A computationally efficient hybrid formulation for viscoelastic–viscoplastic polymer solids and structures under large numbers of loading cycles - 2025
Efficient computations; Hybrid formulation; Large number of loading cycles; Polymer solids and structures; Computationally efficient; Efficient computation; High cycle; Hybrid formulations; Large number of loading cycle; Loading cycles; Polymer (solid); Polymer structure; Viscoelastics; Viscoplastic; Modeling and Simulation; Materials Science (all); Condensed Matter Physics; Mechanics of Materials; Mechanical Engineering; Applied Mathematics
Abstract :
[en] The numerical simulation of the high cycle response of solids and structures made of thermoplastic polymers is challenging because those materials exhibit a complex viscoelastic–viscoplastic (VEVP) behavior and even under large numbers of loading cycles, they continue to dissipate energy and feature a frequency dependent response. On the one hand classical simplified methods based on linear elasticity are not applicable, and on the other hand direct structural analyses with VEVP material models are so computationally prohibitive that they are not possible in practice. In this article, a computationally efficient hybrid formulation is proposed. The structure is first computed as being purely VE, using a recently proposed formulation based on Laplace-Carson transform (LCT) and its numerical inversion, and enabling to compute accurate strain and stress fields at a very reduced cost, which is also independent of the number of cycles. Next, the VEVP solution at any points of interest is computed with a time homogenization formulation which uses fast and slow time scales and asymptotic time expansions to compute complete solutions at extremely limited cost. An experimentally identified TPU material and a 3D lattice are used for the numerical simulations. Predictions of the hybrid formulation are compared against reference VEVP solutions and their accuracy verified. Numerical simulations for one million cycles are presented and the low computational cost of the hybrid formulation illustrated. The underlying assumptions of the hybrid formulation linking the VE results with the VEVP calculations are discussed. The proposal lays the foundation for the time and space multiscale modeling and simulation of the high cycle fatigue of thermoplastic solids and structures.
Disciplines :
Materials science & engineering
Author, co-author :
Hun, Darith ; Université de Liège - ULiège > Aérospatiale et Mécanique (A&M)
Haddad, Mohamed ; Université catholique de Louvain, Institute of Mechanics, Materials and Civil Engineering, Louvain-La-Neuve, Belgium
Doghri, Issam ; Université catholique de Louvain, Institute of Mechanics, Materials and Civil Engineering, Louvain-La-Neuve, Belgium
Tsilimidos, Georgios; Université catholique de Louvain, Institute of Mechanics, Materials and Civil Engineering, Louvain-La-Neuve, Belgium
Lackner, Michael; Johannes Kepler University, Institute of Polymer Product Engineering, Linz, Austria
Major, Zoltan; Johannes Kepler University, Institute of Polymer Product Engineering, Linz, Austria
Doppelbauer, Leonhard ; Johannes Kepler University, Institute of Polymer Product Engineering, Linz, Austria ; Johannes Kepler University, Institute of Technical Mechanics, Linz, Austria
Haouala, Sara; Holtec Europe SL, Madrid, Spain
Language :
English
Title :
A computationally efficient hybrid formulation for viscoelastic–viscoplastic polymer solids and structures under large numbers of loading cycles
H2020 - 862015 - MOAMMM - Multi-scale Optimisation for Additive Manufacturing of fatigue resistant shock-absorbing MetaMaterials
Funders :
Erasmus+ EC - European Commission European Union
Funding text :
The present work was conducted and funded within the framework of the European Union\u2019s Horizon 2020 research and innovation programme for the project \u201CMulti-scale Optimisation for Additive Manufacturing of fatigue resistant shock-absorbing MetaMaterials (MOAMMM)\u201D , grant agreement No. 862015 , of the H2020-EU.1.2.1. - FET OpenProgramme. Also, G. Tsilimidos gratefully acknowledges a scholarship of STRAINS , the European Master on Advanced Solid Mechanics ( https://www.master-strains.eu/ ) funded by the Erasmus+ program of the European Union .
Bahloul, A., Doghri, I., Adam, L., Linking a phase field model for polymer crystallization to full-field micromechanical simulations of semi-crystalline polymers. Comput. Mater. Sci., 199, 2021, 110685, 10.1016/j.commatsci.2021.110685.
Bahloul, A., Doghri, I., Adam, L., Mesoscale modelling of polymer powder densification due to thermal sintering. Appl. Math. Model. 114 (2023), 408–422, 10.1016/j.apm.2022.09.022.
