Strain constant volume

Considerably better agreement with the observed stress-strain relationships has been obtained through the use of empirical equations first proposed by Mooney and subsequently generalized by Rivlin. The latter showed, solely on the basis of required symmetry conditions and independently of any hypothesis as to the nature of the elastic body, that the stored energy associated with a deformation described by ax ay, az at constant volume (i.e., with axayaz l) must be a function of two quantities (q +q +q ) and (l/a +l/ay+l/ag). The simplest acceptable function of these two quantities can be written... 

In this equation e is the longitudinal strain and er is the strain in the width (transverse) direction or the direction perpendicular to the applied force It can be shown that when Poisson s ratio is 0.50, the volume of the specimen remains constant while being stretched. This condition of constant volume holds for liquids and ideal rubbers. In general, there is an increase in volume, which is given by... 

The molecular theory of elasticity of polymeric networks which leads to the equation of state, Eq. (28), rests on the following basic postulates Undeformed polymeric chains of elastic networks adopt random configurations or spatial arrangements in the bulk amorphous state. The stress resulting from the deformation of such networks originates within the elastically active chains and not from interactions between them. It means that the stress exhibited by a strained network is assumed to be entirely intramolecular in origin and intermolecular interactions play no role in deformations (at constant volume and composition). 

Fig. 29. Temperature dependence of complex piezoelectric strain constant of composite film of polyester resin and powdered PZT (50% of the volume) polarized at room temperature under a d.c. field of 100 kV/cm. Reproduced from Fukada and Date [Polymer Journal, 1,410 (1970)] by permission of the Society of Polymer...

If one proceeds in this way, it does not cause problems in treating constant-volume deformations (i.e. dry network deformations), but it meets with obstacles in swelling type strains because the logarithmic term HvlnXxXyXz in Eq. (III-9) is absent in the simplified treatment of Eq. (IV-5). A full derivation according to the HFW approach but employing the series distribution Eq. (IV-4), is, however, not available. 

We can, of course, account for a possible front factor in the usual way by converting A into Ax, and v into Av(z8>/0 (compare Eq. III-ll). It might be pointed out that the logarithmic term in Eq. (IV-32) bears no relation to the logarithmic term which appears in the HFW treatment, but is absent in the JG treatment. In fact, Eq. (IV-32) is obtained on the basis of a simplified JG treatment, so that its validity is restricted to constant volume deformations (compare also Eq. III-8). In unidirectional extension or compression Eq. (IV-32) yields for the stress per strained cross-section ... 

Typically, in compression tests a cylindrical piece of the test sample is compressed between smooth plates using a Material Tester. Assuming constant volume, the stress and strain (Hencky strain) are calculated from the force, displacement data. However,... 

A simulation of a collection of like chains under both melt and network conditions was performed to test this hypothesis. In the melt condition, no restrictions were placed on the chain vectors, whereas in the network condition, chain vectors were controlled and subjected to a constant volume extension X. In both cases, ars was found to be diagonal, as expected from symmetry considerations, whereas the principal values were the same, within computation accuracy, both for the strained network where they are independent of X, and for the melt see Fig. 12. 

Wintch R. P. and Kvale C. M. (1991) Open-system, constant-volume, development of slaty cleavage, and strain-induced replacement reactions in the Martinsburg Formation, Lehigh Gap, Pennsylvania. Geol. Soc. Am. Bull. 103, 916-927. 

If the strain takes place at constant volume, then... 

Suppose a person stands on a substantial block of firm cheese and the height of the block diminishes by 1 mm. First let us assume the effect is purely elastic at constant volume and consider what the material s shear modulus might be. Assume height of block = 160 mm. Then strain C] = 1/160 taking vertical as direction 1. Assume horizontal strains 82 and 83 are equal then for constant volume, 83 = 83 = —1/320 and 8j — 83 = 3/320. 

The value of the elastic Poisson ratio used is shown in Table I. However, the analysis was carried out using the assumption that plastic deformation takes place essentially at constant volume that is, with a Poisson ratio of 0.5. The Poisson ratio of any element is the weighted sum of the elastic Poisson ratio and 0.5, weighted via the proportion of elastic and plastic strain. However, it should be noted that this complex behavior leads to considerable difficulty regarding directional loading of the spherical model for elastic-plastic behavior. 

Once the temperature has stabilised, the sample is hydrated at constant volume through the bottom face with deionised water injected at a pressure of 0.6 MPa, while the upper outlet remains open to atmosphere. At the same time, the swelling pressure exerted by the clay is measured by a load cell installed in the loading frame and the vertical deformation of the specimen is measured by two LVDT s. The values of load, strain and water exchange are automatically recorded. The final density may differ slightly from the nominal one due to the small displacement allowed by the equipment. 

Then ej is a simple dilatation of the crystal and the remaining 6 are distortions at constant volume. 62 is a distortion with rotational symmetry about the c-axis and 63 to eg represent various plane shears. For these strains and the similarly defined stresses we have new stiffness coefficients which can be shown to be... 

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