Model to predict the mechanical behaviour of oriented rigid PVC

Transcription

1 Louhborouh University Institutional Repository Model to predict the mechanical behaviour o oriented riid PVC This item was submitted to Louhborouh University's Institutional Repository by the/an author. Citation: HITT and MIROSHNYCHENKO, 005. Model to predict the mechanical behaviour o oriented riid PVC. Plastics, Rubber and Composites, 4(), pp Additional Inormation: This article was published in the journal, Pastics, Rubber and Composites [ c Maney]. It is also available at: Metadata Record: Publisher: c Maney Please cite the published version.

2 This article was published in the journal, Plastics, Rubbers and Composites, 4 (005) MODEL TO PREDICT THE MECHANICAL BEHAVIOUR OF ORIENTED RIGID PVC D.J. Hitt and Dmitri Miroshnychenko IPTME, Louhborouh University, Louhborouh, Leicestershire LE TU, UK Department o Mathematics, University o Glasow, 5 University Gardens, Glasow G 8QW, UK ABSTRACT The mechanical properties o PVC sheets can be modiied substantially by both uniaxial and biaxial stretchin o the material above its lass transition temperature. Previous experimental studies have established a clear pattern in the relationship between tensile properties o oriented PVC products and imposed strains. Several mathematical models have been scrutinised to assess whether the established pattern o behaviour can be modelled and predicted. O these, "the ilament theory", proposed by Turner, emered as the best candidate. The ilament theory has been reined and developed urther into "the composite model". In its present orm it ives a ood correlation between predicted and measured yield stress values o oriented riid PVC sheets and is also capable o predictin the "established pattern" o property dependence upon imposed strain.. INTRODUCTION The mechanical properties o riid PVC can be improved substantially by stretchin at temperatures above the lass transition temperature o the polymer, T, which is typically in the rane 75 C 80 C. As PVC has its peak elonation at 90 C [ 4], the mechanical properties o the oriented product can be optimised by drawin at this temperature. Ater stretchin, the material must be held under restraint to ensure that the oriented structure is "rozen-in" on coolin below T to ambient temperature. Removal o the restraint at the draw temperature will enable the stretched material to recover its oriinal dimensions. Experimental data athered rom previous studies at IPTME [5 8] has shown the tensile strenth properties o oriented PVC sheets are related to the strains imposed at 90 C in a very systematic way. This behaviour is clearly ideally suited to the application o mathematical modellin. In pursuit o this oal several mathematical models rom the literature have been evaluated. O these, one particular treatment, "the ilament theory" proposed by Turner [9], emered as havin the best potential to achieve the oal. This theory has been reined and developed into the composite model by Miroshnychenko et al. [0, ]. To relect the application o "the ilament theory" to oriented PVC and its onoin reinement this paper is divided into three main sections: First, a selection o data is presented that illustrates the systematic pattern o tensile strenth versus imposed strain or oriented riid PVC. Second, the ilament theory is presented in terms o its concept, its key assumptions and its use to predict the properties o oriented PVC. The third section will examine the scope or urther reinements to the ilament theory and composite model and their validation.

3 . THE DEPENDENCE OF MECHANICAL STRENGTH OF ORIENTED RIGID PVC UPON IMPOSED STRAINS From the studies at IPTME [5 8] on the eects o orientation on the properties o riid PVC it has been ound that both yield strenth and tensile strenth ollow the same pattern when properties are plotted aainst planar strain. (This is essentially the product o the draw ratios applied to the material, or example, PVC sheets stretched uniaxially to draw ratios o.0 x.0 and equally biaxially to.7 x.7 have the same planar strain o.0.) Fiures and illustrate the systematic pattern or both tensile properties. In both iures a selection o draw ratios have been included adjacent to data points: or unequal draw ratios the direction o property measurement is indicated by the underlined draw ratio. It can be seen that at a iven planar strain tensile strenth can be ranked accordin to the stretchin mode: uniaxial (machine direction, MD) > unequal biaxial (major draw direction) > equal biaxial > unequal biaxial (minor draw direction) > uniaxial (transverse direction, TD). Note: Brady [] advocated the use o the planar strain parameter to correlate data o oriented products and Fiure is consistent with his indins on the yield stress o uniaxially and biaxially oriented PVC x x.7.4 x.4 YIELD STRENGTH (MPa) x.0.4 x.4.4 x.7 Uniaxial MD Unequal Biaxial MD (TD = constant) Equal Biaxial Unequal Biaxial TD (TD = constant) Uniaxial TD PLANAR STRAIN Fiure : Dependence o yield strenth o oriented PVC-U on strains imposed at 90 C

