The Small Scale Effect on Nonlinear Vibration of Single Layer Graphene Sheets

Transcription

1 World Acdem of Science, Engineering nd Technolog Interntionl Journl of Mechnicl, Aerospce, Industril, Mechtronic nd Mnufcturing Engineering Vol:5, No:6, The Smll Scle Effect on Nonliner Virtion of Single Ler Grphene Sheets E. Jomehzdeh, A.R. Sidi Interntionl Science Inde, Mechnicl nd Mechtronics Engineering Vol:5, No:6, wset.org/puliction/86 Astrct In the present rticle, nonliner virtion nlsis of single ler grphene sheets is presented nd the effect of smll length scle is investigted. Using the Hmilton's principle, the three coupled nonliner equtions of motion re otined sed on the von Krmn geometricl model nd Eringen theor of nonlocl continuum. The solutions of Free nonliner virtion, sed on one term mode shpe, re found for oth simpl supported nd clmped grphene sheets. A complete nlsis of grphene sheets with movle s well s immovle in-plne conditions is lso crried out. The results otined herein re compred with those ville in the literture for clssicl isotropic rectngulr pltes nd ecellent greement is seen. Also, the nonliner effects re presented s functions of geometric properties nd smll scle prmeter. Kewords Smll scle, Nonliner virtion, Grphene sheet, Nonlocl continuum I. INTRODUCTION HE lrge mplitude virtion nlsis hs ecome T incresingl importnt prticulrl in thin-wlled structures such s pltes nd shells nd severl ttempts hve een mde to otin solution for nonliner virtion of such structures. The lrge mplitude virtion nlsis hs ecome incresingl importnt prticulrl in thin-wlled structures such s pltes nd shells nd severl ttempts hve een mde to otin solution for nonliner virtion of such structures. The lrge mplitude virtions of pltes of vrious geometries hve een investigted severl uthors. The generl solutions of the lrge mplitude virtion of thin elstic pltes were otined Chu nd Herrmn [] nd Ymki []. The nonliner free virtion ehvior of rectngulr cross pl lmintes ws investigted Singh et l. [] using direct numericl integrtion. Leung nd Mo [] studied simpl supported rectngulr pltes with movle edges using the Glerkin method. Theoreticl nd eperimentl studies for geometricll nonliner virtions of rectngulr pltes were developed Amili [5]. Nonliner free ismmetric virtion of simpl supported isotropic circulr pltes ws investigted Hterouch nd Benmr [6] using the energ method nd multimode pproch. An nlticl solution ws provided Woo et l. [7] for the nonliner free virtion ehvior of pltes mde of functionll grded mterils. Amili nd Frhdi [8] used the clssicl von Krmn nd the first order sher deformtion theories for E. Jomehzdeh is with the Deprtment of Mechnicl Engineering t Shhid Bhonr Universit of Kermn, Irn (corresponding uthor to provide phone: ; f: ; e-mil: jomehzdeh@hoo.com. A.R. Sidi is with the Deprtment of Mechnicl Engineering t Shhid Bhonr Universit of Kermn, Irn, (e-mil:sidi@mil.uk.c.ir. studing the nonliner forced virtions of isotropic nd lminted composite rectngulr pltes.due to the vst computtionl epenses of nno-structures nlses when using tomic lttice dnmics nd moleculr dnmic simultions, there is gret interest in ppling continuum mechnics for nlsis of nno-structures. Mn studies hve een crried out for virtion nlsis of nno-structures such s nnotues nd nno-pltes. Lim nd He [9] developed von Krmn tpe nonliner model for ultr-thin, elsticll isotropic films with surfce effect. Kitipornchi et l. [] used the continuum plte model for mechnicl nlsis of grphene sheets. Bsed on the continuum mechnics nd multiple-elstic em model, Fu et l. [] investigted the nonliner free virtion nlsis of emedded cron nnotues. Prdhn nd Phdikr [] presented clssicl nd first-order sher deformtion plte theories for virtion of nno-pltes. Their pproch ws sed on the Nvier solution nd for nno-plte with ll edges simpl supported. Ke et l. [] investigted the nonliner free virtion of emedded doule-wlled cron nnotues sed on the Eringen s nonlocl elsticit theor nd von Krmn