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ANNALS  OF  GEOPHYSICS,  VOL.  47,  N.  4,  August  2004 
Key  words dyke injection – dyke arrest – crustal
stresses – crustal layering – surface deformation –
volcanic hazard
1.  Introduction
What are the mechanical conditions that make
it possible for magma to flow from a source
chamber to the surface of a volcano, resulting in
an eruption? Alternatively, what are the necessary
and sufficient conditions for a volcanic eruption
to occur? The answers to these questions are of
fundamental importance in volcanology; not only
because they improve our understanding of how
volcanoes work, but also because they must be
known in order to assess volcanic risks during un-
rest periods in rift zones and volcanic edifices. 
Field observations indicate that almost all
eruptions in volcanic edifices and rift zones are
supplied with magma through fractures that are
generated by magmatic overpressure. It follows
that the main mechanical condition for a vol-
canic eruption to occur is that an overpressured
magma-filled fracture is able to propagate to the
Mailing address: Dr. Agust Gudmundsson, Department
of Structural Geology and Geodynamics, Geoscience Centre,
University of Göttingen, Goldschmidtstrasse 3, 37077 Göt-
tingen, Germany; e-mail: Agust.Gudmundsson@gwdg.de
Local stresses, dyke arrest and surface
deformation in volcanic edifices
and rift zones
Agust Gudmundsson (1) and Sonja L. Brenner (2)
(1) Department of Structural Geology and Geodynamics, Geoscience Centre, University of Göttingen, Germany
(2) Department of Earth Sciences, University of Bergen, Norway
Abstract
Field studies indicate that nearly all eruptions in volcanic edifices and rift zones are supplied with magma through
fractures (dykes) that are opened by magmatic overpressure. While (inferred) dyke injections are frequent during un-
rest periods, volcanic eruptions are, in comparison, infrequent, suggesting that most dykes become arrested at certain
depths in the crust, in agreement with field studies. The frequency of dyke arrest can be partly explained by the nu-
merical models presented here which indicate that volcanic edifices and rift zones consisting of rocks of contrasting
mechanical properties, such as soft pyroclastic layers and stiff lava flows, commonly develop local stress fields that
encourage dyke arrest. During unrest, surface deformation studies are routinely used to infer the geometries of ar-
rested dykes, and some models (using homogeneous, isotropic half-spaces) infer large grabens to be induced by such
dykes. Our results, however, show that the dyke-tip tensile stresses are normally much greater than the induced sur-
face stresses, making it difficult to explain how a dyke can induce surface stresses in excess of the tensile (or shear)
strength while the same strength is not exceeded at the (arrested) dyke tip. Also, arrested dyke tips in eroded or ac-
tive rift zones are normally not associated with dyke-induced grabens or normal faults, and some dykes arrested with-
in a few metres of the surface do not generate faults or grabens. The numerical models show that abrupt changes in
Young’s moduli (stiffnesses), layers with relatively high dyke-normal compressive stresses (stress barriers), and weak
horizontal contacts may make the dyke-induced surface tensile stresses too small for significant fault or graben for-
mation to occur in rift zones or volcanic edifices. Also, these small surface stresses may have no simple relation to
the dyke geometry or the depth to its tip. Thus, for a layered crust with weak contacts, straightforward inversion of
surface geodetic data may lead to unreliable geometries of arrested dykes in active rift zones and volcanic edifices.
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Agust Gudmundsson and Sonja L. Brenner
Fig.  1. 400 m thick Tertiary pile of lava flows, sediments and pyroclastic rocks in the valley of Vidbordsdalur
in Southeast Iceland. View northeast, the general dip of the pile increases by 2-3° from its top to the bottom. The
pile is dissected by dykes, some of which are arrested, that reach thicknesses of up to 13 m.
Fig.  2. Horizontal contacts between flow units and vertical columnar joints in a thick pahoehoe flow. Dykes that
meet with such contacts commonly become offset or arrested. This 20-m-tall wall (see the person for scale) belongs
to a normal fault that dissects the Krafla Volcanic System of the Holocene rift zone in North Iceland (cf. fig. 5).
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Local stresses, dyke arrest and surface deformation in volcanic edifices and rift zones
surface. Field observations, however, also show
that many magma-filled fractures never make it
to the surface but rather solidify as arrested
dykes (Gudmundsson, 2002). Many dykes have
their tips arrested in essentially homogeneous
and isotropic rock bodies, whereas others are ar-
rested in heterogeneous and anisotropic rock
bodies, particularly at contacts in layered rocks.
Most volcanic edifices and rift zones are com-
posed of layered rocks (fig. 1). The layers are nor-
mally with very different mechanical properties.
For example, there are commonly soft (low
Young’s moduli) sedimentary and pyroclastic lay-
ers alternating with stiff (high Young’s moduli) la-
va flows and intrusions. Furthermore, many layer
contacts are weak (non-welded) or even open,
particularly at shallow depths (fig. 2). When a
dyke meets with weak or open contacts, or layers
of contrasting mechanical properties, it normally
becomes either offset (fig. 3) or arrested (fig. 4).
Thus, observational data show that dykes
commonly become arrested or, more generally,
offset on meeting with contacts between layers of
contrasting mechanical properties in volcanic edi-
43
Fig.  3. Dyke becomes offset on crossing a weak scoria contact between two basaltic lava flows, of Tertiary age,
in the palaeorift zone of North Iceland. View north, the dyke is around 1 m thick; the backpack at the lower dyke
segment provides a scale.
Fig.  4. Arrested tip of a basaltic dyke in a vertical exposure in the lava pile of Tenerife (Canary Islands). The
dyke exposure is 3.7 m tall, strikes E-W, and dissects a baked pyroclastic rock layer in the lower part of the pho-
tograph. The dyke is 0.26 m thick near the bottom of the exposure (at a road), and its thickness is similar up to
near the contact between the pyroclastic rock and the basaltic lava flow (on top). There the dyke abruptly tapers
away towards its tip. No dyke-induced joints or faults occur at the dyke tip.
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Agust Gudmundsson and Sonja L. Brenner
Fig.  5. The active volcanic zones of Iceland consist of volcanic systems most of which have associated central
volcanoes (stratovolcanoes). The main volcanic zones are the North Volcanic Zone, the East Volcanic Zone and the
West Volcanic Zone. The active rift zone comprises all these zones except for the southernmost part of the East Vol-
canic Zone which is a propagating rift (cf. Gudmundsson, 2000). FVZ stands for the Fremrinamur Volcanic Sys-
tem, KVS for the Krafla Volcanic System, GVS for the Grimsvotn Volcanic System, and RVS for the Reykjanes
Volcanic System.
fices and rift zones. These data indicate that me-
chanical, rather than thermal, conditions are of pri-
mary importance in controlling the arrest of dykes
or, alternatively, their propagation to the surface as
feeder dykes. These mechanical conditions are re-
flected in the local stress fields associated with in-
dividual layers and contacts in the volcanic edi-
fices. When a dyke meets with a contact between
rock layers of contrasting mechanical properties, it
is the local stress field associated with these layers
that controls whether or not the dyke propagates
through the contact (with or without an offset) or,
alternatively, becomes arrested (Gudmundsson,
2002; Gudmundsson et al., 2002).
Many authors have suggested that a dyke
arrested at a considerable depth below the sur-
face of a volcanic edifice or a rift zone during a
period of unrest may contribute to brittle defor-
mation of the surface (e.g., Pollard and
Holzhausen, 1979; Okada, 1985; Rubin and
Pollard, 1988). Most models used to inverse
surface data to infer dyke geometries assume
that the host rock may be modelled as a homo-
geneous, isotropic half space and imply that the
host-rock stresses are uniform. Although the
use of homogeneous, isotropic half-space mod-
els for analysing ground deformation in volca-
noes has been criticised (e.g., De Natale and
Pingue, 1993), they continue to be routinely
used, presumably because they are simple (e.g.,
Jousset et al., 2003), to help assess volcanic
hazards during periods of unrest. 
Some of the homogeneous, isotropic half-
space models predict that stresses associated
with arrested dykes may induce major grabens
at rift-zone and edifice surfaces (Pollard and
Holz-hausen, 1979). Many tens of arrested
dyke tips have been observed in eroded rift
zones, however, and few, if any, of these are as-
sociated with grabens or normal faults that
could be generated by dyke-induced stresses
ahead of the tips (Gudmundsson, 2002, 2003).
Also, normal faults and grabens tend to be more
common in those parts of eroded rift zones
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Local stresses, dyke arrest and surface deformation in volcanic edifices and rift zones
where dykes are relatively rare, and there are
dykes arrested within a few metres of the sur-
face that do not generate faults or grabens at the
surface (Gudmundsson, 2003).
In this paper the focus is on the effects of lo-
cal stress fields on dyke propagation, dyke ar-
rest, and the surface deformation associated
with arrested dykes in rift zones and volcanic
edifices. The first part of the paper reviews the
results of field observations of several thousand
dykes, primarily in Iceland (fig. 5) and Tenerife
(Canary Islands), focusing on feeder dykes, ar-
rested dyke tips, and associated dyke tips and
surface deformations. The second part of the
paper presents analytical and numerical models
as to how local stress fields in layered rocks af-
fect dyke propagation, dyke arrest and surface
deformation in rift zones and volcanic edifices.
