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\chapter{The strength of the lithosphere}
Constitutive equations specify the relations between stress and strain or the time
derivatives of strain and thus can be used in conjunction with the equations of motion to
relate stresses and displacements. A number of idealized classes of constitutive relations
are recognized. The principal material behaviours are {\it elastic}, {\it plastic} and
{\it viscous}. It should be noted that the constitutive relations appropriate to these
behaviours are idealized and many materials show more complicated stress-strain relations.
For example, a special class of behaviour of relevance to geology is {\it
visco-elasticity} (and visco-plasticity).
\section{A rheological primer}
In simple, isotropic, elastic materials stress and strain are linearly related. An
important aspect of elasticity is that all strain is recovered on relaxation of the
stress. Plastic material shows an elastic response up to a critical stress, termed the
yield stress, at which the material fails. For perfectly plastic behaviour the material
cannot support stresses greater than the yield stress. In viscous materials stresses and
strain rates are related. Thus no deformation is recovered with the relaxation of
stresses. The general form of the constitutive relation for viscous materials is :
\begin{equation} \label{eq:viscosity} \tau = k \, \dot{\gamma}^{\frac{1}{n}}
\end{equation} where $k$ and $n$ are constants, and $\dot{\gamma} $ is the shear strain
rate. For $n$ = 1 the relationship between stress and strain rate is linear and the
material is said to be Newtonian with the viscosity equal to $k$. Materials with $n > 1$
are termed {\it power-law} or {\it shear thinning} fluids. For high values of $n$ the
behaviour of the material may approximate plasticity in as much as a dramatic change in
the rate of deformation occurs with the relatively small increases in shear stress.
\begin{figure} \label{fig:step} \vspace{1.9in} \special{epsf=:gdfigs:chap3:Fig3_1.eps
scale=0.800 } \caption{\protect\scriptsize{Elastic, viscous and visco-elastic response to
a step shear strain applied at time t=0.} }\end{figure}
Visco-elastic materials show time dependent behaviour which is like elastic materials at
short time-scales and like viscous materials at longer time-scales as illustrated in the
response to the application of a step-like shear strain in Figure \ref{fig:step} and in
a creep test in Figure \ref{fig:creep}.
\begin{figure}
\label{fig:creep} \vspace{1.9in} \special{epsf=:gdfigs:chap3:Fig3_2.eps
scale=0.800 } \caption{\protect\scriptsize{Elastic, viscous and visco-elastic response to
a creep test.}} \end{figure}
\section{Background to lithospheric rheology}
The distribution of seismicity in the ocean basins suggests that the ocean lithosphere
deforms primarily by the rigid body translation and rotation of large "plates"; as we
shall see these motions are sustained by the push of the ocean ridges and the pull of
the subducting slabs. The lack of internal deformation of the ocean lithosphere reflects
its strength; it must be strong enough to sustain the stresses arising from these
fundamental driving forces (that is, the ocean lithosphere acts as a stress guide). The
behaviour of the ocean lithosphere contrasts with many parts of the continents wherein
intense, diffuse, internal deformation is manifested by the wide distribution of seismic
events. The existence of diffuse deformation within the continental interiors implies
that the continental lithosphere is, in general, considerably weaker than the oceanic
lithosphere (or that the magnitude of the stresses sustaining deformation of the
continents is larger than for oceans). In addition, differences in the behaviour of the
oceanic and continental lithosphere are governed by differences in their buoyancy. Old
oceanic lithosphere is negatively buoyant allowing eventual subduction and the effective
recycling of the oceanic lithosphere in the convective mantle; one consequence is that
there is no oceanic lithosphere older than about 200 Ma on the surface of the modern
Earth. In contrast the continental lithosphere is always positively buoyant and cannot be
subducted to any significant extent; consequently we have fragments of the continental
lithosphere dating back to about 3.9 Ga.
In later chapters we consider the deformation of the continental lithosphere in both
compression (which results in the development of mountain belts) and in tension (which
results in the development of rifts and the attendant sedimentary basins and flanking
uplifted margins). We will not be so much concerned with the internal geometry of
deformation (which is very much the realm of structural geology) but the mechanical and
thermal consequences of the deformation on the lithospheric scale. Our aim is to develop an understanding of the
controls on some simple and regular, yet, as we shall see, fundamentally important
features of continental orogenic belts, for example, the {\em first-order} controls on
elevation. In order to understand the behaviour of the continents during deformation we
begin with some simple notions concerning the strength or rheology of the continental
lithosphere.
