Engineering Rock Mechanics

76 Strain

ir

(x + u,, y + UY' z + u.)

+ dx + ux*.

+ dy + uy*,

+ dz + uz*

2

Figure 5.7 Change in co-ordinates as a line PQ is strained to P*Qx

z + Si), is strained to Q* which has co-ordinates (x + 6x + u,+, y + @ + uY*,

z + & + u,+). If we now consider holding P in a constant position and Q

being strained to Q*, the normal and shear components of strain can be

isolated.

The infinitesimal longitudinal strain is now considered in the x-

direction. Because strain is 'normalized displacement' (see Section 5.1), if

it is assumed that u, is a function of x only, as in Fig. 5.8, then

E,, = duddx and hence dux = &,,dx.

Considering similar deformations in the y- and z-directions, the normal

components of the strain matrix can be generated as also shown in Fig. 5.8.

Derivation of the expressions for the shear strains follows a similar

course, except that instead of assuming that simple shear occurs parallel to

one of the co-ordinate axes, the assumption is made initially that the shear

strain (expressed as a change in angle) is equally distributed between both

co-ordinate axes, i.e. du = du, if dx = dy. This is graphically illustrated in

Fig. 5.9.

-dU

Exx = -x dx

Figure 5.8 Infinitesimal longitudinal strain.

-Exx 0 0

0 -Eyy 0

0 0 -Ezz

::J dz