Structural Geology
Lecture 1
Introduction:
The parameters of structural geology
(constants, conversion factors, etc)

Structural geology boils down to a study of Newton's famous laws of motion as they pertain to the deformation of rocks within the earth.  Newton's first law of motion states that "Every body persists in its state of rest or of uniform motion in a straight line unless it is compelled to change that state by forces impressed on it".  The very presence of faults and folds in the crust suggests that rocks, seemingly at rest, were once subject to forces that changed their original state by motion of one point relative to another.   Structural geology is the study of the deformation of rocks.  In its simplest form this is a description of present geometries.  A study of the motion causing the geometries within rocks is called kinematics.  A study  of the forces that cause the motion is called dynamics.  The mathematics of structural geology are designed to simplify the study of kinematics and dynamics.

Structural geology is the study of the geometry, kinematics, and dynamics of rock structures.  Geometric analysis is the descriptive or qualitative portion of structural geology.  This portion of structural geology is as the name implies: A study of the size, shape, and orientation of structures.  This portion was covered in classical structural geology courses.  However, in this set of lecture notes the study of geometry will be delayed until a good mathematical base is established.  In the meantime, many of the lab exercises will be devoted to geometric analysis.  One of the most useful tools in geometric analysis is the stereonet which is a qualitative tool that serves the same purpose as vectors within a coordinate system. 

Kinematic analysis requires a mathematical base for a rigorous treatment.  Kinematics, as you learned when taking elementary physics, is a mathematical description of the motion of objects.  In the case of structural geology kinematics is the description of the path that rocks took during deformation.  It is also the mathematical description of the relative position of two infinitesimal points during the deformation of rocks.  Two points can change  by translating together, rotating around each other, or changing in distance relative to one another.  We shall call such a mathematical description deformation mapping.

Dynamics is the study of the forces which caused the deformations studied during kinematic analysis.  In the case of structural geology dynamics includes the study of how rocks react to stress.  For every stress the rocks respond with a finite strain.   In a sense rock structures would not have formed, if rocks had not been subject to a stress.  A study of dynamics starts with the fourth lecture.

Like any physical science, structural geology comes complete with its table of numerical data and symbols for physical properties.  Physics has its table of fundamental and derived physical constants which includes the speed of light (c = 3 x 108 m/sec), Avogadro's number (No = 6.02 x 1023 / mole), and universal gas constant (R = 8.23 joules/(mole)(K°).  A comparable table involves numerical data which depend on circumstances such as geographic location and are, therefore, not strictly physical constants.  Another table consists of a list of common physical properties which are represented by symbols including letters of the Greek alphabet.  These numerical data and material properties form an important component of the language of structural geology. 

Table of symbols for the structural geologist

	Symbol	Name 	Units

	r	density	ML-3

	s	stress	ML-1T-2

	t	shear stress	ML-1T-2

	sn	normal stress	ML-1T-2

	e	strain	dimensionless [LL-1]

	E	Young's Modulus	ML-1T-2

	n	Poisson's ratio	dimensionless

	g	Engineering shear strain	dimensionless

	Pp	pore pressure	ML-1T-2

	f	porosity	dimensionless

	T	temperature	C°

	q	heat flow	JL-2T-1

	k	thermal conductivity	JL-1T-1C°-1

	z	depth	L

Table of numerical data for the structural geologist

	Symbol	Name 	Magnitude

	g	average gravity at sea level	9.8 m/sec2

	rm	mean density of the mantle	4.5 x 103 kg/m2 (4.5 g/cm2)

	rquartz	density of quartz	2.65 x 103 kg/m2 (2.65 g/cm2)

Table of conventions for the structural geologist

	Name	Convention 	

	Principal stresses	s1 > s2 > s3 

Stresses in the crust

	Maximum horizontal stress	SH
	
	Minimum horizontal stress	Sh
	
	Vertical horizontal stress	Sv
	
	Compressional normal stress	positive

	Tensile normal stress	negative	

Table of conversion factors for the structural geologist

Stress and pressure

1 atm = 14.5 psi = 1 bar = 106 dynes/cm2 = 105 N/m2 = 105 Pascals (Pa)

1 MPa = 10 bars = 106 N/m2

pressure applies to a fluid

stress applies to a solid

Table of definitions for the structural geologist

	Name	Definition 	

	A Component of Principal Stress	sii  or  si

	Any Component of Stress	sij

	Differential Stress	sd = s1 - s3

	Maximum Shear Stress	

	Lithostatic Stress	SH = Sh  = Sv

	Hydrostatic Pressure	Pp = Pp = Pp	 

	Mean Stress	

	Deviatoric Stress (3 components)	sm - s1,  sm - s2, sm - s3

	Effective Stress	si - Pp

Table of equations for the structural geologist

Stress and pressure

Pp = rH2Ogz          Sv = rrockgz 

if   rrock = 2.5 x 103 kg/m3,   g = 9.8 m/sec2,   z = 103 m,

then   Sv = 2.5 x 105 kg/m-sec2  =  25 MPa/km

Finally, the average geothermal gradient (dT/dz) within the crust of the earth is about 20°C/km where T is temperature and z is depth.  This gradient can vary from 10°C/km in a glaucophane-schist terrain to 40°C/km.  The low geothermal gradients can occur in the vicinity of crystalline overthrusts where cold crust is depressed.  High gradients occur in a region of magmatic intrusion.  Heat flow (q) at the surface is an indication of geothermal gradient provided the thermal conductivity (K) of the crust is known
	q  =  K(dT/dz).