Interfacial Stress Transfer in a Graphene Monolayer Nanocomposite

Transcription

1 Interacial Stress Transer in a Graphene Monolayer Nanocoposite L. Gong 1, I. A. Kinloch 1, R. J. Young 1, *, I. Riaz 2, R. Jalil 2 & K. S. Novoselov 2, 1 School o Materials, University o Manchester, Grosvenor Street, Manchester, M1 7HS, UK 2 School o Physics & Astronoy, University o Manchester, Oxord Road, Manchester, M13 9PL, UK *E-ail: robert.young@anchester.ac.uk E-ail: kostya@anchester.ac.uk Keywords: Graphene, Coposite Materials, Monolayers, Nanostructures Graphene is one o the stiest known aterials, with a Young s odulus o 1 TPa, aking it an ideal candidate or use as a reinorceent in high-perorance coposites. However, being a one-ato thick crystalline aterial, graphene poses several undaental questions: (1) can decades o research on carbon-based coposites be applied to such an ultiately-thin crystalline aterial? (2) is continuu echanics used traditionally with coposites still valid at the atoic level? (3) how does the atrix interact with the graphene crystals and what kind o theoretical description is appropriate? We have deonstrated unabiguously that stress transer takes place ro the polyer atrix to onolayer graphene, showing that the graphene acts as a reinorcing phase. We have also odeled the behavior using shear-lag theory, showing that graphene onolayer nanocoposites can be analyzed using continuu echanics. Additionally, we have been able to onitor stress transer eiciency and breakdown o the graphene/polyer interace. 1

2 Since graphene was irst isolated in 2004 [1,2] the ajority o the research eort has concentrated upon its electronic properties aied at applications such as in electronic devices. [3,4] A recent study has investigated the elastic echanical properties o onolayers o graphene using nanoindentation by atoic orce icroscopy. [5] It was shown that the aterial has a Young s odulus o the order o 1 TPa and an intrinsic strength o around 130 GPa, aking it the strongest aterial ever easured. Such ipressive echanical properties have ade graphene an obvious candidate or use in high-perorance polyer-based coposites. Carbon nanotubes are under active investigation as reinorceents in nanocoposites [6,7] and it is well established that platelet reinorceents such as exoliated nanoclays [8,9] can be eployed as additives to enhance the echanical and other properties o polyers. Recently it has been deonstrated that polyerbased nanocoposites with cheically-treated graphene oxide as a reinorceent show draatic iproveents in both electronic [10] and echanical [11] properties (thus a 30 K increase in the glass transition teperature is achieved or only a 1% loading by weight o the cheically-treated graphene oxide in a poly(ethyl ethacrylate) atrix). Issues that arise in these nanocoposites systes include the diiculty o dispersion o the reinorcing phases and stress transer at the interace between the dispersed phase and the polyer atrix. It is now well established that Raan spectroscopy can be used to ollow stress transer in a variety o coposites reinorced with carbon based aterials such as carbon ibres [12,13] and single- and double-walled carbon nanotubes. [14-16] Such reinorceents have well-deined Raan spectra and their Raan bands are ound to shit with stress which enables stress-transer to be onitored between the atrix and reinorcing phase. Moreover, a universal calibration has been established between the rate o shit o the G carbon Raan bands with strain [14] that allows the eective Young s odulus o the carbon reinorceent to be estiated. Recent studies have shown that since the Raan scattering ro these carbon-based aterials is resonantly enhanced then strong well-deined spectra can be obtained ro very sall aounts o the carbon aterials, or exaple individual carbon nanotubes either isolated on a substrate [17] or debundled and isolated within polyer nanoibers. [18,19] 2

