Qualitative Analysis for Solutions of a Class of. Nonlinear Ordinary Differential Equations

Transcription

1 Adv. Theo. Appl. Mech., Vol. 7, 2014, no. 1, 1-7 HIKARI Ltd, Qulittive Anlysis fo Solutions of Clss of Nonline Odiny Diffeentil Equtions Juxin Li *, Wei Zho *, **, Cong Ren * nd Jing Zhng * * School of Science, Dlin Ntionlities Univesity, Dlin , Chin ** Stte Key Lbotoy of Stuctul Anlysis fo Industil Equipment, Deptment of Engineeing Mechnics, Dlin Univesity of Technology, Dlin , Chin Copyight 2014 Juxin Li, Wei Zho, Cong Ren nd Jing Zhng. This is n open ccess ticle distibuted unde the Cetive Commons Attibution License, which pemits unesticted use, distibution, nd epoduction in ny medium, povided the oiginl wok is popely cited. Abstct The boundy vlue poblem (BVP) fo clss of nonline odiny diffeentil equtions is exmined. It cn be used to descibe the evesion poblem fo spheicl shell composed of clss of tnsvesely isotopic incompessible Mooney-Rivlin mteils. The solutions of the nonline eqution descibing the eltion mong defomed thickness, initil thickness nd mteil pmetes of the spheicl shell e obtined. The effects of stuctue pmete nd mteil pmetes on the thickness of the eveted spheicl shell e discussed by numeicl exmples. Keywods : incompessible hypeelstic mteil, spheicl shell, evesion, defomtion, implicit solution 1. Intoduction Recently, the evesion poblems of bodies composed of hypeelstic mteils (such s ubbe nd ubbe-like mteils) widely ise fom engineeing design nd eospce fields, nd so on. These poblems cn be descibed s BVP fo clss of

2 2 Juxin Li, Wei Zho, Cong Ren nd Jing Zhng nonline odiny diffeentil equtions. Mny eseches studied the eveted spheicl shell theoeticlly nd expeimentlly. In 191, Amnni [1] fistly consideed the evesion poblem of spheicl shell composed of clss of compessible hypeelstic mteils. Eicksen [2] studied the existence nd uniqueness of the eveted solutions fo spheicl shell composed of incompessible Mooney-Rivlin mteil. Antmn [], Szei [4] gve futhe nlysis fo the existence of the eveted solutions fo compessible spheicl shells. In 199, Jeemih [5] mde compison between the qulittive fetues of the egul nd eveted shells composed of specil compessible mteil unde intenl pessue. It is found tht the stess distibution in both cses is diffeent nd lge pessue cn be sustined by the eveted shell. Fo some incompessible hypeelstic spheicl shells, Chen nd Hughton [6] poved tht, if the mteil stisfied the Bke-Eicksen inequlities, no mtte how the thickness ws, thee ws unique spheiclly symmetic eveted solution, nd they gve sufficient condition of cvity fomtion fo the eveted spheicl shell. Othewise, they lso found tht thicke spheicl shells could undego bifuction on evesion. In this ppe, the evesion poblem of spheicl shell composed of clss of tnsvesely isotopic incompessible Mooney-Rivlin mteils is consideed. The coesponding mthemticl model cn be teted s BVP. The implicit solution is obtined by using the incompessible condition nd the semi-invese method. The effects of stuctue pmete nd mteil pmetes on the thickness of the eveted spheicl shells e discussed by numeicl exmples. 2. Mthemticl model nd Solutions Hee we conside the solution of the evesion poblem fo n incompessible hypeelstic thin-wlled spheicl shell. Since the defomtion is spheicl symmety, the defomtion configution is given by = (R), 0 A R B, 0 < b, θ = π Θ, ϕ = Φ, (1) whee R nd e the dii of undefomed nd defomed shell, nd b e the inne nd oute dii, espectively. Note tht b = (A), = (B). The pincipl stetches e given by d λ = =, λθ = λϕ = = λ, (2) dr R whee λ is pmete descibing the xil stetch te. Since the mteil is

3 Qulittive nlysis fo solutions of clss of nonline ODE incompessible, we get λλλ θ ϕ = 1, then = B R. Moeove, b = B A. Fo diffeent mteils, thee e diffeent defomtion modes. So fo solving this poblem, we need to give specific stin-enegy function. In this ppe, suppose tht the spheicl shell is consist of dil tnsvesely isotopic incompessible Mooney-Rivlin mteil, whose stin-enegy function is [8] μ 1 W = + β ( I ) + β ( I ) + α( λ ), () whee μ > 0 is the she modulus in infinitesiml defomtion, β ( 12 β 12) is dimensionless mteil constnt, α is dimensionless pmete mesuing the degee of nisotopy of the mteil, I nd 2 1 = λ + λ θ + λϕ I = λ λ + λ λ + λ λ 2 θ θ ϕ ϕ e the pincipl invints of the ight Cuchy defomtion tenso. The coesponding pincipl components of the Cuchy stess tenso e given by σ = μ( ) ( + β + 2( β) λ + α) p, σθθ = μλ ( + β + ( β)(( ) + λ )) p, (4) whee p is the hydosttic pessue. In the bsence of body foce, the equilibium diffeentil equtions cn be educed to dσ 2 + ( σ σ θθ ) = 0. d (5) Integting Eq. (5) with espect to, we hve () () 2 σθθ σ σ = σ + d. (6) Assume tht the inne nd oute sufces of the eveted spheicl shell e tction-fee, the boundy conditions e s follows Then fom Eq. (6), we get σ ( ) = σ ( b) = 0. (7) b σθθ σ d = 0. (8)

