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Originally published as:
Porth, R. (2000): A strain-rate dependent force model of lithospheric strength. - Geophysical
Journal International, 141, 3, 647-660,
DOI: 10.1046/j.1365-246x.2000.00115.x
Geophys. J. Int. (2000) 141, 647^660
A strain-rate-dependent force model of lithospheric strength
R. Porth
GeoForschungsZentrum Potsdam, Section 5.3, Recent Stress Field and Earthquake Hazard, Telegrafenberg, D-14473 Potsdam, Germany.
E-mail: [email protected]
Accepted 2000 January 4. Received 1999 November 22; in original form 1998 February 1
S U M M A RY
This study investigates the rate of intraplate deformation, the vertically integrated stress
magnitude and vertical distributions of tectonic stress in continental lithosphere that is
subjected to horizontal tectonic force. The fundamental assumption of this study is that
the magnitude of the imposed tectonic force depends on the rate of deformation. This
modi¢cation of the often applied strength envelope concept accounts for resistance
forces generated externally to the lithospheric section that the calculated vertical stress
distribution is considered to represent. The model presented overcomes the di¤culties
that arose in previous constant strain rate and constant force approaches and thus
proves to be physically more realistic. Results are discussed with emphasis on the e¡ects
related to the strain-rate-dependent force assumption and on di¡erences from the
results of both previous approaches. The strain-rate-dependent force model predicts
that in and adjacent to a weak and fast-deforming lithospheric plate, the integrated
stress magnitude is signi¢cantly reduced due to externally derived resistance forces.
Modelling results suggest that the rate of intraplate deformation in many cases is controlled by the balance of externally derived deformation-driving and externally derived
deformation-resisting forces, but is relatively insensitive to the low internal strength of
the deforming lithosphere. The integrated stress level in regions of active intraplate
deformation, in contrast, is predicted to be largely controlled by the time-independent
brittle strength. The strain-rate-dependent force model provides a geodynamic concept
that allows one to investigate intraplate stress distribution depending on tectonic force
magnitude, lithospheric strength and rate of intraplate deformation.
Key words: continental deformation, intraplate tectonics, lithosphere strength,
rheology, strain rate, stress distribution.
1
IN T ROD U C T I O N
Calculations of 1-D vertical stress distributions, assuming
symmetrical pure shear compression or extension of the lithosphere, have been widely used during the last two decades to
analyse the mechanical behaviour of continental lithosphere
that is subjected to horizontal tectonic forces (e.g. Goetze
& Evans 1979; Brace & Kohlstedt 1980). The 1-D strength
envelope approach commonly divides the continental lithosphere into two or three rheological layers (upper crust, lower
crust, mantle) and applies a simple rheological model that
assumes that the deformation mechanisms within each layer are
controlled by frictional sliding (`Byerlee's law') and thermally
activated dislocation creep (power-law creep). This approach
allows one to derive insights into the mechanical strength
of continental lithosphere from laboratory measurements of
rock deformation (e.g. Kirby 1980). The concept of strength
envelopes has been applied to investigate the in£uence of
ß 2000 RAS
various factors such as temperature, crustal thickness and
lithological structure on stress distributions and lithospheric
strength (e.g. Ranalli & Murphy 1987; Strehlau & Meissner
1987; Ranalli 1991; Burov & Diament 1995; Kohlstedt et al.
1995). For recent reviews and discussions of the strength
envelope concept see Fernändez & Ranalli (1997) and Ranalli
(1997) .
With few exceptions, strength envelopes have been calculated
assuming a constant strain rate (CSR) that is independent
of the lithospheric strength. The CSR assumption allows
a straightforward calculation of the vertical distribution of
deviatoric stress. However, the kinematic concept may lead to
an unrealistic high stress level when applied to stable regions
(see below), which limits applying it to real situations. An
alternative approach keeps the tectonic force imposed on the
lithospheric model constant (Kusznir 1982, 1991; Kusznir &
Park 1984, 1987; Takeshita & Yamaji 1990; Hopper & Buck
1993). The constant force (CF) model overcomes the problem
647
648
R. Porth
of unrealistically high stress level that may arise from the CSR
assumption. However, the strong temperature dependence
and the non-linear stress^strain rate relation of continental
rheology may result in strain rates that are unrealistically
high by several orders of magnitude for long-term intraplate
deformation when applying the CF model to a weak lithosphere
(see below).
Christensen (1992) recognized the problems that may arise
from a constant tectonic force applied to a lithospheric model
and investigated problems of lithospheric extension assuming
a tectonic force that depends on the velocity of extension or
compression. The physical idea behind this concept is that,
considering the lithospheric model as part of a system of
global plate tectonics, the medium external to the section of the
lithosphere under consideration is resistant to deformation
and, hence, counteracts the tectonic force. Porth (1997) and
Porth & Wdowinski (in preparation) discussed the concept of
a velocity-dependent force boundary condition for numerical
models of lithospheric deformation more comprehensively and
applied it to large-scale compressional problems.
The successful implementation of a velocity-dependent
tectonic force in 2-D models of lithospheric deformation
motivated this study to adopt it for the concept of strength
envelopes. For calculations of the vertical distribution of
deviatoric stress, a velocity-dependent force boundary condition translates to a strain-rate-dependent force (SRDF)
assumption. This paper discusses the principal results of
the SRDF strength envelope model and implications for the
dynamics of intraplate deformation. Emphasis is placed on
di¡erences from the previous CSR and CF concepts.
2
G E ODY NA M ICA L F R A M EWOR K
This section provides the physical background for the calculations presented in this paper. It gives an overview of the
two main factors that control continental lithospheric deformation: (1) continental lithospheric rheology and (2) tectonic
forces. The basic rheological laws that are considered in this
study are brie£y discussed and the magnitudes of the most
important forces and of the deviatoric stress available in the
lithosphere are estimated.
2.1 Continental rheology
Tectonic forces imposed on the lithosphere cause deformation.
Material may deform elastically or, when the stress magnitude
exceeds its mechanical strength, by non-reversible brittle
failure or ductile creep. Non-elastic deformation of the continental lithosphere is determined by various time-independent
and time-dependent deformation mechanisms (e.g. Kirby 1983,
1985; Carter & Tsenn 1987; Meissner 1986; Ranalli & Murphy
1987; Ranalli 1995; Kohlstedt et al. 1995). Which mechanism
dominates does not only depend on rock type, physical conditions such as temperature and pressure, and the chemical
and mechanical e¡ects of £uids, but also on strain rate and
deformation history.
In the upper part of the crust, non-elastic deformation is
dominated by frictional sliding on existing faults. The brittle
yield stress (qy ) is largely insensitive to temperature, strain
rate and material but depends on e¡ective pressure and hence
increases with depth. A simpli¢ed form of an empirical relation
between depth (z) and brittle strength (qy ) (Byerlee 1978), often
referred to as Byerlee's law, is given by
qy ~B z .
(1)
The gradient of brittle strength (B) is estimated to be in
the range 20^60 MPa km{1 for compressional failure and
12^25 MPa km{1 for extensional failure (Brace & Kohlstedt
1980). A value of 20 MPa km{1 corresponds approximately
to lithostatic normal stress and hydrostatic pore pressure
on existing faults. Byerlee's law reasonably predicts observed
in situ stresses in boreholes up to depths of a few kilometres
(Byerlee 1978; McGarr et al. 1982; Zoback & Hickman 1982;
Zoback et al. 1993). The assumption of a linear increase of the
brittle strength throughout the entire lithosphere, however, is
most probably a gross oversimpli¢cation (e.g. Carter & Tsenn
1987; Shimada 1993).
