Modelling the Temperature Dependent Biaxial Response of Poly(ether-ether-ketone) Above and Below the Glass Transition for Thermoforming Applications Turner, J. A., Menary, G. H., Martin, P. J., & Yan, S. (2019). Modelling the Temperature Dependent Biaxial Response of Poly(ether-ether-ketone) Above and Below the Glass Transition for Thermoforming Applications. Polymers, 11(6), [1042]. https://doi.org/10.3390/polym11061042

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Download date:23. Jun. 2019 polymers

Article Modelling the Temperature Dependent Biaxial Response of Poly(ether-ether-ketone) Above and Below the Glass Transition for Thermoforming Applications

Josh A. Turner * , Gary H. Menary, Peter J. Martin and Shiyong Yan

School of Mechanical and Aerospace Engineering, Queen’s University, Belfast BT9 5AH, UK; [email protected] (G.H.M.); [email protected] (P.J.M.); [email protected] (S.Y.) * Correspondence: [email protected]; Tel.: +44-(0)28-9097-5523



Received: 15 May 2019; Accepted: 8 June 2019; Published: 12 June 2019


Abstract: Desire to accurately predict the deformation behaviour throughout industrial forming processes, such as thermoforming and stretch blow moulding, has led to the development of mathematical models of material behaviour, with the ultimate aim of embedding into forming simulations enabling process and product optimization. Through the use of modern material characterisation techniques, biaxial data obtained at conditions comparable to the thermoforming process was used to calibrate the Buckley material model to the observed non-linear viscoelastic stress/strain behaviour. The material model was modified to account for the inherent anisotropy observed between the principal directions through the inclusion of a Holazapfel–Gasser–Ogden hyperelastic element. Variations in the post-yield drop in stress values associated with deformation rate and specimen temperature below the glass transition were observable, and facilitated in the modified model through time-temperature superposition creating a linear relationship capable of accurately modelling this change in yield stress behaviour. The modelling of the region of observed flow stress noted when above the glass transition temperature was also facilitated through adoption of the same principal. Comparison of the material model prediction was in excellent agreement with 1 experiments at strain rates and temperatures of 1–16 s− and 130–155 ◦C respectively, for equal-biaxial mode of deformation. Temperature dependency of the material model was well replicated with 1 across the broad temperature range in principal directions, at the reference strain rate of 1 s− . When concerning larger rates of deformation, minimum and maximum average error levels of 6.20% and 10.77% were noted. The formulation, and appropriate characterization, of the modified Buckley material model allows for a stable basis in which future implementation into representative forming simulations of poly-aryl-ether-ketones, poly(ether-ether-ketone) (PEEK) and many other post-yield anisotropic polymers.

Keywords: PEEK; constitutive model; anisotropy; simulation; thermoforming

1. Introduction A semi-crystalline polymer belonging to the family of poly-aryl-ether-ketones, poly(ether-ether-ketone) (PEEK) is an emerging polymer gaining popularity for implementation into a wide range of applications in bulk form and as a matrix material for Short Fibre Reinforced Composites (SFRC) as a result of its superior mechanical qualities combined with high thermal stability and chemical resistance. The anticipated suitability of PEEK for the application of thin walled, thermoformed products is governed by its characteristically large stiffness to mass ratio (2.571 MNm/kg), or specific stiffness,

Polymers 2019, 11, 1042; doi:10.3390/polym11061042 www.mdpi.com/journal/polymers Polymers 2019, 11, 1042 2 of 29 when compared to similar thermoplastics in this field such as poly(ethylene terephthalate) (PET) and polypropylene (PP)—exhibiting stiffness to mass ratios of 1.206 and 1.935 MNm/kg respectively. The industrial practice of thermoforming is widely used for the large-scale manufacture of lightweight, thin-walled polymeric parts with the advantage of producing repeatable, complex products, along with relatively cheap production costs. During the thermoforming process, the thermoplastic material is subject to a range of dynamic multi-axial deformation at strain rates of 1 1–10 s− and higher whilst processing the polymer in its rubber-like state above and below its characteristic glass transition temperature (Tg)[1]. In many applications the relationship between the resultant deformation of the polymeric sheet is generally unknown, leading to local inhomogeneity in final part thicknesses. Trial and error approaches in the hopes of optimising the formed part is often time consuming, causing authors to take a more systematic methodology. This is predominantly achieved through the use of a mathematical material model to represent the deformation behaviour subject to characteristic loading experienced during forming processes. In order for this mathematical model to be accurate in its predictions of the mechanical behaviour of the polymeric material, characterisation at conditions comparable to those experienced during the forming process is encouraged [2–4]—allowing the constitutive model to be calibrated and then implemented into a process simulation [5–7] with the ultimate aim of optimising the forming conditions and/or the formed part. The mechanical properties of PEEK have been widely investigated in previous work. Due to its many applications, authors have rigorously investigated the mechanical response of PEEK subject to various characterisation methods. An extensive review of these is presented by Rae et al. [8]. Mechanical characterisation of PEEK has been primarily focussed on uniaxial stretching [8–12], with authors detailing the strong temperature and deformation rate dependence of specimens. A lack of research concerning the deformation in more than one axis is evident, and has only been investigated recently by the authors [13]. In our previous study, both load-controlled and displacement-controlled testing of thin extruded PEEK specimens were conducted at conditions comparable to the thermoforming process, drawing similar parallels to the temperature and strain rate dependencies found during uniaxial tests as well as noting profound anisotropy when simultaneously stretching in two axes [13]. It is the data from this investigation which serves as the basis for modelling the constitutive biaxial deformation behaviour of PEEK, for the application of thermoforming. Attempts to model the constitutive response of PEEK have only become apparent in the past decade, with mathematical models developed to quantify specific mechanical properties. El-Halabi et al. [14] characterised and modelled the yield stress of PEEK for scaffold cranial implants, through three point bending tests. The data was then used to calibrate homogenized material models and predict local areas of fracture within scaffold designs in finite element models. El-Qoubaa and Othman investigated the discernible influence of deformation rate on the uniaxial tensile and compressive yield stresses over a wide strain rate range [11,15] and temperature range spanning the glass transition [16]. In the first studies a newly proposed constitutive equation was compared with the performance of empirical and physically based models. The modified Eyring equation was shown to be capable of 4 4 1 accurately predicting the compressive yield stress over a strain rate range of 10− –10 s− , and uniaxial 3 3 1 tensile yield stress between 10− –10 s− . Although this provided a fundamental basis for further understanding of their respective applications, during the thermoforming process the polymer is subjected to large, complex strain histories and is typically taken beyond their characteristic yield stress to ensure permanent forming of the product. It would therefore be desirable to understand and be able to model the material response beyond this point. Recent work by Chen et al. [17] recognised the shortcomings of the aforementioned studies, with lack of consideration of the influence of temperature on modelling the post-yield flow stress of PEEK specimens. They initially attempted to model the compressive flow stress above strains of 0.05, given in previous literature [8], using a Johnson Cook (JC) model. It was shown that the JC model could accurately reproduce data at room temperature but at the highest temperature a 38% discrepancy was observable between measured and simulated data. This led to the authors creating a modified Polymers 2019, 11, 1042 3 of 29 version of the JC model, by substituting an altered temperature term and re-calibrating. A maximum deviation of 12% from the experimental flow stress was simulated, with the resultant yield stress 4 2 1 predictions correlating well over the strain rate range of 10− –10 s− and temperatures between 33 and 200 ◦C. Following on from this study, the authors furthered their research into the modelling of uniaxial extension of PEEK around the glass transition [12]. After low strain rate experimental testing 1 (0.01–1 s− ) at temperatures between 100 and 150 ◦C the above-mentioned JC and modified JC models were calibrated as well as a second modified JC model devised to more accurately capture the coupling effect of strain rate and temperature [18], not considered in the previous models. The authors detailed that the original JC model was the most suitable in simulating uniaxial tensile stress-strain behaviour below the glass transition, with the modified models giving maximum deviations of 16% and 21%. Difficulties in modelling flow stress above Tg were documented and hypothesised to be as a result of changing structure within the polymer. It is of particular interest to note that the material model best suited for replicating uniaxial tensile behaviour is unsuitable for modelling compression of PEEK samples. This conclusion further strengthens the argument that in order for an accurate representation of material behaviour, the chosen material model must be calibrated to data at expected modes of deformation experienced during the process under investigation. In order for accurate simulation of material behaviour during deformation an appropriate material model capable of capturing the observed material behaviour during stretching is vital. The physically based, three-dimensional Buckley material model was developed originally to the replicate nonlinear, temperature and strain rate dependent stress/strain behaviour of amorphous polyethylene terephthalate (aPET) near its Tg [19,20]. Several authors have detailed the suitability of the Buckley material model to further model thermoplastics above their respective Tg for many polymer forming processes [7,20–22]. Nixon et al. [23] adopted the Buckley material model for investigation into the forming behaviour of aPET subject to changing external forming parameters. The material model, previously calibrated through displacement-controlled biaxial testing [7], was implemented into a three dimensional finite element simulation of the stretch blow moulding process with the final part shape examined, amongst others. The authors [23] found strong correlation between the experimentally observed final part shapes and those predicted in simulations. By observing the biaxial stress–strain data recorded in a previous study [13] parallels can be drawn between the biaxial deformation behaviour observed for aPET and PEEK, thus warranting the evaluation of this material model to capture the nonlinear, viscoelastic behaviour of PEEK. To the best of the author’s knowledge no previous studies exist in modelling the constitutive biaxial tensile deformation behaviour of PEEK subject to finite deformation. The importance of characterisation through this experimental technique is of particular concern in order to quantify and understand the material behaviour during thermoforming processes. The ultimate aim of this work was to accurately replicate the biaxial stress-strain behaviour of PEEK specimens under a variety of loading conditions through the following research objectives: (a) firstly a suitable material model is identified capable of reproducing the nonlinear viscoelastic constitutive response, (b) the chosen material model is modified to accommodate the post-yield anisotropic behaviour and necking behaviour, (c) then physically calibrated based on previously generated biaxial stress–strain data within the forming window, and (d) the final simulated material behaviour compared to the experimentally measured stress–strain behaviour above and below the glass transition.

