Defonnability

Carpick et al [M] used AFM, with a Pt-coated tip on a mica substrate in ultraliigh vacuum, to show that if the defonnation of the substrate and the tip-substrate adhesion are taken into account (the so-called JKR model [175] of elastic adliesive contact), then the frictional force is indeed proportional to the contact area between tip and sample. Flowever, under these smgle-asperity conditions, Amontons law does not hold, since the statistical effect of more asperities coming into play no longer occurs, and the contact area is not simply proportional to the applied load. 

In tire limit of a small defonnation, a polymer system can be considered as a superjDosition of a two-state system witli different relaxation times. Phenomenologically, tire different relaxation processes are designated by Greek... 

In the last section we considered tire mechanical behaviour of polymers in tire linear regime where tire response is proportional to tire applied stress or strain. This section deals witli tire nonlinear behaviour of polymers under large defonnation. Microscopically, tire transition into tire nonlinear regime is associated with a change of tire polymer stmcture under mechanical loading. 

Cp is tire number of elasticity active chains per volume unit. The comparison between experimental data and tire prediction by (C2.1.20) shows a reasonable agreement up to large defonnation (figure C2.1.16). For large values of X, strain hardening arises because of tire limited extensibility of tire chains or because of shear-induced crystallization. 

Figure C2.1.17. Stress-strain curve measured from plane-strain compression of bisphenol-A polycarbonate at 25 ° C. The sample was loaded to a maximum strain and then rapidly unloaded. After unloading, most of the defonnation remains.

Under compression or shear most polymers show qualitatively similar behaviour. However, under the application of tensile stress, two different defonnation processes after the yield point are known. Ductile polymers elongate in an irreversible process similar to flow, while brittle systems whiten due the fonnation of microvoids. These voids rapidly grow and lead to sample failure [50, 51]- The reason for these conspicuously different defonnation mechanisms are thought to be related to the local dynamics of the polymer chains and to the entanglement network density. 

Atomistically detailed models account for all atoms. The force field contains additive contributions specified in tenns of bond lengtlis, bond angles, torsional angles and possible crosstenns. It also includes non-bonded contributions as tire sum of van der Waals interactions, often described by Lennard-Jones potentials, and Coulomb interactions. Atomistic simulations are successfully used to predict tire transport properties of small molecules in glassy polymers, to calculate elastic moduli and to study plastic defonnation and local motion in quasi-static simulations [fy7, ( ]. The atomistic models are also useful to interiDret scattering data [fyl] and NMR measurements [70] in tenns of local order. 

Here and are elastic constants. The first, is associated with a splay defonnation, is associated with... 

Figure C2.2.11. (a) Splay, (b) twist and (c) bend defonnations in a nematic liquid crystal. The director is indicated by a dot, when nonnal to the page. The corresponding Frank elastic constants are indicated (equation(C2.2.9)).

Here B is again a compressional elastic constant, is a bend elastic constant and tire elastic constant C results from an elliptical defonnation of tire rods (tliis tenn is absent if tire column is liquid). 

If we compare with figure C2.2.I I, we can see that this defonnation involves bend and splay of the director field. This field-induced transition in director orientation is called a Freedericksz transition [9, 106, 1071. We can also define Freedericksz transitions when the director and field are both parallel to the surface, but mutually orthogonal or when the director is nonnal to the surface and the field is parallel to it. It turns out there is a threshold voltage for attaining orientation in the middle of the liquid crystal cell, i.e. a deviation of the angle of the director [9, 107]. For all tliree possible geometries, the threshold voltage takes the fonn [9, 107]... 

Figure C2.2.12. A Freedericksz transition involving splay and bend. This is sometimes called a splay defonnation, but only becomes purely splay in the limit of infinitesimal displacements of the director from its initial position [106]. The other two Freedericksz geometries ( bend and twist ) are described in the text.

The tenn tribology translates literally into the study of nibbing . In modem parlance this field is held to include four phenomena adhesion, friction, lubrication and wear. For the most part these are phenomena that occur between pairs of solid surfaces in contact with one another or separated by a thin fluid film. Adhesion describes the resistance to separation of two surfaces in contact to while friction describes their tendency to resist shearing. Lubrication is the phenomenon of friction reduction by the presence of a fluid (or solid) film between two surfaces. Finally, w>ear describes the irreversible damage or defonnation that occurs as a result of shearing or separation. 