In deeper parts of the lithosphere, due to higher temperatures, ductile deformation mechanisms become increasingly
important. In many tectonically relevant cases, thermally
activated dislocation creep, which depends on temperature
and strain rate, is assumed to be the dominant deformation
mechanism (e.g. Kirby 1983; Carter & Tsenn 1987). For steadystate dislocation creep, the relation between di¡erential stress
(qc ), given by the di¡erence between the maximum principal
stress (p1 ) and the minimum principal stress (p3 ), and strain
rate (_ ) is well represented by the creep law,
1=n
_
Q
,
(2)
exp
qc ~
A
nRT
where T is the absolute temperature, R is the gas constant and
A (pre-exponent), Q (activation energy) and n (power-law
exponent) are material parameters derived from laboratory
creep experiments (e.g. Kirby 1980). In many tectonically
relevant cases, the ductile deformation of the continental lithosphere is controlled by quartz- (upper crust) and feldspar-rich
rocks (lower crust) (Meissner 1986; Carter & Tsenn 1987),
although its structure is very heterogeneous. The best approximation of ductile deformation in the mantle lid is made by
applying wet dunite £ow parameters (Carter & Tsenn 1987).
2.2
Tectonic forces
Large-scale intraplate deformation processes are driven by
lateral density inhomogeneities that are associated with constructive or destructive plate boundaries, or crustal thickening
and/or lithospheric thinning within the lithospheric plates
(Forsyth & Uyeda 1975; Fleitout & Froidevaux 1982; Turcotte
& Schubert 1982, pp. 286; Bott & Kusznir 1984; Richardson
1992). The two dominant types of tectonic forces generated near
plate boundaries are the slab-pull force and the ridge-push
force. The magnitude of the ridge-push force, which depends
on the topography and density distribution of the ridge and
the thickness of the lithospheric plate, is estimated to be in the
range 2^3|1012 N per meter of plate boundary (Richter &
McKenzie 1978; Bott & Kusznir 1984; Bott 1993). The magnitude of the slab-pull force depends on the density and thus the
age of the slab. It can signi¢cantly exceed the magnitude of the
ridge-push force, but may be largely counteracted by resistance
produced by the sinking of the slab into the viscous mantle
(Turcotte & Schubert 1982, pp. 286; Bott & Kusznir 1984). The
net force at trenches is probably comparable to the ridge push
(Turcotte & Schubert 1982, pp. 288).
ß 2000 RAS, GJI 141, 647^660
A lithospheric strength model
An important source of tectonic stress within the plates
is high elevated plateaus. The magnitude of the horizontally
induced compressional force generated by the two largest
active plateaus has been estimated to reach 6|1012 N m{1
near the Altiplano (Froidevaux & Isacks 1984) and nearly
7|1012 N m{1 near the Tibetan plateau (Molnar & LyonCaen 1988). The magnitude of intraplate stress ¢elds has
been predicted by numerical modelling studies to be of the
order of tens of megapascals averaged across a 100-km-thick
lithosphere (Richardson et al. 1979; Stefanik & Jurdy 1992;
Richardson & Coblentz 1994; Coblentz & Richardson 1996),
corresponding to several 1012 N m{1 integrated over the thickness of the lithosphere. The value of 10|1012 N m{1 may
be considered as an upper limit for vertically integrated stress
available in the lithosphere in most tectonic settings, even
though in speci¢c cases the magnitude of tectonic forces may
exceed this value.
3
L I THO S P H E R IC MOD E L
This study applies an often used rheological model that is
controlled by Byerlee's law (eq. 1) and thermally activated
dislocation creep (eq. 2). Following ¢nite element studies of continental lithosphere dynamics (Bassi 1991; Boutilier & Keen
1994), the ductile deformation of the upper crust (0^18 km),
lower crust (18^36 km) and mantle (below 36 km) are assumed
to be controlled by the creep parameters for `wet' quartzite,
anorthosite and `wet' dunite, respectively (for creep parameters
and references see Table 1).
The `wet' (0.1 to 0.2 per cent £uids added) rheology
has been chosen because £uids can be expected to in£uence
ductile deformation in both the upper crust (Kohlstedt et al.
1995) and the mantle (Carter & Tsenn 1987). In this study
B~20 MPa km{1 is chosen as the reference value for the
gradient of brittle strength.
The rheological model applied in this study provides
an upper bound of lithospheric strength, as it neglects other
deformation mechanisms that may be important for lithospheric deformation in some tectonically relevant cases. These
mechanisms include fractional deformation (Shimada 1993),
strain softening at ductile faults (Hobbs et al. 1990), semi-brittle
deformation in the range of the brittle^ductile transition(s)
(Kirby & Kronenberg 1984; Carter & Tsenn 1987; Ross
& Lewis 1989; Long & Zelt 1991) and grain-size-dependent
di¡usion creep in the olivine mantle (Drury et al. 1991). These
deformation mechanisms are often neglected in calculations
of lithospheric strength pro¢les because reliable laws and/or
important factors controlling them (e.g. £uids, grain size, total
strain) are not known quantitatively.
Thermally activated dislocation creep critically depends on
temperature, thus the assumed geotherm has dominant control
on the vertical stress distribution. The studied temperature^
depth functions are the steady-state geotherms with 0 0C at the
surface and surface heat-£ow density (Q) values varying from
50 to 75 mW m{2 (Fig. 1). For the thermal parameters,
thermal conductivity and radiogenic heat production rate, used
to calculate the steady-state geotherms, see Table 1. The base of
the model is de¢ned by T ~1250 0C, which can be considered
to mark the thermal lithosphere^asthenosphere boundary.
The studied geotherms cover the range from that of a thick
shield-like continental lithosphere (at Q~50 mW m{2 , Moho
temperature 430 0C, lithospheric thickness 192 km) to that
of weak and thin continental lithosphere (Q~75 mW m{2 ,
775 0C, 72 km). In the following, the steady-state geotherms are
referred to by the corresponding surface heat-£ow density (Q).
The 1-D strength envelope concept assumes the strain rate to
be independent of depth. For the purpose of this discussion,
three further assumptions are made: (1) the calculated vertical
stress distribution is representative of a laterally extended
3-D section of the lithosphere; (2) the lithosphere deforms in
response to imposed horizontal tectonic forces (parallel to the
x direction) (Fig. 2); (3) the principal stress directions are
parallel to the horizontal (x and y) and vertical (z) coordinates.
From these assumptions it follows that, depending on the
deformation regime, the pxx component of the stress tensor is
either the maximum (p1 ) (compression) or minimum principal
stress (p3 ) (extension). Large-scale deformation of non-elevated
Figure 1. The steady-state geotherms assumed at the lowest
(50 mW m{2 ), highest (75 mW m{2 ) and an intermediate (60 mW m{2 )
surface heat-£ow densities (Q). Surface temperature is 0 0C and
the base of the model is at T ~1250 0C. uc-lc: upper^lower crust
boundary.
Table 1. Material parameters used for the lithospheric model. Thermal parameters: heat production rate (H)
and thermal conductivity (k). Creep parameters: pre-exponent (A), stress exponent (n) and activation energy
(Q). References for creep parameters: (1) Jaoul et al. (1984), (2) Shelton & Tullis (1981), (3) Chopra & Paterson
(1981).
Depth
(km)
Upper crust
0=18
Lower crust 18=36
Mantle
below 36
ß 2000 RAS, GJI 141, 647^660
H
(W m{3 )
k
(W m{1 K{1 †
1:2|10{6
0:6|10{6
0:01|10{6
2:6
2:6
3:2
649
Material
A
(Pa{n s{1 )
quartzite
4:82|10{14
anorthosite 2:06|10{23
dunite
7:94|10{17
n
Q
(J mol{1 )
1:8 150:6|103
3:2 238:0|103
3:35 444:0|103
Ref:
1
2
3
650
R. Porth
x
constant strain rate
W
z
.