2. Constitutive Modelling

2.1. Material Testing Extensive biaxial testing of PEEK samples has been conducted and presented in previous work [13], and hence a brief summary is presented here. Following Dynamic Mechanical Analysis of the PEEK films, of number average molecular weight equal to 45,000 g/mol, a formable temperature window spanning above and below the Tg of 150 ◦C was revealed (130–155 ◦C). Upon realization of this Polymers 2019, 11, 1042 4 of 29 forming window, the 12 µm thick extruded PEEK films were biaxially stretched in the machine (MD) and transverse directions (TD) simultaneously at equal strain rates and displacements, detailed in previousPolymers 2019 work, 11, x [13 ] at the conditions shown in Table1. Samples were stretched using the purpose-built4 of 30 Queen’s Biaxial Stretcher (QBS)—specializing in the replication of temperature, deformation speeds andbuilt modes Queen’s of deformationBiaxial Stretcher equivalent (QBS)—specializing to those experienced in the replication during thermoforming of temperature, and stretchdeformation blow mouldingspeeds and processes. modes of Thedeformation square specimens equivalent were to th heldose by experienced 24 pneumatic during clamps thermoforming around its edges, and stretch shown inblow Figure moulding1, and processes. heated to theThe specifiedsquare specimens temperature were by held two by convection 24 pneumatic heaters: clamps one around placed its above edges, theshown sample in Figure and one 1, and below. heated After to reachingthe specified the desired temperature temperature by two theconvection sample washeaters: simultaneously one placed stretchedabove the in sample two directions and one by below. two servo-controlled After reaching motors, the desired whilst atemperature load cell located the sample centrally was in eachsimultaneously axis recorded stretched the force. in two The directions force and bydisplacement two servo-controlled data were motors, then whilst converted a load to cell true located stress andcentrally nominal in each strain axis as recorded described the by force. O’Connor The forc [24].e and A typically displacement sample data stretched were then equal-biaxially converted to is showntrue stress beside and an nominal unstretched strain counterpartas described in by Figure O’Connor2. Planar [24]. biaxial A typically characterization sample stretched highlighted equal- thebiaxially non-linear is shown viscoelastic beside stress–strainan unstretched behaviour counterpart of the in films, Figure along 1. Planar with post-yieldbiaxial characterization anisotropy at conditionshighlighted equivalent the non-linear to the viscoelastic forming process. stress–strain The true behaviour stress–nominal of the films, strain along data presentedwith post-yield in the companionanisotropy paperat conditions [13] concerning equivalent the to in-plane the forming biaxial process. deformation The behaviourtrue stress–nominal of PEEK provides strain data the foundationpresented in in the which companion the material paper model [13] can concerning be calibrated the inin-plane a step-by-step biaxial approach,deformation similar behaviour to that forof aPETPEEK asprovides presented the by foundation Buckley et al.in [which19,20]. the material model can be calibrated in a step-by-step approach, similar to that for aPET as presented by Buckley et al. [19,20]. Table 1. Specifications of planar biaxial stretching conditions [13]. Table 1. Specifications of planar biaxial stretching conditions [13]. Temperature Range 130–155 ◦C 1 StrainTemperature Rate Range Range 130–1551–16 °C s− Mode ofStrain Deformation Rate Range Equal1–16 Biaxial s−1 (EB) Mode of Deformation Equal Biaxial (EB)

FigureFigure 1. Annotated 1. Annotated photo photo of poly-aryl-ether-ketones, of poly-aryl-ether-ketones, poly(ether-ether-ketone) poly(ether-ether-ketone) (PEEK) (PEEK) film mountedfilm in Queen’s Biaxial Stretchermounted (QBS) [13 in]. Queen’s Biaxial Stretcher (QBS) [13].

Polymers 2019, 11, 1042 5 of 29 Polymers 2019, 11, x 5 of 30

Figure 2. Equal Biaxial deformation of PEEK specimens (a) unstretched and (b) biaxially stretched to a Figure 2. Equal Biaxial deformation of PEEK specimens (a) unstretched and (b) biaxially stretched to ratio of 2 by 2 [13]. a ratio of 2 by 2 [13]. 2.2. Buckley Material Model 2.2. Buckley Material Model It is assumed in the Buckley material model that the total stress is from a contribution of two independentIt is assumed sources: in athe bond-stretching Buckley material and model conformational that the total part—represented stress is from a contribution in 1D by aof parallel two independent sources: a bond-stretching and conformational part—represented in 1D by a parallel network of a spring and dashpot in series. The former is described by a linear elastic spring, to network of a spring and dashpot in series. The former is described by a linear elastic spring, to model model initial linear elasticity, and viscous Eyring formulation for an activated rate process representing initial linear elasticity, and viscous Eyring formulation for an activated rate process representing the the temperature and strain rate dependence of the yield stress. The latter is characterized by an temperature and strain rate dependence of the yield stress. The latter is characterized by an Edwards– Edwards–Vilgis spring and nonlinear dashpot to represent strain hardening and entanglement slippage Vilgis spring and nonlinear dashpot to represent strain hardening and entanglement slippage betweenbetween polymeric polymeric chains chains associated associated withwith beingbeing aboveabove the glass glass transition. transition. The The principal principal values values of of Cauchy stress (σ ) are represented through the addition of the individual contribution of the two Cauchy stress (iσi) are represented through the addition of the individual contribution of the two sourcessources as as shown shown in in Equations Equations (1)–(3): (1)–(3):

b c σi = σi + σi (i = 1, 2, 3), (1) 𝜎 = 𝜎 + 𝜎 (𝑖 = 1, 2,, 3) (1) X3 b b b c σi = s + K i +σ + σ , (2) i ∈ m0 i 𝜎 = 𝑠 + 𝐾 ∑ ∈i=+1 𝜎 + 𝜎 , (2) b b b σm = K ln(J) + σm0, (3) b c 𝜎 =𝐾 ln( 𝐽) + 𝜎, (3) where σi and σi are the bond-stretching and conformational stresses in the principal directions b b respectively. s is the principal deviatoric bond stretching stress, K is the bulk modulus, i is the i ∈ principalwhere true𝜎 and strain 𝜎 in are each the direction, bond-stretchingσb is the and stress conformational at zero strain, stressesσb is the in strainthe principal induced directions hydrostatic m0 m stressrespectively. and J is the 𝑠 determinant is the principal of deviatoric the deformation bond stretching gradient stress, tensor. 𝐾 Deviatoric is the bulk stresses modulus, (sb) ∈ within is the the principal true strain in each direction, 𝜎 is the stress at zero strain, 𝜎 is the strain induced bond stretching arm exhibit viscoelasticity in terms of the deviatoric true strains (e) by Equation (4): hydrostatic stress and 𝐽 is the determinant of the deformation gradient tensor. Deviatoric stresses (𝑠 ) within the bond stretching arm exhibit viscoelade dssticityb sb in terms of the deviatoric true strains (𝑒) 2Gb = + , (4) by Equation (4): dt dt τ b 𝑑𝑒 𝑑𝑠 𝑠 where G is shear modulus arising2𝐺 from = bond + stretching , and τ is the relaxation time—obtained(4) 𝑑𝑡 𝑑𝑡 𝜏 by shifting the linear viscoelastic relaxation time at a reference temperature and structure (τ0∗ ). The shift factors responsible are due to stress (ασ), structure (αs) and temperature (αT) using an Eyring

Polymers 2019, 11, 1042 6 of 29 formulation [25], Vogel–Tammann–Fulcher function [26–28] and Arrhenius equation [29], as shown in Equations (5)–(8):

τ = τ0∗ α0αsαT, (5)  b  Vp∆σm b exp Vsτ RT = oct − ασ  b  , (6) 2RT Vsτoct sinh 2RT    v v   C C  αs = exp , (7) T f T − T∗ T  − ∞ f − ∞    ∆H0 1 1 αT = exp , (8) R T − T∗ b where Vs is the shear activation volume, Vp is the pressure activation volume, τoct is the isotropic b v invariant of the stress state, ∆σm is the mean stress, R is the universal gas constant, C is the Cohen–Turnbull constant, T is the Vogel temperature and ∆H0 is the activation enthalpy. T and T f are ∞ actual and fictive temperature, whilst the reference configurations of these variables are denoted by T∗ and T∗f respectively. In previous work concerning the deformation behaviour of polymers above their characteristic glass transition T f is assumed to equal T [20], but this cannot be assumed for materials deformed below Tg. Post-yield softening is typically observed for this condition when stress is plotted against strain and is captured through variation of fictive temperature with visco-plastic strain, as detailed in Equation (9) [30]:    r εv    j   T = T + ∆T 1 exp   , (9) f f 0 f    v   − − ε0 where T f 0 is the initial fictive temperature, ∆T f is the change in fictive temperature due to rejuvenation, v v ε0 is the strain range in which rejuvenation occurs, εj is the von Mises equivalent viscoplastic strain and r is a fitting parameter of order unity. c In the original material model the conformational stress, σi , is characterised by the hyperelastic Edwards–Vilgis model [31], representative of polymeric chains of finite length physically entangled with one another. The free energy function, Ac, of the Edwards–Vilgis model is defined by Equation (10):

" 2 3 n2 N K T (1+η)(1 α ) P λ c = e B − i A 2 P3 n2 n2 1 α2 λ = 1+ηλ − i=1 i i 1 i !# (10) 3   3 + P + n2 + 2 P n2 ln 1 ηλi ln 1 α λi , i=1 − i=1

23 2 2 1 where Ne is the density of entanglements, KB is Boltzmann’s constant (1.38 10− m kg s− K− ), η is n × the parameter identifying the looseness of entanglements, λi is the network stretches in principal directions and α is the inextensibility of the entanglement network. The constitutive material model was further developed by Adams et al. [31] to incorporate entanglement slippage when above Tg. The viscosity of this dashpot, γ, dictating the onset of ‘lock-up’ of molecular chains, is governed by Equation (11): n ! λmax γ = γ0/ 1 n , (11) − λcrit n n where λmax and λcrit are maximum principal and critical values of the network stretch, with γ0 the initial viscosity of the dashpot. For further insight, an extensive description of the material model has already been detailed elsewhere in previous literature [19,20,32]. Polymers 2019, 11, x 7 of 30