2.  Feeder dykes
For a volcanic eruption to occur, it is nor-
mally necessary that a magma chamber be-
Fig.  6. Part of a volcanic fissure with crater cones in the Grimsvotn Volcanic System of the East Volcanic Zone
(fig. 5). View northeast, this crater row, which has a total length of 27 km (Thordarson and Self, 1993), was
formed during the Laki 1783 eruption, the largest historical eruption in Iceland.
Fig.  7. On solidification, subvertical magma-driven fractures that reach the surface become feeder dykes, like
this 6000-8000-year-old feeder dyke in the canyon of the river Jokulsa a Fjollum, at the eastern margin of Frem-
rinamur Volcanic System of the active rift zone in North Iceland (fig. 5). The dyke is around 6 m thick where the
person is standing, but gradually increases in thickness to around 10 m where it meets the free surface at the time
of eruption and connects with a crater cone (cf. Gudmundsson, 1995).
6 7
1438
Agust Gudmundsson and Sonja L. Brenner
comes ruptured and a magma-filled fracture
forms. But this condition is not sufficient. An
additional condition is that the magma-filled
fracture must be able to propagate to the surface,
through all the many local stress fields that af-
fect its potential pathway. When the magma-
filled fracture reaches the surface, it forms a vol-
canic fissure (fig. 6), an opening-mode fracture
(mode I crack) that subsequently develops crater
cones and is then referred to as a crater row.
On solidification, the magma-filled fractures
that feed crater rows become feeder dykes. One
of the best-exposed feeder dykes in Iceland is the
one that supplied magma to the 50-70 km long
Sveinar-Randarholar crater row in Northeast Ice-
land (fig. 7). This 6000-8000-year-old feeder
dyke is best exposed in the east walls of a 100 m-
deep canyon of a glacial river where it cuts
through the Holocene volcanic rift zone in North-
east Iceland. The dyke is in a visible connection
with one of the crater cones of the crater row,
where it reaches a thickness of around 10 m, but
thins to 5-7 m in the deeper parts of the exposure.
The greater thickness (opening) of the volcanic
fissure where it dissects the palaeosurface of the
rift zone (on which the crater cone rests) is due to
the free-surface effects on the opening-displace-
ment of the fracture (Tada et al., 2000).
Several other feeder dykes are known from
Iceland. Most of them, however, are located in-
side the active volcanic zones, such as the one in
fig. 7. Generally, surprisingly few well-document-
ed feeder dykes have been reported from active
and extinct volcanic areas, indicating that much
more work is needed in this field of volcanology.
By contrast, many arrested dykes have been de-
scribed. The main features of typical arrested
dyke tips are summarised in the next section.
3.  Arrested and offset dykes
As an example of a dyke arrested at the con-
tact between layers of contrasting mechanical
properties, we consider the dyke in fig. 4. This
is an E-W striking basaltic dyke that cuts
through a layer of pyroclastic rock in lower part
of the exposure. At the bottom of the 3.7-m-tall
exposure, the dyke is 0.26 m thick, and main-
tains roughly that thickness up to the contact be-
tween the pyroclastic rock and the basaltic lava
flow at the surface. At the contact, the dyke thins
abruptly and ends vertically in the scoracious
bottom of the basaltic lava flow. As is common
in such situations, the dyke may continue as an
offset segment higher up in the lava pile. How-
ever, there are no visible igneous veins or other
connections between the segment seen to end
vertically in fig. 4 and dyke segments at shal-
lower levels in the pile exposed in this section.
Another example of an arrested dyke occurs
in well-exposed cliffs where the Holo- cene rift
zone of Southwest Iceland enters into the sea
(fig. 8). At the bottom of the 8 m-tall exposure,
the dyke thickness is 0.34 m but decreases on
approaching the contact between the pyroclas-
tic rock and the Holocene basaltic lava flow
above. This lava flow, referred to as Yngri
Stampar, was erupted between 1210-1240 A.D.
(Sigurgeirsson, 1995), so that the dyke must be
of around this age or younger. The dyke tip,
which is rounded, is arrested at the contact. Al-
though the tip is arrested at only 5 m below the
surface of the Holocene rift zone, there is no
visible faulting or graben formation in the lava
flow between the tip and the surface.
Some of the many typical ends of arrested
dykes are indicated in fig. 9a-d. These are based
on field studies of around 14-Ma-old basaltic
dykes in the Tertiary lava pile in Northwest Ice-
land. In thick, comparatively homogeneous and
isotropic lava flows most dykes end vertically by
tapering away (fig. 9a). Similar dyke tips are ob-
served, both in lava flows and in thick pyroclastic
layers, in Tenerife (Marinoni and Gudmundsson,
2000). Although many dykes become arrested on
meeting with subhorizontal contacts (figs. 4 and
8), some become arrested on meeting with in-
clined fractures (fig. 9b). It is common to see the
dyke tip expanding on meeting with a soft layer,
particularly scoria layers at the bottom of stiff la-
va flows, in which case the dyke may be offset
across the contact (fig. 3). Many offset dykes try
several pathways, sending off thin, igneous veins,
before selecting the eventual pathway. Other
dykes expand on meeting with stiff lava flows
(which may also have comparatively high tensile
strengths) and propagate partly into the bottoms
of the lava flows, but fail to propagate through the
lavas and so become arrested (fig. 9c,d).
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Local stresses, dyke arrest and surface deformation in volcanic edifices and rift zones
Fig.  8. Arrested dyke tip exposed in sea cliffs that
cut through the Reykjanes Volcanic System in the
Holocene rift zone of Southwest Iceland (fig. 5).
This basaltic dyke strikes northeast, is subvertical,
and cuts through a layer of pyroclastic rock (volcanic
tuff) in the lower part of the photograph. The dyke
tip is arrested at the contact between the pyroclastic
rock and the Holocene basaltic lava flow that forms
the surface of the rift zone. The dyke thickness is
from 0.34 m, at the bottom of the 8 m tall exposure,
to 0.1 m at the tip. There are no dyke-induced frac-
tures or normal faults above the dyke tip.
Fig.  9a-d. Four arrested dykes in the (14 Ma)
basaltic lava pile of Northwest Iceland. In this
schematic illustration, the dykes are shaded, the sco-
ria layers between the lava flows are hatched, and the
basaltic lava flows themselves are not shaded. The
dykes occur in outcrops which are steep to vertical
cliffs; the indicated scale is approximate. In (a) the
dyke tapers away in the lava pile, in (b) the dyke tip
is arrested at an oblique fracture (indicated by c) in a
thick lava flow, in (c) the dyke has propagated for a
short vertical distance into the bottom of the lava
flow and expanded its arrested tip (cf. fig. 3), and in
(d) the dyke sends off a thin, vertical igneous vein in-
to the bottom of a massive lava flow but otherwise
expands and becomes arrested at the contact between
the soft scoria layer and the stiff basaltic lava flow
(cf. fig. 3).
8
9
a b
dc
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Agust Gudmundsson and Sonja L. Brenner
Fig.  10. When dykes dissect contacts between lava
flows, particularly scoria layers that are still weak when
the dyke injection occurs, they commonly become lat-
erally and vertically offset (cf. fig. 3). Some dyke seg-
ments, however, are connected by thin igneous sheets,
like here. Lateral dyke offsets are also very common,
both in eroded, extinct dyke swarms, like here, as well
as at the surfaces of active rift zones and volcanic edi-
fices, where the feeder dykes are seen as volcanic fis-
sures. For example, the Laki Volcanic Fissure, part of
which is seen in fig. 6, is offset in many places (cf.
Thordarson and Self, 1993).
Dyke offsets across layers of contrasting
mechanical properties, such as illustrated in fig.
3, are very common. Some dyke segments are
connected by thin, igneous veins across the con-
tacts (fig. 10), in which case there is no doubt
that the segments belong to the same dyke. Oth-
er dyke segments, however, have no visible con-
nections; a feature that is particularly common
in lateral sections (Gudmundsson, 1995).
A remarkable feature of some offset dykes
(and sheets) is that they are continuous across the
contact between two layers (lava flows), but a
part of the dyke follows the subhorizontal con-
tact for a while (fig. 11). Examples are known
from Iceland where a dyke changes into a sill at
a contact between layers, follows the contact for
many metres and then continues its vertical
propagation up into the lava pile.
These results indicate that contacts between
layers of contrasting mechanical properties have
commonly strong effects on dyke propagation.
These effects are partly related to the mechanical
weakness of the contact itself, which may open up
Fig.  11. Inclined, 0.5-1 m thick sheet in the Pleistocene lava pile in Southwest Iceland. The contact between
the weak scoria (below) and the basaltic lava flow (on the top) has opened up, so as to allow the sheet to follow
it for several metres until the sheet continues its inclined propagation up through the lava flow on the top. Open-
ing of weak contacts when magma-filled fractures approach them is a common feature, as is modelled in fig. 20.
