SIMPLIFIED APPROACH FOR THE ANALYSIS OF A VISCOELASTIC PLATE IMPACT RESPONSE USING FRACTIONAL DERIVATIVE CONSTITUTIVE EQUATIONS

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1 Copyight 9 by ABCM th Pan-Aeican Congess of Applied Mechanics Januay 4-8 Foz do Iguaçu PR Bazil SIMPLIFIED APPROACH FOR THE ANALYSIS OF A VISCOELASTIC PLATE IMPACT RESPONSE USING FRACTIONAL DERIVATIVE CONSTITUTIVE EQUATIONS Yuiy A. Rossikhin ya@vgasu.vn.u Maina V. Shitikova shitikova@vail.u Vjacheslav V. Shaain shaain@ya.u Voonezh State Univesity of Achitectue and Civil Engineeing -letija Oktjabja 84Voonezh 3946 Russia Abstact. The ipact of a igid body upon an infinite isotopic plate is investigated fo the case when the viscoelastic featues of the plate epesent theselves only in the place of contact and ae govened by the standad linea solid odel with factional deivatives. Thus the poble concens the shock inteaction of the dopping ass and the taget wheein instead of the Hetz contact law the genealized factional-deivative standad linea solid law is eployed as a law of inteaction. The pat of the plate beyond the contact doain is assued to be elastic and its behavio is descibed by the equations of otion which take otay inetia and shea defoations into account. It is assued that tansient waves geneate in the plate at the oent of ipact the influence of which on the contact doain is consideed using the theoy of discontinuities. To deteine the desied values behind the tansvese shea wave font one-te ay expansions ae used as well as the equations of otion of the falling ass and the contact egion. As a esult we ae led to a set of two linea diffeential equations the solution of which is found analytically by the Eule substitution ethod what allows us to obtain the tie-dependence of the contact foce. Nueical analysis shows that axiu of the contact foce inceases tending to the axial contact foce at the factional paaete equal to unity. Keywods: ipact shock inteaction factional deivative viscoelasticity ay ethod. INTRODUCTION Phillips and Calvit (967) wee pobably the fist to investigate the ipact esponse of a viscoelastic infinitely extended plate. They used the Hetz s contact law in its heeditay fo. This poble is an iediate extension of Zene s appoach fo the dynaic igid spheical-indente poble fo the case of a thin elastic plate (Zene 94). The othe appoach to the poble when viscosity is included duing ipact is based on eplacing Hetz s contact equation by the Maxwell equation connecting the contact foce with the defoation of a viscoelastic eleent located between the ipacto and the taget. This appoach was ipleented by Hael (976) fo the analysis of aicaft ipact on a spheical shell. Gonsovskii et al. (97) when investigating the ipact of a viscoelastic od against a igid baie suggested to descibe the heeditay featues of the od s ateial by a factional deivative odel which was witten in an equivalent fo in tes of Boltzann-Voltea elationships with a factional exponent as a weakly singula kenel. Rossikhin and Shitikova (8) genealizing the appoaches descibed in Gonsovskii et al. (97) and Hael (976) poposed to use a factional deivative Maxwell odel fo the analysis of the ipact plate esponse when its equations of otion take the otay inetia and shea defoations into account. It is assued that a tansient wave of tansvese shea is geneated in the plate and that the eflected wave has insufficient tie to etun to the location of the sping s contact with the plate befoe the ipact pocess is copleted. To deteine the desied values behind the tansvese-shea wave font one-te ay expansions ae used as well as the equations of the ipacto and the contact egion. The solution to this poble was found analytically by the Laplace tansfo ethod and the tie-dependence of the contact foce was obtained. In the pesent pape the siplified appoach developed by Rossikhin and Shitikova (9) fo the analysis of fee daped vibations of factional deivative oscillatos is ipleented fo the analysis of the plate esponse to the ipact by the dopping ass wheein instead of the Hetz contact law the genealized factional-deivative standad linea solid law is eployed as a law of shock inteaction.. PROBLEM FORMULATION AND GOVERNING EQUATIONS Let a igid cylindical body of ass and adius with the initial velocity V ipact a cicula Uflyand-Mindlin plate of infinite extent (this assuption is intoduced due to the shot duation of contact inteaction in ode to ignoe eflected waves) with thickness h. In othe wods it is assued that the ipacto will bounce fo the taget befoe the eflected waves have a tie to each the place of contact. Geneally speaking the pocedue to be developed in this pape allows one to conside the influence of the eflected waves on the duation of contact in the case they appoach the contact place befoe the teination of the inteaction pocess. Howeve this question will not be consideed hee.

