Andrä, H.

Köbler, J.

Magino, N.

Schneider, M.

Welschinger, F

English

Welschinger, F.

KITopen

Received: 19 June 2020 Accepted: 12 November 2020DOI: 10.1002/pamm.202000091A multi-scale fatigue-damage model for fiber-reinforced polymersNicola Magino1,∗, Jonathan Köbler1, Heiko Andrä1, Matti Schneider2, and Fabian Welschinger31 Fraunhofer Institute for Industrial Mathematics ITWM, Department of Flow and Material Simulation, Kaiserslautern,Germany2 Karlsruhe Institute of Technology (KIT), Institute of Engineering Mechanics, Karlsruhe, Germany3 Robert Bosch GmbH, Corporate Sector Research and Advanced Engineering, Renningen, GermanyExperimental studies of Chebbi et al. [1] on fatigue loading of fiber-reinforced polymers have shown that there is a phaseof stable stiffness decrease prior to growing fatigue cracks. Modeling this stiffness degradation is an essential step in under-standing fatigue effects of these materials.The constitutive behavior of short-fiber reinforced polymers depends on numerous factors, such as fiber-volume content,the aspect ratio of the fibers, the fiber-orientation tensor and the loading direction. Accounting for these influence factors ona purely experimental basis is very time and resource demanding.As a remedy, we follow a multi-scale approach for simulating the fatigue-damage evolution in short-fiber reinforcedpolymers. Using a simple damage model for the polymer matrix, the model inherently accounts for the influence of the fibermicro-structure through homogenization. We show that the stiffness degradation predicted by this model is of anisotropicnature and depends strongly on loading direction and fiber-orientation tensor.Due to its specific structure, the model permits a straightforward model-order-reduction strategy and can be efficientlyemployed for component-scale simulations, see Köbler et al. [3].© 2021 The Authors Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH GmbH1 Fatigue-damage model for the polymer matrixWe present a material model for the matrix of a fiber-reinforced polymer material. The fibers are considered to be purelylinear elastic. The matrix-material model is formulated in the framework of generalized standard materials proposed byHalphen-Nguyen [2]. The free energy potential ψ and the dissipation potential φ are defined asψ(ε,D) =12[η + (1−D)2]ε : C : ε+ w `2 ‖∇D‖2 and φ(D′) = 12αD′2,where ε denotes the strain tensor and the relative residual stiffness η > 0 is introduced for numerical reasons, D is the scalardamage variable ranging from 0 (undamaged) to 1 (fully damaged), C is the initial stiffness tensor of the material, w is amaterial parameter, ` is a length-scale parameter and α governs the speed of the damage evolution.The model is a priori formulated in cycle space. We denote by (.)′ ≡ ddN differentiation w.r.t. the cycle variable N .The model is applied to different fiber orientations realized as representative volume elements that were generated by theSAM algorithm, see Schneider [5]. A typical representative volume element for a uni-directional (left) and a planar-isotropicfiber orientation, respectively, is shown in figure 1.The model is discretized in time and space, and solved by a fast Fourier-transform based approach, see Köbler et al. [4], [3].a) Uni-directional fiber structure b) Planar-isotropic fiber structureFig. 1: Two unit cells with different fiber orientations∗ Corresponding author: e-mail nicola.magino@itwm.fraunhofer.de, phone +49 (0)63131600-4494This is an open access article under the terms of the Creative Commons Attribution License, which permits use,distribution and reproduction in any medium, provided the original work is properly cited.PAMM · Proc. Appl. Math. Mech. 2020;20:1 e202000091. www.gamm-proceedings.com 1 of 2https://doi.org/10.1002/pamm.202000091 © 2021 The Authors Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH GmbH2 of 2 Section 8: Multiscales and homogenization2 Stiffness degradation for different stress amplitudes and fiber orientationsAfter carrying out several verification steps, a representative volume with an edge length of twice