U.P.B. Si. Bull., Series B, Vol. 76, Iss., 4 ISSN 454 33 PREDICTION OF STRESS-STRAIN BEHAVIOR FOR IMPROVEMENT OF MECHANICAL PROPERTIES OF POLYMERS Afsaneh BAREKAT, Hossein HOSSEINI A nonlinear visoelasti rheologial model is implemented for developing the proess model in thermoforming. This model desribes deformation proess of a sheet during inflation. In this paper by using of a non-linear visoelasti model relationship between stress-strain during the proess is obtained. By using of presented mathematial model it is not only possible to estimate wall-thikness distribution of produts, but also evaluate the kinetis of thermoforming proess to prevent from produt instability suh as warpage and rupture of sheet. Keywords: Mehanial Properties, Visoelasti model, stress-strain behavior. Introdution Polymer proessing for prodution of all forms of polymeri artiles has found a great plae in hemial industries. Thermoforming proess is one of the most popular tehniques in this field. It applies to thermoplasti sheet as a filmforming tehnique for various pakaging appliations suh as medial devies, food ontainers, and pharmaeutials [-8]. Wide appliations of thermoforming are due to its high performane, simpliity, ompatness and relatively low-ost equipment. These issues make it possible to produe omplex, large-sale onfigurations and free form shapes of produts. In thermoforming, a heated plasti sheet is strethed into a mold avity by applying pressure and eventually diret mehanial loading are used [9, ]. Upon ontating of a sheet with the old surfae of the mold, the sheet deformation is terminated. The forming sequene indues a thikness variation in the final part. Besides wall-thikness variation, other problems faing the thermoforming industry are mainly physial instabilities during inflation rupture of sheet and warpage exhibited in the final parts. There are many ways to streth sheets: vauum, air pressure and mehanial aids suh as implementation of a plug. For inreasing the quality of produts suh as narrow wall-thikness tolerane or elimination of frozen-in stresses, a ombination of mehanial and vauum or pressure forming methods may be Leturer, Department of Chemistry, ollege of Chemial Engineering, Mahshahr branh, Islami Azad University, Mahshahr, Iran. e-mail: afsaneh.barekat@yahoo.om Phd, Department of Chemial Engineering, Abadan branh, Islami Azad University, Abadan, Iran. e-mail: pedram465@yahoo.om
78 Afsaneh Barekat, Hossein Hosseini implemented. The proess initially involves the usage of mehanial prestrething with plug and then vauum or pressure forming is applied. Sine quality of final produts is haraterized by their minor wallthikness variation, so evaluation of this property provides estimation to physial properties of polymeri produts suh as strength and toughness. Many researhes have been arried out to investigate the thermoforming proess both analytially and numerially [-4]. However, there is lak of literature about deformation proesses in thermoforming and its effet on wall-thikness uniformity and frozen-in stresses, there is a need for further researh. Present study was onduted to simulate deformation proesses in thermoforming proesses.. Rheologial modeling As mentioned above, warpage predition is very important due to proessing onstraints. It may finally ause system failure. Proper desription of the problem is diretly dependent on the orret seletion of an appropriate rheologial model [5]. Leonov developed a theoretial model in this area [6]: σ + pδ W W e f / θ G (T ) exp{- β w s/ G (T)}. [( -I δ /3) Ws-( -- I δ / 3 ) Ws] () d dt + ω ω γ e ( e e ) ( e e ) γ ω f f () Now, with equations () and for pure shear proess ( dt ), the seond and third equations of system equations () will be as follows: + 4γ θ ( T ) F ( ) + γ θ T F ( ) ( ) ( ) + d ij
