Lame’s elastic constants

Apart from E, G, and u, the so-called Lame s elastic constants A and /u are sometimes used. Their relation to the other elastic constants is... 

Lame s elastic constant Poisson s ratio density notch radius dislocation density normal stress... 

Equation (3) is the equation of equilibrium of the porous medium. In this equation, it is assumed that the medium is non-linearly elastic, and G (Pa) and A (Pa) are Lame s constants of elasticity and P is the coefficient of volumetric thermal expansion of the solid matrix. G and A, and also A d the bulk modulus can also be expressed as functions of the... 

The coefficients A, /r are called Lame s constants. Then the response of the linear elastic solid, called the Hookean solid, is written as... 

Elastic deformation in isotropic materials is fully characterized in terms of two elastic constants, such as elastic modulus E and Poisson s ratio V, or in terms of the Lame constants, yu (the shear modulus) and A. For an isotropic material, the various elastic constants are related by... 

Other forms of the generalized Hooke s law can be found in many texts. The relation between various material constants for linear elastic materials are shown below in Table 2.1 where E, G and v are previously defined and where K is the bulk modulus and X is known as Lame s constant. 

The two Lame constants occurring in Equations (22) through (26) are one possible choiee of elastic constants which can be used in the case of isotropie materials. Depending on the application in question, other elastic constants can be more advantageous, e.g. the tensile modulus (Yoimg s modulus) E (imits [GPa]), the shear modulus G (imits [GPa]), the bulk modulus K (imits [GPa]) and the Poisson ratio V (dimensionless). Some of these constants are preferable from the practical point of view, since they can be relatively easily determined by standard test procedures E and G ), while others are preferable from the theoretical point of view, e.g. for micromechanical calculations (G and K). Note, however, that even in the case of isotropic materials always two of these elastic constants are needed to determine the elastic behavior completely. 

The various moduli and other parameters that can be formed out of Lame s constants in elastic theory may be generalized to the viscoelastic case by simply substituting fi (co), A (co) for //, A, and, if one wishes to return to the time representation, inverting the Fourier transform. An important example is v(o ), given by... 

The terminologies of Young s modulus, Poisson s ratio, and Lame s constants are still used here, however, it by no means implies that the woiic is in the scope of linear elasticity. On the other hand, when the strains involved are small, the solutions will reduce to the linear elastic ones. 

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. 

Consider the following boundary value problem express the displacement vector field U in some domain V in terms of the values of U and of its normal derivative d J/dn on the inner side of the surface S bounding this domain. The constant elastic parameters of a homogeneous medium, Cp and Cj, are assumed to be known. The external volume forces F are distributed within some domain D, which is located inside V D C V), so the field U in Z satisfies the Lame 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. 

The two fundamental constants used in elasticity theory are the Lame constants A and jut, and the commonly used material parameters can be expressed in terms of these two constants. However, it is convenient to introduce into engineering practice a third materials constant, Poisson s ratio, V, which strictly is valid for simple stress fields where there are no shear components and only a single main tensile stress. The second Lame constant is then equivalent to the shear modulus G which then becomes related to the Young s modulus by the expression ... 

While in the Kuster and Toksoz model randomly distributed cracks are assumed and an isotropic effect results, Hudson s concept results in an anisotropy effect caused by the oriented fractures. For a single crack set, the first correction terms are given in Table 6.17. Please note that in Eq. (6.109), the correction term is added, but Table 6.17 shows that the correction term is negative—thus, elastic properties decrease with fracturing, where As> fis are Lame constants of the solid host material (background material) the crack density is... 

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