3 Raan spectroscopy has also been eployed to characterise the structure and deoration o graphene. It has been deonstrated that the technique can be used to deterine the nuber o layers in graphene ils [20] (see also Supporting Inoration [21]). Graphene onolayers have characteristic spectra in which the G band (also tered the 2D band) can be itted with a single peak, whereas the G band in bilayers is ade up o 4 peaks [20], which is a consequence o the dierence between the electronic structure o the two type o saples. Several recent papers have established that the Raan bands o onolayer graphene shit during deoration. [22-25] In these reports the graphene has been deored in tension by either stretching [22,23] or copressing [24] it on a PDMS substrate [22] or a PMMA bea. [23,24] It is also ound that the G band both shits to lower wavenuber in tension and undergoes splitting. The G band undergoes a shit in excess o -50 c -1 /% strain which is consistent with it having a Young s odulus o over 1 TPa [14]. A recent study [25] o graphene subjected to hydrostatic pressure has shown that the Raan bands shit to higher wavenuber or this ode o deoration and that the behavior can be predicted ro knowledge o the band shits in uniaxial tension. In this present study we have used Raan spectroscopy to onitor stress transer in a odel coposite consisting o a thin polyer atrix layer and a echanically cleaved single graphene onolayer using the stress-sensitivity o the graphene G band. An optical icrograph o the specien is shown in Figure 1(a) where the approxiately diaond-shaped 12 µ 30 µ graphene onolayer is indicated and Figure 1(b) shows a scheatic diagra o the specien. Raan spectra were obtained initially ro the iddle o the onolayer and Figure 2(a) shows the position o the G band beore deoration, at 0.7 % strain and then unloaded. It can be seen ro Figure 2(b) that there is a large stress-induced shit o the G band. There was a linear shit o the band up to 0.4 % strain when the stepwise deoration was halted to ap the strain across the onolayer. It was then loaded up to 0.5 % and 0.6 % strain when urther apping was undertaken and inally the specien was unloaded ro 0.7 % strain. It can be seen that there was soe relaxation in the specien ollowing each o the apping stages so that the band shits becae irregular. In addition, the slope o the unloading line ro the highest strain level is 3

4 signiicantly higher than that o the loading line. The slope o the unloading line is ~ -60 c -1 /% strain, siilar to the behavior ound or the deoration o a ree-standing onolayer on a substrate. [22,23] Moreover, the G band position ater unloading is at a higher wavenuber than beore loading. This behavior is consistent with the graphene undergoing slippage in the coposite during the initial tensile deoration and then becoing subjected to in-plane copression on unloading. Mapping the local strain in along a carbon iber in a polyer atrix allows the level o adhesion between the iber and atrix to be evaluated. [12,13] In a siilar way apping the strain across the graphene onolayer enables stress transer ro the polyer to the graphene to be ollowed. Figure 3 shows the local strain in the graphene onolayer deterined ro the stressinduced Raan band shits at 0.4 % atrix strain. The laser bea in the spectroeter was ocused to a spot around 2 µ which allows a spatial resolution o the order o 1 µ on the onolayer by taking overlapping easureents. Figure 3(a) shows the variation o axial strain across the onolayer in the direction parallel to the strain axis. It can be seen that the strain builds up ro the edges and is constant across the iddle o the onolayer where the strain in the onolayer equals the applied atrix strain (0.4 %). This is copletely analogous to the situation o a single discontinuous iber in a odel coposite when there is good bonding between the iber and atrix. [12,13] This behavior has been analyzed using the well-established shear-lag theory [27-29] where it is assued that there is elastic stress transer ro the atrix to the iber through a shear stress at the iber-atrix interace. It is relatively easy to odiy the analysis or platelet rather than iber reinorceent (see Supporting Inoration [21]). It is predicted ro shear-lag analysis or the platelet that or a given level o atrix strain, e, the variation o strain in the graphene lake, e, with position, x, across the onolayer will be o the or e = e x cosh ns l 1 cosh( ns / 2) (1) where 4

