124
5 Mechanical Properties
(a)
(b)
Fig. 5.32 Ratio − ⊥ // for GaN under symmetric biaxial stress. In a θ denotes the angle of the c-axis with respect to
the surface normal, b is a three-dimensional visualization, showing the in-plane isotropy
Fig. 5.33 Strains c
(dashed lines) and ⊥
(solid lines) for
Al 0.17 Ga 0.83 N/GaN (blue)
and Mg 0.3 Ga 0.7 O/ZnO
(red) as a function of the
interface tilt angle θ with
respect to [00.1]
and the surrounding matrix. The solution will be given in terms of a surface integral of the boundary
of the inclusion, which is fairly easy to handle. Several disconnected inclusions can be treated by a
sequence of surface integrals.
The strain distribution for the inner and outer parts of a sphere with radius ρ 0 is given (in spherical
coordinates) by
in
ρρ =
2
3
0
1 − 2ν
1 − ν
=
in
θθ =
in
φφ
(5.71)
out
ρρ =
2
3
0
1 + ν
1 − ν
ρ 0
ρ
3
= −2
out
θθ = −2
out
φφ ,
(5.72)
where ρ denotes the radius, ν the Poisson ratio, and 0 the relative lattice mismatch of the inclusion
and the matrix. The radial displacements are
u
in
ρ =
2
3
0
1 − 2ν
1 − ν
ρ
(5.73)
u
out
ρ =
2
3
0
1 − 2ν
1 − ν
ρ
3
0
1
ρ 2 .
(5.74)