ε = 10-15 s-1
crust
0
modelled section
mantle
B = 20 MPa km-1
Q = 60 mW m-2
(a)
(b)
25
0
25
tectonic force
lithosphere-asthenosphere
boundary
Figure 2. Simple sketch to illustrate assumptions made in this
study. The calculated 1-D stress distributions are considered to be
representative of a laterally uniform lithosphere that is subjected
to externally derived tectonic (buoyancy) forces. The strain rate is
assumed to be independent of depth and lateral position within the
lithospheric section considered, which extends over the length W.
Large-scale deformation of non-elevated continental lithosphere is
primarily compressive (as shown in this sketch); this study, however, is
independent of the tectonic deformation regime.
continental lithosphere is primarily compressive. This discussion,
however, is independent of the deformation regime. Di¡erential
stress (q~p1 {p3 ) as a function of the horizontal strain rate
(_xx ) is calculated from dislocation creep law (eq. 2), but is
limited to the brittle yield stress (eq. 1)
z [km]
(uniform deformation)
75
50
10-16
60
50
10-15
.
ε [s-1] = 10-14
75
75
Q [mW m-2] = 50
100
100
101
102
103
100
100
101
102
103
(σ1 - σ3) [MPa]
Figure 3. Vertical distributions of di¡erential stress (p1 {p3 ) at constant strain rate for di¡erent geotherms (a) and di¡erent geologically
relevant strain rates (b). The stress level within the entire depth range is
predicted by the CSR model to increase with decreasing surface heat
£ow density, and hence temperature, and increasing strain rate. The
gradient of brittle yield strength (B) is 20 MPa km{1 (grey line).
(4)
Temperature is the most important controlling factor. As
temperature increases, material becomes more ductile and
allows di¡erential stress to relax, which causes the depth of
the brittle^ductile transition in the upper crust to decrease
(Fig. 3a). At a constant strain rate of 10{15 s{1 (corresponding
to 100 per cent stretching in 30 Myr) and low temperatures
(Q~50 mW m{2 geotherm), the calculated stress level is at a
maximum near the brittle^ductile transition in the mantle at
about 50 km depth, where shear stresses exceed 1 GPa. At low
temperatures, the model predicts brittle failure in the lower
crust and the mantle. At high temperatures (Q~75 mW m{2
geotherm) the strongest part of the lithospheric model is the
upper crust, to where brittle failure is con¢ned. As ductile
strength depends on strain rate, this parameter also has an
important in£uence on the vertical distribution of stress (Fig. 3b).
However, the variation of strain rate between tectonically low
(10{16 s{1 ) and very high (10{14 s{1 ) has signi¢cantly less e¡ect
on stress magnitudes than the studied temperature variation.
The tectonic force per unit length (Ft ) required to maintain
lithospheric deformation can be calculated from integrating
di¡erential stress (q) over the thickness of the lithosphere (L),
…L
Ft ~
q(,_ z) dz .
(5)
This kinematic concept allows a straightforward calculation
of the vertical stress pro¢le according to temperature and
material from eq. (3).
The main results of the CSR model are as follows
(Fig. 3). Within the brittle deformation regime deviatoric stress
increases with depth but decreases in deeper parts of each
rheological layer as temperature increases. Brittle failure is
the governing deformation mechanism in the upper parts of the
upper crust and possibly of the lower crust and of the mantle.
Stronger parts of the lithosphere are separated, and mechanically decoupled, by weak layers at the base of the upper and
lower crust.
In Fig. 4 the integrated stress magnitude (Ft ), which is
often loosely referred to as the strength of the lithosphere,
is plotted against surface heat-£ow density (Q) for constant
strain rates for di¡erent tectonically relevant magnitudes.
At high temperatures, the integrated stress is relatively low
(<2|1012 N m{1 ) even at high strain rates, but Ft strongly
increases with decreasing temperature. At intermediate temperatures (Q~60 mW m{2 geotherm), Ft has the magnitude of the ridge-push force (3|1012 N m{1 ) at a strain rate
of 10{16 s{1 and reaches the upper limit of realistic values
when the strain rate is 10{14 s{1 . At low temperatures
(Q <55 mW m{2 geotherms), Ft signi¢cantly exceeds the
q~min(qy , qc ) .
(3)
In the following, the horizontal strain rate (_xx ) is referred to
simply as the strain rate (_ ).
4 C O N STA N T ST R A I N R AT E A N D
C O N STA N T FORCE MOD E L
This section brie£y discusses vertical lithospheric stress pro¢les
predicted by the CSR and CF models. The most important
results of these two types of model are summarized and the
problems that may arise in both approaches are discussed. This
provides the background for discussing results predicted by the
SRDF model and allows us to compare them to the results of
the two previous models.
4.1
Constant strain rate (CSR) model
The CSR model assumes tectonic deformation to occur at a
prede¢ned strain rate (_ o ) that is constant with variations of
lithospheric strength
_ _ o ~const .
~
0
ß 2000 RAS, GJI 141, 647^660
A lithospheric strength model
Ft [1012 N m-1]
25
20
15
10-15
10
.
ε [s-1] = 10-14
10-16
5
0
50
55
60
Q [mW
65
70
75
m-2]
Figure 4. The magnitude of integrated di¡erential stress (Ft ),
which corresponds to the force needed to maintain deformation at the
indicated strain rate (_ ), as a function of the surface heat £ow density
(Q). Dashed grey line is the magnitude of the ridge-push force
(3|1012 N m{1 ); solid grey line is the upper magnitude limit of
tectonic forces available in the lithosphere (10|1012 N m{1 ). At low
temperatures the CSR model predicts unrealistically high integrated
stress levels, which shows the limits of the kinematic concept.
magnitude of tectonic forces available in the lithosphere. Stress
distributions with Ft exceeding 10|1012 N m{1 , like in the
{15 {1
_
CSR model with Q~50 mW m{2 and ~3|10
s , are
not realistic.
Modelling results showing geologically unrealistic stresses
mean that the assumed lithosphere cannot be deformed at the
given strain rate. The problem can be avoided by restricting the
CSR model to cases that lead to realistic stress levels. However,
the fact that unrealistic results may occur when applying the
CSR model to strong lithosphere is a major shortcoming of the
kinematic concept.
viscous deeper parts. Due to this e¡ect, strain rate increases
with time, which may lead to tectonic deformation at very high
strain rates. A further important result is that in a stable region
a steady-state strain rate may not be reached on geological
timescales and thus the distribution of stress is governed by
transient e¡ects.
In contrast to the CSR model, in the CF assumption the
di¡erential stress magnitude at a given depth depends on the
entire strength pro¢le of the lithosphere because the integrated
lithospheric strength controls the rate of deformation. The
relation between tectonic force (Ft ) and strain rate (_ ) for the
lithospheric model depends on the rheology assumed, which is
controlled by power-law creep and Byerlee's law. Power-law
creep (2) predicts the strain rate to depend exponentially on
stress, _ ! qn , where n for most rocks is 2^5 (e.g. Carter &
Tsenn 1987), while brittle deformation is time-independent
(n??). These two constitution laws lead to a non-linear
relation between tectonic force (integrated strength) and strain
rate in problems of lithospheric dynamics (Sonder & England
1986). Because of the non-linear and relatively complex relation
between imposed tectonic force and strain rate, an iteration
process has to be applied to ¢nd the strain rate at which
the integrated stress (Ft ) has the prede¢ned value (Fto ) (see
Appendix A).
In the CF assumption, temperature has two opposing e¡ects
on stress distributions. At higher temperatures material is
more ductile, which, on the one hand, allows di¡erential stress
to relax more easily. On the other hand, the low viscosity leads
to higher strain rates, which increases stress. As a result, stress
distributions di¡er signi¢cantly less in the CF model with
temperature variations (Fig. 5) compared to the CSR model.
