Knauss-Emri model, nonlinear viscoelasticity

The Knauss-Emri model captures some of the nonlinear stress relaxation response of materials and looks like linear viscoelasticity in the reduced time variables, and hence is relatively straightforward to implement. However, the observation that material nonlinearities occur in shearing deformations as well as in compression, where the free-volume mechanisms predict decreasing mobility suggest that the model is limited in its usefulness (164,165). 

The Knauss-Emri Model. There have been several works in the literature in which volume- or ee-ooZume-dependent clocks were used to describe the nonlinear viscoelastic response of polymeric glasses. The chief success among these is the ICnauss-Emri model (163) in which the reduced time was defined in terms of a shift factor that depended on temperature, stress, and concentration of small molecules in such a way that the responses depended on the free volume induced by each of these parameters. For an isothermal single phase and homogeneous material, the equations are... 

A related issue is that the modulus is a viscoelastic property, as evidenced by the temperature/strain-rate dependence, and that for most poljnners (at least those without a large beta transition near the alpha transition) time-temperature superposition of, for example, the shear relaxation modulus is valid (80). Further, G Sell and McKenna (81) have shown that the 5neld stress vs strain rate also seems to obey time-temperature superposition. Hence there is a correlation between the viscoelastic properties and the yield response of pol5uners, though one that is not generally stated explicitly. We note that some of the models mentioned previously, such as those of Caruthers group (41,42), Tervoort and co-workers (40), and Knauss and Emri (35), are (nonlinear) viscoelastic models that have yield arising due to the nonlinear response induced by the material clock (see Viscoelasticity). 

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. 

Strain-Induced Dilatation. An alternative view of yield in poljnners comes fi-om the fact that a tensile strain induces a hydrostatic tension in the material and a corresponding increase in the sample volume. This in turn translates to an increase in the free volume, which increases the poljmaer mobility and effectively lowers the glass-transition temperature (Tg) of the poljmaer (alternatively it can be looked upon as increasing the free volume to the value it would have at the normal measured Tg). The increased mobility results in a lowering of the yield stress. Knauss and Emri (35) used an integral representation of nonlinear viscoelasticity with a state-dependent variable related to free volume to model the yield behavior, with the free volume a function of temperature, time, and stress history. This model uses the concept of reduced time (see Viscoelasticity), where application of a tensile stress causes a volume dilatation and consequently causes the material time scale to change by a shift factor related to the magnitude of the applied stress. Yield occurs because the free-volume shift factor causes the molecular mobility to increase in such a way that yield can occur. 

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