Bulk deformations

The bulk modulus K of polymer liquids is normal in the sense that  

1 Note that normal liquids cannot sustain tensile stresses, so this question cannot arise for them. It is for this reason that the reader may not have considered the question of tensile stress in liquids before. 

Pressures up to atmospheres are encountered in injection moulding, and this leads to a fractional volume decrease of 

This is to say a 10% change in volume. This volume decrease raises the shear viscosity, as described below. Note also that if a mould is filled at 1000 atmospheres it will contain 10% more polymer than if it were filled at 1 atmosphere this fact plays a central role in injection moulding. 

During its passage from z to z + dz, its cross-section changes from A to A + dA and its length from / to / + dl. Since its volume remains constant, it follows that 

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Fabrication. After the preheat or homogenization step, the ingots may be fabricated directly. Often, however, the preheated ingots are reheated in a separate operation before the first metal working operation. Bulk deformation temperatures usually range from about 350 to 450°C. 

The concept of affine deformation is central to the theory of rubber elasticity. The foundations of the statistical theory of rubber elasticity were laid down by Kuhn (JJ, by Guth and James (2) and by Flory and Rehner (3), who introduced the notion of affine deformation namely, that the values of the cartesian components of the end-to-end chain vectors in a network vary according to the same strain tensor which characterizes the macroscopic bulk deformation. To account for apparent deviations from affine deformation, refinements have been proposed by Flory (4) and by Ronca and Allegra (5) which take into account effects such as chain-junction entanglements. 

If we were to have an isolated polymer chain with a single nuclear spin attached to each segment (the marked chain) crosslinked into an unmarked network, the second moment of the NMR line of that spin species would carry information relating to the separation of chain segments, and to their relative orientation with respect to the field direction. If the network were to be subjected to a bulk deformation, these geometrical parameters would be altered, and hence we would expect a corresponding change in the value of the experimentally measured... 

These moment studies have been performed on polymer systems such as polyethylene (or on penetrants in polymer systems) in which the interacting spins (protons or fluorines) reside on the same or on adjacent atoms. This allows essentially no freedom of variation in the internuclear vectors upon deformation of the network. The primary informational content therefore relates to independent segmental orientation distributions. By placing single spins on alternate segments, there should be much greater sensitivity to changes in the chain extension upon bulk deformation. 

The most common cause of it is the neglect of 3-dimensional effects as compared with those in two dimensions. Thus, all stresses in a loaded wire or ribbon are disregarded in the shrinkage method, Section III. 1. The work of deformation leading to rupture is a bulk effect which does not receive its due consideration in the calculation of fracture energy, Section III.3. Bulk deformations associated with thermal etching, Section III.4, demand more attention than was alloted to them by many scientists. The method of bubbles, Section III.5, is invalid both because of the above neglect (that is, that of the volume stresses around the bubble) and because of another popular error, namely an erroneous treatment of capillary pressure Pc. 

The above derivation of the effective Hamiltonian is only complete when, for some reasons, the uniform strains of the crystal are not relevant. This is clearly the case for crystals with low concentration of Jahn-Teller impurities. Contrary to that, bulk deformations often arise in its low-symmetry structural phases of Jahn-Teller crystals [2,11]. The uniform strains describing the bulk deformations of the crystal cannot be reduced to a combination of phonon modes, as it was first pointed out by... 

This crude distinction between adhesive and cohesive wear mechanisms is probably oversimplified in the sense that it neglects many aspects of the interactions between bulk deformation modes and interface rheology. It has, however, the merit of making a clear distinction between wear processes which can, to some extent, be related to known bulk failure properties and... 

Both banded (Tc = 52°C) and unbanded (Tc = 60°C) spherulitic morphologies had essentially identical stress-strain curves despite a difference in crystallinity of 8% and variations in spherulite size for these two crystallization conditions. These changes in crystallinity and spherulite size might compensate sufficiently to allow similar bulk deformation behavior. However, the sample crystallized at 52 °C should have smaller spherulites and thinner lamellae than the sample crystallized at 60 °C because of a greater probability of tie molecules. This, combined with its lower crystallinity, should allow more ductile behavior for the 52° C crystallized sample. The fact that both specimens deform similarly indi-... 

Initially the local removal rate increases until the planarization time for structure a. is Because the level of the dense area is higher than the other, the bulk deformation is larger resulting in an increased pressure. At the planarization time for structure a the local removal for structure a is reduced. As a result the level differences and the bulk deformations are reduced. This results in a drop of the removal rate. When structure b is completely planarized, the removal rate is discontinue like in the case of perfect pad bending. Afterwards the local removal rate for structure b drops gradually till its final value. 

For the top pad with the in-between thickness (50 mils) the removal rate increases initially slightly, reaches its maximum and then it drops as well. Its maximumvalue is smaller than for the thick top pad. Based on figure 3 it is concluded this pad bends more than the thick top pad. The bulk deformations are smaller and the observed increase in local removal rate is lower. 

The pad stack determines the bending of the top pad. Two extreme case are considered perfect pad bending and no pad bending. A typical example for perfect pad bending is when a thin top pad is put on a soft bottom pad. In this case the top pad bends easily and it was concluded that the average pressure on an unit area is equal to the nominal pressure. This results in a large oxide thickness variation after CMP. In the case of a thick top pad on a soft top pad the top pad does not bend. This introduces bulk deformations in the top pad. In this case the oxide thickness variation after CMP is limited. The experiments confirm that a real stacked pad is always in between these two extreme cases. 

Regime II occurs at applied velocities corresponding to stresses above the yield point (cr > <7y). The apparent motion then involves a combination of slip and bulk deformation. In this regime, the increase of the slip velocity with the applied velocity is relatively slow (Vg indicating that at large stresses, the effect of slip... 

The strand chains in the network deform affinely with the bulk deformation due to the constraint of crosslinks ... 

According to the affine deformation postulate for the bulk deformation A, a symmetrical rank two tensor, the end-to-end distance vector deforms from Ro to... 

This balance between interfacial propagation and bulk deformation has been described for linear elastic materials [56] and results from the competition between two mechanisms the velocity of propagation of an interfacial crack, which is controlled by the critical energy release rate Gc, and the bulk deformation, which is controlled by the cavitation stress and hence essentially by the elastic modulus E or G. In the hnear elastic model, the key parameter is the ratio GJE, which represents the distance over which an elastic layer needs to be deformed before being fuUy detached from the hard surface. This model has been verified experimentally for elastic gels [57]. 

Although the standard test conditions arc arbitrary, they were generally chosen to represent the deformations met in service. The bending and compression shear rubber tests correspond to tire sidewall flexing and the bulk deformations of a tire respectively. Other bending tests were intended to relate to the flexing of shoe materials or belting, and... 

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