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J. metamorphic Geol., 2006
doi:10.1111/j.1525-1314.2006.00644.x
The effects of porphyroblast growth on the effective viscosity of
metapelitic rocks: implications for the strength of the middle crust
W.G. GROOME, S.E. JOHNSON AND P.O. KOONS
Department of Earth Sciences, University of Maine, Orono, ME 04469, USA ([email protected])
ABSTRACT
Numerical models are used to examine the effects of porphyroblast growth on the rheology of
compositionally layered rocks (metapelites and metapsammites) and by extension the middle crust during
prograde metamorphism. As porphyroblast abundance increases during prograde metamorphism,
metapelitic layers will strengthen relative to porphyroblast-free metapelitic units, and potentially relative
to quartzofeldspathic metapsammitic units. As metapelitic layers become stronger, the integrated
strength of compositionally layered successions increases, potentially causing large volumes of midcrustal rock to strengthen, altering the strain-rate distribution in the middle crust and affecting the
geodynamic evolution of an orogenic belt. The growth of effectively rigid porphyroblasts creates strength
heterogeneities in the layer undergoing porphyroblast growth, which leads to complex strain-rate
distributions within the layer. At the orogen scale, the strengthening of large crustal volumes (on the order
of thousands of cubic kilometres) changes the strain-rate distribution, which may change exhumation
rates of high-grade metamorphic rocks, the geothermal structure and the topography of the orogen. The
presence of a strong zone in the middle crust causes strain-rate partitioning around the zone, suppressed
uplift rates within and above the zone and leads to the development of a basin on the surface.
Key words: metamorphic strengthening; geodynamics; numerical modelling; porphyroblasts; rheology;
strain-rate partitioning.
INTRODUCTION
The rheological structure of the lithosphere is a firstorder control on the distribution of strain within an
orogen, which can in turn affect processes such as the
localization of metamorphism (e.g. Kerrich et al.,
1977; Rubie, 1983; Brodie & Rutter, 1985; Koons
et al., 1987; Frueh-Green, 1994; Baxter & DePaolo,
2004), the exhumation of deep crustal rocks (e.g. Platt,
1986; Koons, 1987; Beaumont et al., 2001; Zeitler
et al., 2001; Jamieson et al., 2002; Koons et al., 2002)
and the topographic evolution of an orogen (e.g. Koons, 1989, 1995; Williams et al., 1994; Carminati &
Siletto, 1997). Much of our knowledge concerning the
rheology of earth materials comes from laboratory
experiments conducted on monomineralic aggregates
(natural and synthetic) under pressure, temperature
and strain rate conditions not generally expected in
natural settings (e.g. Tullis & Yund, 1977; Arzi, 1978;
Paquet et al., 1981; Ji & Zhao, 1993; Shea & Kronenberg, 1993; Farver & Yund, 1999; Renner et al.,
2000; Hirth et al., 2001; Rosenberg, 2001; Stunitz &
Tullis, 2001; Mecklenburgh & Rutter, 2003; Tenthorey
& Cox, 2003). In this paper, rock ÔstrengthÕ is used to
characterize the resistance to deformation of a rock,
which can be described in several ways, including the
yield stress (r1 ) r3) at failure for plastic deformation,
_ for viscous deformation
the effective viscosity (r=e)
(either Newtonian or non-Newtonian), or the elastic
2006 Blackwell Publishing Ltd
modulus (r/e) for elastic deformation. Broadly,
strength can be viewed as the resistance to deformation
of a rock unit, and, unless otherwise stated, this broad
definition of strength is used.
Understanding the strength of polymineralic rocks is
problematic and has been approached in four main
ways: (1) analytical treatments viewing rocks as composites of individual minerals with known strengths,
such that the bulk strength of the rock is a function of
the volume fraction and relative strengths of different
phases (e.g. Burg & Wilson, 1987; Jordan, 1988;
Handy, 1990; Tullis et al., 1991; Ji & Zhao, 1993;
Treagus, 2002; Ji, 2004); (2) experiments using analogue materials such as normcamphor (e.g. Bons &
Cox, 1994; Bons & Urai, 1994) and silicone putty (e.g.
Treagus & Sokoutis, 1992), and two-phase mixtures of
weak minerals such as halite and calcite (e.g. Jordan,
1987; Ji et al., 2001); (3) numerical experiments
studying the effects of different volume fractions and
relative strengths of constituent materials (e.g. Tullis
et al., 1991; Bons & Cox, 1994; Treagus, 2002; Johnson
et al., 2004); and (4) field measurements to constrain
the relative strengths of different rock types deformed
under natural conditions (e.g. Smith, 1977; Lisle et al.,
1983; Kanagawa, 1993; Treagus, 1999; Treagus &
Treagus, 2002; Kenis et al., 2005; Groome & Johnson,
2006).
During orogenesis, parts of the middle crust
will periodically strengthen and weaken during
1
2 W.G. GROOME ET AL.
metamorphism (e.g. Beach, 1980; Rubie, 1983; Brodie
& Rutter, 1987; Koons et al., 1987; De Bresser et al.,
2001; Handy et al., 2001). The strength of a rock
during metamorphism will change if: (1) reactants have
a different strength than the products, which alters the
ductile strength of the rock (e.g. Arzi, 1978; Beach,
1980; Poirier, 1982; Brodie & Rutter, 1987; Handy
et al., 2001); (2) product minerals are fine-grained
enough for the deformation mechanism to change,
decreasing differential stress or increasing strain rate,
which alters the ductile strength of the rock (e.g.
Brodie & Rutter, 1987; Koons et al., 1987; De Bresser
et al., 2001; Stunitz & Tullis, 2001); and (3) fluid
pressure changes, which alters the brittle strength of
the rock (e.g. Etheridge et al., 1983; Sibson, 2003).
Weakening or strengthening can occur if product
phases have different strengths than reactant phases.
An example of this process occurs during partial
melting, when relatively weak partial melt is produced,
leading to an overall weakening of the rock (e.g. Arzi,
1978). A solid-state example of weakening by the
replacement of relatively strong minerals by relatively
weak minerals occurs during the serpentinization of
mafic rocks, when relatively strong olivine grains are
replaced by relatively weak serpentine grains (Barnes
et al., 2004). Conversely, strengthening should be
expected during the prograde growth of garnet, which
is relatively strong (e.g. Wang & Ji, 1999), in metapelitic rocks because the relative volume fraction of
strong garnet increases.
Weakening can occur if reaction products are finer
grained than the reactants because the dominant
deformation regime in the zones of reaction may
switch from intracrystalline creep mechanisms to grain
size-dependent creep mechanisms, which generally
accommodate higher strain rates (e.g. Passchier &
Trouw, 1996). However, a situation may develop in
which the grain size is maintained near the transitional
grain size between intracrystalline and intercrystalline
creep mechanisms (e.g. De Bresser et al., 2001). During
continued reaction, grain size may increase, shifting
the dominant deformation mechanism into the intracrystalline creep regime, causing strengthening (e.g.
Brodie & Rutter, 1987; Koons et al., 1987; De Bresser
et al., 2001).