5 n = 2G E t T (2) and G is the atrix shear odulus, E is the Young s odulus o the graphene lake, l is the length o the graphene lake in the x direction, t is the thickness o the graphene, T is the total resin thickness and s is the aspect ratio o the graphene (l/t) in the x direction. The paraeter n is accepted widely as an eective easure o the interacial stress transer eiciency, so ns depends on both the orphology o the graphene lake and the degree o interaction it has with the atrix. The curve in Figure 3(a) is a it o Equation (1) to the experiental data using the paraeter ns as the itting variable. A reasonable it was ound or ns 20 at e = 0.4 % (see Supporting Inoration [21]) showing that the interace between the polyer and graphene reained intact at this level o strain and that the behavior could be odeled using the shear-lag approach. The variation o shear stress, τ i, at the polyer-graphene interace is given by (see Supporting Inoration [21]) x sinh ns l τ i = ne e (3) cosh( ns / 2) and the axiu value o τ i at the edges o the sheet or ns = 20 is ound to be ~2.3 MPa (see Supporting Inoration [21]). Equation (1) shows that the distribution o strain in the graphene onolayer in the x direction in the elastic case depends upon length o the onolayer, l. It can be seen ro Figure 1 that the lake tapers to a point in the y direction and so the axial strain in the iddle o the onolayer was also apped along the y direction (see Supporting Inoration [21]). The strain is airly constant along ost o the onolayer but alls to zero at the tip o the lake, y = 0. The distribution o axial graphene strain in the iddle o the onolayer at e = 0.4 % was deterined using Equation (1), taking into account the changing width by varying l (and hence s). It was ound that there is also excellent agreeent between the easured and predicted variation o iber strain 5

6 with position on the onolayer using ns = 20, validating the use o the shear lag analysis (see Supporting Inoration [21]). When the atrix strain was increased to e = 0.6 % a dierent distribution o axial strain in the graphene onolayer was obtained as shown in Figure 3(b). In this case there appears to be an approxiately linear variation o the graphene strain ro the edges to the centre o the onolayer up to 0.6 % strain ( = e ) and a dip in the iddle down to around 0.4 % strain. In this case it appears that the interace between the graphene and polyer has ailed and stress transer is taking place through interacial riction. [29] The strain in the graphene does not all to zero in the iddle o the lake, however, showing that the lake reains intact unlike the behavior o carbon ibers undergoing racture in the ragentation test. [12,13] The interacial shear stress, τ i, in this case can be deterined ro the slope o the lines in Figure 3(b) using the orce balance equation (see Supporting Inoration [21]) de dx τ i = (4) E t which gives an interacial shear stress in the range MPa or the lines o dierent slope. There are iportant iplications ro this study or the use o graphene as a reinorceent in nanocoposites. The quality o iber reinorceent is oten described in ters o the critical length, l c the paraeter is sall or strong interaces and is deined as 2 the distance over which the strain rises ro the iber ends to the plateau level. [29] It can be seen ro Figure 3(a) that the strain rises to about 90 % o the plateau value over about 1.5 µ ro the edge o the lake aking the critical length o the graphene reinorceent o the order o 3 µ. It is generally thought that in order to obtain good iber reinorceent the iber length should be ~10l c. Hence, relatively large graphene lakes (>30 µ) will be needed beore eicient reinorceent can take place. One process or eiciently exoliating graphene to single layers reported recently produced onolayers o no larger than a ew icrons across. [30,31] The relatively poor level o adhesion between the graphene and polyer atrix is also relected in the low level o interacial shear stress, τ i, 6

