Inelastic plates

Models of inelastic plates are introduced here, which are analysed in Chapters 2, 3 and 5. 

By the constitutive law (1.4), from (1.21)-(1.25) we obtain the model of a plate under the creep condition  

Substitution of these equalities into (1.30) yields the equation 

For the viscoelastic law (1.6), instead of (1.24), (1.25) the same arguments guarantee a validity of the following equations for the vertical displacements w  

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Thus, the relations (1.36) or (1.37) describe the interaction between a plate and a punch. To derive the contact model for an elastic plate, one needs to use the constitutive law (1.25). Contact problems for inelastic plates are derived by the utilizing of corresponding inelastic constitutive laws given in Section 1.1.4. 

The lug plate can either fail elastically or inelastically. Plates in which the following is true will fail inelastically ... 

Shear-stress-shear-strain curves typical of fiber-reinforced epoxy resins are quite nonlinear, but all other stress-strain curves are essentially linear. Hahn and Tsai [6-48] analyzed lamina behavior with this nonlinear deformation behavior. Hahn [6-49] extended the analysis to laminate behavior. Inelastic effects in micromechanics analyses were examined by Adams [6-50]. Jones and Morgan [6-51] developed an approach to treat nonlinearities in all stress-strain curves for a lamina of a metal-matrix or carbon-carbon composite material. Morgan and Jones extended the lamina analysis to laminate deformation analysis [6-52] and then to buckling of laminated plates [6-53]. 

Viscoelasticity was introduced in Section 11.5. A polymer example may be useful by way of reeapitulalion. Imagine a polymer melt or solution confined in the aperture between two parallel plates to which it adheres. One plate is rotated at a constant rate, while the other is held stationary. Figure 11-3la shows the time dependence of the shear stress after the rotation has been stopped, r decays immediately to zero for an inelastic fluid but the decrease in stress is much more gradual if the material is viscoelastic. In some cases, the residual stresses may... 

J. Eckert, R. Varma, L. Diebolt M. Reid (1997). J. Electrochem. Soc., 144, 1895-1899. Effects of cycling conditions of active material from discharged Ni positive plates smdied by inelastic neutron scattering spectroscopy. 

In summary, the simplified inelastic analysis rules as indicated in subsections NB 3327.6 and NB 3228.3 of the ASME Boiler and Pressure Vessel Code Section III have been critically appraised. The first rule is shown to be equivalent to a correction factor, K, to be applied to local thermal stresses, and is based on an analysis involving a modified Poisson s ratio. For a simplified situation of thermal stress in a plate with a through the thickness temperature gradient (perfect biaxiality) the solution using NB 3227.6 are comparable to the existing solutions in the literature. However, the solutions obtained using finite-element methods and a different form of Poisson s ratio than that specified in NB 3227.6 (Eq. (11.1)) typically yield higher values of K. ... 

Polymer melts undergo large deformations in steady shear, and nonlinear effects associated with the dynamics of the polymer chains are expected. The existence of a finite normal stress is one manifestation of such nonlinearity. Suppose we shear a liquid between two parallel plates, one of which is moved relative to the other. The shear stress is equal to the ratio of the force required to move the plate at a given rate to the plate area, and it is from this measurement that we obtain the viscosity. With a polymeric liquid we find that a finite force is required to keep the spacing between the plates constant in the absence of such a force the plates will move apart. A stress normal to the direction of shear is not observed for inelastic liquids. (A similar phenomenon is well known in the mechanics of solids. If a rod of... 

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