Structural Geology
Lecture 10
An Introduction to Stress
(Tensor Analysis)

Hooke's Law

When a body changes shape we say that the body is strained.  Strain, a deformation, is measured as the ratio of the change in shape of a body to its initial shape.  Previous figures illustrates strain using a long rod with an initial length (L).  When the rod is stretched the additional length (i.e., displacement) of the rod is DL.  Strain (e) is DL divided by L.  If the rod is deformed elastically, it will return to the length, L, when the forces are removed from the end of the body.

Another aspect of elasticity which we will examine is the relationship between the amount of force and the amount of strain.  Figure 10-1 shows an unloaded spring and the same spring loaded with one, two, and three weights, respectively.  The spring gets longer in equal increments with the addition of comparable increments of weight.  Remember that a weight is a force.  Relationship between change in force and change in stain is characteristic of an elastic material.

Figure 10-1

It is accurate to imagine that the atomic forces which govern the atomic distances in a lattice act just like springs which return to their natural shape regardless of whether they were subject to a push or a pull.  The law that governs the action of a spring is called Hooke's Law.   We can write the equation for the Hooke's Law of a spring as follows:

  force = spring constant  X  displacement 	
	(10-0)

Equation 10-0 says that the 'stretch' or displacement of a spring is proportional to the force on the spring.  This elastic law can be illustrated in a plot of strain (displacement) versus stress (force) (Figure 10-2).  Such a plot is a straight line and the relationship between force and displacement is linear.   The constant which specifies the relationship between the force on the spring and its displacement is called the spring constant.  Often rock deformation is given in units of stress (force per unit area) and strain (change in length divided by initial length).  In this case Hookes Law is written as:

stress = Young's modulus X strain	
	(10-00)

Figure 10-2

In Figure 10-2 stress (force) is plotted as the independent variable as it should be.  However, geologists are an odd bunch who by convention always plot stress (force) on the vertical axis as if it were the dependent variable.  Don't be fooled by this convention in the following plots.

Measurements of strain of rocks usually involve very small displacements.  This is because most rocks are very stiff relative to a rubber band.  In fact, rocks are so stiff that strain is measured using the unit, microstrain.  A strain of 10-1 is a 10% strain which means that a body of one meter in length will shorten by 10 centimeters.  A strain of 10-3 is a 0.1% strain which means that a body of one meter in length will shorten by 1 millimeter.  A microstrain is 10-6 or a 0.0001% strain which means that a body of one meter in length will shorten by one micron (one micrometer = 10-3 millimeter).  

If we wish to measure very small strains we must use strain gauges hitched to a strain indicator.  Strain gauges are small wires wound like an accordion (Figure 10-3).  The strain gauge wires are glued to the surface of specimen so that when the specimen stretches, the strain gauge wires also stretch.  If the wires are stretched the cross sectional area of the wires gets smaller.  The electrical resistance of the wire increases if the cross sectional area decreases.  The strain gauge operates on the principle that the change in resistance of the accordion-like wire is equivalent to the strain of the sample.  A strain indicator measures strain in the strain gauge wires by measuring the change in resistance of the wire as the specimen stretches. 

                 Figure 10-3

Stress Vector -- Mohr's diagram is a convenient graphical representation of state of stress within the crust of the earth.  It works well for the two dimensional representation of stress but in three dimensions it becomes less convenient.  However, some of the rules that apply in two dimensions hold for three dimensions.  One of the more important rules that was demonstrated in two dimensions was that the sum of stresses was invariant or unchanged by the rotation of coordinate axes.  In this lecture we are going to examine the more general case for rotation of coordinate axes with the stress transformation.  To introduce the stress transformation we appeal to the concept of a stress vector.

