Principal stretches

FIGURE 2 la The three normal vibrational modes of 11,0. Two of these modes are principally stretching motions of the bonds, but mode v2 is primarily bending, (b) The four normal vibrational modes of C02. The first two are symmetrical and antisymmetrical stretching motions, and the last two are perpendicular bending motions. 

FT-IR study revealed that no damage of the Co-POM structure occurred when its TBA-salts dissolved in MeCN were used for the immobilization. As one can judge from Figure 2, the IR spectra after subtraction of the peaks due to NH2-X exhibited the principal stretching modes of the Keggin Co-POM imit (956, 888, 818, 752, 720 cm" ). 

In principle, W can be determined from Eq. (2) if principal stresses at are measured as functions of applied principal stretch ratios X,-. However, since bW/bJ/ rather than W itself are more directly connected with the stress-strain relations [see Eq. (11)], their determination from the measurements of at and X,- is more feasible than that of W. 

The tensor E E, called the Piola tensor (Astarita and Marrucci 1974), is closely related to B. In an extensional deformation, E E is exactly equal to B. B, a symmetric tensor, contains information about the orientation of the three principal axes of stretch and about the magnitudes of the three principal stretch ratios, but no information about rotations of material lines that occurred during that deformation. Thus, for example, from the Finger tensor alone, one could not determine whether a deformation was a simple shear (which has rotation of material lines) or a planar extensional deformation (which does not). The Finger tensor B(r, f) describes the change in shape of a small material element between times t and t, not whether it was rotated during this time interval. 

The distortional stretches can be obtained from the applied principal stretches by... 

The uniaxial extension is illustrated in Fig. 8. If the Xi axis is aligned with the principal stretching direction, it can be noted that two fluid particles located on this axis will get further apart as the deformation proceeds. It is convenient to use a coordinate system in which the axes are oriented in the directions of the principal strain axes. For this choice, the strain rate components always have the general form... 

For flexible (mbbery) adhesives which show Tg far below room temperature, hyperelastic material models are generally used. In the hyperelastic regime, standard solutions are available which use various types of potential functions [7]. The flexible adhesive systems investigated were best fitted by a potential function which is formulated in terms of the principal stretches Aj originally suggested by Ogden [Eq. (1)] [8]. 

For our adhesive systems, a first-order approximation (n=l) was sufficient to achieve a good fit. The material was treated as incompressible. The temperature-dependent parameters n and a were determined by fitting theoretical nominal stress-strain relationships following from Eq. (1) to the experimental data. The nominal stress-strain relationships are calculated by differentiating Eq. (1) with respect to the principal stretches. For simple deformation modes, e.g., uniaxial tension, uniaxial compression, and simple shear, the principal stretches and their relationships are easily obtained by geometrical considerations. For a reliable determination of the values of fi and a, these calculated stress-strain relationships were fitted to experimental data which were measured in the corre-... 

As a result, the principal stretch ratios can be expressed in terms of R as follows ... 

In plane-strain tension the principal stretch is in the 2-direction. Thus,... 

Expressions for timescales associated with stretching and molecular diffusion for nonreacting systems are available in the literature [112]. A fluid element undergoes shear in two-dimensional chaotic flow with the principal stretching direction Xi and compression direction X2- Molecular diffusion becomes important after a time... 

Figure 3.7 Development of principal stretches and clockwise rigid-body rotation angle in simple shear.

Other forms of the Landel-Valanis functions u are of course possible. Recently, Darijani, Naghdabadi and Kargarnovin [16] explored a number of possibilities, involving polynomial, logarithmic and exponential functions. In this scheme, strain energy functions are constructed using a set of basic functions of the principal stretches. These are ... 

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