𝜆 𝛾= 𝛾 1− , (11) 𝜆 where 𝜆 and 𝜆 are maximum principal and critical values of the network stretch, with 𝛾 Polymers 2019, 11, 1042 7 of 29 the initial viscosity of the dashpot. For further insight, an extensive description of the material model has already been detailed elsewhere in previous literature [19,20,32]. 2.3. Existing Capabilities of Buckley Material Model 2.2. Existing Capabilities of Buckley Material Model Following calibration to the model described in Section 2.1, the specifics of which are detailed later,Following the simulated calibration equal biaxial to the stress–strain model described behaviour in Section in both 2.1, material the specifics directions of which at a temperature are detailed of 1 130later,◦C the and simulated a strain rateequal of biaxial 1 s− is stress–strain shown in Figure behavi3. Twoour in major both shortcomingsmaterial directions are noticeable. at a temperature Firstly, itof is130 evident °C and that a strain stress rate levels of 1 post-yields−1 is shown in in the Figure machine 3. Two direction major sh (MD)ortcomings are significantly are noticeable. greater Firstly, than thoseit is evident observed that in stress the transverselevels post-yield direction in the (TD)—equal machine direction to a 20% (MD) difference are significantly in total stress greater at failure than betweenthose observed the two in principal the transverse directions. direction The current (TD)—equal capability to a of 20% the traditionaldifference Buckleyin total stress material at failure model isbetween unable the to reproduce two principal this directions. phenomenon, The assuming current capability isotropic deformationof the traditional after Buckley yielding. material Second, model is the inabilityis unable to to accurately reproduce capture this phenomenon, the increased assuming dip in stress isotropic after yielding,deformation known after as yielding. necking, Second, shown inis 1 Figurethe inability3 with to biaxial accurately deformation capture atthe the increased highest dip strain in stress rate (16 after s − yielding,). In the currentknown as model necking, no concern shown isin givenFigure to 3 thewith potential biaxial deformation evolution of at yield the highest stress behaviour strain rate with (16 s strain−1). In the rate. current It is therefore model no necessary concern thatis given modifications to the potential be made evolution in order of foryield accurate stress behaviour representation with ofstrain the equalrate. It biaxial is therefore deformation necessary of PEEKthat modifications specimens. be made in order for accurate representation of the equal biaxial deformation of PEEK specimens.

Figure 3. ComparisonComparison between simulated and experimental experimental Equal Biaxial (EB) deformation behaviour 1 of PEEK in machine andand transversetransverse directionsdirections atatstrain strain rates rates of of 1 1 and and 16 16 s s−-1 at a temperature of 130 ◦°C.C. 2.4. Modification to the Material Model 2.3. Modification to the Material Model As mentioned, one such limitation of the traditional Buckley material model is that associated in the As mentioned, one such limitation of the traditional Buckley material model is that associated calculation of isotropic conformational stresses by the current Edwards–Vilgis model. To accommodate in the calculation of isotropic conformational stresses by the current Edwards–Vilgis model. To the observed post-yield anisotropic behaviour during biaxial stretching of PEEK specimens, the accommodate the observed post-yield anisotropic behaviour during biaxial stretching of PEEK anisotropic Holzapfel–Gasser–Ogden (HGO) material model, initially developed to imitate the specimens, the anisotropic Holzapfel–Gasser–Ogden (HGO) material model, initially developed to anisotropic mechanical response of arterial walls during inflation tests [33], was chosen. imitate the anisotropic mechanical response of arterial walls during inflation tests [33], was chosen. The isochoric strain energy function of the HGO model is calculated through superposition of an The isochoric strain energy function of the HGO model is calculated through superposition of isotropic part, ψiso, associated with the non-collagenous matrix, and an anisotropic part due to the an isotropic part, 𝜓, associated with the non-collagenous matrix, and an anisotropic part due to the orientated fibres, ψaniso. The total strain energy per unit volume, ψ, is detailed by Equation (12): orientated fibres, 𝜓. The total strain energy per unit volume, 𝜓, is detailed by Equation (12):

ψ = ψiso + ψaniso

k1 n h 2 2io = µ(I1 3)iso + (exp k2 (1 ρ)(I1 3) + ρ(I4 1) 1) , (12) − k2 − − − − aniso Polymers 2019, 11, x 8 of 30

𝜓= 𝜓 + 𝜓

= 𝜇(𝐼 −3) + (exp𝑘( 1 − 𝜌)(𝐼 −3) + 𝜌(𝐼 −1) −1), (12) Polymers 2019, 11, 1042 8 of 29 where 𝐼 is the first invariant of the right Cauchy–Green tensor, 𝜇 and 𝑘 are parameters given in MPa, and 𝑘 and 𝜌 are dimensionless parameters. 𝐼 is a pseudo-invariant as defined by Equation where(13): I1 is the first invariant of the right Cauchy–Green tensor, µ and k1 are parameters given in MPa, and k2 and ρ are dimensionless parameters. I4 is a pseudo-invariant as defined by Equation (13): 𝐼 = 𝜆 cos 𝛽+ 𝜆 sin 𝛽 𝜆 2 2 2 ,2 (13) I4 = λ1 cos β + λ2 sin β λ1, (13) where 𝛽 is the fibre angle measured between the MD axis and the direction of fibre alignment, with where β is the fibre angle measured between the MD axis and the direction of fibre alignment, with λ1 𝜆 and 𝜆 equal to the total stretches in principal directions. The stress components in each direction and λ2 equal to the total stretches in principal directions. The stress components in each direction are are then calculated by Equation (14): then calculated by Equation (14):

𝛿𝜓 δψ δψ 𝛿𝜓 𝜎 = 𝜆 , 𝜎 = 𝜆 . (14) σ11𝛿𝜆= λ1 ,σ22 =λ2 .𝛿𝜆 (14) δλ1 δλ2

AlthoughAlthough clearlyclearly nono fibres,fibres, collagenouscollagenous oror otherwise,otherwise, areare presentpresent withinwithin thethe biaxiallybiaxially stretchedstretched PEEKPEEK samples,samples, oneone cancan confidentlyconfidently hypothesisehypothesise that the observableobservable didifferencesfferences inin stressstress valuesvalues seenseen post-yieldpost-yield betweenbetween thethe MDMD andand TDTD areare duedue toto thethe molecularmolecular alignmentalignment ofof polymericpolymeric chainschains inin thethe underlyingunderlying microstructuremicrostructure of thethe material.material. As a resultresult ofof thethe analogyanalogy betweenbetween thethe polymericpolymeric chainschains andand the fibres, fibres, the the fibre fibre angle angle can can be be defined defined as th asat that between between the MD the MDand direction and direction of proposed of proposed chain chainalignment. alignment. AA representationrepresentation ofof thethe modifiedmodified BuckleyBuckley materialmaterial modelmodel cancan bebe seenseen inin FigureFigure4 4.. ItIt is is with with this this proposedproposed modelmodel configurationconfiguration thatthat calibrationcalibration will be performedperformed using the extensive displacement- controlledcontrolled biaxialbiaxial datadata providedprovided inin previousprevious workwork [[13]13] overover thethe widewide temperaturetemperature andand strainstrain raterate windowwindow associatedassociated withwith thethe real-lifereal-life thermoformingthermoforming process.process.

Figure 4. Spring and dashpot assembly of the proposed Buckley material model including modifications toFigure the conformational 4. Spring and part. dashpot assembly of the proposed Buckley material model including modifications to the conformational part. 3. Fitting of Constitutive Model 3. Fitting of Constitutive Model Calibration of the proposed material model was conducted using previously detailed biaxial data 1 at strainCalibration rates between of the 1proposed and 16 s −materialand temperatures model was ofconducted 130–155 ◦usingC, for previously equal biaxial detailed and constant biaxial −1 widthdata at modes strain ofrates deformation. between 1 and From 16 Equation s and temperatures (4) it can be seen of 130–155 that the °C, shear for equal modulus, biaxial Gb ,and is necessary constant inwidth modelling modes the of deformation. initial stiffness From response Equation exhibited (4) it can in the be trueseen stress–nominalthat the shear modulus, strain curves. Gb, isAssuming necessary incompressibilityin modelling the and initial isotropy, stiffness correct response for both exhibited material directionsin the true subject stress–nominal to small strains, strain the Gcurves.b of a givenAssuming material incompressibility can be commonly and isotropy, calculated correct as a thirdfor bo ofth the material Young’s directions modulus. subject Between to small a nominal strains, 1 strainthe Gbrange of a given of 0–0.02 material the Young’scan be commonly modulus ofcalculated PEEK samples as a third at 130 of the◦C Young’s and a strain modulus. rate of Between 1 s− was a found to be 750 MPa, corresponding to a value of 250 MPa for the shear modulus. It is noted that the tensile elastic modulus of PEEK specimens appears to be unaffected by temperature, deformation rate and material orientation [13]. Polymers 2019, 11, x 9 of 30 nominal strain range of 0–0.02 the Young’s modulus of PEEK samples at 130 °C and a strain rate of 1 s-1 was found to be 750 MPa, corresponding to a value of 250 MPa for the shear modulus. It is noted thatPolymers the2019 tensile, 11, 1042 elastic modulus of PEEK specimens appears to be unaffected by temperature,9 of 29 deformation rate and material orientation [13]. 3.1. The Eyring Process in the Bond Stretching Component 3.1. The Eyring Process in the Bond Stretching Component The dependency of the yield stress of thin PEEK specimens as a function of temperature and The dependency of the yield stress of thin PEEK specimens as a function of temperature and strain rate, is evident from previous literature [13] and highlighted in Figure5. strain rate, is evident from previous literature [13] and highlighted in Figure 5.