a fiber length was found tobe sufficient for studying the primary stiffness degradation (loss in the effective moduli of several 10 %), see Köbler et al. [4].A fiber is resolved by 128 voxels per length or, at a chosen aspect ratio of 20, by 6.4 voxels across the fiber diameter.Subsequently, a study on the fiber-orientation influence on the stiffness degradation was conducted. The results for theabove shown uni-directional and planar-isotropic structures are shown in figure 2 and 3.Figure 2 shows the dependence of the effective Young’s modulus decrease in x-, y- and z-direction subjected to loading in x-direction. The considered second-order fiber orientations interpolate between a uni-directional (µ = 0) and a planar-isotropic(µ = 1) second-order fiber-orientation tensor viaA(µ) = µe1 ⊗ e1 + (1− µ)12(e2 ⊗ e2 + e3 ⊗ e3), µ ∈ [0, 1]. (1)The evolution curves for different values of µ lie in between the extreme structures, i.e., for µ = 0 and µ = 1, and dependcontinuously on the parameter µ.Figure 3 depicts the evolution of the Young’s modulus bodies of the planar-isotropic and the uni-directional structure forthree different cutting planes. The decrease in loading direction, i.e. the x-direction, is the most significant for both scenarios.The degradation processes is thus found to be of anisotropic nature.0 0.5 1 1.5 2 2.5 3 3.5 4 4.52468101214number of cycles in 106 ×NEeffinGPaµ Eeffx Eeffy Eeffz0.000.250.500.751.00Fig. 2: Decrease of the absolute Young’s modulus underloading in x-direction for fiber orientations parametrizedbetween uni-directional and planar-isotropic03060901201501802102402703003300 5 10 15E[GPa]first cyclecycle 2 · 106cycle 3.5 · 106a) x-y-plane, UD03060901201501802102402703003300 5 10 15E[GPa]first cyclecycle 2 · 106cycle 3.5 · 106b) z-x-plane, UD03060901201501802102402703003300 2 4E[GPa]first cyclecycle 2 · 106cycle 3.5 · 106c) y-z-plane, UD03060901201501802102402703003300 2 4E[GPa]first cyclecycle 2 · 106cycle 3.5 · 10603060901201501802102402703003300 2 4 6 8E[GPa]first cyclecycle 8.9 · 105cycle 1.3 · 106d) x-y-plane, PISO03060901201501802102402703003300 2 4 6 8E[GPa]first cyclecycle 8.9 · 105cycle 1.3 · 106e) z-x-plane, PISO03060901201501802102402703003300 2 4 6 8E[GPa]first cyclecycle 8.9 · 105cycle 1.3 · 106f) y-z-plane, PISO03060901201501802102402703003300 2 4 6 8E[GPa]first cyclecycle 8.9 · 105cycle 1.3 · 106Fig. 3: Evolution of the Young’s modulus body of the uni-directional (UD,top) and the planar-isotropic (PISO, bottom) structure for different cuttingplanesHow to incorporate the presented material model into a multi-scale simulation framework is discussed in Köbler et al. [3]. Thisdata-driven method enables efficient computations on the component scale, predicting the stiffness degradation for engineeringapplications on component scale. This is part of ongoing research.Acknowledgements We thank the Robert Bosch GmbH for financial support. MS thanks the German Research Foundation (DFG) forsupport within the International Research Training Group “Integrated engineering of continuous-discontinuous long fiber reinforced polymerstructures” (GRK 2078).Open access funding enabled and organized by Projekt DEAL.References[1] Chebbi, E., Mars, J., Wali, M. and Dammak, F. Periodica Polytechnica Mechanical Engineering 60, 247–255 (2016).[2] Halphen, B., Nguyen, Q. S. Journal de mécanique 14, 39-63 (1975)[3] Köbler, J., Magino, N., Andrä, H., Welschinger, F., Müller, R., Schneider, M. Comput. Meth. Appl. Mech. Eng. (submitted, 2020)[4] Köbler, J., Schneider, M., Ospald, F., Andrä, H, Müller, R. Comput. Mech. 61, 729-750 (2018)[5] Schneider, M. Comput. Mech. 59, 247–263 (2017)© 2021 The Authors Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH GmbH www.gamm-proceedings.com

A multi-scale fatigue-damage model for fiber-reinforced polymers

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A multi‐scale fatigue‐damage model for fiber‐reinforced polymers

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A multi-scale fatigue-damage model for fiber-reinforced polymers

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