Bulk stress

Fig. 16 a. Stress dependence of the molar fraction of the P form (x ) and b) bulk stress-strain curve measured for uniaxially oriented polybutyleneterephtalate [83]... 

The stress is simply defined as the force divided by the area over which it is applied. Pressure is a compressive bulk stress. When we hang a weight on a wire, we are applying an extensional stress and when we slide a piece of paper over a gummed surface to reach the correct position, we are applying a shear stress. We will focus more strongly on this latter stress because most of our instruments are designed around this format. The units of stress are Pascals. 

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 ... 

The particle stress A, defined as the contribution to the bulk stress arising from the presence of the suspended particles in the fluid (Batchelor, 1970), possesses the symmetric and traceless deviatoric form... 

In a sheared suspension, the effects are two-fold. First, the expression for bulk stress itself must be modified. Second, the probability density is affected since the continuity equation for the latter must be replaced by a convection-diffusion equation. As a consequence, the distinction between open and closed trajectories loses some of its meaning. Batchelor (1977) gives the equivalent viscosity of a sheared suspension subject to strong Brownian motion as... 

Howells (1974) restricted his attention to fixed particles, extending the method of Childress (1972) by considering a given number of particles chosen from an infinite set. This partly self-consistent scheme furnishes terms valid in the small-solids concentration limit. In a very readable paper, Hinch (1977) combined some of the above procedures in formulating an averaged-equation approach to particle interactions, providing expressions for the bulk stress, average sedimentation velocity, and effective permeability in suspensions and fixed beds. 

In addition to the microstructural geometrical features described above, macroscopic, dynamical, rheological properties of the suspensions are derived by Brady and Bossis (1985). Dual calculations are again performed, respectively with and without DLVO-type forces. When such forces are present, an additional contribution (the so-called elastic stress) to the bulk stress tensor exists. In such circumstances, the term (Batchelor, 1977 Brady and Bossis, 1985)... 

Some authors call this tensor the surface (or interfacial) tension tensor. The interfacial tensor is an excess quantity, (hence the superscript o) and acts in two dimensions (its SI units are N m as compared with N m for bulk stresses). Equation [3.6.14) applies to an isolated Interface. In reality isolation is of course impossible the interface is in contact and at mechanical equilibrium with the bulk. Otherwise the interface would accelerate, slow down, or display shear with respect to the adjacent bulk. An alternative way of formulation would be to retain the bulk tensor [3.6. Ij of which five components are zero in the interface. 

The relaxation behavior, or the transient behavior of cellulosic liquid crystalline solutions upon the cessation of steady flow, is unique for LCPs. There are two kinds of relaxation. The bulk stresses relax quickly while the structures relax over a much longer time. Mewis and Moldenaers suggested that two levels of structures exist.Stress relaxation reflects fast relaxation at the molecule level and is independent of the previous shear rate. Structure relaxation reflects the gradual change of the textures. This slow process is unique in... 

In a granular material there are primarily three mechanisms by which the bulk stresses are generated. These are (i) dry friction, (ii) transport of momentum by particle translation like in dilute gases, and (hi) momentum... 

Here, T is the total bulk stress in the fluid above the surface element A evaluated in the limit as we approach the interface, and T is the stress in the second fluid, evaluated as we approach the interface from below. The negative sign associated with T n is a consequence of the fact that the unit normal from the interface into the second fluid is —n. 

To discuss the macroscopic (or bulk ) properties of a suspension, it is necessary to specify the connection between local variables at the particle scale andmacroscopic variables at the scale L. One plausible choice, in view of the relationship between continuum and molecular variables in Chap. 2, is to assume that the macroscopic variables are just volume averages of the local variables. In particular, we assume in the discussion that follows that the macroscopic (or bulk) stress can be defined as a volume average of the local stress in the suspension, namely,... 

The expression (7-189) for the bulk stress is, of course, valid for arbitrary concentrations of particles, but the volume integrals over Vp are exceedingly difficult to evaluate in general because the value for a particular particle depends on the complete configuration of particles in the suspension. For a dilute suspension of identical particles, on the other hand, the problem simplifies immensely, because the integral over Vp is exactly the same for all particles, and the expression (7-189) can be replaced with... 