Figure 5. Influence of strain rate on resultant stress-strain behaviour at temperatures of 130 and 150 ◦C Figurein the machine5. Influence direction of strain (MD). rate on resultant stress-strain behaviour at temperatures of 130 and 150 °C in the machine direction (MD). One such way of modelling the deformation rate dependency of yield stresses is through the use of theOne Eyring such activation way of modelling rate process. the deformation Buckley et al. rate [20 dependency] previously of concluded yield stresses that theis through contribution the use of ofbond the stretchingEyring activation on the totalrate process. stress can Buckley be described et al. [20] by Equationpreviously (15): concluded that the contribution of bond stretching on the total stress can be described by Equation! (15): p  σy   √2 1 ξ + ξ2µ V  6R  1 dλ  0 s  = p ln + ln − , (15) T 2(1 + ξ)V + √2 1 ξ + ξ2V  λy dt  (2 ξ)RT  p s − 𝜎 6𝑅 − 1 𝑑𝜆 √21− 𝜉+ 𝜉 𝜇𝑉 = ln +ln , (15) 𝑇 𝜆 𝑑𝑡 (2− 𝜉)𝑅𝑇 where Vs and V2p(1+are 𝜉 shear)𝑉 + and √2 pressure1− 𝜉+ activation 𝜉 𝑉 volumes, σy is the true stress at yield, T is the absolute temperature, λ is the total stretch, R is the universal gas constant, µ0 is the limiting viscosity and ξ is a parameter concerned with the biaxial stretching conditions, governed by Equation (16): where 𝑉 and 𝑉 are shear and pressure activation volumes, 𝜎 is the true stress at yield, 𝑇 is the 𝜆 1 dλ𝑅 θ dλ 𝜇 absolute temperature, is the total stretch, 2 =is the universal1 gas constant, is the limiting viscosity and 𝜉 is a parameter concernedλ2 withdt theλ1 biaxdt ial stretching conditions, governed by Equation (16): 2θ + 1 ξ = , ξ [0, 1], (16) 1 𝑑𝜆θ + 2𝜃 𝑑𝜆∈ = where θ is a ratio of the in-plane natural strain𝜆 𝑑𝑡 rate.𝜆 Values𝑑𝑡 for Vs and Vp were computed using a similar methodology as Dooling et al. [21]. Using previous work by Rae et al. [8], the ratio of uniaxial tensile and compressive yield stresses of PEEK were plotted2𝜃 + 1 and shown in Figure6. The relationship between the 𝜉= ,𝜉 ∈0,1, (16) activation volumes and ratio of tensile and𝜃+2 compressive yield stresses are given by Equation (17) [34]:

σyc 1 + √2Vp/Vs = . (17) σyt 1 √2Vp/Vs − Polymers 2019, 11, x 10 of 30 where 𝜃 is a ratio of the in-plane natural strain rate. Values for 𝑉 and 𝑉 were computed using a similar methodology as Dooling et al. [21]. Using previous work by Rae et al. [8], the ratio of uniaxial tensile and compressive yield stresses of PEEK were plotted and shown in Figure 6. The relationship between the activation volumes and ratio of tensile and compressive yield stresses are given by Equation (17) [34]:

𝜎 1+ √2 𝑉⁄𝑉 = . (17) Polymers 2019, 11, 1042 𝜎 1− √2 𝑉⁄𝑉 10 of 29

Figure 6. Tensile and compressive yield stresses plotted against natural log of true strain rate [11]. Figure 6. Tensile and compressive yield stresses plotted against natural log of true strain rate [11]. 1 From Figure3, at a chosen reference strain rate of 1 s − , the ratio of yield stresses is 1.26—close to theFrom value Figure of 1.3 widely3, at a chosen discerned reference for many strain glassy rate polymersof 1 s−1, the [21 ratio]. From of yield Equation stresses (17) is the1.26—close ratio of activationto the value volumes of 1.3 widely is calculated discerned to be for 0.0821. many From glassy Equation polymers (15) [21]. when From the Equation equal-biaxial (17) the true ratio yield of σy stressactivation over volumes temperature, is calculatedT , is plotted to be against0.0821. From the natural Equation logarithm (15) when of the the true equal-biaxial strain rate true at yield, yield   𝜎 ln 1 dλ , a linear relationship is noted as observed in Figure7. Taking an average of the gradients of the stressλy dt over temperature, 𝑇, is plotted against the natural logarithm of the true strain rate at 1 Eyringyield, ln plots for, eacha linear temperature, relationship equal is tonoted 0.0167 as MPa observed K− , the in activationFigure 7. volumesTaking an can average be calculated of the 3 3 1 4 3 1 from Equation (15), with values of 1.714 10− m mol− and 1.407 10− m mol− determined for Polymersgradients 2019 of, 11 the, x Eyring plots for each temperature,× equal to 0.0167 ×MPa K−1, the activation volumes11 of 30 Vs and Vp respectively. can be calculated from Equation (15), with values of 1.714 × 10−3 m3 mol−1 and 1.407 × 10−4 m3 mol−1 determined for 𝑉 and 𝑉 respectively.

Figure 7. Eyring plots for EB deformation of PEEK between temperatures of 130–150 ◦C, at a nominal Figure 7. Eyring plots for EB1 deformation of PEEK between temperatures of 130–150 °C, at a nominal strain rate range of 1–16 s− . strain rate range of 1–16 s−1.

The temperature effect on decreasing yield stress is incorporated into the bond stretching component of the Buckley material model through the limiting viscosity, 𝜇, is given by Equation (18):

𝜆 𝑅𝑇 √2𝑉𝜎 2𝑉𝜎 𝜇 = √2 sinh exp , (18) 𝜆 𝑉 6𝑅𝑇 3𝑅𝑇 where 𝜎 is the bond stretching stress. Traditional methods for determining this variable require the identification of a rubber-like plateau within isometric true stress-temperature plots. It is at the temperature of this plateau that the contribution to total stress by bond stretching is assumed to decrease to zero and the total stress is now solely a product of the conformational arm. In this work the true stresses were established at a nominal stretch of 1.2, just after yielding, for EB deformation at a nominal strain rate of 1 s−1 in order to minimise any potential self-heating effects. The isometric stress against temperature is shown in Figure 8.

Polymers 2019, 11, x 11 of 30

Figure 7. Eyring plots for EB deformation of PEEK between temperatures of 130–150 °C, at a nominal Polymersstrain2019 rate, 11, range 1042 of 1–16 s−1. 11 of 29

The temperature effect on decreasing yield stress is incorporated into the bond stretching The temperature effect on decreasing yield stress is incorporated into the bond stretching component of the Buckley material model through the limiting viscosity, 𝜇, is given by Equation (18):component of the Buckley material model through the limiting viscosity, µ0, is given by Equation (18):   √ b ! 2V σb λ RT 2V sσ 2𝑉 𝜎p  µ0 = √𝜆2𝑅𝑇. sinh√2𝑉𝜎 exp , (18) 𝜇 = √2 Vsinhs 6RTexp  3RT,  (18) 𝜆 𝑉λ 6𝑅𝑇 3𝑅𝑇 where σb is the bond stretching stress. Traditional methods for determining this variable require where 𝜎 is the bond stretching stress. Traditional methods for determining this variable require the the identification of a rubber-like plateau within isometric true stress-temperature plots. It is at the identification of a rubber-like plateau within isometric true stress-temperature plots. It is at the temperature of this plateau that the contribution to total stress by bond stretching is assumed to temperature of this plateau that the contribution to total stress by bond stretching is assumed to decrease to zero and the total stress is now solely a product of the conformational arm. In this work decrease to zero and the total stress is now solely a product of the conformational arm. In this work the true stresses were established at a nominal stretch of 1.2, just after yielding, for EB deformation at a the true stresses were established1 at a nominal stretch of 1.2, just after yielding, for EB deformation nominal strain rate of 1 s− in order to minimise any potential self-heating effects. The isometric stress at a nominal strain rate of 1 s−1 in order to minimise any potential self-heating effects. The isometric against temperature is shown in Figure8. stress against temperature is shown in Figure 8.

Figure 8. Isometric stress-temperature plot between a temperature range of 130–150 ◦C, at a strain rate 1 of 1 s− .

As can be seen in Figure8, the total true stress levels follow an almost linear decrease with increasing temperature with a lack of an apparent plateau region evident. In this case the bond stretching stress is assumed to be fully relaxed at the highest temperature of 155 ◦C, therefore the rubber-like stress, σc, assumed to be independent of temperature was considered to be equal to the total stress at a temperature of 155 ◦C (6.77 MPa). The bond stretching stresses can then be calculated by subtracting the rubber-like stress from the total stress (Equation (1)). Values of σb at each temperature were substituted into Equation (18) to calculate the limiting viscosity for each temperature in the forming window, shown in Table2. Modelling this viscosity on specimen temperature is accounted for by the previously mentioned Vogel–Fulcher–Tammann and Arrhenius functions, as shown in Equation (19):    C C ∆H ∆H  =  v v + 0 0  µ0 µ0∗ exp . (19) T f T − T∗ T RT − RT∗  − ∞ f − ∞ A least squares curve fitting procedure was performed to characterise these parameters, with

µ0∗ equal to 92.66 MPa—the limiting viscosity at the reference temperature of the characterisation window, T∗, of 130 ◦C (Table1). For this fitting procedure, for temperatures below the glass transition, T f is preliminarily considered to be equal to the previously identified Tg of PEEK (150 ◦C), until Polymers 2019, 11, x 12 of 30

Figure 8. Isometric stress-temperature plot between a temperature range of 130–150 °C, at a strain rate of 1 s−1.

As can be seen in Figure 8, the total true stress levels follow an almost linear decrease with increasing temperature with a lack of an apparent plateau region evident. In this case the bond stretching stress is assumed to be fully relaxed at the highest temperature of 155 °C, therefore the rubber-like stress, 𝜎, assumed to be independent of temperature was considered to be equal to the total stress at a temperature of 155 °C (6.77 MPa). The bond stretching stresses can then be calculated by subtracting the rubber-like stress from the total stress (Equation (1)). Values of 𝜎 at each temperature were substituted into Equation (18) to calculate the limiting viscosity for each temperature in the forming window, shown in Table 2.