Predition of stress-strain behavior for improvement of mehanial properties of polymers 79 ( ) F exp ( β 7.8 ) where From solving of equations system (3) we will have: γ θ ( ) ( ) T F (4) But in pratie there is a problem for appliation of Eq.(). This problem W W arises due to the hoie of seleting strain energy funtion ( I,I ). Most researhers use Mooney-Rivlin potential, but there are differenes between experimental and theoretial results for predition of stress and strain. Results of reent researh show that in various kinematial deformations, the following potential an be used []: W.5G ( I + I 6) From the first equation of the Rheologial model () with ondition (5) and elasti tensors in equation (), following equation for stress an be developed: σ G T ( ) (6) By using the strain energy funtion presented in Eq. (5), in onjuntion with the model in Eq. (), a mathematial desription for the deformation proess ourring in thermoforming an be developed. 3. Wall thikness distribution and deformation proess Consider a polymeri sheet with radius (r3) is heated for prodution of an axisymmetri artile. It is deformed by movement of a plug with radius of rp at onstant veloity Vp, in diretion of z axis. The implemented material is assumed to be inompressible and isotropi. The deformation proess is arried out under isothermal ondition. The deformed sheet ould be onsidered a thin shell, thus the hot polymer an be modeled as a membrane. Therefore, the bending resistane of the hot sheet is ignored and the material thikness is assumed to be small in omparison to dimensions of the material. Three different streth ratios involving in deformation proess are as follows: dξ λ dξ λ r r λ h h 3 ; ; (5) (7)
8 Afsaneh Barekat, Hossein Hosseini where λ, λ and λ3 are the prinipal streth ratios in the meridional, radial and thikness diretions of the membrane, respetively. They are related together by the inompressibility ondition λ λ λ3 and ξ, ξ are length of meridian in deformed and strainless sheet. r, r :: radii of deformed and strainless sheet, respetively. h, h :: thikness of the sheet after and before deformation, respetively. Mehanial pre-strething is a planar strething (pure shear). Therefore, the following onditions exist: λ ; σ 3 ; 3 λ λ (8) With respet to the onditions and the tensors in equation (), the following expression an be written: e ε ; ω ; ; (9) whereε :: rate of deformation in longitudinal diretion. The primary and seondary invariants of tensor are resulted from equation (9) as: I I + + () By utilizing equations (9) and (), following form of equation () an be developed. ( ) + T G p,5 δ σ () ( ) + exp f e T 4 β θ ()
Predition of stress-strain behavior for improvement of mehanial properties of polymers 8 Parameter p is resulted from ondition (8): expressions (9) and () into equation (): σ 3. By substituting ( ) [ β ( + ) ] d ε exp dt 4 ( T ) θ (3) This differential equation defines kinetis of elasti strain during the deformation proess of visoelasti media. The deformation rate is defined as follows: dε H d ln λ dλ ε dt dt λ dt H where ε is Henky strain. There are two different strains: ~ H t ε ln + Total strain ( visous and elasti deformations) a Elasti strain ( where H H ε e λ ( ) ): e exp εe a r~ r r~ з p ~ t t V θ (T ) p ( T ) r~ p Finally, from equation and ondition shown by equation 4 following relationship an be derived. σ σ G ( T ) ( ) (7) For desribing deformation proess at any moment of thermoforming the following onditions an be used. at z~ z~ K r~ r~ ( ) : ~ ф z K θ θ ( ), ~ ф z K, r~ ( ~ ф( z~ K ) λ λ ( z~ K ) λ λ z K ), r~ ( z~ K ) (8) where: r~ ф ( z~ ) :: profile equation in orner surfae of the mold, θ (4) (5) (6)
8 Afsaneh Barekat, Hossein Hosseini θ ф dr ~ ф arоtg d ~ z ( z~ ), ( z~ ) ( z~ ), λ λ :: streth-ratios in the previous deformation adjoin phase, r~ :: funtion of sheet surfae profile in the previous adjoin phase, Quality riteria for wall-thikness variation an be defined as: hmin δ h max (9) h h where min and max are the minimum and maximum wall-thikness of produt, respetively. As non-uniform variations of polymeri sheet thikness during plug-assist vauum forming proess ours in the both first and seond stages, the final distribution funtion of wall-thikness using Eq. (8) and onsidering inompressibility of polymer ( λ λλ3 ) it takes the following form: h ( ) ( r~ ) h r~ ф λ ( r~ ф ) λ ( r~ ф ) () ~ ( ) where h ~ r is funtion of wall-thikness distribution in polymeri sheet at the end of plug-assist proess. By using the seond relationship in Eq. (7), Eq. () an be modified to: ( r~ ) ~ h ф h ( r~ ф ) h 4 λ ( r~ ф ) + b r~ ф where: h : initial thikness of the polymeri sheet, b λ ( z ) λ ( r~ ) λ ( r~ ) ф ф () 4. Results and disussion Variation of stress versus shear rate an be seen in Fig. for ABS-sheet at temperature Т4 С. There is a good agreement between theoretial results from equations (4-6) and experimental results (points).