Fluids liberated during dehydration metamorphism
may remain trapped in the rock at grain boundaries,
causing weakening, and possibly brittle fracture, as the
pore pressure increases (e.g. Hubbert & Rubey, 1959;
Sibson, 2003). This is probably a transient process,
because as the pore pressure increases and the rock
weakens, deformation will partition into the weakened
zone, which will serve to liberate the fluids, thus
strengthening the rock again until fluid pressure can
build up to weaken the rock (e.g. Etheridge et al.,
1983; Selverstone et al., 1991; Tobisch et al., 1991;
Handy et al., 2001; Sibson, 2003). Fluid-enhanced
weakening may also occur during the retrograde
metamorphism of relatively anhydrous rocks if the
fluids catalyse reactions that produce relatively weak
minerals (e.g. Rubie, 1983; Brodie & Rutter, 1987;
Koons et al., 1987; Selverstone et al., 1991; FruehGreen, 1994; Moecher & Wintsch, 1994; Barnes et al.,
2004).
In this paper, we explore the effects of porphyroblast
growth in layered sequences on the effective viscosity
structure and strain-rate partitioning at the grain and
bed scale, and address the strengthening effects of
porphyroblast growth on lithosphere-scale strength
profiles and the geodynamics of orogens.
THEORETICAL BACKGROUND
The strength of polyphase aggregates at given pressure
and temperature conditions is generally considered to
be controlled by the volume fraction, distribution and
relative strength of the constituent phases (e.g. Jordan,
1988; Handy, 1990; Tullis et al., 1991; Ji & Zhao, 1993;
Takeda, 1998; Ji & Xia, 2002; Treagus, 2002; Takeda
& Obata, 2003; Ji, 2004; Johnson et al., 2004). Two
theoretical bounds, relating the elastic strength of an
aggregate to the volume fraction and relative strengths
of the constitutive phases, are generally considered to
define the maximum and minimum strength of an
aggregate (e.g. Handy, 1990; Tullis et al., 1991; Ji &
Zhao, 1993; Takeda, 1998; Ji & Xia, 2002; Ji, 2004).
The Voigt bound assumes that all grains within an
aggregate will experience the same strain, and that the
distribution of differential stresses will differ from
grain to grain such that strong grains will accumulate
higher differential stresses to achieve the same strain as
weak grains, which deform at lower differential stress
(e.g. Handy, 1990; Tullis et al., 1991; Ji & Zhao, 1993;
Ji & Xia, 2002):
ra ¼ /s rs þ /w rw ;
ð1Þ
where r is the differential stress in the aggregate (a),
strong phase (s) and weak phase (w), and / is the
volume fraction of a given phase. The Reuss bound
assumes that all grains within the aggregate experience
the same differential stress and that strain in individual
grains will be different depending on the bulk modulus
of each grain (e.g. Handy, 1990; Tullis et al., 1991; Ji &
Zhao, 1993; Ji & Xia, 2002):
/s /w 1
þ
:
ð2Þ
ra ¼
rs rw
Strictly, the Voigt and Reuss Bounds are defined for
homogeneous elastic materials deformed at low elastic
strain; however, the two bounds can be reformulated
for homogeneous viscous materials by assuming that
the bulk modulus (elastic stress/elastic strain) is analogous to viscosity (flow stress/viscous strain rate) (e.g.
Tullis et al., 1991; Ji & Zhao, 1993; Ji & Xia, 2002). In
this paper, the following Voigt and Reuss formulations
are used for viscous materials (Fig. 1):
2006 Blackwell Publishing Ltd
Aggregate viscosity
)
VB
1(
75
0.
50
5
0.
5
0
2
0.
–0
.2
5
–0
.5
Aggregate viscosity
5
.7
–0
1
0
–1
0.25
0.5
0.75
Volume fraction of strong phase
5
5
.2
–0
5
–0.
–0.75
B)
(R
1.0
0
5
0.2
0.5
1 (V
10
0.7
5
100
B)
EFFECTIVE VISCOSITY OF PORPHYROBLASTIC ROCKS 3
–1 (RB)
1
0
0.25
Volume fraction of strong phase
0.5
Fig. 1. (a) Plot of the Voigt (VB) (Eq. 3a) and Reuss (RB) (Eq. 3b) bounds and the generalized mixing rule of Ji (2004) (Eq. 4) for an
arbitrary two-phase mixture with two orders of magnitude difference in viscosity between the weak and strong phase. Dashed lines
show the effects of different J values in Eq. 4. (b) Close-up of the light grey box on panel a. Note that the rate of viscosity increase with
increasing volume fraction of the strong phase is a function of the J exponent.
ga ¼ / s gs þ / w gw ;
ga ¼
/s /w
þ
gs gw
ð3aÞ
1
;
ð3bÞ
where g is the effective viscosity:
r
g¼ :
ð3cÞ
e_
Ji (2004) provided a generalized mixing rule that
describes the behaviour of two phase mixtures:
(
"
#)1=J
Ms J
1
;
ð4Þ
Ma ¼ Mw 1 þ Vs
Mw
where M is a specific mechanical property (e.g. viscosity) of the composite (a), weak (w) or strong (s) phase,
J is a constant dependent on the specific microstructure, and V is the volume fraction of the strong grain
(Fig. 1). The parameter J can be varied to account for
specific microstructural features (e.g. grain shape,
connectivity and continuity) in the aggregate. For instance, J 0 approximates the geometrical mean
(similar to the formulation of Tullis et al. (1991), J ¼
±0.5 approximates aggregates with spherical grains,
and J ¼ ±0.25 approximates aggregates with grains of
arbitrary shape (Ji, 2004). The sign of J describes
whether the aggregate consists of strong grains in a
weak matrix (J > 0) or weak grains in a strong matrix
(J < 0). The Voigt and Reuss limits can be described
with J ¼ 1 and J ¼ )1, respectively.
In natural systems, the strength of a polyphase
aggregate is considered to lie between the Voigt and
Reuss bounds, with the Reuss bound being an
approximation of strength for a material with strong
grains embedded in a weak matrix and the Voigt
bound being an approximation of strength for a
material with weak grains embedded in a strong matrix
(e.g. Arzi, 1978; Handy, 1990; Ji & Zhao, 1993; Ji
2006 Blackwell Publishing Ltd
et al., 2001; Treagus, 2002). However, divergence from
the theoretical bounds is observed in experiments,
particularly if the volume fraction of the embedded
phase approaches c. 40% (e.g. Arzi, 1978; Jordan,
1987, 1988; Tullis et al., 1991; Ji & Zhao, 1993; Treagus, 2002; Takeda & Obata, 2003), at which point the
strength–volume fraction trend varies between the two
theoretical bounds. In a system where volume fraction
of the strong inclusion phase increases with time, the
transition from Reuss- to Voigt-type behaviour corresponds to the point at which the system becomes an
interconnected network of strong inclusions, and the
aggregate strength unavoidably trends towards the
strength of the strong phase (e.g. Ji & Xia, 2002). Finally, the Voigt and Reuss bounds only describe the
theoretical strength evolution trends of homogeneous
aggregates in which the embedded phase is evenly
distributed through the aggregate volume.
