5.2 Grundbegriffe

5.2.1 Elastizitätstheorie der Versetzungen

General Remarks

Elastizitätstheorie ist schwierig und mit Versetzungen noch schwieriger, da der Versetzungskern nicht mit Elastizitätstheorie behandelt werden kann. Nur im unendlich-großen Kristall können einfache Lösungen bestimmt werden, die aber zu unendlich großen Energien führen.
Die Ergebnisse für alle praktischen Zwecke sind aber trotzdem ganz einfach. Insbesondere ergeben sich einfache Formeln für die Linienenergie einer Versetzung und für Kräfte auf Versetzungen.
Wir bedienen uns hier der Darstellung in: Introductions to Dislocations, D. Hull and D. J. Bacon; 3rd Edition (Int. Series on Mat. Science and Technology, Vol. 37) Pergamon Press; Seiten 71 ff.
Erinnert sei auch an die Einführung in die Materialwissenschaft I von früher.
The atoms in a crystal containing a dislocation are displaced from their perfect lattice sites, and the resulting distortion produces a stress field in the crystal around the dislocation. If we look at the picture of the edge dislocation, we see that the region above the inserted half-plane is in compression - the distance between the atoms is smaller then in equilibrium; the region below the half-plane is in tension.
The dislocation is therefore a source of internal stress in the crystal. In all regions of the crystal except right at the dislocation core, the stress is small enough to be treated by conventional (linear) elasticity theory. Moreover, it is generally sufficient to use isotropic theory, simplifying things even more.
If we know the elastic field, i.e. the relative displacement of all atoms, we can calculate the force a dislocation exerts on other dislocations, or, more generally, any interaction with elastic fields from other defects or from external forces. We also can then calculate the energy contained in the elastic field produced by a dislocation.

Basics of Elasticity Theory

The first element of elasticity theory is to define the displacement field u(x,y,z), where u s a vector that defines the displacement of atoms or, since we essentially consider a continuum, any point in a strained body from its position in the unstrained state by the elastic deformation.

Displacement of P to P' by displacement vector u
The displacement vector u is then given by
u=[ux, uy, uz]
The components ux , uy , uz represent projections of u on the x, y, z axes, as shown above.
The vector field u, however contains not only rigid body translations, but at some point (x,y,z) all the summed up displacements from the other parts of the body. If, for example, a long rod is just elongated along the x-axis, the resulting u field would be
ux=const · x.
Since we are only interested in the local deformation, we resort to the strain e, defined by the nine components of the strain tensor.
Applied to our case, the nine components of the strain tensor are directly given in terms of the first derivatives of the displacement components; we obtain the normal strains
exx = (dux)/(dx),
eyy = (duy)/(dy),
ezz = (duz)/(dz),
and the shear strains
eyz = ezy = 1/2(du/dz + du/dy),
ezx = exz = 1/2(du/dx + du/dz),
exy = eyx = 1/2(du/dy + du/dx)
Within our basic assumption of linear theory, the magnitude of these components is 1. They represent the fractional change in length of elements parallel to the x, y and z axes respectively. The physical meaning of the normal- and shear strains is shown in the following illustration
A small area element ABCD in the xy plane has been strained to the shape AB'C'D' without change of area. The angle between the sides AB and AD initially parallel to x and y respectively has decreased by 2exy. By rotating, but not deforming, the element as in (b), it is seen that the element has undergone a simple shear. The simple shear strain often used in engineering practice is 2exy, as indicated.
The volume V of a small volume element is changed by strain to (V+DV)=V(1+exx ) (1+eyy) (1+ezz). The fractional change in volume D, known as the dilatation, is therefore
D = DV/V = (exx + eyy + ezz)
and D is independent of the orientation of the axes x, y, z.

