Elastic stiffnesses

The contribution to the stress from electromechanical coupling is readily estimated from the constitutive relation [Eq. (4.2)]. Under conditions of uniaxial strain and field, and for an open circuit, we find that the elastic stiffness is increased by the multiplying factor (1 -i- K ) where the square of the electromechanical coupling factor for uniaxial strain, is a measure of the stiffening effect of the electric field. Values of for various materials are for x-cut quartz, 0.0008, for z-cut lithium niobate, 0.055 for y-cut lithium niobate, 0.074 for barium titanate ceramic, 0.5 and for PZT-5H ceramic, 0.75. These examples show that electromechanical coupling effects can be expected to vary from barely detectable to quite substantial. 

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... 

Hardness is a somewhat ambiguous property. A dictionary definition is that it is a property of something that is not easily penetrated, spread, or scratched. These behaviors involve very different physical mechanisms. The first relates to elastic stiffness, the second to plastic deformation, and the third to fracturing. But, for many substances, the mechanisms of these are closely related because they all involve the strength of chemical bonding (cohesion). Thus discussion of the mechanism for one case may provide some understanding of all three. 

These equations are all that is needed to describe a creep test at constant stress, but to describe tensile (or compression) tests, the machine being used must be taken into account because the elastic stiffness of the machine plays an important role. See Gilman and Johnston (1962). 

Since these structures are formed by filling the open spaces in the diamond and wurtzite structures, they have high atomic densities. This implies high valence electron densities and therefore considerable stability which is manifested by high melting points and elastic stiffnesses. They behave more like metal-metalloid compounds than like pure metals. That is, like covalent compounds embedded in metals. 

Since the elastic stiffness is related to the electronegativity difference density (Gilman, 2003) so is the hardness. Thus, like the covalent solids, the hardnesses of the alkali halides depends on the strength of the chemical bonding within them. 

The structures of the prototype borides, carbides, and nitrides yield high values for the valence electron densities of these compounds. This accounts for their high elastic stiffnesses, and hardnesses. As a first approximation, they may be considered to be metals with extra valence electrons (from the metalloids) that increase their average valence electron densities. The evidence for this is that their bulk modili fall on the same correlation line (B versus VED) as the simple metals. This correlation line is given in Gilman (2003). 

Entropy versus temperature data give values for 0S, so values for g can be obtained from Equation 10.3. These values depend on valence electron densities just as the elastic stiffnesses do. 

Equation (5.2) also implies that a crystalline solid becomes mechanically unstable when an elastic constant vanishes. Explicitly, for a three-dimensional cubic solid the stability conditions can be expressed in terms of the elastic stiffness coefficients of the substance [9] as... 

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

Resistance-Deflection Function. The resistance-deflection function establishes the dynamic resistance of the trial cross-section. Figure 4a shows a typical design resistance-deflection function with elastic stiffness, Kg (psi/in), elastic deflection limit, Xg (in) and ultimate resistance, r.. (psi). The stiffness is determined from a static elastic analysis using the average moment of inertia of a cracked and uncracked cross-section. (For design... 

Paper made on a paper machine exhibits quite different properties in the x and y directions (the machine and cross machine directions), an example of which is a difference in stiffness which can be demonstrated by plotting the specific elastic stiffness in the x-y plane as a function of the machine direction and cross machine direction co-ordinates in the form of a polar diagram (Figure 4.7). 

Compute the effective "bilinear" elastic stiffness and deflection to determine the natural period, ductility ratios, and hinge rotations/... 

Composites provide an atPactive alternative to the various metal-, polymer- and ceramic-based biomaterials, which all have some mismatch with natural bone properties. A comparison of modulus and fracture toughness values for natural bone provide a basis for the approximate mechanical compatibility required for arUficial bone in an exact structural replacement, or to stabilize a bone-implant interface. A precise matching requires a comparison of all the elastic stiffness coefficients (see the generalized Hooke s Law in Section 5.4.3.1). From Table 5.15 it can be seen that a possible approach to the development of a mechanically compatible artificial bone material... 

When the quasi-elastic method is used, the viscoelastic resultant moment can be approximated by substituting the time-dependent stiffnesses for elastic stiffnesses in Equation 8.30 and making use of the convolution integral. The resulting moments are... 

All crystals are anisotropic many other structures also have elastic anisotropy. The propagation of elastic waves in anisotropic media is described by the Christoffel equation. This still depends on Newton s law and Hooke s law, but it is expressed in tensor form so that elastic anisotropy may be included. The tensor description of elastic stiffness was summarized in 6.2, especially eqns (6.23)—(6.29). The Christoffel equation is... 

In many cases considerable simplification is possible, because of the constraints imposed on the number of independent elastic constants. For cubic symmetry, for which the elastic stiffness tensor has only three independent constants as given in (6.29), the elements of T, are given in Table 11.1(b), and for hexagonal symmetry the elements are given in Table 11.1(c). If c12 = Cn - 2c44 were to be substituted in Table 11.1(b) the isotropic elements would... 

The beauty of the differential UFM approach is that the absolute value of the contact stiffness of a nanoscale contact at a known force level F is directly measured in terms of the ultrasonic vibration amplitude and the applied force, and is practically independent of the adhesion or other contact parameters. The contact geometry would need to be known in order to determine the elastic stiffness of the sample. 

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. 

The group-theoretical stiffness parameters can be expressed in the conventional (cubic) elastic stiffness coefficients ... 

In performing such experiments on isotropic materials, one is accustomed to express the elastic stiffness parameters in the experimentally more readily accessible technical parameters E (Young s modulus) and v (Poisson ratio). The relative change in length, in the direction of the tensile stress a is, by definition, given by (Al/t)i — a/E, whereas v = (Af/ )x/( A / )u. For several magnetostrictive films and substrates, E and v values are listed in table 1. Some useful relations are ... 

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