Other formulations proposed for describing the
strength of two-phase aggregates consider the effects of
clast shape (e.g. Jordan, 1988; Handy, 1990; Tullis
et al., 1991; Treagus, 2002), power law constitutive
relationships for the phases (e.g. Tullis et al., 1991; Ji &
Zhao, 1993; Ji et al., 2001), the presence of a truly rigid
phase (e.g. Duva, 1984; Yoon & Chen, 1990; Ravichandron & Seetharaman, 1993; Treagus, 2002) and
the effects of layer anisotropy and orientation (e.g.
Treagus, 1993; Ji et al., 2001). The strength–volume
fraction relationships of these formulations all lie between the end-member Voigt and Reuss bounds,
indicating that these limits are good descriptions for
the maximum and minimum strengths of two-phase
aggregates, respectively.
Based on theoretical considerations, the following
generalizations can be made about the strength of
polyphase rocks:
1. the strength of a two-phase aggregate will lie between the end-member strengths of the constituent
phases;
4 W.G. GROOME ET AL.
2. a two-phase aggregate will become stronger if the
volume fraction of the strong phase increases;
3. the shape of the strong or weak phase will affect the
strength of an aggregate, with square inclusions
resulting in stronger aggregates than circular inclusions
(e.g. Tullis et al., 1991; Treagus, 2002);
4. the presence of a truly rigid phase will strengthen an
aggregate more than the presence of a strong deformable phase (e.g. Duva, 1984);
5. layered successions compressed perpendicular to
layering will be stronger than homogeneously mixed
two-phase aggregates (e.g. Ji & Xia, 2002); and
6. layered successions deformed in layer-parallel simple
shear will be weaker than successions deformed in
layer-perpendicular simple shear (e.g. Treagus, 1993).
In nature, phyllosilicate-rich pelitic layers in turbidite couplets are considered to be weak because they
generally record higher finite strains (e.g. Treagus,
1988, 1993, 1999 and references therein). Based on
foliation refraction measurements from sedimentary
and low-grade metasedimentary rocks, Treagus (1999)
estimated that psammitic units have between two and
10 times higher effective viscosities than pelitic units
and Kenis et al. (2005) estimated that pelitic units had
effective viscosities approximately two to five times
lower than psammitic units based on mullion shape
measurements. Experimental deformation of mica-rich
rocks suggests that mica is quite weak when sheared
parallel to {001}, relative to quartz at low temperatures (e.g. Shea & Kronenberg, 1993; Tullis & Wenk,
1994), and numerical models (e.g. Johnson et al.,
2004) indicated that rocks experience significant
weakening when mica grains become interconnected
to form a foliation. These observations suggest that
the relative weakness of low-grade, phyllosilicate-rich,
pelitic rock is due at least in part to the weakness of
mica.
In general, prograde metamorphism of pelitic
rocks results in a decrease in the volume fraction of
weak hydrous minerals, such as clays, and a corresponding increase in relatively strong minerals,
such as garnet (e.g. Bucher & Frey, 1994). In many
cases, the volume fraction of relatively strong porphyroblastic minerals (i.e. staurolite, garnet, andalusite, kyanite) can be quite high (20–30%), which
should significantly alter the strength of these rocks
(see Eq. 3a,b). If the volume fraction of effectively
rigid porphyroblasts increases sufficiently, pelitic
layers should become stronger than interlayered
psammitic layers, which typically have inappropriate
bulk compositions for the growth of porphyroblastic
phases.
NATURAL EXAMPLE
The White Mountains region of the New England
Appalachians (Fig. 2) preserves field evidence for the
strengthening of metapelitic layers relative to metapsammitic layers in amphibolite facies metaturbidites.
During prograde low-pressure, high-temperature
metamorphism, large (up to 15 cm long) andalusite
porphyroblasts grew in the metapelitic layers
(Fig. 3a), causing these layers to become strong relative to interlayered, porphyroblast-free metapsammitic layers during a period of isoclinal, km-scale
nappe folding. Based on field relationships and porphyroblast-matrix microstructure relationships, it
appears that andalusite growth was prior to or synchronous with the early stages of nappe folding (e.g.
Eusden et al., 1996; Groome & Johnson, 2006). The
foliation developed during the nappe-stage folding
anastamoses around andalusite porphyroblasts, and
pressure shadows are ubiquitously developed along
the margins of andalusite porphyroblasts (Fig. 3b,c),
supporting the interpretation that these large porphyroblasts were present during the early deformation
(e.g. Johnson, 1999; Vernon, 2004). Additionally,
andalusite porphyroblasts locally define a mineral
lineation that lies in the plane of the fold axis (e.g.
Eusden et al., 1996; Rodda, 2005). Later regional
metamorphism led to the pseudomorphing of andalusite porphyroblasts to aggregates of blocky and
fibrolitic sillimanite, muscovite and staurolite (e.g.
Wall, 1988; Allen, 1992; Eusden et al., 1996; Groome
& Johnson, 2006). However, andalusite porphyroblasts do not appear to have undergone intracrystalline deformation prior to being pseudomorphed,
suggesting that they behaved as essentially rigid
objects during nappe-stage folding.
The relative strengths of metapelitic and metapsammitic layers were determined using the foliation
refraction technique of Treagus (1999) (Groome &
Johnson, 2006). To use this technique, it is assumed
that the difference in finite shear strain recorded in two
rock types is a function of the effective viscosities of
the two layers, such that the unit with higher effective
viscosity will record lower finite shear strain (e.g.
Treagus, 1983, 1988, 1999). Using geometric arguments, Treagus (1999) proposed that the ratio of
bedding–foliation angles for two layers (hA/hB) is a
function of the effective viscosity contrast between the
layers (gA/gB) via:
cA tan wA tan hB gB
¼
¼
;
cB tan wB tan hA gA
ð5Þ
where c is the shear strain, w is the shear angle and
subscripts A and B refer to two rock layers with contrasting viscosity (Fig. 3d–f).
Using foliation refraction angles, Groome &
Johnson (2006) estimated that the porphyroblast-rich
metapelitic layers in the amphibolite facies metaturbidites of the White Mountains region had effective viscosities approximately two to three times
greater than the interlayered metapsammitic layers. In
unmetamorphosed pelite–psammite successions, the
psammitic layers are expected to have higher effective
viscosities because they have a higher volume fraction
2006 Blackwell Publishing Ltd
Pe
ab
od
yR
.
US
Rte
16
EFFECTIVE VISCOSITY OF PORPHYROBLASTIC ROCKS 5
Peabody Granite
anite
d Gr
Mig Front
kfor
Bic
3
St
ut
St O
Ou
t
Sil In
Fro
nt
Mig
St Out
Sil
In
1
R.