geometric nonlinerit using differentil qudrture method. Dong nd Lim [] studied the nonliner free virtions of nno-em with simpl supports oundr conditions sed on nonlocl elsticit theor. Murmu nd Prdhn [5] developed single elstic em model for thermo-mechnicl virtion of single-wlled cron nnotue emedded in n elstic medium sed on nonlocl elsticit theor. Nonliner free virtion of single-wlled cron nnotues sed on the Timoshenko em model ws studied Yng et l. [6]. An elstic continuum pproch for modeling the nonliner virtion of doule-wlled cron nnotues under hrmonic ecittion ws investigted Hww nd Qhtni [7].Prcticll ll of the prolems in mechnics re nonliner nd lineriztion is commonl n pproimtion. Therefore, the infinitesiml deformtion model is invlid, nd geometricll nonliner model is evidentl needed. In this pper, the lrge mplitude virtion of single ler grphene sheet is studied sed on the nno-plte model. Considering the smll scle effect in constitutive reltions nd using the von Krmn nonliner model, the governing equtions of motion re otined in form of two coupled nonliner prtil differentil equtions. The free virtion nlsis re presented for oth simpl supported nd clmped nno-pltes with movle nd immovle in-plne conditions. The effects of nonlocl prmeter, oundr condition nd spect rtio on the nonliner virtion ehvior of grphene sheets re discussed in detils. Interntionl Scholrl nd Scientific Reserch & Innovtion 5(6 99 scholr.wset.org/999.8/86

2 World Acdem of Science, Engineering nd Technolog Interntionl Journl of Mechnicl, Aerospce, Industril, Mechtronic nd Mnufcturing Engineering Vol:5, No:6, Interntionl Science Inde, Mechnicl nd Mechtronics Engineering Vol:5, No:6, wset.org/puliction/86 II. CONSTITUTIVE RELATIONS OF NONLOCAL CONTINUUM According to nonlocl elsticit theor, the stress t reference point X is considered to e function of the strin field t ever point X ' in the od. The nonlocl stress tensor σ t point X cn e epressed s σ K( X ' X, τ σ '( X ' dx ' ( V where σ ' is the clssicl stress tensor nd K( X ' X is the Kernel function represents the nonlocl modulus. While the constitutive equtions of clssicl elsticit is n lgeric reltion etween stress nd strin tensors, tht of nonlocl elsticit involves sptil integrls which represent weighted verges of contriutions of the strin of ll points in the od to the stress t the given point. Eringen [8] showed tht it is possile to represent the integrl constitutive reltion in n equivlent differentil form s ( μ σ σ ' ( where μ ( e is the nonlocl prmeter, n internl chrcteristic length nd e constnt. Also, is the Lplcin opertor. III. FORMULATION Consider thin nno-plte of totl thickness h with dimension for modeling the single ler grphene sheet. The origin of the Crtesin coordinte sstem is locted in the middle of the plte. Since the grphene sheet is ssumed to hve lrge mplitude motion, the von Krmn tpe strindisplcement reltions re used s ε ε + zκ ε ε + zκ ( γ γ + zκ, where u w + ( ε κ v w ε + (, κ ( γ u v κ + + w Here the time dependent vriles u, v nd w re the midplne displcement components in the, nd z directions respectivel. As it cn e seen, the strin-displcement reltions re nonliner with respect to trnsverse displcement. According to Hmilton's principle, the equtions of motion of the nno-plte cn e given t δ Ldt (5 t where L is the Lgrngin nd t is the time vrile. Epressing the Lgrngin prmeter sed on the von Krmn theor nd considering the smll scle effect from Eq. ( in constitutive reltions, the non-liner equtions of motion for nno-plte cn e otined s follows N N & + ( μ ( Iu&& I (6 N N & + ( μ ( I v&& I (6 M M M ( N + N + ( N + N + P (6c u&& v&& w&& w&& ( μ ( I w&& + I( + I ( + where dot denotes differentition with respect to time nd P is the lod cting on the plte in z direction. The resultnt forces ( N, N, N nd resultnt moments ( M, M, M cn e defined in terms of strins s ν N ν ε ν N ε N Eh h h γ ν (7 M ν κ M h h ν κ M ( ν h κ where E nd ν re respectivel the Young modulus nd Poisson rtio of the