Engineering Rock Mechanics

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.