Engineering Rock Mechanics

41 2 Appendix A: Stress and strain analysis

P moves to P’, Q moves to Q. The vector P’Q’ may have a different
magnitude and direction to the vector PQ. Is it possible for us to determine
P‘Q knowing PQ and the general form deformation in this body takes?
Providing we make some assumptions, yes.
Assume that displacement varies with position in the body-it is a
function of x and y. Then say that:

y + dy + v + dv)

U

dT r/ P (x + u, y + v) Q (x + dx, y + dy)

P(X.Y) D

function describing displacements in the x-direction = u(x, y)

function describing displacements in the y-direction = v(x, y).

([x + dx] + [u + du], [Y + dyl + [v + dvl).

initial X X initial Y Y

co-ordinate displacement co-ordinate displacement

Both u and v are functions of x and y, and so calculating the derivatives is

awkward: the functions are surfaces, not curves, and each derivative

contains components due to dx and dy. We can calculate du (and similarly

dv) like this:

Gradient

au

=- ax

Change in

u as

x changes

=au,

ax

Change in

u as

y changes

=-dy au

aY

UA Gradient

X -

x x+dx Y Y+dY y

(a) Constant y (b) Constant x