2
hic
North
t
rus
Th
1 = Staurolite zone (St + Gt + Bt + Qz + Ms)
2 = Lower sillimanite zone (Sil + St + Bt + Gt +
Qz + Ms)
3 = Upper sillimanite zone (Sil + Bt + Gt + Qz +
Ms)
3
3
rp
mo
Peak metamorphic zones
St
Ou
t
West Peabody
AR
MW
eta
Extent of pseudomorphed andalusitebearing metaturbidites
Two-mica granite
2
3
Legend
Undivided migmatite
st-m
Po
Mig
Fron
Mt Washington
t
O
ut
St Out
St
3
t
ron
F
Mig
2
3
2.5 km
W60
W65
W70
W75
N50
PQ
White
Mountains
Region
NB
ME
ON
NY
N45
NS
VT
NH
MA
CT RI
PN
200 km
NJ
of relatively strong quartz and feldspar as well as a
generally coarser grain size. Using foliation refraction
angles, Treagus (1999) estimated that unmetamorphosed (or low grade) metapsammites had effective
viscosities two to 10 times greater than metapelites,
and Kenis et al. (2005) estimated that porphyroblastfree metapelitic units were up to five times weaker
than interlayered metapsammitic units based on
mullion-shape characteristics. In the White Mountains
region, metapelitic units contain up to c. 25–30%
andalusite porphyroblasts, and Groome & Johnson
(2006) hypothesized that the high volume fraction of
effectively rigid porphyroblasts led to the metapelitic
units having higher effective viscosities than
metapsammitic units. The numerical modelling presented below was designed to test the hypothesis
that prograde metamorphism resulting in the growth
of effectively rigid porphyroblasts preferentially in
2006 Blackwell Publishing Ltd
Fig. 2. Location map and simplified metamorphic
assemblage map for the study area in the eastern
White Mountains region of New Hampshire. NY,
New York State; VT, Vermont; NH, New Hampshire; ME, Maine; NB, New Brunswick; PQ,
Quebec; NS, Nova Scotia; MA, Massachusetts; CT,
Connecticut; RI, Rhode Island; NJ, New Jersey;
PN, Pennsylvania; MWAR, Mount Washington
Auto Road.
metapelitic bulk compositions could lead to a strength
reversal relative to unmetamorphosed pelite–psammite successions.
GRAIN- AND BED-SCALE NUMERICAL
EXPERIMENTS
Grain- and bed-scale models were constructed to explore the effects of porphyroblast growth on the
evolving strength of metapelitic layers, the evolving
strength contrast between porphyroblast-rich metapelitic layers and interbedded metapsammitic layers, and
the changing strain-rate partitioning within the metapelitic layer as porphyroblast abundance increases. A
linear viscous, plane strain, finite element formulation
is used to investigate the problems outlined. The model
domain consists of a three-layer system analagous to a
single metaturbidite couplet (Fig. 4).
6 W.G. GROOME ET AL.
5 cm
(a)
(b)
And
And
5 mm
(d)
(c)
S1
St
Pel
Psa
Pel
And
S4
Pseud
P-Sha
dow
1m
5 mm
(f)
(e)
ηA
ψA
el
θP
θA
S1
Psa
sa
Pel
Pe
l
Ps
a
θP
1m
ηB
θB
γA
ψB γ B
=
tanψ A tan θ B η B
≈
=
tanψ B tan θ A η A
Fig. 3. (a) Photograph of a typical andalusite-rich (And) pelitic unit. (b) Photomicrograph of a partially pseudomorphed andalusite
showing anastamozing S1 foliation and a synchronous pressure shadow, indicating that the porphyroblast was present during the
formation of this porphyroblast. (c) Sketch of the photograph in panel b. (d) Sketch of an outcrop from near the summit of Mt
Washington showing refracting foliation through a pelite (Pel) – psammite (Psa) layered succession. (e) Photograph of a refracting
foliation through a psammite–pelite layered succession. The larger bedding-foliation angles in the pelitic layers indicate that they had
higher effective viscosities during the development of this foliation. (f) Schematic diagram showing a refracting foliation with the angles
used to estimate relative effective viscosities.
Modelling environment
Analytical expressions for the strength of aggregates
are useful when dealing with a constant volume fraction of two or more phases. However, to explore the
evolving strength of a rock undergoing metamorphic
strengthening or weakening reactions during which the
relative volume fractions of strong and weak phases
changes, a numerical approach is more appropriate.
Furthermore, numerical treatments allow us to
investigate the effects of changing mineralogy on
strain-rate partitioning during syn-deformational
growth. Ideally, to numerically investigate the rheological effects of porphyroblast growth on both the
2006 Blackwell Publishing Ltd
EFFECTIVE VISCOSITY OF PORPHYROBLASTIC ROCKS 7
Undeformed
No porphyroblasts
Deformed
SSR
1.0
Psammite
Pelite
Psammite
0.0
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
Log10viscosity
Shear strain rate
Fig. 4. Model results for the porphyroblast-free experiment with a viscosity contrast of 2:1 (gpsa:gpel). Far left panel: One-dimensional
profile of the viscosity structure through the centre of the model. Note the log10 viscosity scale. Middle panels: Shear strain-rate
contour maps and the model geometry in the undeformed (left) and deformed (right) states. Deformation was by dextral simple shear,
as indicated by the shear arrows. Far right panel: One-dimensional profile of the shear strain rate distribution through the model.
evolving strength and strain-rate partitioning within a
given unit, we would use a formulation that: (1) allows
for an arbitrary geometry of individual grains that can
have irregular shapes and sizes; (2) allows the volume
fractions of strong and weak grains to change with
time; (3) stores information, such as effective viscosity,
position and crystallographic orientation, for each
grain; and (4) can couple the geometric evolution with
a mechanical response to an imposed stress or displacement field. The ELLE microstructural modelling
platform satisfies these requirements.
ELLE is an open-source, Linux-based code that can
be used to model the two-dimensional microstructural
evolution of a system by executing a number of specific
processes in sequence for small time steps (e.g. Jessell
et al., 2001). Processes that have been modelled in the
ELLE environment include Taylor–Bishop–Hill lattice
rotations (e.g. Piazollo, 2000), grain growth via surface
energy driving forces (e.g. Jessell et al., 2001), subgrain
rotation recrystallization (e.g. Piazollo, 2000), grainboundary diffusion (e.g. Park et al., 2004), porphyroblast growth (this paper), and nucleation processes.
Each process is a separate open-source code and can be
modified to suit the user’s requirements, and a particular ELLE model is controlled by a central script that
calls each sub-routine in sequence (e.g. Jessell et al.,
2001).
The microstructure of a system is simulated using a
geometry consisting of several nodes that contain
specific information such as viscosity, chemical concentration, lattice orientation and position. Individual
grains are defined by a group of bounding nodes that
can hold information such as mineralogy and viscosity.
Porphyroblast growth is modelled using a routine that
allows the surface area of selected grains to increase by
a finite amount per time step in order to ultimately
form large subcircular grains. The growth routine
simulates porphyroblast growth from initially dispersed nuclei in the pelite matrix. In order to simplify
our modelling and focus on the rheological effects of
porphyroblast growth, local chemistry is not consi 2006 Blackwell Publishing Ltd
Table 1. Parameters used in grain-scale numerical models.
Grain
g1 a
g2
g3
Pelite
Psammite
Porphyroblast
1
2
100
1
4
100
1
6
100
a
Dimensionless viscosity in Eq. 6.
dered, nor do we prescribe specific porphyroblastforming reactions to control the growth of porphyroblasts. Conservation of volume is attained by allowing
grains along the margins of the growing porphyroblast
to decrease in size as the porphyroblast grows. Porphyroblast grains were assigned a normalized viscosity
100 times that of the matrix grains surrounding them,
making the porphyroblasts effectively rigid during
deformation (Table 1).