The temperature variations with variations of the surface heat£ow density are much higher in the mantle than in the crust
(Fig. 1). Hence, as temperature rises, the strength in the mantle
B = 20 MPa km-1
constant force
Fot = 3 x 1012 N m-1
Constant force (CF) model
To overcome limits of the CSR concept, Kusznir (1982, 1991),
Kusznir & Park (1984, 1987), Takeshita & Yamaji (1990) and
Hopper & Buck (1993) discussed stress distributions in continental lithosphere assuming a tectonic force that is constant
with (temporal) variations of lithospheric strength and the rate
of deformation,
Ft ~Fto ~const .
(6)
The CF concept is based on the assumption that tectonic forces
able to drive large-scale tectonic deformation are renewable;
that is, that they persist despite ongoing stress relaxation in the
lithosphere (Bott & Kusznir 1984).
The models of Kusznir (1982, 1991) and Kusznir & Park
(1984, 1987) assume the lithosphere to behave as a brittle^
viscoelastic material. The incorporation of elastic deformation
leads to time-dependent behaviour with `memory' e¡ects typical
of Maxwell rheology. Kusznir (1982, 1991) and Kusznir &
Park (1984, 1987) discussed the time evolution of vertical
stress distributions in response to a Heaviside (step) load of
horizontal tectonic force. They showed that, as time develops,
stress ampli¢es in the upper lithosphere due to relaxation in the
ß 2000 RAS, GJI 141, 647^660
0
Q = 60 mW m-2
(a)
(b)
25
z [km]
4.2
651
0
25
75
50
60
1
50
3
Q [mW m-2] = 50
75
75
Fot [1012 N m-1] = 6
increasing
temperature
100
100
100
101
102
103
100
101
102
103
(σ1 - σ3) [MPa]
Figure 5. Vertical distributions of di¡erential stress (p1 {p3 ) with the
constant force assumption for di¡erent geotherms (a) and di¡erent
geologically relevant magnitudes of the tectonic force (b). The stress
level is predicted by the CF model to decrease within the mantle and
to be ampli¢ed in the upper crust with increasing surface heat £ow
density. Increasing Fto increases the stress level within the entire
depth range. The gradient of brittle yield strength (B) is 20 MPa km{1
(grey line).
652
R. Porth
reduces signi¢cantly more than in the crust. As the integral
over the strength envelope is, by de¢nition, constant, crustal
stresses must amplify when mantle stresses decrease. As a
consequence, the upper crustal stress magnitude and consequently the depth of the brittle^ductile transition increase
with increasing temperature (Fig. 5a).
At the lowest temperatures studied (Q~50 mW m{2
geotherm), the assumed lithospheric model cannot be deformed
at a tectonically signi¢cant rate (_ < 10{19 s{1 ) (Fig. 6). At
the Q~60 mW m{2 geotherm, a tectonic force of the same
magnitude as the ridge-push force (3|1012 N m{1 ) is required
to cause slow but signi¢cant tectonic deformation (10{16 s{1 ).
A tectonic force twice as great causes lithospheric deformation
that is two orders of magnitude faster. The predicted strain
rate of 10{14 s{1 is the upper limit for large-scale intraplate
deformation. At the highest studied temperatures, a tectonic
force with a magnitude of 6|1012 N m{1 leads to strain
rates that exceed 10{11 s{1 , which corresponds to 100 per cent
stretching in 3 kyr. Such high strain rates are unrealistically
high by several orders of magnitude for long-term lithospheric
compression or extension extending laterally over several
hundred kilometres. When the imposed tectonic force has
a magnitude of 1012 N m{1 , however, the predicted strain
rate has a reasonable value even at the highest temperatures
studied.
The dynamic CF concept has some important advantages
over the kinematic CSR concept: (1) the rate of deformation
depends on the mechanical strength of the lithosphere; (2) the
prede¢ned tectonic force can be constrained to physically
realistic magnitudes; and (3) it allows one to compare stress
pro¢les and strain rates of two neighbouring sections of
continental lithosphere with di¡erent strengths because both
sections will deform at di¡erent strain rates but, due to
equilibrium, at the same stress level. However, unrealistically
high strain rates show that the CF assumption is not realistic
when applied to a weak lithosphere. Hence, it may be useful to
modify the force-controlled concept.
5 ST R A I N - R AT E - D E PE N D E N T FORC E
( S R D F ) MOD E L
This section introduces the SRDF model. The physical concept
is provided and estimates for the parameters that determine
the model are given. The SRDF model is then applied in the
investigation of the in£uence of temperature and brittle
strength on lithospheric deformation.
5.1 Concept
The physical rational behind the SRDF concept can be
understood when viewing the modelled (laterally uniform)
lithospheric section as part of a bigger system: the global
system of mantle and lithosphere dynamics. When the lithospheric section considered deforms, the medium external to it
must also deform. The physical basis of the SRDF model is to
consider the resistance forces required to deform the external
medium. The principal assumption made by the SRDF concept
is that the net force (Ft ) imposed on the lithospheric model
is given by externally generated tectonic forces (Fto ) that
are reduced by externally generated resistance forces (Fre )
(Fig. 7):
Ft ~Fto {Fre .
(7)
The concept of considering external resistance has previously been used in 2-D ¢nite element models of lithospheric
deformation (Christensen 1992; Porth 1997). In these studies
the force (Ft ) applied along the boundary of a 2-D lithospheric
section depends linearly on the velocity of horizontal compression or extension of the section (the indenting velocity, Ui ).
For a 2-D lithospheric model with uniform deformation, the
indenting velocity is given by
Ui ~W _ ,
(8)
where W is the horizontal extent of the model. Thus, adopting
the velocity-dependent force boundary condition for strength
envelopes leads to a tectonic force that depends linearly on
strain rate. In this form, the strain-rate-dependent net force
Ft
modelled section
(uniform deformation)
Fto
Figure 6. The rate of deformation (_) at the indicated constant
tectonic force (Fto ) as a function of surface heat-£ow density (Q).
The solid grey line is the upper limit for the rate of long-term intraplate deformation. At high temperatures, the CF model predicts
unrealistically high strain rates, which shows the limits of this
approach.
Fre
external medium
Figure 7. Sketch to illustrate the physical concept of the SRDF
model. Considering the lithospheric model as part of the global system
of mantle and lithosphere dynamics, the dash pot (Newton body)
simulates in a simple way the external medium.
ß 2000 RAS, GJI 141, 647^660
A lithospheric strength model
(9)
For the tectonic force assumed at zero strain rate (Fto ),
reasonable values are given by the relatively well-known magnitudes of most important tectonic forces (e.g. Bott & Kusznir
1984). Estimates of reasonable values of Re can be derived
from maximum observed rates of intraplate deformation. The
maximum possible strain rate (_ m ) occurs when the strength
of the modelled lithosphere and therefore its resistance to
deformation vanishes (Ft ?0). The highest possible strain rate
is given by
_m ~
Fto
,
Re
(10)
and assuming uniform deformation extending over the length
W , the highest possible indenting velocity (Uim ) is given by
Uim ~W
Fto
.
Re
(11)
The horizontal extent of major zones of active continental
deformation is typically several hundreds to thousands of
kilometres (e.g. England & Jackson 1989). Assuming Uim ~
20 mm yr{1 to be an upper bound for intraplate deformation
at a high tectonic force magnitude (Fto ~ 6 |1012 N m{1 )
extending over W ~ 500 km (normal to the direction of
deformation) yields an estimate of Re ~4.7|1027 N s m{1 .
For the investigated values of Fto ~1, 3 and 6|1012 N m{1 ,
the resulting values for the maximum possible strain rate (_ m )
are 3.17, 6.35 and 12.7|10{16 s{1 , respectively.