Table 2. Bond-stretching stresses and limiting viscosities for different temperatures at strain of 0.2 at 1 s−1, for EB deformation. Temperature, Total True Conformational Bond-Stretching Limiting 𝒄 𝒃 T (°C) Stress, σ (MPa) Stress, 𝝈 (MPa) Stress, 𝝈 (MPa) Viscosity, 𝝁𝟎 (MPa) 130 34.10 6.77 27.33 92.66 135 28.36 6.77 21.59 39.68 Polymers 1402019 , 11, 1042 22.87 6.77 16.10 17.15 12 of 29 145 18.09 6.77 11.32 8.08 150 12.29 6.77 5.52 2.74 temperatures above this in which the fictive temperature is assumed to be equal to the specimen 155 6.77 6.77 0 1 0 temperature. Curve fitting resulted in values of ∆H0 = 301.2 kJ mol− , T = 465 K, and Cv = 399 K, Modelling this viscosity on specimen temperature is accounted for by∞ the previously mentioned producing the plot of limiting viscosity against temperature as seen in Figure9. Vogel–Fulcher–Tammann and Arrhenius functions, as shown in Equation (19): Table 2. Bond-stretching stresses and limiting viscosities for different temperatures at strain of 0.2 at 1 s 1, for EB deformation. ∗ 𝐶 𝐶 ΔH ΔH − 𝜇 =𝜇 exp − ∗ + − ∗ . (19) 𝑇 − 𝑇 𝑇 − 𝑇 𝑅𝑇 𝑅𝑇 Total True Stress, Conformational Bond-Stretching Limiting Viscosity, Temperature, T (◦C) c b σ (MPa) Stress, σ (MPa) Stress, σ (MPa) µ0 (MPa) A least squares curve fitting procedure was performed to characterise these parameters, with 𝜇∗ 130 34.10 6.77 27.33 92.66 equal to 92.66135 MPa—the limiting 28.36 viscosity at the 6.77 reference temperature 21.59 of the characterisation 39.68 window, 𝑇140∗ , of 130 °C (Table 22.87 1). For this fitting 6.77 procedure, for 16.10 temperatures below 17.15 the glass 145 18.09 6.77 11.32 8.08 transition, 𝑇 is preliminarily considered to be equal to the previously identified Tg of PEEK (150 150 12.29 6.77 5.52 2.74 °C), until temperatures above this in which the fictive temperature is assumed to be equal to the 155 6.77 6.77 0 0 −1 specimen temperature. Curve fitting resulted in values of ΔH = 301.2 kJ mol , 𝑇 = 465 K, and 𝐶 = 399 K, producing the plot of limiting viscosity against temperature as seen in Figure 9.

Figure 9. Plot of limiting viscosity against temperature showing non-linear curve fitting to Vogel– Fulcher–Tammann and Arrhenius functions.

3.2. Evolution of Fictive Temperature in the Bond Stretching Part

As mentioned in the previous section the evolution of fictive temperature, T f , is important in capturing post-yield stress-softening effects when near the glass transition—the evolution of this variable being primarily dependent on the four parameters as outlined in Equation (9). Initial fitting 1 of these to reference EB data at a temperature of 130 ◦C and a deformation rate of 1 s− , yielded v values of T f 0 = 419 K, ∆T f = 7 K, ε0 = 0.2, and r = 1.6. As observed in previous work the biaxial stress–strain curves of PEEK [13], with increasing strain rate creating a larger and broader decrease in post-yield stress levels (Figure 10) whilst increasing temperature is correlated with a decrease in these stresses. Previous work in modelling the deformation behaviour of glassy polymers has noticed the complexity of capturing the evolution of T f with increasing strain rate [30]. By probing Equation (9) the parameter ∆T f was identified as that responsible for the magnitude of the drop observed in post-yield stresses with increasing values of ∆T f producing elevated values of T f —resulting in a softer material response after yielding and thus, lower associated true stress levels. Individual values of ∆T f were then manually fitted to each test case of varying combinations of temperature and strain rate to best represent the experimental data, with Figure 10 showing this dependence of ∆T f on strain rate and temperature. It is clear from the Figure that with increased strain rate the value of ∆T f is seen to Polymers 2019, 11, x 13 of 30

Figure 9. Plot of limiting viscosity against temperature showing non-linear curve fitting to Vogel– Fulcher–Tammann and Arrhenius functions.

3.2 Evolution of Fictive Temperature in the Bond Stretching Part

As mentioned in the previous section the evolution of fictive temperature, 𝑇, is important in capturing post-yield stress-softening effects when near the glass transition—the evolution of this variable being primarily dependent on the four parameters as outlined in Equation (9). Initial fitting of these to reference EB data at a temperature of 130 °C and a deformation rate of 1 s−1, yielded values of 𝑇 = 419 K, ∆𝑇 = 7 K, 𝜀 = 0.2, and 𝑟 = 1.6. As observed in previous work the biaxial stress– strain curves of PEEK [13], with increasing strain rate creating a larger and broader decrease in post- yield stress levels (Figure 10) whilst increasing temperature is correlated with a decrease in these stresses. Previous work in modelling the deformation behaviour of glassy polymers has noticed the complexity of capturing the evolution of 𝑇 with increasing strain rate [30]. By probing Equation (9) the parameter ∆𝑇 was identified as that responsible for the magnitude of the drop observed in post- yield stresses with increasing values of ∆𝑇 producing elevated values of 𝑇—resulting in a softer material response after yielding and thus, lower associated true stress levels. Individual values of

∆𝑇 were then manually fitted to each test case of varying combinations of temperature and strain Polymers 2019, 11, 1042 13 of 29 rate to best represent the experimental data, with Figure 10 showing this dependence of ∆𝑇 on strain rate and temperature. It is clear from the Figure that with increased strain rate the value of ∆𝑇 isincrease, seen to correlatedincrease, correlated with an increased with an dipincreased in stresses dip associatedin stresses afterassociated yielding after occurs, yielding with occurs, the opposite with thebeing opposite true with being increasing true with specimen increasing temperature–as specimen temperature shown when – as individuallyshown whenplotted individually in Figure plotted 11. in Figure 11.

Figure 10. Observed variation drop in post-yield stress with increasing strain rate at a temperature of PolymersFigure 2019 ,10. 11 , Observedx variation drop in post-yield stress with increasing strain rate at a temperature 14of of 30 130 C. 130 °C.◦

Figure 11. ∆Tf against temperature for EB stretching of PEEK. Figure 11. ∆Tf against temperature for EB stretching of PEEK. One such method of potential implementation of this ∆T f evolution with strain rate and temperature is through the method of time-temperature superposition (TTS).∆𝑇 Equation (20) demonstrates the One such method of potential implementation of. this evolution with strain rate and temperaturecalculation of is a shiftthrough factor, theα, withmethod associated of time-temperature strain rate, ε: superposition (TTS). Equation (20) demonstrates the calculation of a shift factor, 𝛼, with .associated.  strain rate, 𝜀: C ε ε 1 − re f log10(α) = . . , (20) 𝐶𝜀C2−+ 𝜀ε εre f 𝑙𝑜𝑔(𝛼) = − , (20) 𝐶 + 𝜀 − 𝜀

−1 where 𝐶 and 𝐶 are constants, and 𝜀 is the reference strain rate chosen as 1 s . The constants in Equation (20) were fitted in order to shift the temperature data above a strain rate of 1 s−1 in Figure

10 intending to create a linear relationship of ∆𝑇 with shifted temperature. Values of −0.08915 and 5.905 for 𝐶 and 𝐶 respectively were derived resulting in Figure 12—a master curve of ∆𝑇 against shifted temperature, taking into account the rate of deformation. A strong linear relationship is exhibited in which an R2 value of 0.9637 is exhibited. This relationship is defined by Equation (21), with shifted temperature in Kelvin:

∆𝑇 = −0.2525 . 𝑇 + 109. (21)

Polymers 2019, 11, 1042 14 of 29

. 1 where C1 and C2 are constants, and εre f is the reference strain rate chosen as 1 s− . The constants in 1 Equation (20) were fitted in order to shift the temperature data above a strain rate of 1 s− in Figure 10 intending to create a linear relationship of ∆T with shifted temperature. Values of 0.08915 and 5.905 f − for C1 and C2 respectively were derived resulting in Figure 12—a master curve of ∆T f against shifted temperature, taking into account the rate of deformation. A strong linear relationship is exhibited in which an R2 value of 0.9637 is exhibited. This relationship is defined by Equation (21), with shifted temperature in Kelvin: ∆T = 0.2525 . T + 109. (21) Polymers 2019, 11, x f − shi f ted 15 of 30

Figure 12. ∆Tf against shifted temperature following use of time-temperature superposition. Figure 12. ∆Tf against shifted temperature following use of time-temperature superposition. 3.3. Calibration of HGO Model in the Conformational Part 3.3. Calibration of HGO Model in the Conformational Part As previously highlighted in Equation (1) the total stress is a result of the addition of two components:As previously the bond highlighted stretching in stress Equation and conformational (1) the total stress stress. is In a theresult traditional of the Buckleyaddition materialof two components:model the conformational the bond stretching stress stress is predicted and conformati by an isotropiconal stress. Edwards–Vilgis In the traditional formulation, Buckley material but as modelalready the discussed conformational these current stress methods is predicted are unsuitable by an isotropic for the biaxial Edwards–Vilgis deformation formulation, behaviour observed but as alreadywith the discussed extruded PEEKthese sheetscurrent used methods in our studies—whichare unsuitable for clearly the demonstratebiaxial deformation some orientation behaviour in observedthe MD, demonstratedwith the extruded through PEEK the sheets larger stressused responsein our studies—which in the MD direction clearly compared demonstrate to the some TD. orientationIn order to modelin the theMD, post-yield demonstrated deformation through behaviour the larger the stress simulated response bond stretchingin the MD stress direction must comparedbe subtracted to the from TD. the Intotal order experimental to model the stress, post-yield using deformation the characterised behaviour parameters the simulated derived inbond the stretchingprevious section. stress must These be equal-biaxial subtracted rubber-likefrom the to stressestal experimental are shown stress, in Figure using 12 in the the characterised machine and 1 parameterstransverse directions,derived in atthe the previous reference section. temperature These equal-biaxial of 130 ◦C and rubber-like strain rate stresses of and 1are s− shown. A least in Figuresquares 12 curve in the fitting machine procedure and transverse was then directions, conducted at to the best reference fit the parameters temperature in of Equation 130 °C and (12) strain to the rateexperimental of and 1 s− conformational1. A least squares stresses, curve fitting giving procedure values of wasµ = 1.00thenMPa, conductedk = 2.16 to best MPa, fitk the= parameters1.50 10 2 , 1 2 × − inρ =Equation0.585 and (12)β = to4.02 the experimental10 11 0. The co generatednformational conformational stresses, giving stresses values by theof 𝜇 characterised = 1.00 MPa, HGO𝑘 = × − −2 −11 1 2.16model MPa, is shown 𝑘 = 1.50 in Figure × 10 , 13𝜌 at = 0.585 a temperature and 𝛽 = 4.02 of 130 × 10◦C and⁰. The strain generated rate of 1conformational s− . stresses by the characterised HGO model is shown in Figure 13 at a temperature of 130 °C and strain rate of 1 s−1.

Figure 13. Non-linear curve fitting of Holzapfel–Gasser–Ogden (HGO) model to conformational stresses at a temperature of 130 °C and strain rate of 1 s−1

Polymers 2019, 11, x 15 of 30

Figure 12. ∆Tf against shifted temperature following use of time-temperature superposition.

