dy
dr
¼ cos
π
2
¼ 0
As a result, we can write
du
dr
dv
dr
2
6
4
3
7
5 ¼
E xx E xy
E yx E yy
! 1
0
!
¼
E xx
E yx
!
Thus, for AB
E xx ¼
du
dr
, E yx ¼
dv
dr
It is important to point out that these equations are only valid for small
deformations.
Similarly, for AD we can write
Assuming γ 2 is very small:
dx
dr
¼ cos
π
2
¼ 0,
dy
dr
¼ cos 0 ¼ 1
du
dr
dv
dr
2
6
4
3
7
5 ¼
E xx E xy
E yx E yy
! 0
1
!
¼
E xy
E yy
!
du
dr
¼ E xy
dv
dr
¼ E yy
2.7.2.2 Small Strain in Local Coordinates
Displacement of corner point C and C
0 is plotted in detail Figs. 2.22, 2.23, and 2.24.
Strain along local axis r can be written as
E rr ¼
E xx dx þ γ xy dy
À
Á
cos θ þ E yy dy sin θ
dr
where dr is the initial length of AC in local coordinates,
36
2 Stress and Strain in Continuum
dr
¼ cos
π
2
¼ 0
As a result, we can write
du
dr
dv
dr
2
6
4
3
7
5 ¼
E xx E xy
E yx E yy
! 1
0
!
¼
E xx
E yx
!
Thus, for AB
E xx ¼
du
dr
, E yx ¼
dv
dr
It is important to point out that these equations are only valid for small
deformations.
Similarly, for AD we can write
Assuming γ 2 is very small:
dx
dr
¼ cos
π
2
¼ 0,
dy
dr
¼ cos 0 ¼ 1
du
dr
dv
dr
2
6
4
3
7
5 ¼
E xx E xy
E yx E yy
! 0
1
!
¼
E xy
E yy
!
du
dr
¼ E xy
dv
dr
¼ E yy
2.7.2.2 Small Strain in Local Coordinates
Displacement of corner point C and C
0 is plotted in detail Figs. 2.22, 2.23, and 2.24.
Strain along local axis r can be written as
E rr ¼
E xx dx þ γ xy dy
À
Á
cos θ þ E yy dy sin θ
dr
where dr is the initial length of AC in local coordinates,
36
2 Stress and Strain in Continuum