2 Copyight 9 by ABCM th Pan-Aeican Congess of Applied Mechanics Januay 4-8 Foz do Iguaçu PR Bazil To descibe the pocess of the shock inteaction of the ipacto with the taget with due account fo the viscoelastic featues we shall use the genealized factional-deivative standad linea solid law instead of the Hetz contact law (Fig. ). At the oent of ipact shock waves ae geneated in the plate which then popagate along the plate with the velocities of tansient elastic waves. Figue. Schee of the shock inteaction of a falling ass and a plate The dynaic behaviou of an Uflyand-Mindlin plate behind the tansient elastic wave fonts is descibed in the pola coodinates by the following equations (Rossikhin and Shitikova 8): Q + Q = ρ hw ( ϕ ) 3 W Q = Kμh B M ρ h B B M M + + Q = B M = D σ + whee and φ ae the pola adius and angle espectively () B B Mϕ = D + σ () M and M ϕ ae the bending oents is the shea foce B is the angula velocity of otation of the noal to the plate s iddle suface in the -diection W = w is the velocity of plate s deflection D is the cylindical igidity ρ is the density K is the shea coefficient μ is the shea odulus σ is Poissons s atio and an ovedot denotes the tie deivative. The equations of otion of the ipacto and the contact aea (Fig. ) w ( + w ) = F (3) Q ρhπ w = π Q = + F (4) subjected to the initial conditions w = w = w = w = V t= t= t= t= (5) should be added to Eqs. () and () whee w and w ae the displaceents of the uppe and lowe points of the buffe espectively. The contact foce F is connected with the diffeence in displaceents Δ w= w wof the buffe s uppe and lowe ends by the genealized standad linea solid odel (Rossikhin and Shitikova 997) with Rieann-Liouville deivative ε + τσ ( + τ D ) F = E ( + D+ ) Δ w (6) t d F( t t ) D+ F = dt dt (7) Γ( ) t whee τ ε and τ σ ae the elaxation and etadation ties espectively E is the elaxed elastic odulus and ( < ) is the ode of the factional deivative (factional paaete). The factional deivative epesentation in the fo of (7) can be utilized in the cases when the tansient pocesses can be ignoed (Rossikhin and Shitikova 9).

3 Copyight 9 by ABCM th Pan-Aeican Congess of Applied Mechanics Januay 4-8 Foz do Iguaçu PR Bazil Equation (6) can be ewitten as + τ D σ + F() t = E Δ w() t = E ( E E) w() t Δ (8) + τε D+ + τε D+ whee τ τ ε σ = EE and E is the agnitude of the non-elaxed odulus of elasticity. 3. METHOD OF SOLUTION Following Rossikhin and Shitikova (8) we shall assue that duing the ipact pocess tansvese foces and shea defoations pedoinate in the plate s stess-defoed state in vicinity of the contact spot (the contact egion of plate and buffe inteaction) and use one-te ay expansions fo the desied functions outside the igid contact spot what allows us to obtain in the vicinity of the contact egion and on its bounday the following elationship between the () shea foce the velocity of the quasi-tansvese wave G and the velocity of plate s deflection W i.e. the dynaic condition of copatibility: () Q = ρg hw G μ =. (9) ρ () K Consideing (5) (8) and (9) Eqs. (3) and (4) can be ewitten as ( ) ( )( τ ε + ) ( w = Bw +Ω w w Ω Ω + D w w ) (a) w + w = ω ( w w ) ( ω ω )( + τ ε D+ ) ( w w ) (b) whee Ω = E M Ω = EM M () = ρπ h B = G ω E = and ω = E. The solution of Eqs. () we shall seek with a help of the Eule substitution w = ce λt w = ce λt () whee c and c ae abitay constants. Substituting () in Eqs. () yields + ( λτ ) σ + ( λτσ) c λ + Bλ+Ω c Ω = (a) + ( λτε) + ( λτε) + ( λτ ) σ + ( λτσ) c λ ω c + λ + ω + ( λτε) + ( λτε) = (b) Fo the syste () possesses nontivial solution it is a need to vanish to zeo its deteinant i.e. λ λτσ + Bλ+Ω Ω λτε + ( ) + ( λτ ) + ( ) + ( λτ ) λτσ + λτε σ ε + ( ) + ( λτσ) λ ω λ ω + ( ) + ( λτ ) ε =. (3) As a esult we ae led to the chaacteistic equation λ λ τε λ λ τε λ ω λ ω τε λ ω λ ω τ f ( ) = + + B + B + C + C + B + B ε = (4)