3.3. Calibration of HGO Model in the Conformational Part As previously highlighted in Equation (1) the total stress is a result of the addition of two components: the bond stretching stress and conformational stress. In the traditional Buckley material model the conformational stress is predicted by an isotropic Edwards–Vilgis formulation, but as already discussed these current methods are unsuitable for the biaxial deformation behaviour observed with the extruded PEEK sheets used in our studies—which clearly demonstrate some orientation in the MD, demonstrated through the larger stress response in the MD direction compared to the TD. In order to model the post-yield deformation behaviour the simulated bond stretching stress must be subtracted from the total experimental stress, using the characterised parameters derived in the previous section. These equal-biaxial rubber-like stresses are shown in Figure 12 in the machine and transverse directions, at the reference temperature of 130 °C and strain rate of and 1 s−1. A least squares curve fitting procedure was then conducted to best fit the parameters in Equation (12) to the experimental conformational stresses, giving values of 𝜇 = 1.00 MPa, 𝑘 = Polymers 2019, 11, 1042 15 of 29 −2 −11 2.16 MPa, 𝑘 = 1.50 × 10 , 𝜌 = 0.585 and 𝛽 = 4.02 × 10 ⁰. The generated conformational stresses by the characterised HGO model is shown in Figure 13 at a temperature of 130 °C and strain rate of 1 s−1.

Figure 13. Non-linear curve fitting of Holzapfel–Gasser–Ogden (HGO) model to conformational Figure 13. Non-linear curve fitting of Holzapfel–Gasser–Ogden (HGO) model to conformational stresses at a temperature of 130 C and strain rate of 1 s 1 stresses at a temperature of 130 °C◦ and strain rate of 1 s−−1 3.4. Modelling Entanglement Slippage in the Conformational Part In an extension to the constitutive model, Adams et al. [32] considered the inclusion of slippage to model the deformation behaviour of aPET above Tg. The total conformational stretch is considered to be multiplicative combination of network, λn, and slippage stretches, λs, expressed by Equation (22):

n s λi = λi λi (i = 1, 2, 3). (22)

Biaxial stretching of PEEK at all strain rates and temperatures equal to and above the Tg creates a prolonged region of flowing behaviour creating a delay in strain hardening from occurring. It is assumed that the conformational stresses simulated by the HGO model experience no slip. From Equation (22) the slippage stretch can be calculated by division of the total stretch by the network 1 stretch at equal levels of true stress, as highlighted in Figure 14 for a strain rate of 1 s− at a temperature of 150 ◦C. By plotting the slippage stretch against total stretch, a parameter known as the critical network n stretch, λcrit, can be found—the value at which entanglement slippage is considered to have fully arrested, ceasing flow behaviour and allowing the strain hardening process to occur. When a polynomial n is fitted to the data, as shown in Figure 15, λcrit is defined as the point where the slippage stretch rate is equal to zero and derived when the gradient of the polynomial is horizontal. The critical network stretch is then found by division of the total stretch by the slippage stretch at this point, as n defined by Equation (22). Plotting λcrit as a function of temperature and strain rate yields the plot observed in Figure 16, with increasing temperature and deformation rate inducing a proportional 1 effect on the critical network stretch. Data at a temperature of 155 ◦C regarding strain rates above 4 s− proved difficult due to dynamic oscillatory effects appearing on the stress–strain plots as a result of dramatically softened material behaviour and is therefore not included. Polymers 2019, 11, x 16 of 30

3.4 Modelling Entanglement Slippage in the Conformational Part In an extension to the constitutive model, Adams et al. [32] considered the inclusion of slippage to model the deformation behaviour of aPET above Tg. The total conformational stretch is considered to be multiplicative combination of network, 𝜆, and slippage stretches, 𝜆, expressed by Equation (22):

𝜆 =𝜆 𝜆 (𝑖 = 1, 2, 3). (22)

Biaxial stretching of PEEK at all strain rates and temperatures equal to and above the Tg creates a prolonged region of flowing behaviour creating a delay in strain hardening from occurring. It is assumed that the conformational stresses simulated by the HGO model experience no slip. From Equation (22) the slippage stretch can be calculated by division of the total stretch by the network stretch at equal levels of true stress, as highlighted in Figure 14 for a strain rate of 1 s−1 at a temperature of 150 °C. By plotting the slippage stretch against total stretch, a parameter known as the critical network stretch, 𝜆, can be found—the value at which entanglement slippage is considered to have fully arrested, ceasing flow behaviour and allowing the strain hardening process to occur. When a polynomial is fitted to the data, as shown in Figure 15, 𝜆 is defined as the point where the slippage stretch rate is equal to zero and derived when the gradient of the polynomial is horizontal. The critical network stretch is then found by division of the total stretch by the slippage stretch at this point, as defined by Equation (22). Plotting 𝜆 as a function of temperature and strain rate yields the plot observed in Figure 16, with increasing temperature and deformation rate inducing a proportional effect on the critical network stretch. Data at a temperature of 155 °C regarding strain ratesPolymers above2019, 114 ,s 1042-1 proved difficult due to dynamic oscillatory effects appearing on the stress–strain16 of 29 plots as a result of dramatically softened material behaviour and is therefore not included.

Figure 14. Comparison between conformational stresses in the MD of EB stretching of PEEK specimens Figure 14. Comparison between conformational1 stresses in the MD of EB stretching of PEEK for strain rates between 1 and 16 s− at a temperature of 150 ◦C, and simulated HGO network stresses, specimens for strain rates between 1 and 16 s-1 at a temperature of 150 °C, and simulated HGO accurate for conformational stresses simulated at temperatures of 130–145 ◦C. Polymersnetwork 2019, 11 stresses,, x accurate for conformational stresses simulated at temperatures of 130–145 °C. 17 of 30

1 FigureFigure 15. 15. SlippageSlippage stretch stretch plotted plotted against against total st stretchretch for strain rates between 44 andand 1616 ss−-1, ,with with fitted fitted polynomialspolynomials of of order order 3.

Polymers 2019, 11, 1042 17 of 29 Polymers 2019, 11, x 18 of 30

FigureFigure 16. (16.a) Critical(a) Critical network network stretch stretch against against temperaturetemperature and and (b ()b critical) critical networ networkk stretch stretch against against 1 shiftedshifted temperature temperature following following time-temperature time-temperature superpositionsuperposition (TTS), (TTS), between between strain strain rates rates of 1–16 of 1–16 s−1 s− for EBfor stretching EB stretching in the in the machine machine direction. direction.

AdoptionAdoption of theof the TTS TTS method method was was chosen, chosen, similar to to that that used used in inSection Section 3.2 3.2to create to create a linear a linear relationshiprelationship between between shifted shifted temperature temperature and criticaland crit networkical network stretch. stretch. Following Following a least a least squares squares curve fit curve fit of the data shown in Figure 16, values of 0.2132 and 6.469 were characterized for 𝐶 and 𝐶 of the data shown in Figure 16, values of 0.2132 and 6.469 were characterized for C1 and C2 respectively 2 in Equationrespectively (20). in A Equation linear relationship (20). A linear of relationship R2 value equal of R value to 0.986 equal is givento 0.986 in isEquation given in Equation (23), relating (23), the critical network stretch and shifted temperature with little variation seen between material directions: Polymers 2019, 11, 1042 18 of 29

Polymers 2019, 11, x 19 of 30 The slippage viscosity as a function of temperature, γ0, as seen in Equation (23) is modelled by a

Vogel–Tammann–FulcherThe slippage viscosity relationship as a function [26 –of28 temperature,]: 𝛾, as seen in Equation (23) is modelled by a Vogel–Tammann–Fulcher relationship [26–28]: ! Cs Cs = γ0 γ0∗ exp . (23) ∗ 𝐶T T −𝐶T∗ T 𝛾 =𝛾 exp − −∞ ∗ −. ∞ (23) 𝑇− 𝑇 𝑇 − 𝑇 Similar to Equation (7), Cs and T are the Cohen–Turnbull constant and Vogel temperature ∞ governing the temperature dependency of the dashpot. is the slippage viscosity at the reference Similar to Equation (7), 𝐶 and 𝑇 are the Cohen–Turnbullγ0∗ constant and Vogel temperature ∗ temperature,governingT the∗ and temperature must also dependency be direction of dependantthe dashpot. due𝛾 is to the anisotropy slippage viscosity observed at the when reference stretching ∗ samples.temperature, These parameters 𝑇 and must were also fittedbe direction to post-T dependantg conformational due to anisotropy stresses observed between when deformation stretching rates samples.1 These parameters were fitted to post-Tg conformationals stresses between deformation rates of 1–16 s− , with the values of best fit equal to C = 3296.50 K, T = 5.95 K and γ0∗ = 1.68 MPa and −1 ∞ ∗ 0.20 MPaof 1–16 for s the, with machine the values and of transverse best fit equal directions. to 𝐶 = 3296.50 Fits for K, the 𝑇 conformational= 5.95 K and 𝛾 =stresses 1.68 MPa in and both the 0.20 MPa for the machine and transverse directions. Fits for the conformational stresses in both the MD and TD at a temperature of 150 ◦C are shown in Figure 17. MD and TD at a temperature of 150 °C are shown in Figure 17.

1 FigureFigure 17. EB17. conformationalEB conformational stresses stresses atat aa temperaturetemperature of of 150 150 °C◦C and and strain strain rate rate of 1 of s− 1 following s− following fittingfitting of slippage of slippage constants. constants.

4. Modified4. Modified Model Model Prediction Prediction and and Discussion Discussion TheThe characterised characterised material material parameters parameters ofof the Bu Buckleyckley material material model, model, including including modifications, modifications, are listedare listed in Table in Table3. 3. The biaxial deformation behaviour can be predicted by the modified Buckley material model by employing the above characterised constants under differing combinations of experimental variables typical to those observed within the processing window. A comparison of the modified model with the original Buckley material model, the capabilities of which previously detailed in Figure3, is now shown in Figure 18. The accuracies of the predicted material deformation behaviour compared with those observed experimentally are quantitatively analysed through calculation of the normalised root mean square error (NRMSE). This error, expressed as a percentage, is calculated by Equations (24) and (25): " # RMSE % NRMSE = 100%, (24) ymax y × − min s Pn 2 = (yˆt yt) RMSE = i 1 − , (25) n Polymers 2019, 11, 1042 19 of 29

where yˆt is the simulated stress value as predicted by the model and yt is the experimental stress value over the total number of strain data points, n. Table4 details the %NRMSE values comparing the predictions of the modified model to the original Buckley model shown in Figure 18.

