Background
Geological Background
Figure 1 Schematic diagram with magmatic structures (taken from http://earth.geol.ksu.edu/sgao/g100/plots/)

A dyke is a magma-filled fracture that cuts across bedding or other rock structure. In other words, it is a discordant tabular intrusive body. A sill is a concordant tabular intrusive body, so sills form when magma exploits the planer weakness between sedimentary beds or other foliations (Winter, 2001). Both terms are used regardless of whether the material is making up the body in liquid magma or solidified rock. Figure 1 shows a schematic block diagram in which both structures are drawn.

Figure 2 shows some pictures of how dykes appear in the field.

Dykes are the most important magma transportation system on earth. They occur in a variety of geological and tectonic settings like mid-ocean ridges, sedimentary basins and granitic shields. Individual dykes may transport a small amount of magma and cause little deformation of the host rock, but the effect of many dykes combined can be large. They can locally reshape the surface and interior of Earth’s crust. This happens for example at oceanic spreading ridges, where dykes accommodate great crustal extension and become one of the principal building blocks of new crust. Another example is that dykes feed the small fissure eruptions that eventually built shield volcanoes, the largest mountains on earth (Pollard, 1987).
Figure 2 Pictures of some dykes taken in the field.
From top to bottom:
a) a big basaltic dyke (the black rock) in the Grand Canyon (Arizona, USA) (taken from http://earth.geol.ksu.edu/sgao/g100/plots/)
b) and c) mafic dykes, quarter is used for scale. At Scoodic Point Acadia National Park (Maine, USA) Both from S.E. Johnson


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Why use (layered) Gelatin as an Analog? 

Gelatin has been used by several workers in experiments on fluid-filled fracture propagation (Takada, 1990; Muller et al., 2001; Ito & Martel, 2002; Menand & Tait, 2002; Watanabe et al., 2002). Several advantages for using gelatin as an analog for the crust in these experiments are that the material is transparent, brittle at room temperature, the typical length of propagating fractures is on the order of centimeters, the elastic parameters can be controlled by varying the concentration of gelatin in water (Takada, 1990; Rivalta et al., 2004) and the material has photoelastic properties allowing us stresses in the gelatin to be viewed by using polarizing sheets (Muller et al., 2001).

The layering in the gelatin can be related to geology in the following way: dyke ascent from a high rigidity to a low rigidity can be expected at crust-mantle or basement-sediment boundaries. The opposite (ascent from a low to a high rigidity) can occur, for example, if a more compact solidified magmatic intrusion or sill is lying on more compliant volcanic deposits or sediments (Rivalta et al, 2004).

The quality of gelatin is usually expressed as its Bloom Strength. This is the force in grams required to press a 12.5 mm diameter plunger into 112 g of a standard 6.66% gelatin gel at 10˚C.

The relation between the Bloom Strength (B) and the concentration (C) can roughly be expressed as:

Where k is constant (Cole, 2005).

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Fracture Theory

In nature, dykes are formed when a fracture is dilated by magma. The primary requirements for dyke emplacement is that the minimum principal stress in the host rock and the host-rock tensile strength are exceeded by the magma pressure. The difference between magma pressure and initial host-rock stress is called the driving pressure, which provides the fundamental driving force in the dyke emplacement (Hoek, 1994).
In gelatin experiments this driving force can be achieved by injecting the gelatin (the “host rock”) with a sufficient amount of liquid (the “magma”, in my case air) which will, due to buoyancy (caused by a density difference between the gelatin and the air), eventually propagate through the gelatin.
Figure 3 Schematic illustration of the radial propagation
from the front (left) and from a cross sectional view (right).
Taken from Menand & Tait (2002).
During propagation two processes can be distinguished. First the fracture will grow radially until a certain shape is reached, then the fracture will start to propagate vertically (Maaløe, 1987; Menand & Tait, 2002). This is illustrated in figure 3. Mention that in the cross sectional view the fracture develops a head thicker than its tail. It starts propagating radially, but eventually propagates vertically.
During propagation two processes can be distinguished. First the fracture will grow radially until a certain shape is reached, then the fracture will start to propagate vertically (Maaløe, 1987; Menand & Tait, 2002). This is illustrated in figure 3. Mention that in the cross sectional view the fracture develops a head thicker than its tail. It starts propagating radially, but eventually propagates vertically.
Figure 4 Dimensions of a fracture. Taken from Menand & Tait (2002).

