3.9 Change in Length of a Linear Element—Linear Components
65
γ xy =
∂u
∂ y
+
∂v
∂ x
, γ yz =
∂v
∂z
+
∂w
∂ y
, γ zx =
∂w
∂ x
+
∂u
∂z
(3.21b)
and ε P Q ∼ = ε P Q = ε x l
2
+ ε y m
2
+ ε z n
2
+ γ xy lm + γ yz mn + γ zx nl
(3.22)
If, however, the line element is parallel to X-axis, then l = 1, m = 0, n = 0 and
the linear strain is
ε P Q = ε x =
∂u
∂ x
Similarly, for element parallel to Y-axis, then l = 0, m = 1, n = 0 and the linear
strain is
ε P Q = ε y =
∂v
∂ y
and for element parallel to z-axis, then l = 0, m = 0, n = 1 and the linear strain is
ε P Q = ε z =
∂w
∂z
The relations expressed by Eqs. (3.21a) and (3.21b) are known as the strain–
displacement relations of Cauchy.
3.10 Strain Transformation
If the displacement components u, v and w at a point are represented in terms of
known functions of x, y and z, respectively, in Cartesian co-ordinates, then the six
strain components can be determined by using the strain–displacement relations
given below.
ε x =
∂u
∂ x
,
ε y =
∂v
∂ y
,
ε z =
∂w
∂z
γ xy =
∂u
∂ y
+
∂v
∂ x
,
γ yz =
∂v
∂z
+
∂w
∂ y
and γ zx =
∂w
∂ x
+
∂u
∂z
If at the same point, the strain components with reference to another set of coordinates axes x
, y
and z
are desired, then they can be calculated using the concepts
of axis transformation and the corresponding direction cosines. It is to be noted
that the above equations are valid for any system of orthogonal co-ordinate axes
irrespective of their orientations.
Hence,