The generalized Kelvin chain-based model for an orthotropic viscoelastic material
Tóm tắt
We propose a constitutive material model to describe the rheological (viscoelastic) mechanical response of timber. The viscoelastic model is based on the generalized Kelvin chain applied to the orthotropic material and is compared to the simple approach given by standards. The contribution of this study consists of the algorithmization of the viscoelastic material model of the material applied to the orthotropic constitutive law and implementation into the FEM solver. In the next step, the fitting of the input parameters of the Kelvin chain is described, and at least a material model benchmark and comparison to the approach given by standards were done. The standardized approach is based on the reduction of the material rigidity at the end of the loading period using a creep coefficient, whereas the loading history state variables are not considered when establishing the result for a specific time step. The paper presents the benefits of the rheological model. It also demonstrates the fitting algorithm based on particle swarm optimization and the least squares method, which are essential for the use of the generalized Kelvin chain model. The material model based on the orthotropic generalized Kelvin chain was implemented into the FEM solver for the shell elements. This material model was validated on the presented benchmark tasks, and the influence of the time step size on the accuracy of model results was analyzed.
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Tài liệu tham khảo
Bažant, Z.P.: Constitutive equation of wood at variable humidity and temperature. Wood Sci. Technol. 19, 159–177 (1985). ISSN 1432-5225
Bažant, Z.P., Jirásek, M.: Fundamentals of linear viscoelasticity. In: Bažant, Z.P., Jirásek, M. (eds.) Creep and Hygrothermal Effects in Concrete Structures. Solid Mechanics and Its Applications, pp. 9–28. Springer, Dordrecht (2018). ISBN 978-94-024-1136-2. https://doi.org/10.1007/978-94-024-1138-6_2
Bažant, Z.P., Wu, S.T.: Dirichlet series creep function for aging concrete. J. Eng. Mech. Div. 99(2), 367–387 (1973). https://doi.org/10.1061/JMCEA3.0001741. ISSN 0044-7951
Bengtsson, C., Kliger, R.: Bending creep of high-temperature dried spruce timber. Holzforschung 57(1), 95–100 (2003). https://doi.org/10.1515/HF.2003.015. ISSN 0018-3830
Bengtsson, R., Afshar, R., Gamstedt, E.K.: An applicable orthotropic creep model for wood materials and composites. Wood Sci. Technol. 56(6), 1585–1604 (2022). https://doi.org/10.1007/s00226-022-01421-x. ISSN 0043-7719
Distéfano, N.: On the identification problem in linear viscoelasticity. Z. Angew. Math. Mech. 50(11), 683–690 (1970). https://doi.org/10.1002/zamm.19700501106. ISSN 00442267
EN 1995-1-1:2004 Eurocode 5: Design of timber structures – Part 1-1: General - Common rules and rules for buildings. The European Union Per Regulation 305/2011 (2004)
Endo, V.T., de Carvalho Pereir, J.C.: Linear orthotropic viscoelasticity model for fiber reinforced thermoplastic material based on Prony series. Mech. Time-Depend. Mater. 21, 199–221 (2017). https://doi.org/10.1007/s11043-016-9329-8
Fortino, S., Mirianon, F., Toratti, T.: A 3D moisture-stress FEM analysis for time dependent problems in timber structures. Mech. Time-Depend. Mater. 13(4), 333–356 (2009). https://doi.org/10.1007/s11043-009-9103-z. ISSN 1385-2000
Ghosh, K., Lopez-Pamies, O.: On the two-potential constitutive modeling of dielectric elastomers. Meccanica 56(6), 1505–1521 (2021). https://doi.org/10.1007/s11012-020-01179-1. [cit. 2024-01-08]. ISSN 0025-6455
Ghosh, K., Shrimali, B., Kumar, A., Lopez-Pamies, O.: The nonlinear viscoelastic response of suspensions of rigid inclusions in rubber: I—Gaussian rubber with constant viscosity. Journal of the Mechanics and Physics of Solids [online], 154 (2021). https://doi.org/10.1016/j.jmps.2021.104544. [Cit. 2024-01-08]. ISSN 00225096
Hanhijärvi, A., Hunt, D.: Experimental indication of interaction between viscoelastic and mechano-sorptive creep. Wood Sci. Technol. 32(1), 57–70 (1998). https://doi.org/10.1007/BF00702560. ISSN 0043-7719