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Local stresses, dyke arrest and surface deformation in volcanic edifices and rift zones
when the dyke tip approaches. The dyke may fol-
low the contact as a sill for a while or end alto-
gether when meeting with the contact (Gud-
mundsson et al., 2002). Partly, however, these ef-
fects have to do with local stresses which may de-
velop at, and in the vicinity of, contacts between
rock layers of contrasting mechanical properties.
These stress fields are often unfavourable to dyke
propagation and may encourage dyke arrest.
4.  Stress fields favouring dyke arrest
Most dykes never reach the surface to sup-
ply magma to volcanic eruptions, but rather be-
come arrested. This conclusion is based on the-
oretical considerations (Gudmundsson, 2002)
and is supported by many field studies. For ex-
ample, in dyke profiles in Iceland and Tenerife
arrested dykes are much more common than
feeder dykes (Marinoni and Gudmundsson,
2000; Gudmundsson, 2002). Similar results are
obtained from ophiolites and dyke swarms ex-
posed in the walls of large fracture zones (Pal-
lister, 1981; Pedersen, 1986; Dilek et al., 1998;
Karson, 1998). Also, estimates of the magma
balance in active volcanoes indicate that the
volume of intruded magma greatly exceeds
that of erupted magma. For instance, the esti-
mated ratio of erupted to intruded volumes of
magma in the Krafla Central Volcano in North
Iceland (fig. 5) for the period 1975-1984 is
0.30, whereas that for Etna for the period 1980-
1995 is 0.13 (Harris et al., 2000).
On a regional scale, the general decrease in
stiffness (Young’s modulus) of the crust as the
surface of a volcanic rift zone is approached
(Palmason, 1971) indicates that most dykes
injected from magma reservoirs become ar-
rested on their way to the surface (Gudmunds-
son, 2002). As regards individual volcanic ed-
ifices and parts of volcanic rift zones, howev-
er, the arrest of dykes can be mostly explained
in terms of two related factors: stress barriers
and mechanical anisotropy. As used here, a
stress barrier is any layer with a local stress
field that is unfavourable to dyke propagation,
whereas a mechanical anisotropy covers dis-
continuities such as joints and weak or open
contacts that may arrest dykes.
More generally, the term stress barrier in-
cludes any rock body where the stress field is
unfavourable to the type of fracture propagation
under consideration, whether the fractures are
shear fractures (faults), such as in earthquakes,
or hydrofractures, such as mineral-filled veins,
inclined sheets, sills or dykes (Gudmundsson,
2002). Here the term stress barrier is used in the
restricted sense of a rock layer with a stress
field unsuitable for dyke propagation, and thus
encouraging dyke arrest. For vertical dyke
propagation, a stress barrier is thus a layer with
a locally high horizontal compressive stress.
Stress barriers may develop in many ways.
One is the gradual decrease in stiffness with al-
titude in rift zones. Another is the fall off in in-
tensity of the (relative or absolute) tensile
stresses generated around a pressured magma
chamber with increasing distance from that
chamber. Then dyke injection is favoured in the
vicinity of the chamber while at a certain dis-
tance from the chamber the stress field favours
dyke arrest. The third way for stress barriers to
form is when rock layers of contrasting me-
chanical properties meet. Depending on the
loading conditions, either the soft layers (dur-
ing extension) or the stiff layers (during com-
pression) may then act as stress barriers (cf.
Gudmundsson and Brenner, 2001).
4.1. Homogeneous, isotropic crust
Many magma chambers, particularly in the
later stages of their development, approach the
geometries of spheres and cylinders (Gud-
mundsson, 2002). We shall here consider the
analytical solutions for the stress fields around
ideal spherical and cylindrical magma cham-
bers to illustrate how dyke injection may be
favoured in the vicinity of the chambers while
dyke arrest is encouraged at greater distances
from the chambers.
Consider first a spherical magma chamber
with radius R1. To simplify the problem, we
consider the chamber to be deep-seated and lo-
cated within a larger rock sphere, of radius R2,
where R2 >> R1. Geologically, this means that
the chamber radius is small in comparison with
its depth below the free surface of the associat-
1442
Agust Gudmundsson and Sonja L. Brenner
ed volcanic edifice. The chamber ruptures and
initiates a dyke or and inclined sheet, depend-
ing on the stress situation and location of rup-
ture (Gudmundsson, 1998), when the following
conditions are satisfied:
Pt = σ3 + T0 (4.1)
or
pl + pe = σ3 + T0. (4.2)
In this notation, pe = Pt − pl, is the excess mag-
matic pressure in the chamber, that is, the differ-
ence between the total magma pressure, Pt, at the
time of chamber rupture and the lithostatic stress
pl. Field observations indicate that dykes and
sheets are mostly extension fractures (Gud-
mundsson, 1995), meaning that they form in
planes that are perpendicular to the local mini-
mum principal compressive (maximum tensile)
stress σ3. For a dyke injection to occur, the ex-
cess pressure, pe, in the chamber must reach the
local in situ tensile strength, T0, of the chamber
walls at the rupture sites. Following the conven-
tion in structural geology and rock mechanics,
compressive stress is considered positive and
tensile stress negative. However, in many of the
applications in this paper, we take the symbols to
represent absolute values, in which case the mi-
nus signs can be omitted.
Consider a chamber subject to a total magma
pressure Pt at its boundary R1 while there is litho-
static stress pl at the outer radius R2 of the spher-
ical elastic body within which the chamber is lo-
cated (in reality, a crustal segment). Using the to-
tal magma pressure Pt as defined in eq. (4.1), the
radial stress σr is given by (Saada, 1983)  
r t
33
P
r
R p
r
R1= + -v 1 1
l
f fp p
R
T
S
S
V
X
W
W. (4.3)
Similar equations can be presented for the ten-
sile stresses σθ and σφ (Saada, 1983; Gud-
mundsson, 2002). For these stresses, the spher-
ical polar coordinates (r, θ, φ) are used, r being
the radius vector, θ the angle between the radius
vector and a fixed axis, and φ the angle meas-
ured around this axis.
A simpler presentation of eq. (4.3), and the
related equations for σθ and σφ, can be obtained
if the excess magma pressure pe, as defined in
eq. (4.2), is used instead of the total pressure Pt.
Then eq. (4.3) reduces to  
r
3
ep r
R
=v 1f p (4.4)
and the equation for the tensile stresses σθ and
σφ (which are equal because of spherical sym-
metry) becomes  
3
zi r
R
2= =-v v
1ep f p
. (4.5)
Equations (4.4) and (4.5) show that the inten-
sity of the stress field associated with a deep-
seated spherical magma chamber falls off in-
versely as the cube of the distance r from the
chamber (fig. 12).
Similar results are obtained in case the mag-
ma chamber is modelled as a vertical cylinder.
Such a model is appropriate, for example, for
magma chambers that, on solidification, become
plugs or necks (if small) or stocks (if large). Many
volcanic edifices, particularly those that develop
swarms of radial dykes, are likely to have cylin-
drical magma chambers. Consider a cylindrical
chamber of radius R1 located in a cylindrical rock
body of radius R2, where R2 >> R1. Geologically,
the solid cylindrical rock body is just that part of
the crust surrounding the cylindrical magma
chamber which is of interest in the problem. As in
the case of a spherical magma chamber, the rock
body hosting the chamber is, for mathematical
convenience, assumed to have the same shape as
the chamber.
Using polar coordinates, with radius vec-
tor r and polar angle θ, we next consider a
cylindrical magma chamber subject to a total
magma pressure Pt, as defined in eq. (4.1), at
its radius, R1. The cylindrical rock body host-
ing the chamber is subject to an external total
horizontal compressive stress σΗ at R2. The
state of stress in the host rock is taken to be
isotropic so that the maximum, σΗ, and mini-
mum, σΗ, horizontal compressive stresses are
equal. The radial stress σr away from the
1443
Local stresses, dyke arrest and surface deformation in volcanic edifices and rift zones
chamber can then be obtained from the equa-
tion (Saada, 1983)
22
r t HP r
R
r
R1= + -v v1 1f fp p
R
T
S
S
V
X
W
W. (4.6)
As in the case of the spherical magma cham-
ber, a simpler presentation of eq. (4.6), and the
related equation for the tangential or circum-
ferential stress σθ, can be obtained if one sub-
stitutes the excess magma pressure pe for the
total magma pressure Pt. Then σΗ is taken to
be equal to the lithostatic stress in the crust
outside the chamber, so that σΗ = σh = pl. Then
eq. (4.6) reduces to  
er
2
P
r
R
=v 1f p
. (4.7)
Similarly, the circumferential stress is given by
(Gudmundsson, 2002)  
ei
2
P
r
R
=-v 1f p
. (4.8)
Equations (4.7) and (4.8) show that, for a verti-
cal cylindrical magma chamber, the intensity of
the stress field generated by the excess pressure
pe falls off with the square of the distance r
from the chamber (fig. 13).
From eqs. (4.1) and (4.2) it follows that the
excess pressure pe in a chamber at the time of rup-
ture and dyke injection is normally equal to the
local tensile strength, that is, pe = T0. In situ tensile
strengths of most rocks are low, the typical values
being 0.5-6 MPa (Haimson and Rummel, 1982;
Schultz, 1995; Amadei and Stephansson, 1997). It
follows that, at chamber rupture and initiation of
a dyke or an inclined sheet, the tensile stresses at
the margin of the chamber are small and diminish
rapidly with increasing distance along the poten-
tial pathway of the propagating dyke.
