Schapery model

The aL of Kevlar 49 (obtained by them) gave excellent agreement with measured values for unidirectional Kevlar/epoxy composites in terms of the Schapery model, which is an improvement over the reported axial aL of Kevlar fiber and those of Kevlar epoxy composites reported by Strife and Prewo [90]. 

The Schapery Model. One of the earliest models of the nonlinear viscoelastic response of pol5nners to use the concept of a reduced time is due to Schapery (147-149). The model is based on thermodynamic considerations and has a form similar to the Boltzmann superposition principal described previously. The model time dependences, except for the shift factors, are the same as those obtained in the linear response regime. Hence, the model is relatively easy to implement and to determine the relevant material parameters. It results in a generalization of the generalized superposition principal developed by Leaderman (150). 

An interesting aspect of the Bemstein-Shokooh model is that the structure of the nonlinear equations is somewhat different from either the Schapery model or the Zapas strain-clock model. In the Bernstein-Shokooh model, the... 

Verify that the x function of Eq. 11.36 for creep of a nonlinear viscoelastic material in simple tension that can be represented by the Schapery model using a power law for creep. (Assume a constant Poisson s ratio.)... 

The comparison of calculated and experimental data of the creep curves showed a good correlation. After comparing the calculated results it can be concluded that the viscoelastic behavior of the technical fabric can be described by the one-integral model. In warp and weft directions the numerical curve fitting resulted in a difference of 3.5-9.9% and 0.3-2.6% respectively. Therefore, the Schapery model with the power function characterized more accurate creep behavior in weft direction than warp direction. Also the power function described the strain evaluation better than the exponential function. This research concluded that both the linear and nonlinear viscoelastic identifications based on different material models can be brought together and the results of linear characterization can be applied to the nonlinear description of the material. 

The creep and recovery behaviour of an UHMWPE was studied in the region of small xmiaxial deformations by Zapas and Crissman [152]. These results are used to illustrate the capability of the Schapery model to represent the viscoelastic/viscoplastic behaviour of UHMWPE. Creep and recovery experiments were carried out on specimens under creep stresses in the range 1-8 MPa. In Figure 7.9 are plotted the creep compliances obtained, showing to be stress dependent above 1 MPa. Using the appropriate values for the model parameters, the strain under creep and creep-recovery loading conditions were very well captured as shown in Figure 7.10. 

The quasielastic method as developed by Schapery [26] is used in the development of the viscoelastic residual stress model. The use of the quasielastic method is motivated by the fact that the relaxation moduli are required in the viscoelastic analysis of residual stresses, whereas the experimental characterization of composite materials is usually in terms of the creep compliances. An excellent account of the development of the quasielastic method is given in [27]. The underlying restriction in the application of the quasielastic method is that the compliance response of the material shows little curvature when plotted versus log time [28]. Harper [27] shows excellent agreement between the quasielastic method and direct inversion for AS4/3510-6 graphite/epoxy composite. For most graphite/thermoset systems, the restrictions imposed by the quasielastic method are satisfied. 

There is strong interest to analytically describe the fzme-dependence of polymer creep in order to extrapolate the deformation behaviour into otherwise inaccessible time-ranges. Several empirical and thermo-dynamical models have been proposed, such as the Andrade or Findley Potential equation [47,48] or the classical linear and non-linear visco-elastic theories ([36,37,49-51]). In the linear viscoelastic range Findley [48] and Schapery [49] successfully represent the (primary) creep compliance D(t) by a potential equation ... 

The principal limitation of Wright s static displacement model is that it does not consider the accumulation of deformations due to the passage of a number of waves. This problem has been approached by Schapery and Dunlap (1978), modeling the soil as a linearly viscoelastic material. Their analysis also included the effect of energy adsorption of the seafloor on the wave characteristics. 

The Schapery-Zapas-Crissman (SZC) model that was described in an earlier section was used to model the constitutive behavior of a [90]i6 specimen. Under conditions of uniaxial loading transverse to the fiber direction at constant temperature, the SZC constitutive model takes the form ... 

Table 12.2 Material properties for the Schapery-Zapas-Crissman constitutive model...