The lateral extent of the deforming area (W ) and the
maximum estimate of the indenting velocity (Ui ) are adopted
for the onset of large-scale Andean compression in the
late Oligocene (Isacks 1988). In other examples, a somewhat
di¡erent value of Re may be more appropriate, but Re should
be of the same order of magnitude for all cases of large-scale
intraplate deformation. Reasonable values of Re decrease
with the lateral extent of the assumed 2-D model, because in
narrow zones of deformation higher strain rates may occur
compared to larger extending areas. Values of Re ?? simulate
a practically rigid external medium that does not allow the
modelled lithosphere to deform, and lead to a CSR model with
_
~0.
A value of Re ~0 simulates an inviscous external medium
without any e¡ect on the force balance, leading to a CF model
with Ft ~Fto .
Although transient e¡ects may be important in some geologically relevant cases (e.g. Kusznir 1991), this study neglects
elastic deformation and assumes steady-state deformation.
Compared to linear viscoelastic material, the non-linear viscoelastic lithospheric rheology causes the strain rate to approach
the steady-state value relatively fast in early stages after the
imposed force changes (Govers 1993). Furthermore, transient
e¡ects are most important at low forces and for a stable
lithosphere, and for a quantitative analysis of `memory' e¡ects
the entire deformation history must be known. Thus, it seems
justi¢ed to neglect elastic deformation and, hence, transient
e¡ects in a study that focuses on long-term intraplate deformation and principal e¡ects related to the SRDF assumption.
ß 2000 RAS, GJI 141, 647^660
Results
In this section the in£uence of two factors that largely control
the mechanical behaviour of the lithosphere is investigated
using the SRDF model: (1) temperature, which controls timedependent ductile deformation; and (2) time-independent
brittle strength. This section focuses on the principal results
predicted by the SRDF model and emphasizes di¡erences from
the results of the previous CSR and CF models.
5.2.1 Temperature
At low temperatures (Q < 55 mW m{2 geotherms) the lithospheric model does not deform at geologically signi¢cant strain
rates (_ < 10{16 s{1 ) when subjected to a tectonic force of a
geodynamically realistic magnitude (Fig. 8a). Thus, the e¡ect
of strain-rate-dependent external resistance forces is negligible
and the SRDF model predicts strain rates and magnitudes of
integrated stress very similar to the CF model (Fig. 8a). Both
force-controlled models predict lithospheric deformation at
a realistic strain rate driven by tectonic force of a reasonable magnitude. In contrast, as has been discussed above, the
assumption of deformation at a constant strain rate with
tectonically signi¢cant magnitude leads to vertically integrated
-12
SRDF
.
log10 ε [s-1]
Ft ~Ft (_)~Fto {Re _ .
5.2
CF
.
εm
-14
Fot
6
3
1
-16
6
Fot [1012 N m-1] = 1
3
(a)
-18
50
55
60
65
70
75
10
Ft [1012 N m-1]
(Ft ) is determined by two independent scalar parameters,
representing the externally derived tectonic force (Fto ) and the
strength of the external medium (Re ),
653
10-16
8
10-15
SRDF
.
ε [s-1] = 10-14
CSR
6
4
(b)
2
0
50
55
60
65
70
75
Q [mW m-2]
Figure 8. (a) The rate of deformation (_) at di¡erent values of Fto
plotted against surface heat-£ow density (Q), as predicted by the CF
model (grey lines), and the SRDF model (black lines). The horizontal
grey line is the upper limit for the rate of long-term intraplate deformation. Increasing surface heat £ow density to the highest values
studied, the SRDF model predicts that the strain rates approach
asymptotically the maximum possible value (_ m ) (shown by the short
horizontal lines). (b) The magnitude of integrated di¡erential stress
(Ft ), shown within the realistic range of forces available in the lithosphere, plotted against surface heat-£ow density, as predicted by
the SRDF model (black lines) and the CSR model (grey lines). The
horizontal grey line marks the magnitude of the ridge-push force
(3|1012 N m{1 ).
654
R. Porth
stress magnitudes in the low temperature range that exceed
the magnitude of tectonic forces available in the lithosphere
(Fig. 8b).
The importance of external resistance forces and, hence, the
di¡erence between integrated stress magnitudes predicted by
the SRDF and the CF model increases with temperature and
tectonic force. External resistance becomes signi¢cant when
{16 {1
_
the strain rate becomes geologically signi¢cant (*10
s ),
o
which occurs, depending on Ft , at surface heat-£ow density
values between 55 and 65 mW m{2 . At Q~60 mW m{2 , the
SRDF model predicts the net force (Ft ) to be reduced to about
60 per cent of the zero strain rate value (Fto ) when Fto is
6|1012 N m{1 , but the decay of the predicted tectonic force
magnitude is still relatively small (*15 per cent) when Fto
is 3|1012 N m{1 and negligible when Fto is 1|1012 N m{1 .
Accordingly, in the low to moderate temperature range
(50 ¦ Q ¦ 60 mW m{2 geotherms), calculated stress^depth
distributions in the SRDF model are relatively similar to the
results of the CF model, especially at low and intermediate
tectonic force levels (Fig. 9). When temperature rises from
low to moderate, the SRDF model predicts, similarly to the
CF model, that tectonic stress will amplify in the upper crust
due to stress relaxation in deeper parts of the lithosphere.
Accordingly, the depth of the brittle^ductile transition is predicted by both force-controlled models to be at a greater depth
at intermediate temperatures than at high temperatures. This
is, however, not of great tectonic signi¢cance because of the
low strain rates.
When temperature rises, the reduction of the imposed net
force in the SRDF model limits the predicted rate of intraplate
deformation to geodynamically realistic values, even when the
studied lithospheric model is weak. At high temperatures
the strain rate asymptotically approaches the maximum possible
strain rate (_ m ). In the CF model, where external resistance
forces are neglected and the deformation driving force is
counteracted by resistance forces due to the (internal) mechanical strength of the considered lithosphere only, the strain
rate reaches the upper bound of the tectonically realistic values
at relatively moderate surface heat-£ow density values.
At Q~65 mW m{2 , the net force (Ft ) predicted by the
SRDF model is reduced to one-third of the zero strain rate
value (Fto ) when Fto is 6|1012 N m{1 . The reduced net force of
approximately 2|1012 N m{1 causes the lithospheric model
to deform with a realistic strain rate of about 10{15 s{1. When
the tectonic force is kept constant at 6|1012 N m{1 , the
modelled lithosphere deforms two orders of magnitude faster.
The strain rate of about 10{13 s{1 predicted by the CF model
corresponds to 100 per cent stretching in 0.3 Myr, which is not
realistic for intraplate tectonics extending horizontally over
several hundred kilometres. The large di¡erence between the
strain rate values predicted by the two force-controlled models
re£ect the mechanical instability of continental lithosphere,
which is controlled by non-linear viscoplastic rheology, at high
temperatures. At Q~65 mW m{2 , the rate of deformation is
relatively low (*10{17 s{1 ) when Fto is 1|1012 N m{1 , thus
external resistance is not important and both force-controlled
strain rate dependent force
strain-rate-dependent
force
z [km]
Fot = 6 x 1012 N m-1
Fot = 3 x 1012 N m-1
B = 20 MPa km-1
Fot = 1 x 1012 N m-1
0
0
25
25
50
50
Q [mW
75
m-2]
50
60
75
100
100
100
101
102
103 100
Q = 50 mW m-2
z [km]
75
101
102
103 100
Q = 60 mW m-2
101
102
103
(a)
Q = 75 mW m-2
0
0
25
25
50
50
Fot [1012 N m-1]
6
3
1
75
100
75
100
100
101
102
103 100
101
102
(σ1 - σ3) [MPa]
103 100
101
102
103
(b)
Figure 9. Vertical distributions of di¡erential stress (p1 {p3 ) with the SRDF assumption for (a) di¡erent values of the zero strain rate force (Fto ) and
(b) di¡erent values of the surface heat-£ow density (Q). Note that the stress level in the crust is highest for the intermediate temperature
(Q~60 mW m{2 geotherm). The gradient of brittle yield strength (B) is 20 MPa km{1 (grey line).