Consequently, in host rocks that can be mod-
elled as homogeneous and isotropic, the stress
Fig.  12. Decrease in intensity (magnitude) of the tensile circumferential stress σθ and the compressive radial
stress σr with increasing distance from the margin of a spherical magma chamber subject to an excess magmat-
ic pressure pe. The variation is seen in a vertical plane. The stresses are made dimensionless through dividing by
the magmatic excess pressure and the distance is made dimensionless by dividing the radius vector r (distance)
by the radius of the magma chamber R1.
Fig.  13.  Decrease in intensity (magnitude) of the tensile circumferential stress σθ and the compressive radial
stress σr with increasing distance from the margin of a (vertical) cylindrical magma chamber subject to an ex-
cess magmatic pressure pe. The variation is seen in a horizontal plane. The stresses are made dimensionless
through dividing by the magmatic excess pressure and the distance is made dimensionless by dividing the radius
vector r (distance) by the radius of the magma chamber R1.
1312
R σpe
R
r
σr
pe
pe
R
σ
pe
R
r
σr
pe
pe
1444
Agust Gudmundsson and Sonja L. Brenner
fields associated with ideal spherical and cylin-
drical magma chambers may commonly satisfy
the conditions for dyke injection and propaga-
tion in the vicinity of the chambers while at cer-
tain limited distances from the chambers satisfy
the conditions for dyke arrest. Many dykes may
thus propagate for only a short distance from the
source chamber until they enter a stress barrier
in which they become arrested.
4.2. Layered crust with a gradual increase
in stiffness with depth 
Although the variation in layer stiffness is
likely to be irregular in rift zones and volcanic ed-
ifices, seismic studies generally indicate an aver-
age increase in stiffness with increasing crustal
depths (Palmason, 1971). Here we use these re-
sults, together with analytical and numerical
models, to show that this increase in stiffness
with depth favours dyke arrest at shallow depths.
We consider flat, sill-like, elongated magma
reservoirs, such as presumably underlie large
parts of the rift-zone volcanic systems in Ice-
land (Gudmundsson, 2000). The conditions of
rupture and dyke injection (eqs. (4.1) and (4.2))
are presumably commonly reached through re-
duction in σ3 in the roof of a reservoir as a con-
sequence of buildup of relative or absolute ten-
sile stress generated by plate movements (plate
pull). The buildup of tensile stress can be in-
ferred from the one-dimensional Hooke’s law
in elasticity which relates tensile stress σ and
strain ε through the equation (Timoshenko and
Goodier, 1970)  
σ = Eε (4.9)
where E is the stiffness (Young’s modulus) of
the crust. In the convention used in this paper,
both the tensile stress and strain should be neg-
ative, but to simplify the discussion they are
here considered as absolute values and the mi-
nus signs are dropped.
The part of a rift zone that undergoes tensile
strain during plate movements (plate pull) corre-
sponds closely to the width u of the associated
volcanic system(s) (figs. 5 and 14). If the associ-
ated spreading rate is denoted by v (fig. 14), the
buildup rate of tensile stress dσ / dt in the rift zone
(when there is no shape-related stress concentra-
tion in the roof of the sill-like magma reservoir) is  
dt
d
u
vE
=
v
. (4.10)
The rate dσ / dt has the units of 1/yr, and the
time ti for the tensile stress in the crust to reach
its tensile strength T0 is  
00
i /
t
d dt
T
vE
uT
= =v . (4.11)
Although eq. (4.11) applies in the case of a sill-
like magma reservoir where there is no tensile
stress concentration in the chamber roof, the re-
sults can easily be extended to magma reser-
voirs of different shapes where there will be
stress concentration in the reservoir roof (Gud-
mundsson, 1998). Here, however, the focus is
on sill-like reservoirs.
The dyke-injection frequency associated
with the magma reservoir, if, is the reciprocal of
ti, and is, from eq. (4.11), given by  
0
i
t uT
vE1
= =f
i
. (4.12)
Fig.  14. The part of a rift zone that is subject to ten-
sile strain during plate movements (plate pull) with a
total spreading (opening) rate of v (half rate v/ 2) has
the width u (cf. Gudmundsson, 1990). In this model,
the crustal stiffness (Young’s modulus, E) increases
linearly with depth; the surface modulus being Esurface.
u
v v
E E
1445
Local stresses, dyke arrest and surface deformation in volcanic edifices and rift zones
Equation (4.12) applies to a single magma reser-
voir and, in particular, to a deep-seated magma
reservoir such as may exist beneath most of the
rift-zone volcanic systems in Iceland (Gud-
mundsson, 1995). These reservoirs normally
feed the eruptions in the rift-zone volcanic sys-
tems outside volcanic edifices, that is, outside
stratovolcanoes or central volcanoes (fig. 5).
Using the simple model of a general increase
in stiffness with increasing crustal depth, the
above equations can be applied to the dyke-injec-
tion frequency in a rift zone outside its central vol-
canoes (fig. 5) and compared with those obtained
from extinct volcanic systems in Iceland. Extinct
volcanic systems are represented by a typical dyke
swarm in the Tertiary lava pile of East Iceland (fig.
15), for which the width of the source reservoir
may be taken as 10 km (Gudmundsson, 1995).
In the Tertiary, as in South Iceland today (fig.
5), the rift zone was split in two parts, each one
taking up roughly half the spreading rate. Be-
cause, in addition, many of the dyke swarms are
parallel, or overlap to a large degree (fig. 5), the
spreading rate of a single swarm is of the order of
0.5 cm yr–1, that is, around one-fourth of the total
spreading (opening) rate in Iceland (Gudmunds-
son, 2000). The roof of the reservoir is taken to
be located in a host rock with a static stiffness
(Young’s modulus) of E = 67 GPa (Gudmunds-
son, 1990). In the model of Staples et al. (1997),
this stiffness implies that the reservoir is located
in the lowermost part of the crust, but in Palma-
son’s (1971) model the reservoir is located at the
boundary between the crust and upper mantle.
With a typical in situ tensile strength of T0 = 3
MPa, eq. (4.12) yields the dyke-injection fre-
quency if ≈ 0.01yr–1, meaning that one dyke is in-
jected from the reservoir every 100 years.
These theoretical results can be compared
with the number of dykes observed in a typical
Tertiary dyke swarm in Iceland such as the
Alftafjördur Dyke Swarm in East Iceland (fig.
15; Gudmundsson, 1995). The dykes in this
swarm are mostly exposed at crustal depths of
0.5-1.3 km below the initial top of the lava pile,
which has been eroded by the glaciers of the Ice
Age. At this crustal depth, the estimated static
stiffness is around 10 GPa (Gudmundsson,
1990), or about 15% of the stiffness of the roof
of the reservoir (67 GPa) used in the above cal-
culations. From eqs. (4.9) and (4.12) we can
thus infer that only 15% of the dykes injected
from the magma reservoir reach this crustal lev-
el. It follows that during a typical lifetime of a
volcanic system (and its associated source
reservoir) of 0.3-1 Ma (Gudmundsson, 2000),
between 450 and 1500 dykes should occur in
the Alftafjördur Swarm at the present exposure.
This is in good agreement with the estimated to-
tal number of exposed dykes in the Alftafjördur
Swarm, 1000 to 1500 dykes (Gudmundsson,
1995). Thus, the gradual increase in stiffness
with crustal depth (Gudmundsson, 1990) may
be a good first-approximation explanation for
the number of regional dykes found at a partic-
ular crustal depth in an extinct volcanic system.
Fig.  15. Dyke swarms and associated central volca-
noes (stratovolcanoes, edifices) in East Iceland. The
main dyke swarms and central volcanoes indicated
are Alftafjordur (A), Breiddalur (B), Reydarfjordur
(R), and Thingmuli (T), (cf. Walker, 1963 and Gud-
mundsson, 1995).
1446
Agust Gudmundsson and Sonja L. Brenner
4.3. Layered crust with abrupt stiffness
changes with depth
Next we consider a shallow chamber in a
layered crust and ignore the stress effects of the
deep-seated source reservoir. This we do be-
cause the focus is here on the stresses in the
crust above the shallow chamber where the
stress-effects of the deeper reservoir are negli-
gible. For this purpose, we made several bound-
ary-element models, using the BEASY (1991)
program, of a magma chamber with a circular
cross section in a layered crust where the layer
stiffness increases gradually with depth (fig.
16). The boundary-element method in general,
and the BEASY program in particular, are de-
scribed by Brebbia and Dominguez (1992), the
BEASY homepage (www.beasy.com), BEASY
(1991), and Jing and Hudson (2002).
The model in fig. 16 is of unit height (vertical
dimension) and width (horizontal dimension),
and all its dimensions are given as fractions of
these units. A magma chamber of circular cross
section, with a diameter of 0.30 units, has its cen-
tre at a depth of 0.45 units from the top of the
model (the rift-zone surface) and thus 0.55 units
from the bottom of the model. For a typical scal-
ing, where the rift zone is 10 km thick, the cham-
ber is 3 km in diameter with a centre at 4.5 km,
and a top at 2 km, depth below the surface: these
figures are similar to those observed or inferred
for active and extinct crustal magma chambers in
Iceland (Gudmundsson, 1990, 1998).
