MICROMECHANICS OF COMPOSITES WITH INTEGRATED SHEAR THICKENING FLUIDS R. Cristian Neagu, Christian Fischer, Pierre-Etienne Bourban, and Jan-Anders E. Månson Laboratoire de Technologie des Composites et Polymères (LTC) Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland cristian.neagu@epl.ch ABSTRACT In the current paper, a micromechanical model was used to predict the damping o a composite material containing shear thickening luids (STFs) at the ibre-matrix interaces. Predictions o the model and dynamic mechanical analysis results are in concert. The damping o the composites was improved signiicantly. The dynamic properties exhibited a strong dependence on both requency and applied external load amplitude. Damping peaks appeared which coincided with the thickening o the STF at the matrix-rod interace. The location o the peaks depends on the onset o thickening and post-thickening rheological behaviour o the STF. This work shows that a micromechanics approach can be useul or an appropriate choice o microstructural design and properties o STFs in order to control the stiness and damping behaviour o composites. 1. INTRODUCTION 1.1 Damping in composite materials Damping is an important parameter or vibration control, noise reduction, atigue endurance or impact resistance and has become increasingly important over the last ew decades in a wide range o ields, including aeronautics, automotives and sport. Fibre-reinorced composites possess damping mechanisms very dierent rom those in conventional metals and alloys. At the micromechanical level, the energy dissipation is induced by dierent mechanisms such as the viscoelastic nature o the matrix, the damping at the ibre-matrix interace/interphase or the damping due to damage [1]. At the laminate level, damping is strongly dependent on the layer constituent properties as well as layer orientations, interlaminar eects, and stacking sequence [2]. Passive methods to enhance damping include surace damping treatments and constrained layer damping. Active methods involve integration o adaptive materials into composite structures to better control vibration and stiness. Examples are laminates containing pre-deormed shape memory alloy wires, electro-rheological and/or magneto-rheological luids which provide means o modiying damping properties via external stimuli [2]. These approaches have in common that they require an external power source to be activated. A way to avoid the need or an external power source is to use a material that changes its properties according to the loading conditions. Such behaviour is characteristic o shear-thickening luids (STFs). Fischer et al. [3] demonstrated that STFs can be integrated in dynamically loaded structures to tailor their damping and stiness properties. In vibrating beam tests on model sandwich structures containing layers o STF, the vibration properties were signiicantly modiied, due to simultaneous increase in stiness and damping. Thereore, it would be desirable to investigate the potential o integrating STFs at the microscale o polymer composites to create new materials with a unique unctionality, i.e. with load-controlled adaptive dynamic stiness-damping properties. A possible application are alpine skis where STFs can be used to damp low requencies at high amplitude deormation known to deteriorate the ski-snow contact.
1.2 Properties o shear thickening luids STFs are typically composed o a concentrated stabilized dispersion o rigid sub-micron particles in a carrier luid and show a marked increase in viscosity beyond a critical shear rate [4, 5]. The shear-thickening eect is due to the build-up o large hydroclusters when the critical shear rate is reached, provoking the dramatic increase in the suspension s viscosity [4]. Beyond the end-point o the transition the viscosity may become approximately independent o or may decrease with increasing shear rate. Due to the reversibility o the phenomenon, the viscosity instantaneously goes back to initial values when releasing the shear stress. STFs have very interesting dynamic properties. Lee and Wagner [4] demonstrated rom stress strain Lissajous plots that a dramatic increase in viscous dissipation occurs at the shear-thickening transition. Fischer et al. [5] showed that the pre- and post-transition viscosity o a colloidal suspensions o used silica in polypropylene glycol subjected to oscillating shear is independent o the applied strain amplitude in requency sweeps. Frequency sweeps corresponding to dierent set strain amplitudes showed a unique power-law decrease in the viscosity with requency, both below and above the shearthickening transition. Similar results were obtained or monodispersed silica particles in polyethylene glycol [6]. It was also shown that the critical shear strain or thickening can be estimated using the shear stress rom steady state measurements and the complex viscosity over the entire investigated requency range. 