The “critical” shape the fracture has to achieve before it starts to rise has been investigated by a number of workers (Maaløe, 1987; Takada, 1990; Dahm, 2000; Menand & Tait, 2002) and depends mostly on the elastic properties, or strength, of the host material. But also the density difference between the liquid and the host material and the volume and velocity by which the liquid is injected play a role (Menand & Tait, 2002).
Figure 4 shows the typical shape that develops with its dimensions.

The critical length that is necessary for propagation is given by:

In which Kc is the fracture toughness, Δρ is the density difference between the solid and the liquid and g is the gravity acceleration (Weertman, 1971). Kc can be expressed as:

where γs is the surface energy of gelatin and is thought to be dependent only on the temperature (Griffith, 1920 as quoted by Menand & Tait, 2002), G is the shear modulus and ν is Poissons ratio. G can also be expressed as:

where E is Youngs modulus (Menand & Tait, 2002).

The equation given for the ellipsoidal cross section of a penny-shaped crack is given by:

In which r is the distance from the center of the crack (Maaløe, 1987).

As stated previously, a dyke can propagate as long as it achieves a "critical" pressure. In order to determine this critical pressure the fracture toughness is compared to the stress intensity (K), which, for an ellipsoid crack, is given by:

The crack will propagate if K>Kc so the pressure required for crack propagation is then given by (Maaløe, 1987):

Other pressures present during the propagation of the fracture are as follows.

-The elastic pressure required to open the fracture, Pe, which is equivalent to

in which w is the width of the fracture.

-The hydrostatic (or buoyancy) pressure, Ph, present due to the density difference between the solid and the fluid. This pressure is equivalent to:

Δρgl

-The viscous pressure drop caused by the flow in the fracture:

in which η is the viscosity of the fluid and u is the average velocity of the fluid inside the fissure (Lister, 1990).

Fractures are believed to open in the direction of the least compressive stress and propagate in directions perpendicular to the least compressive stress (Martel, 2001).


Basically there are three stresses applied to the surrounding material when a fracture propagates. At the tip of the fracture a tensile stress is working that opens up the gelatin (or country rock). The stress contours of the tensile stress working in the horizontal direction (perpendicular to the propagation direction of the fracture) are illustrated in figure 5.
The typical pattern made by the maximum compressive stresses and the shear stresses caused by a propagating fracture are shown in figure 6.

Figure 5 Contours of the tensile stress component operating perpendicular to the travelling fracture. Taken from Clemens & Mawer (1992).
Figure 6a)A schematic drawing of a bouyancy driven fracture with the coordinates used in b) and c). b)The arrows indicate the direction of the maximum compressive stress and the contours show the shear stress around a two dimensional buoyancy driven crack. The shear stress is normalized by the average liquid excess pressure. c)Shows the same as b) but in a different direction. Taken from Watanabe et al. (2002).

Lateral fracture propagation is believed to happen at the neutral buoyancy level (LNB) (Lister, 1990; Lister & Kerr, 1991) where the magma is in gravitational equilibrium. I will not go further into the theory of that here. 

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Photoelastic Stress

Gelatin is a birefringent material with photoelastic properties. Thus, if it is placed between polarizers and light is guided through it while stress is applied to the gelatin, a pattern of stress distribution will be visible. The principle of photoelasticity is explained below.

Light is formed by electromagnetic waves. A light source emits radiating energy, which propagates in all directions and contains a whole spectrum of vibrations of different frequencies (measurement of the number of times that a repeated event occurs per unit time) or wavelengths. Only a part of those wavelengths can be detected by humans. The different wavelengths that are detected by the human eye are interpreted by the brain as colors, ranging from red at the longest wavelengths of about 700 nm (lowest frequencies) to violet at the shortest wavelengths of about 400 nm (highest frequencies). The intervening frequencies are seen as orange, yellow, green, cyan, blue and indigo.
Light emitted by a ‘normal’ light source is white, which means that all visible wavelengths are present in that beam. Natural light is generally unpolarized: all planes of propagation being equally probable.

If light is leaded through a polarizing filter only the waves that vibrate parallel to (in the same direction as) the privileged axis of the filter will be transmitted (see figure 7). This transmitted beam is called polarized light or “plane polarized” because the vibration is contained in one plane. If another polarizing filter is placed in its way, complete extinction of the beam can be obtained if the axes of the two filters are perpendicular to one another.

