Bulk strain

Now, in rheological terminology, our compressibility JT, is our bulk compliance and the bulk elastic modulus K = 1 /Jr- This is not a surprise of course, as the difference in the heat capacities is the rate of change of the pV term with temperature, and pressure is the bulk stress and the relative volume change, the bulk strain. Immediately we can see the relationship between the thermodynamic and rheological expressions. If, for example, we use the equation of state for a perfect gas, substituting pV = RTinto a = /V(dV/dT)p yields a = R/pV = /Tand so for our perfect gas ... 

When large non-linear deformations are made, we need to make additional assumptions within the tube model on how the tube itself deforms with the bulk strain. The simplest, and original assumptions are that ... 

It is of interest to think about relaxing the assumptions (i) and particularly (ii) introduced at the beginning of Sect. 6.1, although hard experimental tests for specific assumptions of the deformation of the tube constraint itself can never be confined to rheology alone, but will involve at least careful analysis of neutronscattering experiments [46,63]. Not only might the tube diameter depend on the local strain, the localising field described by the tube may well take on an anisotropy consistent with the symmetry of the bulk strain. For discussions of how tube variables deform with strain see [67,68]... 

A dimensionless quantity for the change in volume, V, per unit volume. A synonymous term is bulk strain. An example of volume strain can be seen in bodies experiencing hydrostatic pressure. Volume strain is usually symbolized by 9 thus, 9 = AVIVq. 

In addition to the tensile and shear moduli, a compressive modulus, or modulus of compressibility, K, exists to describe the elastic response to compressive stresses (see Fignre 5.7). The compressive modulus is also sometimes called the bulk modulus. It is the proportionality constant between the compressive stress, CTc, and the bulk strain, represented by the relative change in bulk volume, AV/Vo-... 

The left-hand side is a compliance parameter or bulk strain normalized for the centrifugal, gravitational, and inertial stresses exerted on the material during spheronization. The volume shape factor of pellets became closer to that of a sphere as the compliance of the extrudate increased, when measured in a creep test (56). 

This paper reviews the results of investigations into low-frequency mechanical and high-frequency (ultrasonic) vibration effects upon flowable polymeric systems, primarily, on molten commercial thermoplastics. We tried to systematize possible techniques to realize vibration in molding of polymers. Theoretical and experimental corroboration is provided for major effects obtained at cyclic (shear and bulk) strains of molten polymers and compositions based thereon. It is demonstrated that combined stress of polymeric media is attained under overlapping vibrations and this results in a decreased effective viscosity of the melts, a drop i the pressure required to extrude them through molding tools, increased critical velocities of unstable flow occurrence and a reduced load on the thrust elements of extruder screws. 

British thermal unit 112 bulk modulus 12 bulk strain 12 Burgers vector 36... 

The bulk strains , i.e. strains on the whole unit cell, are defined using the following relationship ... 

The elastic constants have already been defined in Section 4.2 as the second derivative of energy at zero basis strain with respect to bulk strain normalized to the cell volume. However, when minimizing to a specified pressure or to a specific temperature using lattice dynamics, pressure and temperature corrections must be considered (Barron and Klein, 1965 Garber and Granato, 1975 Wall et al., 1993). 

Such a microcrack formation occurs as soon as the strain enhancement at any microfibril end is high enough for material separation. The higher the bulk strain the higher the number of such defects which are deformed so much that a microcrack can be opened. Their number increases almost exponentially with strain as can be concluded from the dependence of radical concentration on bulk strain. But the sample itself is still strong and will hold the load up to the point where the microcrack coalescence yields the first critical size crack which will start to grow catastrophically and will make the sample fail. 

Fig. 2. Depiction of the three independent types of bulk strain and their associated elastic constants, kn for splay, k22 for twist and k33 for bend.

Magnetostriction coefficients have also been deduced from measurements of interplanar lattice spacings by X-ray diffraction (Darnell, 1%3 Finkel and Belovol, 1974). However, it is open to question whether the microscopic strain determined in a multidomain, (and sometimes powdered or polycrystalline) sample is equivalent to the bulk strain in a single domain crystal measured by either of the first two methods. Again the determination of Af and A involves an extrapolation from the paramagnetic region, which is a source of some uncertainty. 

Diffraction spectra indicated only a-Si3N4, with no reflections apparent from the grain boundary phase, an H-phase apatite glass [I], indicating no crystallization of said phase. A SiC-blade high-temperature extensometer monitored the bulk strain of the sample during the entire experiment. 

The strain at a point of the surface S is understood to be consistent with the limiting value of bulk strain Cjj as the observation point approaches the surface from within R. The surface strain expressed as a three-dimensional tensor field over the surface with outward unit normal n, is... 

It is assumed that the system free energy is again given by (8.1) where Us = Us 0, e) depends on the elastic strain field and on the local orientation of the material surface in a characteristic way for a given material. The extensional strain of the surface is related to the bulk strain field through (8.124). The process of elastic equilibration is expected to be very rapid compared to shape equilibration, which requires diffusive transport of material. Consequently, it is assumed from the outset that the elastic field is always in mechanical equilibrium for any surface shape. This expectation is enforced by requiring the variation in free energy due to a kinematically admissible small perturbation 6ui in the displacement field from its equilibrium distribution to be stationary, that is, to vanish to first order in the perturbation. 

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