7 deterined - carbon ibers coposites have values o τ i an order o agnitude higher (~ MPa). [12,13] However, in the graphene coposite, interacial stress transer will only be taking place though van der Waals bonding across an atoically sooth surace. The eiciency o reinorceent is also relected in the value o the paraeter ns (= 20) in the shear lag analysis used to it the experiental data. Since the graphene is so thin, the aspect ratio s will be large (12 µ/0.35 n = ) aking n sall ( ). This value o n is a actor o 4 saller than that deterined by putting the values o G ~ 1 GPa, E ~ 1 TPa and t/t (~ 0.35 n/100 n) into Equation (2) (n ~ ), showing a possible liitation o the shear-lag analysis. [28] Nevertheless, the paraeter n deterined experientally can be eployed to onitor the eiciency o stress transer across the graphene-polyer interace, which in this case appears to be less than ideal. This present study has iportant iplications or the use o graphene as a reinorceent in coposites. As well as deonstrating or the irst tie that it is possible to ap the deoration o graphene onolayer in a polyer coposite using Raan spectroscopy, a nuber o other issues also arise. Firstly, it is quite rearkable that a spectru can be obtained ro a reinorceent only one ato thick, allowing the echanics o nano-reinorceent to be probed directly. Secondly, it appears that the continuu echanics approach is also valid at the atoic level - a question widely asked in the ield o nanocoposites - and that the coposite icroechanics developed or the case o iber reinorceent is also valid at the atoic level or graphene onolayers. We expect that our technique will be used widely in the evaluation o graphene coposites. This present study has concentrated upon pristine, untreated graphene. Cheical odiication [10] o the surace or edges ay signiicantly strengthen the interace between the graphene and a polyer, reducing the critical length and increasing n. Our technique should allow the eect o cheical odiication to be evaluated. Moreover, i graphene is to be used in devices in electronic circuits, it will have to be encapsulated within a polyer. The technique will also allow the eect o encapsulation upon residual stresses in the aterial to be probed. 7

8 Experiental The specien was prepared using a 5 thick poly(ethyl ethacrylate) bea spin-coated with 300 n o SU-8 epoxy resin. The graphene, produced by the echanical cleaving o graphite, was deposited on the surace o the SU-8. This ethod produced graphene with a range o dierent nubers o layers and the onolayers were identiied both optically [26] and using Raan spectroscopy (see Supporting Inoration [21]). A thin 50 n layer o PMMA was then spin-coated on top o the bea so that the graphene reained visible when sandwiched between the two coated polyer layers as shown in Figure 1(a). The PMMA bea was deored in 4-point bending and the strain onitored using a strain gauge attached to the bea surace. A well-deined Raan spectru could be obtained through the PMMA coating using a low-power HeNe laser (1.96 ev and < 1 W at the saple in a Renishaw 2000 spectroeter) and the deoration o the graphene in the coposite was ollowed ro the shit o the G band [22-25]. The laser bea polarization was always parallel to the tensile axis. _[1] K. S. Novoselov, A. K. Gei, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos I. V. Grigorieva, A. A. Firsov, Science, 2004, 306, 666 _[2] A. K. Gei, K. S. Novoselov, Nature Materials, 2007, 6, 183 _[3] K. S. Novoselov, A. K. Gei, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, A. A. Firsov, Nature, 2005, 438, 197 _[4] Y. B. Zhang, Y. W. Tan, H. L. Storer, P. Ki, Nature, 2005, 438, 201 _[5] C. Lee, X. D. Wei, J. W. Kysar, J. Hone, Science, 2008, 321, 385 _[6] E. T. Thostenson, Z. F. Ren, T. W. Chou Coposites Science and Technology, 2001, 61, 1899 _[7] R. H. Baughan, A. A. Zakhidov, W. A. de Heer, Science, 2002, 297, 787 8

