Elastic, defined

Regarding foam stability, it has been recognized that the surface tension under film deformation must always change in such a way as to resist the deforming forces. Thus, tension in the film where expansion takes place will increase, while it will decrease in the part where contraction takes place. A force exists that tends to restore the original condition, which is film elasticity, defined as... 

Gelation. A sol becomes a gel when it can support a stress elastically, defined as the gelation point or gelation time. tg. A sharp increase in viscosity accompanies gelation. A sol freezes in a particular polymer structure at the gelation point. This frozen-in structure may change appreciably with time, depending on the temperature, solvent, and pH conditions or on removal of solvent. 

These subjects will be described explicitly in the subsequent chapters. It is evident that both the mechanical properties and the exchange of matter at interfaces are of general interest for an improved understanding of the foaming and emulsification processes. Experimental access to these properties of adsorption layers is given by relaxations experiments which provide simultaneously information about the exchange of matter and the dilational interfacial elasticity defined by... 

Interfacial relaxation methods are typically based on a perturbation of the equilibrium state of an interface by small changes of the interfacial area. The ratio of the amplitudes of surface tension and relative area changes gives the modulus of elasticity , defined as... 

Although E drops significantly as T is raised above Tg, K changes relatively little, so that K E and, from Eq. (9), v 0.5. Volume changes may hence be considered negligible compared with other types of deformation. This justifies the use of the Helmholtz free energy in the thermodynamic analysis of rubber elasticity, defined by Eq. (11). 

The intensity of ground motion is increased by the surface soft soil deposits and this phenomenal is called amplification. It is known that the extent of amplification changes basically with the thickness of the surface soil (//) and the velocity of S-wave propagation (F ). The S(econdary)-wave means a propagation of shear deformation and the theory of elasticity defines its velocity by... 

Acoustic emission is a naturally occurring phenomenon within materials, and the term Acoustic Emission is used to define the spontaneous elastic energy released within material or by a process, in the form of transient elastic waves. (2)... 

As for crystals, tire elasticity of smectic and columnar phases is analysed in tenns of displacements of tire lattice witli respect to the undistorted state, described by tire field u(r). This represents tire distortion of tire layers in a smectic phase and, tluis, u(r) is a one-dimensional vector (conventionally defined along z), whereas tire columnar phase is two dimensional, so tliat u(r) is also. The symmetry of a smectic A phase leads to an elastic free energy density of tire fonn [86]... 

Stress relaxation time, obtained from rheograms based on viscometric flows, is used to define a dimensionless parameter called the Deborah number , which quantifies the elastic character of a fluid... 

Incorporation of viscosity variations in non-elastic generalized Newtonian flow models is based on using empirical rheological relationships such as the power law or Carreau equation, described in Chapter 1. In these relationships fluid viscosity is given as a function of shear rate and material parameters. Therefore in the application of finite element schemes to non-Newtonian flow, shear rate at the elemental level should be calculated and used to update the fluid viscosity. The shear rale is defined as the second invariant of the rate of deformation tensor as (Bird et at.., 1977)... 

Here /, r and, v are unequal integers in the set 1, 2, 3. As already mentioned, in the thin-layer approach the fluid is assumed to be non-elastic and hence the stress tensor here is given in ternis of the rate of deforaiation tensor as r(p) = riD(ij), where, in the present analysis, viscosity p is defined using the power law equation. The model equations are non-dimensionalized using... 

The elastic and viscoelastic properties of materials are less familiar in chemistry than many other physical properties hence it is necessary to spend a fair amount of time describing the experiments and the observed response of the polymer. There are a large number of possible modes of deformation that might be considered We shall consider only elongation and shear. For each of these we consider the stress associated with a unit strain and the strain associated with a unit stress the former is called the modulus, the latter the compliance. Experiments can be time independent (equilibrium), time dependent (transient), or periodic (dynamic). Just to define and describe these basic combinations takes us into a fair amount of detail and affords some possibilities for confusion. Pay close attention to the definitions of terms and symbols. 

The various elastic and viscoelastic phenomena we discuss in this chapter will be developed in stages. We begin with the simplest the case of a sample that displays a purely elastic response when deformed by simple elongation. On the basis of Hooke s law, we expect that the force of deformation—the stress—and the distortion that results-the strain-will be directly proportional, at least for small deformations. In addition, the energy spent to produce the deformation is recoverable The material snaps back when the force is released. We are interested in the molecular origin of this property for polymeric materials but, before we can get to that, we need to define the variables more quantitatively. 

It is necessary to establish some conventions concerning signs before proceeding further. When the applied force is a tensile force and the distortion is one of stretching, F, dL, and dw are all defined to be positive quantities. Thus dw is positive when elastic work is done on the system. The work done by the sample when the elastomer snaps back to its original size is a negative quantity. 

Figure 3.6 Definition of variables to define the shear deformation of an elastic body.

We have to stress that the analysed problems prove to be free boundary problems. Mathematically, the existence of free boundaries for the models concerned, as a rule, is due to the available inequality restrictions imposed on a solution. As to all contact problems, this is a nonpenetration condition of two bodies. The given condition is of a geometric nature and should be met for any constitutive law. The second class of restrictions is defined by the constitutive law and has a physical nature. Such restrictions are typical for elastoplastic models. Some problems of the elasticity theory discussed in the book have generally allowable variational formulation... 

The elasticity of a fiber describes its abiUty to return to original dimensions upon release of a deforming stress, and is quantitatively described by the stress or tenacity at the yield point. The final fiber quaUty factor is its toughness, which describes its abiUty to absorb work. Toughness may be quantitatively designated by the work required to mpture the fiber, which may be evaluated from the area under the total stress-strain curve. The usual textile unit for this property is mass pet unit linear density. The toughness index, defined as one-half the product of the stress and strain at break also in units of mass pet unit linear density, is frequentiy used as an approximation of the work required to mpture a fiber. The stress-strain curves of some typical textile fibers ate shown in Figure 5. 

Fig. 5. The boundary between elastic and plastic zones at the crack tip. Terms are defined in text.

Equations 1 to 3 enable the stresses which exist at any point across the wall thickness of a cylindrical shell to be calculated when the material is stressed elastically by applying an internal pressure. The principal stresses cannot be used to determine how thick a shell must be to withstand a particular pressure until a criterion of elastic failure is defined in terms of some limiting combination of the principal stresses. 

The use of the single parameter, K, to define the stress field at the crack tip is justified for brittle materials, but its extension to ductile materials is based on the assumption that although some plasticity may occur at the tip the surrounding linear elastic stress field is the controlling parameter. 

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