Poisson’s effect

Since a buried pipe movement is resisted by the surrounding soil, a tensile load is produced within the pipe. The internal longitudinal pressure load in the pipe is independent of the length of the pipe. Thus, Poisson s effect must be considered when designing any length of pipe, whether long or short that is part of a buried pipe system. Buried pipes are influenced by friction with their surrounding media. 

The bar impact test, which is a variation of the plate impact test, produces a one-dimensional compressive square stress pulse which neglects the effects of lateral inertia caused by Poisson s effect. This testing procedure was developed in the laboratory of the author and his colleague,30 and is described here in some detail. 

The first term represents the elastic component that is related to Poisson s effect by the equation = (1 — 2vei) 3 introduced above, where 3 is the elastic component of axial tme strain. The second term, P, corresponding to plastic shear, is usually considered to be zero in metals, but we showed elsewhere (32) that it can be slightly negative in some polymers, due to the compaction of macromolecular chains subjected to strain-induced orientation. The last term , measures the contribution of cavitation and/or crazing to the macroscopic volume change of the tensile specimen (33). 

Figure 3.47 Comparison of (a) the programmed specimen and (b) the recovered specimen (side view) and (c) the recovered specimen (isometric view) (the face marked X is perpendicular to the tension direction, the face marked C is perpendicular to the compression direction, and the face marked 40/25 is the free direction subjected to Poisson s effects only). Source [59] Reproduced with permission from the American Society of Civil Engineers...

This entropy driven shape recovery can be further validated by Rgure 3.48 (d). The prestrain level in both the tension direction and compression direction is the same (25%). Therefore, it is expected that the two directions will recover simultaneously. This is exactly the ease. From Figure 3.48 (d), both directions start to recover at a temperature of about 50 °C. For the free direetion, it is mainly affected by the Poisson s effect up to a temperature of about 68 °C. After that, this direction also recovers to its permanent shape. Overall, this case suggests that one-stage 2-D programming can be replaced by two-stage 1-D programming. 

Transverse normal stresses could be higher due to Poisson s effects, resulting in a higher interfacial shear strength... 

Pesce and McKenna (140) applied equation 114 with the VL function obtained from torsional data and successfiilly described the imiaxial data for the same polycarbonate material to within approximately 15% up to the yield point (140). The results are shown in Figure 59. Importantly, the successful description required both a knowledge of w X) and the actual values of the lateral contraction of the material—the equivalent in the nonlinear range of the Poisson s effect in linear elasticity (or viscoelasticity). 

Both Eqs. (11.1) and (11.2) account for the effect of transverse strain on plastic strain intensity factor characterized by the modified Poisson s ratio, V. In Eq. (11.1), this is accounted for by the ratio Sy/Sa, whereas in Eq. (11.2) the ratio Eg/E serves the same purpose as will be shown later. The modified Poisson s ratio in each case is intended to account for the different transverse contraction in the elastic-plastic condition as compared to the assumed elastic condition. Therefore this effect is primarily associated with the differences in variation in volume without any consideration given to the nonlinear stress-strain relationship in plasticity. Instead the approaches are based on an equation analogous to Hooke s law as obtained by Nadai. Gonyea uses expression (rule) due to Neuber to estimate the strain concentration effects through a correction factor, K, for various notches (characterized by the elastic stress concentration factor, Kj). Moulin and Roche obtain the same factor for a biaxial situation involving thermal shock problem and present a design curve for K, for alloy steels as a function of equivalent strain range. Similar results were obtained by Houtman for thermal shock in plates and cylinders and for cylinders fixed to a wall, which were discussed by Nickell. The problem of Poisson s effect in plasticity has been discussed in detail by Severud. Hubei... 

Several equations can be used to calculate the result of Poisson s effect on pipe in the longitudinal direction in terms of stress or strain. The following equation provides a solution for a straight run of pipe in terms of strain. Thus, the longitudinal strain in pipe due to internal pressure is... 

The present section deals with the review and extension of Schapery s single integral constitutive law to two dimensions. First, a stress operator that defines uniaxial strain as a function of current and past stress is developed. Extension to multiaxial stress state is accomplished by incorporating Poisson s effects, resulting in a constitutive matrix that consists of instantaneous compliance, Poisson s ratio, and a vector of hereditary strains. The constitutive equations thus obtained are suitable for nonlinear viscoelastic finite-element analysis. 

Electrostatic held components in parallel with the polarization direction induce normal mode actuation, see Eq. (4.19). Not visible in the constitutive equation, but explainable by a behavior corresponding to Poisson s effect included in the piezoelectric constants, the signs of strains or stresses parallel and transverse to the applied electrostatic fields are opposed. The deformation of a piezoelectric cube subjected to electrostatic fields in the direction of polarization is shown in Figure 4.3. 

After failure of adhesion, the interfacial shear stress r-, = pp, increases with increasing pull-out distance [46], but the normal stress from the matrix acting on the fiber will produce a reduction in cross-sectional area due to Poisson s effect resulting in a reduction of interfacial normal stresses. 

The Euler-Bemoulli theory assumes the vertical deflection is constant across any cross section and that cross sections remain plane after deformation, which means that the shear deformation is neglected. On the other hand, the classical rod theory considers only axial displacements and neglects lateral contractions due to Poisson s effect. Under these assumptions, the displacement field for a straight beam is given by ... 

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