Straining nonlinear elastic stress response

Nonlinear Elastic Stress Response to the Sinusoidal Straining. When a nonlinear elastic body whose modulus varies by Equation 4 is subjected to cyclic straining, the stress response would be ... 

Referring to Figure 5, we can take the minimum strain point as the reference point and apply Equation 5 to establish the nonlinear elastic stress response (Tei y) to the sinusoidal straining ... 

It is instructive to describe elastic-plastic responses in terms of idealized behaviors. Generally, elastic-deformation models describe the solid as either linearly or nonlinearly elastic. The plastic deformation material models describe rate-independent behaviors in terms of either ideal plasticity, strainhardening plasticity, strain-softening plasticity, or as stress-history dependent, e.g. the Bauschinger effect [64J01, 91S01]. Rate-dependent descriptions are more physically realistic and are the basis for viscoplastic models. The degree of flexibility afforded elastic-plastic model development has typically led to descriptions of materials response that contain more adjustable parameters than can be independently verified. 

When compression and/or shear rise sufficiently, the symmetrical elasticity of the unit cell collapses, and the cell walls begin to buckle (Figure 6.26). The response to increased stress no longer produces a linear strain, and the material begins to respond to nonlinear elastic behavior, characterized by region B in Figure 6.22. 

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

Basically, the time dependence of the nonlinear behavior is considered to be in a separable form, where the viscoelasticity is accounted for by a relaxation function that is independent of stress or strain, while the effects of large deformations are incorporated in a reference potential. Simo [51] developed a nonlinear viscoelastic model based on a free energy with uncoupled volumetric and deviatoric parts. The time-dependent effects are contained in the deviatoric stress component, while volumetric stress response is assumed to be elastic. 

The size of the interface-comer fracture process zone is not known, but one can estimate the extent of yielding. Fig. 13 shows three different predictions for the interface-corner yield zone at joint failure. Epoxy yielding is rate- and temperature-dependent and is thought to be a manifestation of stress-dependent, nonlinear viscoelastic material response. A precise estimate of the size of the interface-comer yield zone is, of course, totally dependent on the accuracy of the epoxy constitutive model. This constitutive model must be valid at the extremely high strain and hydrostatic tension levels generated in the region of an interface comer. Unfortunately, accurate epoxy models of this type are not readily available. Nevertheless, simpler material models can be used to provide some insights. The cmdest yield zone prediction shown in Fig. 13 uses a linear-elastic adhesive model to determine when the calculated effective stress exceeds the epoxy s yield... 

In particular it can be shown that the dynamic flocculation model of stress softening and hysteresis fulfils a plausibility criterion, important, e.g., for finite element (FE) apphcations. Accordingly, any deformation mode can be predicted based solely on uniaxial stress-strain measurements, which can be carried out relatively easily. From the simulations of stress-strain cycles at medium and large strain it can be concluded that the model of cluster breakdown and reaggregation for prestrained samples represents a fundamental micromechanical basis for the description of nonlinear viscoelasticity of filler-reinforced rubbers. Thereby, the mechanisms of energy storage and dissipation are traced back to the elastic response of tender but fragile filler clusters [24]. 

Even in the apparently linear range, the response to stress should be considered as viscoelastic rather than elastic. Most polymers that behave in a linear, viscoelastic manner at small strains (< 1 %) behave in a nonlinear fashion at strains of the order of 1 % or more. However, in a fibrous composite, the resin may behave quite differently than it would in bulk. Stress and strain concentrations may exceed the limiting values for linearity in localized regions. Thus the composite may exhibit nonlinearity (Ashton, 1969 Trachte and DiBenedetto, 1968), as is the case with particulate-filled polymers (Section 12.1.2). Although nonlinearity at low strains is characteristic, Halpin and Pagano (1969) have predicted constitutive relations for isotropic linear viscoelastic systems, and verified their prediction using specimens of fiber-reinforced rubbers. 

Odemark s transformation method has been widely used for pavement response analyses (Ullidtz 1987). Comparisons of measured and calculated stresses, strains and deflections have shown that the simple combination of Odemark s transformation with Boussinesq s equations (modified for nonlinearity) results in an agreement between measured and calculated values that is as good as that obtained with the finite element method. Linear elastic methods normally result in rather poor agreement (Ullidtz 1999). 

The utility of the K-BKZ theory arises from several aspects of the model. First, it does capture many of the features, described below, of the behavior of polymeric melts and fluids subjected to large deformations or high shear rates. That is, it captures many of the nonlinear behaviors described above for steady flows as well as behaviors in transient conditions. In addition, imlike the more general multiple integral constitutive models (108,109), the experimental data required to determine the material properties are not overly burdensome. In fact, the information required is the single-step stress relaxation response in the mode of deformation of interest (72). If one is only interested in, eg, simple shear, then experiments need only be performed in simple shear and the exact form for U I, /2, ) need not be obtained. Furthermore, because the structure of the K-BKZ model is similar to that of finite elasticity theory, if a full three-dimensional characterization of the material is needed, some of the simplilying aspects of finite elasticity theories that have been developed over the years can be applied to the behavior of the viscoelastic fluid description provided by the K-BKZ model. One such example is the use of the VL form (98) of the strain energy function discussed above (110). The next section shows some comparisons of the material response predicted by the K-BKZ theory with actual experimental data. 

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