Section 1.2
Solid Mechanics Part II
Kelly
9
1.2 The StrainDisplacement Relations
The strain was introduced in Part I: §3.6.
Expressions which relate the displacements of
material particles to the strains for a continuously varying strain field are derived in what
follows.
1.2.1
The StrainDisplacement Relations
Normal Strain
Consider a line element of length
x
Δ
emanating from position
)
,
(
y
x
and lying in the
x

direction, denoted by
AB
in Fig. 1.2.1.
After deformation the line element occupies
B
A
′
′
, having undergone a translation, extension and rotation.
Figure 1.2.1: deformation of a line element
The particle that was originally at
x
has undergone a displacement
)
,
(
y
x
u
x
and the other
end of the line element has undergone a displacement
)
,
(
y
x
x
u
x
Δ
+
.
By the definition of
(small) normal strain,
x
y
x
u
y
x
x
u
AB
AB
B
A
x
x
xx
Δ
−
Δ
+
=
−
′
=
)
,
(
)
,
(
*
ε
(1.2.1)
In the limit
0
→
Δ
x
one has
x
u
x
xx
∂
∂
=
ε
(1.2.2)
This partial derivative is a
displacement gradient
, a measure of how rapid the
displacement changes through the material, and is the strain
at
)
,
(
y
x
.
Physically, it
represents the (approximate) unit change in length of a line element, as indicated in Fig.
1.2.2.
x
•
•
A
•
•
B
A
′
B
′
)
,
(
y
x
u
x
)
,
(
y
x
x
u
x
Δ
+
x
x
x
Δ
+
y
*
B
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