Elastic Green Function

The dynamical elastic and inelastic scattering ofhigh-energy electrons by solids may be described by three fundamental equations [5]. The first equation determines the wave amplitude G ( r, r, E), or the Green function, at point r due to a point source of electrons at r in the averaged potential (V (r)) ... 

From the standpoint of the continuum simulation of processes in the mechanics of materials, modeling ultimately boils down to the solution of boundary value problems. What this means in particular is the search for solutions of the equations of continuum dynamics in conjunction with some constitutive model and boundary conditions of relevance to the problem at hand. In this section after setting down some of the key theoretical tools used in continuum modeling, we set ourselves the task of striking a balance between the analytic and numerical tools that have been set forth for solving boundary value problems. In particular, we will examine Green function techniques in the setting of linear elasticity as well as the use of the finite element method as the basis for numerical solutions. 

As noted above, the elastic Green function yields the elastic fields induced by a point load. The ideas here resemble those from the setting of electrodynamics where point sources of charge are the basis of the Green function approach. Recall from earlier that the equilibrium equations that govern the continuum are given by... 

To construct the elastic Green function for an isotropic linear elastic solid, we make a special choice of the body force field, namely, f(r) = fo5(r), where fo is a constant vector. In this case, we will denote the displacement field as Gik r) with the interpretation that this is the component of displacement in the case in which there is a unit force in the k direction, fo = e. To be more precise about the problem of interest, we now seek the solution to this problem in an infinite body in which it is presumed that the elastic constants are uniform. In light of these definitions and comments, the equilibrium equation for the Green function may be written as... 

This result will serve as the cornerstone for many of our developments in the elastic theory of dislocations. We are similarly now in a position to obtain the Green function in real space which, when coupled with the reciprocal theorem, will allow for the determination of the displacement fields associated with various defects. 

Eqn (2.92) is the culmination of our efforts to compute the displacements due to an arbitrary distribution of body forces. Although this result will be of paramount importance in coming chapters, it is also important to acknowledge its limitations. First, we have assumed that the medium of interest is isotropic. Further refinements are necessary to recast this result in a form that is appropriate for anisotropic elastic solids. A detailed accounting of the anisotropic results is spelled out in Bacon et al. (1979). The second key limitation of our result is the fact that it was founded upon the assumption that the body of interest is infinite in extent. On the other hand, there are a variety of problems in which we will be interested in the presence of defects near surfaces and for which the half-space Green function will be needed. Yet another problem with our analysis is the assumption that the elastic constants... 

In mathematical terms, the implementation of the eigenstrain concept is carried out by representing the geometrical state of interest by some distribution of stress-free strains which can be mapped onto an equivalent set of body forces, the solution for which can be obtained using the relevant Green function. To illustrate the problem with a concrete example, we follow Eshelby in thinking of an elastic inclusion. The key point is that this inclusion is subject to some internal strain e, such as arises from a structural transformation, and for which there is no associated stress. 

The elastic Green function is a useful tool in the analysis of a variety of problems in the elasticity of defects in solids. In this problem, we consider the various steps in the derivation of the elastic Green function for an infinite, isotropic solid. In particular, flesh out the details involved in the derivation of eqn (2.90) and carry out the Fourier inversion to obtain eqn (2.91). 

Within the context of the elastic Green function, the reciprocal theorem serves as a jumping off point for the construction of fundamental solutions to a number of different problems. For example, we will first show how the reciprocal theorem may be used to construct the solution for an arbitrary dislocation loop via consideration of a distribution of point forces. Later, the fundamental dislocation solution will be bootstrapped to construct solutions associated with the problem of a cracked solid. 

The basic idea is to exploit the fact that the displacements associated with the point force are already known, and correspond to the elastic Green function. Further, the displacements associated with the dislocation are prescribed on the slip plane. We can rewrite the surface integral that spans the slipped region as... 

Mathematical Theory of Dislocations and Fracture by R. W. Lardner, University of Toronto Press, Toronto Canada, 1974. This book treats a variety of interesting problems in dislocation theory without shying away from mathematically sophisticated treatments using the elastic Green function. A reference that I return to repeatedly. 

Edge dislocation fields using the elastic Green function... 

As already remarked, once the body force distribution has been specified, by invoking the elastic Green function, the displacement fields are immediately obtained as the integral... 

To compute the displacements and strains associated with an inclusion explicitly, we must now carry out the operations implied by eqn (10.7). To make concrete analytic progress, we now specialize the analysis to the case of an elastically isotropic solid. As a preliminary to the determination of the elastic fields, it is necessary to differentiate the elastic Green function which was featured earlier as... 

Using the concept of the elastic Green function, the displacement field in eq (4) for any array of point forces relative to its... 

With the use of the stationary elastic Green function, the mechanical displacement in the eguivalent elastic composite solid in the presence of another field u is then... 

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