1.3 Scope and aim In this work, a micromechanics approach is used to study the eect o a STF coating on the damping o the composite. A micromechanical model developed or unidirectional composites (UD) composites with Newtonian viscous interaces [7] is adapted to UD composites with STF at the interace between the ibre and matrix. The novelty lies in accounting or the particular nonlinear rheological properties o the viscous interace, i.e. o the STF, which depend both on the interacial shear strain and the vibration requency. Furthermore, the model is validated with experiments. 2. MICROMECHANICS A simple analytical micromechanical model is developed here. The model can be used to predict the viscoelastic properties o composites with STF coated ibres at dierent vibration requencies. The modelling approach is based on the composite cylinder assemblage (CCA) model o Hashin [8, 9], which was modiied by He and Liu [7] to composites with Newtonian viscous interaces. The model is used to estimate the average steady-state response o the composite with a viscous interace with viscosity changing depending on the interacial shear strain and vibration requency (i.e. STF). The analysis is constrained to UD composites under longitudinal shear deormation. The ibres (usually the sti phase) are considered elastic, while viscoelastic properties are assumed or the matrix. The viscoelastic nature o the matrix cannot be ignored or many matrix materials e.g. elastomers, where the ratio between the storage modulus, G, and the loss modulus, G, can be close to one. The CCA model, schematically depicted in Figure 1a, assumes a spatial arrangement o randomly distributed composite cylinders o dierent cross-sections, with diameters ranging rom inite to ininitesimal size. Each o the composite cylinders is made o a long inner circular ibre surrounded by an outer concentric matrix shell. The ratio o the diameter o the ibre and the matrix shell is constant and related to the ibre volume
raction o the composite. It can be shown that all but the transverse eective properties o the CCA are those o one composite cylinder [9]. Hence, the overall response o the composite can be considered as the same as that o a composite cylinder shown in Figure 1b, consisting o a ibre surrounded by a matrix layer, with an outer radius b = a (V ) 1/2 where, a is the ibre radius and V the ibre volume raction. Figure 1: (a) Fibre composite subjected to a periodic longitudinal shear load, and (b) composite cylinder model with a viscous ibre-matrix interace with shear-thickening properties. The interace between the ibre and the matrix is assumed to consist o a thin viscous STF layer with negligible thickness h. As a result, the ibre and the matrix slip with respect to each other when a shear orce is applied, ollowing the law * τ h δi = (1) η where δi is the rate o sliding o the matrix relative to the ibre, i.e. the dot denoting derivative with respect on time, τ * is the shear stress on the interace, and η the viscosity o the STF. The overall mechanical response o the composite cylinder can be determined in terms o the volume averaged stress and strain. A periodic longitudinal shear load T 3 = τ 0 e iωt n 2 is applied to the composite (Figure 1a), where n 2 is the x 2 component o the outward unit normal to the surace o the composites, i is the square root o the negative unit, t is time, τ 0 and ω are the amplitude and the angular requency o the external load, respectively. This stress state transormed to cylindrical coordinates (r,θ,z) leads to one nonzero stress component on the boundary o the composite cylinder (Figure 1b), i.e. τ rz = τ 0 e iωt sinθ, where θ is the angle between directions x 1 and r. Accordingly, the composite cylinder is in a state o anti-plane deormation with the displacement component, u z = u z (r,θ;t). The solution or the stress and displacement ield can be obtained by considering the equilibrium equations, and the ibre and matrix constitutive relations in He and Liu [7]. The boundary conditions at the ibre and matrix interace needed to solve the problem include continuity o the stress component τ rz m τ ( r = a) = τ ( r = a) (2) rz rz where, superscript m and denote matrix and ibre, respectively. The displacement at the interace, however, sustains a discontinuity δ i m δ i = uz ( r = a) uz ( r = a) (3) The solution or the steady state is desired and thereore it is assumed that the solutions to the displacement ields can be written I i t u r, θ = w ( r) e ω sinθ (4) z ( ) I
where I = m, and wi ( r) is an unknown unction o r. Similarly, the solution to the displacement discontinuity at the interace can be assumed to be i t δ ( θ) = δ e ω sinθ (5) i amp where δ amp is the maximal value o interacial sliding. This remains to be determined together with wi ( r) in Equation (4) using the boundary conditions, Equations (2)-(3) and τ rz = τ 0 e iωt sinθ at r = b, and the equilibrium and constitutive relations given in He and Liu [7]. Once the displacement ield is known, the shear stress at the interace can then be computed rom * * G δ m amp 2τ 0 iωt τ = (1 * * V ) e sin * θ (6) 1 + α + (1 α ) V a Gm where α * = G * / G, G * m is the elastic shear modulus o the ibres and G m is the * complex shear modulus o the matrix. Thus, ' ' Gm = Gm (1 + itan δ m), where G m is the storage modulus and tan δ m the loss actor o the matrix. The shear strain in the STF layer is assumed to be uniorm, i.e. γ = δ amp /h. The assumption that the shear strain is uniorm around the ibre (i.e. no θ dependence and STF is in the same state around the ibre) aects the viscosity, and in a more accurate model, this has to be taken into consideration. The shear strain can be obtained by combining Equation (1) with Equations (5)-(6) as 2τ 0a 1 iω x γ = (7) * 2 h(1 V) Gm 1 + ω x 2 where x is a parameter given by * * 1 + α + (1 α ) Va x = η (8) * (1 V ) Gm h It is known that the STF viscosity depends both on γ and ω, and that η can vary over several orders o magnitude as the strain-requency combination is changed [5]. Nevertheless, or a given γ, the linear relation in Equation (1) can be applied and η may be considered a unction o ω. STFs show a unique strain independent power-law decrease in the viscosity as unction o requency, both below and above the shearthickening transition which occurs at a critical shear strain γ c [5, 6]. It is thereore assumed that the STF viscosity η as unction o the requency, i.e. = ω / 2π, can be described by kl ηl η ( ) = (9) kh ηh below and above the transition state, respectively. At the onset o thickening when the shear strain has reached the critical value corresponding to a critical requency, c, the viscosity changes abruptly and is assumed to ollow a relation k ( ) l 1 η( ) = η, ηη kl 1 ktr 1 l c c l h c 1 ( ktr kh ) i.e. until the viscosity reaches the post-transition value given in Equation (9). The coeicients, η l, η h, k l, k h and k tr to describe the unctions in Equation (9)-(10) and γ c () can be obtained rom oscillatory rheology measurements at dierent shear strain amplitudes as in Fischer et al. [5]. Finally to obtain the dynamic eective shear modulus o the composite, the periodic average stress τ 32 and average strain γ 32 are computed using the relations or the stress and displacement ield in the composite cylinder [7]. The complex eective shear modulus o the composite is then given by G * = τ / γ = G ' ( 1+ itanδc), where c 32 32 c (10)
' G c and tanδ c, are is the storage modulus and the damping o the composite, respectively. Analytical expressions are obtained, but or the case o a viscoelastic matrix, the expressions become large and diicult to comprehend. However, when the matrix and ibre both are elastic, simple analytical expressions are obtained as shown here or the damping o the composite tanδ = 4Vω x c 2 2 4V + 1+ 2 V + V (1 ) (1+ω ( α α ) x 2 ) (11) where α = G / G, and G m m is the elastic shear modulus o the matrix. Equation (11) reveals that the eective damping o a composite with a viscous interace depends on several actors such as V, ω, α, a/h, η, and G m. These parameters together with the amplitude o the applied external orce F a are needed as input to the modelling. 3. EXPERIMENTS Dynamic mechanical analysis (DMA) was carried out to experimentally validate the model s predictive capability. Composites containing STF at the interace between its two constituent phases were designed particularly or this purpose. The STF was composed o spherical monodisperse particles (KE-P50, Nippon Shokubai Co.) with an average particle size o 500 nm. The carrier luid was polyethylene glycol (PEG) (H(OCH 2 CH 2 ) n OH) with an average molar mass o 200 g/mol (Sigma-Aldrich). Ater adding the carrier luid to the dry powder, the suspension was roll-mixed over night, beore putting it in a vacuum chamber or several hours to eliminate air bubbles. The solids concentration was set to 67.75 %w/w. The rheological properties o the STF, i.e. η() in Equation (9)-(10), were determined with rheological measurements perormed on a strain-controlled Advanced Rheometric Expansion System (ARES) rheometer (Rheometric Scientiic). Dynamic requency sweeps were carried out using 25 mm diameter parallel plates with a gap o 400 µm. Composite samples (10 7 3 mm), shown in Figure 2a, were made o a silicone matrix (Elastosil M4470, Wacker Inc.) and glass ibre-epoxy rods (Swiss Composites). The sti rods were use as reinorcing phase since there exists no common way to (i) coat conventional reinorcing ibres with STFs, and (ii) control interaces o the order o magnitude o the ibre radius. Silicone samples with holes were moulded and then illed with STF. Ater the glass ibre-epoxy rods were gently introduced into the STF illed holes, rom one side o the sample while the other side was sealed in order to guarantee luid at the interace between the silicone and the rods. The compliant matrix was chosen to be able to induce signiicant shear deormations at the interace and obtain thickening under the used testing conditions. For an applied shear stress, the shear strain in the STF layer, given in Equation (7), depends largely on the ratio o the STF layer thickness by the ibre radius, a/h. Considering this and that the model used here is valid only or an interace thickness negligible compared to the ibre radius, i.e. Equation (1), rods with a radius o a = 1 mm were chosen and the STF layer thickness was approximately h = 50 μm. The obtained volume raction o the ibres was V = 0.25.
