THEORETICAL AND EXPERIMENTAL ANALISYS OF ELASTOMERS

Transcription

1 THEORETICAL AND EXPERIMENTAL ANALISYS OF ELASTOMERS Roque M.R. IST, DEM, Av. Rovsco Pas, Lsboa, Portugal, e-mal: Abstract: The work developed and presented n ths dssertaton had as prmary objectve, the study and analyss of several consttutve models for large deformatons, ether hperelastc or vscoelastc, and t s applcaton on elastomers. In the theoretcal scope, the performance of the man consttutve models used for these type of materals, was evaluated. Ther applcaton was accomplshed by usng data from unaxal, baxal and plane deformaton expermental work. The mechancal behavor of elastomers was analyzed under dstnct modes of deformaton, where vscoelastcty was characterzed by tenson relaxaton and the varaton of the stran rate. The Mullns Effect, characterstc of elastomers, was also evaluated usng contnuous cycles of compresson and decompresson. The fnte element modelng, employed n plastc deformaton of metal wth the ad of elastomers, was equally analyzed. In order to acheve that goal, the large deformaton theory and the hperelastc and vscoelastc consttutve models were used. The correlaton between the theoretcal prevsons and expermental data, reproducng pure modes of deformaton, ether hperelastc or vscoelastc, has dsclosed to be good. Keywords: Elastomers, Expermentaton, Fnte element method, Hperelastcty, Large deformatons, Vscoelastcty.

2 . Introducton The elastomers are, actually, materals of extreme mportance, ether from the technologcal pont of vew or commercal. The success of these materals can be explaned by several aspects, to detach: the rased elastcty and capacty of deformaton that they possess; the low raw materal wastefulness for beng produced n ts fnal form (wthout removal of materal), and ts presence n practcally all the actvty sectors as the automoble, aerospace, cvl constructon, medcne, footwear, sport, among others. However, beyond the recognzed advantages of elastomers, these also have some lmtatons. One of whch s ts low resstance at extreme temperatures. When these are subjected to hgh temperatures, a fast degradaton of ts molecular structure s verfed, on the other hand, under the effect of low temperatures the same one becomes rgd. As t was referenced, the elastomers are materals wth several applcatons, and n ths sense t s of all nterest to know and to understand the way how these materals behave. Flexble tool materals, s one of the applcatons of elastomers and are used n a varety of metal-formng operatons, these ncludng bendng, formng, drawng of varously shaped cups, tube bulgng, embossng, percng and blankng. The use of flexble tool materals elmnates ether the male or the female de and avods the necessty for careful algnng of conventonal matng des. However, n most formng processes usng flexble tools the formng loads are hgh n comparson wth those n equvalent conventonal processes. Tube bulgng s one excepton where formng pressures are comparable to those obtaned n hydraulc or mechancal-bulgng processes. Furthermore, n tube bulgng usng a flexble tool t s well known that frcton between the tube and the elastomers rod helps to buld up axal compressve stress on the tube, whch n turn helps to elmnate the need to devse means of applyng axal compressve force ndependently.. Elastomer Behavor The most mportant characterstc of an elastomer s ts large deformaton capacty, even when t s under the acton of small stresses (Fgure.). From the graph represented n fgure (.) t s verfed that the curves are nonlnear, and therefore t s not possble to defne the Young's modulus (E) of the materal except n the regon of low stran. The rased capacty of deformaton and low modulus of elastcty, are opposte to the propertes of a metal.

3 Nomnal Stran Fgure. Natural rubber expermental data [] from three dfferent assays The mechancal response of elastomers presents some nterestng characterstcs. For example, when an elastomerc test specmen s subjected to smple tenson from ts vrgn state, unloaded, and then reloaded, the stress requred on reloadng s less than that on the ntal loadng for strans up to the maxmum stran acheved durng the ntal loadng. Ths stress softenng phenomenon s known as the Mullns effect and reflects damage ncurred durng prevous loadng. In fgure., four contnuous load cycles are represented (charge/dscharge) to exemplfy ths behavor. The hysteress, verfed n the unloadng phase, s frut of the vscoelastcty present the elastomer and s proportonal to the maxmum stran suffered. Another mportant aspect s related wth the stress relaxaton process, that s, when the deformaton level s kept constant, the force exerted by the elastomer n order to oppose the mposed deformaton dmnshes wth the tme passng untl t reaches an equlbrum state (Fgure.3). The stress relaxaton s proportonal to the suffered deformaton and also to the stran rate. Ths rate nfluences the behavor of the elastomer n such a way that the tenson ncreases wth the ncrease of the stran rate. Nomnal Stran Fgure. Unaxal cyclc stress-stran behavor for the chloroprene rubber [].

