Breemen, van, L.C.A.

Govaert, L.E.

Meijer, H.E.H.

English

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 Multi-relaxation time viscoplastic modeling ofthermorheological simple materialsCitation for published version (APA):Breemen, van, L. C. A., Govaert, L. E., & Meijer, H. E. H. (2007). Multi-relaxation time viscoplastic modeling ofthermorheological simple materials. Poster session presented at Mate Poster Award 2007 : 12th Annual PosterContest.Document status and date:Published: 01/01/2007Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)Please check the document version of this publication:• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publicationGeneral rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal.If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverneTake down policyIf you believe that this document breaches copyright please contact us at:openaccess@tue.nlproviding details and we will investigate your claim.Download date: 17. Nov. 202312/department of mechanical engineering PO Box 513, 5600 MB Eindhoven, the NetherlandsMulti-relaxation time viscoplastic modeling ofthermorheological simple materialsL.C.A. van Breemen, L.E. Govaert and H.E.H. MeijerEindhoven University of Technology, Department of Mechanical EngineeringintroductionOver several decades, an in-house constitutive model forglassy polymers was developed [1, 2], that accurately cap-tures the yield and post-yield response of glassy polymers.Despite these capabilities, several problems exist due to thenature of the single-mode modeling:2 inaccurate description of the pre-yield regime.2 incapability to qualitatively predict elastic recovery.σ = κ(J − 1)I + ...GB̃de +GrB̃dDp =σds2η(τ , p, T, S)η = η0 · aσ · ap · aSaFigure 1 Experiment and simulation using the single-modemodel foruni-axial compression (left); governing constitutive relations (right).solutionAn elegant solution to this problem was proposed by [1],hence a multi-mode approach is employed.σ = κ(J − 1)I + ...n∑i=1(GiB̃dei) +GrB̃dDpi =σdsi2ηi(τ , p, T, S)Figure 2 Experiment and simulation using the multi-mode model foruni-axial compression (left); modified constitutive relations (right).This is a straightforward extension to multi-relaxation times,with the special feature that the non-linearity is coupled tothe total stress.material characterizationAs model material, polycarbonate was selected, and valuesfor material parameters were taken from literature [2]. Themoduli and initial viscosities of the modes were obtainedfrom an intrinsic deformation curve as depicted in figures 1and 2. The stress up to yield can be described with the wellknown Boltzmann integral.By taking into account the non-linearity of the stress in theviscosities, the moduli and corresponding viscosities can becalculated.σ(t) =n∑i=1[Eiǫ̇∫ t−∞exp(−ψ − ψ′λi)dt′]; ηi = λi · Eiwhereψ =∫ t−∞dt′′aσ[σ(t′′)]ψ′=∫ t′−∞dt′′aσ[σ(t′′)]aσ =τ/τ0sinh(τ/τ0)10−810−41001041081012101610201024102810−1100101102103104time [s]relaxation modulus [MPa]  rejuvenatedAs receivedAnnealed @ 100 °CAnnealed @ 120 °CFigure 3 Relaxation moduli versus time.Similar to the single-mode approximation [2], a unique reju-venated relaxation time spectrum can be defined that accu-rately describes the pre-yield range in different strain ratesand thermal history.resultsThe multi-mode model accurately describes the pre-yieldregime and is therefore capable of calculating loading andunloading curves for flat-tip indentation (Figure 4).0 1 2 3 4 501020304050displacement into surface [µm]force [mN]  single modemulti modeexperimentsFigure 4 Flat-tip indentation using the single andmulti-modemodel.conclusionThe newmulti-mode implementation is capable of describingthe non-linear viscoelastic pre-yield regime accurately, whilethe post-yield behavior of the single-modemodel remains un-affected.References:[1] TERVOORT, T.A.: Constitutive modelling of polymer glasses. Finite, non-linear viscoelastic behaviour of polycarbonate (Eindhoven University ofTechnology 1996)[2] KLOMPEN, E.T.J.: Mechanical properties of solid polymers: Constitutivemodelling of long and short term behaviour (Eindhoven University ofTechnology 2005)

Multi-relaxation time viscoplastic modeling of thermorheological simple materials

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Breemen, LCA Lambèrt  van

Govaert, LE Leon

Meijer, HEH

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Multi-relaxation time viscoplastic modeling of thermorheological simple materials

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