Dislocations displacement field

This problem is a bit nasty, but will illustrate the way in which the Volterra formula can be used to derive dislocation displacement fields, (a) Imitate the derivation we performed for the screw dislocation, this time to obtain the displacement fields for an edge dislocation. Use the Volterra formula (as we did for the screw dislocation) to obtain these displacements. Make sure you are careful to explain the logic of your various steps. In particular, make sure at the outset that you make a decent sketch that explains what surface integral you will perform and why. 

The behavior of an edge dislocation is more complicated since its displacement field produces both shear and normal stresses. The solution consists of the superposition of two terms, each of which behave relativistically with limiting velocities corresponding to the speed of transverse shear waves and longitudinal waves, respectively [2, 4, 5]. The relative magnitudes of these terms depend upon v. 

The significance of this result will be appreciated later (for example in section 8.4.2) when we see that it can be used for the construction of fundamental solutions for the elastic displacement fields of defects such as dislocations. 

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. 

A dislocated crystal has a distinct geometric character from one that is dislocation-free. From both the atomistic and continuum perspectives, the boundary between slipped and unslipped parts of the crystal has a unique signature. Whether we choose to view the material from the detailed atomic-level perspective of the crystal lattice or the macroscopic perspective offered by smeared out displacement fields, this geometric signature is evidenced by the presence of the so-called Burgers vector. After the passage of a lattice dislocation, atoms across the slip plane assume new partnerships. Atoms which were formerly across from... 

For many purposes, we will find that antiplane shear problems in which there is only one nonzero component of the displacement field are the most mathematically transparent. In the context of dislocations, this leads us to first undertake an analysis of the straight screw dislocation in which the slip direction is parallel to the dislocation line itself. In particular, we consider a dislocation along the X3-direction (i.e. = (001)) characterized by a displacement field Usixi, X2). The Burgers vector is of the form b = (0, 0, b). Our present aim is to deduce the equilibrium fields associated with such a dislocation which we seek by recourse to the Navier equations. For the situation of interest here, the Navier equations given in eqn (2.55) simplify to the Laplace equation (V ms = 0) in the unknown three-component of displacement. Our statement of equilibrium is supplemented by the boundary condition that for xi > 0, the jump in the displacement field be equal to the Burgers vector (i.e. Usixi, O" ") — M3(xi, 0 ) = b). Our notation usixi, 0+) means that the field M3 is to be evaluated just above the slip plane (i.e. X2 = e). 

Our initial foray into the elastic theory of dislocations has revealed much about both the structure and energetics of dislocations. From the continuum standpoint, the determination of the displacement fields (in this case uf) is equivalent to solving for the structure of the dislocation. We have determined a generic feature of such fields, namely, the presence of long-range strains that decay as r. Another... 

In this case f )(+) refers to the traction associated with the dislocation just above the slip plane while t (—) refers to the value of the traction just below the slip plane. However, since the displacement fields (u ) associated with the point force are continuous across the slip plane, Similarly, the tractions... 

Our current ambition is to elucidate one of the microscopic mechanisms that has been charged with giving rise to solution and precipitate hardening. The argument is that by virtue of the elastic fields induced by an obstacle there will be a force on a dislocation which the dislocation must overcome in its motion through the crystal. As a first step towards modeling this phenomenon, we imagine the obstacle to be a spherical disturbance within the material. As was already demonstrated in chap. 7, such an obstacle produces spherically symmetric displacement fields of the form Ur = Ar + bjr. ... 

Recall from our review of continuum mechanics that the divergence of the displacement field is a measure of the volume change associated with a given deformation. In this case, it is the volume change associated with the dislocation fields at the obstacle that gives rise to the interaction energy. 

Figure 5.4. Simplified representation of an edge dislocation, (a) Drawing a path around the core of the dislocation helps visualize the Burger vector of that dislocation, (b) Representation of the displacement field of the atoms surrounding the dislocation...

Figure 5.4b shows the displacement field associated with the dislocation described in Figure 5.4a. The presence of a dislocation causes atomic displacements which are too significant to be overlooked, even for atoms located relatively far away. Thus, in a crystal containing a certain dislocation density, the positions of the atoms are altered by displacements 8 that correspond to the sum of elementary displacements u associated with each of the dislocation.

It is convenient to separate the displacement field u(r) into a smoothly varying part U(,(r) and a singular part Up(r) due to dislocations,... 

The displacement field of an edge-dislocation in the SmA phase has been calculated by De Gennes in the framework of the elastic continuum theory... 

We usually derive the displacement field of a screw dislocation because it is much easier than deriving the edge dislocation. 

The displacement field for any dislocation falls off as r as we move away from the dislocation. 

It is also interesting to analyze the displacement fields for the two types of dislocations. For the screw dislocation, the displacement field on the xy plane is given by... 

Figure 10.6. The and Ux components of the displacement fields for the screw and edge dislocations, in units of the Burgers vectors. The values of the x and y variables are also scaled by the Burgers vectors. For negative values of x in the case of the screw dislocation, or of y in the case of the edge dislocation, the curves are reversed they become their mirror images with respect to the vertical axis. For these examples we used a typical value of V = 0.25.

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