Physics 927
E.Y.Tsymbal
1
Section 4: Elastic Properties
Elastic constants
Elastic properties of solids are determined by interatomic forces acting on atoms when they are
displaced from the equilibrium positions. At small deformations these forces are proportional to the
displacements of atoms. As an example, consider a 1D solid. A typical binding curve has a minimum
at the equilibrium interatomic distance R
0
:
Expanding the energy at the minimum in the Taylor series we find:
0
0
2
2
0
0
0
2
(
)
(
)
(
)
...
R
R
U
U
U R
U
R
R
R
R
R
R
∂
∂
=
+
−
+
−
+
∂
∂
(4.1)
At equilibrium
0
0
R
U
R
∂
=
∂
, so that
2
0
1
(
)
2
U R
U
ku
=
+
,
(4.2)
where we defined
0
2
2
1
2
R
U
k
R
∂
=
∂
and
0
u
R
R
=
−
is the displacement of an atom from equilibrium
position
R
0
. Differentiating Eq.(4.2),
U
F
R
∂
=
−
∂
, we obtain force
F
acting on an atom :
F
ku
=
−
.
(4.3)
The constant
k
is an interatomic force constant. Eq.(4.3) represents the simplest expression for the
Hooke’s law
showing that the force acting on an atom,
F
, is proportional to the displacement
u
. This
law is valid only for small displacements and characterizes a
linear region
in which the restoring force
is linear with respect to the displacement of atoms.
The elastic properties are described by considering a crystal as a homogeneous continuum medium
rather than a periodic array of atoms. In a general case the problem is formulated as follows:
(i) Apply forces, which are described in terms of
stress
σ
, and determine displacements of atoms
which are described in terms of
strain
ε
.
(ii) Define elastic constants
C
relating stress
and strain
, so that
=
C
.
U
R
R
0
U
0
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentPhysics 927
E.Y.Tsymbal
2
Example
: In 1D case,
F
ku
=
−
, where
u
is a change in the crystal length under applied force
F
. We can
therefore write
F
kL
u
C
A
A
L
σ
ε
−
=
=
=
,
(4.4)
where
A
is the area of the cross section, and
L
is the equilibrium length of the 1D crystal. The stress
is defined as the force per unit area and the strain
is the dimensionless constant which describes the
relative displacement (deformation).
In a general case of a 3D crystal the stress and the strain are tensors which are defined as follows.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '11
 STAFF
 Force, Elastic Properties, elastic constants

Click to edit the document details