Nonlinear responses, stress-strain curves

Here m is the usual small-strain tensile stress-relaxation modulus as described and observed in linear viscoelastic response [i.e., the same E(l) as that discussed up to this point in the chapter). The nonlinearity function describes the shape of the isochronal stress-strain curve. It is a simple function of A, which, however, depends on the type of deformation. Thus for uniaxial extension,... 

Figure 3. Stress-strain curves in nonlinear elastic and nonlinear viscoelastic responses...

Fig. 14.12 shows the stress—strain—normalized resistance plot for the specimen with 12-mm notch spacing. We know that the nominal strain at fracture for the composite material is around 0.0147. Due to the notches, the strain concentration would be three times that of the smooth specimen. Thus any damage around the notch should start at one-third of the applied strain on the smooth specimen. This is indeed the case as seen in Fig. 14.13. A sharp change in the slope of the stress—strain curve is seen at a nominal strain of 0.005 = l/3eu- Before this knee the resistivity variation is nonlinear with respect to the applied strain. With the onset of damage at e = 0.005 (at the edges of the notches) a sharp increase in resistivity is seen. After this point the resistivity response is linear with the applied strain. Another jump in resistivity can be seen, probably due to damage initiation at the other notch, however, the stress—strain diagram does not detect this. After the second jump the sensor responds very...

Typical examples of tensile (isochronous) linear and nonlinear stress-strain diagrams for elastic and viscoelastic materials are shown in Fig, 10.1. For elastic materials, the response is time independent, so there is a single curve for multiple times and the nonlinearity is apparent as a deviation of the stress-strain response from linear. For linear viscoelastic materials, the isochronous response is linear, but the effective modulus decreases with time so that the stress-strain curves at different times are separated from one another. When a viscoelastic material behaves nonlinearly, the isochronous stress-strain curves begin to deviate from linearity at a certain stress level. Fig. 10.2 shows creep compliance data for an epoxy adhesive as a function of stress level for various time intervals after initial loading. 

Other less well-known types of nonlinearities include interaction and intermode . In the former, stress-strain response for a fundamental load component (e.g. shear) in a multi-axial stress state is not equivalent to the stress-strain response in simple one component load test (e.g. simple shear). For example. Fig. 10.3 shows that the stress-strain curve under pure shear loading of a composite specimen varies considerably from the shear stress-strain curve obtained from an off-axis specimen. In this type of test, a unidirectional laminate is tested in uniaxial tension where the fiber axis runs 15° to the tensile loading axis. A 90° strain gage rosette is applied to the specimen oriented to the fiber direction and normal to the fiber direction and thus obtain the strain components in the fiber coordinate system. Using simple coordinate transformations, the shear response of the unidirectional composite can be found (Daniel, 1993, Hyer, 1998). At small strains in the linear range, the shear response from the two tests coincide. 

Bias-induced reverse piezoelectric response Broadband dielectric spectroscopy (BDS) Dielectric permittivity spectrum Dielectric resonance spectroscopy Elastic modulus Ferroelectrets Electrical breakdown Acoustic method Characterization Dynamic coefficient Interferometric method Pressure and frequency dependence of piezoelectric coefficient Profilometer Quasistatic piezoelectric coefficient Stress-strain curves Thermal stability of piezoelectricity Ferroelectric hysteresis Impedance spectroscopy Laser-induced pressure pulse Layer-structure model of ferroelectret Low-field dielectric spectroscopy Nonlinear dielectric spectroscopy Piezoelectrically generated pressure step technique (PPS) Pyroelectric current spectrum Pyroelectric microscopy Pyroelectricity Quasistatic method Scale transform method Scanning pyroelectric microscopy (SPEM) Thermal step teehnique Thermal wave technique Thermal-pulse method Weibull distribution... 

This is a curious result, as it indicates that a nonlinear property can be calculated from linear data, but it has been found to describe accurately the response of polymeric liquids at sufficiently low shear rates. The low-shear-rate limiting behavior indicated by the above equations, which involves monotone increasing functions, is always shown in plots of nonlinear data the nonlinear responses involve overshoots in the material functions, but should always start out at short times, when the strain is still small, by following the low-shear-rate, LVE curve. Then, as the shear rate increases, the nonlinear data fall below the linear envelopes at shorter and shorter times [44,45]. These features can be seen in Fig. 10.9, which shows the data of Menezes and Graessley [46] for shear and first normal stress difference in start-up of simple shear. The dashed lines are calculated from the linear spectrum using Eqs.4.8 and 10.49. As expected, the... 

Extensional flows yield information about rheological behavior that cannot be inferred from shear flow data. The test most widely used is start-up of steady, uniaxial extension. It is common practice to compare the transient tensile stress with the response predicted by the Boltzmann superposition principle using the linear relaxation spectrum a nonlinear response should approach this curve at short times or low strain rates. A transient response that rises significantly above this curve is said to reflect strain-hardening behavior, while a material whose stress falls... 

This chapter is devoted to the molecular rheology of transient networks made up of associating polymers in which the network junctions break and recombine. After an introduction to theoretical description of the model networks, the linear response of the network to oscillatory deformations is studied in detail. The analysis is then developed to the nonlinear regime. Stationary nonhnear viscosity, and first and second normal stresses, are calculated and compared with the experiments. The criterion for thickening and thinning of the flows is presented in terms of the molecular parameters. Transient flows such as nonhnear relaxation, start-up flow, etc., are studied within the same theoretical framework. Macroscopic properties such as strain hardening and stress overshoot are related to the tension-elongation curve of the constituent network polymers. 

Transient Response Constant Rate of Deformation. The constant rate of deformation response of the linear viscoelastic material was discussed above. From that discussion, the stress-deformation response looks nonlinear even when the material is linear viscoelastic. For the nonlinear material the response will not be simply described by the linear viscoelastic laws. However, the curves will look similar at low strain rates. At higher strain rates, a stress overshoot is observed. 

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