2 Lame constant

Note 3 The Lame constant, (/), is related to the shear modulus (G) and Young s modulus (E) by the equation... 

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) ... 

The first Lame constant (A) has no physical interpretation. However, both Lame constants are related to other elastic moduli. To see this, recall that the Young s modulus, E, is defined as the ratio of normal stress to normal strain. Hence, for an elastically isotropic body, E is given by (cn-Ci2)(cn-b 2ci2)/(cn-I-C12), or /r(3A-b 2/r)/(A-b/r). It should be emphasized that the Young s modulus is anisotropic for all crystal classes, including the cubic class, so this relation would never apply to any monocrystal. 

In a similar fashion, the rigidity modulus, G, for an elastically isotropic solid is given by 0-4/84 = C44 = 0-5/85 = C55 = cTs/8g = cgg = i(cn - C12) = /r, or C44, which represents a shape change without a volume change. Therefore, the second Lame constant (fi) is the shear modulus for an elastically isotropic body. The Lame constants may also be related directly to the bulk modulus, B, for an elastically isotropic body, which can be obtained through the relations /r = ( )(B - A) and = B - ( )G. 

In terms of the Lame constants, write the equations relating the six stresses to the strains for a polycrystalline material. 

Using Eq. 10.14 with Table 10.3, it is easily shown that the stresses can be written in terms of the two Lame constants as ... 

Equating real and imaginary parts of Equation 3.58 yields velocity and attenuation changes resulting from the SAW/film interaction. With elastic films, the intrinsic elastic moduli are real, resulting in real 0 and imaginary liy/ko, so that Aa/ko = 0. Substituting the ( > expressions written in terms of the Lame constants (A, fi) from Table 3.3 into Equation 3.58 yields the Tiersten formula [52]... 

Here the Kelvin-Voigt model is assumed to adequately describe the viscoelastic properties of the elastomer and the Lame constants can be written to include the characteristic relaxatiog times of the material. Theg become the operators A... 

The components of the modulus tensor Cy have traditionally been expressed in terms of the Lame constants X and G. Specifically,... 

The three quantities, the Young s modulus E, shear modulus G and the Poisson s ratio vare related to each other for the case of an isotropic body like glass such that there are only two independent moduli A and ju known as Lame constants ... 

Figure 1. A planet of radius R with density q, rigidity p and Lame constant A is tidally distorted by an amount SR due to a tide-raising mass M at distance a.

However, Love (1911) had earlier derived a more general result, valid for arbitrary compressibility. Compressibility is characterized by the Lame constant A A increases as a material gets more incompressible to a limiting case of A —> oo for an incompressible body. The equation governing the deformation is re-derived in the appendix, derivation of the Governing Equation. 

The elastic properties are expressed by the Lame constants A and p, which connect stress and strain in Hook s law. For many rocks, these constants are almost equal therefore we use A = p, denoting fi as the rigidity. An elastic solid with this property is called a Poisson solid. Because the strain is dimensionless, fj, has the same dimension as the stress. In the present study, we use p = 30GPa. 

Where, the is the displacement function is the Lame-constant, (S,y is the Kronecker function, G is the shear modulus and the is the free term for the stress. 

In a last step the model is completed by an appropriate choice of the free energy function. The basic idea is to enhance the free energy of a Hnear elastic material [Eq. (18)] in such a way that the variation of the effective stiffness on the stractural parameter is obtained, i.e., we assume that the Lame constants depend on k in the form of Eq. (19). 

Band gap energy Applied force density Lame constant Carrier generation rate Planck s constant, h/2ir Heat flux vector Radiation intensity Current density Boltzman Constant MOS transistor channel length Mass... 

In some texts, the Lame constants, A and fi, are used. These constants are equal to c,2 and c, respectively. In some cases, it is convenient to write Hooke s Law in a form specific for isotropic materials. For example, using the Lame constants. 

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