2.7.1.2 Pure Shear Strain
The shear strain is defined as the change in initial right angle at point A (Fig. 2.14):
γ XY ¼
π
2
À Ψ ¼ θ 1 þ θ 2
For small angles [less than 0.5 radian]
tan θ 1 ¼ θ 1 and tan θ 2 ¼ θ 2
tan θ 1 ¼ θ 1 ¼
ΔV
ΔX
and tan θ 2 ¼ θ 2 ¼
ΔU
ΔY
As a result, we can write
γ XY ¼ θ 1 þ θ 2 ¼
ΔV
ΔX
þ
ΔU
ΔY
:
Since we define our strain at a point in the limit dimensions of the differential
element, as ΔX and ΔY approach zero. Hence, we can define strain in differential
form as
E X ¼
∂U
∂X
E Y ¼
∂V
∂Y
γ XY ¼
∂U
∂Y
þ
∂V
∂X
where U is the displacement in X axis and V is the displacement in Y axis.
ΔX
ΔY
C
B
A
D
C’
B’
D’
ΔV
ΔU
θ1
θ2
Ψ
Fig. 2.14 Shear strain
definition in two dimensions
30
2 Stress and Strain in Continuum