The magma chamber is located in a crust
where stiffness increases in abrupt steps from 5
GPa in the topmost layer to 100 GPa in the bottom
layer. To emphasise the importance of changes in
stiffness, and because variations in Poisson’s ratio
are commonly much less than those in Young’s
modulus (Bell, 2000), all the layers have the same
Poisson’s ratio: 0.25. The only loading in the mod-
el is a magmatic excess pressure (defined in eq.
(4.2)) of 10 MPa in the chamber. To avoid rigid
body rotation and translation, the model is fas-
tened, using the condition of no displacements
(BEASY, 1991), in all the corners.
The results show that tensile stresses are
highest around the magma chamber (only con-
tours between 1 and 10 MPa are shown) and
fall off with increasing distance from the cham-
ber, in agreement with eqs. (4.3) to (4.8). A re-
markable feature is that there are no significant
tensile stresses transferred to the low-stiffness
top layer A of the model. This indicates that this
layer would normally act as a stress barrier to
the dykes propagating from the magma cham-
ber. Thus, during periods of volcanic unrest
with dyke and sheet injection, layer A might ar-
rest many of, or all, the dykes.
Even if the volcanic zone hosting the mag-
ma chamber in fig. 16 is subject to extension, it
is still difficult for dykes to propagate through
layer A and to the surface to feed a volcanic
eruption. This is seen in fig. 17 where the mod-
el in fig. 16 has been subjected to crustal exten-
sion of 3 MPa, a value that is thought to be typ-
ical during rifting episodes in the upper part of
the volcanic zones in Iceland. In this model, the
region of high tensile stress (in excess of 10
Fig.  16. Boundary-element model showing the con-
tours, in MPa, of the maximum principal tensile stress
σ3 around a magma chamber in a layered crust of a rift
zone or a volcanic edifice. The chamber is subject to
magmatic excess pressure pe = 10 MPa, similar to the
laboratory tensile strengths of common rocks. The stiff-
nesses of the host-rock layers, also based on laboratory
measurements (Hansen et al., 1998; Bell, 2000; Nilsen
and Palmstrom, 2000), are as follows: A = 5 GPa; B =
20 GPa; C = 30 GPa; D = 50 GPa; E = 70 GPa; F = 80
GPa; G = 90 GPa; and H = 100 GPa.
A
B
C
D
E
F
G
H
pe
1447
Local stresses, dyke arrest and surface deformation in volcanic edifices and rift zones
MPa) around the magma chamber is much larg-
er than that in the model in fig. 16 and extends
all the way through layers C (above the cham-
ber) and E (below the chamber). There are ten-
sile stresses throughout the lower part of the
crust (the lowermost part of the model is fas-
tened, using the no-displacement conditions,
which explains the fall-off in tensile stress in
layer H, despite is high stiffness). The high ten-
sile stresses in layers C and B, above the mag-
ma chamber, indicate that dykes would normal-
ly easily propagate through those layers. Again,
however, the dykes would tend to become ar-
rested in the soft surface layer A.
The results of the models in figs. 16 and 17
thus generally support the analytical results that
during extension and rifting dyke injection is
encouraged in stiff layers, but dyke arrest is en-
couraged in soft layers (Gudmundsson and
Brenner, 2001). These results should not, how-
ever, be taken to indicate that no dykes can
propagate through soft layers. Tensile stresses
are not needed in these layers for at least some
dykes to be able to propagate to the surface.
However, the lack of significant tensile stress-
es, even during periods of active rifting (fig.
17), indicates that only few dykes, at most,
would be able to propagate through the surface
layer A so long as it has such a low stiffness.
Thus, the stress situation in a layered crust
where the stiffness gradually increases with
depth and, in particular, where there is a very
soft layer at the surface, encourages dyke arrest
in the uppermost part of the crust.
5.  Surface effects of arrested dykes
Surface deformation in volcanic rift zones
and edifices is routinely used to infer the
geometry and shape of emplaced magma-
filled fractures, sheets and dykes during peri-
ods of unrest (e.g., Jousset et al., 2003). The
methods and models used to make these infer-
ences are very important for assessing vol-
canic hazards and must be highly reliable.
Analytical solutions for surface effects of
an extension fracture (such as a dyke) have ex-
isted for many years; one of the oldest is the
one by Isida (1955). For a vertical extension
fracture subject to tension or internal fluid
overpressure there will be surface tensile
stress concentration to either side of the frac-
ture, and no concentration right above the
fracture tip. Later papers that deal explicitly
with the surface effects of emplaced dykes in-
clude those of Pollard et al. (1983), Okada
(1985), Rubin and Pollard (1988), Bonafede
and Olivieri (1995), Bonafede and Danesi
(1997), and Bonafede and Rivalta (1999a,b).
Generally, these works agree with the results
of Isida (1955), except that the early work of
Pollard and Holzhausen (1979) indicated a
significant graben formation above the dyke,
implying subsidence of the ground surface
above the dyke tip, but this conclusion was
changed in later papers (Pollard et al., 1983;
Rubin and Pollard, 1988).
Except for the models by Bonafede and Ri-
valta (1999a,b), all these models assume a ho-
mogeneous, isotropic elastic material. There are
two main problems with these types of models
Fig.  17. Boundary-element model showing the con-
tours, in MPa, of the maximum principal tensile stress
around a magma chamber in a layered crust of a rift
zone or a volcanic edifice. The model and layering is
the same as in fig. 16 except that in addition to the
magmatic excess pressure of 10 MPa there is a uni-
form horizontal tension, as indicated by the horizontal
arrows, of 3 MPa. This tension would apply to plate
pull in rift zones and volcano spreading in edifices.
A
B
C
D
E
F
G
H
pe
1448
Agust Gudmundsson and Sonja L. Brenner
for volcanic edifices and rift zones. The first is
that it is difficult to explain how the dykes be-
came arrested in the first place. Whether the
dyke is modelled as an elliptic hole or a mathe-
matical crack (with a linear or constant magma
overpressure), so long as there is any signifi-
cant overpressure in the dyke, there will be very
high tensile stresses at the dyke tip (normally
much higher than any induced surface stresses),
and difficult to see how the dyke should be-
come arrested.
The second difficulty is that while the ho-
mogeneous and isotropic model may be a rea-
sonable approximation at great depths in an old
crust, it is definitely a poor model of the upper-
most part of the crust in active rift zones and
edifices. As was discussed earlier, such a crust
normally consists of alternating pyroclastic lay-
ers and lava flows (fig. 1), of strongly contrast-
ing mechanical properties and with weak or
even open layer contacts (fig. 2).
To illustrate the first difficulty, consider the
simplest model of an arrested dyke in a verti-
cal section, namely an elliptical hole of a dip
(height) dimension 2a. If the radius of curva-
ture of the dyke tip is rc = b2 / a, where the
dyke thickness (opening) is 2b, and the dyke is
subject to a constant magmatic overpressure
P0, then the maximum tensile (minimum com-
pressive) principal stress σ3 at the dyke tip is
03
aP
r
2 1
/
c
1 2
=- -v f p
R
T
S
SS
V
X
W
WW
. (5.1)
where the magmatic overpressure is defined as
the total magma pressure minus the dyke-nor-
mal compressive stress.
For dykes that end in vertical sections,
typical radii of curvature of their tips are a
few centimetres (figs. 4, 7 and 8). When the
dyke dip dimension a reaches many kilome-
tres, as is common, eq. (5.1) yields theoreti-
cal tensile stresses at the dyke tip as ∼ 103-104
MPa. In case the radius of curvature refers
only to individual dyke segments, with
lengths from tens of metres to a hundred me-
tres (Gudmundsson, 1995), rather than whole
dykes, eq. (5.1) still yields theoretical dyke-
tip tensile stresses of ∼ 102-103 MPa. The the-
oretical tensile stresses at the tips of dykes, as
estimated from eq. (5.1), are thus from a hun-
dred to several thousand times greater than
situ tensile strengths (0.5-6 MPa) of common
host rocks.
A dyke can also be modelled as a two-di-
mensional mathematical crack (Gudmunds-
son, 2002; Gudmundsson et al., 2002). If the
mathematical crack that the dyke occupies is
located on the vertical y-axis and defined by
x = 0, − a ≤ y ≤ a, then the magma overpres-
sure is given by the even function p(y) =
= p(–y); this ensures that the magma over-
pressure is the same on the dyke walls above
and below the horizontal x-axis. Based on
field observations, a dyke should normally be
modelled as an extension fracture (a mode I
crack). For a uniform magmatic overpressure
p(y) = P0, the dyke-normal stress inside the
dyke is the maximum principal tensile stress
σ3 = − P0, for 0 ≤ y ≤ a, whereas outside the
dyke tips, for y > a, the tensile stress σ3 (y, 0)
is (Sneddon, 1973; Maugis, 2000; Gud-
mundsson, 2002)
03 /1 2P y y
1 1
=- -v 2 2( )y a-> H. (5.2)
It follows from eq. (5.2) that when y approach-
es  (from outside the dyke tip), the tensile stress
σ3 at the tip becomes infinite.
