Stiffness constants

Key Words —Nanotubes, mechanical properties, thermal properties, fiber-reinforced composites, stiffness constant, natural resonance. 

Since most SWNTs have diameters in the range of 1 -2 nm, we can expect them to remain cylindrical when they form cables. The stiffness constant of the cable structures will then be the sum of the stiffness constants of the SWNTs. However, just as with MWNTs, the van der Waals binding between the tubes limits tensile strength unless the ends of all the tubes can be fused to a load. In the case of bending, a more exact... 

The calculated and experimental values of the equilibrium lattice constant, bulk modulus and elastic stiffness constants across the M3X series are listed in Table I. With the exception of NiaGa, the calculated values of the elastic constants agree with the experimental values to within 30 %. The calculated elastic constants of NiaGa show a large discrepancy with the experimental values. Our calculated value of 2.49 for the bulk modulus for NiaGa, which agrees well with the FLAPW result of 2.24 differs substantially from experiment. The error in C44 of NiaGe is... 

Among the metals of the Periodic Table, osmium has the highest bulk modulus (412 GPa), and shear stiffness constants of C44 = 270GPa and C66 = 268 GPa. (Pantea et al., 2008).The corresponding values for diamond are B = 433 GPa and C44 (111) = 507 GPa. Although the bulk modulus of Os is about 95% that of diamond, the indentation hardness is only about 3% of diamond s hardness. In other words dislocations move readily in Os but not in diamond. 

Figure 5.2 Temperature dependence of the isothermal elastic stiffness constants of aluminium [10].

Wx or Wf Let us suppose that/is the control parameter. In this case the JE and GET, Eqs. (40) and (41), are valid for the work, Eq. (96). How large is the error that we make when we apply the JE using Wx instead This question has been experimentally addressed by Ciliberto and co-workers [97, 98], who measured the work in an oscillator system with high precision (within tenths of fesT). As shown in Eq. (99), the difference between both works is mainly a boundary term, A xf). Fluctuations of this term can be a problem if they are on the same order as fluctuations of Wx itself. For a harmonic oscillator of stiffness constant equal to k, the variance of fluctuations mfx are equal to k8(x ), that is, approximately on the order of k T due to the fluctuation-dissipation relation. Therefore, for experimental measurements that do not reach such precision, Wx or Wf is equally good. 

The summation sign has been included for emphasis. In the first line of (6.30) there do not appear to be any repeated subscripts, but that is because they have been subsumed in the abbreviated notation. If the stiffness tensor elements are written out in full, as in (6.24), this is immediately apparent. The selection rules in (6.30) correspond directly to the components of stress and strain that are related by each of the three stiffness constants. 

Here, Q are the elastic stiffness constants, are the thermal stress coefficients, and gkj and are the direct and converse piezoelectic stress coefficients, respectively. The superscript , on Pk, p k, and Xki indicates that these quantities are now defined under the conditions of constant strain. 

TABLE 2 Lattice and elastic stiffness constants ol binary nitride compounds. 

Here K is the stiffness constant. The values of the parameters generally used in Eq. (2.5) are given in Table 2.2. These values lead to a = 4.2 eV/A and the electron-phonon coupling constant k = 0.19. However using Eq. (2.5), the value of Ao comes out to be only 0.53 eV which is considerably smaller than the experimental value of 0.7 eV. Eq. (2.5) is approximate because the electron-electron interactions have been neglected. Also the optical absorption experiments do not give a reliable value of the gap because the absorption can occur by processes other than electronic excitation. 

Such an antiferromagnetic ferrimagnetic transition has been observed (120,122,609) in the system Mn2-xCrzSbo.95ln0.o5. Apparently the decrease in lattice parameter with increasing Cr content is sufficient that a Te is observed for x > 0.03, Te increasing from about 120°K to nearly 400°K at x = 0.2. For x = 0.1, the change in lattice parameter at Tt is Ac = 0.014 A. From equation 178 and reasonable estimates of the stiffness constant R it follows that... 

The constants s and c ( = 1 /s) are known as the elastic compliance constant and the elastic stiffness constant, respectively. The elastic stiffness constant is the elastic modulus, which is seen to be the ratio of stress to strain. In the case of normal stress-normal strain (Fig. 10.3a) the ratio is the Young s modulus, whereas for shear stress-shear strain the ratio is called the rigidity, or shear, modulus (Fig. 10.36). The Young s modulus and rigidity modulus are the slopes of the stress-strain curves and for nonHookean bodies they may be defined alternatively as da-/ds. They are requited to be positive quantities. Note that the higher the strain, for a given stress, the lower the modulus. 

Because stress and strain are vectors (first-rank tensors), the forms of Eqs. 10.5 and 10.6 state that the elastic constants that relate stress to strain must be fourth-rank tensors. In general, an wth-rank tensor property in p dimensional space requires p" coefficients. Thus, the elastic stiffness constant is comprised of 81 (3 ) elastic stiffness coefficients,... 

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) ... 

From Table 10.3, there are six independent elastic-stiffness constants Cn, C12, C13, C33, C44, and C66. Substitution of these relations into Eq. 10.21, and using the relations given in Eqs. 10.22-10.24, gives ... 

Use the following values of the elastic-stiffness constants and the elastic-comphance constants (Kisi and Howard, 1998) for tetragonal zirconia monocrystals to determine the Voigt-Reuss-HiU averages for the Young s modulus, E, the shear modulus, G, and the bulk modulus, B. 

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