Elasticity theory links the strain experienced in a volume element to the forces acting on this element. The forces act on the surface of the element and are expressed as stress, i.e. as force per area. Stress is propagated in a solid because each volume element acts on its neighbors.
A complete description of the stresses acting therefore requires not only specification of the magnitude and direction of the force but also of the orientation of the surface, for as the orientation changes so, in general, does the force.
Consequently, nine components must be defined to specify the state of stress. This is shown in the illustration below.

Volume element with the component of the stress. Since on any surface an arbitrary force vector can be applied, it must be decomposed into 3 vectors at right angels to each other. Since we want to keep the volume element at rest (no translation and no rotation), the sum of all forces and moments must be zero, which leaves us with 6 independent components.
The component sij, where i and j can be x, y or z, is defined as the force per unit area exerted in the +i direction on a face with outward normal in the +j direction by the material outside upon the material inside. For a face with outward normal in the –j direction, i.e. the bottom and back faces shown in Fig. 3(b), sij is the force per unit area exerted in the –i direction. For example, syz acts in the positive y direction on the top face and the negative y direction on the bottom face.
The six components with i j are the shear stresses. (It is customary in dislocation studies to represent the shear stress acting on the slip plane in the slip direction of a crystal by the symbol t.
As mentioned above, by considering moments of forces taken about x, y and z axes placed through the centre of the cube, it can be shown that rotational equilibrium of the element, i.e. net couple=0, requires
 szy = szyszx = sxzsxy = syx.
It therefore does not matter in which order the subscripts are written.
he three remaining components sxx, syy, szz are the normal components. From the definition given above, a positive normal stress results in tension, and a negative one in compression. The effective pressure acting on a volume element is therefore
p = -1/3(sxx + syy + szz)
For some problems, it is more convenient to use cylindrical polar coordinates (r, q, z). The stresses are still defined as above, and are shown below. The notation is easier to follow if the second subscript j is considered as referring to the face of the element having a constant value of the coordinate j.

The relation between stress and strain is taken to be linear, as in most "material laws" (take, e.g. "Ohms law", or the relation between electrical field and polarization); it is called "Hooke´s law".
Each strain component is linearly proportional to each stress; in full generality for anisotropic media we have, e.g.
exx=a11s11 + a22s22 + a33s33 + a12s12 + a13s13 + a23s23.
For isotropic solids, however, only two independent aij remain and Hookes law can be written as
sxx = 2Gexx + l(exx + eyy + ezz)
syy = 2Geyy + l(exx + eyy + ezz)
szz = 2Gezz + l(exx + eyy + ezz)
sxy = 2Gexy,   syz = 2Geyz,   szx = 2Gezx .
The two remaining material parameters l and G are known as Lamé constants, but G is more commonly known as the shear modulus.
It is customary to use different elastic moduli, too. But there are always only two independent constants; if you have more, some may expressed by the other ones.
Most frequently used, and most useful are Young's modulus, E, Poisson's ratio,n, and the bulk modulus, K. These moduli refer to simple deformation experiments:
Under uniaxial, normal loading in the longitudinal direction, Young´s modulus E is the ratio of longitudinal stress to longitudinal strain and Poisson's ratio n is minus the ratio of lateral strain to longitudinal strain. The bulk modulus K is defined to be – p/DV (p=pressure, DV=volume change);. Since only two material parameters are required in Hooke's law, these constants are interrelated by the following equations
E=2G·(1 + n),
n=l /2·(l + G), 
K=E/3·(1 - 2n)
.
Typical values of E and n for metallic and ceramic solids lie in the ranges 40-600 GNm-2 and 0.2 – 0.45 respectively.
A material under strain contains elastic energy - it is just the sum of the energy it takes to move atoms off their equilibrium position at the bottom of the potential well from the binding potential. Since energy is the sum over all displacement time the force needed for the displacement, we have
Strain energy per unit volume=one-half the product of stress times strain for each component. The factor 1/2 comes from counting twice by taking each component. Thus, for an element of volume dV, the elastic strain energy is
dEel=1/2 dV Si=x,y,z Sj=x,y,zsij eij
For polar coordinates we would get a similar formula.

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