Predition of stress-strain behavior for improvement of mehanial properties of polymers 83 Fig. shows kinetis of development of total and elasti strains. As shown there is a good agreement between theoretial ( Eqs. 5 and 6) and experimental data for ABS at T4 С Fig.. variation of stress vs. shear rate for ABS at Т4 С, β 3.55, G.635 MPa ;θ.3 se.( ): equation 4; points: experimental data. Fig.. Kinetis of development of total and elasti strains for ABS sheet. ( model ; points: experimental data. β 3.55 ;θ.3 se; a 3.6 ) theoretial Fig. 3. Comparison of model simulation from equation ( ) and experimental ( ) wallthikness distribution for ABS (h mm)
84 Afsaneh Barekat, Hossein Hosseini By implementing Eq. (), wall-thikness distribution for whole sheet an be determined. For evaluation of the theoretially developed model, simulated wall-thikness distribution is ompared with the experimental one (Fig. 3). The experimental data are for ABS sheet that is used widely in various industries suh as food pakaging. As shown, the theoretial and experimental results demonstrate a very satisfatory agreement. The following alulation method enables us to have a quantitative evaluation of produt quality even at designing step. Followings are required data in order to perform this method: a) Geometri parameters of produt suh as radius of sheet and depth of the artile. b) Thikness of the used polymer sheet. ) Obtained results out of thermoforming proess modeling, aording to the desription of deformation proesses mentioned at the beginning of this paper. d) Relaxation time and flexibility parameter of maromoleular hains. By the use of above-mentioned information, it is possible to find following results: It is possible to speify aumulated stresses in meridional and radial diretions at plug-assisted stage and by results of deformation proess modeling. h ~ ( r ) To find ф equation and then we have minimum & maximum amounts of thikness. To alulate wall-thikness distribution by equation 9. To alulate total and elasti strains for preventing from sheet frature during thermoforming proess. To find variation of stress versus shear rate at any moment of thermoforming proess for preventing from warpage during appliation of polymeri artiles. By using of presented mathematial model in this paper it is not only possible to estimate wall-thikness distribution of produt but also evaluate the kinetis of thermoforming proess when neessary. It is important when we want to stop the deformation proess at speial points of mold level at vauum forming stage. For solving this problem, it is neessary to speify only the mutual relation between boundary ontat points of polymer sheet, mold surfae and deformation proess time. There is the following equation for vauum thermoforming time: t pro V μ s arti Vzag k + k RT k + k ()
Predition of stress-strain behavior for improvement of mehanial properties of polymers 85 Now, it is possible to have a suitable relation for kinetis of thermoforming proess by equation of. It is well known that the stage of vauum-forming with a pre-strethed sheet ours too quikly for experiening of relaxation proesses in the polymeri sheet. This will result in this fat that almost all aumulated deformations in polymeri sheet within this stage are elasti. Consequently, the minimization of the aumulation may be realized only by minimizing of the general deformations aumulated in polymeri sheet during this stage. For pratial purposes, this means that the profile of the pre-strethed sheet should be maximally approximated by the profile of the final produt whih ould be assured based on the appliation of a plug with the respetive radius. In ontrast to the seond stage, the stage of plug-assisted an be regulated in the sense that it is tehnially possible to ontrol the motion of the plug. This reates a pratial opportunity that at this stage to organize the relaxation proess of elasti deformations aumulated in a polymeri sheet during the proess of plug-assist forming. The essene of the simplest of many variants used in the realization of this proess lies in the reation of a relaxation pause period between the first and the seond stage. During this period the aumulated elasti deformations are ompletely or partially relaxed and onsequently minor wallthikness variation. So, beause of relaxation pause after plug-assist stage and also implementation of two stage thermoforming proess have minor wallthikness variation and frozen-stresses and onsequently better mehanial properties of thermoformed artiles. 5. Conlusion Beause of implementation of two-stage ombinational thermoforming proess and also relaxation pause after plug-assist stage has been observed minor wall-thikness variation and onsequently better mehanial properties of thermoformed artile. The stage of vauum-forming with a pre-strethed sheet happens quikly for experiening of relaxation proesses in the polymeri sheet. This will result that almost all aumulated deformations in polymeri sheet within this stage are elasti. In ontrast to the seond stage, the stage of plugassisted an be regulated in the sense that it is tehnially possible to ontrol the motion of the plug. This reates a pratial opportunity at this stage to organize the relaxation proess of elasti deformations aumulated in a polymeri sheet during the proess of plug-assist forming.
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