Lame’ equations

Sandig A.M., Richter U., Sandig R. (1989) The regularity of boundary value problem for the Lame equations in a polygonal domain. Rostock. Math. Kolloq. 36, 21-50. 

For straight metal pipe under internal pressure the formula for minimum reqiiired w thickness is applicable for D /t ratios greater than 6. Tme more conservative Barlow and Lame equations may also be used. Equation (10-92) includes a factor Y varying with material and temperature to account for the redistribution of circumferential stress which occurs under steady-state creep at high temperature and permits slightly lesser thickness at this range. 

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... 

The calculation of the strength resp. the admissible internal pressure varies with the wall-thickness thick-walled hollow cylinders are calculated by neglecting the radial stress (equal to the pressure) which is small compared to the tangential. On the other side the thick-walled hollow cylinders are calculated with the Lame equations (1833). 

Since Eq. (10) has been solved analytically for a symmetric double layer system [4,13] we shall restrict ourselves to this case. The solution, Eq. (21), can then be inserted in Eq. (71) which leads to a homogeneous Lame equation for which linearly independent solutions have been derived [14] ... 

The vector form of the equations of motion (13.26) is called the Lame equation. The constants Cp and Cs have clear physical meaning. We will see below that equation (13.26) characterizes the propagation of two types of so called body waves in an elastic medium, compressional and shear waves, while the constants Cp and c are the velocities of those waves respectively. We will call them Lame velocities. 

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.,... 

Generalized Kirchhoff integral formulae for the Lame equation and the vector wave equation... 

Now, following Zhdanov (1988) and Zhdanov et al. (1988), we generalize the Kirchhoff formula for vector fields characterizing arbitrary elastic oscillations of the medium, that is for fields satisfying the Lame equation (13.29) or the vector wave equation (13.31). 

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... 

In particular, assuming that the domain D with the external forces is located outside the volume V, we arrive at the generalized Kirchhoff integral formula for the Lame equation (Zhdanov, 1988) ... 

Indeed, let us consider again the model shown in Figure 13-4b. We assume that the elastic field U(r,f) satisfies the Lame equation (13.201) everywhere in full space, while the sources of the wavefield are located inside the domain CV, the complement of domain V for the full space. We can write the integral representation (13.120) of the elastic wavefield within a domain Dr, bounded by a sphere Or of radius r as ... 

Thus, we have the unique solution of the Lame equation (13.201) for the infinite elastic space, provided by integral formulae (13.207) and (13.206) ... 

Consider a 3-D elastic medium. The propagation of elastic waves in the frequency domain can be described by the Lame equation (13.34)... 

Operator Al denotes the nonlinear forward modeling operator, given by the Lame equation (15.230) and radiation conditions (13.202) - (13.204). This operator can be calculated, for example, from the general integral representation of the elastic field in the frequency domain (13.97), which we write here in the form... 

Frechet derivative for the elastic forward modeling operator Let us assume that the Lame velocities are known and fixed everywhere in space with the exception of some local domain D. We can find the equations for the Frechet derivative by applying the variational operator to both sides of the corresponding Lame equation (15.230)... 

Theoretical Model of Modification of Polymer Membrane Surfaces in After-Glow of Oxygen-Containing Plasma of Non-Polymerizing Gases Lame Equation... 

To describe the influence of non-equilibrium plasma-treatment and cross-links on the permeability and selectivity of the gas-separating polymer membranes, let us first calculate the energy required for formation in an unbounded elastic medium of a hollow sphere with radius R corresponding to the radius of a penetrating molecule. Solving the Lame equation (9-90) in the absence of external forces (/ = 0) and for radial displacement Mr = R, the compression energy of the elastic medium can be found in this case as (Landau Lifshitz, 1986)... 

The equations shown on the following worksheet for determining shell thicknesses, are all based on Lame equations. Reviewing the equations for various shell stresses show that they are in complete agreement with the ASME equations and yield the exact same results. The following equations serve to illustrate this point ... 

The ratio of fi/pi may be conveniently plotted again.sl K as shown by Maccary ahd Fey (194) and as iridi< ated in Fig. 14.5. The determination of shell thickness using the Lame equation involves calculation by successive approximation. The same calculation using the membrane e(jua-tion is more convenient, being a direct calculation, but is limited in its application to v sels in whicli l/d, is equal to or le than 0.10. 

For a given internal pressure p and as long as the elastic limit of the material is not exceeded at any point in the wall, the elastic stress may be represented as three main stresses at right angles to each other. These are given by the LAME equations as functions of radius and the ratio of the external to the internal diameters [6, 7, 8]. 

In the central part of the fits, the pressures agreed closely with predictions obtained by the Lame equations. At the edges, however, stress concentrations were observed. 

On substituting these relations into Eq. (14), the stress equilibrium equations can be replaced by a pair of second-order differential equations, the Lame equations ... 

Numerical solutions are also possible and in view of the complexity of the analytical solutions often desirable. One method is to replace the Lame equations (16) by a set of difference equations for points on an array over an r-z section of the cylinder. The boundary conditions are then used to remove undefined points at the boundaries. The solution is obtained by iteration (23) or by solving the equations directly by a matrix technique (24). The other common method is the use of finite elements, which have been applied widely to axisymmetric elastic and thermoelastic problems (23). This technique breaks the r-z section of the cylinder into regions or elements where the properties and conditions can be assumed to be approximately uniform. At the junctions of the elements, the nodes, displacements, and forces are defined. These displacements and forces are connected, using the elastic and thermal properties of the material, by minimizing the energy of the system. A set of linear equations in terms of the displacements is then obtained by matching the nodal forces and displacements from element to element, together with the boundary conditions. The set of linear equations is then solved in the same way as for the finite difference approximation. 

These in turn substituted into the stress equilibrium, Eq. (36), produce the Lame equation including nonelastic strains ... 

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