Hanhijärvi, A., Mackenzie-Helnwein, P.: Computational analysis of quality reduction during drying of lumber due to irrecoverable deformation. I: orthotropic viscoelastic-mechanosorptive-plastic material model for the transverse plane of wood. J. Eng. Mech. 129(9), 996–1005 (2003). https://doi.org/10.1061/(ASCE)0733-9399(2003)129:9(996). ISSN 0733-9399
Holzer, S.M., Loferski, J.R., Dillard, D.A.: A review of creep in wood: concepts relevant to develop long-term behavior predictions for wood structures. Wood Fibre Sci. 21(4), 376–392 (1989). ISSN 7356161
Huč, S., Svensson, S.: Coupled two-dimensional modeling of viscoelastic creep of wood. Wood Sci. Technol. 52(1), 29–43 (2018). https://doi.org/10.1007/s00226-017-0944-3. ISSN 0043-7719
Hunt, D.G., Shelton, C.F.: Progress in the analysis of creep in wood during concurrent moisture changes. J. Mater. Sci. 22(1), 313–320 (1987). https://doi.org/10.1007/BF01160586. ISSN 0022-2461
Jirásek, M., Zeman, J.: Přetváření a porušování materiálů: dotvarování, plasticita, lom a poškození. ČVUT Publishing, Praha (2006). ISBN 80-01-03555-7
Kaliske, M.: A formulation of elasticity and viscoelasticity for fibre reinforced material at small and finite strains. Comput. Methods Appl. Mech. Eng. 185(2–4), 225–243 (2000). https://doi.org/10.1016/S0045-7825(99)00261-3. ISSN 00457825
Kennedy, J., Eberhart, R.: Particle swarm optimization. In: Proceedings of ICNN’95 – International Conference on Neural Networks, pp. 1942–1948. IEEE (1995). ISBN 0-7803-2768-3. https://doi.org/10.1109/ICNN.1995.488968
Lawson, J.D.: An order five Runge–Kutta process with extended region of stability. SIAM J. Numer. Anal. 3(4), 593–597 (1966). https://doi.org/10.1137/0703051. [cit. 2024-01-08]. ISSN 0036-1429
Liu, T.: Creep of wood under a large span of loads in constant and varying environments. Holz Roh- Werkst. 52(1), 63–70 (1994). https://doi.org/10.1007/BF02615022. ISSN 0018-3768
Lockett, F.J.: Nonlinear viscoelastic solids. XI + 195. S. m. Fig. London/New York 1972. Academic Press. Z. Angew. Math. Mech. 54(4), 288–288 (1974). https://doi.org/10.1002/zamm.19740540422. ISSN 00442267
Milch, J., Tippner, J., Sebera, V., Brabec, M.: Determination of the elasto-plastic material characteristics of Norway spruce and European beech wood by experimental and numerical analyses. Holzforschung 70(11), 1081–1092 (2016). https://doi.org/10.1515/hf-2015-0267
Nowacki, W.: Theorie des Kriechens. Lineare Viskoelastizität. Franz Deuticke, Wien (1965)
Ozyhar, T., Hering, S., Niemz, P.: Viscoelastic characterization of wood: time dependence of the orthotropic compliance in tension and compression. J. Rheol. 57(2), 699–717 (2013). https://doi.org/10.1122/1.4790170. ISSN 0148-6055
Pister, K.S.: Mathematical modeling for structural analysis and design. Nucl. Eng. Des. 18(3), 353–375 (1972). https://doi.org/10.1016/0029-5493(72)90108-2. ISSN 00295493
Ranta-Maunus, A.: The viscoelasticity of wood at varying moisture content. Wood Sci. Technol. 9(3), 189–205 (1975). https://doi.org/10.1007/BF00364637. ISSN 0043-7719
Shi, Y., Eberhart, R.C.: Parameter selection in particle swarm optimization. In: Porto, V.W., Saravanan, N., Waagen a, D., Eiben, A.E. (eds.) Evolutionary Programming VII. Lecture Notes in Computer Science, pp. 591–600. Springer, Berlin (1998). ISBN 978-3-540-64891-8. https://doi.org/10.1007/BFb0040810
Svensson, S., Toratti, T.: Mechanical response of wood perpendicular to grain when subjected to changes of humidity. Wood Sci. Technol. 36(2), 145–156 (2002). https://doi.org/10.1007/s00226-001-0130-4. ISSN 0043-7719
Toratti, T.: Creep of timber beams in a variable environment. Dissertation thesis, Helsinki University of Technology (1992)
Toratti, T., Svensson, S.: Mechano-sorptive experiments perpendicular to grain under tensile and compressive loads. Wood Sci. Technol. 34(4), 317–326 (2000). https://doi.org/10.1007/s002260000059. ISSN 0043-7719
Vidal-Sallé, E., Chassagne, P.: Constitutive equations for orthotropic nonlinear viscoelastic behaviour using a generalized Maxwell model application to wood material. Mech. Time-Depend. Mater. 11(2), 127–142 (2007). https://doi.org/10.1007/s11043-007-9037-2. ISSN 1385-2000