Modelling of fibrous polymeric composites in the viscoelastic range
Andrzej P. Wilczyński, Marian Klasztorny Politechnika Warszawska, Instytut Mechaniki i Konstrukcji, ul. Narbutta 85, 02-524 Warszawa
Annals 2 No. 3, 2002 pages 97-102
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abstract An analytical method for viscoelastic modelling of a fibre-reinforced resin-matrix composite has been developed, based on the Wilczynski’s reinforcement theory, the HWKK rheological model for resins and the elastic-viscoelastic analogy. The following assumptions have been adopted: 1) the matrix is a linear viscoelastic isotropic material, described by the HWKK model [6]; 2) the fibres are made of a linear elastic monotropic material, with the direction of monotropy coinciding the fibre’s axis; 3) the fibres are uniformly distributed in a hexagonal scheme; 4) the matrix - fibres contact is protected during loading; 5) an ultra-thin intermediate layer between the matrix and the fibre is neglected; 6) the homogenised composite is modelled as a linear viscoelastic monotropic material, with the direction of monotropy coinciding the fibres’ alignment. The Wilczynski’s reinforcement theory, used in the method of viscoelastic modelling, is based on four tasks of the Lamé type and Hill’s assumption. Constitutive equations of viscoelasticity, describing the HWKK model of a resin matrix, are formulated in the shear- -bulk description (Eqs (1)-(5)), using three generating functions, i.e. a fractional exponential function and two normal exponential functions. This model, reflecting short-lasting and long-lasting first-rank reversible creep, is described by 9 material constants [6], collected in Table 1 for two basic resins (epoxy and polyester). The approximate constitutive equations for the homogenised composite, described by 5 elastic constants and 13 viscoelastic constants, have been formulated as Eqs (6), (7). An analytic method, for deriving the viscoelastic constants of the final material, is composed of five stages: 1) analytic derivation of the elastic compliances of the composite; 2) analytic derivation of the shear and bulk complex complianes of the matrix; 3) analytic derivation of the exact complex compliances of the composite, using the elastic - viscoelastic analogy; 4) analytic derivation of the approximate complex compliances of the composite; 6) analytic derivation of 10 viscoelastic constants of the homogenised composite. The results of viscoelastic modelling of a composite are verified by comparison of the approximate and the exact complex compliances of the composite. All stages have been described in detail (Eqs (9)-(18)). A computer aided algorithm for estimation of the composite’s material constants has been formulated, programmed and tested on selected materials. The results of modelling, with the obtained high accuracy, for the VHDPE Tenfor SN1A polyethylene/Epidian 53 epoxy composite are presented (Fig. 1).