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5.2.2 Stress Field of a Straight Dislocation

The elastic distortion around astraight screw dislocation of infinite length can be represented in terms of acylinder of elastic material deformed as defined byVolterra. The followingillustration shows the basic geometry.

A screw dislocationproduces the deformation shown in the left hand picture. This can be modeled bythe Volterra deformation mode as shown in the right hand picture - except forthe core region of the dislocation, the deformation is the same. A radial slitwas cut in the cylinder parallel to thez-axis, and the free surfacesdisplaced rigidly with respect to each other by the distanceb,the magnitude of the Burgers vector of the screw dislocation, in thez-direction.

In the core region the strain is verylarge - atoms are displaced by about a lattice constant. elasticity theory thus is not a validapproximation there, and we must exclude the core region. We then have noproblem in using the Volterra approach; we just have to consider the coreregion separately and add it to the solutions from linear elasticity theory.

The elastic field in the dislocatedcylinder can be found by.First, it is noted that there are nodisplacements in thex andy directions, i.e.u_x =u_y = 0 .

In thez-direction, thedisplacement varies smoothly from0 tob as the angle goes from0 to2. This can be expressed as

u_z	=	b · θ 2π	=	b 2π	· tan^–1(y/x)	=	b 2π	· arctan (y/x)

Using theequations for the strainwe obtain thestrainfield of a dislocation:

ε_xx	=	_yy = ε_zz = ε_xy = ε_yx = 0

ε_xz	=	ε_zx = –	b 4π	·	y x² +y²	= –	b 4π	·	sin θ r

ε_yz	=	ε_zy =	b 4π	·	x x² +y²	=	b 4π	·	cos θ r

The correspondingstress field is alsoeasily obtained from therelevantequations:

σ_xx	=	σ_yy = σ_zz = _xy = σ_yx = 0

σ_xz	=	σ_zx = –	G ·b 2π	·	y x² +y²	= –	G ·b 2π	·	sin θ r

σ_yz	=	σ_zy =	G ·b 2π	·	x x² +y²	=	G ·b 2π	·	cos θ r

Incylindricalcoordinates, which are clearly better matched to the situation, thestress can be expressed via the following relations:

σ_rz	=	σ_xycosθ + σ_yz sinθ

σ_{θ z}	=	– σ_xz sinθ + σ_yz cosθ

Similar relations hold for the strain. We obtain the simple equations:

ε_{θ z}	=	ε_zθ =	b 4πr

σ_{θ z}	=	σ_zθ =	G · b 2πr

The elastic distortion contains notensile or compressive components and consists of pure shear._zacts parallel to thez axis in radial planes of constant and_z acts in the fashion of atorque on planes normal to the axis. The field exhibits complete radialsymmetry and the cut thus can be made on any radial plane = constant. For a dislocation ofsign, i.e. a left-handed screw,the signs of all the field components are .

There is, however, a serious problemwith these equations:

The stresses and strains are proportional to1/r and therefore 0 as shown in theschematic picture on the left.

This makes no sense and therefore the cylinderused for the must be to avoidr - values that aretoo small, i.e. smaller than the core radiusr₀.

Real crystals,of course, do (usually) contain hollow dislocation cores. If wewant to include the dislocation core, we must do this with a more advancedtheory of deformation, which means a non-linear atomistic theory. There are,however, ways to avoid this, provided one is willing to accept a bit ofempirical science.

The picture simply illustrates that strain andstress are, of course, smooth functions ofr. The fact thatlinear elasticity theory can not cope with the core, does not mean that thereis a real problem.

How large is radiusr₀ or the extension of thedislocation core? Since the theory used isonly valid for small strains, we may equate the core region with the regionwere the strain is larger than, say,10%. From the equationsabove it is seen that the strain exceeds about0,1 or10% wheneverrb. A reasonable value for thedislocation core radiusr₀ therefore lies in the rangeb to4b, i.e.r₀ 1 nm in mostcases.

The stress field of an edgedislocation is somewhat more complex than that of a screw dislocation, but canalso be represented in an isotropic cylinder by theVolterra construction.

Using the same methodology as in thecase of a screw dislocation, we replace the edge dislocation by the appropriatecut in a cylinder. The displacement and strains in thez-direction are zero and the deformation is basically a"plane strain".

It is not as easy as in the case of the screwdislocation to write down thestrain field, but thereasoning follows the same line of arguments. We simply look at theresults:

σ_xx	= –	D · y	3x² + y² (x² + y²)²

σ_yy	=	D · y	x² – y² (x² + y²)²

σ_xy	=	σ_yx =	D · x	x² – y² (x² + y²)²

σ_zz	=	ν · (σ_xx + σ_yy)

σ_zz	=	σ_zx = σ_yz = σ_zy = 0

We used the abbreviationD =Gb/2π (1 – ν).

The stress field has, therefore, bothdilational and shear components. The largest normal stress is_xx which acts parallel to the Burgersvector. Since the slip plane can be defined asy = 0, the maximumcompressive stress (_xx isnegative) acts immediately above the slip plane and the maximum tensile stress(_xx is positive) acts immediatelybelow the slip plane.

The effective pressure (given by thesum over the normalcomponents of the stress) is

p	=	2 · (1 + ν) ·D 3	·	y x² + y²

We thus have compressive stress above the slipplane and tensile stresses below - just as deduced from thequalitative picture of anedge dislocation; graphical representation of thestress field of an edgedislocation is shown in the link.

For edge dislocations (and screwdislocations too), the sign of the stress- and strain componentsif the sign of the Burgers vector is reversed.

Again, we have to leave out thedislocation core; the core radius again can be taken to be about1b -4b

We are left with the case of amixed dislocation.This is not a problem anymore. Since we have a linear isotropic theory, we canjust take the solutions for the edge- and screwcomponent of the mixeddislocation and superimpose, i.e. add them.

As far as "simple" elasticity theorygoes, we now have everything we can obtain. If better descriptions are needed,the matter becomes extremely complicated! But thankfully, this simpledescription is sufficient for most applications.

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