Table 3. Characterised constants for modified Buckley material model through biaxial testing methods.

Property Value Unit Shear Modulus, Gb 250 MPa Bulk Modulus, Kb 6250 MPa 3 3 1 Shear Activation Volume, Vs 1.74 10− m mol− × 4 3 1 Pressure Activation Volume, Vp 1.41 10 m mol × − − Reference Viscosity, µ0∗ 92.7 MPa 1 Activation Enthalpy, ∆H0 301 kJ mol− Limiting Temperature, T 465 K Bond Stretching Part Viscosity Constant, Cv ∞ 399 K Initial Fictive Temp, T f 0 419 K Reference Fictive Temp, T∗f 423 K C1 T 0.0892 − shi f ted − C T 5.90 2 − shi f ted Strain Induced Increase, ∆T 0.253 T + 109 K f − × shi f ted Rejuvenation Strain Range, εv 0.0058 SR + 0.181 0 × Fitting Parameter, r 1.60 µ 1.00 MPa k1 2.16 MPa k2 0.0150 ρ 0.585 β 4.02 10 11 0 × − Conformational Part 1.68 MPa (MD) Reference Slippage Viscosity, γ 0∗ 0.20 MPa (TD) Polymers 2019, 11, x Limiting Temperature, T 5.94 K 21 of 30 Viscosity Constant, Cv ∞ 3296 K n C1 λcrit 0.213 where 𝑦 is the simulated stress value as predicted− n by the model and 𝑦 is the experimental stress C2 λcrit 6.47 value over the total number of strain data points,− 𝑛. Tablen 4 details the %NRMSE values comparing Critical Network Stretch, λcrit 0.0128 Tshi f ted 4.30 K the predictions of the modified model to the original Buckley model shown× in− Figure 18.

Figure 18. Comparison between simulated and experimental EB deformation behaviour of PEEK in Figure 18. Comparison between simulated and experimental EB1 deformation behaviour of PEEK in machine and transverse directions at strain rates of 1 and 16 s− at a temperature of 130 ◦C. machine and transverse directions at strain rates of 1 and 16 s-1 at a temperature of 130 °C.

Table 4. Comparison in observed root mean squared errors (%RMSE) between predicted and observed stress–strain behaviour, for the modified and original Buckley material model, at a temperature of 130 °C.

Machine Direction Transverse Direction Model Strain Rate (s−1) Strain Rate (s−1) 1 16 1 16 Original 5.874% 19.91% 21.42% 36.08% Modified 5.294% 10.73% 11.87% 5.340%

The improved ability of the proposed modified Buckley material model is in Table 4. At a strain rate of 1 s−1, the original and modified Buckley material model configurations are comparable in the machine direction, it is only when probing larger strain rates and material directions that the superiority of the modified model is demonstrated. In the transverse direction the associated errors with the original Buckley model is markedly greater, with the addition of the HGO hyperelastic element proving pivotal in replicating the post-yield anisotropic response of the PEEK films subject to biaxial loading. Previous use of the material model also showed no consideration of modelling the rate dependency of the post-yield dip in stress associated with necking below the Tg, as shown in the machine direction at a strain rate of 16 s−1 with an error of almost 20%. When dealing with a combination of these two factors of high strain rates in the transverse direction, the error values seem to be compounded—with %NRMSE reaching values as large as 36%, warranting the adoption of the HGO conformational element and TTS for the accurate modelling of the specific material deformation behaviour. Figure 19 demonstrates the temperature dependence of equal-biaxial deformation in the machine and transverse directions respectively, at the reference strain rate of 1 s−1. The suitability of the modified Buckley model is clear from the Figure with initial stiffness, yielding, necking, and strain hardening stages of the biaxial deformation behaviour seen to be simulated well with only slight

Polymers 2019, 11, 1042 20 of 29

Table 4. Comparison in observed root mean squared errors (%RMSE) between predicted and observed

stress–strain behaviour, for the modified and original Buckley material model, at a temperature of 130 ◦C.

Machine Direction Transverse Direction Model 1 1 Strain Rate (s− ) Strain Rate (s− ) 1 16 1 16 Original 5.874% 19.91% 21.42% 36.08% Modified 5.294% 10.73% 11.87% 5.340%

The improved ability of the proposed modified Buckley material model is in Table4. At a strain 1 rate of 1 s− , the original and modified Buckley material model configurations are comparable in the machine direction, it is only when probing larger strain rates and material directions that the superiority of the modified model is demonstrated. In the transverse direction the associated errors with the original Buckley model is markedly greater, with the addition of the HGO hyperelastic element proving pivotal in replicating the post-yield anisotropic response of the PEEK films subject to biaxial loading. Previous use of the material model also showed no consideration of modelling the rate dependency of the post-yield dip in stress associated with necking below the Tg, as shown in the machine direction at 1 a strain rate of 16 s− with an error of almost 20%. When dealing with a combination of these two factors of high strain rates in the transverse direction, the error values seem to be compounded—with %NRMSE reaching values as large as 36%, warranting the adoption of the HGO conformational element and TTS for the accurate modelling of the specific material deformation behaviour. Figure 19 demonstrates the temperature dependence of equal-biaxial deformation in the machine 1 and transverse directions respectively, at the reference strain rate of 1 s− . The suitability of the modified Buckley model is clear from the Figure with initial stiffness, yielding, necking, and strain hardening stages of the biaxial deformation behaviour seen to be simulated well with only slight inaccuracies shown by underestimations of predicted yield stress levels at temperatures of 135 and 140 ◦C—16.3% and 10.1% deviation respectively. This correlates with the plot shown in Figure9, where the modelled limiting viscosity governing the yield stress at these temperatures is less than the experimental values causing stress levels after yield to be slightly lower than those seen experimentally. Figure 20 shows the %NRSME values over the temperature range of 130–155 ◦C at a strain rate of 1 1 s− . The average error between predicted and actual stress plots were 6.92% and 10.52% in principal directions respectively, showing simulations were more accurate in the machine direction. A minimum error of 5.29% at 130 ◦C and maximum of 14.20% at a temperature of 140 ◦C is in agreement with the above hypothesis concerning the modelling of limiting viscosity at this temperature. Post-yield stress disparities are visually correlated appropriately with the observed biaxial deformation behaviour of PEEK specimens, highlighting the applicability and future promise of the HGO model in capturing anisotropic effects at all temperatures considered. Simulation of slippage at temperatures at the reference strain rate above the glass transition is captured effectively through use of TTS. Although minor deviation is only observed once above a strain of 1, overall the characterised material model is seen to qualitatively agree with the temperature dependency at all stages of the equal-biaxial deformation behaviour of PEEK. Polymers 2019, 11, x 22 of 30 inaccuracies shown by underestimations of predicted yield stress levels at temperatures of 135 and 140 °C—16.3% and 10.1% deviation respectively. This correlates with the plot shown in Figure 9, where the modelled limiting viscosity governing the yield stress at these temperatures is less than Polymersthe experimental2019, 11, 1042 values causing stress levels after yield to be slightly lower than those21 seen of 29 experimentally.

Figure 19. Predicted EB deformation behaviour by modified Buckley model for temperatures between Figure 19. Predicted EB deformation behaviour by modified Buckley model for temperatures between 130 and 155 ◦C in the (a) machine direction and (b) transverse direction. 130 and 155 °C in the (a) machine direction and (b) transverse direction.

Figure 20 shows the %NRSME values over the temperature range of 130–155 °C at a strain rate of 1 s−1. The average error between predicted and actual stress plots were 6.92% and 10.52% in principal directions respectively, showing simulations were more accurate in the machine direction. A minimum error of 5.29% at 130 °C and maximum of 14.20% at a temperature of 140 °C is in agreement with the above hypothesis concerning the modelling of limiting viscosity at this temperature.

Polymers 2019, 11, x 23 of 30

Post-yield stress disparities are visually correlated appropriately with the observed biaxial deformation behaviour of PEEK specimens, highlighting the applicability and future promise of the HGO model in capturing anisotropic effects at all temperatures considered. Simulation of slippage at temperatures at the reference strain rate above the glass transition is captured effectively through use of TTS. Although minor deviation is only observed once above a strain of 1, overall the characterised Polymers 2019, 11, 1042 22 of 29 material model is seen to qualitatively agree with the temperature dependency at all stages of the equal-biaxial deformation behaviour of PEEK.

Figure 20. Normalised root mean squared error between simulated and experimentally observed Figure 20. Normalised root mean squared error between simulated and experimentally observed stress–strain behaviour between temperatures of 130 and 155 C, at a strain rate of 1 s 1. stress–strain behaviour between temperatures of 130 and 155◦ °C, at a strain rate of 1 s−−1. Figures 21–23 show the predicted true stress–nominal strain behaviour of PEEK specimens Figure 21–23 show the predicted1 true stress–nominal strain behaviour of PEEK specimens between strain rates of 1–16 s− , at temperatures of 130–150 ◦C in both directions. Again, a good between strain rates of 1–16 s−1, at temperatures of 130–150 °C in both directions. Again, a good qualitative fit is exhibited by the characterised material model in replicating the aforementioned qualitative fit is exhibited by the characterised material model in replicating the aforementioned four four stages of the deformation behaviour at elevated strain rates. Any differences observed between stages of the deformation behaviour at elevated strain rates. Any differences observed between simulated and measured behaviour is typically focussed on modelling yield stress levels at the highest simulated and measured1 behaviour is typically focussed on modelling yield stress levels at the strain rate of 16 s− , with maximum percentage differences of 6.28%, 12.2% and 12.7% recorded at highest strain rate of 16 s−1, with maximum percentage differences of 6.28%, 12.2% and 12.7% the yield points of 130, 140, and 150 C, respectively. By reviewing the Eyring plots in Figure7 it is recorded at the yield points of 130, 140,◦ and 150 °C, respectively. By reviewing the Eyring plots in important to note that the values for Vs and Vp, responsible for the magnitude in yield stresses, were Figure 7 it is important to note that the values for 𝑉 and 𝑉 , responsible for the magnitude in yield calculated from an average of the individual gradients of the temperature plots. As the mean value of stresses, were calculated1 from an average of the individual gradients of the temperature plots. As the 0.0167 MPa K− was chosen, which is less than the value of the gradients at 130 and 140 ◦C but greater mean value of 0.0167 MPa K−1 was chosen, which is less than the value of the gradients at 130 and 140 than that at 150 ◦C, this explains the simulated underestimation in yield stresses at a strain rate of °C but1 greater than that at 150 °C, this explains the simulated underestimation in yield stresses at a 16 s− shown at temperatures of 130 and 140 ◦C, yet an overestimation for 150 ◦C at the highest strain strain rate of 16 s−1 shown at temperatures of 130 and 140 °C, yet an overestimation for 150 °C at the rate. Although a larger value for the gradient to calculate the activation volumes would accommodate highest strain rate. Although a larger value for the gradient to calculate the activation volumes would a more representative prediction for temperatures 130 and 140 ◦C, an average of all temperatures is accommodate a more representative prediction for temperatures 130 and 140 °C, an average of all used in order to ensure accurate modelling over the entire processing window. temperatures is used in order to ensure accurate modelling over the entire processing window.

