Strain in 3D

Finite strain in 3D

Strain ellipsoid

When we move into 3 dimensions we can use analogous measures of strain to those in two dimensions.

The 3 D equivalent of the strain ellipse is the strain ellipsoid. This is the product of deformation applied to a unit sphere.

Cross-sections of the strain ellipsoid are strain ellipses (these can include circular cross-sections)

In general, the way we determine the 3D strain ellipsoid is to saw up rocks to define surfaces on which we find 2D strain ellipses. Then recombine the ellipses into an ellipsoid.

Strain axes and principal planes of strain

The orientation of the strain ellipsoid is indicated by the directions of three mutually perpendicular strain axes, which are, in general, the only three lines that are mutually perpendicular before and after deformation.

The strains along the strain axes are the three principal strains. The principal stretches are S1 > S2 > S3 or X, Y, and Z

Lines of no finite extension typically lie in a cone

The three strain axes are poles to three principal planes of strain, which are, in general, the only three planes that suffer zero shear strain.

Dilation

The volume dilation is given by 1+Δ = S1S2S3

Strain ellipsoid shapes

Axial ratios

In 2 dimensions we could specify the shape (distortion component) of the strain ellipse with a single number, the strain ratio.

In 3 dimensions that's not enough. There are many different shapes of strain ellipsoid

Typically we use two strain ratios to indicate the shape of the ellipsoid. They are conventionally

a = S1/S2 = X/Y

b = S2/S3 = Y/Z

Notice that the minimum value of a and b is 1.0, from the definition.

Flinn plot

We can make a plot of a against b, on which the shape of any strain ellipsoid is represented as a point. This is known as a Flinn plot. Note that the origin is at (1,1) not (0,0) because the minimum value of a strain ratio is 1 by definition.

An alternative plot, known as the logarithmic Flinn plot or Ramsay plot, shows log(a) against log(b). Its origin is at (0,0).

Measures of strain ellipsoid shape

The ratio a/b, known as k, is an indication of the overall symmetry of the strain ellipsoid. Values of k greater than 1 characterize ellipsoids with one long axis and two shorter ones S1 >> S2 > S3, informally known as cigars. Values of k less than 1 characterize ellipsoids with two long axes and one shorter one S1 > S2 >> S3, informally known as pancakes.

If S1 >> S2 = S3 then the strain is described as axially symmetric constriction. k is infinite.

If S1 = S2 >> S3 then the strain is described as axially symmetric flattening. k is zero.

Plane strain

Between the field of cigars (constriction) and the field of pancakes (flattening) is a line where k=1.

If k=1 and we additionally know there is no volume change, then it is easy to prove that

Under these circumstances the only movements of particles (particle paths and flow lines) due to distortion are in the S1 S3 plane. We can represent this strain as if it were a 2D strain. This type of strain is called plane strain and is a common assumption in section balancing.

Progressive strain and flow in 3D

Rotation and strain in 3D

Rotational deformation is much more complex in 3D than in 2D.

In 2D progressive deformation, rotation is about an axis perpendicular to the plane of section; any rotation axis is perpendicular to two strain axes.

In 3D, the rotational component of deformation can be perpendicular to two of the strain axes and therefore parallel to the third. When this is the case, the deformation is described as monoclinic. The 'before' and 'after' shapes of the strain ellipsoid have the same symmetry as a monoclinic crystal.

However, the rotation axis can be different from any of the strain axes. If this is the case, the strain symmetry is very low and the deformation is triclinic. Progressive triclinic strains (where rotation and distortion occur concurrently but on different axes) are extremely complicated to work with and require techniqes beyond the scope of this course.

Strain paths

The logarithmic Flynn plot, otherwise known as the Ramsay plot, comes into its own in dealing with strain histories. It turns out that is a constant incremental strain is applied, the finite strain follows a straight line path on the Ramsay plot.