\begin{figure} \label{fig:rthree} \vspace{2.2in} \hspace{1.7in}
\special{epsf=:gdfigs:chap3:Fig3_3.eps scale=0.600 } \caption{ \protect\scriptsize{ The
stress needed to deform the lithosphere and the failure mode depend upon the depth, the
strain rate, the thermal structure as well as the mineralogical composition of the
deforming rocks. At shallow depths (low confining pressures and low temperatures)
failure occurs in the brittle mode, at deeper levels (high confining pressures and high
temperatures) deformation occurs in a ductile fashion. The strength of rocks is strongly
dependent on the temperature and the strain rate. Curve 1 is appropriate to low strain
rates or high geotherms, while curve 2 is appropriate to lower geotherms or higher strain
rates. The brittle-ductile transition can be viewed as the depth at which failure mode
switches, and coincides approximately with the base of the seismogenic zone. } } \end{figure}
\section{A model lithosphere}
The above discussion implies that the lithosphere has finite strength! That is rocks are
able to sustain a finite deviatoric stress (or stress difference) by elastic deformation,
all of which is recoverable on the relaxation of the stress. At stress in excess of the
critical deviatoric stress rocks will undergo a permanent deformation by processes such
as brittle fracture, dislocation creep, pressure solution, the actual deformation
mechanism depending on the temperature, strain rate and composition of the rock. The
finite elastic strength of the lithosphere implies the general rheological response is
plastic, and since the response of the lithosphere can be shown in many cases to be time
dependent \marginpar{{\scriptsize For example, in flexural basins}}the lithosphere
exhibits a complex form of visco-plasticity. The complete description of the rheological
behaviour of such a lithosphere represents a formidable challenge. However it is possible
to achieve the general form of a visco-plastic response by specifying a lithospheric
rheology governed by a combination of two simple failure mechanisms, as described below.
\begin{figure} \label{fig:rfour} \vspace{2.2in} \hspace{0.7in}
\special{epsf=:gdfigs:chap3:Fig3_4.eps scale=0.600 } \caption{\protect\scriptsize{The
strength of the lithosphere is given by the area under the $\tau$ vs depth curves. The
strength is a function primarily of the vertical compositional structure of the
lithosphere, the geotherm and the strain rate. (a) is appropriate to a low geotherm or
high strain rate, i.e. strong lithosphere, where the strongest part of the lithosphere is
the brittle upper mantle. (b) is appropriate to a high geotherm or low strain rate (i.e.,
a weaker lithosphere) where there is no brittle mantle. Note that the proportion of the
total strength of the lithosphere concentrated in the crust (especially near the
brittle-ductile transition) increases with increasing geotherm. }} \end{figure}
Active seismicity in the continents is, by and large, restricted to depths less than about
15 kms. Since seismic energy represents the release of elastic energy at failure along a
discrete fault plane, the lower limit of seismicity can be thought of as the
brittle-ductile transition. The constitutive law describing failure in the brittle mode
by a frictional sliding mechanism is frequently called Byrelee's law: \begin{equation}
\label{eq:friction1} \tau = c_{o} + \mu\, \sigma_{n} \end{equation} where
$c_{o}$ is the cohesion, $\tau$ is the shear stress required for failure ($\tau =
\left(\sigma_1 - \sigma_3 \right)/2$), $\sigma_{n}$ is the normal stress on the failure
plane and $\mu$ is the coefficient of friction. Since the normal stress acting across the
failure plane increases with depth the stress needed to cause brittle failure also
increases with depth (as a linear function of depth for the case where the density is
constant with depth).
The constitutive laws describing deformation in the ductile regime have
a power-law form:
\begin{equation} \label{eq:creep1} \frac{\left(\sigma_1 - \sigma_3 \right)}{G} =
\left(\frac{\dot\gamma}{A}\right)^{\frac{1}{n}} \: exp \left(\frac{\left(Q +
P\,V^{*}\right)}{R\,T\,n}\right) \end{equation} where $G$ is the shear modulus (units of
Pa), $A$ is a material constant with units s$^{-1}$, $\dot\gamma$ is the shear strain
rate, $Q$ is a material constant known as the activation energy of units J mol$^{-1}$,
$V^{*}$ is the activation volume in m$^{3}$ mol$^{-3}$, and $n$ is a dimensionless
material constant known as the power law exponent. The exponential term
in Eqn \ref{eq:creep1} expresses the inverse exponential dependence of strength on
temperature which obeys an Arhenius relationship. The PV* term is usually small compared to Q and
thus Eqn \ref{eq:creep1} can be approximated by: \begin{equation} \label{eq:creep2}
\frac{\left(\sigma_1 - \sigma_3 \right)}{G} =
\left(\frac{\dot\gamma}{A}\right)^{\frac{1}{n}} \: exp\left(\frac{Q}{R\,T\,n}\right) \end{equation}
Equations \ref{eq:creep1} and \ref{eq:creep2} show that the strength of rocks to creep
decreases exponentially with increasing temperature and increases with the strain rate.