ß 2000 RAS, GJI 141, 647^660
A lithospheric strength model
5.2.2
Brittle strength
So far, the in£uence of temperature, which has a strong
control on the time-dependent ductile deformation mechanism predominantly in deeper parts of the lithosphere has
been investigated. The second deformation mechanism considered in this study is brittle failure, which controls the
time-independent strength predominately in the upper crust.
Brittle strength of the lithosphere may vary with time or
region, for example, due to the varying in£uence of £uids. It
is generally higher in compressional than extensional environments because of additional tectonic (normal) stresses on
existing faults. In this section the in£uence of the gradient of
brittle strength (B), within the geodynamically reasonable
range of 10¦B¦60 MPa km{1 (Brace & Kohlstedt 1980), is
investigated using SRDF assumption.
In Fig. 10(a) the strain rate is plotted against surface heat£ow density for B~10, 20 (reference) and 60 MPa km{1 for
both the SRDF and the CF assumptions (Fto ~3|1012 N m{1 ).
ß 2000 RAS, GJI 141, 647^660
.
log10 ε [s-1]
-12
SRDF
CF
Fot = 3 x 1012 N m-1
-14
-16
B [MPa km-1}
10
20
60
(a)
-18
50
55
60
65
70
75
10
(b)
Ft [1012 N m-1]
models predict a similar strain rate of a realistic magnitude. At
high temperatures, the CSR model also predicts reasonable
values of the integrated di¡erential stress, even for tectonically
high strain rates.
Increasing the temperature further (Q > 65 mW m{2
geotherms) does not signi¢cantly in£uence the stress level and
rate of deformation predicted by the SRDF model. In the high
temperature range the rate of deformation is largely controlled
by the balance of external deformation-driving and external
resisting forces and is not greatly a¡ected by the low internal
strength of the lithospheric model. The net force is low and
the strain rate is close to the maximum possible value. The
mechanical strength of the model at high temperatures is
dominated by the time-independent brittle strength of the
upper crust and, hence, depends only slightly on the assumed
temperature distribution. Accordingly, at high temperatures Ft
is relatively independent of Fto . The integrated stress approaches
values of about 1|1012 N m{1 at the highest investigated
temperatures. Consequently, the Fto ~1|1012 N m{1 model
is not signi¢cantly a¡ected by simulated external resistance
forces in the SRDF assumption, and the stress level predicted
by the CF model is also of realistic magnitude.
In contrast to the CF model, the SRDF model predicts
the stress level in the crust to decrease when temperature
is increased from moderate to high (60 to 75 mW m{2
geotherms) (Fig. 9). Accordingly, in the SRDF model the
change from brittle to ductile deformation in the upper crust is
at a shallower depth at high temperatures than at moderate
temperatures. The ampli¢cation of tectonic stress in the upper
crust, due to stress relaxation in the upper mantle, is counteracted by the reduction of stress due to external resistance,
which becomes increasingly important when internal resistance
becomes low. The external medium takes up a large part of
the deformation-driving forces and prevents the strong stress
ampli¢cation predicted by the CF model. Due to these e¡ects,
the SRDF model predicts the depth of the brittle^ductile
transition to decrease with increasing temperature in the low
to moderate temperature range but to increase when temperature rises further. This leads to a relatively complex relation
between temperature and depth extent of the zone of brittle
failure.
655
8
SRDF
Fto= 3 x 1012 N m-1
6
CSR
.
ε = 10-15 s-1
4
2
0
50
55
60
65
70
75
Q [mW m-2]
Figure 10. (a) The rate of deformation (_) plotted against surface
heat-£ow density (Q), as predicted by the CF model (grey lines) and the
SRDF model (black lines) at di¡erent values of the gradient of brittle
strength (B). The horizontal grey line is the upper limit for the rate
of long-term intraplate deformation. (b) The magnitude of integrated
di¡erential stress (Ft ), shown within the realistic range of forces
available in the lithosphere, plotted against surface heat-£ow density
as predicted by the SRDF model (black lines) and the CSR model
(grey lines) at di¡erent values of the gradient of brittle strength (B). The
horizontal grey line marks the magnitude of the ridge-push force
(3|1012 N m{1 ).
At low temperatures, where both force-controlled models
predict very similar strain rates, the in£uence of brittle strength
is small compared to that of the assumed temperature^depth
distribution. When temperature is low, the upper mantle is the
strongest part of the lithosphere. The model predicts signi¢cant
brittle failure to be con¢ned to the upper crust for all studied
values of B (Fig. 11). However, varying the upper crustal
stress level with the gradient of brittle strength has a signi¢cant
e¡ect on stress distribution in the mantle also. When the upper
crust is strong and capable of taking up a signi¢cant part of
the imposed tectonic force, the stress level in the mantle is
lower because the integrated stress is prede¢ned in the CF
assumption and only slightly reduced by external resistance
in the SRDF assumption. When the upper crust is weak, the
deviatoric stress is predicted to amplify in the lower lithosphere.
With the CF assumption, the in£uence of brittle strength
becomes increasingly important when temperature rises and
the ductile strength of the lithospheric model is reduced
(Fig. 10a). In the SRDF model, however, the strain rate is
largely controlled by the balance of external deformationdriving forces and resisting forces when the internal strength of
the investigated lithosphere is low. Therefore, the in£uence of
brittle strength on strain rate in the SRDF assumption is also
small at high temperatures. Hence, the strain rate is insensitive
to variations of brittle strength (B) in the entire temperature
range. The magnitude of integrated stress, in contrast, is
signi¢cantly in£uenced by the gradient of brittle strength in
656
R. Porth
strain rate dependent force
strain-rate-dependent
z [km]
Q = 50 mW m-2
Q = 60 mW m-2
Fot = 3 x 1012 [N m-1]
Q = 75 mW m-2
0
0
25
25
60
50
75
20
50
75
B [MPa km-1] = 10
100
100
100
101
102
103 100
101
102
103 100
101
102
103
(σ1 - σ3) [MPa]
Figure 11. Vertical distributions of di¡erential stress with the SRDF assumption for di¡erent values of the gradient of brittle strength (B) and
surface heat-£ow density (Q) (Fto ~3|1012 N m{1 ). The grey lines show the brittle yield strength for di¡erent values of B.
the SRDF assumption when temperature is high, because
most of the imposed tectonic force is taken up by the brittle
upper crust. Integrated stress is higher by approximately a
factor two at B~60 MPa km{1 (1.7|1012 N m{1 ) than at
B~20 MPa km{1 (0.9|1012 N m{1 ). The di¡erence in the
integrated stress results from the di¡erent stress level in the
brittle upper crust (Fig. 11). In contrast to cold geotherms,
at intermediate and high temperatures (60 and 75 mW m{2
geotherms), stress in the lower crust and mantle is largely
insensitive to the gradient of brittle strength assumed.
6
D I SC US S I O N
This study applies a simple rheological model to investigate the
vertically integrated stress level, the rate of intraplate deformation and the vertical distribution of deviatoric stress in
continental lithosphere that is subjected to horizontal tectonic
forces. Because of the uncertainties involved in extrapolating
laboratory-derived deformation laws to tectonic deformation at
the lithospheric scale, this study focuses on the principal results
related to the basic assumption of a strain-rate-dependent
tectonic force. Some of the results are very di¡erent from those
predicted by the previous CSR and CF models, which has some
important implications for the dynamics of intraplate deformation. These results are valid despite the simple rheological
model. This study does not attempt to apply results to a speci¢c
region of intraplate deformation, as a quantitative investigation of intraplate deformation requires a comprehensive and
careful description of the region. However, in this discussion
some of the principal results will be compared to ¢rst-order
observations made in regions of active intraplate deformation.