4 Fgura.3 Stress relaxaton at dfferent stran rates, for the ntrle rubber []. 3. Large Deformaton Theory 3. Introducton In ths secton t wll be evaluated the way how the elastomer deforms. Not makng any assumpton concernng the dmenson of the deformaton, the resultant stran tensor s vald for any stuaton. However, ths tensor s nonlnear leadng to a complex analyss. Therefore, f the deformaton s small (typcally nferor to 3-4%), an analyss for small deformatons can be used. In the case of elastomers, these present large deformatons (extensons larger than 5%) and therefore t s necessary to derve measures for large deformatons []. 3. Deformaton Gradent Tensor The frst step to defne measures for large deformatons, s to fnd a relaton between the reference or undeformed confguraton and the deformed confguraton of the body (Fgure 3.). Snce the length of dx' can change when gong to the deformed confguraton as well as ts orentaton, we can say that dx' deforms nto dx. The queston then becomes how to relate dx n the deformed confguraton nto dx' of the reference confguraton. Ths can readly be done through the chan rule as: dx x = dx x ' j j ' (3.) Equaton (3.) gves a relatonshp between a materal vector n the undeformed and deformed confguraton. We defne the mappng tself as the deformaton gradent tensor:

5 dx F j = (3.) dx ' where F s the deformaton gradent tensor. It s known from the rules of ndex notaton that F s a second order tensor, snce t has two ndependent ndces. It s also mportant to note that F s not symmetrc. j Undeformed Deformed 3.3 st Pola-Krchoff Stress Fgure 3. Infntesmal materal vector, dx. Cauchy's stress tensor s defned n the deformed confguraton and s thus not practcal to use for large deformaton analyss or expermental measures. Therefore, t must be developed an alternatve stress tensor. The st PK stress s defned such that the total force resultng from the st PK stress multpled by the normal and area n the reference confguraton s the same as the total force resultng from the Cauchy stress tmes the normal and area n the deformed confguraton. If we denote the total force over the nfntesmal area n both confguratons as dp, the Cauchy stress as σ, the nfntesmal deformed area as da, we have: dp = ( σ j n j )da (3.3) Recall that the stress tmes the normal s the tracton force, whch s defned per unt area. The same total force can be generated n the reference confguraton usng the st PK stress T, the normal n' and the nfntesmal area da' as: The queston becomes how the st dp = ( T j n' j )da' (3.4) PK stress s related to the Cauchy stress. To determne ths, we need to wrte the normal and deformed area of the deformed confguraton n terms of the normal and area of the reference confguraton, whch can be done usng the Nanson formula. The forces, defned from stresses, can be equated n two dfferent confguratons: dp ( T n ) da = ( σ )da = j ' j ' j n j (3.5) ( ) ' da' n j da = J F kj n k (3.6)

6 dp ( ' ) da' = σ ( ) ' da' = T j n j j J Fkj n k (3.7) After some manpulaton, the followng relashonshp s attaned: Tj = σ k J ( F ) jk [ T] = J[ σ][ F] T (3.8) 4. Hperelastcty and Vscoelastcty 4. Hperelastcty The mechancs of materals, specally the mechancs of elastomers, s suffcently complex and some hypotheses can appear n the dervaton of a consttutve model. A consttutve model s typcally a phenomenologcal mathematcal model used to descrbe the relaton between the tenson and the extenson n a materal. The phenomenologc models consst of unknown parameters whose determnaton needs expermental data. The expermental data s later approached usng, for example, the method of the mnmum square. It s hghly desrable that the used expermental data s retreved from several assays: unaxal tenson/compresson, baxal and plane deformaton or pure shear. Ths way s the only that can lead to a more accurate consttutve model capable of approachng the deformaton modes that the elastomer wll suffer durng ts lfe cycle. 4. Hperelastc Consttutve Models In order to ft the expermental data (Fgure.), three dstnct hperelastc models were used: Monney-Rvln model: Ogden model: W N = C j = 0,j = 0 ( I ) ( I ) j 3 3 (4.) 8-Chan model: where, N µ α α α W = λ + λ + λ 3 3 = (4.) α N C = W µ I = λ m 9 C =,C =,C3 =,C4 =,C5 = (4.3) W T j = (4.4) λ

7 4.3 Hperelastc Model Comparson As t was prevously stated, the analyzed hperelastc models derve from dfferent formulatons, and then t s natural that they present behavors and dstnct fttngs between t. Thus, t s necessary to select the model that better approaches the expermental results, n such a way that the materal behavor s reproduced wth the mnmum possble error. To llustrate the prevously dsplayed, the hperelastc fttngs wll be argued, havng the expermental data of the natural rubber n fgure. as base. In ths way, t s revealed n fgure 4. the fttngs for the models of Mooney-Rvln [5], Ogden [8] and Arruda-Boyce [9], respectvely for the unaxal, baxal assay and the combnaton of the unaxal, baxal and plane deformaton assays. To refer that, n ths last case, the potental energy functon, W, assume a more general form, because t s retreved from a set of expermental assays that cover dfferent states of deformaton. Nomnal Stran Nomnal Stran Mooney-Rvln Ogden (N=) Ogden (N=3) Arruda-Boyce C C α.9063 µ.3996 α µ Nomnal Stran α µ µ α.3499 µ α λ m Fgure 4. Fttng usng unaxal, baxal and plane deformaton expermental data.