(a) Figure 2: (a) Composite samples made o a silicone matrix and glass ibre-epoxy rods coated with STF. (b) Shear clamp used in dynamic mechanical testing. The dynamic behaviour o both the matrix material and the composite samples was experimentally determined. Dynamic mechanical data were obtained with a DMA Q800 (TA Instruments) using the shear sandwich mode at various requencies,, and orce amplitudes, F a. The shear sandwich clamp o the DMA is shown in Figure 2b. The shear stress amplitude τ 0 (Figure 1a) relates to the external applied orce amplitude, F a, as τ 0 = F a /2A, where A is the area o the longitudinal section o the sample. Both requency sweeps at constant amplitude and stress sweeps at constant requency were made or a range o requency = 0.1-100 Hz and or orce amplitudes F a = 0.1, 0.3, 0.6, 0.9, 1, 1.1, 1.2, 1.5 N. 4. EXPERIMENTAL RESULTS 4.1 Rheology o shear thickening luids Figure 3a shows the dynamic viscosity o the STF, η(), at dierent strain amplitudes. The results are consistent with other experiments on the same material system [6]. The pre- and post-transition viscosity, shown in Figure 3b and Figure 3c, respectively, is as expected independent o the applied strain amplitude in requency sweeps. A unique power-law decrease in the viscosity with requency is seen both below and above the shear-thickening transition (Figure 3a-c). Figure 3b-c show the corresponding it rom which the coeicients in Equation (9) were determined. It should be mentioned that the STFs showed a brittle post-transition behaviour, oten leading to racture o the specimens. Because the viscosities in some cases were out o the range o a conventional rheometer ater thickening, the post-transition behaviour at high shear strain amplitudes (Figure 3a and Figure 3c) could not be properly investigated. The transition behaviour seen in Figure 3d can be considered to ollow a similar trend line irrespective o the shear strain at which the thickening occurs, i.e. in log-log scale the curves are almost parallel with a positive slope. The power law exponent k tr in Equation (10), i.e. the slope o the transition lines in Figure 3a, was determined by itting to each transition line. An average value o k tr = 2.2 was obtained. The coeicients that describe the pre- and post-transition behaviour are shown in Figure 3bc. In Figure 3d the critical shear strain or thickening as unction o the requency is shown. The values, taken rom Fischer [6], are based on numerous measurements on a STF having the same characteristics as the one used in this work. A linear relation in log-log scale is obtained and the corresponding it shown in Figure 3d was used to determine the critical transition requency c or any given strain amplitude. (b)
(a) (b) (c) (d) Figure 3: (a) The dynamic viscosity o a STF o 67.75 %w/w silica particles in PEG. (b) Pre-transition and (c) post-transition behaviour. (d) The critical shear strain or thickening. 4.2 Dynamic mechanical analysis The dynamic properties o the matrix material were determined irst. The silicone matrix was tested at F a = 0.5, 1.0 N. It was seen that the shear storage modulus G m and the shear damping ratio tanδ c did not vary signiicantly with and F a. A mean value o * Gm = 0.37(1 + i0.04) MPa was obtained rom these experiments. It should be mentioned that results o the DMA testing were strongly dependent on clamping conditions. Two extreme clamping conditions, i.e. the minimum required and a very high clamping orce, were tested or the neat silicone matrix. The average storage modulus over a requency range rom 0.1 to 100 Hz, increased with as much as 40% at higher clamping orce testing conditions. The dierence in the damping ratio was lower, however the average value o over the requency range decreased almost by 20% when a high clamping orce was applied. Figure 4a-b shows DMA results o the composite with STF at the matrix-rod interace. The straight horizontal lines correspond to the values o the neat silicone matrix. It can be seen in Figure 4a-b that the composite dynamic properties exhibited a strong dependence on both F a and, in