In order to model the mechanical effects of
porphyroblast growth, ELLE was coupled with the
two-dimensional finite element code BASIL (Barr &
Houseman, 1996), which solves for incompressible,
plane-strain deformation using linear and non-linear
viscous constitutive relationships:
_
sij ¼ 2ge;
ð6aÞ
1n
g ¼ 0:5BE_ n ;
ð6bÞ
e_ ¼
@vy @vx
þ
;
@x
@y
ð6cÞ
where sij is deviatoric stress, g is the effective viscosity,
B is a material constant, E_ is the second invariant of
the strain rate tensor, n is a stress exponent that can
vary from 1 to 3, e_ is the shear strain rate and v is
velocity. In the experiments presented below, a constant displacement rate is used on the upper and lower
bounds of the model domain to solve for shear stress.
In order to concentrate the modelling on the rheological effects of changing porphyroblast abundance,
we have chosen to use a linear-viscous constitutive
8 W.G. GROOME ET AL.
relationship (n ¼ 1) so that the effects of strain hardening can be ignored at this stage, though this may not
provide a satisfactory description for the deformation
behaviour of natural rocks in the middle crust.
layer-parallel simple shear (e.g. Treagus, 1993), which
allows us to asses the end-member strengthening effects of porphyroblast growth.
Porphyroblast-free models
Model results
Two groups of models are presented here: (1) porphyroblast-free models and (2) porphyroblast growth
models. All models consist of a three-layer succession
with a model pelite layer between two model psammite
layers (Fig. 4). The starting microstructure for the
entire system is homogeneous, with most grains being
subcircular, similar to a foam-textured quartzite. A
simplified microstructure was used because we did not
want to impose an arbitrary geometry on the model
with mica, quartz and feldspar grains in varying proportions depending on layer bulk chemistry (i.e.
quartz-feldspar-dominated psammite layers and micadominated pelite layers). The only difference between
model pelite and model psammite layers is the bulk
viscosity of that layer (Table 1). To establish the bulk
viscosity of a given layer, all grains within that layer
were assigned the same viscosity. This modelling
assumption was made because we wanted to focus on
the bulk strengthening effect of porphyroblast growth
and decided to simplify the pelite layer rheology to a
two-phase mixture – bulk matrix viscosity plus porphyroblasts. Furthermore, as porphyroblast growth in
our models is not governed by chemical processes, we
do not need to assign individual grains in the pelite
layer to be mica, quartz or feldspar. The bulk viscosity
contrast between model pelite and psammite layers
ranged from 2 to 6 (gpsa:gpel) in the experiments
reported here. In the porphyroblast model group,
porphyroblast viscosity was two orders of magnitude
greater than the pelite matrix, making the model porphyroblasts essentially rigid. In our natural example,
microstructural evidence suggests that the andalusite
porphyroblasts were effectively rigid during deformation, but as little is known about the flow properties of
andalusite, we could not use effective viscosity estimates based on experimental literature to constrain the
viscosity of our model porphyroblasts.
Model porphyroblast shapes are ultimately subcircular; however, the andalusite porphyroblasts in our
natural example are elongate. We chose to simplify the
model porphyroblast shapes to be subcircular (i.e.
garnet-like) so that we could focus our results on the
strengthening effects of porphyroblast growth without
consideration for grain shape. Previous numerical
modelling assessing the role of clast shape suggests that
elongate clasts may increase the strength of aggregates
more if they are oriented at an angle to the flow field
than if the clasts are oriented parallel to the flow field
(e.g. Treagus, 2002). In all experiments, deformation is
layer-parallel dextral simple shear. The layer-parallel
simple shear boundary condition was chosen because
layered rocks are considered to be weakest during
Porphyroblast-free layered systems were deformed to a
bulk shear strain of c. 1.0 to illustrate the strain partitioning behaviour expected in a porphyroblast-free
layered succession during layer-parallel simple shear
(Fig. 4). The strain partitioning in these models is
controlled by the viscosity ratios between the model
pelite and psammite layers. The model with the lowest
viscosity contrast between pelite and psammite
(gpsa:gpel ¼ 2) had the least amount of strain partitioning between the two layers. Furthermore, within
each bed, strain rates are homogeneously distributed in
these models (Fig. 4). These models serve as a reference frame to compare with the porphyroblast growth
models presented below.
Porphyroblast growth models
Porphyroblast growth experiments were conducted to
examine the evolving viscosity structure of our
layered model during porphyroblast growth in the
pelite layer. In these models, porphyroblast abundance varies from <1% to >70% of the model pelite
layer (Fig. 5). Porphyroblast locations were handselected in the starting geometry to be widely dispersed in the model pelite layer, in order to simulate
diffusion-controlled nucleation spacing (e.g. Porter &
Easterling, 1992; Vernon, 2004). Model porphyroblast
growth occurs by constant surface area expansion,
and is not governed by any chemical processes because we only wanted to assess the mechanical effects
of porphyroblast growth. In our natural example,
porphyroblast modal abundance is on the order of
25–30%, as determined visually in the field and by
photographic analysis.
The bulk viscosity of the model pelite layer increases
by more than an order of magnitude between 1% and
70% porphyroblast abundance (Figs 6 & 7). The
evolving strength trend in these models lies close to the
Reuss bounding limit, consistent with what would be
predicted for aggregates consisting of a strong phase
embedded in a weak matrix undergoing layer-parallel
simple shear (e.g. Handy, 1990; Treagus, 1993, 2002; Ji
et al., 2001). Of particular note is the nearly fourfold
increase of pelite layer viscosity between 0 and 30%
porphyroblast abundance. The increasing porphyroblast abundance in the pelitic layer also leads to
changes in strain-rate partitioning, both within the
pelitic bed and within the layered system as a whole
(Fig. 7). Within the pelitic layer, the highest strain
rates are in grains immediately adjacent to the growing
porphyroblasts, with strain rates increasing in these
porphyroblast-marginal grains as the adjacent porphyroblasts get larger. As the pelitic layers strengthen,
2006 Blackwell Publishing Ltd
EFFECTIVE VISCOSITY OF PORPHYROBLASTIC ROCKS 9
Initial geometry
10% Porphyroblast
Psammite
Pelite
Psammite
20% Porphyroblast
30% Porphyroblast
Psammite
Pelite
Psammite
40% Porphyroblast
50% Porphyroblast
Psammite
Pelite
Psammite
60% Porphyroblast
70% Porphyroblast
Psammite
Pelite
Fig. 5. Initial geometries for the porphyroblast deformation experiments showing
porphyroblast abundances within the pelitic
beds ranging from 0 to 70%. Porphyroblasts
are black.
higher strain rates are recorded in the interlayered
psammite (Fig. 7).
The relative strength of porphyroblast-rich model
pelite layers and psammite layers is strongly dependent
on the relative viscosities of the two layers prior to
2006 Blackwell Publishing Ltd
Psammite
porphyroblast growth (Fig. 6). At low initial viscosity
contrasts (gpsa:gpel ¼ 2), the pelitic layer becomes as
viscous as the psammite layer at porphyroblast abundance of c. 11%, and is twice as viscous at a porphyroblast abundance of c. 30%. At an intermediate
10 W.G. GROOME ET AL.
the increasing viscosity of the pelitic layers during
porphyroblast growth causes an increase in the integrated viscosity of the layered succession.
