Geoscience Reference
In-Depth Information
x
δ
x
O L M
Strain
u
O L M
x
+
u
δ
x+
δ
When a body is subjected to stresses, the resulting deformations are called
strains
. Strain is
defined as the relative change (i.e., the fractional change) in the shape of the body. First,
consider a stress that acts in the
x
direction only on an elastic string (Fig. A2.2). The point L
on the string moves a distance
u
to point L
after stretching, and point M moves a distance
u
Figure A2.2.
u
to point M
. The strain in the
x
direction, termed
e
xx
,isthen given by
+
change in length of LM
original length of LM
e
xx
=
L
M
−
LM
=
LM
x
+
u
−
x
=
x
u
x
=
(A2.1)
In the limit when
x
→
0, the strain at L is
∂
u
∂
x
(A2.2)
e
xx
=
To extend the analysis to two dimensions
x
and
y
,wemust consider the deformation
undergone by a rectangle in the
x
−
y
plane (Fig. A2.3).
y
Points L, M and N move to L
,M
and N
with coordinates
N'
L
=
(
x
+
u
,
y
+
v)
L
=
(
x
,
y
)
,
x
x
N
+
∂
u
∂
v
M'
M
=
δ
2
M
=
(
x
+
d
x
,
y
)
,
+
x
+
u
x
x
,
y
+
v
+
x
∂
∂
x
y
δ
y
+
∂
u
∂
v
L
'
δ
N
=
1
N
=
(
x
,
y
+
d
y
)
+
u
y
y
,
y
+
y
+
v
+
y
∂
∂
L
δ
x
M
x
The strain in the
x
direction,
e
xx
,isgiven by
e
xx
=
change in length of LM
original length of LM
Figure A2.3.
x
+
∂
u
∂
x
x
−
x
=
x
∂
u
=
(A2.3)
∂
x
Likewise, the strain in the
y
direction is
change in length of LN
original length of LN
e
yy
=
∂
v
∂
y
(A2.4)
=
These are called the
normal strains
, the fractional changes in length along the
x
and
y
axes.
For three dimensions,
e
zz
=
∂
z
is the third normal strain.
As well as changing size, the rectangle undergoes a change in shape (Fig. A2.3). The right
angle NLM is reduced by an amount
w
/∂
1
+
2
called the
angle of shear
,where
1
+
2
=
∂
v
x
+
∂
u
(A2.5)
∂
∂
y
(We assume that products of
x
and so on are small enough to be ignored, which is
the basis of the
theory of infinitesimal strain
.) The quantity which measures the change in
∂
u
/∂
x
,
∂
v
/∂