nno-plte. Also, the inerti prmeters ( I, I, I cn e epressed in term of densit of the plte, ρ, s follow h / ( I, I, I ρ (, z, z dz (8 h / As it cn e concluded, the prmeter I is identicll zero for smmetric nno-plte with respect to z is. Rju et l. [9] showed tht the effect of longitudinl or inplne inerti on lrge mplitude virtion of thin-wlled structures is negligile. Vnishing the in-plne inerti terms, it cn e seen tht two first equtions (6 will e ectl stisfied if stress function ϕ is defined such s N, N, N (9 Sustituting these reltions into Eq. (6c nd epressing the resultnt moments in terms of trnsverse displcement ield D w + ( I I ( μ ( w&& + w& P w w w ( + where D denotes the fleurl rigidit of the nno-plte ( Eh D. Eq. ( is nonliner fourth order prtil ( ν differentil eqution in terms of trnsverse displcement nd Interntionl Scholrl nd Scientific Reserch & Innovtion 5(6 995 scholr.wset.org/999.8/86

3 World Acdem of Science, Engineering nd Technolog Interntionl Journl of Mechnicl, Aerospce, Industril, Mechtronic nd Mnufcturing Engineering Vol:5, No:6, Interntionl Science Inde, Mechnicl nd Mechtronics Engineering Vol:5, No:6, wset.org/puliction/86 stress function, nd it needs to e ugmented with comptiilit eqution. Eliminting u nd v from Eq. ( nd epressing the in-plne strin components in terms of stress function, one cn otined the comptiilit eqution s w w w ϕ Eh ( ( Eqs. ( nd ( re the strting equtions for nlzing the nonliner virtion nlsis of grphene sheets. As it cn e seen these equtions re nonliner prtil differentil equtions with totl degree of eight. Thus, the nonliner ehvior cn e considered s two kinds, one from the inplne forces due to the nture of edge restrins nd the other due to the interction of displcement components ffected in the in-plne differentil equtions s well s comptiilit conditions. Also, the in-plne displcement components cn e epressed in terms of w nd ϕ help of Eqs. (, (7 nd (9 the following reltions u v ( ν Eh ( ν Eh ( ( d d IV. FREE VIBRATION ANALYSIS ( ( For lrge mplitude free virtion nlsis of grphene sheets, the eternl lod is ssumed to e zero ( P. Four oundr conditions re considered for the nno-plte s ll edges simpl supported oundr condition with either movle or immovle in-plne edges nd ll edges clmped oundr condition with either movle or immovle inplne edges. i. An ll edges simpl supported (SSSS grphene sheet Let us consider simpl supported nno-plte with the following oundr conditions Simpl supported with movle in-plne edge (SSSS / d t ± / / / d t ± ( Simpl supported with immovle in-plne edge (SSSS u t ± ( v t ± The movle in-plne edge is the edge which is kept stright distriution of norml stresses nd therefore the resultnt stresses on the edge is zero. The nonliner free virtion response of the simpl supported nno-plte cn e otined introducing the following dmissile function for trnsverse deflection nπ mπ w hw ( tcos( cos( ( where n nd m re the numers of hlf cosine wves in the nd directions respectivel nd W (t is function of time onl. It is ovious tht Eq. ( stisfies the first two oundr conditions in Eqs. ( nd ( of ech edge. Sustituting Eq. ( in the right side of Eq. (, the generl solution for stress function ϕ cn e otined s ϕ C m cos(nπ / + n cos(mπ / (5 C should e determined Eh + C + W ( t n m The integrl constnt C nd wheres the two lst conditions (in-plne oundr conditions of Eqs. ( or ( re stisfied. It is es to show tht for movle simpl supported nno-plte (SSSS, the constnt coefficients C nd C should e equl to zero nd for immovle simpl supported nno-pltes (SSSS these coefficients re s follows Eh π ( + ν C W ( t 6( ν (6 Eh π ( + ν C W ( t 6( ν The trnsverse displcement w nd the stress function ϕ cn e introduced from Eqs. ( nd (5 into the sic Eq. ( nd, in ccording to the Glerkin procedure, the integrl Γ( w, ϕ Ψ da (7 A cn e computed over the re of the nno-plte. Γ ( w, ϕ is the nonliner eqution ( nd Ψ is the sptil prt of dmissile function (. A simple ut length clcultion of the ove integrl leds to modl time differentil eqution of Duffing s tpe, for oth cses, which cn e written s d W ( t + α W ( t + βw ( t (8 dt where the prmeters α nd β re defined in terms of grphene sheet properties s D( + α π (9 ( + h π + h π ( + μπ + μπ ρh Eh ( + β π for SSSS ( + h π + h π ( + μπ + μπ ρ (9 9D( ν + ν ν β π for SSSS ( + h π + h π ( + μπ + μπ ρh (9c For liner virtion of the grphene sheet in which term β W (t cn e neglected, the corresponding liner nturl frequenc is given ω l α. For periodic motion with Interntionl Scholrl nd Scientific Reserch & Innovtion 5(6 996 scholr.wset.org/999.8/86