Barkey, M., Socie, D., Hsia, K., A yield surface approach to the estimation of notch strains for proportional and nonproportional cyclic loading. J. Eng. Mater. Technol. 116 (1994), 173–180, 10.1115/1.3225814.
Bird, R.B., Carreau, P.J., A nonlinear viscoelastic model for polymer solutions and melts—I. Chem. Eng. Sci. 23:5 (1968), 427–434, 10.1016/0009-2509(68)87018-6.
Buczynski, A., Glinka, G., An analysis of elasto-plastic strains and stresses in notched bodies subjected to cyclic non-proportional loading paths. Eur. Struct. Integr. Soc. 31 (2003), 265–283, 10.1016/S1566-1369(03)80015-8.
CM3, A., CM3 Finite Element Solver for Multiphysics Problems. 2024, Computational and Multiscale Mechanics of Materials laboratory, Uliège, Belgium URL https://gitlab.onelab.info/cm3/cm3Libraries.git.
Cobian, L., Rueda-Ruiz, M., Fernandez-Blazquez, J., Martinez, V., Galvez, F., Karayagiz, F., Lück, T., Segurado, J., Monclus, M., Micromechanical characterization of the material response in a PA12-SLS fabricated lattice structure and its correlation with bulk behavior. Polym. Test., 110, 2022, 107556, 10.1016/j.polymertesting.2022.107556 URL https://www.sciencedirect.com/science/article/pii/S0142941822000812.
Cognard, J.-Y., Ladevèze, P., A large time increment approach for cyclic viscoplasticity. Int. J. Plast. 9:2 (1993), 141–157, 10.1016/0749-6419(93)90026-M.
Findley, W.N., Lai, J.S., Onaran, K., Creep and Relaxation of Nonlinear Viscoelastic Materials - with an Introduction to Linear Viscoelasticity. 1989, Dover Publications, New York.
Frank, G.J., Brockman, R.A., A viscoelastic–viscoplastic constitutive model for glassy polymers. Int. J. Solids Struct. 38:30–31 (2001), 5149–5164, 10.1016/S0020-7683(00)00339-5.
Gravouil, A., Combescure, A., Multi-time-step explicit–implicit method for non-linear structural dynamics. Internat. J. Numer. Methods Engrg. 50:1 (2001), 199–225, 10.1002/1097-0207(20010110)50:1<199::AID-NME132>3.0.CO;2-A.
Guennouni, T., Sur une méthode de calcul de structures soumises à des chargements cycliques: l'homogénéisation en temps. ESAIM Math. Model. Numer. Anal. 22:3 (1988), 417–455 URL http://www.numdam.org/item/M2AN_1988__22_3_417_0/.
Haouala, S., Doghri, I., Modeling and algorithms for two-scale time homogenization of viscoelastic-viscoplastic solids under large numbers of cycles. Int. J. Plast. 70 (2015), 98–125, 10.1016/j.ijplas.2015.03.005.
Hasan, O., Boyce, M., A constitutive model for the nonlinear viscoelastic viscoplastic behavior of glassy polymers. Polym. Eng. Sci. 35:4 (1995), 331–344, 10.1002/pen.760350407.
Heczko, J., Kottner, R., Modeling of material damage using finite elements and time homogenization in case of finite strain. Appl. Math. Comput. 319 (2018), 264–273, 10.1016/j.amc.2017.03.014.
Herbland, T., Une méthode de correction élastoplastique pour le calcul en fatigue des zones de concentration de contraintes sous chargement cyclique multiaxial non proportionnel. (Ph.D. thesis), 2009, École Nationale Supérieure des Mines de Paris URL https://pastel.hal.science/tel-00479991.
Hoffmann, M., Seeger, T., A generalized method for estimating multiaxial elastic-plastic notch stresses and strains, Part 1: Theory. J. Eng. Mater. Technol. 107 (1985), 250–254, 10.1115/1.3225814.
Hössinger-Kalteis, A., Lackner, M., Major, Z., Karayagiz, F., Nopper, H., Lück, T., Design method for individualised 3D printed lattice shoe midsoles. Proc. Inst. Mech. Eng. Part L: J. Materials: Des. Appl., 2024, 10.1177/14644207241291232 14644207241291232.