9 _[8] P. C. LeBaron, Z. Wang, T. J. Pinnavaia, Applied Clay Science, 1999, 15, 11 _[9] Y. Kojia, A. Usuki, M. Kawasui, A. Okada, Y. Fukushia, T. Kurauchi, O. Kaigaito, Journal o Materials Research, 1993, 8, 1185 _[10] S. Stankovich, D. A. Dikin, G. H. B. Doett, K. M. Kohlhaas, E. J. Ziney, E. A. Stach, R. D. Piner, S. T. Nguyen, R. S. Ruo, Nature, 2006, 442, 282 _[11] T. Raanathan, A. A. Abdala, S. Stankovich, D. A. Dikin, M. Herrera-Alonso, R. D. Piner, D. H. Adason, H. C. Schniepp, X. Chen, R. S. Ruo, S. T. Nguyen, I. A. Aksay, R. K. Prud'hoe, L. C. Brinson, Nature Nanotechnology, 2008, 3, 327 _[12] Y. L. Huang, R.J. Young, Coposites Science and Technology, 1994, 52, 505 _[13] P. W. J. van den Heuvel, T. Peijs, R. J. Young, Coposites Science and Technology, 1997, 57, 899 _[14] C. A. Cooper, R. J. Young, M. Halsall, Coposites Part A-Applied Science and Manuacturing, 2001, 32, 401 _[15] M. Lucas, R. J. Young, Physical Review B, 2004, 69, _[16] S. Cui, I. A. Kinloch, R. J. Young, L. Noé, M. Monthioux, Advanced Materials, 2009, 21, 3591 _[17] Jorio, A. R. Saito, J. H. Haner, C. M. Lieber, M. Hunter, T. McClure, G. Dresselhaus, M. S. Dresselhaus, Physical Review Letters, 2001, 86, 1118 _[18] P. Kannan, S. J. Eichhorn, R. J. Young, Nanotechnology, 2007, 18, _[19] P. Kannan, R. J. Young, S. J. Eichhorn, Sall, 2008, 4, 930 _[20] A. C. Ferrari, J. C. Meyer, V. Scardaci, C. Casiraghi, M. Lazzeri, F. Mauri, S. Piscanec, D. Jiang, K. S. Novoselov, S. Roth, A. K. Gei, Physical Review Letters, 2006, 97, _[21] Supporting Inoration _[22] M. Y. Huang, H. Yan, C. Y. Chen, D. H. Song, T. F. Heinz, J. Hone, Proceedings o the National Acadey o Sciences, 2009, 106,

10 _[23] T. M. G. Mohiuddin, A. Lobardo, R. R. Nair, A. Bonetti, G. Savini, R. Jalil, N. Bonini, D. M. Basko, C. Galiotis, N. Marzari, K. S. Novoselov, A. K. Gei, A. C. Ferrari, Physical Review B, 2009, 79, _[24] G. Tsoukleri, J. Parthenios, K. Papagelis, R. Jalil, A. C. Ferrari, A. K. Gei, K. S. Novoselov, C. Galiotis, Sall, 2009, 5, 2397 _[25] J. E. Proctor, E. Gregoryanz, K. S. Novoselov, M. Lotya, J. N. Colean, M. P. Halsall, Physical Review B, 2009, 80, _[26] P. Blake, E. W. Hill, A. H. C. Neto, K. S. Novoselov, D. Jiang, R. Yang, T. J. Booth, A. K. Gei, Applied Physics Letters, 2007, 91, _[27] H. L. Cox, British Journal o Applied Physics, 1952, 3, 72 _[28] J. A. Nairn, Mechanics o Materials, 1992, 13, 131 _[29] A. Kelly, N. H. Macillan, Strong Solids, 3 rd Edition, Clarendon Press, Oxord, 1986 _[30] P. Blake, P. D. Briicobe, R. R. Nair, T. J. Booth, D. Jiang, F. Schedin, L. A. Ponoarenko, S. V. Morozov, H. F. Gleeson, E. W. Hill, A. K. Gei, K. S. Novoselov, Nano Letters, 2008, 8, _[31] Y. Hernandez, V. Nicolosi, M. Lotya, F. M. Blighe, Z. Y. Sun, S. De, I. T. McGovern, B. Holland, M. Byrne, Y. K. Gun'ko, J. J. Boland, P. Niraj, G. Duesberg, S. Krishnaurthy, R. Goodhue, J. Hutchison, V. Scardaci, A. C. Ferrari, J. N. Colean, Nature Nanotechnology, 2008, 3,

11 a) b) Figure 1. Single onolayer graphene coposite. a) Optical icrograph showing the onolayer graphene lake investigated. b) Scheatic diagra (not to scale) o a section through the coposite 11

12 a) G' Band Unloaded Intensity (a.u.) 0.7% strain Relaxed Raan wavenuber (c -1 ) b) Loading Unloading Raan wavenuber (c -1 ) slippage region Strain (%) Figure 2. Shits o the Raan G band during loading and unloading o the onolayer graphene coposite. a) Change in the position o the G band with deoration. b) Shit o the G band peak position as a unction o strain. (The blue circles indicate where the loading was halted to ap the strain across the lake). 12

