Abstract
Schapery’s nonlinear viscoelastic model is written in incremental form, and three different approximations of nonlinearity functions in the time increment are systematically analysed with respect to the convergence rate. It is shown that secant slope is the best approximation of the time shift factor, leading to significantly higher convergence rate. This incremental form of the viscoelastic model, Zapas’ model for viscoplasticity, supplemented with terms accounting for damage effect is used to predict inelastic behaviour of material in stress controlled tests. Then the incremental formulation is inverted to simulate stress development in ramps where strain is the input parameter. A comparison with tests shows good ability of the model in inverted form to predict stress–strain response as long as the applied strain is increasing. However, in strain controlled ramps with unloading, the inverted model shows unrealistic hysteresis loops. This is believed to be a proof of the theoretically known incompatibility of the stress and strain controlled formulations for nonlinear materials. It also shows limitations of material models identified in stress controlled tests for use in strain controlled tests.
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References
Beijer, J.G.J., Spoormaker, J.L.: Solution strategies for FEM analysis with nonlinear viscoelastic polymers. Comput. Struct. (2002). doi:10.1016/S0045-7949(02)00089-5
Crochon, T., Schönherr, T., Li, C., Levesque, M.: On finite-element implementation strategies of Schapery-type constitutive theories. Mech. Time-Depend. Mater. (2010). doi:10.1007/s11043-010-9115-8
Giannadakis, K., Varna, J.: Analysis of nonlinear shear stress–strain response of unidirectional GF/EP composite. Composites, Part A, Appl. Sci. Manuf. (2014). doi:10.1016/j.compositesa.2014.03.009
Giannadakis, K., Mannberg, P., Joffe, R., Varna, J.: The sources of inelastic behavior of glass fibre/vinylester non-crimp fabric [45/−45]s laminates. J. Reinf. Plast. Compos. (2011). doi:10.1177/0731684411412644
Haj-Ali, R.M., Muliana, A.H.: Numerical finite element formulation of the Schapery non-linear viscoelastic material model. Int. J. Numer. Methods Eng. (2004). doi:10.1002/nme.861
Henriksen, M.: Nonlinear visco-elastic stress analysis—a finite element approach. Comput. Struct. 18, 133–139 (1984)
Kumar, R.S., Talreja, R.: A continuum damage model for linear viscoelastic composite materials. Mech. Mater. 35, 463–480 (2003)
Lai, J., Bakker, A.: 3-D Schapery representation for non-linear viscoelasticity and finite element implementation. Comput. Mech. 18, 182–191 (1996)
Lou, Y.C., Schapery, R.A.: Viscoelastic characterization of a nonlinear fiber-reinforced plastic. J. Compos. Mater. 5, 208–234 (1971)
Luk-Cyr, J., Crochon, T., Li, C., Levesque, M.: Interconversion of linearly viscoelastic material functions expressed as Prony series: a closure. Mech. Time-Depend. Mater. (2013). doi:10.1007/s11043-012-9176-y
Marklund, E., Eitzenberger, J., Varna, J.: Nonlinear viscoelastic viscoplastic material model including stiffness degradation for hemp/lignin composites. Compos. Sci. Technol. (2008). doi:10.1016/j.compscitech.2008.03.011
Nordin, L.-O., Varna, J.: Nonlinear viscoplastic and nonlinear viscoelastic material model for paper fiber composites in compression. Composites, Part A, Appl. Sci. Manuf. (2006). doi:10.1016/j.compositesa.2005.02.015
Papanicolaou, G.C., Zaoutsos, S.P., Kontou, E.A.: Fiber orientation dependence of continuous carbon/epoxy composites nonlinear viscoelastic behaviour. Compos. Sci. Technol. (2004). doi:10.1016/j.compscitech.2004.05.005
Poon, H., Ahmad, M.F.: A material point time integration procedure for anisotropic, thermo rheologically simple, viscoelastic solids. Comput. Mech. 21, 236–252 (1998)
Pupure, L., Varna, J., Joffe, R., Pupurs, A.: An analysis of the nonlinear behavior of lignin-based flax composites. Mech. Compos. Mater. (2013). doi:10.1007/s11029-013-9330-x
Rozite, L., Varna, J., Joffe, R., Pupurs, A.: Nonlinear behavior of PLA and lignin-based flax composites subjected to tensile loading. J. Thermoplast. Compos. Mater. (2013). doi:10.1177/0892705711425846
Schapery, R.A.: On characterization of nonlinear viscoelastic materials. Polym. Eng. Sci. 9, 295–310 (1969a)
Schapery, R.A.: Further development of a thermodynamic constitutive theory: stress formulation. Purdue University report No. 69-2 (1969b)
Schapery, R.A.: Nonlinear viscoelastic and viscoplastic constitutive equations based on thermodynamics. Mech. Time-Depend. Mater. 1, 209–240 (1997)
Varna, J.: Characterization of viscoelasticity, viscoplasticity and damage in composites. In: Guedes, R.M. (ed.) Creep and Fatigue in Polymer Matrix Composites, pp. 514–542. Woodhead Publishing Materials, London (2011)
Varna, J., Oldenbo, M.: An incremental 2D constitutive model accounting for linear viscoelasticity and damage development in short fiber composites. Int. J. Numer. Methods Eng. (2005). doi:10.1002/nme.1411
Varna, J., Rozite, L., Joffe, R., Pupurs, A.: Nonlinear behavior of PLA based flax composites. Plast. Rubber Compos. Process. Appl. (2012). doi:10.1179/1743289811Y.0000000007
Zapas, L.J., Crissman, J.M.: Creep and recovery behavior of ultra-high molecular weight polyethylene in the region of small uniaxial deformations. Polymer 25, 57–62 (1984)
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Varna, J., Pupure, L. & Joffe, R. Incremental forms of Schapery’s model: convergence and inversion to simulate strain controlled ramps. Mech Time-Depend Mater 20, 535–552 (2016). https://doi.org/10.1007/s11043-016-9311-2
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DOI: https://doi.org/10.1007/s11043-016-9311-2