4 00 80 STRENGTH AT BREAK (MPa) x.0.4 x.0.4 x.4.4 x.4.4 x.7.4 x.7.4 x.4 Uniaxial MD Unequal Biaxial MD (TD = constant) Equal Biaxial Unequal Biaxial TD (TD = constant) Uniaxial TD PLANAR STRAIN Fiure : Dependence o strenth at break o oriented PVC-U on strains imposed at 90 C It must be pointed out that the uniaxially oriented sheets were produced under a constant width stretchin mode, i.e., in eect a low strain was applied in the TD so that the draw ratio in this direction was maintained at.0. The patterns o behaviour can be understood in terms o the physical structure o PVC that is well known. This eectively consists o a three-dimensional amorphous network o lon chains linked toether by very small crystallites, o around 0nm in size, that typically account or 0% o the structure. When the material is stretched above T the lon chains are proressively straihtened and become loosely alined in the direction o the applied draw. At the same time the crystallites are drawn into the plane o the sheet. Naturally, the reatest chain alinment, and consequently the reatest enhancement in tensile strenth, will occur in the MD o a material that has been hihly oriented uniaxially. There is, o course a down side, little chain alinment is produced in the TD and this results in very poor tensile strenth relative to the MD. In PVC products this dierential in mechanical perormance may be advantaeous or a serious problem.

5 Replication o the patterns o Fiures and became the prime objective o work to develop a mathematical model. Allied to this the model needs to reconise the actors that modiy the yield and stress-strain behaviour o riid PVC. A tensile sample o unoriented riid PVC at ambient temperature will yield by the ormation o a neck. This is relected by a sharp drop in the measured load-extension curve: the yield point. The necked reion then rows until the whole aue lenth o the sample has underone neckin. This is then ollowed by uniorm deormation o the aue lenth until ailure occurs. However, as the test temperature approaches T, yield behaviour and subsequent deormation is modiied siniicantly (Fiure ) C STRESS (MPa) C 75 C 90 C STRAIN (%) Fiure : Stress - strain curves or PVC-U 05 C A number o other important points can be taken rom Fiure. It shows how PVC deorms in a rubberlike manner at temperatures above T. Also, it neatly illustrates how elonation at break (strain) passes throuh a maximum at around 90 C. The reduction in elonation at 05 C is related to the network structure o PVC. At 00 C the least preect crystallites in the network are melted and so the ability o the network to sustain loadin is diminished and elonation is reduced. This eect continues to reduce the elonation at break o PVC up to temperatures ~ 40 C to 50 C, ater which elonation at break rises aain as suicient destruction o the PVC network has occurred to allow viscous low [4]. In the orientation o PVC to enhance properties, whether by uniaxial or biaxial stretchin just above T, the drawin process is terminated beore the elonation at break is reached. This has the eect o introducin a residual state o stress into the polymer network. In its simplest orm it is assumed that two mechanisms operate to determine behaviour: elastic stretchin o the lon chains and a viscoelastic resistance to the relative motion o the chains. When the material is stretched above T the elastic chain stresses dominate (as the viscoelastic stresses have relaxed), such that i the applied orce is removed, the material will recover to its oriinal dimensions. However, i the material is cooled under restraint and the applied orce is then removed below T, the viscoelastic resistance dominates and acts to oppose recovery so that the sheet remains in the stretched state. This means that to produce yieldin o the oriented product (at ambient 4

6 temperature) it is now necessary to apply a stress that exceeds the yield stress o the unoriented material (Fiure 4). I suiciently hih strain is imposed on riid PVC by orientation above T, no neckin is observed on subsequent testin o tensile samples at ambient temperature, and the yield point is seen as little more than an inlection point in the load extension curve BIAXIALLY ORIENTED TO.75 x.75 STRESS (MPa) x.5 UNORIENTED STRAIN (%) Fiure 4: Uniaxial stress strain curves at ambient or riid PVC sheets biaxially oriented at 90 C. FILAMENT THEORY The content o this section has been drawn rom the published work o Miroshnychenko et al. [0, ]. Much o the previous work on the mathematical modellin o the uniaxial and/or biaxial deormation o polymers concerns itsel with modellin o the deormation process at a temperature above T. While this remains an essential element or any model to predict stressstrain behaviour o a polymer, the work reported here has attempted to take the modellin a stae urther. That is, to develop a model whereby the mechanical behaviour o riid PVC at room temperature can be predicted ater the polymer has been stretched to a prescribed planar deormation above T. To achieve the objective the basis o the model has to be capable o determinin the residual stress and ideally o on to replicate the various yieldin processes underone by the polymer. For the determination o the residual stress several models reported in the literature were considered [0]. From these, two particular treatments, by Turner [9] and by Arruda and Boyce [, 4], were selected or more riorous evaluation usin experimental data on the deormation o rubberlike materials o Treloar [5] and o Kawabata et al. [6]. The composite model, based on the ilament theory o Turner, ave the best correlation, and the ilament theory has been used as the oundation o the model to predict the yield stress o oriented PVC... Sinle ilament ormulation 5