13 a) % 0.4 ns = 20 Strain (%) y x Position, x (µ) b) % 0.6 Strain (%) x y Position, x (µ) Figure 3. Distribution o strain in the graphene in the direction o the tensile axis (x) across a single onolayer. a) Variation o axial strain with position across the onolayer in the x-direction at 0.4% atrix strain (The curve itted to the data is Equation (1)). b) Variation o graphene strain in the direction o the tensile axis (x) across a single onolayer at 0.6% atrix strain. (The solid lines are itted to the data to guide the eye. The dashed curve is the shear-lag it to the data in Figure 3(a) at 0.4% strain.) 13

14 SUPPORTING INFORMATION Interacial stress transer in a graphene onolayer nanocoposite L. Gong 1, I. A. Kinloch 1, R. J. Young* 1, I. Riaz 2, R. Jalil 2, and K. S. Novoselov 2 SI.1 Characterisation o the Graphene using Raan Spectroscopy S1 Raan spectroscopy has been eployed to ollow the deoration o the graphene in the polyer coposite. Fig. SI.1 shows that the technique can also be used to dierentiate between lakes o graphene with dierent nubers o layers G Mechanically-Cleaved Graphene Intensity (a.u.) G' ultilayer Layers 1 Layer Raan Wavenuber (c -1 ) Figure SI.1 Raan spectra or dierent layer lakes o graphene S2, S3 SI.2 Shear Lag Analysis or a Graphene Single Monolayer In the case o discontinuous graphene lakes reinorcing a coposite atrix, stress transer ro the atrix to the lake is assued to take place through a shear stress at the lake/atrix interace as shown in Fig. SI.2. Beore deoration parallel lines perpendicular to the lake can be drawn beore deoration ro the atrix through the lake. When the syste is subjected to axial stress, σ 1, parallel to the lake axis, the lines becoe distorted 1 School o Materials, University o Manchester, Grosvenor Street, Manchester, M1 7HS, UK 2 School o Physics and Astronoy, University o Manchester, Oxord Road, Manchester, M13 9PL, UK *E-ail: robert.young@anchester.ac.uk E-ail: kostya@anchester.ac.uk 1

15 since the Young s odulus o the atrix is uch less than that o the lake. This induces a shear stress at the lake/atrix interace. The axial stress in the lake will build up ro zero at the lake ends to a axiu value in the iddle o the lake. The unior strain assuption eans that, i the lake is long enough, in the iddle o the lake the strain in the lake equals that in the atrix. Since the lakes have a uch higher Young s odulus it eans that the lakes carry ost o the stress in the coposite. Figure SI.2 Deoration patterns or a discontinuous lake in a polyer atrix Figure SI.3 Balance o stresses acting on an eleent o length, dx, o the lake o thickness, t, in the coposite 2

16 The relationship between the interacial shear stress, τ i, near the lake ends and the lake stress, σ, can be deterined by using a orce balance o the shear orces at the interace and the tensile orces in a lake eleent as shown in Fig. SI3. The ain assuption is that the orces due to the shear stress at the interace, τ i, is balanced by the orce due to the variation o axial stress in the lake, dσ, such that i the eleent shown in Fig. SI.3 is o unit width τ dx = tdσ (SI.1) i and so dσ dx τ i = (SI.2) t Figure SI.4 Model o a lake within a resin used in shear-lag theory. The shear stress,τ, acts at a distance z ro the lake centre. The behaviour o a discontinuous lake in a atrix can be odelled using shear lag theory in which it is assued that the lake is surrounded by a layer o resin at a distance, z, ro the lake centre as show in Fig. SI.4. The resin has an overall thickness o T. It is assued that both the lake and atrix deor elastically and the lake-atrix interace reains intact. I u is the displaceent o the atrix in the lake axial direction at a distance, z, then the shear strain, γ, at that position is be given by 3