Figure 10-4

To understand stress in three dimensions, we start with a force vector f acting on the surface of a rock or acting across a boundary (Fig. 10-4).  This vector is also a stress vector where

	s  =  lim Df/DS                                                    	(10-1)                

where the lim DS approaches 0.  Take any small surface element of area  dS containing a point P  within the stressed rock. A unit vector l can be drawn perpendicular to the area dS at the point P.  Then, the force transmitted across the area is sdS.  Note that stress multiplied by area is a force. The force is exerted on the rock on the positive side of the area (defined by the orientation of l) is resisted by stress within the rock on the negative side of the surface dS.  While the force on the positive side is defined by a first rank tensor (the three components of the force vector), stress on the negative side of the surface is expressed using a second rank tensor (the nine components of the stress tensor).

We now want to examine the variation of sdS as the orientation of l is changed within the stressed rock.  In this exercise the surface will be maintained so that the unit vector l will always pass through the point P  on the surface dS.  For this exercise we need to assume that the stress is homogeneous, that there are no body-forces, and that the body is in equilibrium. By equilibrium we mean to say that the rock is not presently being deformed.  The force sdS does not change with orientation of l.  Stress can be treated as a vector acting on a plane, P.  By examing Fig. 10-1 it can be seen that the force vector, f, can be resolved (by using the dot product) into a component parallel and a component normal to the surface of the plane, P.  Likewise, if f/dS is treated as a vector, it can be resolved into two components: sn, a normal stress, and t, a shear stress.  While the magnitude and direction of the stress vector does not change upon reorientation of the surface, the magnitude and direction of the  sn, a normal stress, and t, a shear stress changes.  s = f/dS is known as the stress vector acting on a certain plane.

Figure 10-5

Reaction of the solid to a stress vector  --  Now we examine a solid piece of the rock in the shape of a tetrahedron with corners ABCO (Fig. 10-5).  Because the rock is in equilibrium no surface of the tetrahedron is moving relative to any other surface. Let the surface defined by ABC be the surface element dS discussed above.  The force s(f) transmitted across dS is s (f) times (area ABC).  The forces on the three faces at right angles may be each denoted by three components sij so that we have nine components in all.  Each face has two components parallel to the surface and one component normal to the surface.  Note that these components are parallel to the three components l1, l2, and, l3 of the unit vector l.  Remember that a stress multiplied by an area is a force.  Resolving forces parallel to Ox we have 

	f1 (ABC)   =  s11(BOC)  +  s12(AOC)  +  s13(AOB)      	(10-2)                

Dividing the area of each side (e.g. BOC) by the area of the face (ABC) gives the component of the unit vector l in the direction normal to the side.  Thus:

	f1  =   s11l1  +  s12l2  +  s13l3

	f2  =   s21l1  +  s22l2  +  s23l3

	f3  =   s31l1  +  s32l2  +  s33l3.	(10-3)

Hence,	
fi   =  s ijlj         	(10-4)                                                            

where both fi and lj are three components of the force vector f and the unit directional vector l. 

Mohr circle construction - We start with the principal axes of a tensor

	(10-5)

and consider the transformation of the components of stress by a clockwise rotation about the principal axis s3.  For this rotation the direction cosine matrix is the following

	(10-6)

We now transform sij using the direction cosines to obtain

	(10-7)

According to the transformation we generate four components.  For example,

where  s11  =  s1  and  s22  =  s2  so that

	(10-8)

These equations may be rewritten to the classic equations (1-29a&b) for the Mohr circle stress analysis.   From the Mohr's Circle analysis we note the property of invariance.

	(10-9)

Field tensors verses matter tensors -  Stress are strain are examples of field tensors whereas the tensors which measure crystal properties such as magnetic susceptibility are matter tensors.  Matter tensors must conform to crystal symmetry whereas stress is not a crystal property but is akin to a force impressed on the crystal.  Both field and matter tensors have similar special forms.

Field Tensor	Symmetry of Matter Tensor

	Hydrostatic Stress		Isometric

	Uniaxial Stress

	Biaxial Stress		Tetragonal

	Triaxial Stress		Orthorhombic
 
	Pure Shear		Monoclinic

			Triclinic