7 This was proposed by Turner [9] to describe the orientation in polymers. He arued that a sinle ilament could be used to represent the averae spatial deormation underone by the chains within a polymer network. Initially, beore stretchin, the ilament is assumed to be the lonest diaonal within a cube (Fiure 5(a)) and ater drawin ilament orientation (Fiure 5(b)) can be represented by a unit vector n = cosα n + cosα n + cosα n, () that characterises the direction o the ilament with respect to n, n and n, which deine the stretched rame. Fiure 5: A sinle ilament in (a) unstretched and (b) drawn coniurations In this coniuration, the anle, α i, between the ilament and a direction o stretch is iven by λi cosα i = ( i =,, ), () λ + λ + λ and the ilament extension, e, is deined as ollows λ + λ + λ e =. () The tension in the ilament, t, ives rise to the stresses and based upon experimental data Turner proposed an empirical relation or the ilament tension 6

8 E e = t, (4) t + -e /emax p in terms o ilament elasticity parameter (or ilament modulus), E, and limit o ilament extensibility, e max. It is clear that the irst term in (4) accurately expresses what happens in practice: at low stretches the ilament tension is directly proportional to the ilament extension and then the ilament tension rises steeply as the ilament extension approaches its maximum. The second term in (4), t p, has been included here ad hoc. It is the pre-existin tension actin upon the ilament that may arise due to actors other than the applied deormation (e.., processin), which will contribute towards the prediction o the yield stress in oriented PVC. Principal tensile orces t i can be determined directly rom their bulk resolution in the deormed coniuration (Fiure 5 (b)) t n + t n + t n = t n, (5) substitutin the expression (4) or the ilament tension. Thus, true tresses or the ilament theory can be expressed in the orm σ σ σ t = = λ t λ λ t = = λ t λ λ t = = λ t λ λ cosα, cosα, cosα. (6).. Application o the ilament theory to predict yield stress in oriented PVC To predict the yield stress o an oriented material the anisotropy resultin rom the stretchin above T has to be taken into account (equal biaxial drawin is a speciic case o this treatment). Durin stretchin above T tension is applied to a ilament, which is rozen in the material as a residual stress on coolin. It is assumed that stresses exceedin the residual stress are now required to produce yieldin in the oriented material. I the initial yield stress o an unoriented material is equal to Y, then ater orientation the principal yield stresses are Y = Y + σ, Y Y = Y + σ, = Y + σ, (7) where σ stand or the rozen in stresses. It is the deviatoric part o overall stress σ and it can be written as 7

9 σ = σ /( σ + σ + σ ), σ = σ /( σ + σ + σ ), σ = σ /( σ + σ + σ ). (8) At this stae, the constitutive equation can be taken in the orm σ = p I +, (9) σ in which p is a hydrostatic pressure, I is the identity tensor, and σ is the ilament stress iven by (6). Hence, these deviatoric stresses (8) can be calculated usin (9) and the ilament theory () (6) and then used to produce values or Y, Y and Y throuh (7). Furthermore, the principal yield stresses (7) can be combined to ive the yield stress in a particular direction θ as Y = Y cos θ + Y sin θ. (0) θ ο ο Yield values in more amiliar directions can be determined as θ = 90 and θ = 0 correspond to the MD and the TD respectively. Thus, i required, the yield strenth can be calculated in any direction relative to the main draw directions. Experimental data on yield stress o oriented PVC A series o PVC-U sheets (0mm by 0mm squares o mm thickness) were uniaxially stretched usin at 90 C and 50mm/min to dierent draw ratios. Ater drawin the sheet was cooled to ambient under restraint. Tensile testpieces were then cut rom the various sheets and yield properties were measured. The ilament theory () (6) depends upon knowlede o a number o physical properties o the material needed to apply the ormula (0). In this case, these have been derived [0] throuh a weihted statistical analysis o the experimental data on yield stress in MD direction shown in Table. This resulted in values o e max =.6, E =.96MPa and t p =6.97MPa or the limit o ilament extensibility, the ilament modulus and the pre-tension respectively. Table also provides a value or the initial yield stress, i.e., Y =7.7MPa. Table : Yield data or uniaxially oriented PVC-U Pre-stretch in MD Number o tests Yield stress, (MPa) Extension at yield, (mm) Unstretched MD TD