particular in the low requency range rom 0 to 20 Hz. The peaks observed at low or the high F a, in Figure 4b, suggested that the STF had thickened and thus signiicantly aected the composites overall damping, increasing tanδ c compared to the results obtained at low F a. The viscosity o the STF (Figure 3a) increases with the shear strain amplitude. When F a is increased, the shear strain in the STF layer, c. Equation (7), also increases which translates into a viscosity increase. The remarkable increase in damping ratio indicates that the STF is in transition, or posttransition, state where it is know that a great amount o viscous dissipation takes place [4].
(a) (b) Figure 4: Experimentally determined dynamic properties o the composite containing STF in terms o the (a) storage modulus, and (b) loss tangent. A closer look at Figure 4a-b reveals and an increase in G c and a decrease in tanδ c with at constant F a. At constant, a decrease in G c and an increase in tanδ c with increasing F a, was observed. An increase in G c with requency at constant F a could be due to thickening o the STF which improves the stress transer between the rods and the matrix. However, the increase is unrealistically high, especially at low F a, where the composite samples become stier than the neat matrix samples (Figure 4a) which would be the case i the rod-matrix bonding was close to perect. Moreover, the trend o a decrease in G c with increasing F a, is not consistent with the rheological properties o the STF i.e. viscosity increase with F a at constant requency due to larger shear strain amplitude at the interace. The values o G c, especially or low F a, cannot be considered reliable. Possible sources o error induced by the experimental setup, i.e. clamping device, are among others (i) non-uniorm interace thickness, due to sample deormation, and (ii) riction, due to direct contact between the matrix and the rods created by sample deormation which reduces the STF layer thickness to zero where maximum shear strain occurs (Figure 2b). Moreover boundary eects, due to the particular coniguration o the sample (Figure 2a), could lead to complex stress states and unexpected results. 5. COMPARISON BETWEEN MODEL PREDICTIONS AND EXPERIMENTS The model was run with values o F a corresponding to the experimental ones over a requency rage rom 0.01 to 100 Hz. Simulations were perormed with the input data given in Table 1. Table 1. Input data used or modelling. Microstructure Constituent properties a h V G G * m η() (mm) (µm) (%) (GPa) (MPa) η l k l η h k h k tr 1 50 25 30 0.37(1+i0.04) 2.8 0.64 1600 1.5 2.2 The predicted G c and tanδ c or silicone based STF composites is shown in Figure 5a-b. For tanδ c good qualitative and quantitative agreement with experimental results was obtained (c. Figure 4b and Figure 5b). The predicted G c is as expected lower than the one o the matrix material due to imperect interace. A slight increase in G c with at constant F a is seen in the low requency range. As opposed to the experimental results,
predicted G c increased with F a at constant requency due to increase in shear strain at the interace, and hence STF viscosity increase. It was mentioned previously that experimental values o G c at low F a cannot be considered reliable. Predicted G c values are comparable to the experimental ones only at high orce amplitudes, e.g. F a = 1.5 N. (a) (b) Figure 5: Predicted dynamic properties o the composite containing STF in terms o the (a) storage modulus, and (b) loss tangent. Figure 6a shows the predicted and the experimentally determined tanδ c or three dierent F a. The peaks in the model, or the higher F a correspond to thickening o the STF (Figure 3a). Thereore, the model conirmed that the damping peaks coincided with the thickening o the STF at the matrix-rod interace. Since the thickening onset o STFs occurs at lower as shear strain is increased (Figure 3d), the damping peak was moved to lower requencies with increasing F a (Figure 5b and Figure 6a). Furthermore, the act that the post-transition viscosity signiicantly decreases with