Elastic constants Young

Isotropic polymers have a single value of , in which case the following relationships, between the four elastic constants (Young s modulus), G (shear modulus), K (bulk modulus) and v (Poisson s ratio) apply ... 

Fig. 7. Relations between elastic constants and ultrasonic wave velocities, (a) Young s modulus (b) shear modulus (c) Poisson s ratio and (d) bulk...

The elastic constants of bulk amorphous Pd-Ni-P and Pd-Cu-P alloys were determined using a resonant i rasound spectroscopy technique. The Pd-Ni-P glasses are slightly stiffer than the Pd-Cu-P glasses. Within each alloy system, the Young s modulus and the bulk modulus show little change with alloy composition. 

The variation in wall thickness and the development of cell wall rigidity (stiffness) with time have significant consequences when considering the flow sensitivity of biomaterials in suspension. For an elastic material, stiffness can be characterised by an elastic constant, for example, by Young s modulus of elasticity (E) or shear modulus of elasticity (G). For a material that obeys Hooke s law,for example, a simple linear relationship exists between stress, , and strain, a, and the ratio of the two uniquely determines the value of the Young s modulus of the material. Furthermore, the (strain) energy associated with elastic de-... 

Here E is Young modulus. Comparison with Equation (3.95) clearly shows that the parameter k, usually called spring stiffness, is inversely proportional to its length. Sometimes k is also called the elastic constant but it may easily cause confusion because of its dependence on length. By definition, Hooke s law is valid when there is a linear relationship between the stress and the strain. Equation (3.97). For instance, if /q = 0.1 m then an extension (/ — /q) cannot usually exceed 1 mm. After this introduction let us write down the condition when all elements of the system mass-spring are at the rest (equilibrium) ... 

Most polymers are applied either as elastomers or as solids. Here, their mechanical properties are the predominant characteristics quantities like the elasticity modulus (Young modulus) E, the shear modulus G, and the temperature-and frequency dependences thereof are of special interest when a material is selected for an application. The mechanical properties of polymers sometimes follow rules which are quite different from those of non-polymeric materials. For example, most polymers do not follow a sudden mechanical load immediately but rather yield slowly, i.e., the deformation increases with time ( retardation ). If the shape of a polymeric item is changed suddenly, the initially high internal stress decreases slowly ( relaxation ). Finally, when an external force (an enforced deformation) is applied to a polymeric material which changes over time with constant (sinus-like) frequency, a phase shift is observed between the force (deformation) and the deformation (internal stress). Therefore, mechanic modules of polymers have to be expressed as complex quantities (see Sect. 2.3.5). 

The coefficients Cn are called elasticity constants and the coefficients Su elastic compliance constants (Azaroff, 1960). Generally, they are described jointly as elasticity constants and constitute a set of strictly defined, in the physical sense, quantities relating to crystal structure. Their experimental determination is impossible in principle, since Cu = (doildefei, where / i, and hence it would be necessary to keep all e constant, except et. It is easier to satisfy the necessary conditions for determining Young s modulus E, when all but one normal stresses are constant, since... 

Several measures exist of tine strength of these bonds. One is the size of the elastic constants—for most solids. Young s modulus is about 1011 newtons per square meter. The other is the frequency of vibration of the atoms—values around 1013 to 1014 Hz are found... 

In various experiments different elastic constants are being determined with a torsion pendulum, for instance, the shear modulus, G, is measured, with creep or vibrations in elongation or in bending the Young s modulus, (tensile modulus), E. For an isotropic material the relation between E and G is as follows ... 

From the repulsive term some important vibrational data, which allow the estimation of some mechanical properties of solids, can be determined. For example, in Chapter 10, it will be shown that the Young s modulus (an elastic constant) of a solid can be approximated by treating the interatomic bonds as springs. The restoring force for small... 

The applied stress is uniaxial and the elastic constant needed in Eq. 10.6, c, is simply Young s modulus ... 

Figure 9.3 Elastic constants of an anisotropic fiber the longitudinal Young s modulus of fiber, or, the transverse Young s modulus or j., and the principal shear modulus, G 2 or Not shown are the two Poisson s ratios i/jj or the longitudinal Poisson s ratio of the fiber and or the transverse or inplane Poisson s ratio of the fiber cross-section.

Near the contact, the vertical arrows at the dashed contour schematically represent the surface forces which cause an additional deformation of the elastic sphere thus increasing the contact radius from aH (Hertz) to aJKR (JKR). The contact radius for the JKR model is a function of the external load, the work of adhesion, the radius of the contacting sphere (or the reduced radii of the contacting spheres, if two spheres are in contact) and the elastic constant K (a combination of the Young s moduli and the Poisson s ratios of the contacting materials), defined as... 

FIGURE 1.10. Shear modulus G, bulk modulus K, Young s modulus E, and Poisson s ratio v of tungsten vs. temperature, as calculated fium single-crystal elastic constants (Gj, E, and v ) [1.31], and from measurements on polyciystalline tungsten (G, K, E, and Vj). [1.30] Taken from Metals Handbook [1.40]. o... 

Elastic constants of solids can be related to the fundamental interaction energy terms. In fact it can be shown easily that the bulk modulus, K scales as the energy density, Ulr for an ionic material interacting through Bom-Mayer potential (see later in this section). Thus the molar volume which increases with the presence of larger ions in glasses can be expected to cause a decrease in Young s modulus. 

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