Modelling of the tme contact area between surfaces requires consideration of the defonnation that occurs at the peaks of asperities as they come into contact with mating surfaces. Purely elastic contact between two solids was first described by H Hertz [7], The Hertzian contact area (A ) between a sphere of radius r and a flat surface compressed under nonnal force N is given by... 

In reality most solids in contact under macroscopic loads undergo irreversible plastic defonnation. This is caused by the fact that at high nonnal forces the stresses in the bulk of the solid below the contact points exceed the yield stress. Under these conditions the contact area expands until the integrated pressure across the contact area is equal to the nonnal force. Since the pressure is equal to the yield strength of the material cr, the plastic contact area is given by... 

Thus, under conditions of plastic defonnation the real area of contact is proportional to the nonnal force. If the shear force during sliding is proportional to that area, one has the condition that the shear force is proportional to the nonnal force, thus leading to the definition of a coefficient of friction. 

The separation of two surfaces in contact is resisted by adhesive forces. As the nonnal force is decreased, the contact regions pass from conditions of compressive to tensile stress. As revealed by JKR theory, surface tension alone is sufficient to ensure that there is a finite contact area between the two at zero nonnal force. One contribution to adhesion is the work that must be done to increase surface area during separation. If the surfaces have undergone plastic defonnation, the contact area will be even greater at zero nonnal force than predicted by JKR theory. In reality, continued plastic defonnation can occur during separation and also contributes to adhesive work. 

After the substitution of Cauchy stress via Equation (3.20) and the viscous part of the extra stress in terms of rate of defonnation, the equation of motion is written as... 

Figure 3.2 Defonnation of a fluid particle along its trajectory in a contracting flow...

The inverse of the Cauchy-Green tensor, Cf, is called the Finger strain tensor. Physically the single-integral constitutive models define the viscoelastic extra stress Tv for a fluid particle as a time integral of the defonnation history, i.e. 

SECINV. Calculates the second invariant of the rate of defonnation tensor at the integration points within the elements. 

Al) Freezing of bonds and angles defonns the phase space of the molecule and perturbs the time averages. The MD results, therefore, require a complicated correction with the so-called metric tensor, which undermines any gain in efficiency due to elimination of variables [10,17-20]. 

Thermal Stress. The formula for thermal stress can be rearranged to calculate the tolerable temperature gradient for keeping defonnation within arbitrary limits. 

The most frequently applied mechanical manometers in ventilation applications are fluid manometers, bur the following types are also used. The Bourdon tube is a small-voiume tube with an elliptic cross-section bent to the shape of a circular arc, the C-type. One end is open to the applied pressure while the other end is closed. The pressure inside the tube causes an elastic defonnation ot the tube and displaces the closed end, which is then converted, by means of a linkage mechanism, into the movement of a pointer. The Bourdon tube may be of a spiral or helical design as well. 

The stress-strain relations for some special cases of biaxial defonnation are derived from Eqs. (13) to (15) in the following way. Strip biaxial extension of incompressible material is defined as the mode of deformation in which one of the Xj, say X2, is kept at unity, while the other, Xt, varies. This deformation is also called pure shear . We have for it ... 

There are several important things to note. The first is that elastic deformation is a reversible process, but plastic deformation and brittle fracture are not. More importantly, plastic deformation and viscoelastic behavior are kinetic phenomena time is important, and they can be affected by press speed. In reality, most materials exhibit both plastic and brittle behavior, but specific materials can be classified as primarily plastic or primarily brittle. For example, microcrystalUne cellulose defonns primarily by a plastic deformation mechanism calcium phosphate de-fonns primarily by a brittle fracture mechanism lactose is in the middle [8]. 

For most solids, one can neglect the difference between Pp f (ap f/3 for an isotropic body) and the coefficient of thermal expansion at constant P is usually used. Therefore, we may use P and a without subscripts. Assuming that E and p are independent of temperature and ignoring the change in lateral dimensions during defonnation (i.e. we take the Poisson s ratio p = 0, because this simplification gives effects of only the second order of smallness), one can arrive at relations similar to Eqs. (17)—(21). To do this, it is necessary to replace in Eq. (16) the volume deformation e by e, the modulus K by E and a by p (see Fig. 1). For the simple deformation of a Hookean body the characteristic parameter r is also inversely dependent on strain, viz. r = 2PT/e and sinv = —2PT. It is interesting to note that... 

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