17 du γ = (SI.3) dz The shear odulus o the atrix is deined as G = τ/γ hence du dz τ = (SI.4) G The shear orce per unit length carried by the atrix is transitted to the lake surace though the layers o resin and so the shear strain at any distance z is given by du dz τ G i = (SI.5) This equation can be integrated using the liits o the displaceent at the lake surace (z = t/2) o u = u and the displaceent at z = T/2 o u = u T u u T τ i du = G T / 2 t / 2 dz (SI.6) τ i hence u T u = ( T t) 2G (SI.7) It is possible to convert these displaceents into strain since the lake strain, e and atrix strain, e, can be approxiated as e du /dx and e du T /dx. It should be noted that this shear-lag analysis is not rigorous but it serves as a siple illustration o the process o stress transer ro the atrix to a lake in a graphene-lake coposite. In addition, τ i is given by Equation (SI.2) and so dierentiating Equation (SI.7) with respect to x leads to e e 2 tt d σ = 2G dx 2 since T >> t. Multiplying through by E gives (SI.8) d σ 2 2 dx 2 n = σ 2 t ( e E ) where n = 2G E t T (SI.9) This dierential equation has the general solution σ = E e nx + C sinh t + nx D cosh t where C and D are constants o integration. This equation can be sipliied and solved i it is assued that the boundary conditions are that there is no stress transitted across the lake 4

18 ends, i.e. i x = 0 in the iddle o the lake where σ = E e then σ = 0 at x = ±l/2. This leads to C = 0 and E D = cosh e ( nl / 2t) The inal equation or the distribution o lake stress as a unction o distance, x along the lake is then cosh( nx / t) σ = E e 1 cosh( nl / 2t ) (SI.10) Finally it is possible to deterine the distribution o interacial shear stress along the lake using Equation (SI.2) which leads to τ = ne i e sinh( nx / t) cosh( nl / 2t) It is convenient at this stage to introduce the concept o lake aspect ratio, s = l/t so that the two equations above can be rewritten as x cosh ns l σ = E e 1 (SI.12) cosh( ns / 2) or the axial lake stress and as τ = ne i e x sinh ns l cosh( ns / 2) or the interacial shear stress. It can be seen that the lake is ost highly stressed, i.e. the ost eicient lake reinorceent is obtained, when the product ns is high. This iplies that a high aspect ratio, s, is desirable along with a high value o n. SI.3 Fit o Experiental Data o the Graphene Monolayer to the Shear Lag Analysis The experiental data on the variation o graphene strain across the onolayer lake are itted to the shear lag analysis derived above in Fig. SI.5. It can be seen that the its o the theoretical shear-lag curves to the strain distribution are sensitive to the value o ns chosen. Likewise the value o interacial shear stress at the lake ends is very sensitive to the values o ns chosen. 5

19 a ns = 10, 20, % 0.4 Strain (%) Position, x (µ) Interacial Shear Stress (MPa) 6 ns = 10, 20, b Position, x (µ) Figure SI.5 a. Distribution o strain in the graphene in direction o the tensile axis across a single onolayer at 0.4% strain. The curves are its o Equ. SI.12 using dierent values o paraeter ns. b. Variation o interacial shear stress with position deterined ro Equ. SI.13 or the values o ns used in a. Figure SI.6 shows the its o Equ. SI.12 to the vertical strain distribution across the graphene onolayer lake as it tapers to a point at y = 0. It can be seen that the its are very sensitive to the value o ns eployed with the best it being or ns = 20. 6

20 ns = 10, 20, % 0.4 Strain (%) Position, y (µ) Figure SI.6 Distribution o strain in the graphene in direction o the tensile axis across a single onolayer at 0.4% strain showing the variation o ibre strain with position across the onolayer in the vertical direction. The curves were calculated ro Equ SI.12 using dierent values o ns. Reerences 1. Ferrari A. C. et al., Raan spectru o graphene and graphene layers, Physical Review Letters, 97, (2006) 2. Kelly, A. & Macillan, N.H., Strong Solids, 3 rd Edition, Clarendon Press, Oxord, Young, R.J. & Lovell, P.A., Introduction to Polyers, 3 rd Edition, Chapter 24, CRC Press, London, in press 7