Stiffness coefficients 282

Steam turbine, 53, 146, 282-92, 179 back pressure, 282 blade deposits, 479 condensing, 282 efficiency, 288 extraction, 282 induction-type, 282 paitial admission, 288 rating, 290 reliability, 478 selecuon variable, 275, 285 speed, 278 stage losses, 286 steam temperatures, 284 steam velocity, 288 trip and throttle valve. 479 Step unloading system, 80 Stiffness coefficients, 385 Stodola slip, 153, 155 Stonewall, 186 Straight labyrinth. seal leakage, 532... 

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... 

Note Of the coefficients used in rotor lateral analysis, those for damping in annular clearances have the highest uncertainty and are therefore usually the first to be adjusted. The stiffness coefficients of annular clearances typically have low uncertainty and, therefore, should be adjusted only on the basis of supporting data. Adjustments of bearing coefficients require specific justification because the typical values are based on reliable empirical data. 

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... 

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

The stiffness coefficients can be derived from the compliances using the following relations which hold for either open- or closed-circuit conditions and also with the symbols c and s interchanged ... 

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,... 

Unlike stress and strain, which are field tensors, elasticity is a matter tensor. It is subject to Neumann s principle. Hence, the number of independent elastic coefficients is further reduced by the crystal symmetry. The proof is beyond the scope of this book (the interested reader is referred to Nye, 1957), here the results will merely be presented. For example, even with triclinic crystals, the lowest symmetry class, there are only 21 independent elastic-stiffness coefficients ... 

TABLE 10.3. The Independent Elastic-Stiffness Coefficients for Each Crystal Class. If an Unlisted Coefficient is not Related to a Listed One by Transpose Symmetry (c,y= Cy/), it is Zero-Valued... 

Upon inspection of Table 10.3, it can be seen that there are twelve nonzero-valued elastic-stiffness coefficients. Some of these are related by the crystal class and some by transpose symmetry, with the result that there are only six independent coefficients Cn = C22 C12 C13 = C23 C33 C44 = C55 Cee- All other components are zero-valued. Hence, the matrix with all the nonzero independent coefficients designated as such is straightforwardly written as ... 

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) ... 

By using the relations between the elastic-stiffness coefficients in the cubic class from Table 10.3 in Eq. 10.19, the Voigt approximation of the Young s modulus is obtained for a material with cubic symmetry ... 

Using the relations between the elastic-stiffness coefficients from Table 10.3 in Eqs. 10.20 and 10.26, one may also derive the Voigt and Reuss approximations for the rigidity modulus of a cubic monocrystal. These are given by Eqs. 10.35 and 10.36, respectively ... 

The EAM analytical potentials (Eq. 10.46) are multi-variable functions. Their second derivatives yield accurate estimates for the elastic-stiffness coefficients. However, calculating the second derivative of a potential with terms beyond the pair... 

One recent exampie of the fiexibiiity of RUS is its use to determine the eiastic stiffness coefficient of thin Aims through free-vibration resonance frequencies of a fiim-substrate-iayered soiid and to measure deformation distributions on the vibrating specimen by laser-Doppier interferometry [87]. 

The xc contribution to the so called spin-stiffness coefficient a c is also approximated within the LDA of [90]. 

The elastic (or stiffness) coefficients (moduli) are defined as the first derivatives of the tensions with respect to strain... 

Thus one would expect from a (6x6) matrix of the elastic stiffness coefficients (c,y) or compliance coefficients (sy) that there are 36 elastic constants. By the application of thermodynamic equilibrium criteria, cy (or Sjj) matrix can be shown to be symmetrical cy =cji and sy=Sji). Therefore there can be only 21 independent elastic constants for a completely anisotropic solid. These are known as first order elastic constants. For a crystalline material, periodicity brings in elements of symmetry. Therefore symmetry operation on a given crystal must be consistent with the representation of the elastic quantities. Thus for example in a cubic crystal the existence of 3C4 and 4C3 axes makes several of the elastic constants equal to each other or zero (zero when under symmetry operation cy becomes -cy,). As a result, cubic crystal has only three independent elastic constants (cu== C22=C33, C44= css= and Ci2=ci3= C2i=C23=C3i=C32). Cubic Symmetry is the highest that can be attained in a crystalline solid but a glass is even more symmetrical in the sense that it is completely isotropic. Therefore the independent elastic constants reduce further to only two, because C44=( c - C i)l2. 

Note that the stiffness coefficients are now evaluated at the strained reference state (as signified by the notation dEtot/ Ria)v) rather than for the state of zero strain considered in our earlier treatment of the harmonic approximation. To make further progress, we specialize the discussion to the case of a crystal subject to a homogeneous strain for which the deformation gradient is a multiple of the identity (i.e. strict volume change given by F = A,I). We now reconsider the stiffness coefficients, but with the aim of evaluating them about the zero strain reference state. For example, we may rewrite the first-order term via Taylor expansion as... 

B. Audoin, and C. Bescond, Measurement by laser-generated ultrasound of four stiffness coefficients of an anisotropic material at elevated temperatures, J. Nondestructive Eval. 16(2) 91-100 (1997). 

Then ej is a simple dilatation of the crystal and the remaining 6 are distortions at constant volume. 62 is a distortion with rotational symmetry about the c-axis and 63 to eg represent various plane shears. For these strains and the similarly defined stresses we have new stiffness coefficients which can be shown to be... 

The distance d is an index of stiffness of the hber. The stiffness coefficient G (ratio of applied force to bending dehection) may be calculated from d using this expression ... 

The stiffness coefficient is directly proportional to fiber linear density. The data plotted in Figure 8-22 provide a correlation coefficient of 0.97 and an index of determination of 0.94 [74]. This demonstrates that 94% of the variation in stiffness (in this experiment) is accounted for by variation in linear density, and as theory predicts, stiffness increases with fiber diameter. Theory predicts a fourth-power dependence between fiber stiffness and diameter for a perfectly elastic system. 

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