Page 58 - Rock Mechanics For Underground Mining

STRESS AND INFINITESIMAL STRAIN

free body is outward, if the outward normal to the surface is directed outward relative
to the co-ordinate origin, and conversely. The sense of positive stress components,
defined in this way, is illustrated in Figures 2.1c and 2.11, for Cartesian and polar
co-ordinate systems. This convention has been followed in this introductory mate-
rial since important notions such as traction retain their conceptual basis, and since
practically significant numerical methods of stress analysis are usually developed
employing it.
States of stress occurring naturally, and generated and sustained in a rock mass by
excavation activity, are pervasively compressive. If the usual engineering mechanics
convention for stresses were followed, all numerical manipulations related to stress
and strain in rock would involve negative quantities. Although this presents no con-
ceptual difficulties, convenience and accuracy in calculations are served by adopting
the following convention for stress and strain analysis in rock mechanics:
(a) positive force and displacement components act in the positive directions of the
co-ordinate axes;
(b) contractile normal strains are taken as positive;
(c) compressive normal stresses are taken as positive;
(d) the sense of positive shear stress on a surface is inward relative to the co-ordinate
origin, if the inward normal to the surface acts inwards relative to the co-ordinate
origin, and conversely.
The senses of positive stress components defined by this convention, for Cartesian
and polar co-ordinate systems, and biaxial and triaxial states of stress, are shown in
Figure 2.12. Some minor changes are required in some of the other general relations
developed earlier, and these are now defined.

2.12.1 Stress-traction relations
If the outward normal to a surface has direction cosines ( x , y , z ), traction compo-
nents are determined by

t x =−( xx x + xy y + zx z ), etc.

2.12.2 Strain-displacement relations
Strain components are determined from displacement components using the expres-
sions

∂u x
ε xx =−
∂x

∂u y ∂u x
xy =− + , etc.
∂x ∂y
2.12.3 Differential equations of equilibrium
The change in the sense of positive stress components yields equations of the form
∂ xx ∂ xy ∂ zx
+ + − X = 0, etc.
∂x ∂y ∂z
All other relations, such as strain compatibility equations, transformation equations
and stress invariants, are unaffected by the change in convention.
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