Polymeric materials have relatively large thermal expansion. However, by incorporating fillers of low a in typical plastics, it is possible to produce a composite having a value of a only one-fifth of the unfilled plastics. Recently the thermal expansivity of a number of in situ composites of polymer liquid crystals and engineering plastics has been studied [14,16, 98, 99]. Choy et al [99] have attempted to correlate the thermal expansivity of a blend with those of its constituents using the Schapery equation for continuous fiber reinforced composites [100] as the PLC fibrils in blends studied are essentially continuous at the draw ratio of 2 = 15. Other authors [14,99] observed that the Takayanagi model [101] explains the thermal expansion. 

Other models have also been proposed in addition to Schapery s model. Chamis [108] has given an expression for ajj in the longitudinal direction identical to that of Schapery. The aj in the transverse direction due to Chamis [108] is... 

Fig. 60. Comparison of experimental (circles) creep and recovery behavior of a glass-reinforced phenolic resin with the predictions from the Schapery (147-149) nonlinear viscoelastic model. o Experimental Data A Predicted Recovery Data (Nonlinear Theory). After Schapery (147), with permission.

E. Krempl and K. Ho, in R. A. Schapery, ed.. An Overstress Model for Solid Polymer Deformation-Behavior Applied to Nylon-66 in Time Dependent and Nonlinear Effects in Polymers and Composites, ASTM STP-1357, W Conshohocken American Society Testing and Materials (Series American Society for Testing and Materials Special Technical Publication, Vol 1357) 2000, pp. 118-137. 

Schapery [16, 17] has used the theory of the thermodynamics of irreversible processes to produce a model that may be viewed as a further extension of Leaderman s. Schapery continued Leaderman s technique of replacing the stress by a function of stress /(a) in the superposition integral, but also replaced time by a function of time, the reduced time ip. The material is assumed to be linear viscoelastic at small strains, with a creep compliance function of the form [17]... 

He showed how the use of double logarithmic plots of the recovery strain against time, obtained for different stress levels, could be related to one another by shift factors the shift factors then could be related simply to gi and. The technique of step loading combined with Equation (10.22) has been used also by Crook [18] and Lai and Bakker [19]. Schapery s model has been applied to nitrocellulose, fibre-reinforced phenolic resin and pol3dsobutylene [17], polycarbonate [18] and high-density polyethylene [19]. 

Botha, Jones, and Brinson,Henriksen,(22) Becker et a/.,(23) and Yadagiri and Papi Reddy(24,25) reported results of viscoelastic finite-element analysis of adhesive joints. Henriksen used Schapery s(26) nonlinear viscoelastic model to verify the experimental results of Peretz and Weitsman(27) for an adhesive layer. The work of Becker et is largely... 

Schaffer and Adams< 2) carried out a nonlinear viscoelastic analysis of a unidirectional composite laminate using the finite-element method. The nonlinear viscoelastic constitutive law proposed by Schapery<26) was used in conjunction with elastoplastic constitutive relations to model the composite response beyond the elastic limit. 

A review of the literature reveals that previous finite-element analyses of adhesive joints were either based on simplified theoretical models or the analyses themselves did not exploit the full potential of the finite-element method. Also, several investigations involving finite-element analyses of the same adhesive joint have reported apparent contradictory conclusions about the variations of stresses in the joint.(24,36) while the computer program VISTA looks promising (see Table 1), its nonlinear viscoelastic capability is limited to Knauss and Emri.(28) Recently, Reddy and Roy(E2) (see also References 37 and 38) developed a computer program, called NOVA, based on the updated Lagrangian formulation of the kinematics of deformation of a two-dimensional continuum and Schapery s(26) nonlinear viscoelastic model. The free-volume model of Knauss and Emri(28) can be obtained as a degenerate model from Schapery s model. 

Lefebvre et developed a generalized Fickean diffusion model using the free-volume concept. A finite-element model that accounts for Schapery s nonlinear viscoelastic constitutive relation(25) and the nonlinear diffusion model of Lefebvre et was discussed by Roy and Reddy. ( 4,45)... 

A review of the theoretical basis, finite-element model, and sample applications of the program NOVA are presented. The updated incremental Lagrangian formulation is used to account for geometric nonlinearity (i.e., small strains and moderately large rotations), the nonlinear viscoelastic model of Schapery is used to account for the nonlinear constitutive behavior of the adhesive, and the nonlinear Fickean diffusion model in which the diffusion coefficient is assumed to depend on the temperature, penetrant concentration, and dilational strain is used. Several geometrically nonlinear, linear and nonlinear viscoelastic and moisture... 

Park S, Schapery R (1997) A viscoelastic constitutive model for particulate composites with growing damage. Int J Solids Struct 34 931-947... 

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