10 The predicted yield stress values are plotted as continuous lines in Fiure 6. The line or properties in the MD o uniaxially oriented sheet lies close to the measured yield properties (solid squares): an expected result as the data was used to enerate the material parameters or the ilament theory. There is reasonably ood correlation between the predicted and measured yield values (unilled squares) in the TD or uniaxially oriented sheet. It appears rom Fiure 6 that measured yield values at a planar strain o.5 are inconsistent with the other measured data. This may have been due experimental error (unlikely as the results o 9 tests contributed to the calculated averae) or more likely the oriinal sheet had relatively poor properties prior to stretchin (a actor that has been experienced on a number o occasions durin the work on orientation o PVC at IPTME). Discardin the uniaxial data point in the MD or the pre-stretch o.5 may well have improved the quality o the predicted yield properties. However, even with its inclusion the really positive result that can be seen rom Fiure 6 is that the ilament theory reproduces the systematic pattern o yield stress as a unction o planar strain. The core line o the plot represents the predicted yield stress or equally biaxially oriented PVC and its branches correspond to the predicted yield stress o unequally biaxially oriented sheets in both the major and minor draw directions. Note: as well as determinin properties in MD and TD (Table ), ο ο yield stress was also measured in the directions o θ = 60 and θ = 0 over a similar rane o uniaxial draw ratios [0]. Yield stress values predicted by equation (0) correlated well with this experimental data YIELD STRESS (MPa) x.4.4 x.4.4 x.7.4 x.0.4 x.0.4 x.7 Uniaxial MD Unequal Biaxial MD Equal Biaxial Unequal Biaxial TD Uniaxial TD PLANAR STRAIN Fiure 6: Prediction o yield stress or oriented riid PVC: and represent measured yield values o uniaxially oriented sheets in the MD and TD respectively. 4. COMPOSITE MODEL AND FURTHER DEVELOPMENTS In a recent paper, Miroshnychenko et al. [] have developed the ilament theory urther by couplin it to the approach o Poisson's unction [7] to orm thus the "composite" model. In the development o the composite model a dierent expression or ilament tension (4) has been proposed [] and in its inal reined orm it can be written 9

11 E λ t =, () -λ / λmax where λ is the ilament stretch (with λ max bein its maximum value) iven by λ + λ + λ λ =. () This expression () conveys the asymptotic behaviour o polymer material in the lare-tension limit more neatly than (4). The concept o the composite model is to have a linear element, σ p, that accurately describes the deormation o a polymer network at low strains (the approach o Poisson's unction) coupled with a non-linear element, σ, that beins to exert an inluence at moderate stretches and then dominates at lare deormations (the ilament theory). Thus, the overall stress, σ, can be expressed in the additive orm σ = p I + σ p + σ. () The composite model [] allows us to predict the stress strain relations or elastomeric materials more accurately than either the ilament theory (9) alone or the eiht-chain model []. 4.. Approach o Poisson's unction Turner and Brennan [7] considered the stresses in the area o low strain: startin rom the wellknown stress strain relationships o linear elasticity σ ν ( σ + σ ) = E( λ ), σ ν ( σ + σ ) = E( λ ), σ ν ( σ + σ ) = E( λ ), (4) where σ i and λ i are the principal true stresses and principal stretches respectively, and E is Youn's modulus. Poisson's ratio is one o the core concepts o linear elasticity, and or incompressible materials λ λλ =, it is constant, i.e., ν =/. In order to extend the applicability o (4) into inite elasticity, Turner and Brennan invoked the concept o a Poisson's unction, which corresponds to a variable Poisson's ratio. This concept is not new and its relation to Poisson's ratio has been reported [8]. For thin polymer sheets, one stress (say, σ ) can be considered as neliible and taken to be zero. Ater some manipulation o the equations the stress-strain relations can be written in terms o the dierence o two stresses σ σ p p σ σ p p λ λ = E, + ν λ λ = E, + ν (5) 0

