Elastic properties stiffness

The important elastic properties of a material undergoing deformation under static tension are stiffness, elastic strength and resilience. For a material obeying Hooke s law, the modulus of elasticity, E (= o/e), can be taken to be a measure of its stiffness. The elastic... 

Another property that is related to chemical hardness is polarizability (Pearson, 1997). Polarizability, a, has the dimensions of volume polarizability (Brinck, Murray, and Politzer, 1993). It requires that an electron be excited from the valence to the conduction band (i.e., across the band gap) in order to change the symmetry of the wave function(s) from spherical to uniaxial. An approximate expression for the polarizability is a = p (N/A2) where p is a constant, N is the number of participating electrons, and A is the excitation gap (Atkins, 1983). The constant, p = (qh)/(2n 2m) with q = electron charge, m = electron mass, and h = Planck s constant. Then, if N = 1, (1/a) is proportional to A2, and elastic shear stiffness is proportional to (1/a). 

The results indicated that cross-link formation increased the bulk modulus of the system. As noted, cross-linking was a pressure-induced effect that was facilitated by a change in the coordination at zinc when the pressure reached 5 GPa. The observation that stiffening of the film is a pressure-induced phenomenon is consistent with the differences in the measured elastic properties of films found on the tops of the asperities and those found in the valleys between asperities as mentioned above. Basically, in real systems, pressures high enough to form stiff cross-linked films are achieved on top of the asperities, but they are not encountered between the asperities. 

The shape of the force versus indentation curve depends on surface adhesive and elastic properties. Variations in these parameters affect the ultrasonically induced deflection. Conversely, the variations in the shape of the ultrasonically induced normal deflection contain information on surface adhesive and elastic properties. Figure 13.3 illustrates how the threshold amplitude should depend on the normal force value. If the normal force is set at a higher value F2 > Fi, then the threshold amplitude a2 = h2) needed to reach the pull-off point should be higher than the threshold amplitude (fli = hi) for Fi. If the threshold amplitude values (fli and a2) are measured for two different normal force values (Fi and F2), the contact stiffness is... 

Figure 13.5(d) presents experimental stiffness measurements using differential UFM for three high modulus surfaces sapphire, Si(100) and LiF(lOO) (Dinelli et al. 2000b). The samples were probed with the same silicon tip on a V-shaped cantilever (nominally cantilever stiffness was kc - 2.8 nN nm 1,and radius of curvature R = 10 nm). The surface RMS roughness of the surfaces was less than 0.2 nm over a few square micrometres for all three samples. The relative difference between the three sets of data reveals that the elastic properties of these three materials can be distinguished by differential UFM the relative independence of the applied force may indicate the fact that the tip had been flattened by extended contact with such hard samples. 

Ultrasonic force microscopy is able to give contrast from samples over a wide range of elastic properties, from stiff crystalline materials like semiconductors... 

Fibrin is a viscoelastic polymer, which means that it has both elastic and viscous properties (Ferry, 1988). Thus, the properties of fibrin may be characterized by stiffness or storage modulus (representing its elastic properties) and creep compliance or loss modulus/loss tangent (representing its inelastic properties). These parameters will determine how the clot responds to the forces applied to it in flowing blood. For example, a stiff clot will not deform as much as a less stiff one with applied stress. 

It can be seen from equations (8.10) and (8.11) that the contribution to the hardening comes mainly from the layers with a higher elastic modulus. However, differences in the elastic properties between the layers will cause the loops in the stiffer layers to be pulled across the interfaces, for the same reason that loops in the less stiff layers are repelled by the interfaces, greatly diminishing their contribution to the overall flow stress. 

The general mechanisms by which the NMF components influence SC functionality have been studied extensively. From a physical chemistry perspective the specific ionic interaction between keratin and NMF, accompanied by a decreased mobility of water, leads to a reduction of intermolecular forces between the keratin fibers and increased elastic behavior. Recent studies have emphasized that it is the neutral and basic FAA36 in particular that are important for helping keratin acquire and maintain its elastic properties. Consistent with these observations Sakai et al.37 reported that the ratio of acidic amino acids to total amino acids correlated to the resonant frequency a measure of skin stiffness. 

With elastically anisotropic materials the elastic behavior varies with the crystallographic axes. The elastic properties of these materials are completely characterized only by the specification of several elastic constants. For example, it can be seen from Table 10.3 that for a cubic monocrystal, the highest symmetry class, there are three independent elastic-stiffness constants, namely, Cn, C12, and C44. By contrast, polycrystalline aggregates, with random or perfectly disordered crystallite orientation and amorphous solids, are elastically isotropic, as a whole, and only two independent elastic-stiffness coefficients, C44 and C12, need be specified to fully describe their elastic response. In other words, the fourth-order elastic modulus tensor for an isotropic body has only two independent constants. These are often referred to as the Lame constants, /r and A, named after French mathematician Gabriel Lame (1795-1870) ... 

As a result, one may be able to choose from a set of materials, adjusting the shear stiffness by adjusting the layer thickness H2. (This is something of an oversimplification, because the value of H2 influences Y.) This ability to adapt a viscoelastic material to a particular application can allow the inclusion of a range of candidate materials having desirable properties aside from the dynamic elastic properties G2 and T 2 ... 

The viscous and elastic properties of orientable particles, especially of long, rod-like particles, are sensitive to particle orientation. Rods that are small enough to be Brownian are usually stiff molecules true particles or fibers are typically many microns long, and hence non-Brownian. The steady-state viscosity of a suspension of Brownian rods is very shear-rate- and concentration-dependent, much more so than non-Brownian fiber suspensions. The existence of significant normal stress differences in non-Brownian fiber suspensions is not yet well understood. 

If brought into contact, the (theoretical) distance assuming an LJ interaction potential would be equal 1.12a. Elastic deformation of the sphere and the flat surface have not been considered (infinite stiffness was assumed for the bodies). In such cases, the contact ideally is a point contact. However, if the Young s modulus (modulus of elasticity) of one of the bodies (or both) has a finite value, then the contact point becomes a contact circle with a radius a. The value of the contact radius a depends in such cases on the elastic properties of the spheres, on the Young s moduli E and E2, and on the Poisson s ratios q and v2, of the two contacting materials, respectively. The value of the contact radius a for two spheres pressed together can be calculated from the following formula ... 

For cubic symmetry materials, three independent elastic properties that are orientation dependent are required to describe the mechanical behavior of the material. This anisotropy effect increases significantly the number of the nonzero elements in the FE stiffness matrix leading to alteration in the calculated stress components and the wave speed. In order to test these anisotropy effects, we plot the wave profiles of three different orientations and compare it with the isotropic behavior with a loading axis in the [001] directions as shown in Fig 8. We observed that under the same loading condition, the peak stress of [111] and [Oil] orientations are slightly higher than those of the [001] which is lower that that of isotropic material. Furthermore, wave speed varies moderately with orientation with the fastest moving wave in the [ 111 ] followed by [011 ], isotropic medium and [001 ] respectively. 

Equations (5.25)-(5.28) demonstrate the importance of the cantilever stiffness with respect to the sample hardness. But to access quantitatively the work of adhesion and the elastic properties of the sample we have to know the spring constant accurately, since Pgff equals kZ with the cantilever deflection, or spring extension needed to unstick the tip. Unfortunately, because of the poor accuracy of the spring constant of the cantilever, rather than quantitative experiments, only comparative studies can be carried out. 

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