For some dykes, the magmatic overpressure
p(y) may not be uniform but rather vary inside
the dyke. For example, the overpressure may
vary from a maximum value of P0 at the vertical
dyke centre by a linear gradient p1y (here treating
y as an absolute value) according to the equation
p(y) = P0 + p1y . (5.3)
Then it follows (Gudmundsson et al., 2002)
that the maximum principal tensile stress σ3 at
the dyke tip is:
1-
3 /1 22 2( ) sin
p
y a
ya
y y
a2
=-
-
-v r
1
R
T
S
S
V
X
W
W (5.4)
so that when y approaches a, the tensile stress
σ3 at the dyke tip is again infinite. The mathe-
1449
Local stresses, dyke arrest and surface deformation in volcanic edifices and rift zones
matical crack model thus indicates that in a
small region around the dyke tip the tensile
stress in the host rock is infinite, representing a
stress singularity. Non-elastic (permanent) de-
formation and microcracking at the dyke tip re-
lieve part of the stress (thereby making the
stress less than infinite); nevertheless, the math-
ematical crack models in eqs. (5.2) and (5.3)
predict very high tensile stresses at a typical
dyke tip in a homogeneous, isotropic crust.
It follows from eqs. (5.1), (5.2) and (5.4) that,
irrespective of whether the dyke is modelled as
an elliptic hole or a mathematical crack, for
dykes with any significant overpressure in a ho-
mogeneous and isotropic host rock there will be
potentially very high tensile stresses at the dyke
tips. A dyke with any significant overpressure
should, therefore, not become arrested in such a
crust. Thus, for a dyke tip to become arrested at
a certain crustal depth, rather than propagating to
the surface of the rift zone, the dyke overpressure
must either be zero or, alternatively, the host rock
must be heterogeneous and anisotropic so as to
have stress barriers, transverse discontinuities, or
an abrupt change in layer stiffnesses. These fea-
tures are common in a layered crust, but then the
stress fields associated with arrested dyke tips
may be very different from that suggested for a
homogeneous, isotropic crust. 
We shall illustrate stress fields generated by
arrested dykes in a layered crust, and in particu-
lar the stresses at the surface of the associated rift
zone (or a volcanic edifice), using the boundary-
element model BEASY (1991). The magmatic
overpressure in all the models has a linear varia-
tion from 10 MPa at the dyke bottom to 0 MPa at
the dyke tip. Consider first the effects of stress
barriers, that is, layers where the stress field is
unfavourable for the propagation of a particular
type of fracture. For dykes, stress barriers are, as
explained above, layers where there is an unusu-
ally high dyke-normal compressive stress.
These barriers usually suppress much of, or
all, the tensile stresses associated with the dyke
tip and allow very small, if any, stresses to be
transferred to the surface. Consider, for exam-
ple, a barrier with 7 MPa compressive stress, as
has been measured in Iceland (Haimson and
Rummel, 1982), at 1 km depth whereas the dyke
tip is at 3 km depth below the surface (fig. 18).
All the layers have the same stiffness, 40 GPa.
The maximum tip tensile stress is 50 MPa (the
highest contour shown is 5 MPa) but no tensile
stress is transferred to the surface.
Consider next the effects of variation in stiff-
ness in a layered crust on the surface stresses as-
sociated with an arrested dyke (fig. 19). Here the
horizontal compressive stress increases linearly
with depth, from 0 to 20 MPa at 0.8 km depth, in
accordance with the increase in overburden pres-
sure (vertical stress). The dyke tip is at 1 km
depth, and above is a layer (200 m thick) with ze-
ro compressive stress, supposed to be due to an
earlier extension or to be a soft sediment acting
as a soft inclusion that takes on little or no hori-
zontal compressive stress. This layer just magni-
fies the dyke-tip tensile stress, but does not oth-
A
B
A
Fig.  18. Boundary-element model showing the con-
tours, in MPa, of the maximum principal tensile
stress σ3 around a dyke tip located at 3 km below the
surface of a rift zone or a volcanic edifice. Layer B is
a stress barrier with excess horizontal compressive
stress of 7 MPa; this layer extends from a depth of 1
km to 2 km below the surface. The stiffnesses and
Poisson’s ratios of layers A and B are uniform, 40
GPa and 0.25, respectively. The contours of σ3 flat-
ten out under the stress barrier and no dyke-induced
tensile stresses reach the surface. In this and the
models in figs. 19 and 20, the dyke thickness (open-
ing), as obtained from the numerical models, is not to
scale when compared with the crustal thickness.
1450
Agust Gudmundsson and Sonja L. Brenner
erwise affect the conclusions of the model calcu-
lations. Layers A and C are moderately stiff, 40
GPa, corresponding to common basaltic lava
flows, whereas layer B is soft, 5 GPa, correspon-
ding to many young hyaloclastites or sedimenta-
ry layers. It is a large-scale version of the layer-
ing in fig. 7 from the Reykjanes Peninsula.
Even if the maximum calculated dyke-tip
tensile stresses are 102 MPa, very little tensile
stresses are transferred to the soft layer and the
surface layer is unstressed (fig. 19). There are
thus no dyke-induced tensile stresses at the sur-
face. The marginal tensile-stress values are due
to the fastening of the model at the corners and
have no physical meaning.
Consider next the effects of a discontinuity,
say a weak contact between layers in the upper
part of a rift zone (or a volcanic edifice). Dykes
tend to open up such contacts and are very com-
monly affected by them; many dykes become ar-
rested at the open contacts, change into sills, or
become offset across them. A dyke may follow
the contact for several metres (fig. 11), occasion-
ally tens of metres, and then leave the contact to
continue its vertical propagation into the lava pile
above. In this model (fig. 20), the layering is the
same as in the last model (fig. 19), but there is a
weak discontinuity, a contact, modelled as an in-
ternal spring with stiffness 6 MPa/m, between
layers A and B. The contact is 8 km wide, similar
to the width of many Holocene volcanic systems
in Iceland (fig. 5). To estimate the maximum ef-
fects that the contact may have on the surface
stresses, we subject the layers neither to overbur-
den pressure nor horizontal compressive stress.
Effectively, this would mean that in the upper-
most 800 m of the rift zone extension had reduced
all compressive stresses to zero. This boundary
condition is not very likely to be commonly satis-
fied, but is used here so as to maximise the po-
tential surface effects of the arrested dyke.
The central part of the contact opens up
(fig. 20). However, even if the calculated
dyke-tip tensile stress is 174 MPa, there are
very little tensile stresses at the surface above
the dyke itself (fig. 20). The main tensile
stresses at the surface, occur above the lateral
ends of the contact, at a distance of 4 km to ei-
ther side of the dyke tip. There are tensile
stresses are low, only 2 MPa and unlikely to
lead to much fracture formation. Similar re-
sults were obtained in several other models:
there are little or no tensile stresses in a large
surface area above the dyke itself, only above
the lateral ends of the weak contact.
A
B
C
Surface distance, x/d
Su
rfa
ce
 s
tre
ss
, σ
3
Fig.  19. Boundary-element model of a dyke with a
tip at 1 km depth below the surface of a rift zone or a
volcanic edifice where the horizontal stress increases
linearly (with overburden pressure) from 0 MPa to 20
MPa at 0.8 km depth, as indicated by horizontal ar-
rows (cf. Gudmundsson, 2003). Layer B is soft (E =
5GPa), whereas the other layers are moderately stiff
(E = 40 GPa). The contours show the values between
1 and 10 MPa of the maximum principal tensile stress.
The upper diagram shows that there is no dyke-in-
duced tensile stress at the surface above the dyke (the
margin stresses are irrelevant because they relate to
the non-physical boundary conditions in the model).
This is because layer B, which takes up very little ten-
sile stress, hinders stress transfer to the surface layer
A. The lower diagram shows the tensile stress con-
centration at the dyke tip, the maximum value being =
102 MPa (cf. eqs. (5.1), (5.2) and (5.4)). 
1451
Local stresses, dyke arrest and surface deformation in volcanic edifices and rift zones
The results in this section indicate, first, that
the tensile stress at a dyke tip is normally much
greater than any dyke-induced tensile stress at
the surface. It is thus not clear why the tip of a
Fig.  20. Boundary-element model of a dyke with a
tip at 1 km depth below the surface of a rift zone or
a volcanic edifice (cf. Gudmundsson, 2003). The up-
per contact between layers A and B, at 0.4 km depth,
is weak and modelled as an internal spring with a
stiffness of 6 MPa/m. This contact opens up when
the dyke tip approaches. The contours show the val-
ues between 1 and 10 MPa of the maximum princi-
pal tensile stress σ3. The upper diagram shows that
there is little dyke-induced tensile stress at the sur-
face above the dyke. However, there are higher ten-
sile stresses at the margins but these occur above the
lateral ends (at roughly 4 km to either side of the
dyke) of the weak contact at 0.4 km depth. The low-
er diagram shows the tensile stress concentration at
the dyke tip, the maximum value being σ3 = 174
MPa (cf. eqs. (5.1), (5.2) and (5.4)).
dyke that is supposed to be able to induce large
tensile (and shear) stresses at the surface should
become arrested. Secondly, the results show that
for an active rift zone or a volcanic edifice,
straightforward inversion of geodetic data to in-
fer the geometry and depth to an emplaced dyke
may yield very unreliable results.
