At this point we skip a family of strains which is most important in engineering applications: bending and twisting strains. They occur in I-beams, flagpoles, and torsion bar suspensions for cars. It turns out that if we ignore the shape of the sample and focus on the displacement field inside, the bending and twisting strains can be represented with a combination of (position dependent) shear strains.
The displacement fields of both shear and tensile strains may be represented in two Cartesian (xy) dimensions. This is not true for twisting strains, which need an angle and a distance along the axis of twist; i.e., cylindrical coordinates. Since the shear strains are sufficient, and easier for our purposes, we will use them, and not twisting strains.
|
|
| A trailer hitch shears the pin that holds the two pieces |
| together in two places. |
When scissors cut paper, they cause the paper to undergo a shear strain so large that the paper yields; it comes apart where it is strained. The figure at the right shows another example of a sample being sheared. The pin is pulled to the left in the middle, and the right at the top.
|
| Two objects being sheared, and a shear displacement |
| field |
A laboratory example of a simple shear is shown in the figure. The upper figure shows a book resting on a table. A hand from above pushes the top cover of the book to the right. The static friction between the table and the bottom cover pushes to the left, and the acceleration remains at zero. The distortion of the book is called a simple shear distortion.
Question: What has been left out of this discussion? (answer later)
The block shown is sheared in a similar way, and measurements are included. Also shown is the displacement field. Note that the displacement is always in the x direction, but now the variation in the displacement occurs in the y direction.
It is clear (see the left edge of the block, for example) that the displacement in the x direction is proportional to the y component of the position. We can write the displacement field for this strain as:
To calculate the x displacement per unit (vertical) length, we use the ratio
Another measure of the shear strain is the slope of the edge of the block:
We see once again that the two methods of measuring strain agree, and define the simple shear strain as
The unusual notation, with the xy in the superscript is used because there is another way to define the shear strain, which gives a significantly different answer. This alternate definition is called the pure shear strain. The motivation for defining pure shear is the mistake we made in discussing simple shear above.
|
| Hands must produce two forces each to cause shear without |
| rotation. |
As with the tensile strain, it helps to think explicitly about the forces applied to all sides of the object. Looking again at the book example, we imagine holding it between two hands, as shown. If the top hand pushes only to the right and the bottom pushes only to the left, the book will experience a clockwise torque, and an angular acceleration. This torque was ignored in the first look at a sheared book.
|
| Anti-torque forces cause strains also |
The cubical block shear requires a similar set of forces to shear without rotational acceleration. To produce a net torque of zero on the cube, and to produce a net force of zero, the magnitude of each force shown on the cube must be the same. The figure makes it clear that the anti-torque (B) forces will also produce a shear strain.
|
| The net displacement field resulting from the sum of two |
| displacement fields. |
Displacement fields combine in the same way that displacements do: by vector addition. Since the magnitude of the (B) forces is the same as the (A) forces, for the cube, we expect the same shear strain from the (B) forces. (Remember, we have assumed that our materials are isotropic.) The only difference is that the (B) forces will produce a displacement field that points up, rather than to the right.
The two displacement fields and their sum are sketched in the figure. The net distortion of the block is clearly different from the simple shear distortion. The upper right hand corner moves right and up, instead of just right.
Mathematically the addition of the two displacement fields is
.
Because this strain has been constructed with no rotations, it is called a pure shear. We must decide how to measure this pure shear strain.
From the picture, one obvious way to characterize the strain is to calculate fractional change in the length of the diagonal. That is
where L is the length of the edge of the original cube.
This is the same answer that we would get by measuring the strain as the rise, per unit length, of the bottom edge of the cube:
or the rightward shift per unit height:
These individual shifts have a flaw: They are not zero if the block is simply rotated, rather than strained.
We could sum those two shifts, and characterize the strain by 2c. The sum has the advantage of being zero when the cube is rotated, so that it automatically eliminates rotations from consideration, as we desire. Since it is c which appears in the pure shear displacement field, we choose to measure the pure shear strain as c rather than 2c. We just agree to divide this sum by two.
The differential method of determining the shear strain is written:
This definition also successfully discriminates against rotations. We will count this partial derivative as out fundamental definition of pure shear.
|
| A pure shear rotates into a simple shear. |
We can make the pure shear distortion look like the simple shear distortion. This is useful when we want to compare the two different descriptions of the same strain in a solid. We imagine creating a pure shear, with displacement field
and then rotating the distorted cubical sample back down to the x axis, so that it becomes a simple shear with displacement field of the form
Our question is this: How is b related to c?
The answer is only simple in the case of small distortions. In that case, the tangent of the angle of rotation is approximately equal to the angle, in radian measure. In this case, rotating the distorted cube down simply doubles the length the displacement vectors of the left edge of the cube, as sketched in the figure. The new displacement field will be
Since the simple shear is b, and the pure shear is c, we have this disappointing result: For the same distortion of the sample (ignoring rotation), the simple shear measure of strain is twice the pure shear measure of strain.
This factor of two haunts elastic calculations from time to time.
Return to index.
Return to survey.
Go forward to Poisson's ratio.