Hughes, T.J., Stewart, J.R., A space-time formulation for multiscale phenomena. J. Comput. Appl. Math. 74:1–2 (1996), 217–229, 10.1016/0377-0427(96)00025-8.
Hun, D.A., Haddad, M., Doghri, I., Modeling and simulation of viscoelastic solids under large numbers of loading cycles. Mech. Adv. Mater. Struct. 30:8 (2023), 1542–1558, 10.1080/15376494.2022.2037170.
Johnsen, J., Clausen, A.H., Grytten, F., Benallal, A., Hopperstad, O.S., A thermo-elasto-viscoplastic constitutive model for polymers. J. Mech. Phys. Solids 124 (2019), 681–701, 10.1016/j.jmps.2018.11.018.
Krairi, A., Doghri, I., Robert, G., Multiscale high cycle fatigue models for neat and short fiber reinforced thermoplastic polymers. Int. J. Fatigue 92 (2016), 179–192, 10.1016/j.ijfatigue.2016.06.029.
Lesser, A., Changes in mechanical behavior during fatigue of semicrystalline thermoplastics. J. Appl. Polym. Sci. 58:5 (1995), 869–879, 10.1002/app.1995.070580504.
Lucarini, S., Cobian, L., Voitus, A., Segurado, J., Adaptation and validation of FFT methods for homogenization of lattice based materials. Comput. Methods Appl. Mech. Engrg., 388, 2022, 114223, 10.1016/j.cma.2021.114223 URL https://www.sciencedirect.com/science/article/pii/S0045782521005545.
Magri, M., Lucarini, S., Lemoine, G., Adam, L., Segurado, J., An FFT framework for simulating non-local ductile failure in heterogeneous materials. Comput. Methods Appl. Mech. Engrg., 380, 2021, 113759, 10.1016/j.cma.2021.113759.
Major, Z., Lackner, M., Hössinger-Kalteis, A., Lück, T., Characterization of the fatigue behavior of SLS thermoplastics. Procedia Struct. Integr. 34 (2021), 191–198, 10.1016/j.prostr.2021.12.028 URL https://www.sciencedirect.com/science/article/pii/S2452321621002699, The second European Conference on the Structural Integrity of Additively Manufactured Materials.
Mandel, J., Cours de mécanique des milieux continus. 1994, Gauthier-Villars 1966/re-edited 1994, J. Gabay.
MATLAB, J., 2022, The MathWorks Inc., Natick, Massachusetts, United States URL https://www.mathworks.com.
Miled, B., Doghri, I., Delannay, L., Coupled viscoelastic–viscoplastic modeling of homogeneous and isotropic polymers: Numerical algorithm and analytical solutions. Comput. Methods Appl. Mech. Engrg. 200:47–48 (2011), 3381–3394, 10.1016/j.cma.2011.08.015.
Mortazavian, S., Fatemi, A., Fatigue of short fiber thermoplastic composites: A review of recent experimental results and analysis. Int. J. Fatigue 102 (2017), 171–183, 10.1016/j.ijfatigue.2017.01.037.
Oskay, C., Fish, J., Fatigue life prediction using 2-scale temporal asymptotic homogenization. Internat. J. Numer. Methods Engrg. 61:3 (2004), 329–359, 10.1002/nme.1069.
Schapery, R.A., A method of viscoelastic stress analysis using elastic solutions. J. Franklin Inst. 279:4 (1965), 268–289, 10.1016/0016-0032(65)90339-X.
Shaw, M.T., MacKnight, W.J., Introduction to Polymer Viscoelasticity. 2018, John Wiley & Sons.
Smith, M., ABAQUS. 2009, Dassault Systèmes Simulia Corp, United States.
Uchida, M., Tada, N., Micro-, meso-to macroscopic modeling of deformation behavior of semi-crystalline polymer. Int. J. Plast. 49 (2013), 164–184, 10.1016/j.ijplas.2013.03.007.
Van Dommelen, J.v., Parks, D., Boyce, M., Brekelmans, W., Baaijens, F., Micromechanical modeling of the elasto-viscoplastic behavior of semi-crystalline polymers. J. Mech. Phys. Solids 51:3 (2003), 519–541, 10.1016/S0022-5096(02)00063-7.
Wang, S., Tang, S., He, C., Wang, Q., Cyclic deformation and fatigue failure mechanisms of thermoplastic polyurethane in high cycle fatigue. Polymers, 15(4), 2023, 899, 10.3390/polym15040899.
