Dislocation in an anisotropic continuum

3.4 Dislocation in an anisotropic continuum

Steeds [72] developed a treatment to derive the dislocation energy within the anisotropic elasticity. According to this treatment, the dislocation is considered straight and extended to infinity along the z-axis. This assumption simplifies the problem to a plane strain problem where no quantity depends on the z-coordinate, so

∂ ---= 0. ∂z

(3.17)

The Burgers vector of the dislocation is b. Displacements are given by the functions u_x, u_y and u_z. The displacements u_x and u_y correspond to the edge component of the considered dislocation whereas u_z corresponds to the screw component. The strain components are

	ε_xx = ,ε_xz = ,
	ε_zz = 0,ε_xy = ,	(3.18)
	ε_yy = ,ε_yz = .

The previous equations relate the six strain components to the three components of the lattice displacement and this implies that relations exist between the various ε_ij. These relations are known as the compatibility equations [45]:

	+ = 2,	(3.19a)
	- = 0.	(3.19b)

Hooke’s law for an anisotropic material is written as:

( ) ( ) ( ) εxx s11 s12 s13 s15 s15 s16 σxx || εyy || || s21 s22 s23 s24 s25 s26|| || σyy|| | εzz | | s31 s32 s33 s34 s35 s36| | σzz| || || = || || || || . | 2εyz| | s41 s42 s43 s44 s45 s46| | σyz| ( 2εxz) ( s51 s52 s53 s54 s55 s56) ( σxz) 2εxy s61 s62 s63 s64 s65 s66 σxy

(3.20)

The infinite straight dislocation lies along the z-axis and therefore ε_zz = 0. This last relation yields an additional condition:

	0 = ε_zz = s₂₁σ_xx + s₂₂σ_zz + s₂₃σ_yy + s₂₄σ_yz + s₂₅σ_xy + s₂₆σ_xz,
	σ_zz = ∕s₂₂.	(3.21)

Steeds [72] expressed the stresses as a function of the Airy functions, F and ϕ

	σ_xx = ,σ_xz = -,
	σ_xy = -,σ_yz = -,
	σ_yy = .	(3.22)

Substituting the expressions (3.21) and (3.22) into the compatibility equations (3.19) yields

	+ s₁₂ + s₂₁+
	+
	s₆₆ + s₁₅ + s₂₅ + s₆₄ =

	+
	s₁₆ + s₂₆ + s₆₁ + s₆₂+
	s₁₄ + s₂₄ + s₆₅,	(3.23a)

	s₁₁ + s₁₂ + s₂₁ + s₂₂+
	+
	s₆₆ + s₁₅ + s₂₅ + s₆₄ =
	+
	s₁₆ + +
	s₂₆ + s₆₁ + s₆₂ + s₁₄ + s₂₄ + s₆₅.	(3.23b)

Solutions for the previous equations with cylindrical symmetry have the following form [72]:

F = Bg

,ϕ = Cf

(3.24)

where g and f are functions of a linear combination of the coordinates x and y. Substituting (3.24) into (3.23) and eliminating B and C yields a sextic equation for the parameter p:

	p⁶-
	2p⁵ +
	p⁴-
	2p³ +
	p²-
	2p +
	S₂₂S₄₄ - S₂₄² = 0	(3.25)

where S_lm = s_lm - ((s3ls3m)∕s33)

. Roots of this equation occur as pairs of complex conjugates p₁ and p₁^*, p₂ and p₂^*, and p₃ and p₃^*. Therefore the most general solution for the functions F and ϕ are

F = ∑ _n=1³ ,		(3.26a)
ϕ = ∑ _n=1³ ,		(3.26b)

where n = 1, 2, 3, ζ_n = x + p_ny, and g_n and f_n denote, respectively, the functions g and f belonging to the root p_n. In order to satisfy the physical expectations that the stress field decays as 1/r, it is required that

(3.27)

Substituting (3.26) and (3.27) into (3.22) causes the stress components to become

	σ_xx = ∑ _n=1³,
	σ_xy = -∑ _n=1³,
	σ_yy = ∑ _n=1³,
	σ_xz = ∑ _n=1³,
	σ_yz = -∑ _n=1³,	(3.28)

The displacement components are obtained combining (3.18) and (3.26):

u_x =	∫ ε_xxdx =
∑ _n=1³
	,	(3.29a)

u_y =	∫ ε_yydz =
∑ _n=1³
	,	(3.29b)

u_z =	∫ ε_xzdx =
∑ _n=1³
	.	(3.29c)

In order to evaluate the quantities B_n and C_n and their complex conjugates, six relationships are required.

The first set of boundary conditions is derived from the force equilibrium state of the media, i.e., the condition of zero net force on the dislocation [72], expressed as

_S ∑ _jσ_ijn_jdS = 0,

(3.30)

for an arbitrary cylindrical surface S enclosing the dislocation line. n_j denotes components of the outer normal to the integration surface S. For an infinite straight dislocation along the z-axis, the previous equation becomes

_S ∑ _jσ_ijn_jdS = 0, (j = 1,2 )

(3.31)

It is assumed that the cylinder S has a height

with a circular base

on the xy-plane. Based on these assumptions, the previous equation yields for i = x

0 =

dS =

∫ _Cd

(3.32)

and when substituting the expression for F from (3.26) and using (3.27) one gets [72]

[ ]
∑3 * * *
(Bnpn ln ζn + Bnp nlnζn)
n=1

= 0.

(3.33)

The start and the end point of the closed integration loop

differ by the argument Δθ_n = 2π. Since the logarithm of a complex argument can be written as

ln ζ_n = ln r_n + iθ_n,		(3.34a)
ln ζ_n^* = ln r_n^*- iθ_n,		(3.34b)

equation (3.33) becomes

∑ _n=1³

[Bnpn (lnrn + i2π - lnr ) + B *np*n (ln r*n - i2π - ln r*n)]

= 0.

(3.35)

Simplifying, one gets

∑ _n=1³

= 0.

(3.36)

Analogously, another two conditions are obtained from equation (3.33) for i = y

∑ _n=1³

= 0,

(3.37)

and for i = z

∑ _n=1³

= 0.

(3.38)

The second set of boundary equations is provided by the displacement relations. The integral of the displacement acquisitions along the Burgers circuit encircling the dislocation line must be equal to the Burgers vector b:

∮ du = b.

(3.39)

Substituting equations (3.29) for the displacements components into the last relation yields a set of three equations of the form b_i∕2πi = Q_niB_n - Q_ni^*B_n^*. More explicitly the last three equations become

b_x =	2πi∑ _n=1³,	(3.40a)
b_y =	2πi∑ _n=1³
	,	(3.40b)
b_z =	2πi∑ _n=1³.	(3.40c)

The stress field of the dislocation is fully described by equations (3.28) where constants p_n are the roots of equation (3.25) and constants B_n are the solution of the system of linear equations (3.36), (3.37), (3.38) and (3.40).

The calculation of the dislocation energy is performed by substituting the stress components in equation (3.15) along the cut surface described by the coordinate r such that x = r sin φ and y = r cos φ. The components of the outer normal n are n_x = cos φ, n_z = 0 and n_y = sin φ. The inner and outer cut-off radii are r_c and R, respectively. Then one obtains

	= ∫ _{r_c}^R ∑ _n=1³
	+ =
	= ∑ _n=1³ ln =
	= K ln ,	(3.41)

where K is the so called pre-logarithmic coefficient of the dislocation energy. K depends on the elastic constants, the Burgers vector b and the particular direction of the dislocation line with respect to the crystallographic axes.

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