6.  Discussion
The results of field observations and analyti-
cal and numerical models presented in this paper
suggest that the conditions for sheet and dyke ar-
rest or, alternatively, their propagation to the sur-
face, are primarily controlled by the local stress
fields in the rift zone or volcanic edifice. Most
volcanic edifices consist of layers of contrasting
mechanical properties, such as soft pyroclastic
layers and stiff lava flows. The present, as well
as earlier (Gudmundsson, 2002, 2003), numeri-
cal models indicate that such edifices commonly
develop stress fields that encourage dyke arrest.
Surface deformation studies are routinely
used to infer the geometry of dykes that become
arrested during periods of volcanic unrest (Pol-
lard et al., 1983; Jousset et al., 2003). Most mod-
els used in inversion of the surface data to infer
dyke geometries assume that the rock hosting the
dykes may be modelled as a homogeneous,
isotropic half space and imply that the host-rock
stresses are uniform. Our results indicate that, for
such a host rock, it remains to be explained how
a dyke is able to induce large tensile (and shear)
stresses and cause fracturing at the surface of a
rift zone while becoming arrested – given that the
dyke-tip tensile stresses would normally be much
greater than any induced surface stresses and thus
greater than the in situ tensile strength of the host
rock. Also, some theoretical models based on
these assumptions predict that arrested dykes
may generate major grabens at the surfaces of rift
zones and volcanic edifices. In this paper we
summarise field observations showing that ar-
rested dyke tips in eroded rift zones are normally
not associated with grabens or normal faults that
could be generated by dyke-induced stresses. Our
field results also show that some dykes that are
arrested within a few metres of the surface do not
generate faults or grabens at the surface (fig. 8).
A
B
C
Surface distance, x/d
Su
rfa
ce
 s
tre
ss
, σ
3
1452
Agust Gudmundsson and Sonja L. Brenner
Our numerical models and field data suggest that
this discrepancy between some theoretical mod-
els and field observations may be explained by
the mechanical layering of the crust.
The numerical models presented here and in
earlier papers show that abrupt changes in
Young’s moduli, layers with relatively high dyke-
normal compressive stresses (stress barriers), and
weak, horizontal contacts have large effects on
local stresses in rift zones and volcanic edifices.
In particular, the models in figs. 16 and 17, and
earlier numerical models (Gudmundsson, 2002;
Gudmundsson et al., 2002), indicate that during
periods of magma-chamber inflation and associ-
ated ground deformation in volcanic areas there
may be faulting (with associated seismicity and
changes in the hydrothermal fields) in stiff layers
while all the injected dykes become arrested in
soft layers (so that no eruptions occur). Observa-
tions of this kind are well known from many ac-
tive volcanoes, particularly calderas (Bjornsson,
1984; De Natale et al., 1997).
As to ground deformation related to dykes,
the numerical models considered (figs. 18-20)
indicate that the surface tensile stresses induced
by arrested dykes are commonly too small to
lead to significant fault or graben formation at
the rift-zone surface. The only significant dyke-
induced surface tensile stresses (2 MPa) in
these models are for a dyke tip arrested at 1 km
depth below the surface of a rift zone with a
weak contact at 400 m depth and subject to ex-
tension (fig. 20). These tensile stresses, howev-
er, peak above the ends of the weak horizontal
contact which, in the model considered, occur
at distances of 4 km to either side of the dyke
and do not show any simple relation to the dyke
geometry or the depth to the dyke tip. Thus, for
a layered crust with weak contacts, straightfor-
ward inversion of surface geodetic data may
lead to unreliable geometries of dykes.
In this paper, we focus on dyke arrest due to
local, unfavourable stresses. There are, howev-
er, various ways by which the stress conditions
favouring dyke arrest in a volcanic edifice or a
rift zone may be changed to conditions favour-
ing dyke propagation to the surface and thus the
formation of feeder dykes. These include i) stiff
surface layers, ii) gradual increase in stiffness
with burial, and iii) stress-field homogenisation.
When new lava flows are erupted, particular-
ly thick basaltic lava flows, the stiffness of the up-
permost part of the crust increases. As a conse-
quence, layer A in the models in figs. 16 and 17
would become relatively stiff, and thus be less
likely to arrest dykes. However, once an overpres-
sured dyke has propagated to the surface through
the surface layer, in particular during a rifting
episode with the formation or development of ten-
sion fractures and normal faults, the stiff lava flow
A at the surface of the rift zone may act as a stress
barrier for subsequent dykes (Gudmundsson and
Brenner, 2001). This follows because dyke injec-
tion, tension fracture formation, and normal fault-
ing all increase the temporary horizontal com-
pressive stress in the layer; and this compressive
stress must be relaxed (e.g., through plate pull)
before the next dyke can propagate through the
layer to the surface to feed an eruption.
One reason that the stiffnesses of layers in-
crease with time and burial in active volcanic ar-
eas is hydrothermal alteration and precipitation
of secondary minerals. Hydrothermal alteration
generally encourages welding together (healing
and sealing) of layer contacts, making the entire
pile gradually approach that of a single, thick
layer. As a consequence of these processes, not
only does the layer stiffness increase, but also its
density; as is well known, density and stiffness
are positively related (Jaeger and Cook, 1979).
In particular, the initially soft scoria (pyroclastic)
layers at the contacts between lava flows are
common pathways for hydrothermal water and
thus the sites of much secondary mineralisation.
These, initially soft, layers gradually approach
the stiffnesses of the adjacent stiffer parts of the
lava flows. It follows that the hydrothermal al-
teration makes the lava pile in an active volcanic
zone gradually more uniform in stiffness, there-
by encouraging stress-field homogenisation.
Stress-field homogenisation means that the
stress field along a potential pathway of a frac-
ture, such as a dyke, gradually becomes uniform
(Gudmundsson and Homberg, 1999). In a vol-
canic zone, stress-field homogenisation is, as in-
dicated above, partly achieved by hydrothermal
alteration of the lava pile above the magma
sources. Partly, however, the stress-field ho-
mogenisation is achieved through dyke injection
and faulting of the lava pile. For example, fault-
1453
Local stresses, dyke arrest and surface deformation in volcanic edifices and rift zones
ing reduces the stress differences along the po-
tential pathway of a dyke. And previous dyke in-
jections make the stress field more uniform along
the pathways of subsequent dykes. It is proposed
that during a stress cycle, which corresponds to
the time between successive eruptions (a volcanic
cycle), a magma chamber gradually homogenises
the surrounding stress field until one or more po-
tential dyke pathways have been homogenised all
the way to the surface, in which case one or more
eruptions may occur. The homogenised condition
is thus not satisfied during most volcanic unrest
periods which explains why, even if they lead to
dyke and sheet injection, most such periods do
not result in volcanic eruptions.
7.  Conclusions
In the model presented in this paper there are
two basic mechanisms by which dykes and
sheets become arrested in volcanoes: stress bar-
riers and rock anisotropy, such as those associ-
ated with discontinuities. These mechanisms are
related, particularly through changes in stiff-
nesses. Such changes are common at contacts
(discontinuities) between different rock types.
This model explains, first, why many dykes
become arrested (commonly with blunt tips) at
contacts and, second, why dykes commonly ta-
per away in homogeneous layers. The model in-
dicates that during most volcanic unrest periods
with sheet or dyke injections, the probability of
an eruption is low. Only during those rare un-
rest periods when the stress field has been ho-
mogenised all the way to the surface are erup-
tions to be expected.
Some theoretical models predict that arrested
dykes may generate major grabens at rift-zone sur-
faces. But most arrested dyke tips (even very shal-
low ones) in rift zones do not generate normal
faults or grabens. Most of these models assume the
crust to be homogeneous and isotropic. Because
dyke-tip tensile stresses are much larger than the
dyke-induced surface stresses, it is unclear in such
models why the tip of a dyke supposed to generate
large surface stresses becomes arrested.
When crustal layering is taken into account,
numerical models show that abrupt changes in
Young’s moduli, stress barriers and weak con-
tacts greatly affect the dyke-induced stresses.
The numerical models presented in this paper in-
dicate that layering commonly suppresses dyke-
induced tensile (and shear) stresses at the sur-
face, making fracture, fault and graben forma-
tion less likely. Weak contacts lower surface
stresses and transfer the stress peaks to the later-
al contact ends, in which case straightforward
inversion of surface geodetic data to infer
geometries of arrested dykes may yield unreli-
able results. It is concluded that local stress fields
unfavourable to dyke propagation to the surface
(and eruption) may change to favourable stress
fields through several processes, the most impor-
tant of which is stress-field homogenisation.
Acknowledgements
This work was supported through several grants
from the Norway Research Council, the European
Commission (through the Prepared project), as well
as through a PhD grant from Statoil (to Agust Gud-
mundsson) for Sonja L. Brenner. We thank Sergio
Vinciguerra for helpful review comments.