8 4.4 Ogden-Roxburgh and Bergström-Boyce models In the Ogden-Roxburgh model [3] the Mullns effect s modeled by the modfed stran energy functon (assumng ncompressblty), gven by equaton (4.5): ( λ,η ) ηw ( λ) φ ( η ) W = d + (4.5) φ m m + βw d W m W d m m + βw d ( η) = exp d + ( η) W m r π d (4.6) The functon φ ( η), denomnated by damage functon, s a contnuous functon of the damage varable η. After descrbng the man formulaton of the Ogden-Roxburgh model, s nterestng to verfy ts applcaton wth the ad of the commercal program ABAQUS. Ths applcaton s ntended to model the behavor of the chloroprene rubber subjected to contnuous load cycles, shown n fgure.. Fgure 4. shows the approach made by ABAQUS to the expermental data, havng been gven the dscharge curves for the calbraton of the model. Nomnal Stran α = α = r = β = µ = µ = m = Fgure 4. Applcaton of the Ogden-Roxburgh model. The Bergstrom-Boyce model, equaton (4.7), s based on the mcromechancal behavor of the materal and as such s vald for any state of deformaton [4] :

9 m & C τ λ v C λ v B =, B B (4.7) τˆ where ˆ ˆm C C /τ > 0, C e m > 0. The parameters Ĉ, C and m should be expermentaly determned and τ B s gven by the Frobenus norm of the devatorc stress wch acts on network B. True Stran Fgure 4.3 Compressão unaxal cíclca a duas velocdades de deformação. Once the model parameters are determned, s necessary to evaluate ts capacty to ft the expermental data. Wth ths objectve n mnd, fgure 4.4 presents two approaches obtaned by the commercal program ABAQUS. True Stran Fgure 4.4 Unaxal compresson at two stran rates (ABAQUS). Materal parameters used: µ A = 0. 65, µ B = , λ A λ B m = m =. 635, C ˆ = , C =, m =

10 5. Numercal Smulaton - Tube Bulgng Once the consttutve model s characterzed, s useful and nterestng, to be able to use t n a fnte element analyss. Thus, and appealng to ABAQUS, the hperelastc models descrbed earler wll be used to smulate a tube bulgng experment. To demonstrate the capactes of the models, the smulaton elapses n compresson/decompresson cycles. (a) (b) (c) (d) Fgure 5. Trdmensonal vew of the Von Mses stress n the tube:. (a) Undeformed tube; (b) 6 mm compresson; (c) mm compresson; (d) 7 mm compresson (fnal). Punch dsplacement [mm] Fgure 5. Punch force for the dfferent hperelastc models.

11 6. Conclusons The work developed and presented n ths artcle belongs to the elastomers technology and had as man objectve the comparson and applcaton of several consttutve models, such as hperelastc and vscoelastc. Thus, the mathematcal theory of large deformatons was analyzed and ts results were later appled n the determnaton of the tensons for each consttutve model. To smulate the applcaton of the models n a fnte element analyss, the commercal program ABAQUS was used n a tube bulgng experment under dfferent condtons of deformaton and speed. The program as revealed to be adequate n the smulaton of operatons that nvolve elastomers, allowng the analyss of smple hperelastc deformatons and n the evaluaton of Mullns and tme effects, beng ths last parameter of extreme mportance n some practcal cases. The elastomers were modeled usng unaxal, baxal and plane deformaton expermental data. The mportance of the expermental data used became clear, ether by the performed type of expermental work or by the dversty of deformaton modes used and ts later contrbuton n the determnaton of the consttutve model parameters. Ths work stll allowed to nqure some lmtatons/advantages of the potental energy functons studed, ether as the success of expermental data aproxmaton or as practcal applcaton. 7. References [] [] Bosold Mechancs: [3] ABAQUS 6.6 documentaton: Mullns effect n elastomers. [4] Bergström J. S., Large Stran Tme-Dependent Behavor of Elastomerc Materals, 999. [5] Treloar L. R. G., The Physcs of rubber elastcty, 975. [6] Do M. e Edwards S. F., The theory of polymer dynamcs, 986. [7] Thruvarudchelvan S. e Travs F. W., Tube bulgng wth a urethane rod, Journal of Materals Processng Technology, 3, pp , 990. [8] Ogden, R. W., and D. G. Roxburgh, A Pseudo-Elastc Model for the Mullns Effect n Flled Rubber, Proceedngs of the Royal Socety of London, Seres A, vol. 455, pp , 999. [9] Arruda E. M., Boyce M. C., A three-dmensonal consttutve model for the large stretch behavor of elastomers, J. Mech. Phys. Solds, 4, pp , 993