12 in which the Poisson's unction, ν, is iven by ν λ λ + λ λ =. (6) However, due to settin σ to zero, the Poisson's unction (6) is asymmetrical (i.e., it is disadvantaed in the λ stretch direction). Thus, Miroshnychenko et al. [] proposed that ν could be approximated by the interpolation ormula ν λ + λ + λ, (7) which is symmetrical in all three principal stretches and ollows closely the limitin behaviour o (6). 4.. Constitutive equations or the composite model The stresses arisin rom the approach o Poisson's unction (5) and rom the ilament theory (6) constitute the composite model () and can be stated in terms o the dierence o two stresses λ λ E λ σ σ = E + + ν λ / λ λ λ E λ σ σ = E + + ν λ / λ max max λ λ, λ + λ + λ λ λ. λ + λ + λ (8) The model now depends upon knowlede o three material parameters: the elastic modulus, E, the ilament modulus, E, and the maximum ilament stretch, λ max, whilst ν is not a material constant but a variable (7) dependin upon the three principal stretches. The model exhibits non-linear elastic behaviour at hih strains: an intrinsic property o elastomeric material. The composite model has been able to accurately reproduce the experimental data o Treloar [5] and Kawabata et al. [6]. However, to date it has yet to be tested with data on the properties o oriented PVC. Further improvement to the composite model (8) can be achieved by proposin a relationship between the elastic modulus, E, o the linear element, and two other parameters, E and λ max, o the non-linear part. Such a relationship would relect common macromolecular oriin (nature) o linear elastic deormation and o orientation and limitin extensibility in polymer chains. One such relation was proposed in [], E = E λ max, (9) and the composite model (8) complemented by (9) has been named "the ilament model" and it has shown considerable promise. 5. DISCUSSION AND CONCLUSIONS

13 It can be seen that the equations o the ilament theory are quite simple mathematically and the theory has complied with the prime objective o the work: it can be used to predict the systematic variation o yield stress o oriented riid PVC as a unction o imposed planar strain. The use o the theory does depend upon knowin certain properties o the material. However, these can be derived rom the yield properties o the unoriented material and rom a small number o uniaxially oriented sheets. This provides the prospect o bein able to enerate a complete understandin o how any combination o draw ratios miht aect the yield strenth o any oriented PVC ormulation rom the results o a limited number o experiments. This could be extremely valuable when selectin draw ratios or biaxially oriented products, such as pipes, as combinations that produce poor mechanical strenth in the minor draw direction can be identiied and avoided. To have complete conidence in the ilament theory, its predictive capability needs to be tested with urther sets o yield data or a rane o oriented riid PVC ormulations. At the same time it would be extremely interestin to evaluate the reined composite model (8) and to assess whether the composite and ilament models improve the accuracy o predictions or the yield stress in oriented PVC. From that point, the next step would be to address the question: can the composite and ilament models predict the strenth at break values or oriented PVC as a unction o imposed planar strain? Intuitively, the answer would be airmative, as the tendency or strenth at break (Fiure ) is similar to the tendency or yield strenth (Fiure ). Acknowledements: The authors would like to thank Proessor Marianne Gilbert (IPTME, Louhborouh University) and Dr Tony Green (Department o Mathematical Sciences, Louhborouh University) or their uidance and valuable contributions durin the research at IPTME. Also, the support iven to the project by Dr Tony Day (Hydro Polymers Ltd) and by Dr Les Holloway (Wavin Industrial Products Ltd) was reatly appreciated. 6. Reerences. S.H. Pinner, Plastics, 8-85, May 965, and 0-6, June A.J. DeVries and C. Bonnebat, Polymer Enineerin and Science, 6, 9-00, B. Terselius, PhD Thesis, The Royal Institute o Technoloy, Stockholm, D.J. Hitt and M. Gilbert, Materials Science and Technoloy, 8, , D.J. Hitt, Proceedins o the IOM International Conerence, PVC 96 "new perspectives", Brihton, 996, p M. Gilbert, Z. Liu and D.J. Hitt, Polymer Enineerin and Science, 7, , D.J. Hitt and M. Gilbert, Plastics, Rubber and Composites, 9, 49-60, D.J. Hitt and M. Gilbert, Journal o Applied Polymer Science, 89, , D.M. Turner, Workshop on Deormation Modellin or Solid Polymers, Oxord University, July 997, p D. Miroshnychenko, MPhil Thesis, Louhborouh University, UK, 00.. D. Miroshnychenko, W.A Green and D.M. Turner, Journal o the Mechanics and Physics o Solids, 5, , T.E. Brady, Polymer Enineerin and Science, 6, , E.M. Arruda and M.C. Boyce, Journal o the Mechanics and Physics o Solids, 4, 89-4, M.C. Boyce and E.M. Arruda, Rubber Chemistry and Technoloy, 7, 504-5, L.R.G. Treloar, Transactions o the Faraday Society, 4, 59-70, 944.

14 6. S. Kawabata, M. Matsuda, K. Tei and H. Kawai, Macromolecules, 4, 54-6, D.M. Turner and M Brennan, Plastics and Rubber Processin and Applications, 4, 8-88, M.F. Beatty and D.O. Stalnaker, Journal o Applied Mechanics, 5, 807-8, 986.