# Shear Strain

Steven Dutch, Professor Emeritus, Natural and Applied Sciences, University of Wisconsin - Green Bay

## Simple Shear

 Perhaps the most familiar illustration of shear is the movement of rocks on opposite sides of a fault as shown here. Because this type of shear is the easiest to visualize, it is called simple shear.

Imagine when the fault starts moving we draw a line at right angles to the fault. As the fault slips, the line rotates (and also lengthens), and angle A increases. However, angle A will never reach 90 degrees unless the slip on the fault is infinite.

 We can define shear strain exactly the way we do longitudinal strain: the ratio of deformation to original dimensions. In the case of shear strain, though, it's the amount of deformation perpendicular to a given line rather than parallel to it. The ratio turns out to be tan A, where A is the angle the sheared line makes with its original orientation. Note that if A equals 90 degrees, the shear strain is infinte.

Note that we are not concerned about the line changing length. That's longitudinal strain. With shear strain we are only concerned about the change in angles.

## Pure Shear

 Any time an object is deformed, shear occurs. For example, in the top row a block is deformed without changing area. It looks like the only deformation involved is compression and extension. However, if we examine the diagonals of the block (bottom row) we see that there is indeed shear because the angle between the diagonals changes. This sort of shear is called pure shear.

Pure shear is harder to see than simple shear because there is no stationary frame of reference. Imagine that you have planted your feet firmly along one of the diagonals of the block. As the block deforms, you see the other diagonal rotate just as you did with simple shear. To an outside observer, you also rotate, but from your perspective the two situations look identical.

Even the principal strain directions look the same. In the simple shear case above, the major and minor axes of the deforming ellipse rotate clockwise as strain progresses. The same thing happens under pure shear as well.

### Directions of Greatest Shear

If we inscribe a square at different orientations in a block and deform it, we can see that the square oriented at 45 degrees to the principal strains is sheared the most.

## Simple and Pure Shear Compared

The blocks above have undergone shear strains of 0, 0.5, 1.0 and 1.5. The top row has undergone simple shear, the bottom row pure shear. Angles 90-A and 90+A (shown for the shear=0.5 case) are the same in each corresponding pair of diagrams. Note that the ellipses are the same shape.

Simple Shear:
One direction remains constant and everything else rotates relative to it. Approximates the situation in a shear zone.
Pure Shear
Directions of greatest compression and extension are constant. The major and minor axes of the deforming ellipse remain constant. All other lines rotate.

### An Unkind Parting Thought

Neither of the situations described in simple and pure shear is likely to occur exactly in nature. Most of the time stress directions and magnitudes change over time and everything in a deforming body of rock rotates. Shear may be clockwise at one time and counterclockwise at others. The real complexity of deformation in rocks has led some geologists to claim: "Pure shear is pure nonsense and simple shear is simple nonsense". However, like bodies falling without air resistance or sliding without friction, they are useful ideal concepts and first approximations to complex reality.

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Created 26 February 1999, Last Update 1 March 1999