OROGEN-SCALE NUMERICAL EXPERIMENTS
B)
(V
50
0.
75
1
0
0.
25
0.
5
ata
–0
.25
Normalized viscosity
100
ld
nta
–0
.5
_–
0.
75
e
rim
e
xp
E
Figure b
1
0
–1
0.25
0.5
B)
(R
0.75
1.0
0
ta
al d
a
ent
9
1 (VB)
10
0.5
Porphyroblast abundance (areal fraction)
erim
7
Exp
6
ηpel:ηpsa = 6.0
WM
Modelling environment
.5
5
ηpel:ηpsa = 4.0
4
–0
Normalized viscosity
8
3
ηpel:ηpsa = 2.0
2
1
0.0
Preliminary three-dimensional numerical models were
constructed to explore the effects of metamorphic
strengthening reactions in the middle crust on the
strain-rate distribution and topography of a collisional
orogen. The problem domain for this series of models
is a 400 · 450 km crustal block with a thickness of
50 km (Fig. 8). The models provide information about
the velocity and displacement fields, which are then
used to calculate strain rates within the problem
domain. The models consist of an upper crust described by a pressure-dependent plasticity constitutive
relationship and a lower crust described by a temperature-dependent plasticity constitutive relationship
constrained by empirical flow law data for wet
quartzite (Carter & Tsenn, 1987). The models presented here are preliminary, but serve to illustrate the
orogen-scale strain-rate distribution resulting from
heterogeneously distributed metamorphic strengthening in the middle to lower crust, as well as the effects of
these metamorphic reactions on the topography of an
orogen, without considering the effects of erosion.
B)
–1 (R
0.1
0.2
0.3
0.4
0.5
0.6
Porphyroblast abundance (areal fraction)
Fig. 6. (a) Normalized viscosity-porphyroblast volume fraction
graph showing the Voigt and Reuss theoretical limits, calculated
from the viscosity contrast between matrix and porphyroblast
grains, and the trend of the experimental data. (b) Detail of the
region outlined in panel a. The initial viscosity contrasts between
pelitic and psammitic beds are indicated to compare the viscosity
of the porphyroblast-rich layer with the psammite layer. WM
indicates the estimated porphyroblast abundance in the study
area.
viscosity contrast (gpsa:gpel ¼ 4), the pelitic layer does
not become as viscous as the psammitic layer until
porphyroblast abundance exceeds 30%. Finally, at
high viscosity contrasts (gpsa:gpel ¼ 6), the pelitic layer
does not become as viscous as the psammitic layer until
porphyroblast abundance exceeds 40%. At porphyroblast abundance >40%, the viscosity of the pelitic
layer increases rapidly, which is consistent with predictions based on the theoretical Reuss bound.
Regardless of the initial viscosity contrast between
porphyroblast-free pelitic layers and psammitic layers,
Three-dimensional orogen-scale models were developed using the numerical code FLAC3D, which was
modified to accommodate large strains (e.g. Koons
et al., 2002; Upton et al., 2003; Johnson et al., 2004).
Materials in the models are represented by polyhedral
elements within a three-dimensional grid using an
explicit, time-marching solution and a form of dynamic relaxation. Each element responds according to
a prescribed linear and non-linear stress–strain law in
response to applied forces or kinematic boundary
conditions. The pressure-dependent strength of the
upper crust in our models is based on the Mohr–
Coulomb constitutive relationship:
rc ¼ rn tan / þ C;
ð7Þ
where rc is the critical shear stress at failure, rn is the
normal stress, / is the internal angle of friction and C
is cohesion. The models presented here have a 10-km
thick upper crust, the strength of which is described by
Eq. 7. The relatively thin Mohr–Coulomb upper crust
was used to simulate the effects of an elevated geothermal gradient, which would decrease the depth
to the brittle-ductile transition. The temperaturedependent strength of the middle and lower crust is
based on the Von Mises failure criterion:
rc ¼ K/ ;
ð8Þ
where K/ is the shear strength of the material. The
temperature-dependent weakening of the ductile part of
2006 Blackwell Publishing Ltd
EFFECTIVE VISCOSITY OF PORPHYROBLASTIC ROCKS 11
(a)
Undeformed
10% Porphyroblasts
Deformed
SSR
1.0
Psammite
Pelite
Psammite
0.0
0.0
0.0 0.2 0.4 0.6 0.8 1.0
(b)
0.2 0.4 0.6 0.8 1.0
Shear strain rate
Log10viscosity
Undeformed
20% Porphyroblasts
Deformed
SSR
1.0
Psammite
Pelite
Psammite
0.0
0.0
0.0 0.2 0.4 0.6 0.8 1.0
(c)
0.2 0.4 0.6 0.8 1.0
Shear strain rate
Log10viscosity
Undeformed
30% Porphyroblasts
Deformed
SSR
1.0
Psammite
Pelite
Psammite
0.0
0.0
0.0 0.2 0.4 0.6 0.8 1.0
(d)
0.2 0.4 0.6 0.8 1.0
Shear strain rate
Log10viscosity
Undeformed
40% Porphyroblasts
Deformed
SSR
1.0
Psammite
Pelite
Psammite
0.0
0.0 0.2 0.4 0.6 0.8 1.0
Log10viscosity
0.0
0.2 0.4 0.6 0.8 1.0
Shear strain rate
Fig. 7. Model results for porphyroblast abundances of 10% (a), 20% (b), 30% (c) and 40% (d) with an initial viscosity contrast of 2:1
(gpsa:gpel). The panels are the same as in Fig. 5. Note the higher strain rates in the psammitic layer as porphyroblast abundance
increases in the pelitic layer.
the crust is characterized by decreasing the value of K/
with increasing depth in accordance with the flow stress
for wet quartzite from Carter & Tsenn (1987). A perfectly plastic approximation of the power law flow law
for wet quartzite is used because in our models there is
2006 Blackwell Publishing Ltd
less than a threefold difference in strain rate from the
reference strain rate of 10)14 used to approximate the
flow stress for wet quartzite (Carter & Tsenn, 1987), so
a perfectly plastic constitutive relationship is a reasonable approximation (e.g. see Chapple, 1978).
12 W.G. GROOME ET AL.
400
km
Zone of
metamorphic
strengthening
15 km depth
50 km
∆σ
BDT=10 km
Elastic
backstop
Elastic lithospheric mantle
450 km
50 km
∆σ
∆σ
Elastic lithospheric mantle
Elastic
backstop
Zone of
metamorphic
strengthening
In order to model the effects of a crustal volume that
has undergone metamorphic strengthening, we assign a
40 · 100 · 5 km (X, Y, Z) crustal volume to have a
Von Mises yield stress four times greater than the
surrounding material, which is slightly greater than the
amount of strengthening recorded in the bed-scale
models at c. 30% porphyroblast. A fourfold increase in
bulk strength (treated in this model as yield stress)
would correspond to a fourfold difference in effective
viscosity between porphyroblast-free pelitic units and
psammitic units, such that 30% porphyroblast abundance would result in pelitic units being as viscous as
psammitic units, and a fourfold increase in bulk
effective viscosity of this region (Fig. 6). The X- and
Z-dimensions of our rheological heterogeneity are
intended to represent the size of the zone of metamorphic strengthening in eastern New Hamsphire and
Western Maine during the early Acadian Orogeny,
prior to overprinting by extensive migmatization (e.g.