Polymers 2019, 11, 1042 23 of 29 Polymers 2019, 11, x 24 of 30

Figure 21. Predicted EB deformation behaviour by modified Buckley model for strain rates between 1 Figure 21.1 Predicted EB deformation behaviour by modified Buckley model for strain rates between and 16 s− , at a temperature of 130 ◦C in the (a) machine direction and (b) transverse direction. 1 and 16 s−1, at a temperature of 130 °C in the (a) machine direction and (b) transverse direction.

Polymers 2019, 11, 1042 24 of 29 Polymers 2019, 11, x 25 of 30

Figure 22. Predicted EB deformation behaviour by modified Buckley model for strain rates between 1 Figure 22.1 Predicted EB deformation behaviour by modified Buckley model for strain rates between and 16 s− , at a temperature of 140 ◦C in the (a) machine direction and (b) transverse direction. 1 and 16 s−1, at a temperature of 140 °C in the (a) machine direction and (b) transverse direction.

Polymers 2019, 11, 1042 25 of 29 Polymers 2019, 11, x 26 of 30

Figure 23. Predicted EB deformation behaviour by modified Buckley model for strain rates between 1 Figure 23.1 Predicted EB deformation behaviour by modified Buckley model for strain rates between and 16 s− , at a temperature of 150 ◦C in the (a) machine direction and (b) transverse direction. 1 and 16 s−1, at a temperature of 150 °C in the (a) machine direction and (b) transverse direction. 1 Figures 24–26 show the calculated %NRSME values over the strain rate range of 1–16 s− at temperature intervals of 130, 140, and 150 ◦C in machine and transverse directions. The figures demonstrate the quantitative agreement of the modified model over the broad processing range, with maximum error levels not seen to exceed 15%. In general, the deformation behaviour was modelled more accurately in the machine direction—the maximum average percentage difference in principal directions being equal to only 4%.

Polymers 2019, 11, x 27 of 30 Polymers 2019, 11, x 27 of 30 Figure 24–26 show the calculated %NRSME values over the strain rate range of 1–16 s−1 at temperatureFigure 24–26intervals show of the130, calculated 140, and 150%NRSME °C in machinevalues over and the transverse strain rate directions. range of The 1–16 figures s−1 at demonstratetemperature theintervals quantitative of 130, agreement 140, and of150 the °C modifi in machineed model and over transverse the broad directions. processing Therange, figures with maximumdemonstrate error the levelsquantitative not seen agreement to exceed of 15%. the modifi In general,ed model the deformation over the broad behaviour processing was range, modelled with moremaximum accurately error inlevels the machinenot seen todirection—the exceed 15%. maxiIn general,mum average the deformation percentage behaviour difference was in modelledprincipal directionsmorePolymers accurately2019 being, 11, 1042 equalin the tomachine only 4%. direction—the maximum average percentage difference in principal26 of 29 directions being equal to only 4%.

Figure 24. Normalised root mean squared error between simulated and experimentally observed Figure 24. Normalised root mean squared error between simulated and experimentally observed stress–strainFigure 24. Normalised behaviour betweenroot mean strain squared rates oferror 1–16 between s−1 at a temperature simulated and of 130 experimentally °C. observed stress–strain behaviour between strain rates of 1–16 s 1 at a temperature of 130 C. stress–strain behaviour between strain rates of 1–16 s−1 at a temperature of 130 °C.◦

Figure 25. Normalised root mean squared error between simulated and experimentally observed Figure 25. Normalised root mean squared error between1 simulated and experimentally observed stress–strain behaviour between strain rates of 1–16 s− at a temperature of 140 ◦C. stress–strainFigure 25. Normalised behaviour betweenroot mean strain squared rates oferror 1–16 between s−1 at a temperature simulated and of 140 experimentally °C. observed stress–strain behaviour between strain rates of 1–16 s−1 at a temperature of 140 °C.

Polymers 2019, 11, 1042 27 of 29 Polymers 2019, 11, x 28 of 30

Figure 26. Normalised root mean squared error between simulated and experimentally observed Figure 26. Normalised root mean squared error between1 simulated and experimentally observed stress–strain behaviour between strain rates of 1–16 s− at a temperature of 150 ◦C. stress–strain behaviour between strain rates of 1–16 s−1 at a temperature of 150 °C. 5. Conclusions 5. Conclusions In this work the constitutive biaxial deformation behaviour of thin PEEK specimens was successfullyIn this modelledwork the extendingconstitutive extensive biaxial displacement-controlleddeformation behaviour biaxialof thin tests PEEK over specimens a temperature was 1 windowsuccessfully of 130–155modelled◦C extending and strain extensive rate range displaceme of 1–16nt-controlled s− . The capability biaxial tests of the over original a temperature Buckley materialwindow of model 130–155 was °C shown and strain to capture rate range linear of 1–16 elastic, s−1. andThe capability yielding behaviour of the original associated Buckley with material that observedmodel was during shown in-plane to capture biaxial linear tests, elastic, although and yi couldelding not behaviour account forassociated anisotropic with e ffthatects observed revealed afterduring yielding in-plane between biaxial tests, the principal although directions.could not account The Holzapfel–Gasser–Ogden for anisotropic effects revealed hyperelastic after yielding model wasbetween seen the to successfully principal directions. replicate theThe anisotropic Holzapfel– stress-strainGasser–Ogden response hyperelastic of arterial model tissues, was andseen as to a result,successfully was substituted replicate the in placeanisotropi of thec Edwards–Vilgis stress-strain response hyperelastic of arterial element tissues, within and the as conformational a result, was arm.substituted The post in place yield of divergence the Edwards–Vilgis in true stresses hyperela observedstic element in equal-biaxial within the testsconformational of PEEK specimens arm. The waspost successfullyyield divergence modelled, in true resulting stresses in anobserved almost 10%in equal-biaxial decrease in thetests error of PEEK levels associatedspecimens withwas thesuccessfully inability tomodelled, replicate resulting this post-yield in an almost anisotropy 10% indecrease the original in the model, error atlevels a temperature associated ofwith 130 the◦C. Wheninability concerning to replicate the this discernible post-yield dependencies anisotropy in of the the original magnitude model, in post-yieldat a temperature stress declinesof 130 °C. at When large strainconcerning rates, the as discernible suggested independenci previouses literature, of the magnitude the time-temperature in post-yield stress superposition declines at principal large strain has shownrates, as promise suggested in e inffectively previous computing literature, the the evolution time-temperature of fictive temperaturesuperposition subject principal to the has forming shown temperaturepromise in effectively and deformation computing rate. the evolution of fictive temperature subject to the forming temperatureUsing the and modified deformation model, rate. following step-by-step definition of the 26 parameters and relationships, it wasUsing shown the that modified the simulated model, equal-biaxial following step-by- deformationstep behaviourdefinition comparedof the 26 qualitativelyparameters welland torelationships, the experimental it was data shown at temperatures that the simulated and strain equal-biaxial rates analogous deformation to the forming behaviour process. compared It was possiblequalitatively to predict well to the the initial experimental stiffness, data yielding, at temperatures necking, and and anisotropic strain rates strain analogous hardening to the stagesforming of deformationprocess. It was over possible the wide to temperature predict the window initial andstiffness, strain rateyielding, range. necking, Errors concerning and anisotropic the simulated strain hardening stages of deformation over the wide temperature window and strain rate range. Errors magnitude of yield stress, presented at temperatures of 135 and 140 ◦C at the reference strain rate and concerning the simulated magnitude of yield stress,1 presented at temperatures of 135 and 140 °C at at temperatures at the highest strain rate of 16 s− , is presumed to be as a result of the modelled limiting viscositythe reference due strain to averaged rate and gradients at temperatures of Eyring at plots. the highest The proposed strain rate modified of 16 s material-1, is presumed model to is seenbe as to a beresult in good of the overall modelled quantitative limiting agreementviscosity due with to the av displacement-controllederaged gradients of Eyring biaxial plots. data The in proposed principal directions,modified material showing model minimum is seen and to be maximum in good ov errorserall ofquantitative 4.74% and agreement 14.2% respectively—insignificant with the displacement- controlled biaxial data in principal directions, showing minimum and maximum errors of 4.74% and 14.2% respectively—insignificant when compared to the 36.1% error associated with the prediction

Polymers 2019, 11, 1042 28 of 29 when compared to the 36.1% error associated with the prediction of the original Buckley material 1 model at a temperature of 130 ◦C and strain rate of 16 s− , in the transverse direction. Considering the applicability of the characterised model in replicating the biaxial deformation of PEEK films, as the first of its kind in literature, the use of the modified Buckley material model is promising for future implementation into finite element simulations for the prediction of material deformation during forming. In addition, not only is the modified Buckley material model solely applicable for the modelling of PEEK films but following appropriate characterisation, can provide a stable basis for the replication of the biaxial forming behaviour of any amorphous polymeric materials exhibiting post-yield anisotropic behaviour as a result of material fabrication or processing.

Author Contributions: Conceptualization and Methodology, J.A.T., G.H.M. and S.Y.; Investigation and Writing—original draft, J.A.T.; Supervision and Writing—review and editing, G.H.M., P.J.M. and S.Y. Funding: This research was funded by the Department for the Economy (DfE). Conflicts of Interest: The authors declare no conflict of interest.

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