Hempel, P.

Kotlik, M.

Seelig, Th.

English

KITopen

Institute of MechanicsContinuum mechanical modelingof processing induced initial anisotropyin the finite strain deformationof amorphous thermoplastic polymers15. International Conference onDeformation, Yield and Fracture of PolymersKerkrade, April 2012Philipp Hempel, Matthias Kotlik, Thomas SeeligInstitute of MechanicsKaiserstr. 12D-76131 KarlsruheTel.: +49 (0) 721/ 608-42071Fax: +49 (0) 721/608-47990E-Mail: info@ifm.kit.eduwww.ifm.uni-karlsruhe.deContinuum mechanical modeling of processing induced initialanisotropy in the finite strain deformation of amorphousthermoplastic polymersP. Hempel, M. Kotlik, Th. SeeligInstitute of Mechanics, Karlsruhe Institute of Technology (KIT), D-76131 Karlsruhe, GermanySummary: This work deals with modeling the mechanical behavior ofthermoplastic polymers in the finite strain regime over a wide range oftemperatures. Thereby, special emphasis is put on the incorporation ofan initial anisotropy in terms of ”frozen-in” molecular orientation whichresults from a preceding manufacturing process. A computational exam-ple is discussed which considers an injection molded plate undergoinginhomogeneous deformation (buckling) during re-heating.IntroductionManufacturing processes of amorphous thermoplastic polymers at elevated temperatures (e.g.injection molding) and subsequent rapid cooling result in an anisotropic microstructure due to”frozen-in” molecular orientation [5]. This state of pre-orientation in the solid material affectsthe subsequent finite strain deformation behavior in terms flow strength and hardening and maygive rise to stress-free deformations of a component during re-heating.This ”memory-effect” can be observed at the injection molded plate in Fig. 1a. After uniformlyheating the plate above the glass transition temperature it deforms and buckles due to the pre-oriented microstructure (Fig. 1c). To get some indication of the microstructure one can use po-larized light to visualize (via birefringence) regions with high molecular orientation and residualstress (Fig. 1b).a) b) c)Figure 1: a) Injection molded plate, b) plate under polarized light, c) deformed plate after re-heating above glass transition temperatureThe aim of this work is to model the mechanical behavior of thermoplastic polymers in the finitestrain regime over a wide range of temperatures (above glass transition) with consideration of aprocessing induced initial anisotropy. Therefore, experimental test data at different temperaturesand strain rates are necessary in order to describe the stress-strain response correctly. Uniaxialtension tests on PMMA at different strain rates and temperatures (below and above glass transi-tion which for PMMA is Tg ≈ 105◦ C) were performed. These tension tests were evaluated usingdigital image correlation [4] which allows to measure local strain fields needed to determine thetrue stress-strain response of the material (Fig. 2). At moderate temperatures the material showsa glassy behavior with an elastic regime followed by inelastic deformation including progressivehardening. At higher temperatures, near to or above the glass transition, the material is in therubbery state. Elastic strains then are much larger and the separation of elastic and inelasticdeformation is less obvious. Furthermore, a rate dependency is observed which is more pro-nounced at high temperatures so that the glass transition is not a fixed temperature but is ratedependent as well.a)ε̇ = 10−3s−1b)ε̇ = 10−1s−1Figure 2: Uniaxial tension tests at different temperatures and strain rates: a) ε̇ = 10−3s−1, b)ε̇ = 10−1s−1Continuum mechanical modelingThe material model presented here is based on that developed by Boyce and co-workers [1-3].Starting with the kinematics, the widely used multiplicative split of the deformation gradient intoan elastic and plastic part is utilizedF = FeFp (1)where Fp defines the stress free configuration after elastic unloading (referred to as the interme-diate configuration). Physically, Fp describes permanent molecular orientation in the material.In order to include an initial anisotropy, the inelastic deformation gradient is complemented byan initial deformation gradient Fi (= Vi) so that the network deformation tensor Fn is defined as[2]Fn= FpFi . (2)This network deformation tensor is used in the following to describe the entropic resistance ofthe molecular network against plastic flow (see below).The split of the deformation gradient (1) leads to an additive decomposition of the velocity gra-dient tensor into an elastic and plastic part L = Le+ Lp with the inelastic velocity gradient in thecurrent configuration Lp = Fe L̂pFe−1 and in the intermediate configuration L̂p= ḞpFp−1. Thelatter can be decomposed as L̂p= D̂p+ Ŵpwith the inelastic rate of deformation tensor D̂pandthe inelastic spin tensor Ŵp, both in the intermediate configuration. While D̂pis constitutivelyprescribed via a flow rule, the inelastic spin tensor can be computed as Ŵp=W −W[D + D̂p]with the forth