Volume Strain Dilatation

The concept of volume strain is defined as the volume change per unit volume of AV 

The term on the left, often denoted as A, represents the volume strain, e. AV, the normal strain components, refer to cases in which the strains are small. Basically, when a rectangular parallelepiped of initial volume V with sides a, b, c is deformed to a value of V, as shown in Fig. 1.38, the following relations hold  

When linear strains are small, as they are in the elastic region, their products can be ignored, rendering Eq. (1.90) into  

In this case, is equal to the sum of the strains on the left side of Eq. 1.85a. In terms of the mean strain, it is expressed as  

K is known as the bulk modulus and also as the volumetric modulus of elasticity and is given by  

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Predictions of the mechanical response of filled elastomers are further aggravated by the phenomenon of strain dilatation. As soon as dilatation commences, the tensile stress lag behind elongation, the degree of dilatation for a given composite being roughly a measure for the deviation from the expected mechanical response. Dilatation increases with particle size and volume fraction of filler—it decreases somewhat if the filler is bonded to the matrix. Farris (16,17) showed that dilatation can account well for the mechanical behavior of solid propellants and his equation ... 

The CDM has two additional features that allow it to represent fracture in rocks. First, there is a brittle/ductile transition pressure. Above this pressure, the rock behaves as an elastic/plastic ductile solid, the failure surface is independent of the level of damage, and the damage is not allowed to increase, even if the failure surface is exceeded. Second, the CDM allows for non-vanishing plastic volume strain to approximate the dilatancy observed in certain laboratory experiments on oil shale. 

In the elastic range, the Poisson s ratio, Vei = —/, can be used to relate the amplitude of transverse constriction with axial strain. The volume strain depends on this material coefficient by = (1 — 2vei)e - Similarly, for nonlinear materials, the tangent Poisson s ratio Vt = —d i/d 3 may be introduced at large strain, so that coefficient is also obtained from the instantaneous slope of the volume strain versus axial strain from (1 — 2vt ) = d v/d 3. In this paper, the latter slope will be called dilatation rate (or damage rate ) and denoted by the variable A. 

There is experimental evidence, for many rubber-toughened polymers, that the rubber particles cavitate early in the deformation. The degree of cross-linking is kept relatively low in the polybutadiene phase of ABS to aid cavitation, and sometimes silicone oil is added for the same reason. Figure 4.12 shows both the conventional stress-strain curve and the volumetric strain versus tensile strain for rubber-modified polystyrene. When the polystyrene yields, the volume strain increases at a higher rate. Majority of the dilatational strain is due to cavitation in the rubber phase. 

When it comes to volume strain, all the specimens show different degrees of dilatation behavior and the dilatancy rate slows down with the increase of axial strain. The dilatation also increases with the decrease of the confining pressure. When the confining pressure is low. 

When it comes to volume strain shown in Figure 8(b), all the specimens show dilatation behavior. The shear dilatancy of specimen with flexible lateral boimdary is much more obvious than specimen with rigid lateral boundary, and the gap between two volume strain curves is bigger and bigger with the increase of axial strain. On the other hand, the reduction of volume strain can be seen in the specimen with rigid lateral boimdary while that of flexible boundary specimen can hardly be seen. 

Here/o (=/i of equation 49 of Chapter 11) is the fractional free volume at zero strain. For reasonable values of parameters near Tg, it can be estimated that a tensile strain of 1% would shift the time scale by about one logarithmic decade. In a simpler formulation, the ratio j6ty/j8 may be taken as unity. Experiments of this sort have been report for poly(methyl methacrylate), copolymers consisting largely of polyacrylonitrile, polycarbonate, and an acrylonitrile-butadiene-styrene copolymer blend the effects of strain dilatation on tensile stress relaxation, torsional stress relaxation, and combined tensile and torsional stress relaxation have been compared. - ... 

For uniaxial (hexagonal) symmetry the 6 strain components are subdivided in two (invariant) one-dimensional subsets (indicated by the superscript a, and subscripts 1 and 2 for the volume dilatation and the axial deformation, respectively), and two different two-dimensional subsets, indicated by y for deformations in the (hexagonal) plane, and by e for skew deformations. These modes are also depicted in fig. 3. In this case, the magnetostriction can be expressed as... 

A new ultrasonic technique for studying dewetting and cumulative internal damage in solid propints has been reported (Refs 17 20). This technique yields volume-dilatation data on proplnt in tension, and on damage in uniaxial compression and shear strain fields. Estimates of vacuole size and number density arising from dewetting can be made, as well as can the time dependent void growth at constant strain be observed... 

Equation (55) differs from Eq. (39) by the term (—yaT) and hence (AU/W)V T should be independent of extension ratio X, since (—yaT) is not a function of deformation. Equation (54) leads also to the following expression for the strain-induced volume dilation... 

Interchain changes arising from the deformation of polymer networks are a result of strain-induced volume dilation. Typical results obtained by various methods (Fig. 6) demonstrate that the statistical theory predicts the volume dilation and... 

Fig. 6. The relative strain-induced volume dilation of polymer networks as a function of strain. A - NR, B — EPR. O and -calorimetric data 241 + — di-...

On the other hand, Kilian 50) having analysed the strain-induced volume dilation 24 91 using the van der Waals equation of state (Fig. 6) emphasized that only pressure dependence of the interchain parameter, a, is required for a full explanation of the relative volume changes. He arrived at a conclusion that non-crystalline rubbers are anisotropic equilibrium liquids and a higher compressibility of NR was only necessary for fitting the extension data. Hence, on using the van der Waals approach, there is no need of postulating volume dependence of the front factor as proposed by Tobolsky and Shen. 

We have developed expressions for each component of the normal strain rate //, which are interpreted as relative elongation (contraction) rates in each of the coordinate directions. It will be useful in later derivations of the conservation equations to relate the volumetric dilatation (1/V)(DV/Dt) to the strain field. Consider a cylindrical differential element dV = rdrdOdz. After a short time interval dr, the element has strained in all three dimensions, resulting in an altered volume, Fig. 2.10. To first order, the relative volume change has three components as can be seen geometrically from the figure,... 

Most nucleation processes are accompanied by a net volume change. In these cases, AG is altered by the strain energy A clasl. The dilatational part of this energy term can be expressed as... 

Solution. First, calculate the energy change contributed by the excess vacancies which are eliminated to relieve the strain due to the dilatation eh. If V is the cluster volume, AV = 3enV. The number of vacancies required is then N = 3e iV7n and the free-energy change due to the removal of these vacancies is therefore... 

The concept of stress-induced dilatation affecting the relaxation time or rate has been suggested by others (5, 6, 7, 8). The density of most solids decreases under uniaxial stress because the lateral contraction of the solid body does not quite compensate for the longitudinal extension in the direction of the stress, and the body expands. The Poisson ratio, the ratio of such contraction to the extension, is about 0.35 for many polymeric solids it would be 0.5 if no change in density occurred, as in an ideal rubber. The volume increase, AV, accompanying the tensile strain of c, can be described by the following equation ... 

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