Lame moduli

It should be noted that these equations were obtained from Eq. (4.76) by substituting the Lame moduli in terms of the tensile modulus E and the Poisson ratio v, by means of the relations of Table 4.1, as is usual in elasticity theory (see also Problem 4.12). 

The elastic constants (elastic moduli) A, and ju are called Lame constants (or Lame moduli, units [GPa]). Switching over from matrix notation to tensor notation the elasticity tensor is... 

In Table 1 are listed conversions between Lame coefficients, Young s modulus, Poisson s ratio, /, and C,/k/ for an isotropic elastic homogeneous medium. 

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

In this equation is the deviator and a is the spherical part of the stress tensor <7, eij is the strain deviator and e the volumetric part of the strain tensor ij, K = (2M + 3A) /3 is bulk modulus with M and A corresponding to the familiar Lame coefficients in the theory of elasticity, while r) and n can be termed the viscous shear and bulk moduli. 

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. 

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

Let us assume that the z axis corresponds to the principal axis of the rod. In this case, the only non-null component of the strain tensor is When Lame coefficients are expressed in terms of the tensile modulus and Poisson ratio [see Eq. (4.102)], the relationship between the stress and strain tensors is given by... 

The coefficients A and jj, were introduced by Lame and named after him. They are related to elastic parameters of a medium, the Young s modulus E and the Poisson s ratio cr, by the formulae (Love, 1944 Udias, 1999)... 

Shear modulus of a solid (transverse elastic constant or Lame-coefficient) Newtonian viscosity of a fluid Shear stress... 

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 ... 

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... 

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. 

Here p is the Lame constant and equivalent to the shear modulus. Vs is the... 

Again, Poisson s ratio, Lame s constant and the bulk modulus were given earlier, respectively, as ... 

A theoretical treatment can be developed for each of these film types with respect to its properties, involving, in the nonconductive case, the thickness /f, the mass density q, X (the Lame constant). and p (the shear modulus). 

For the special case of the sphere, an analysis based on the classical linear theory of elasticity (Lame formulae) yields the following expression for the incremental modulus E namely , = 3l2)Vd oJdV. This same result is obtained from the expression for on setting a = b, and y = 1. 

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. 

Table 2.1 Relation between various elastic constants. X and G are often termed Lame constants and K is the bulk modulus.

Lame s Modulus, X Shear Modulus, G Young s Modulus, E Poisson s Ratio, V Bulk Modulus, K... 

Since modulus of the outer metallic cylinder is much larger than that of the inner polymer cylinder, it is reasonable to assume the outer shell is rigid. This assumption provides a further simplitication to the problem. In this section, we develop the fully elastic solution based upon the classic Lame solution. Subsequently we will consider the viscoelasticity of the polymer and invoke the correspondence principle to solve the viscoelastic problem. 

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. 

In a similar way as the Young modulus E and the Poisson ratio V are connected to the uniaxial extension test, the shear modulus G and the bulk modulus K are connected to simple shear and isotropic deformation (i.e. dilatation or compression). Note that, accidentally, it turns out that the shear modulus G equals the second Lame constant ju. Since for isotropic materials only two of the elastic constants are independent, the knowledge of any pair of them is sufficient to calculate the other constants and thus to describe the elastic behavior of isotropic materials completely. For easy reference in this chapter we list the most important interrelations between the elastic constants ... 

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