4 4 Juxin Li, Wei Zho, Cong Ren nd Jing Zhng Substituting Eq. (4) into Eq. (8), we hve b d ( ( ) ) + β + β λ α( ) = 0 λ. (9) Fo convenience, we intoduce the following nottions Then we obtin A δ =, B m =. (10) B b 1 δ + m M A δ 1 = = ( ), m η = = ( ) b 1 δ + m 1, B + 1 d dλ B + dλ λ = = ( 1), = =. (11) R λ λ 1+ λ λ Using the bove nottions, we ewite Eq. (9) s α ( β) λ ( + α + β) + ( + α + β) + ( β) + ln( λ+ 1) 4 2 λ 4 2 λ λ α = ( ln(( λ ) ) ctn(2 ( λ ))) 0 M m. (12) Obviously, Eq. (12) descibing the finite defomtion of the eveted spheicl shell is nonline eqution with espect to δ nd m. Fom Eqs. (11), (12), we cn get the eltions mong the stuctue pmete δ, the mteil pmetes α, β nd the thickness η of the eveted spheicl shell.. Numeicl simultions Figs. 1-2 show the effects of the initil thickness δ, the mteil pmetes α nd β on the finite defomtion of the eveted spheicl shell composed of the incompessible mteil ().

5 Qulittive nlysis fo solutions of clss of nonline ODE O 5 () Fig. 1 Reltion cuves of δ ~ (b) m, η fo diffeent vlues of α ( β = 0.2 ) () (b)( Fig. 2 Reltion cuves of δ ~ m, η fo diffeent vlues v of β ( α = 2 ) Fom Fig. 1(), it is seen tht, fo the given vlues of β, the inne dius off the eveted spheicl shell inceses with the incesing initil thickness δ, nd the incesing eltions e line l ppoximtely. Menwhile, the inne dius off the eveted spheicl shell inceses withh the incesing the mteil pmete α, tht is to sy, the influence of the mteil pmete α on the inne dius nd thickness of the eveted thickness is significnt. Fom Fig. 1( (b), it is seen tht, fo the given vlues of β, the thickness of the eveted spheicl shell inceses with the incesing initil

6 6 Juxin Li, Wei Zho, Cong Ren nd Jing Zhng thickness δ. Howeve, the influence of the mteil pmete α on the thickness of the eveted thickness is not obvious. Fom Fig. 2(), it is seen tht, fo the given vlues of α, the inne dius of the eveted spheicl shell inceses with the incesing initil thickness δ. Moeove, the effect of the mteil pmete β on the inne dius of the eveted spheicl shell is not obvious. Fom Fig. 2(b), it is seen tht, fo the thickness of the eveted spheicl shell, thee e some simil popeties to the inne dius in Fig. 2(). 4. Conclusions The nonline equtions descibing the finite defomtion fo n eveted spheicl shell composed of dil tnsvesely isotopy Mooney-Rivlin mteil is consideed. The implicit solution is obtined by using the incompessible condition nd the semi-invese method. The conclusions show tht (1) The thinne the initil spheicl shell is, the bigge the defomed inne dius is. (2) The thinne the initil spheicl shell is, the thinne the eveted spheicl shell is. () The influence of the pmete α mesuing the degee of nisotopy of the mteil on the inne dius nd thickness of the eveted thickness is significnt. (4) The influence of the mteil pmete β on the inne dius nd thickness of the eveted thickness is not obvious. Acknowledgement This wok ws suppoted by the Ntionl Ntul Science Foundtion of Chin (No ), the Pogm fo Lioning Excellent Tlents in Univesity (No. LR ) nd the Fundmentl Resech Funds fo Centl Univesities (No. DC ). Refeences [1] G.S. Amnni. Defomzioni finite dei solidi elstici isotopi, II. Nuovo Cimento, 10(191), [2] J.L. Eicksen. Invesion of pefectly elstic spheicl shell. A. Angew Mth Mech, 5(1955),

7 Qulittive nlysis fo solutions of clss of nonline ODE 7 [] S.S. Antmn. The evesion of thick spheicl shells. Ach Rt Mech Anlysis, 70(1979), [4] A.J. Szei. On the eveted stte of spheicl nd cylindicl shells. Q Appl Mth, 48(1990), [5] G.M. Jeemih. Infltion nd evesion of spheicl shells of specil compessible mteil. Jounl of Elsticity, 0(199), [6] D.M. Hughton, Y.C. Chen. On the evesion of incompessible elstic spheicl shells. ZAMP, 50(1999), [7] D.M. Hughton, Y.C. Chen. Asymptotic bifuction esults fo the evesion of elstic shells. ZAMP, 54(200), [8] P. Chdwick. The existence nd uniqueness of solutions of two poblems in the Mooney-Rivlin theoy fo ubbe. Jounl of Elsticity, 2(1972), Received: My 1, 2014