Lame’s equations

These are the fundamental equations for the design of thick cylinders and are often referred to as Lame s equations, as they were first derived by Lame and Clapeyron (1833). The constants A and B are determined from the boundary conditions for the particular loading condition. 

In this approach, the horizontal coordinates x[t] and y[t] both satisfy linear, second order equations. The exact trajectories satisfy Lame s equation of order two, which is extremely difficult to analyse. The corresponding equation for the linear approximation is Mathieu s equation, which is known to have both periodic and aperiodic solutions. 

To overcome the limitations in the Lame theory when applying to orthotropic composite repair systems, a new modified version of Lame s equations (MVLE) was developed by the authors to undertake the analysis of infinitely long orthotropic cyhnders subjected to internal and/or external pressure. The hoop and radial stresses at any point in the wall cross section of an orthotropic cylinder at radius r are given by the following equations ... 

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 l..jime hoop-siress equation indicates that the maximum stress occurs at the famer surface of the vessel. By shrink-lilting concentric shells together the inner shells are placed in residual compression so that the initial cfimpressive h(M>p stress must be relieved by the internal pressure before hoop tensile stresses are developed. Therefore the maximum hoop tensile stress as determined by Lame s relaUon-ship is apprecial>ly reduced with the result that there is a reduction of the total wall thickness retfuired l,o cnnf,ain the pressure when the vessel-wall thickness is designed with a speci/ied allowable stress. 

Equation 8.1 is shown in Fig. 5.6 as being very similar to Lame s Eq. 5.9 for thick vessels. Disregarding external pressures, s. 5.9 and 5.10 become... 

The transformation into other moduli (for example compressional modulus, Lame s numbers) follows the equations given in Table 6.1. 

Nicaise S. (1992) About the Lame system in a polygonal or polyhedral domain and coupled problem between the Lame system and the plate equation. 1. Regularity of the solutions. Ann. Scuola Norm. Super. Pisa, Serie IV 19, 327-361. 

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

Reversible bimolecular reactions such asA + B C + D can be solved exactly by the method of separation of variables and the ordinary differential equations in the variable s are Lame equations. This makes the evaluation of the Fourier-type coefficients very difficult since derivative formulas and orthogonality conditions do not seem to exist or at least are not easily used. In addition to this, even if such formulas did exist, it seems unlikely that numerical results could be easily obtained. It does turn out, however, that these reversible bimolecular processes can be solved exactly and conveniently in the equilibrium limit, and this was done by Darvey, Ninham, and Staff.14... 

Now let us consider the Green s tensor for the system of equations of dynamic elasticity theory, the vector form of which is called a Lame equation. We will call this tensor an elastic oscillation tensor G or Green s tensor for the Lame equation. As in the case of the vector wave equation, discussed above, the components of the elastic oscillation tensor describe the propagation of elastic waves generated by a point pulse force. In other words, it satisfies the following Lame equation (see equation (13.29)) ... 

Using Green s tensor G (r, t) for the Lame equation, we can express the solution of this equation for an arbitrary right-hand side F (r, t) as the convolution of the Green s tensor G (r,<) with the function F (r, t), i.e.,... 

Note that both representations (13.96) and (13.97) hold true for a medium with variable Lame parameters (Aki and Richards, 2002). However, in a general case one should use the corresponding Green s tensors for the Lam6 equation with variable coefficients. 

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... 

This equation allows one to estimate the compression of the UPD adlayer at AE = 0. For many Me UPD systems, the excess binding energy A Mcads-S Meads-Me is in the order of lO J (corresponding to about 10 J mole ), so that eq. (3.35) predicts a substantial compression of several percent since the 3D Lame coefficients are usually... 

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. 

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