order tensorW once the assumption of a symmetric elastic deformation gradientFeT= Fe is made [3].The inelastic part of the rate of deformation tensor D̂pin the intermediate configuration is deter-mined by the deviatoric part of the driving stress tensor Σ̂ − b̂ and the plastic shear strain rate γ̇paccording toD̂p= γ̇pΣ̂′ − b̂′|Σ̂′ − b̂′|, γ̇p = γ̇p0exp−AsT1 −|Σ̂′ − b̂′|s56(3)with material parameters A, s, γ̇p0and the temperature T .The driving stress Σ̂− b̂ is the difference of the Mandel stress Σ̂ = det(F)FeσFe−1 (with σ beingthe Cauchy stress tensor) and the backstress b̂ (Fig. 3a). The backstress tensor is determinedfrom the Arruda-Boyce eight-chain-model [1]b̂ =CR√N3λcL−1(λc√N)B̂n(4)with the left Cauchy-Green tensor B̂n= FnFnT , the inverse of the Langevin-function L, therubbery modulus CR, the average number N of rigid links between entanglements and the meanchain stretch λc =(13trB̂n)12. In (4) use is made of the network deformation gradient tensor Fn(2) so that an initial backstress can be prescribed. The Cauchy stress σ depends on the elasticleft Cauchy-Green tensor Be = FeFeT and is determined from a hyperelastic Neo-Hooke model.In addition, the model is modified to capture a wide range of temperatures. Therefore, the shearmodulus, the material parameter A in (3) and the molecular network parameter N are takentemperature dependent as suggested in [1]. In Fig. 3b the stress-strain response of the model isshown in comparison with experimental data.a) b)Figure 3: a) Rheological model, b) stress-strain response of the model (dashed lines) and exper-imental data (solid lines) for a strain rate of ε̇ = 0.1s−1Computational exampleAs a computational example the injection molded plate shown in Fig. 1a is investigated. Anindication of the ”frozen-in” molecular orientation (and stretch) is provided by inspecting theplate under polarized light utilizing birefringence (Fig. 1b). This is used as a first guess forthe distribution and magnitude of the initial molecular orientation where high or low valuesprevail in bright or dark regions, respectively. Mapping of this information to the FE model isaccomplished by assuming a corresponding distribution of the maximum principal stretch λimax(eigenvalue of Vi, detVi = 1). For simplicity five sections with different constant values of λimaxare considered. The direction of the orientation is assumed to be radial from the injection point(”gate”; see Fig. 1a). The plate is subjected to a temperature which is taken spatially uniformand increases with time. With increasing temperature the material flow resistance decreases andenables the ”frozen-in” molecular stretch (and backstress) to relax. This spatially non-uniformre-deformation causes buckling of the plate. The deformed FE model is depicted in Fig. 3b andshows a qualitatively good agreement with the real plate in Fig. 1c.a) b)Figure 4: FE analysis of injection molded plate: a) distribution of initial molecular stretch, b)deformation after heating to 120◦ CConclusionsIn this work a constitutive model for amorphous thermoplastic polymers is set up which incor-porates an initial anisotropy and is able to describe the stress-strain response even above theglass transition. The model is used successfully to simulate a re-heating process of an injectionmolded plate. The resulting deformation of the plate is in good agreement with that observed inthe real experiment. The correlation between the deformation of the real plate and the deforma-tion obtained in the simulation evidences that the assumed initial molecular orientation and itsmapping to the computational model is reasonable. However, while the flow field in the presentcase is quite obvious, it is hardly conceivable that this heuristic method is applicable to morecomplicated geometries. In that case one could use, for example, form filling simulations to mapthe initial state of the material to the FE model. Another subject of future work is the modifi-cation of the material model to capture the effect of strain rate dependent hardening at elevatedtemperatures caused by reptation of molecules in the entanglement network.References[1] Arruda, E. M.; Boyce, M. C.; Jayachandran, R., Mechanics of Materials 1995, 19(2-3), 193-212[2] Boyce, M.C.; Parks, D.M.; Argon, A.S., International Journal of Plasticity 1989, 6, 593-615[3] Boyce, M.C.; Weber, G.G.; Parks, D.M., Journal of the Mechanics and Physics of Solids 1989,37(5),647-665[4] Limess, Digital Image Correlation. Q-400 Operation Manual, Krefeld 2009[5] Struik, L.C.E., Internal stresses, dimensional instabilities and molecular orientations in plastics, JohnWiley & Sons Ltd, Chichester 1990

Continuum mechanical modeling of processing induced initial anisotropy in the finite strain deformation of amorphous thermoplastic polymers

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Continuum mechanical modeling of processing induced initial anisotropy in the finite strain deformation of amorphous thermoplastic polymers

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