3.4 Components of Strain
59
Hence,
tan β = ∂v/∂ x ≈ β
(3.4)
Similarly,
tan α = ∂u/∂ y ≈ α
(3.5)
Therefore, the change in angle between the two sides AB and AC due to
deformation is:
γ xy = α + β = (∂u/∂ y + ∂v/∂ x)
(3.6)
Similarly, in other two directions, it can be proved that
γ yz = (∂v/∂z + ∂w/∂ y)
(3.7)
γ zx = (∂w/∂ x + ∂u/∂z)
(3.8)
Here, ε x , ε y and ε z denote normal strains in X, Y and Z directions and
γ xy , γ yz and γ zx denote shear strains in XY, YZ and ZX planes.
3.5 Strain Tensor
Just as the state of stress at a point is described by nine-term array, the strain can be
represented tensorially as below:
ε i j =
1
2
∂u i
∂ x j
+
∂u j
∂ x i
(i, j = x, y, z)
(3.9)
The factor 1/2 in Eq. (3.9) is used to represent the strain transformation equations
in indicial notation. The longitudinal strains are obtained when i = j and the shear
strains are obtained when i = j.
Therefore, it can be expressed as
ε xy =
1
2
γ xy , ε yz =
1
2
γ yz and ε zx =
1
2
γ zx
(3.10)
Hence, the corresponding strain tensor is given by