In my experiments circular polarizers were used. Circularly polarized light consists of two perpendicular waves of equal amplitude and a 90° difference in phase. The light illustrated in figure 8 is right-circularly polarized.

If you could see the tip of the electric field vector, it would appear to be moving in a circle as it approached you (click here for a 3D animation; this animation is taken from http://www.physics.upenn.edu/courses/gladney/phys151/lectures/lecture_apr_09_2003.shtml).
If while looking at the source, the electric vector of the light coming toward you appears to be rotating clockwise, the light is said to be right-circularly polarized. If the electric vector is rotating counterclockwise, it is called left-circularly polarized light. The electric field vector makes one complete revolution as the light advances one wavelength toward you.
Figure 8 Circular polarization (image taken from http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/polclas.html)
Circularly polarized light may be produced by passing linearly polarized light through a quarter-wave plate at an angle of 45° to the optic axis of the plate (see figure 9). If linearly polarized light is incident on a quarter-wave plate at 45° to the optic axis, then the light is divided into two equal electric field components. One of these is retarded by a quarter wavelength by the plate. This produces circularly polarized light. Incident circularly polarized light will be changed to linearly polarized light.
Figure 9 Quarter-wave plate (image taken from http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/polclas.html)

Next to planer and circular, elliptical polarization is another form of polarization. Elliptical polarized light consists of two perpendicular waves of unequal amplitude which differ in phase by 90°.

Light propagates in a vacuum or in air at a speed C of 3 x 1010 cm/ sec. In other transparent bodies, the speed V is lower and the ratio C/V is called the index of refraction (n). In a homogeneous body, this index is constant regardless of the direction of propagation or plane of vibration. Certain materials, like plastics, photoelastic coatings and gelatin behave isotropically when unstressed but become optically anisotropic when stressed. The change in index of refraction is a function of the resulting strain.

If a polarized beam α propagates through a strained gelatin body of thickness t, where X and Y are the directions of principal strains (εx and εy) at the point under consideration, the light vector splits and two polarized beams are propagated in planes X and Y (see figure 10).
Figure 10 Schematic drawing of a beam propagating through a stressed body (image taken from http://www.vishay.com/company/brands/measurements-group/guide/pstress/tech/tn702/702a.htm)

The basic relation for strain measurement using photoelasticity is:

Which is obtained from the equations:

and

In these equations δ is the relative retardation (or phase difference) between the two beams vibrating along the directions X and Y with a speed of Vx and Vy respectively. K is a constant called the “strain-optical coefficient” and characterizes a physical property of the material.

Due to the relative retardation the two waves are no longer in phase when emerging from the gelatin. The analyzer will transmit only one component of each of these waves (the one parallel to the axis of the analyzer). These waves will interfere and the resulting light intensity will be a function of the retardation, and of the angle between the analyzer and the direction of principal strains (β-α).
By using circular polarizers this last factor is eliminated.

In the case of a plane polariscope, the intensity of light (I) emerging will be:

or (for circular polarizers):

In which a is the amplitude and λ is the wavelength of the light.

With plane polarization the light intensity diminishes when either sin term goes to zero, therefore we have two possible fringe patterns of points where the light is extinguished:
isochromatics, which indicate areas of constant stress magnitude, and isoclinics, which indicate principal stress directions.

As stated before, when using circular polarizers the waves propagating in the principal strain (or stress) direction are eliminated. So isoclinics do not show up when using circular polarizers.

The isochromatic color pattern indicates areas of constant stress magnitude. The stress indicated by the colors is the differential stress (σ1-σ3), which can be linearly related to the maximum shear stress (Martel, 2001) since:

  

For thin materials (like photoelastic coatings) a Michel-Levy chart can be used to determine the birefringence (Δn) by comparing the colors from the chart with the colors obtained in the fringes.

Information for this section on photoelasticity is derived from:
http://www.vishay.com/company/brands/measurements-group/guide/pstress/tech/tn702/702a.htm
http://www.doitpoms.ac.uk/tlplib/photoelasticity/index.php
http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/polclas.html
http://www.physics.upenn.edu/courses/gladney/phys151/lectures/lecture_apr_09_2003.shtml

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