It can, in fact, be proven that a dilute suspension of rigid spheres will always be Newtonian at the first 0(C) correction to the bulk stress, with an effective viscosity given by (7 196) of the form... 

BATCHELOR, G.K. 1977. The effect of Brownian motion on the bulk stress in a suspension of spherical particles. ]. Fluid Mech. 83, 97—117. 

The morphological feature is in accordance with that of binary blends of iPP and true diblock iPP-b-EPR copolymers (63). For the immiscible blends showing phase separation, the EP domains have no ability to affect the lamellar or crystalline morphology of iPP. It follows that the strength of iPP lamellae is essentially independent of the addition of EP copolymers although the magnitude of bulk stress is reduced by the addition of EP copolymers with a lower modulus. 

Better candidates for the study of acoustic emission are composite materials. The extremely brittle polyvinyltoluene sample which showed easily detectable acousic emission (6) was indeed to some extent such a composite material since it was a sample used for scintillation counting of nuclear radiation. The crystalline particles of the inorganic scintillator embedded in the rather rigid polymer matrix differ enough in elastic properties from those of the matrix that a substantial stress enhancement occurs on the interface between the two components. One has about twice the bulk stress on the poles and one third on the equator of a perfectly rigid spherical particle. Such a stress increase in the poles leads rather early to adhesion failure of the particle-matrix boundary and to microcrack formation. This finally makes the sample fail at small strain-to-fracture, cb = 0.5%. The microcracks act as nuclei for crazing. The opening of a fissure between the particle and the matrix is sufficiently... 

Fig. 1.7 Actuation mechanism of a liquid crystal elastomer. The application of an electric field results in the realignment of intrinsically polarized liquid crystal mesogens. The mesogens are either grafted to elastomer chains or incorporated within them. The elastomer chains prevent the free flow of the mesogens and couple their motion to bulk stresses and strain...

These promising indications paved the way for the above-mentioned Nature Chemistry paper [43]. Here, the mechanophore chosen was gDBC, embedded as previously shown in polybutadiene, and extruded at 40°C for 30 min and 50 rpm (corresponding to bulk stresses of 0.15-0.25 MPa), leading to a 7% conversion of the gDBCs into their open form and a reductimi in the MW from 780 to 560 kDa. Extruding in the presence of a dicarboxylate, the ditetrabutylammonium salt of... 

Even in the apparently linear range, the response to stress should be considered as viscoelastic rather than elastic. Most polymers that behave in a linear, viscoelastic manner at small strains (< 1 %) behave in a nonlinear fashion at strains of the order of 1 % or more. However, in a fibrous composite, the resin may behave quite differently than it would in bulk. Stress and strain concentrations may exceed the limiting values for linearity in localized regions. Thus the composite may exhibit nonlinearity (Ashton, 1969 Trachte and DiBenedetto, 1968), as is the case with particulate-filled polymers (Section 12.1.2). Although nonlinearity at low strains is characteristic, Halpin and Pagano (1969) have predicted constitutive relations for isotropic linear viscoelastic systems, and verified their prediction using specimens of fiber-reinforced rubbers. 

In contrast with the mathematical dividing surface, which has an isotropic interfacial tension (7, the forces acting on the material interface are not isotropic. They are characterized by the interfacial stress tensor, a, which is a two-dimensional counterpart of the bulk stress tensor, P. The two-dimensional analogue of the momentum balance equation (1) written in the bulk is called the interfacial momentum balance equation. Note, that the inter-phase exchanges momentum also with the contiguous bulk phases and the corresponding balance equation reads (Slattery 1990, Edwards et al. 1991) ... 

In fact, this type of separatim process is bound to happen if a thin layer of elastic material, like sticky tape adhesive, is trapped between two more rigid layers which are pulled apart. The stress within the thin layer is a bulk stress... 

Fig. 14.10. Interface stresses, plotted versus temperature at atmospheric bulk stresses in the adjoining crystal and melt phases ( ), Ttyy ( ), and Ttxy (A).

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