increasing (Figure 3c) clearly aected the damping behaviour o the silicone matrix composite, which ollowed a similar behaviour. (a) (b) Figure 6: (a) Loss actor vs. requency as determined by DMA experiments compared with model predictions. (b) Stiness-loss map or the composite containing STF or = 0.1-100 Hz and F a = 1.5 N. It can be concluded that damping peaks appear and their location depends on the onset o thickening and post-thickening behaviour. With the behaviour o the STF used in this particular case, the maximum increase in damping will occur at low requencies or high orce amplitudes. Once the STF is thickened at low requencies, the viscosity changes with higher requencies leading to a decrease in damping as can be seen in Figure 6a. Hence the increase in damping due to thickening o STF at higher requencies will not be so dramatic, and it will ollow the corresponding viscosity change in the transition state o the STF rom a lower bound (e.g. at low orce
amplitude o 0.1 N in Figure 6a) to an upper bound given by the behaviour in the thickened state (e.g. curves or high orce amplitude in Figure 6a). The simulations show that the eect o a viscous interace on the damping o composites is considerable. Figure 6b shows a normalized stiness-loss map over the tested requency range and F a = 1.5 N. The great improve in damping is striking, however the stiness is decreased. Nevertheless, the stiness-damping behaviour can be tailored by properly adjusting the interace viscosity, i.e. η(), the ratio a/h, the ibre volume raction and the matrix stiness, independently or jointly, or a certain vibration requency. 6. CONCLUSIONS A micromechanical model developed or composite materials with a STF at the ibrematrix interace showed good agreement with experimental DMA results. The damping o the composites was improved signiicantly. The dynamic properties exhibited a strong dependence on both requency and applied external load amplitude. Damping peaks appeared which coincided with the thickening o the STF at the matrix-rod interace. The location o the peaks depends on the onset o thickening and postthickening rheological behaviour o the STF. Composite materials with STF integrated at the ibre matrix interace oer the possibility o tailoring the stiness-damping behaviour or a certain vibration requency by properly adjusting the STF viscosity, the constituent mechanical properties and the microstructure, independently or jointly. ACKNOWLEDGEMENTS The authors grateully acknowledge the Sports and Rehabilitation Engineering (SRE) program o the EPFL or inancial support, Jérôme Durivault or help to validate the model and Sebastien Lavanchy or technical support. REFERENCES 1- Chandra, R., Singh, S. P., and Gupta, K., Damping studies in iber-reinorced composites - a review, Composite Structures, 1999;46:41-51. 2- Finegan, I. C., and Gibson, R. F., Recent research on enhancement o damping in polymer composites, Composite Structures, 1999;44:89-98. 3- Fischer, C., Braun, S. A., Bourban, P.-E., Michaud, V., Plummer, C. J. G., and Månson, J.-A. E., Dynamic properties o sandwich structures with integrated shear-thickening luids, Smart Materials and Structures 2006;15:1467-1475. 4- Lee, Y. S., and Wagner, N. J., Dynamic properties o shear thickening colloidal suspensions, Rheologica Acta, 2003;42:199-208. 5- Fischer, C., Plummer, C. J. G., Michaud, V., Bourban, P.-E., and Månson, J.-A. E., Pre- and posttransition behaviour o shear thickening luids in oscillating shear, Rheologica Acta, 2007;46:1099-1108. 6- Fischer, C., 2007. Adaptive composite structures or tailored human-materials interaction., Thèse EPFL, no 3890. Dir.: Jan-Anders E. Månson, Pierre-Etienne Bourban. 7- He, L. H., and Liu, Y. L., Damping behavior o ibrous composites with viscous interace under longitudinal shear loads, Composites Science and Technology, 2005;65:855-860. 8- Hashin, Z., Analysis o composite materials - A survey, Journal o Applied Mechanics, 1983;50:481-505. 9- Hashin, Z., Analysis o properties o iber composites with anisotropic constituents, Journal o Applied Mechanics, 1979;46:543-550.