Page 52 - Rock Mechanics For Underground Mining

STRESS AND INFINITESIMAL STRAIN

strain by

⎡ ⎤
ε xx − /3 xy zx
[ ] = ⎣ xy ε yy − /3 yz ⎦
ε zz − /3
zx yz
Plane geometric problems, subject to biaxial strain in the xy plane, for example,
are described in terms of three strain components, ε xx , ε yy , xy .

2.9 Strain compatibility equations

Equations 2.35 and 2.36, which define the components of strain at a point, suggest
that the strains are mutually independent. The requirement of physical continuity of
the displacement field throughout a continuous body leads automatically to analytical
relations between the displacement gradients, restricting the degree of independence
of strains. A set of six identities can be established readily from equations 2.35 and
2.36. Three of these identities are of the form
2 2 2
∂ ε xx ∂ ε yy ∂ xy
+ =
∂y 2 ∂x 2 ∂x∂y
and three are of the form
2
∂ ε xx ∂ ∂ yz ∂ zx ∂ xy
2 = − + +
∂y∂z ∂x ∂x ∂y ∂z
These expressions play a basic role in the development of analytical solutions to
problems in deformable body mechanics.

2.10 Stress-strain relations

It was noted previously that an admissible solution to any problem in solid mechanics
must satisfy both the differential equations of static equilibrium and the equations of
strain compatibility. It will be recalled that in the development of analytical descrip-
tions for the states of stress and strain at a point in a body, there was no reference
to, nor exploitation of, any mechanical property of the solid. The way in which
stress and strain are related in a material under load is described qualitatively by its
constitutive behaviour. A variety of idealised constitutive models has been formu-
lated for various engineering materials, which describe both the time-independent and
time-dependent responses of the material to applied load. These models describe re-
sponses in terms of elasticity, plasticity, viscosity and creep, and combinations of these
modes. For any constitutive model, stress and strain, or some derived quantities, such
as stress and strain rates, are related through a set of constitutive equations. Elasticity
represents the most common constitutive behaviour of engineering materials, includ-
ing many rocks, and it forms a useful basis for the description of more complex
behaviour.
In formulating constitutive equations, it is useful to construct column vectors
from the elements of the stress and strain matrices, i.e. stress and strain vectors
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