4 Copyight 9 by ABCM th Pan-Aeican Congess of Applied Mechanics Januay 4-8 Foz do Iguaçu PR Bazil whee C = + M. The zeo oot of Eq. (3) has been discaded since it does not appea in the exact solution. Nueical analysis of the chaacteistic Eq. (4) shows that it possesses two pais of coplex conjugate oots λ. Substituting the poposed oots λ into the set of Eqs. () and dopping one of its equations due to thei linea dependence we obtain j j c = ς( λ j ) c M M ς ( λj) = + + B λ j ( j = 34). (5) Based on the afoesaid the solution of Eqs. () takes the fo () t () t () t () = λ + λ λ + + λ w c e c e c e c e t (6a) λ w c ( ) e c ( ) e c ( ) e c ( ) e t (6b) () t () t () t () = ςλ λ + ςλ λ λ + ςλ + ςλ () whee an oveba denotes the coplex conjugate of the coesponding value and c abitay coplex constants. If we ewite the coplex values appeaing in (6) in thei geoetical fo i.e. () () c c and () c ae () c = a e ϕ () i c = a e ϕ () i c = a e ϕ () i c = a e ϕ i i λ = λ = e ψ i λ = e ψ i λ = e ψ i e ψ iφ Re ςλ ( ) = ςλ ( ) = R e Φ i M B sinψ M M B cosψ tan Φ = + + R M M cosψ M = B then elationships (6) can be witten in the fo αt αt = ω + ϕ + ω ϕ w a e cos( t ) a e cos( t+ ) (7a) αt αt ω ϕ ω ϕ w = a R e cos( t+ +Φ ) + a R e cos( t+ +Φ ) (7b) whee a a ϕ and ϕ ae eal abitay constants which ae deteined fo the initial conditions (5). Substituting (7) into the initial conditions (5) we find a cosϕ + a cosϕ = (8a) acos( ϕ + ψ ) + a cos( ϕ + ψ ) = (8b) arcos( ϕ +Φ ) + ar cos( ϕ +Φ ) = (8c) arcos( ϕ +Φ + ψ ) + arcos( ϕ +Φ + ψ ) = V. (8d) Solution of Eqs. (8) gives us the unknown constants R (cosψ cos Φ ) + R cosφ R cosψ tanϕ = R sin Φ R sinψ (9a)

5 Copyight 9 by ABCM th Pan-Aeican Congess of Applied Mechanics Januay 4-8 Foz do Iguaçu PR Bazil R(cosΦ cos ψ ) + Rcosψ RcosΦ tanϕ = Rsinψ Rsin Φ a cosϕ (9b) = a (9c) cosϕ { [ ] [ ]} a = V R cos( Φ + ψ) tanϕsin( Φ + ψ) R cos( Φ + ψ) tanϕsin( Φ + ψ). (9d) cosϕ To deteine the contact foce F() t it is a need to substitute (6) into (8). As a esult we obtain ρσ αt ρσ αt Ft () = E av e cos( ω t+ ϕ+ u+ χσ χε) + av e cos( ω t+ ϕ + u + χσ χε ) () ρε ρε whee σε ( σε ) cos ( σε ) ρ = + τ ψ + τ M M cosψ M = sinψ V B B tan χ σε σε ψ τσε ( τ ) sin = + ( ) cos ψ tan u M B sinψ =. + + cos ( M M B ψ ) The tie-dependence of the contact foce calculated by Eq. () is pesented in Fig. whee the agnitudes of the factional paaete ae indicated by figues nea the coesponding cuves. Nueical analysis shows that axiu of the contact foce inceases tending to the axial contact foce at the factional paaete equal to unity. The duation of contact of colliding bodies also inceases with the incease in the factional paaete. Figue. The tie-dependence of the contact foce 4. DISCUSSION The ipact of a ass on a viscoelastic sping based on a igid foundation was investigated in Chen and Lakes (99). The pocess of ipact was appoxiated as one half cycle of fee decay oscillation of the one-diensional ass-viscoelastic buffe syste. In the pesent pape the viscoelastic sping is ebedded into an elastic foundation (Fig. ). If B and C in Eq. (4) i.e. if we suppose the foundation to be igid then we ae led to the chaacteistic equation fo the factional oscillato based on the factional deivative standad linea solid odel + + ε + + = λ τ λ ω λ ω τε. () This equation has been investigated in detail in Rossikhin and Shitikova (997).

