4 1 4 Questions 5.7 -5.7 0: strain and the theory of elasticity
Q5.6 Explain clearly why an isotropic rock has two independent elastic
constants, a transversely isotropic rock has five independent constants
and an orthotropic rock has nine independent constants (compared to
the general anisotropic case where there are twenty-one independent
elastic constants).
45.7 Each of the following four rock masses is to be modelled using
elasticity theory. State whether you think that an isotropic rock assump-
tion is justified, or whether one of the anisotropic assumptions would be
more appropriate. Assume that the fractures have a significant effect on
the rock deformability.
(a) A limestone with effectively three fracture sets, i.e. the bedding with
mean strata thicknesses of 1 m, plus fracture set 1 (perpendicular to
the bedding) with two fractures/m, and fracture set 2 (perpendicular
to both the bedding and fracture set 1) with five fractures/m.Fracture set 1: dip direction 089"; dip 50"; frequency 2.9/m.
Fracture set 2: dip direction 278"; dip 88"; frequency 1.3/m.
Fracture set 3: dip direction 224"; dip 08"; frequency 0.9/m.
Fracture set 4: dip direction 169"; dip 23"; frequency 2.l/m.
Fracture set 5: dip direction 113"; dip 70"; frequency 0.7/m.(b) A welded volcanic tuff with five fracture sets.(c) A strong sandstone with almost no fractures present.
(d) A granodiorite with three fracture sets.
Fracture set 1: dip direction 314"; dip 35"; frequency 1.2/m.
Fracture set 2: dip direction 048"; dip 43"; frequency 1.3/m.
Fracture set 3: dip direction 089"; dip 79"; frequency 0.9/m.45.8 (a) At the time of writing this book, most elastic analyses that have
been conducted for rock engineering design purposes have assumed that
the rock is perfectly isotropic with two elastic constants. Why do you
suppose that is, given that most rock masses are clearly not isotropic?
(b) Conversely, no elastic analysis for rock mechanics has been con-
ducted assuming that the rock mass is fully anisotropic with 21 elastic
constants? Why is that?
(c) In this context, what do you think will happen in future analyses?Q5.9 (a) How do you know if a material is elastic?
(b) How is time taken into account in the theory of elasticity?Q5.10 When a rock specimen fails in a compressive test in the laborat-
ory or when an in situ rock mass fails due to natural compression (see
Fig. Q5.1), is 'stress' or 'strain' the cause of failure?