Pseudo-elastic method

As reviewed viscoelastic behavior relates to deformations that are dependent on time under load and the temperatiu e. Therefore, when structural components are to be designed using plastics it must be remembered that the standard engineering equations that are available (Figs 2.31 and 2.32) have been derived under the assumptions that (1) the strains are small, (2) the modulus is constant, (3) the strains are independent of the loading rate or history and are immediately reversible, (4) the material is isotropic, and (5) the material behaves in the same way in tension and compression. 

These equations cannot be used indiscriminately. Each case must be considered on its merits, with account being taken of the plastic behavior in time under load, mode of deformation, static and/or dynamic loads, service temperature, fabrication method, environment, and others. The traditional engineering equations are derived using the relationship that stress equals modulus times strain, where the modulus is a constant. The moduli of many plastics are generally not a constant. Several approaches have been reviewed permitting use of these type 

In the PE method time-dependent property values for the modulus (include secant modulus) are selected and substituted into the standard equations. This approach is sufficiently accurate if the value chosen for the modulus takes into account the projected service life of the product and/or the limiting strain of the plastic. This approach is not a straightforward solution applicable to all plastics or even to one plastic in all its applications. This type of evaluation takes into consideration the value to use as a safety factor (SF). If no history exists a high value will be required. In time with service condition inputs, the SF can be reduced if justified (Chapter 7). 

Determining a secant modulus is usually based on 1% strain or that is 0.85% of the initial tangent modulus (Fig. 2.4). However, for many plastics, particularly the crystalline TPs, this method is too restrictive, so in most practical situations the limiting strain is decided in consultation between the designer and the plastic material s manufacturer. Once the limiting strain is known, design methods based on its static and/or dynamic load becomes rather straightforward. 

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The reader will note that the pseudo-elastic method is conservative. The stress analysis uses a modulus that is really appropriate only to the most highly strained regions of the design, and applies it to the whole component. Elsewhere, strains are lower, and the creep modulus is greater than that used in the analysis. This results in a small but unavoidable element of over-design a component designed in this way will be somewhat thicker and more complex (e.g. because of ribbing) than strictly it needs to be to meet the specification. 

Abstract The paper presents results of an experimental study of thermal effects on the mechanical behaviour of a saturated clay, with emphasis on the determination of the onset of yielding. The study was performed on CM clay (Kaolin) using a temperature-controlled u-iaxial apparatus. Applied temperatures were between 22 °C and 90 °C. Various methods are used to identify the yield points (pseudo-elastic limit) and to define the shape of the yield surface in the invariant stress space p (effective mean pressure)- q (deviatoric stress). Yield surface obtained at 90 °C is compared with results at ambient temperature. Based on this comparison, thermo-mechanical yielding is discussed and yield limit evolution with temperature is presented. 

It is wasteful of material to design a product to be in the linear viscoelastic region. The pseudo-elastic design method, for non-linear viscoelastic materials, gives a more reasonable design. The process requires an initial design,... 

The problems of exact design for a viscoelastic polymer with non-linear properties are severe. For example, in Figure 8.1 a) the stress-strain curve is linear only at the smallest strains (below 0.2%). Most plastic parts are designed to operate at strains well above 0.2%, and in this case exact stress analysis is impossible. In practice, a safe approximate procedure known as the pseudo-elastic design method is used. The salient features of the method, which is veiy straightforward to apply, are as follows ... 

This is an example of strain-limited design. We apply the pseudo-elastic design method, specifying the duration of loading as S hours (which is 18000 seconds). In order to determine the modulus, we need the isochronous stress-strain curve for 18000 s. Substituting in the equation,... 

The pseudo-elastic design method may be used for components submitted to intermittent loading, provided that the intervals during which the material is unloaded are suffident to allow virtually complete recovery. Some manufacturers provide recovery data that enable the validity of this assumption to be tested. Altemativefy, the Boltzmann superposition prind-ple may be used to determine whether the assumption gives a satisfactory q>proximatk>n (see Oiapter 4). If not, or if die 1 is varying in a more complex manner, a more complete anafysis of deflection behaviour based upon the Boltzmaim prindple may be necessary. Linearity can be assumed for strains up n> about 0.005. 

The two-network method has been carefully examined. All the previous two-network results were obtained in simple extension for which the Gaussian composite network theory was found to be inadequate. Results obtained on composite networks of 1,2-polybutadiene for three different types of strain, namely equibiaxial extension, pure shear, and simple extension, are discussed in the present paper. The Gaussian composite network elastic free energy relation is found to be adequate in equibiaxial extension and possibly pure shear. Extrapolation to zero strain gives the same result for all three types of strain The contribution from chain entangling at elastic equilibrium is found to be approximately equal to the pseudo-equilibrium rubber plateau modulus and about three times larger than the contribution from chemical cross-links. 

If both walls are spaced far apart, the pressures on one wall are not influenced by the presence of the other. For low-frequency input motions with frequency less than half the fundamental frequency of the unrestrained backfill, VJAW (Fj is the soil shear wave velocity), the pseudo static conditions are governed (i.e., the dynamic amplification is negligible). For this range of frequencies, wall pressures with plane strain assumption can be obtained from elastic solution for the case of a uniform, constant horizontal acceleration applied throughout the soil. The dynamic earth pressures obtained from this method must be... 

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