Strain relationship between volume

Fig. 2. Relationship between volume strain and axial strain during creep of ASA polymer and polypropylene copolymer at 20 °C., showing difference in deformation mechanism...

Figure 3.20. Relationship between volume strain A F/Fand longitudinal strain AL/L for creep of HiPS and blends of HiPS with PPO (from 12.5 to 50% ABS) (Bucknall et u/., 1972a). Note that the fraction of deformation due to crazing as opposed to shear drops from about 100 % in the case of HiPS alone to about 60% for a 50-50 blend with PPO. It is interesting that the HiPS, itself a craze-prone polymer, promotes shear yielding in the somewhat ductile PPO.

Figure 3.21. Relationship between volume strain AF/F and longitudinal strain for creep of a high-impact ABS resin, showing mechanism of creep as a function of strain at five different stresses. (Bucknall and Drink water, 1973.)...

Fig. 8.5. Relationship between volume changes and strains, observed for creep experiments on PP and ASA. Work of Bucknall [84]...

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 principal coordinates provide an extraordinarily useful conceptual framework within which to develop the fundamental relationships between stress and strain rate. For practical application, however, it is essential that a common coordinate system be used for all points in the flow. The coordinate system is usually chosen to align as closely as possible with the natural boundaries of a particular problem. Thus it is essential that the stress-strain-rate relationships can be translated from the principal-coordinate setting (which, in general, is oriented differently at all points in the flow) to the particular coordinate system or control-volume orientation of interest. Accomplishing this objective requires developing a general transformation for the rotation between the principal axes and any other set of axes. 

In the solid state deformation, the nonlinear viscoelastic effect is most clearly shown in the yield behavior. The activation volume tensor is a key parameter. In addition to the well known dependence of yield stress on temperature and strain rate, the functional relationships between yield, stress field, and physical aging are presented. 

The power of the theory developed by Shih et al. (1) lies in the fact that it is possible to experimentally determine if a system is in the strong- or weak-link. The strain at the limit of linearity increases as a function of the volume fraction of network material for the weak-link regime while it decreases for the strong-link regime. Below we derive expressions for the relationship between the strain at the limit of linearity and network material volume fraction. 

The concept of direct attack mechanisms as direct contact phenomena requiring intimate physical contact between the mineral surface and the organism requires some precise definition. Vanselow (1976) has thrown some light on the possible physical interactions between cells and minerals in studies on the effects of dilution on the rates of oxygen uptake by T. ferrooxidans strains in the presence of synthetic covellite. The dilution of a slurry will lower the rate of copper sulfide oxidation per unit volume, and the relationship between the dilution factor, and the factor by which the oxidation rate is lowered, will depend upon the nature of the physical interaction between the cells and the mineral particles. Three principal situations were postulated, namely, that in which the oxidation was carried out by cells... 

Here Ct is a tetragonal shear, Co is an orthorhombic shear and, for small strains, Ca is the volume strain. Symmetry-adapted strains for the sequence of phase transitions Pm3m < i>P4/mbm < Cmcm are shown for NaTaOs in Figure 5, though, for reasons which are given later, a different orientation relationship between crystallographic x-, y-and z-axes with respect to reference X-, Y- and Z-axes was chosen for this system. 

The total dipole strength induced by a rigid body in a straining flow is approximately the sum of (7.7) and (7.17). The relationship between drag and volume flux is now no longer valid for when the wake vorticity is partially or completely annihilated, and is now determined by the local strain rate through... 

The response of unvulcanized black-filled polymers (in the rubbery zone) to oscillating shear strains (151) is characterized by a strong dependence of the dynamic storage modulus, G, on the strain amplitude or the strain work (product of stress and strain amplitudes). The same behavior is observed in cross-linked rubbers and will be discussed in more detail in connection with the dynamic response of filled networks. It is clearly established that the manyfold drop of G, which occurs between double strain amplitudes of ca. 0.001 and 0.5, is due to the breakdown of secondary (Van der Waals) filler aggregation. In fact, as Payne (102) has shown, in the limit of low strain amplitudes a storage modulus of the order of 10 dynes/cm2 is obtained with concentrated (30 parts by volume and higher) carbon black dispersions made up from low molecular liquids or polymers alike. Carbon black pastes from low molecular liquids also show a very similar functional relationship between G and the strain amplitude. At lower black concentrations the contribution due to secondary aggregation becomes much smaller and, in polymers, it is always sensitive to the state of filler dispersion. 

FIG. 30. Relationship between loaf volume and storage and loss moduli for commercial glutens rehydrated, after freeze drying, to 65% moisture (frequency = 10 rad/sec, strain = 0.25 LeGrys et al., 1981). 

Relationship between elasticity and orientational order As remarked in chapter I, a uniformly oriented film of nematic liquid crystal may be prepared by prior treatment of the surfaces with which it is in contact. If the preferred orientation imposed by the surfaces is perturbed, let us say by a magnetic field, a curvature strain will be introduced in the medium. The theory of such a deformation will be discussed at length in 3.2 for the present it will suffice to state some of the important results. The free energy per unit volume of the deformed medium relative to the state of uniform orientation is... 

In an ideal elastic solid, a one-to-one relationship between stress and strain is expected. In practice, however, there are often small deviations. These are termed anelastic effects and result from internal friction in the material. Part of the strain develops over a period of time. One source of anelasticity is thermoelasticity, in which the volume of a body can be changed by both temperature and applied stress. The interaction will depend on whether a material has time to equilibrate with the surroundings. For example, if a body is rapidly dilated, the sudden... 

Figure 7.4 Stress-strain relationships for dough of good and poor bread-making flours (left) and the relationship between strain hardening index and baked loaf volume (right). (Data from Dobraszczyk, B. J. 1999. In Bubbles in food, ed. G. M. Campbell, C. Webb, and S. S. Pandiella, 173-182. St. Paul, MN AACC International.)...

The relationship between the mechanical stresses and strains and electrical surface charges and polarization is given by the piezoelectric coefficients d and e. These parameters are defined in terms of the stress elasticity modulus (or = Ex), the electric field strength O (in Vm ) and polarization P (dipole moment per unit volume in C/m ) ... 

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