6 Copyight 9 by ABCM th Pan-Aeican Congess of Applied Mechanics Januay 4-8 Foz do Iguaçu PR Bazil Equation () possesses two coplex conjugate oots and in contast to the chaacteistic equation fo the standad linea solid odel with odinay deivatives lacks the eal negative oots at any agnitude of τ ε. In this case the solution in the fo of () is valid only fo sall viscosity ( τ ε is sall); othewise the solution should contain one oe te which is elated to the elaxation-etadation pocesses and defines the dift of the position of equilibiu. To study the foced vibations of the factional standad linea solid odel oscillato the loss tangent tanδ ( E E )sinψ tanδ = E E E E ( ωτε) + ( ωτε) + ( )cosψ () can be used. As it was suggested by Chen and Lakes (99) the solution fo the foced vibations can be ewitten in tes of tanδ in ou case as well and it is possible to investigate its influence on the contact foce duing ipact. Howeve the pesence of the elastic foundation seveely coplicates the atte since it esults in the asyetic plot of the contact foce (Fig. ). The ode of the factional paaete also leads to the asyetic chaacte of the tie-dependence of the contact foce. 5. CONCLUSION The ipact of a igid body upon an infinite isotopic plate is investigated fo the case when the viscoelastic featues of the plate epesent theselves only in the place of contact and ae govened by the standad linea solid odel with factional deivatives. Due to the shot duation of contact inteaction the eflected waves ae not taken into account in othe wods it is assued that the ipacto will bounce fo the taget befoe the eflected waves have a tie to each the place of contact. The solution of the stated poble is found analytically by the Eule substitution ethod avoiding the difficulties of the Laplace tansfo ethod duing the invesion fo the fequency doain to the tie doain. The siplified appoach poposed in this pape is fully justified in the case of sall viscosity when elaxationetadation pocesses pass in a viscoelastic plate athe fast and the dift of the position of syste s equilibiu influences weakly the daped vibations () occuing aound this equilibiu position. If viscosity is not sall then the given appoach allows one at least to estiate qualitatively the value of the axial contact foce and the duation of contact of colliding bodies. 6. ACKNOWLEDGEMENTS The eseach descibed in this publication has been ade possible in pat by the joint Gant fo the Russian Foundation fo Basic Reseach No.7--9-HHC-a and the National Science Council of Taiwan No.96WFA REFERENCES Chen C.P. and Lakes R.S. 99. Design of Viscoelastic Ipact Absobes: Optial Mateial Popeties Intenational Jounal of Solids and Stuctues Vol. 6 pp Hael J Aicaft Ipact on a Spheical Shell Nuclea Engineeing and Design Vol. 37 pp Gonsovkii V.L. Meshkov S.I. and Rossikhin Yu.A. 97. Ipact of a Viscoelastic Rod onto a Rigid Taget Mechanics of Solids Vol. 8 pp Phillips J.W. and Calvit H.H Ipact of a Rigid Sphee on a Viscoelastic Plate ASME Jounal of Applied Mechanics Vol. 34 pp Rossikhin Yu.A. and Shitikova M.V Application of Factional Calculus to Dynaic Pobles of Linea and Nonlinea Heeditay Mechanics of Solids Applied Mechanics Reviews Vol. 5 No. pp Rossikhin Yu.A. Shitikova M.V. 8. Factional-Deivative Viscoelastic Model of the Shock Inteaction of a Rigid Body with a Plate Jounal of Engineeing Matheatics Vol Rossikhin Yu.A. and Shitikova M.V. 9. New Appoach fo the Analysis of Daped Vibations of Factional Oscillatos Shock and Vibation Vol.6 No. 4 pp Zene C. 94. The Intinsic Inelasticity of Lage Plates Physical Reviews Vol. 59 pp RESPONSIBILITY NOTICE The authos ae the only esponsible fo the pinted ateial included in this pape.