4 World Acdem of Science, Engineering nd Technolog Interntionl Journl of Mechnicl, Aerospce, Industril, Mechtronic nd Mnufcturing Engineering Vol:5, No:6, Interntionl Science Inde, Mechnicl nd Mechtronics Engineering Vol:5, No:6, wset.org/puliction/86 mplitude w, the nonliner nturl frequenc of Duffing Eq. (8 cn e epressed in term of Jcoi elliptic function s [] π α + βw ω nl ( βw K ( α + βw where K is the complete elliptic integrl of the first kind defined s π / dθ K ( m ( msin θ ii. An ll edges clmped (CCCC grphene sheet The oundr conditions for clmped edges nno-plte m e identified s Clmped movle edge (CCCC / w d t ± w / / / d t ± ( Clmped immovle edge (CCCC w u t ± ( w v t ± Assume tht solution of trnsverse displcement cn e epressed in the following form nπ mπ w hw ( tcos ( cos ( ( It is es to show tht the function ( ectl stisfies the two first oundr conditions in clmped edges. Doing the sme procedure s the previous cse, the Duffing s eqution cn e otined s Eq. (8 with different coefficients. V. NUMERICAL RESULTS In order to verif the ccurc of the present formultions, the results re compred with the ville results in literture for specil cse of clssicl rectngulr pltes in which the nonlocl effect is neglected ( μ. Fig. presents comprtive stud of the non-liner to liner time period rtio versus the non-dimensionl mplitude for squre simpl supported rectngulr plte for oth immovle nd movle edges with ν.. It cn e oserved tht the results of the present stud re sufficientl ccurte with the results in Ref. []. For numericl results, the following properties re ssumed for the grphene sheet E 5.86eV / A, ν. 9, h. 7 A, ρ.mg / A ( In order to recognize the hrdening or softening stte of the grphene sheet, the prmeter β is depicted in Fig. versus the nonlocl prmeter μ. It cn e sid tht the coefficient of W (t is lws positive nd therefore the grphene sheet hs hrd stiffness. Also, it cn e concluded tht s the smll scle effect increses the hrdening stiffness of the nno-plte rpidl decreses. i.e. the smll scle effect mkes the nnoplte more fleile s the nonlocl model m e viewed s toms linked elstic springs while the locl continuum model ssumes the spring constnt to tke on n infinite vlue. The nonliner to liner frequencies rtio versus mplitude is plotted to stud the effects of phsicl properties of grphene sheets. Bckone curves (nonliner to liner frequencmplitude curve re plotted for ll edge simpl supported nd ll edges clmped grphene sheets in Fig. nd, respectivel. It is found tht the ckone curves do not depend on the nonlocl prmeter. Also, it cn e seen tht the effect of spect rtio on ckone curves of simpl supported nno-pltes is more significnt thn tht of clmped nno-pltes. Besides, it cn e concluded tht the in-plne oundr conditions re ver importnt in nonliner nlsis, s these cn chnge the ckone curves quite significntl. VI. CONCLUSION The free nonliner virtion nlsis of single ler grphene sheets hs een studied nd the smll scle effect on the nonliner ehvior of the nno-pltes hs een investigted. Considering the Eringen nonlocl theor nd von Krmn hpothesis, the nonliner equtions of motion hve een otined using the Hmilton s principle. The solutions for the lrge mplitude virtions of nno-plte hve een found. The effects of nonlocl prmeter, oundr conditions, spect rtio on the nonliner virtion of grphene sheets hve een discussed. It hs een seen tht the smll length scle decreses the nonliner ehvior of the grphene sheets. Also, it hs een shown tht the ckone curves of the grphene sheets do not depend on nonlocl prmeter. T nl /T l present SSSS Ref. [] SSSS present SSSS Ref. [] SSSS..5.5 w Fig. The comprison of ckone curves of rectngulr plte Interntionl Scholrl nd Scientific Reserch & Innovtion 5(6 997 scholr.wset.org/999.8/86

