GEOS 606 Physical Volcanology
Dr. Jonathan Dehn
Alaska Volcano Observatory
X. Fracture, how magma gets out
I.
Magma Chamber Shape
a. Classic shapes: cylinders
i. Had corners to concentrate stress
ii. Dike propagation and evacuation was classically thought to start here. We
now know this is not the case.
b. Spheroids
i. Closer to real case, large mush zones with slush pods
ii. How would stress be concentrated? Cone sheets & Ring Dikes.
iii. This example from the Spanish Peaks Colorado:
c. Given that the real shape is a little more complex where would stresses be
naturally concentrated?
i. Near the top, balance between overburden and regional stresses
ii. What is needed to ensure fracture?
d. What would be the effects of inhomogeneities in the host rock?
i. Concentrate stresses
ii. Zones of weakness
II.
Phase Changes
a. Crystals
i. Solids, provide drag, hence higher viscosity
ii. Where would these concentrate?
b. Gas (volatiles)
i. Much lower density, much higher buoyancy
ii. Where would bubbles concentrate?
c. Remaining melt (fluids)
i. How would its rheology change?
ii. Where would the melt go?
III.
Fracture
a. How does fracture happen?
i. When the stress exceeds the strength of the rock
ii. Fracture is a non-recoverable form of strain
b. How is the “Fracture Toughness” or strength of a material measured?
i. Yield strength is a simple value of force. MPa is usually its unit for
magmatic cases.
ii. Given the unusual units of MPa(m1/2), fracture toughness is a quantitative
way of expressing "a material's resistance to brittle fracture when a crack
is present.". If a material has a large value of fracture toughness it will
probably undergo ductile fracture. Brittle fracture is very characteristic of
materials with a low fracture toughness value.
iii. Fracture toughness is a good way to consider how a fracture (dike) may
propagate.
iv. Four types of fracture toughness:
There are actually four different types of fracture toughness, KC, KIC, KIIC,
and KIIIC. KC is used to measure a material's fracture toughness in a
sample that has a thickness that is less than some critical value, B. When
the material's thickness is less than B, and stress is applied, the material is
in a state called plane stress. The value of B is given in the equation
below. A material's thickness is related to its fracture toughness in the
graph below.
B = minimum thickness to distinguish between KC and K1C
KC = fracture toughness, when the sample has a thickness less than B
sy = yield stress of material
For a material with thickness < B:
KC = fracture toughness, when the sample has a thickness less than B
Y = constant related to the sample's geometry
a = crack length (surface crack), one half crack length (internal crack)
s = stress applied to the material
KIC, KIIC, and KIIIC all represent a material's fracture toughness when a
sample of material has a thickness greater than B. If a stress is applied to a
sample with a thickness greater than B, it is in a state called plane strain.
The differences between KIC, KIIC, and KIIIC, however, do not depend on
the thickness of the material. Instead, KIC, KIIC, and KIIIC are the fracture
toughness of a material under the three different modes of fracture, mode
I, mode II, and mode III, respectively. The different modes of fracture I,
II, and III are all graphically expressed in the figures below:
Mode I fracture
Mode II fracture
Mode III fracture
For materials with a thickness > B:
KIC = fracture toughness, when the sample has a thickness greater than B
Y = constant related to the sample's geometry
a = crack length (surface crack), one half crack length (internal crack)
s = stress applied to the material
KIC values can be used to help determine critical lengths given an applied
stress; or a critical stress values can be calculated given a crack length
already in the material with equations:
Applied stress to cause failure in a material where;
sC = critical applied stress that causes the material to fail
KIC = fracture toughness, when the sample has a thickness less than B
Y = constant related to the sample's geometry
a = crack length (surface crack), one half crack length (internal crack)
Critical crack length to cause failure in a material where;
a = critical crack length (surface crack), one half crack length (internal crack)
s = stress applied to the material
KIC = fracture toughness, when the sample has a thickness less than B
Y = constant related to the sample's geometry
v. Table of fracture toughness values
KIC values for Engineering Materials
Material
K1C MPa (m)1/2
Metals
Aluminum alloy
36
Steel alloy
50
Titanium alloy
44-66
Aluminum oxide
14-28
Ceramic
Aluminum oxide
3-5.3
Soda-lime-glass
0.7-0.8
Concrete
0.2-1.4
Polymers
Polymethyl methacrylate
1
Polystyrene
0.8-1.1
c. Magmatic cases
i. What would be a case where a fracture could form but not propagate?
(think of rip-stop material)
ii. Under what conditions would a fracture propagate virtually unchecked?
iii. Does a “magma chamber” explode due to over pressure or collapse due to
loss of pressure?