REFERENCES
AMADEI, B. and O. STEPHANSSON (1997): Rock Stress and
its Measurement (Chapman and Hall, New York).
BEASY (1991): The Boundary Element Analysis System
User Guide (Computational Mechanics, Boston).
BELL, F.G. (2000): Engineering Properties of Soils and
Rocks (Blackwell, Oxford).
BJORNSSON, A. (1984): Dynamics of crustal rifting in NE
Iceland, J. Geophys. Res., 90, 10151-10162.
BONAFEDE, M. and S. DANESI (1997): Near-field modifica-
tions of stress induced by dyke injection at shallow
depth, Geophys. J. Int., 130, 435-448.
BONAFEDE, M. and M. OLIVIERI (1995): Displacement
and gravity-anomaly produced by a shallow vertical
dyke in a cohesionless medium, Geophys. J. Int.,
123, 639-652.
BONAFEDE, M. and E. RIVALTA (1999a): On tensile cracks
close to and across the interface between two welded
elastic half-spaces, Geophys. J. Int., 138, 410-434.
BONAFEDE, M. and E. RIVALTA (1999b): The tensile dislo-
cation problem in a layered elastic medium, Geophys.
J. Int., 136, 341-356.
BREBBIA, C.A. and J. DOMINGUEZ (1992): Boundary Ele-
ments, an Introductory Course (Computational Me-
chanics, Boston).
DE NATALE, G. and F. PINGUE (1993): Ground deformations
in collapsed caldera structures, J. Volcanol. Geotherm.
Res., 57, 19-38.
1454
Agust Gudmundsson and Sonja L. Brenner
DE NATALE, G., S.M. PETRAZZUOLI and F. PINGUE (1997):
The effect of collapse caldera structures on ground
deformation in calderas, Geophys. Res. Lett., 24,
1555-1558.
DILEK, Y., E.M. MOORES and H. FURNES (1998): Structure
of modern oceanic crust and ophiolites and implica-
tions for faulting and magmatism at ocean spreading
centers, in Faulting and Magmatism at Mid-Ocean
Ridges, edited by W.R. BUCK, P.T. DELANEY, J.A. KAR-
SON and Y. LAGABRIELLE (American Geophysical
Union, Washington), 219-265.
GUDMUNDSSON, A. (1990): Emplacement of dikes, sills and
crustal magma chambers at divergent plate bound-
aries, Tectonophysics, 176, 257-275.
GUDMUNDSSON, A. (1995): Infrastructure and mechanics of
volcanic systems in Iceland, J. Volcanol. Geotherm.
Res., 64, 1-22.
GUDMUNDSSON, A. (1998): Magma chambers modeled as
cavities explain the formation of rift zone central vol-
canoes and their eruption and intrusion statistics, J.
Geophys. Res., 103, 7401-7412.
GUDMUNDSSON, A. (2000): Dynamics of volcanic systems
in Iceland: example of tectonism and volcanism at jux-
taposed hot spot and mid-ocean ridge systems, Ann.
Rev. Earth Planet. Sci., 28, 107-140.
GUDMUNDSSON, A. (2002): Emplacement and arrest of
sheets and dykes in central volcanoes, J. Volcanol. Ge-
otherm. Res., 116, 279-298.
GUDMUNDSSON, A. (2003): Surface stresses associated with
arrested dykes in rift zones, Bull. Volcanol., 65, 606-
619.
GUDMUNDSSON, A. and S.L. BRENNER (2001): How hy-
drofractures become arrested, Terra Nova, 13, 456-462.
GUDMUNDSSON, A. and C. HOMBERG (1999): Evolution of
stress fields and faulting in seismic zones, Pure Appl.
Geophys., 154, 257-280.
GUDMUNDSSON, A., I. FJELDSKAAR and S.L. BRENNER
(2002): Propagation pathways and fluid transport of
hydrofractures in jointed and layered rocks in geother-
mal fields, J. Volcanol. Geotherm. Res., 116, 257-278.
HAIMSON, B.C. and F. RUMMEL (1982): Hydrofracturing
stress measurements in the Iceland research drilling
project drill hole at Reydarfjordur, Iceland, J. Geo-
phys. Res., 87, 6631-6649.
HANSEN, S.E., T. SORLOKK and A.E. JOHANNESSON (1998):
Mechanical Properties of Rocks (Sintef, Trondheim),
(in Norwegian).
HARRIS, A.J.L., J.B. MURRAY, S.E. ARIES, M.A. DAVIES,
L.P. FLYNN, M.J. WOOSTER, R. WRIGHT and D.A.
ROTHERY (2000): Effusion rate trend at Etna and
Krafla and their implications for eruptive mechanisms,
J. Volcanol. Geotherm. Res., 102, 237-270.
ISIDA, M. (1955): On the tension of a semi-infinite plate with
an elliptic hole, Tech. Rep. Fac. Eng., Tokushina Univ.,
5, 75-95.
JAEGER, J.C. and N.G.W. COOK (1979): Fundamentals of
Rock Mechanics (Chapman & Hall, London).
JING, L. and J.A. HUDSON (2002): Numerical methods in rock
mechanics, Inte. J. Rock Mech. Min. Sci., 39, 409-427.
JOUSSET, P., H. MORI and H. OKADA (2003): Elastic models
for the magma intrusion associated with the 2000
eruption of Usu Volcano, Hokkaido, Japan, J. Vol-
canol. Geotherm. Res., 125, 81-106.
KARSON, J.A. (1998): Internal structure of oceanic litho-
sphere: a perspective from tectonic windows, in Fault-
ing and Magmatism at Mid-Ocean Ridges, edited by
W.R. BUCK, P.T. DELANEY, J.A. KARSON and Y.
LAGABRIELLE (American Geophysical Union, Wash-
ington), pp. 177-218.
MARINONI, L.B. and A. GUDMUNDSSON (2000): Dykes,
faults and palaeostresses in the Teno and Anaga mas-
sifs of Tenerife (Canary Islands), J. Volcanol Geot-
herm. Res., 103, 83-103.
MAUGIS, D. (2000): Contact, Adhesion and Rupture of
Elastic Solids (Springer-Verlag, Berlin).
NILSEN, B. and A. PALMSTROM (2000): Engineering Geol-
ogy and Rock Engineering, Handbook 2, The Rock
Mechanics Group, Oslo.
OKADA, Y. (1985): Surface deformation due to shear and
tensile faults in a half-space, Bull. Seismol. Soc. Am.,
75, 1135-1154.
PALLISTER, J.S. (1981): Structure of the sheeted dike com-
plex of the Samail ophiolite near Imbra, Oman, J.
Geophys. Res., 86, 2661-2672.
PALMASON, G. (1971): Crustal structure of Iceland from ex-
plosion seismology, Societas Scientiarum Islandica
Rit, 40, 1-187.
PEDERSEN, R.B. (1986): The nature and significance of
magma chamber margins in ophiolites: examples from
the Norwegian Caledonides, Earth Plan. Sci. Lett., 77,
100-112.
POLLARD, D.D. and G. HOLZHAUSEN (1979): On the me-
chanical interaction between a fluid-filled fracture and
the earth’s surface, Tectonophysics, 53, 27-57.
POLLARD, D.D., P.T. DELANEY, W.A. DUFFIELD, E.T. ENDO
and A.T. OKAMURA (1983): Surface deformation in
volcanic rift zones, Tectonophysics, 94, 541-584.
RUBIN, A.M. and D.D. POLLARD (1988): Dike-induced
faulting in rift zones of Iceland and Afar, Geology, 16,
413-417.
SAADA, A.S. (1983): Elasticity: Theory and Applications
(Krieger, Malabar, Florida).
SCHULTZ, R.A. (1995): Limits on strength and deformation
properties of jointed basaltic rock masses, Rock Mech.
Rock Eng., 28, 1-15.
SIGURGEIRSSON, M.A. (1995): The Younger Stampa Erup-
tion on the Reykjanes Peninsula (in Icelandic), Nat-
turufraedingurinn (Reykjavik), 64, 211-230.
SNEDDON, I.N. (1973): Integral Transform Methods, in
Methods of Analysis and Solutions of Crack Problems,
edited by G.C. SIH (Nordhoff, Leyden), 315-367.
STAPLES, R.K., R.S. WHITE, B. BRANDSDOTTIR, W.
MENKE, P.K.H. MAGUIRE and J.H. MCBRIDE (1997):
Faroe-Iceland Ridge Experiment 1, crustal structure
of Northeastern Iceland, J. Geophys. Res., 102,
7849-7866.
TADA, H., P.C. PARIS and G.R. IRWIN (2000): The Stress
Analysis of Cracks Handbook (Del Research Corpora-
tion, Hellertown, Pennsylvania), 3rd edition.
THORDARSON, T. and S. SELF (1993): The Laki (Skaftar
Fires) and Grimsvotn eruptions in 1783-1785, Bull.
Volcanol., 55, 233-263.
TIMOSHENKO, G.P. and J.N. GOODIER (1970): Theory of
Elasticity (McGraw-Hill, London).
WALKER, G.P.L. (1963): The Breiddalur Central Volcano
Eastern Iceland, Q. J. Geol. Soc. London, 119, 29-63.