5 World Acdem of Science, Engineering nd Technolog Interntionl Journl of Mechnicl, Aerospce, Industril, Mechtronic nd Mnufcturing Engineering Vol:5, No:6, Interntionl Science Inde, Mechnicl nd Mechtronics Engineering Vol:5, No:6, wset.org/puliction/86 β CCCC SSSS CCCC SSSS μ Fig. Influence of nonlocl prmeter on the nonliner ehvior of grphene sheets. h ω nl / ω l SSSS / SSSS / SSSS / SSSS / w Fig. Bckone curves for ll edges simpl supported grphene sheets ω nl / ω l CCCC / CCCC / CCCC / CCCC / REFERENCES [] H. Chu, G. Herrmnn, Influence of lrge mplitudes on free fleurl virtions of rectngulr elstic pltes, Journl of Applied Mechnics,, 956, pp [] N. Ymki, Influence of lrge mplitudes on fleurl virtions of elstic pltes, ZAMM,, 96, pp [] G. Singh, K. Rju, G.V. Ro, Non-liner virtions of simpl supported rectngulr cross-pl pltes, Journl of Sound nd Virtion,, 99, pp. -6. [] A.Y.T. Leung, S.G. Mo, A smplectic Glerkin method for non-liner virtion of ems nd pltes, Journl of Sound nd Virtion, 8, 995, pp [5] M. Amili, Nonliner virtions of rectngulr pltes with different oundr conditions: theor nd eperiments, Computers nd Structures, 8,, pp [6] M. Hterouch, R. Benmr, Geometricll nonliner free virtions of simpl supported isotropic thin circulr pltes, Journl of Sound nd Virtion, 8, 5, pp [7] J. Woo, S.A. Meguid, L.S. Ong, Nonliner free virtion ehvior of functionll grded pltes, Journl of Sound nd Virtion, 89, 6, pp [8] M. Amili, S. Frhdi, Sher deformle versus clssicl theories for nonliner virtions of rectngulr isotropic nd lminted composite pltes, Journl of Sound nd Virtion,, 9, pp [9] C.W. Lim, L.H. He, Size-dependent nonliner response of thin elstic films with nno-scle thickness, Interntionl Journl of Mechnicl Sciences, 6,, pp [] S. Kitipornchi, X.Q. He, K.M. Liew, Continuum model for the virtion of multilered grphene sheets, Phsicl Review B, 7, 75, 5, 6 pges. [] Y.M. Fu, J.W. Hong, X.Q. Wng, Anlsis of nonliner virtion for emedded cron nnotues, Journl of Sound nd Virtion, 96, 6, pp [] S.C. Prdhn, J.K. Phdikr. Nonlocl elsticit theor for virtion of nnopltes, Journl of Sound nd Virtion, 5, 9, pp. 6. [] L.L. Ke, Y. Xing, J. Yng, S. Kitipornchi, Nonliner free virtion of emedded doule-wlled cron nnotues sed on nonlocl Timoshenko em theor, Computtionl Mterils Science, 7, 9, pp.9 7. [] Y.X. Dong, C.W. LIM, Nonliner virtions of nno-ems ccounting for nonlocl effect using multiple scle method, Science in Chin Series E: Technologicl Sciences, 5, 9, pp [5] T. Murmu, S.C. Prdhn, Thermo-mechnicl virtion of singlewlled cron nnotue emedded in n elstic medium sed on nonlocl elsticit theor, Computtionl Mterils Science, 6, 9, pp [6] J. Yng, L.L. Ke, S. Kitipornchi, Nonliner free virtion of singlewlled cron nnotues using nonlocl Timoshenko em theor, Phsic E,,, pp [7] M.A. Hww, H.M. Al-Qhtni, Nonliner oscilltions of doulewlled cron nnotue, Computtionl Mterils Science, 8,, pp.. [8] C. Eringen, On differentil equtions of nonlocl elsticit nd solutions of screw disloction nd surfce wves, Journl of Applied Phsics, 5, 98, pp [9] I.S. Rju, G.V. Ro, K. Rju, Effect of longitudinl or inplne deformtion nd inerti on the lrge mplitude fleurl virtions of slender em nd thin pltes, Journl of Sound nd Virtion, 9, 976, pp. 5-. [] A. Beléndez, D.I. Méndez, E. Fernndez, S. Mrini, I. Pscul, An eplicit pproimte solution to the Duffing-hrmonic oscilltor cuiction method, Phsics Letters A, 7, 9, pp [] J.J. Stoker, Nonliner virtions in Mechnicl nd Electricl Sstems, John Wile & Sons, Inc., New York, 95. w Fig. Bckone curves for ll edges clmped grphene sheets Interntionl Scholrl nd Scientific Reserch & Innovtion 5(6 998 scholr.wset.org/999.8/86