Guidotti, 1989; Solar & Brown, 1999). The 100-km
Y-dimension is somewhat arbitrary and chosen to
eliminate boundary effects when evaluating mechanical data in X-Z cross-sections.
Model results
Two models are presented here, a reference model with
no strengthened zone and a model with a strengthened
zone in the core of the orogen. We use a subducting
elastic lithospheric mantle overlain by a 30-km thick
deformable crustal section, simulating the effects of
continental collision or the accretion of an exotic ter-
Fig. 8. Schematic diagram of the orogenscale model domain. Top: Perspective view
of the problem domain showing the rheological layering and applied boundary conditions. BDT ¼ Brittle-ductile transition.
The dashed white line indicates the crosssection line. Bottom: Cross-section through
the centre of the model showing the position
of the zone of metamorphic strengthening
(white box).
rane. A constant velocity boundary condition is imposed along the base of the elastic slab and a fixed, farfield elastic backstop on the overriding plate, similar to
models described in Koons et al. (2002). Internal
velocities are not prescribed, but are allowed to develop in response to boundary kinematics and internal
rheological definitions. Margin-parallel velocity in
both models is fixed at zero, simulating orthogonal
collision.
A reference model is presented with a homogeneous
10-km-thick upper crust having a pressure-dependent
yield strength underlain by a 20- to 40-km-thick plastic
lower crust having a temperature-dependent yield
strength is used for comparison with a model having a
strengthened zone in the middle crust. The model
topography after c. 10 km of convergence develops
into a uniform, two-sided orogenic wedge, similar to
those described in Koons (1990, 1995) and Willett
et al. (1993) (Fig. 9). A cross-section of the velocity
and vertical displacement fields for this model is shown
in Fig. 10 along with a shear strain-rate map. Three
zones of high shear strain-rate develop, which define
shear zones along the crust-mantle boundary as well as
orogen-bounding shear zones.
The presence of a strengthened zone alters the
topography of the orogenic wedge (Fig. 9) and changes
the strain-rate distribution at depth (Fig. 10). The
strengthened zone is intended to simulate the effects of
a bulk strengthening of a particular stratigraphic succession, although in nature only certain layers within a
compositionally layered succession will undergo
metamorphic strengthening reactions. The size of the
2006 Blackwell Publishing Ltd
EFFECTIVE VISCOSITY OF PORPHYROBLASTIC ROCKS 13
400 km
size of the zone of metamorphic strengthening. However, the models presented here clearly show that
metamorphic strengthening reactions can alter the geodynamic evolution of an orogen.
DISCUSSION
400 km
450 km
450 km
Fig. 9. Model topography for the reference (top) and metamorphic strengthened (bottom) models. The dashed line indicates the position of the cross-sections shown in Fig. 10.
Contour interval ¼ 500 m, 10· vertical exaggeration.
strengthened zone in our model is broadly constrained
by the approximate extent of Devonian-aged andalusite-grade metamorphism in eastern New Hampshire
and western Maine (e.g. Guidotti, 1989), although the
actual size of the zone of strengthening in northern
New England is unknown. The topography of this
model is shown in Fig. 9, and a change in topographic
slope is evident compared with the reference model,
marking the location of the zone of metamorphic
strengthening. The velocity and vertical displacement
fields along with the shear strain-rate map for this
model are shown in Fig. 10. Three high strain-rate
zones are present in this model, defining a shear zone
at the base of the crust as well as two orogen-bounding
shear zones similar to the reference model, but the
strong zone in the middle crust alters the location of
high strain-rate zones within the orogenic wedge.
Comparative diagrams showing the differences in
vertical displacement and shear strain rate between the
reference and strengthened models demonstrate the
effects of a zone of metamorphic strengthening
(Fig. 11). The presence of a strong zone in the middle
crust leads to the development of zones of enhanced
uplift outboard and inboard of the strengthened zone,
which correspond to relatively high shear strain-rate
zones at depth around the strengthened zone. As previously stated, these models are preliminary and are
used to illustrate the possible orogen-scale effects of
mid-crustal metamorphic strengthening; thus, we are
presently unable to address the sensitivity of these
results to geographical location, strength contrast and
2006 Blackwell Publishing Ltd
It is clear from published analytical models and the
numerical models presented here that the growth of
effectively rigid porphyroblasts in metapelitic rocks
will strengthen them relative to unmetamorphosed or
porphyroblast-free pelitic rocks. The extent to which
the porphyroblastic metapelitic rocks strengthen relative to interlayered rocks not undergoing porphyroblast-growth reactions is dependent on the final
porphyroblast abundance in the layer and the initial
relative strengths of the different bulk compositions.
At low porphyroblast abundances, metapelitic layers
are unlikely to become stronger than porphyroblastfree layers unless the initial viscosity contrast is small.
Published estimates of viscosity contrasts between
metapsammitic and porphyroblast-free metapelitic
rocks are on the order of two to 10 (e.g. Treagus,
1999; Kenis et al., 2005). Based on the results presented here, in order for metapelitic layers to become
stronger than interlayered metapsammitic units
during porphyroblast growth, the initial effective viscosity contrast between the two units has to be at
the low end of the published estimates (less than
c. gpsa:gpel ¼ 4).
Using our natural example from eastern New
Hampshire, where the porphyroblastic metapelitic
layers were approximately two to three times more
viscous than the interlayered metapsammitic layers, we
hypothesize that metapsammitic units have effective
viscosities approximately two times greater than porphyroblast-free metapelitic units at amphibolite facies
conditions. In our natural example, the metapelitic
layers have c. 30% andalusite porphyroblasts by volume, which is in the mid-range of the porphyroblast
abundance of our numerical models. In the numerical
models, when porphyroblast abundance is c. 30%, the
model pelite layer is approximately twice as viscous as
the porphyroblast-free layer when the initial viscosity
contrast (gpsa:gpel) is two, but the two layers have the
same viscosity when the initial viscosity contrast
(gpsa:gpel) is four. In the experiment with an initial
viscosity ratio of six, the model pelite layer was still
weaker than the model psammite layer at 30% porphyroblast abundance.
The extent to which porphyroblast growth will
strengthen the middle crust is dependent on the geographical extent of the metamorphic strengthening
reaction. In the field example from eastern New
Hampshire, andalusite-rich metapelitic rocks outcrop
over a minimum area of c. 50 km2. The original extent
of andalusite schist in the study are was probably much
greater, perhaps as much as 100 km2 based on extent
of correlative rocks in the study area, but overprinting
Depth (km)
Depth (km)
Depth (km)
200
0
0
300
300
1
2
Shear strain rate (10–14/s)
X-position (km)
250
0
00
+3
350
350
350
350
400
Elastic
backstop
400
Elastic
backstop
400
Elastic
backstop
400
Elastic
backstop
200
Shear strain rate
150
40
20
0
200
300
300
300
300
X-position (km)
250
X-position (km)
250
–1000
0
00
+1
X-position (km)
250
X-position (km)
250
Vertical displacement (m)
150
40
20
0
200
Velocity field
150
40
20
0
200
Lithospheric mantle
40
150
Ductile lower crust
20
+3000
Strengthened model
Schematic
Brittle upper crust
0
0
350
350
350
350
400
Elastic
backstop
400
Elastic
backstop
400
Elastic
backstop
400
Elastic
backstop
Fig. 10. Cross-sectional profiles through the model domains showing the model schematic, velocity fields, contour maps of the vertical displacement and shear strain-rate maps.
