optical microscope (plane-sphere contact, RTM6-AISI 52100, P = 75 N, T = 90 ◦ C, ı* = ±4 m).
Fig. 6. Temperature dependence of Hertzian contact radius (P = 75 N): () experimental data (post-mortem examination of the fretting scar); (—) and (– – –) theoretical prediction (Eq. (1)) computed using the mechanical properties given by the DMA analysis at 1 and 10 Hz respectively.
is that: first, the normal force P is applied, then a small number of fretting cycles less than 1000 are applied to mark the contact area without cracking or damaging the interface (Fig. 5). In order to maintain a partial slip contact and constrain the sliding zone to the tiny borders of the contact area, very small tangential force amplitudes were applied: less than ±4 m. After the test, the contact radius can easily be measured and the elastic properties (E1 ) extrapolated. This methodology is currently applied, fixing the normal force at 75 N, but varying the temperature from 25 to 220 ◦ C. Table 2 compiles the measured contact radius using this indirect fretting test method and the extrapolated E1 values defined from Eq. (5). Fig. 6 shows the evolution of the measured contact radius (a) versus the temperature. The evolution of the contact radius is smooth, quasi linear, until 200 ◦ C. After this threshold temperature
Table 2 Experimental determination of the contact radius a and extrapolation of the storage modulus E 1 assuming 1 = 0.35 in the range of temperature [25–220 ◦ C], P = 75 N. T, ◦ C
25
90
150
175
200
210
215
220
a, mm E1 (RTM6), GPa (Eq. (5))
0.8 2.2
0.85 1.8
0.9 1.52
0.91 1.47
0.98 1.17
1.15 0.72
1.4 0.4
2.1 0.12
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Fig. 7. Evolution of the elastic modulus defined from the contact analysis (i.e. postmortem examination of fretting scars) and the corresponding value defined from DMA analysis (studied temperature range: 25–220 ◦ C): () 1 Hz; (♦) 10 Hz. (—) Theoretical curve related to a perfect correlation.
a sharp increase can be observed. This explains that the contact area is dependent on the viscoelastic response of the material, i.e. on tan (ı) and Er as illustrated by the DMA analysis. Above 200 ◦ C the storage modulus of the RTM6 resin is sharply reduced, which promotes a sharp increase of the contact radius (Fig. 3). These experimental results are compared to the prediction given by the Hertzian expression (Eq. (1)) imputing the storage modulus evolution, obtained by the DMA experiments at 1 and 10 Hz. Excellent agreement is observed. All the experimental contact radius values are bracketed by the model considering the DMA elastic values analysis established at 1 and 10 Hz. Besides, the interval error remains very small: less than 5% below 200 ◦ C and around 20% for the worst estimation at 220 ◦ C. This suggests that the temperature contact radius variation can be predicted by coupling the Hertzian formalism and accurate mechanical analysis by DMA. This conclusion is confirmed in Fig. 7 where Young’s modulus defined from the DMA analysis (Er ), is plotted versus Young’s modulus extrapolated from the contact analysis (E1 ), for the different studied temperatures. A very good correlation is observed, particularly for the lowest temperatures which correspond to the highest modulus values. This confirms the reliability of this methodology as long as the Hertzian formalism can be considered. Nevertheless, above Tth ≈ 200 ◦ C a sharp increase of the viscous response of the material (Fig. 3) induces a decay of the Hertzian estimation of the elastic properties compared to the reference DMA values. This is confirmed by a larger discrepancy observed for the smallest elastic modulus values (Fig. 7). However, for industrial applications the resin RTM6 is never worked in the range of glass transition temperatures and systematically applied at temperatures lower than 200 ◦ C. Therefore, it appears that the given indirect Hertzian contact analysis can be considered as a useful and practical method to extrapolate the elastic properties of the material under ambient but also high temperature conditions. 4. Summary and conclusion A major objective of this work was to propose a new alternative methodology, applied to epoxy RTM6 resin, to measure the elastic modulus as a function of temperature, using an indirect sphere/plane indentation methodology. This consists in applying a given normal force followed by few tiny partial slip fretting cycles to mark the contact area. The post-mortem contact radius
measurement combined with the Hertzian formalism then allows extrapolating of the elastic properties of the contacting material. To validate this original approach, the elastic values obtained have been compared to reference DMA data. A rather good correlation has been observed. The relative error compared to the reference DMA values remains lower than 5% below 200 ◦ C. However, above 200 ◦ C a decay of the Hertzian estimation is observed due to a significant viscous response of the polymer. DMA appears unquestionably as the reference method to extract mechanical properties of polymer materials, as the tension/compression test is for metals. However, these methods require a large quantity of material to make the specimens, which is quite restrictive when a very small quantity of material is available or if a load analysis is required. Hence, alternative indentation methods have been developed to overcome such limitations. The most extensively applied strategy is to perform instrumented indentation and to analyze both loading and unloading curves [6]. This given alternative method is quite similar to the conventional instrumented indentation. However, it displays two main advantages. 1. Using indentation, the contact area is determined through the estimated normal displacement within the contact which is highly affected by indentation apparatus stiffness. The proposed methodology allows a direct estimation of the elastic contact area, thanks to the surface marking induced by fretting displacement. 2. This contact methodology can easily be adopted for high temperatures and very high temperatures because it is not affected by any thermal dilation effects, unlike the conventional instrumented test systems where the depth indentation displacement is highly affected by any thermal gradient. This alternative approach, however, displays some limitations. Like for the instrumented test system, it can be affected by surface roughness effects [20,21]. This implies a systematic polishing of the surface. Besides, unlike the instrumented test method, this approach is currently not adopted to quantify the plastic and viscosity responses of materials. However, it could be interesting to combine the two methods – the instrumented indentation and the given indirect fretting marking approach – to take advantage of each strategy. Acknowledgements The authors are grateful to Hexcel Composites (France), especially to C. Dauphin, for supplying materials. References [1] J. Teng, K. Sato, In situ observations of fretting wear behaviour in PMMA/steel model, Mater. Des. 25 (2004) 471–478. [2] M. Varenberg, G. Halperin, I. Etsion, Different aspects of the role of wear debris in fretting wear, Wear 252 (2002) 902–910. [3] K. Sanjay, Composites Manufacturing: Materials, Product, and Process Engineering, CRC Press, USA, 2002, pp. 158–175. [4] K.-L. Johnson, Contact Mechanics, Cambridge University Press, 2003, p. 454. [5] J.F. Lamethe, P. Sergot, A. Chateauminois, B.J. Briscoe, Contact fatigue behaviour of glassy polymers with improved toughness under fretting wear conditions, Wear 255 (2003) 758–765. [6] W.C. Oliver, G.M. Pharr, An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation measurements, J. Mater. Res. 7 (6) (1992) 1564–1583. [7] J.D. Ferry, Viscoelastic Properties of Polymers, 3rd ed., Wiley, John & Sons, Incorporated, 1980, 668 pp. [8] O. Vingsbo, S. Soderberg, On fretting maps, Wear 126 (1998) 131–147.
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