Strain/stress isotropic samples

Another interesting result is obtained from the comparison of the scattering of the sheared samples in the z direction to that of an isotropic sample. Fig. 30 shows the Zimm plot of sample C in the directions x and z for a specimen cut in the x-z plane. The corresponding curve for an isotropic sample, which has also been plotted in the same figure, is found to be identical to within experimental error to the curve in the z direction. This result indicates that the position correlations within the chain in the neutral direction of the shear flow are not affected by the flow, at least up to the values of stress and strain used in the present study. In particular. Table 7 shows that the mean square chain dimension in that direction, Rg 2> which has been determined either in the x-z or in the y-z planes for the various samples, is round to be equal to the radius of gyration of an isotropic sample to within experimental error (Rg 2=82 lA). The same result has been found by Lindner in dilute solutions [31]. 

The following sections develop three subjects the classical approximations for the strain/stress in isotropic polycrystals, isotropic polycrystals under hydrostatic pressure and the spherical harmonic analysis to determine the average strain/stress tensors and the intergranular strain/stress in textured samples of any crystal and sample symmetry. Most of the expressions that are obtained for the peak shift have the potential to be implemented in the Rietveld routine, but only a few have been implemented already. 

Strain/Stress in Isotropic Samples - Qassical Approximations... 

The present hypothesis fully describes the hydrostatic strain/stress state in isotropic samples. Indeed, from the refined parameters e, the macroscopic strain and stress e, x can be calculated and also the intergranular strains and stresses Ae,(g), Ax,(g), both different from zero. Note that nothing was presumed concerning the nature of the crystallite interaction, which can be elastic or plastic. From Equations (112) it is not possible to obtain relations of the type (84) but only of the type (86). For this reason a linear homogenous equation of the Hooke type between the macroscopic stress and strain cannot be established. 

More detailed analysis of experimental data can be found in our previous paper [78]. In contradiction to the isotropic sample, the stress-strain depen-... 

These are most easily represented by the equation E = E + iE". where E is the ratio between (the amplitude of the in-phase stress component strain, a/e) and E" is the loss modulus (the amplitude of the out-of-phase component. strain amplitude). Similarly for (7 and K and the ratio between the Young s modulus E and the shear modulus C includes Poisson s ratio u, for an isotropic linear elastic solid with a uniaxial stress. (Poisson s ratio is more correctly defined as minus the ratio of the perpendicular. strain to the plane strain, or for one orthogonal direction 22 which equals the. 3.3 strain if the sample is... 

It was observed empirically by Hooke that, for many materials under low strain, stress is proportional to strain. Young s modulus may then be defined as the ratio of stress to strain for a material under uniaxial tension or compression, but it should be noted that not all materials (and this includes polymers) obey Hooke s law rigorously. This is particularly so at high values of strain but this section only considers the linear portion of the stress-strain curve. Clearly, reality is more complicated than described previously because the application of stress in one direction on a body results in a strain, not only in that direction, but in the two orthogonal directions also. Thus, a sample subjected to uniaxial tension increases in length, but it also becomes narrower and thinner. This quickly leads the student into tensors and is beyond the scope of this chapter. The subject is discussed elsewhere [21-23]. There are four elastic constants usually used to describe a macroscopically isotropic material. These are Young s modulus, E, shear modulus, G, bulk modulus, K, and Poisson s ratio, v. They are defined in Figure 9.2 and they are related by Equations 9.1-9.3. 

Bulk Modulus and Bulk Compressibility. According to Cramer(Ref 5,P 1), one of the important constants of an isotropic elastic solid is the bulk modulus (K) or its reciprocal the bulk compressibility(B). The K is defined as the ratio of stress to strain when the stress is a pressure applied equally on all surfaces of the sample and the strain is the resulting change in volume per unit volume. If a sinusoidally varying pressure is superimposed... 

From the dynamic mechanical investigations we have derived a discontinuous jump of G and G" at the phase transformation isotropic to l.c. Additional information about the mechanical properties of the elastomers can be obtained by measurements of the retractive force of a strained sample. In Fig. 40 the retractive force divided by the cross-sectional area of the unstrained sample at the corresponding temperature, a° is measured at constant length of the sample as function of temperature. In the upper temperature range, T > T0 (Tc is indicated by the dashed line), the typical behavior of rubbers is observed, where the (nominal) stress depends linearly on temperature. Because of the small elongation of the sample, however, a decrease of ct° with increasing temperature is observed for X < 1.1. This indicates that the thermal expansion of the material predominates the retractive force due to entropy elasticity. Fork = 1.1 the nominal stress o° is independent on T, which is the so-called thermoelastic inversion point. In contrast to this normal behavior of the l.c. elastomer... 

