41 8 Appendix A: Stress and strain analysis
Now
so
By convention
E,’ = decrease in lengthhit length in x’ direction
%’ = decrease in lengthbit length in y‘ direction
= increase in angle between x’ and y’ axes.
Thus we need expressions for
aut aV aut aV,
ax’ ay, ayf ax‘
-- I , - and -.
From the previous notes,
uf = u COS 8 + zr sin 8
and
V’ = v cos 8 + u sin 8.
Now, let f = any function of x and y. Then
-__- af - af ax + -- af ?Y
ax’ ax ax, ay ax’
af af
ax aY
= -cos 8 + -sin8
and
-3f af
ax aY
= - sin8 + -cos 8.