The most important result of this study is that the SRDF
model, unlike the previous CSR and CF models, leads to
realistic results in terms of stress magnitudes and strain rates
within the entire strength range of the continental lithosphere.
The SRDF model can be applied to thin and weak continental
lithosphere as well as to thick and strong shield-like lithosphere, which implies that it can also be applied to oceanic
lithosphere of any thermal age. This shows that the SRDF
concept is physically more realistic than the previous CSR
and CF models. Di¡erences from the CSR model demonstrate
the importance of a dynamic force-controlled approach for a
quantitative investigation of intraplate deformation. Di¡erences
from the CF model demonstrate the importance of resistance
forces generated externally to the deforming lithosphere for the
dynamics of intraplate tectonics.
As predicted by the SRDF model, external resistance forces
signi¢cantly reduce the net (driving minus resistance) force
imposed on actively deforming areas and hence the level of
tectonic stress in regions of intraplate deformation. External
resistance becomes important when the rate of intraplate
deformation becomes tectonically signi¢cant. Young continental
lithosphere (< 250 Myr) and areas that have experienced a
thermotectonic event, for example, delamination of mantle lid,
within the last 100 Myr are often characterized by a high heat£ow density (Sclater et al. 1980; Pollack et al. 1993). In these
regions, the uppermost mantle, which in many tectonic settings
is the strongest part of the continental lithosphere, probably
has an extremely low mechanical strength. Thus, external
resistance can be expected to be the dominating factor that
counteracts deformation-driving forces and limits the rate of
deformation in many regions of intraplate deformation. The
results of this study suggest that the rate of large-scale horizontal deformation is largely controlled by other parts of the
global system of mantle and lithospheric dynamics. Because
of equilibrium conditions, reduction of the stress magnitude
in zones of active deformation implies that the stress level in
adjacent stable regions is also reduced.
Some aspects of this result can also be applied to deforming
areas that are dominated by localized (shear) deformation
along major zones of weakness. However, reasonable values
of the parameter representing the external resistance in the
approach presented in this study (Re ) depend on the lateral
extent of the deforming area. A study of narrow zones of
deformation requires a signi¢cantly lower value of Re , which
allows higher strain rates.
The often correlated directions of plate motion and maximum horizontal stress (SHmax ) in the plate interiors suggest
that the forces that drive plate motion also control the tectonic
stress ¢eld in plate interiors (Richardson 1992; Zoback 1992;
Wdowinski 1998) and hence drive intraplate deformation. As
these forces result from lateral density variations they are
renewable; that is, they persist despite ongoing deformation
ß 2000 RAS, GJI 141, 647^660
A lithospheric strength model
(e.g. Bott & Kusznir 1984). Externally derived resistance to
intraplate deformation driven by these forces may be dominated by more stable regions of the same plate surrounding the
deforming area. Large-scale continental deformation, however,
is primarily localized near (constructive) plate boundaries,
extending several hundreds to thousands of kilometres inland,
rather than in the interiors of lithospheric plates (e.g. England
& Jackson 1989; Thatcher 1995; Wdowinski 1998). This suggests
that strain-rate-dependent resistance is a more fundamental
feature of the tectonic forces that drive intraplate tectonics
and, hence, that the e¡ect of these forces is not renewable in
an in¢nitely short time. Di¡erent tectonic forces may be controlled by resistances of di¡erent magnitudes. For example, the
low viscosity at mid-oceanic ridges may lead to signi¢cantly
lower resistances than the drag forces produced by the sinking
of oceanic slabs into the mantle that counteracts the slab-pull
force.
The lithospheric model assumed is tectonically stable at
heat-£ow density values of below approximately 55 mW m{2 .
Variations of the heat-£ow density within the range 50^55 mW
m{2 do not signi¢cantly in£uence the predicted stress magnitude. The strain rate, in contrast, varies with surface heat-£ow
density, but at a tectonically insigni¢cant low level. In the low
temperature range, external resistance, simulated in the SRDF
model, is negligible and both force-controlled models lead to
similar results. Above surface heat-£ow densities of approximately 65 mW m{2 , the mechanical strength of the lithospheric
model assumed is low. At high temperatures, the strain rate
is controlled by the balance of external deformation-driving
and external deformation-resisting forces, leading to a low
net force being imposed on the deforming lithosphere. In the
high temperature range, the mechanical strength of the lithosphere is dominated by the brittle strength of the upper crust,
which is independent of the strain rate and temperature.
Consequently, strain rate, and hence the strain-rate-dependent
tectonic force, does not vary signi¢cantly with surface heat£ow density. At high temperatures the predicted strain rate
is close to the maximum value (_m ), and the SRDF model
leads to similar results to a CF model with _ o ~Fto /Re . The
transition from stable to weak lithosphere occurs in a relatively
narrow temperature range (between 55 and 65 mW m{2 ),
which re£ects the strong temperature dependence and the
non-linear behaviour of continental lithosphere dominated
by viscoplastic rheology (e.g. Sonder & England 1986). The
transition described above within a relatively narrow temperature range suggests that the rate of intraplate deformation in
actively deforming areas is in many cases not signi¢cantly
controlled by the internal mechanical strength of the deforming
lithosphere.
The average surface heat-£ow density of continents has been
estimated to be in the range 57^65 mW m{2 (Sclater et al. 1980;
Pollack et al. 1993); in recently active regions observed values
reach 80 mW m{2 and locally up to 100 mW m{2 or more
(Blackwell et al. 1982; Francheteau et al. 1984; Ranalli 1991;
Springer & FÎrster 1998). Compared to observations, the transition from stable lithosphere to active intraplate deformation
is predicted by the lithospheric model to occur at relatively
moderate heat-£ow density values (Q^55^65 mW m{2 ). This
suggests that the assumed lithospheric rheology, which is
based on Byerlee's law and thermally activated dislocation
creep, underestimates the mechanical strength of continental
lithosphere, although various deformation mechanisms are
ß 2000 RAS, GJI 141, 647^660
657
neglected by the model. Hence, if the sparse heat-£ow density
values available are representative of regions of intraplate
deformation, the results of this study suggest that the creep
parameter of wet dunite derived from laboratory experiments
(Chopra & Paterson 1981) may underestimate the ductile
strength of the continental mantle lid.
The results of this study suggest that the reduction of
tectonic force magnitude in actively deforming areas has an
important e¡ect on the vertical distribution of tectonic stress,
which, in turn, can be expected to have a signi¢cant in£uence
on the style of intraplate tectonics. At low and intermediate
temperatures, the strongest part of the lithosphere is the
uppermost mantle, where most of the imposed tectonic force is
supported. In both force-controlled models, stress ampli¢es
in the crust when it relaxes in the ductile mantle lithosphere.
In the CF model the upper crustal stress magnitude and consequently the depth of the brittle^ductile transition increase
monotonously with increasing temperature at all temperature
levels. The SRDF model, in contrast, predicts the depth of
the brittle^ductile transition to increase with temperature in
the low to moderate temperature range, when strain rates are
low, but to decrease with increasing surface heat £ow in the
moderate to high temperature range, when strain rates become
tectonically signi¢cant. The prediction made by the SRDF
model for the temperature range where tectonically signi¢cant
strain rates occur is consistent with the observation that in
tectonically active regions the maximum depth of earthquakes
decreases with increasing heat-£ow density (Meissner & Strehlau
1982; Sibson 1982; Chen & Molnar 1983; Meissner 1986).
This study investigates the in£uence of two parameter that
largely control the lithospheric strength: the surface heat-£ow
density (Q) and the gradient of brittle strength (B). Both parameters are varied within the plausible range for continental
lithosphere. Lithospheric deformation has been shown to depend
signi¢cantly more on surface heat-£ow density, and hence
temperature, than on brittle strength. An important exception
is that the variation of brittle strength has a more important
in£uence on the integrated stress magnitude at high temperature
than variations of the surface heat-£ow density.
