Elasticity pseudo

The theory relating stress, strain, time and temperature of viscoelastic materials is complex. For many practical purposes it is often better to use an ad hoc system known as the pseudo-elastic design approach. This approach uses classical elastic analysis but employs time- and temperature-dependent data obtained from creep curves and their derivatives. In outline the procedure consists of the following steps ... 

Since these assumptions are not always justifiable when applied to plastics, the classic equations cannot be used indiscriminately. Each case must be considered on its merits, with account being taken of such factors as the time under load, the mode of deformation, the service conditions, the fabrication method, the environment, and others. In particular, it should be noted that the traditional equations are derived using the relationship that stress equals modulus times strain, where the modulus is a constant. From the review in Chapter 2 it should be clear that the modulus of a plastic is generally not a constant. Several approaches have been used to allow for this condition. The drawback is that these methods can be quite complex, involving numerical techniques that are not attractive to designers. However, one method has been widely accepted, the so-called pseudo-elastic design method. 

If the SMA is sufficiently close to Tm, an imposed stress is sufficient to cause pressure-induced austenite —> martensite phase transitions in selected grains of the alloy, relieving the stress through pseudo-elastic deformation of the softer martensite grains. Similarly, if the original austenite-shaped alloy is brought below Tm to convert it to malleable martensite form, many deformations of macroscopic shape leave the martensitic atoms close to their... 

Thixotropic and dilatant behavior of the suspension can be monitored by viscosi-metric measurements as a function of shear rate and the duration of the experiments. Studies with a 25 wt.% suspension of LiChrospher 100, 7 gm in a mixture of dioxane-cyclohexane-n-octanol (45/45/10, v/v/v) have shown that the suspension has rather low thixotropic and pseudo-elastic behavior (Hallmann, 1992). The packing pressure of the axial dynamic column packing technique has a significant influence on the column performance (Tab. 3.13). 

Savinsky )., Havas G., Deak A., Power requirement of anchor and helical ribbon impellers for the case of agitating Newtonian and pseudo-elastic liquids, Chem. Eng. Sci. 31 (1976), p. 507-509... 

Ti(q) and T2(q) are the two relaxation times describing respectively the global diffiision and the internal motions. Ao(q) is the pseudo Elastic Incoherent... 

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,... 

By chance this result is very close to the required one, so the maximum allowed SDR is 20. The conservative nature of the pseudo-elastic calculation provides an inbuilt safety factor. 

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 ... 

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. 

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. 

Above the transition temperature an extraordinary elasticity can be observed in the shape memory alloys. Figure 6.53 shows a uniaxial stress-strain diagram of a pseudo-elastic (or super-elastic) SMA. 

Initially the pseudo-elastic material is in its austenitic phase at room temperature. Initially the material in the austenitic phase deforms like a conventional material linear elastic under load. With increasing loads a stress-induced transformation of the austenitic to the martensitic phase is initiated at the pseudo-yield stress Rpe- This transformation is accompanied with large reversible strains at nearly constant stresses, resulting in a stress plateau shown in Fig. 6.53. At the end of the stress plateau the sample is completely transformed into martensite. Additional loading passing the upper stress plateau causes a conventional elastic and subsequently plastic deformation of the martensitic material. If the load is decreased within the plateau and the stress reaches the lower stress level a reverse transformation from martensite to austenite occurs. Since the strains are fully reversible the material and the sample respectively is completely recovered to its underformed shape. These strains are often called pseudo-elastic because the reversible deformation is caused by a reversible phase transformation and is not only due to a translation of atoms out of their former equilibrium position [74]. 

Fig. 6.53. Stress-strain diagram of pseudo-elastic shape memory alloys...

In this section some examples of precision engineering prototypes are presented that apply electrically heated shape memory actuators as driving elements. Further on flexure hinges of pseudo-elastic SM alloys will be presented. 

SM actuators may have very many different shapes and offer a variety of shape changes (i. e. actuator strokes). This property can be exploited so as to adapt the SM elements shape to the actuating task. As an application example, a miniature parallel gripper with electrically heated SM wires integrated into its mechanical structure was presented. Further on the performance of pseudo-elastic shape memory flexure hinges in parallel robots for micro-assembly tasks was shown. The future opportunity for thin-fllm SM actuators to drive micromechanical systems and devices was demonstrated by a miniature silicon gripper. 

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