Fig. 1. Apparent secant modulus (stress/strain) in tension at 0.1% strain and at various sample aspect ratios measured at a strain rate of 10 4s 1. Isotropic polyethylene sheet at 22 °C (sample cross section 1 mm X 3 mm)...

Figure 14.28 Strain of an isotropic elastic solid, (a) Application of a stress Gq to a sample having an equilibrium separation of atomic planes, h. (b) Potential energy as a function of the separation x. (c) Dependence of cro on x. (From Ref. 35.)...

The description in terms of a substrate that self-diffuses plus components that interdiffuse permits a further distinction it is the substrate that has continuum properties, to which the reasoning in Chapter 11 applies, and for which we use eqn. (12.7) specifically along one direction or another. The interdiffusive effects, here mimicked by the motion of the additive a, do not resemble continuum behavior in isotropic materials, the additive a affects only the volume of a sample-element and cannot affect its shape the additive responds directly to the mean stress and produces only an isotropic change in mean strain. In the cylinder problem treated above, the symmetry and uniformity assumed are such that this distinction leaves the mathematical solution unchanged in form. But if a less regular physical situation were to be treated, the distinction between the behavior of BX and the behavior of the additive would have more noticeable consequences. 

All the results of DSC, elastic recovery, and x-ray measurements on the sample portions formed under characteristic deformation flow components indicate that the complex stress strain field in an extrusion die produces markedly different deformation flow profiles on crystalline-state extrusion of HDPE, but results in no significant effect on the morphology and properties of the extrudates prepared under the conditions used in this work. It is remembered, however, that an earlier electron microscopy study found marked morphological variations across the radius of HDPE extrudates prepared at higher temperatures (134 to 137°C) under the combined effects of chain orientation and pressure (4). The T, of the undeformed isotropic... 

From Hooke s law, the six independent components of the stress tensor can be expressed as a function of the six components of the strain tensor in a symmetrical matrix of order 6 with 21 modulus components for a general anisotropic sample of material. For an isotropic body, there are only two independent components. The mode of deformation will determine which modulus will be measured. 

X=LILq is plotted as a function of the reduced temperature red at constant nominal stress CTn = 2.11xlO N mm . Here Lg is the loaded sample length at Tred l-OS. These results will also be used below to establish a close connection between the strain tensor and the nematic order parameter. It has also been shown that a quadratic stress-strain relation yields in the isotropic phase above the nematic-isotropic phase transition a good description of the data for ele-ongations up to at least 60% [4]. 

Nematic monodomains have been prepared using a variant of the method introduced by Kiipfer and Finkelmann [10, 18] instead of keeping the stress constant in the second crosslinking step the strain was held constant in the second crosslinking step [53]. As for the case studied by Kupfer and Finkelmann, it is found that the monodomain character is retained after heating the sample into the isotropic phase upon return into the nematic phase. Stress strain relation has been measured for two classes of poly-siloxane elastomers. 

Ao. (J = f/Aq. This is called engineering stress, it is easier to measure in practice, because to determine the true stress f/A, one needs to measure both f and A. Stress has dimension of pressure and here it is presented in the units of megaPascal (a useful reminder for the book on molecules 1 MPa= 1- ). Strain is presented as unitless relative elongation of the sample A f o. Panel (a) presents data for an isotropic semi-crystalline polymeric material, specifically — low density polyethylene film. The inset presents a wider interval of strains. Notice that at low... 

The value of the simplified technique for producing isochronous stress-strain curves for non-linear isotropic materials by successive loading and unloading of a single sample have been amply demonstrated over many years and fully described elsewhere.These techniques become even more valuable in studies of anisotropy, where samples may be difficult to obtain in large numbers and where the scope of the problem is much larger. A considerable proportion of work on oriented materials reported in the literature is essentially confined to this measurement and does not include studies of time dependence of behaviour. Detailed work has been carried out validating this procedure for oriented materials by comparison of the isochronous stress-strain data with isochronous sections from families of creep curves. ... 

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