# Using Mohr’s Circle for Plane Strain

Mohr's circle for plane strain can be used for determining principal strains in a body under plane strain.  This article will cover the basics of Mohr's circle in plane strain, and how it is easily created using MEboost.

MEboost also handles bodies under plane stress.  To learn about this scenario, refer to the article on plane stress.

## Mohr's Circle for Plane Strain Basics

When conducting critical plane analysis, Mohr's circle can be used to determine the critical plane where maximum damage occurs.  Let's start with some definitions.

Principal Planes  In an element under plane strain, there are two planes where shear strain is zero.  These are referred to as the principal planes.

Principal Strain A tensile or compressive strain that acts perpendicular to a principal plane.

Consider the element below that is under plane strain.

We need to define a convention for shear strain.  Shear strain yx that results in clockwise rotation is positive, and shear strain yx that results in counterclockwise rotation is negative.  In the element above, we see that shear strain yx is positive.  Also note that shear strain yx = -shear strain xy.  When Mohr's circle is plotted, we will see how this affects the principal plane angles.

To illustrate, let's consider an example.  Strain values are:

• Strain in x direction: 1,000μ
• Strain in y direction: -650μ
• Shear strain: 750μ

Positive strain is tensile, and negative strain is compressive.

To create Mohr's circle, we plot two points: (strain x, -shear strain yx/2) and (strain y, shear strain yx/2).  Shear strain is the vertical axis and strain is the horizontal axis.  A line is drawn between these points, and a circle is drawn that goes through both points.  A Mohr's circle is shown below with labels to illustrate the properties.

### Properties of Mohr's Circle

Mohr space refers to properties of Mohr's circle.  Real space refers to properties of the actual element under stress.

• The principal stresses, ε1 and ε2, are located on the horizontal axis where shear strain is zero.
• Mohr space: the principal planes are 180 degrees apart.  Real space: the principal planes are 90 degrees apart.
• Mohr space: the principal planes are located at an angle of 2Θ from the red line.  Real space: the principal planes are at an angle of θ from the x-direction.
• Maximum shear is at the top and bottom of the circle.
• Mohr space: maximum shear strain/2 occurs 90 degrees from the horizontal axis.  Real space: maximum shear strain/2 occurs at 45 degrees from the principal planes.
• The radius of the circle is maximum shear strain/2.  Alternatively, it is 0.5(ε1 - ε2).

## Principal Planes

The principal planes of the element are shown below.

## Drawing the Circle with MEboost

In the MEboost ribbon, clicking on the Mohr's Circle button will show the Mohr's Circle form.  Using the example above, we enter strain values in the form.  The value and sign for shear strain is for shear strain yx.

Note that strain values must be entered in units of strain that avoid decimal values.  In our example, strains are entered as micro strain.

After clicking the Create button, a report will be created in a new worksheet.

In addition to the chart, the report also shows the values of principal strains, maximum shear strain/2, and the principal plane angle.

## Reference

Riley, W.F., Zachary, L. Introduction to Mechanics of Materials, 1989, John Wiley & Sons, Inc., New York, NY, pp. 430-431.

Excel is a registered trademark of Microsoft Corporation. Used with permission from Microsoft.