E rr ¼ E xx
dx
dr
þ γ xy
dy
dr
cos θ þ E yy
dy
dr
sin θ
Knowing that cos θ ¼
dx
dr , sin θ ¼
dy
dr
Inserting these equations in ε rr yields
E rr ¼ E xx cos
2
θ þ γ xy sin θ cos θ þ E yy sin
2
θ
Using trigonometric relations
cos
2
θ ¼
1
2
1 þ cos 2θ
ð
Þ
sin
2
θ ¼
1
2
1 À cos 2θ
ð
Þ
2 sin θ cos θ ¼ sin 2θ
Strain along r axis can also be expressed as
E rr ¼
E xx þ E yy
2
þ
E xx À E yy
2
cos 2θ þ
γ xy
2
sin 2θ
We could also obtain components of displacement CC
0 in r and s local coordinates by using dot product of vector CC
0 and unit vectors along r and s (Fig. 2.25).
Material coordinate system unit vectors can be represented in terms of spatial
coordinates’ unit vectors as follows:
l ¼ i cos θ þ j sin θ
m ¼ Ài sin θ þ j cos θ
The displacement in r and s direction can be found by dot product:
CC
0
Á l ¼ E xx dx þ γ xy dy
À
Á
i þ E y dyj
Â
Ã
icosθ þ jsinθ
½
Š
θ
θ
θ
Fig. 2.25 Unit vectors in
local (material) r-s
coordinates and spatial
coordinates x,y
38
2 Stress and Strain in Continuum