Structural Geology
Lecture 7
Irrotational vs. Rotational Strain
(another look at strain)

The rotational strain tensor, eij, is invariant with respect to arbitrary, superposed shifts of the point of origin of the coordinate system.  We will discuss the implications of this property in detail when looking at the stress tensor.   For the time, we must accept the fact that any tensor property is independent of the position and orientation of the coordinate system.

Nomenclature for the rotational strain tensor eij is as follows:

The first subscript, identifying the axis (i.e. direction) of a displacement component ui, is permuted from 1 to 3 in each of the columns, the second, identifying the face (i.e. the surface normal to an axis) whose center is displaced by the (u)j, is permuted from 1 to 3 in each row.

We wish to make a distinction between infinitesimal strains and finite strains.  An infinitesimal strain is one for which the final positions are very close to the initial positions of adjacent particles. Infinitesimal strain theory is used in the theory of elasticity where strains are very small.  While infinitesimal strains are independent of actual path of displacement, this is not the case for finite strains.  Strain evident in outcrop is finite.

The rotational strain tensor, eij, applies to infinitesimal strains and is a general (or asymmetric) second rank tensor and can be expressed as the sum to a symmetric and an antisymmetric tensor 

where

and

Here is eij symmetric and wij is antisymmetric.  The symmetric component of the infinitesimal rotational strain tensor, eij, involves only dilatation (the change in volume) and distortion (the change in shape).  Because of symmetry, the component, eij , consists of only six independent variables.  The component, eij , can be regarded as strain proper, or irrotational strain.  The antisymmetric tensor, wij , has only three independent components and involves only the rotational component of the strain tensor.  

With these rather simple definitions the distinctions between simple shear and pure shear become clear.  In a previous lecture we learned that the displacement equations for a pure shear involving overburden compaction were as follows:

		u1  =   0x1  +  0x2 +  0x3
		u2  =   0x1  +  0x2  +  0x3
		u3  =   0x1  +  0x2  -  0.005x3

Note that this time I have indicated a very small overburden compaction.  This was done because I am now dealing with infinitesimal (i.e. very small strains).  The deformation gradient matrix for these equations is

The rotational strain tensor, eij, is

For infinitesimal strain the rotational strain tensor and the deformation gradient are the same.  The irrotational strain tensor, eij, is

The rotational component of strain is

For pure shear there is a remarkable similarity between the rotational strain tensor and the irrotational strain tensor,.  This similarity disappears for the case of simple shear.

Again recall that displacement equations which were previously discussed for the same of simple shear as might be encountered within a fault zone.  For infinitesimal strain these equations were

u1  =   0x1  +  0x2 +  0x3
u2  =   0x1  +  0x2  +  0.005x3
u3  =   0x1  +  0x2  + 0x3

The deformation gradient matrix and the rotational strain tensor for these equations is

The irrotational strain tensor, eij, is

The rotational component of strain is

The distinction between pure shear and simple shear is further clarified by considering the principal strain axes.  If the directions of the principal axes of strain do not change as a result of displacement, then that deformation is termed irrotational strain.  In the case of simple shear the principal axes of strain always differ depending on the amount of shear.  The difference defines the rotational component of strain which is known as rotational strain.  It is important to note that although simple shear is a rotational deformation, there has been no actual rotation in space within a fault zone.  The development of rotational strain does not necessarily imply that the body has to spin physically around some axis.  Because of this lack of real rotation of within a fault zone, some like to refer to the rotation as an internal rotation.  This type of rotation is in contrast to our example of external rotation where bedded sediments being turned on end as given by the displacement equations below.  

		u1  =   0X1 + 0X2 + 0X3
		u2  =   0X1  -  1X2  - 1X3
		u3  =   0X1  +  1X2  -  1X3

Note that in the case of external rotation we are dealing with finite strain where the components of the deformation gradient matrix are large.