This document discusses time-dependent behavior in rocks that occurs over a wide range of strain rates in rock mechanics and engineering applications. It introduces the topic, explaining there are 15 orders of magnitude between high strain rates like explosions and low strain rates like gradual deformation over decades. The document then covers dynamic rock properties, stress waves, time-dependent concepts like creep and relaxation, and rheological models. It discusses the relevance for rock engineering, including concerns over extrapolating short-term test data to designs that must last 1,000 years.

1. Rock dynamics and
timedependent aspects
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2. Hello. Good morning!

3. • In this chapter we will be discussing a variety
of effects related to the different strain rates
that occur throughout the range of rock
mechanics processes and rock engineering
applications.
• Following the introduction, highlighting the
wide time ranges over which these effects are
manifested, we discuss the basic theory of
rock dynamics, obtaining dynamic rock
properties and the relevance of the ideas in
engineering.

4. Introduction
• In Chapter 6, it was noted that the compressive strength is the
maximum stress that can be sustained by a specimen of rock.
Let us now say that the compressive strength is reached at 0.1%
strain, i.e. 0.001. If this strain is developed in 1 p-for example,
during an explosion-the strain rate is 1 x lo3 s-’. If, on the other
hand, this strain is developed over a period of 30 years, the
average strain rate is of the order of 1 x 10-l’ s-’.
• Between these two extremes, there are 15 orders of magnitude
of strain rate, and so, if the rock exhibits any time-dependent
behaviour, we would not expect to be able to use the same rock
properties for an analysis of both cases. In Fig. 13.1, we illustrate
two manifestations of these extremes of strain rate.
• Fig. 13.l(a) shows hackle marks that develop on rock fracture
surfaces formed during high strain rate failure, in this case on
the surface of a blast-induced fracture. Fig. 13.1@) shows the
effect of the gradual deterioration, and subsequent failure, of
the pillars in an old mine in chalk beneath a main road.

5. Stress waves
• Stress waves are the manifestation of dynamic
stress changes.
• They occur when the body is not in static
equilibrium as described so far, and are essentially
sound waves in solid material.
• The differential equations of equilibrium,
represent the fact that, for any given axis, the
resultant force on a body is zero when the body is
in equilibrium.

6. • Considering now that an infinitesimal cube of
material is accelerating, and applying Newton’s
Second Law of Motion, these equations become
the differential equations of motion:

10. Time-dependency
• We noted that no time component is incorporated in
elasticity theory: it is assumed that the stresses and strains
develop instantaneously on loading or unloading. However,
we noted in Chapter 6 when discussing the complete stress-
strain curve, that the exact form of the curve will depend
on the strain rate at which it is determined. It is commonly
observed at rock engineering sites, that the rock continues
to deform after a stress change occurs-e.g. convergence of
well bores and tunnels. So, it is evident that, whilst the
theory of elasticity is of assistance in understanding and
analysing the mechanics of rock masses, a theory is also
required for timedependent effects.
• Words used to describe time-dependent behaviour are
clarified in the glossary below.

11. Glossury of Terms
• Elastic
• Stresses are related to strains in a time-
independent manner (i.e. o = SE, where S is
the elastic compliance matrix). All strain
energy is recoverable. It is assumed, in this
context, that elastic materials remain elastic
and so have infinite strength.

12. • Plastic
• Stresses are related to strains in a time-
independent manner, but the material
undergoes plasticflow when stressed (i.e. do =
I'd&, where P is a 6 x 6 plasticity matrix whose
coefficients are stress- or strain-dependent).
• Deformation continues indefinitely without any
further increase in stress. Strain energy is lost
through permanent plastic straining.
• Generally, plastic behaviour is a function of
distortional strains and deviatoric stresses.

13. • Viscous
• Stresses are related to strain rate (i.e. o = vi,
where 17 is a 6 x 6 viscosity matrix). Generally,
viscous behaviour is also a function of distortional
strains and deviatoric stresses.
• Elastoplasticity
• Time-independent theory combining elasticity
and plasticity: materials behave elastically up to
certain stress states and plastically thereafter.

14. • Viscoelasticity
• A generic term for a time-dependent theory in which
strains are related to stresses and time.
Instantaneously, viscoelastic materials have
effectively infinite strength.
• Viscoplasticity
• Time-dependent behaviour in which the deviatoric
stresses (or distortional strains) give rise to viscous
behaviour, or plastic behaviour if the instantaneous
strength of the material is temporarily exceeded.
• Elastoviscoplasticity
• This is the same as viscoplasticity, except that the
instantaneous response of the material is purely
elastic.

15. • Creep
• Under the action of a constant stress state,
straining continues (see Fig. 6.16).
• Relaxation
• Under the action of a constant strain state, the
stress within a material reduces (also see Fig.
6.16).
• Fatigue
• A generic term generally used to describe the
increase in strain (or decrease in strength) due to
cyclical loading.
• Rheology
• The study of flow.

16. • Rheological models
• These are analogues of different material
behaviour, formed from assemblages of
mechanical components, usually springs,
dashpots and sliders. They assist in
understanding the material behaviour and
allow the formulation of the various
constitutive relations.
• Using just three rheological elements-spring
(or Hookean substance),

17. • The Kelvin model consists of viscous and elastic
elements in parallel.
• Consequently, the strain is identical in each of
the elements and the stress developed in the
material, os, is the sum of the stresses
developed in the elastic and viscous elements,
oE and o,, respectively,

18. • Again, considering a period of constant stress
followed by constant strain,
• which, on rearranging, becomes:

19. • Integration and substitution of C = (l/E)log,g
(because at t = 0, E = 0) yields

20. • The strain that has accumulated at t = fl is
thus

22. • However, for constant strain, d&ldt=O, the
basic differential equation
• reduces to 0 = EE, with the result that

23. Time-dependency in rock engineering
• Engineers have found it convenient to consider
phenomena as either associated with very high strain
rates or very low strain rates. This is because the
process of rock excavation (e.g. by blasting) occurs
rapidly, whereas deformation (e.g. displacement
occurring throughout the life of an excavation) occurs
slowly. In the high strain rate category we include
blasting, vibrations and fatigue; in the low strain rate
category we include creep, subsidence and long-term
displacements. We noted that the strain rates can be
spread over 15 orders of magnitude, with the result
that it is debatable whether any generic time-
dependent model can be valid over such a large
range.

24. • In the newer applications of rock engineering,
such as radioactive waste disposal, the specified
design lives can be large, of the order of 1000
years. Thus, not only is there concern with the
time-dependent behaviour but we have to
consider whether all the rock properties and
mechanisms can be considered to be uniform
over such an extreme time period. This is
exacerbated by the fact that we can only conduct
testing procedures in the range of medium to
high strain rates.

25. • If the rock properties are determined by
geophysical means, at very high strain rates,
we should ask ourselves how valid is it to
apply these to engineering applications of, say,
a billion times greater duration than the test
period? This question has profound
implications for the validity of theoretical
models, test results and the interpretation of
field measurements.
• We are led to the conclusion that engineering
judgement must still play a large part in
determining the type of time-dependent
analysis that is used.

26. Thank you very much