Thus for a given composition, strength must decrease with depth.
In terms of this rheology the brittle-ductile
transition can therefore be understood as the depth where the stress required for failure
is equal for both ductile creep and brittle failure (Figure \ref{fig:rthree}).
The material constants for creep vary significantly with the mineralogical makeup of the
rock. For example, quartz-rich rocks are much weaker than olivine-bearing rocks. Thus
the strength of the lithosphere can be determined only for a specific compositional
structure (as well as thermal structure). To a first approximation we can consider that
the continental lithosphere comprises two layers: a quartz-rich crustal layer and an
olivine-rich mantle layer. For any arbitrary geotherm the strength of such a hypothetical
lithosphere depends only on the material constants appropriate to quartz and olivine
(Figure \ref{fig:rfour}). Permanent deformation of the whole lithosphere at given shear strain
rate, $\dot\gamma$, will only occur when the force applied to the lithosphere exceeds the
strength of the lithosphere, $F_l$, given by (Figure \ref{fig:rfive}b): \begin{equation}
\label{eq:strength} F_l \, = \, \int_{0}^{z_{l}}Ê\left(\sigma_1 - \sigma_3 \right) \, dz \end{equation}
For high strain rates, low geotherms or low applied forces much of the stress in the
lithosphere will be supported at least in part elastically (Figure \ref{fig:rfive}a).
Because of the dependence on strain rate
the behaviour of the lithosphere is time dependent. On short time scales the lithosphere
is able to support forces in an elastic fashion; on longer timescales it will relieve the
applied forces by viscous, permanent deformation. The rheology of the lithosphere may
therefore be considered as visco-plastic.
\begin{figure}[htb] \label{fig:rfive} \vspace{2.2in} \hspace{0.7in}
\special{epsf=:gdfigs:chap3:Fig3_5.eps scale=0.600 } \caption{ \protect\scriptsize{ The
distribution of stress within the lithosphere subject to a horizontal force (in
compression or tension) depends on the magnitude of the force, the strain rate and the
geotherm (for a given composition). An applied force, $F_{d}$, will give instantaneously
a homogeneous stress distribution, $\sigma_{i}$, throughout the lithosphere such that
$\sigma_{i}\, z_{l} = F_{d}$ In those regions where $\sigma_{i}$ exceeds the critical
stress difference for permanent deformation the stresses will be relaxed, with resultant
amplification of the stress in the stronger parts of the lithosphere. Permanent
deformation of the whole lithosphere can only occur when the whole lithosphere fails (b),
that is when the stress concentration everywhere exceeds the stress difference required
for permanent deformation. For lower applied forces the stresses will be supported at
least in part elastically (a). }} \end{figure}
\section{Uncertainties}
The simple model for lithospheric rheology outlined in the previous section is highly
uncertain for a number of reasons, and consequently must be used with some caution.
Firstly, the lithosphere comprises polymineralic aggregates and the extent to which the
whole of the crust can be described by {\em quartz-failure} and the mantle by {\em
olivine-failure} is extremely dubious. The strength-depth curve for more
realistic compositional models of the lithosphere is likely to be much more complex than shown in models shown in Figs
\ref{fig:rfour} \& \ref{fig:rfive}. Secondly, the material constants are determined in
laboratory experiments carried out at strain rates appropriate to human activity
($10^{-5}$--$10^{-8} {\rm s}^{-1}$), as opposed to geological strain rates
($10^{-13}$--$10^{-16} {\rm s}^{-1}$). This is particularly important because only small
changes in the value of the material constants have a large effect on the calculated
strength. Finally, the model excludes a number of deformation mechanisms, such as pressure
solution which are known to be operative at least under some conditions.
Such uncertainties render futile the calculation of the absolute strength of the
lithosphere using this simplistic model. For example, a 10\% uncertainty in the activation
energy for creep corresponds to an uncertainty in the strength of the lithosphere of about
a factor of 2. However, much more significance can be attached to estimates of the
changes in strength accompanying changes in the physical state of the lithosphere using
this model, because at least in this case uncertainties in the material constants cancel.
For example, a change in the thermal regime of the lithosphere corresponding to a
change in the Moho temperature of 100$^{\rm o}$C causes a change in the strength of about
a factor of 2 (Fig. \ref{rsix}).
\begin{figure}[htb] \label{rsix} \vspace{3.2in} \hspace{0.7in}
\special{epsf=:gdfigs:chap3:Fig3_6.eps scale=0.600 } \caption{\protect\scriptsize{
Illustration of the dependence of lithospheric strength calculated using our simple
rheological model on thermal state of the lithosphere, as reflected by the Moho
temperature. }} \end{figure}