Elastic defonnations

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. 

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. 

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. 

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). 

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... 

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]. 

The unique cellular morphologies of foams play a key role in determining their deformation mechanisms [51. They also allow the development of very simple alternative equations based on the mechanical models of beam theory (a branch of civil engineering) combined with scaling concepts, to estimate both the thermoelastic properties and the strengths of foams. Such simple relationships have been developed for foams manifesting elastomeric, elastic-plastic and elastic-brittle responses to mechanical defonnation. While much of this work has focused on the responses of foams to compressive defonnation because of the special importance of this deformation mode in many applications of foams, the responses of foams to tensile and shear deformation have also been considered within this theoretical framework. 

A real material whose behaviour can be modelled in this way initially undergoes irreversible deformation as the stress is applied. This eventually ceases, and the material then behaves effectively as an elastic solid. Release of the stress will cause a rapid return to a less strained state, corresponding to the spring component of the response, but part of the defonnation, arising due to viscous flow in the da.shpot will not disappear. 

The elastic deformation experienced is a result of pulling atoms apart or pushing atoms together and so is directly related to interatomic bonding. If the chemical bonds between the atoms can be accurately described in terms of energies, then the amount of defonnation that will result from a given applied force can be calculated (Section S4.1.7). 

It is leasonaUe to assume that a solid will defonn in an affine manner this is the elastic assun ition. Note that in metal aystals dastk defonnation by dislocation glide on slip planes is not affine. The aystal between the slip planes is not plastically distorted idl the defonnation oocuts at the sl plane. 

In addition, lamination can result in up to 18 elastic coefficients and increased defonnational complexities. But the additional coefficients can all be derived from the four primary coefficients using the concept of rotation and ply-stacking sequence (14, 15). These complications are the result of geometric variables. If the laminate is prelastic coefficients. We shall consid laminates of this nature. 

Axial compression of high modulus polymeric filaments results in the formation of so-called kinkbands. These are regions of sub-micron thickness where the compressive defonnation is concentrated. This mode of failure is characteristic for anisotropic materials. The processes involved in kinkband formation are not yet well understood. In this work the kinkband formation in single aramid filaments is measured as a function of the applied compressive strain. Above a critical compressive strain the kinkband density initially increases yery rapidly until eventually a maximum density is obtained. The experimental results are compared with an elastic stability model. The kinkbands form before elastic instability occurs and are therefore attributed to a plastic deformation process. A model is developed to describe the kinkband density as a function of the applied strain. 

Equations 6.6 and 6.7 are convenient reIationshi S givipg the stresses in two directions rpendicular to each other respective slra i resulting fiwm the stresses. Poison s ratio, and the modulus of elasUt ity of the material in question. It is assumed that the material is homogeneous and the defonnations are all within the elastic limit. 

As the pull-off force is applied, the elastic substrate defonns into the shape shown in Fig. 16b and a crack starts at the edge of the contact and moves through the interface to cause rapid and unstable fracture. The energy balance analysis can be carried out for this geometry by considering the three energy terms involved in the cracking. 

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