150
40
300
X-position (km)
250
0
Shear strain rate
200
–1000
00
20
250
+1
0
150
40
20
0
300
X-position (km)
250
X-position (km)
Vertical displacement (m)
200
Velocity field
150
40
20
0
200
Lithospheric mantle
40
150
Ductile lower crust
20
Depth (km)
Depth (km)
Depth (km)
Reference model
Schematic
Brittle upper crust
0
00
+2
Depth (km)
0
00
+2
Depth (km)
0
14 W.G. GROOME ET AL.
2006 Blackwell Publishing Ltd
EFFECTIVE VISCOSITY OF PORPHYROBLASTIC ROCKS 15
Difference in vertical displacement
100
300
Suppressed uplift
up
lift
100
–2
100
00
00
0
–100
10
200
Depth (km)
up
–1
200
0
20
0
0
0
10
200
100
20
0
0
40
300
200
400
X position (km)
Strengthened zone
Percent difference in shear strain rate
Depth (km)
0
0
–20
20
–20
–40
+40+20
0
0
+20
+4
0
0
40
200
+20
–2 0
+20
+60
+40
+20
0
0
–20
0
metamorphism has obscured the ultimate extent of
andalusite metamorphism.
During Acadian deformation, this 100 km2 region
would have behaved as a lozenge of relatively strong
rock in the middle crust and would probably have
caused strain to partition around it. This mid-crustal
strain partitioning would have affected the geodynamic
evolution of the developing orogen by: (1) potentially
enhancing exhumation rates of deep crustal rocks
during channel flow, which would alter the geothermal
structure of the orogen (e.g. Koons, 1987; Beaumont
et al., 2001; Zeitler et al., 2001; Jamieson et al., 2002;
Koons et al., 2002, 2003); (2) altering the topographic
expression above the zone of active metamorphism
(e.g. Koons, 1995; Petrini et al., 2001; Husson &
Sempere, 2003; Jackson et al., 2004), potentially leading to the development of intramontane sedimentary
basins, similar to those described in the Himalaya
attributed to lithospheric strength heterogeneities (e.g.
England & Houseman, 1985; Neil & Houseman, 1997);
and (3) shifting strain distribution, which may serve to
catalyse metamorphic reactions in peripheral parts of
the crust if they are in thermal disequilibrium (e.g.
Brodie & Rutter, 1985; Rubie, 1986; Koons et al.,
1987; Bell & Hayward, 1991; Bell et al., 2004).
At the grain-scale, the growth of porphyroblasts
leads to heterogeneities in strain-rate distribution
within the layer experiencing porphyroblast growth
2006 Blackwell Publishing Ltd
ed
0
ha
En
Fig. 11. Contour maps of the difference in
vertical displacement (top) and shear strainrate between the reference model and the
metamorphic strengthened model. Top:
Perspective view through the middle of the
model domain showing the difference in
vertical displacement, calculated by subtracting the strengthened model displacement from the reference model. Contour
interval ¼ 50 m. Note the development of
zones of enhanced uplift around the periphery of the strengthened zone and a zone
of suppressed uplift immediately above the
strengthened zone. Bottom: Contour map of
the percent difference of shear strain-rate
between the strengthened model and the
reference model, calculated by dividing the
strain-rate of the strengthened model by the
strain-rate in the reference model and subtracting 100. Contour interval is 20%. Positive values indicate zones where the shear
strain-rate is higher in the strengthened
model than the reference model. The position of the strengthened zone is indicated on
both diagrams.
nc
d
e
nc
ha
lift
En
Yp
osi
tion
(km
)
400
+40
+20
0
0
+20
+20
+40
300
400
X position (km)
(Fig. 12). These heterogeneities lead to the development of strain-rate shadows and high strain-rate zones
around the porphyroblast, which are oriented within
the flow field. The formation of high strain-rate zones
around porphyroblasts can enhance metamorphic
reaction rates by: (1) increasing dislocation densities in
quartz and feldspar, which adds strain energy to the
reaction, lowering the energy barrier for reaction and
increasing reaction rates (e.g. Brodie & Rutter, 1985;
Bell & Hayward, 1991); (2) causing dilatent zones to
develop along the grain boundaries, which would
transiently increase the permeability of the rock,
allowing fluids to either leave or enter the reaction site,
thus increasing reaction rates (e.g. Etheridge et al.,
1983; Brodie & Rutter, 1985; Frueh-Green, 1994;
Graham et al., 1997; Farver & Yund, 1999; Zhang
et al., 2000; Tenthorey & Cox, 2003); and (3) advecting
reactant components to the site of reaction, which
reduces the length scale for diffusive mass transfer (e.g.
Koons et al., 1987).
If a rock body is in disequilibrium with respect to
ambient pressure–temperature conditions, the initial
growth of porphyroblasts may lead to a feedback
relationship between strain partitioning and metamorphic reaction. The development of high strain-rate
zones around effectively rigid porphyroblasts in rocks
in disequilibrium with respect to pressure and temperature could lead to the rapid advection of reactant
16 W.G. GROOME ET AL.
Di
l
HSRZ
Dilation along grain boundary
HSRZ
components to the site of reaction, which reduces the
length scale for diffusional mass transfer and potentially increases the reaction rate (e.g. Koons et al.,
1987). Furthermore, if high strain-rate gradients along
the margins of porphyroblasts lead to the opening of
grain boundaries, increasing permeability, the advection of fluids into or out of the reaction sites may
catalyse reactions (e.g. Etheridge et al., 1983; Rubie,
1986; Graham et al., 1997). Thus, during porphyroblast growth, local heterogeneities may allow for rapid
porphyroblast-producing reactions, which would in
turn lead to rapid changes in rock strength, leading to
shifting strain-rate distribution on a larger scale.
ACKNOWLEDGEMENTS
Funding for this project came from NSF Grant #
EAR0207717 (to S.E.J. and P.O.K.), the Geological
Society of America Grants in Aid Program (to
W.G.G.) and the University of Maine (to W.G.G.).
We thank M. Washburn for field assistance, D. Eusden
for introducing us to the geology of the White
Mountains region, M. Jessell, L. Evans and G.
Houseman for help with ELLE and BASIL, and the
Appalachian Mountain Club, the US Forest Service
Androscoggin Ranger Station and the Mount Washington Auto Road for logistical assistance in the field.
Constructive journal reviews by E. Baxter, S. Ji and
D. Whitney improved this manuscript.
Fig. 12. Shear strain rate contour map and
model geometry for the 20% porphyroblast
abundance experiment and detailed diagrams of select regions showing the high
strain-rate zones around porphyroblasts
(HSRZ) and zones of dilatation (Dil) between porphyroblasts. Also shown is a zone
where grain boundary dilation would likely
occur in a natural rock.
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