In order to keep the study simple and to focus on e¡ects
related to the SRDF assumption, several factors are neglected
that have been shown to have signi¢cant in£uence on the
mechanical behaviour of continental lithosphere. These factors
include thermal de£ections, crustal thickness and deformation
mechanisms such as di¡usion creep in the mantle, powerlaw break down and semi-brittle deformation (see Fernändez
& Ranalli (1997) and references therein). These factors will
in£uence either the time-dependent or time-independent strength
of the lithosphere. As the e¡ect of the variation of both timedependent and time-independent strength has been investigated,
the principal in£uence on intraplate deformation of other e¡ects
of the SRDF assumption can be estimated from this study. For
example, increasing crustal thickness will have a similar e¡ect
on lithospheric strength as increasing temperature.
The model presented fails to predict one important observation made in many regions of continental convergence,
namely, the seismically active uppermost mantle (Chen &
Molnar 1983). Signi¢cant brittle deformation predicted by
the SRDF model used is con¢ned to the upper crust. The stress
level in the uppermost mantle reaches the brittle yield stress
only when the tectonic force is very high and surface heat-£ow
density is very low. Even when the gradient of brittle strength is
658
R. Porth
low and the tectonic force is high, brittle failure in the uppermost mantle of the model occurs only within a small area.
This suggests that the linear relation assumed between depth
and brittle strength is too simple to investigate the problem of
brittle deformation in the uppermost mantle.
Some vertical stress distributions shown for stable lithosphere may be oversimpli¢ed because this study neglects elastic
deformation and assumes steady-state deformation. As has
been shown by Kusznir (1982, 1991), the stress ¢eld in stable
lithosphere may be dominated by transient e¡ects and steadystate stress may not be reached in geological time. However,
neglecting elastic deformation has no in£uence on the main
result for large-scale intraplate deformation, where non-elastic
strain is much larger than elastic strain.
7
C O NCLUS IO N S
The SRDF model presented is an important modi¢cation
of the strength envelope concept, which since the 1970s has
widely been used to evaluate vertical lithospheric stress distributions and factors controlling the mechanical strength of the
lithosphere. It overcomes di¤culties arising in both previous
approaches, the CSR model and the CF model, and hence proves
to be physically more realistic. Di¡erences from the kinematic
CSR concept demonstrate the importance of a dynamic forcecontrolled model, which calculates the strain rate according to
the balance of the net forces assumed and the strength of the
lithosphere studied. Di¡erences from the dynamic CF model
demonstrate the importance of externally derived resistance
of the global system of mantle and lithosphere dynamics for
large-scale intraplate deformation.
When the lithosphere is weak, neglecting external resistance
leads to reasonable results only when the imposed tectonic
force does not exceed 1|1012 N m{1 . The transition from
tectonically stable lithosphere to intraplate deformation at
unrealistically high strain rates, as predicted by the CF model,
occurs within a narrow range of moderate temperatures. This
re£ects the strong temperature dependence and the non-linear
behaviour of the viscoplastic rheology of continental, and
oceanic, lithosphere. Results predicted by the SRDF model
suggest that the rate of intraplate deformation in many tectonically relevant cases is controlled by the balance of externally
derived deformation-driving and deformation-resisting forces
and depends only relatively little on the low internal strength of
the deforming lithosphere. The e¡ect of tectonic forces that
drive large-scale intraplate deformation and plate motion has
been shown not to be renewable at in¢nitely short times. This
implies important consequences for intraplate stress ¢elds of
fast-moving lithospheric plates and plates that contain regions
of active intraplate deformation.
The time-dependent ductile strength of continental lithosphere is negligible for moderate temperatures. Thus, within
the high temperature range, variations of the geotherm do not
signi¢cantly in£uence predicted strain rates and the magnitude of integrated strength. The rate of intraplate deformation
is predicted by the SRDF to be largely independent of timeindependent brittle strength within the entire temperature
range. The integrated stress level in regions of active intraplate deformation, in contrast, is predicted to be largely controlled by the brittle time-independent strength but not to
be signi¢cantly in£uenced by time-independent deformation
mechanisms.
This study assumes a very simple viscoplastic rheology,
which neglects elastic deformation and several deformation
mechanisms that may be important in some geodynamically
relevant cases. While the principal results are valid despite the
above-mentioned simpli¢cation, some more detailed results
may be oversimpli¢ed. Results of this study suggest that more
speci¢c problems such as brittle deformation in the mantle or
time-dependent e¡ects should be investigated using SRDF
models with a more complicated elastoviscoplastic rheology.
The SRDF model allows us to investigate the in£uence of the
parameters controlling the strength of the lithosphere such as
temperature, brittle yield strength and crustal thickness on the
rate of deformation, the mechanical strength and the vertical
stress distribution within the entire strength range of continental, or oceanic, lithosphere. The SRDF model provides the
geodynamic background to relate kinematic observations of
intraplate deformation, derived from geodetic observations
of active intraplate deformation and seismic observations of
the depth distribution of intraplate seismicity, to the net forces
acting. It predicts the depth of the brittle^ductile transition to
depend on the temperature^depth distribution in a relatively
complex way. However, a careful description of the region
yields the potential to place constraints on the driving and
resisting forces controlling a speci¢c region of intraplate deformation using the simple 1-D SRDF model. If 2-D (or 3-D)
e¡ects cannot be neglected, a ¢nite element model that applies
a velocity-dependent force boundary condition that corresponds
to the SRDF assumption should be considered.
AC K NOW L E D G ME N T S
The main part of my funding was provided by the German
Sciences Foundation (SFB 267 `Deformation processes in the
Andes'). I am grateful to Oliver Heidbach, Thomas Kenkmann
and Dietrich Stromeyer for helpful comments and discussions.
The manuscript was improved by the critical review of an
anonymous reviewer. The ¢gures were generated using the free
GMT software (Wessel & Smith 1991).
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A PPE N D I X A : IT E R AT I O N S CH E M E FOR
T H E C F A N D SR D F MOD E L S
The lithospheric model studied assumes a viscoplastic rheology,
dominated by brittle failure according to Byerlee's law (eq. 1)
and thermally activated dislocation creep (eq. 2). Both deformation laws describe a non-linear relation between stress and
strain rates. Furthermore, the depth of the brittle^ductile
transition, where the dominating deformation mechanism
changes, depends on strain rate. Due to these e¡ects the
integral of the stress envelope (Ft ) depends on strain rate in a
relatively complex non-linear way. Thus, in both, the CF and
the SRDF models, an iteration process has to be employed to
¢nd the strain rate at which Ft (_ ) has the prede¢ned value Fto .
A1 Constant force
In the CF model, where
Ft (_ )~Fto ~const
(A1)
has to be satis¢ed, the problem can be solved by the contractive
¢xpoint iteration scheme,
_iz1 ~_i
Fto
.
Ft (_i )
(A2)
In the ¢rst iteration step an appropriate starting value _1 has to
be assumed. The numerical examples were calculated using the
convergence criterion
o
Ft {Ft (_ ) < 10{4 .
(A3)
Fto
A2
Strain-rate-dependent force
When the right-hand side of (A1) is not constant but also
depends on strain rate in the form assumed in this study and
Ft (_ )~Fto {Re _
(A4)
must be satis¢ed, the strain rate in each iteration step follows
from
_iz1 ~_i
Fto
.
(Ft (_i )zRe _i )
(A5)
Accordingly, the convergence criterion for the SRDF model is
o
Ft {(Ft (_ )zRe _ ) < 10{4 .
(A6)
Fto
ß 2000 RAS, GJI 141, 647^660