Mohr's circle

Mohr's circle is a two-dimensional graphical representation of the transformation law for the Cauchy stress tensor.
Mohr's circle is often used in calculations relating to mechanical engineering for materials' strength, geotechnical engineering for strength of soils, and structural engineering for strength of built structures. It is also used for calculating stresses in many planes by reducing them to vertical and horizontal components. These are called principal planes in which principal stresses are calculated; Mohr's circle can also be used to find the principal planes and the principal stresses in a graphical representation, and is one of the easiest ways to do so.
After performing a stress analysis on a material body assumed as a continuum, the components of the Cauchy stress tensor at a particular material point are known with respect to a coordinate system. The Mohr circle is then used to determine graphically the stress components acting on a rotated coordinate system, i.e., acting on a differently oriented plane passing through that point.
The abscissa and ordinate of each point on the circle are the magnitudes of the normal stress and shear stress components, respectively, acting on the rotated coordinate system. In other words, the circle is the locus of points that represent the state of stress on individual planes at all their orientations, where the axes represent the principal axes of the stress element.
19th-century German engineer Karl Culmann was the first to conceive a graphical representation for stresses while considering longitudinal and vertical stresses in horizontal beams during bending. His work inspired fellow German engineer Christian Otto Mohr, who extended it to both two- and three-dimensional stresses and developed a failure criterion based on the stress circle.
Alternative graphical methods for the representation of the stress state at a point include the Lamé's stress ellipsoid and Cauchy's stress quadric.
The Mohr circle can be applied to any symmetric 2x2 tensor matrix, including the strain and moment of inertia tensors.

Motivation

Internal forces are produced between the particles of a deformable object, assumed as a continuum, as a reaction to applied external forces, i.e., either surface forces or body forces. This reaction follows from Euler's laws of motion for a continuum, which are equivalent to Newton's laws of motion for a particle. A measure of the intensity of these internal forces is called stress. Because the object is assumed as a continuum, these internal forces are distributed continuously within the volume of the object.
In engineering, e.g., structural, mechanical, or geotechnical, the stress distribution within an object, for instance stresses in a rock mass around a tunnel, airplane wings, or building columns, is determined through a stress analysis. Calculating the stress distribution implies the determination of stresses at every point in the object. According to Cauchy, the stress at any point in an object, assumed as a continuum, is completely defined by the nine stress components of a second order tensor of type known as the Cauchy stress tensor, :
After the stress distribution within the object has been determined with respect to a coordinate system, it may be necessary to calculate the components of the stress tensor at a particular material point with respect to a rotated coordinate system, i.e., the stresses acting on a plane with a different orientation passing through that point of interest —forming an angle with the coordinate system . For example, it is of interest to find the maximum normal stress and maximum shear stress, as well as the orientation of the planes where they act upon. To achieve this, it is necessary to perform a tensor transformation under a rotation of the coordinate system. From the definition of tensor, the Cauchy stress tensor obeys the tensor transformation law. A graphical representation of this transformation law for the Cauchy stress tensor is the Mohr circle for stress.

Mohr's circle for two-dimensional state of stress

In two dimensions, the stress tensor at a given material point with respect to any two perpendicular directions is completely defined by only three stress components. For the particular coordinate system these stress components are: the normal stresses and, and the shear stress. From the balance of angular momentum, the symmetry of the Cauchy stress tensor can be demonstrated. This symmetry implies that. Thus, the Cauchy stress tensor can be written as:
The objective is to use the Mohr circle to find the stress components and on a rotated coordinate system, i.e., on a differently oriented plane passing through and perpendicular to the - plane. The rotated coordinate system makes an angle with the original coordinate system.

Equation of the Mohr circle

To derive the equation of the Mohr circle for the two-dimensional cases of plane stress and plane strain, first consider a two-dimensional infinitesimal material element around a material point , with a unit area in the direction parallel to the - plane, i.e., perpendicular to the page or screen.
From equilibrium of forces on the infinitesimal element, the magnitudes of the normal stress and the shear stress are given by:
However, knowing that
we obtain
Now, from equilibrium of forces in the direction of , and knowing that the area of the plane where acts is, we have:
However, knowing that
we obtain
Both equations can also be obtained by applying the tensor transformation law on the known Cauchy stress tensor, which is equivalent to performing the static equilibrium of forces in the direction of and.
Expanding the right hand side, and knowing that and, we have:
However, knowing that
we obtain
However, knowing that
we obtain
It is not necessary at this moment to calculate the stress component acting on the plane perpendicular to the plane of action of as it is not required for deriving the equation for the Mohr circle.
These two equations are the parametric equations of the Mohr circle. In these equations, is the parameter, and and are the coordinates. This means that by choosing a coordinate system with abscissa and ordinate, giving values to the parameter will place the points obtained lying on a circle.
Eliminating the parameter from these parametric equations will yield the non-parametric equation of the Mohr circle. This can be achieved by rearranging the equations for and, first transposing the first term in the first equation and squaring both sides of each of the equations then adding them. Thus we have
where
This is the equation of a circle of the form
with radius centered at a point with coordinates in the coordinate system.

Sign conventions

There are two separate sets of sign conventions that need to be considered when using the Mohr Circle: One sign convention for stress components in the "physical space", and another for stress components in the "Mohr-Circle-space". In addition, within each of the two set of sign conventions, the engineering mechanics literature follows a different sign convention from the geomechanics literature. There is no standard sign convention, and the choice of a particular sign convention is influenced by convenience for calculation and interpretation for the particular problem in hand. A more detailed explanation of these sign conventions is presented below.
The previous derivation for the equation of the Mohr Circle using Figure 4 follows the engineering mechanics sign convention. The engineering mechanics sign convention will be used for this article.

Physical-space sign convention

From the convention of the Cauchy stress tensor, the first subscript in the stress components denotes the face on which the stress component acts, and the second subscript indicates the direction of the stress component. Thus is the shear stress acting on the face with normal vector in the positive direction of the -axis, and in the positive direction of the -axis.
In the physical-space sign convention, positive normal stresses are outward to the plane of action, and negative normal stresses are inward to the plane of action .
In the physical-space sign convention, positive shear stresses act on positive faces of the material element in the positive direction of an axis. Also, positive shear stresses act on negative faces of the material element in the negative direction of an axis. A positive face has its normal vector in the positive direction of an axis, and a negative face has its normal vector in the negative direction of an axis. For example, the shear stresses and are positive because they act on positive faces, and they act as well in the positive direction of the -axis and the -axis, respectively. Similarly, the respective opposite shear stresses and acting in the negative faces have a negative sign because they act in the negative direction of the -axis and -axis, respectively.

Mohr-circle-space sign convention

In the Mohr-circle-space sign convention, normal stresses have the same sign as normal stresses in the physical-space sign convention: positive normal stresses act outward to the plane of action, and negative normal stresses act inward to the plane of action.
Shear stresses, however, have a different convention in the Mohr-circle space compared to the convention in the physical space. In the Mohr-circle-space sign convention, positive shear stresses rotate the material element in the counterclockwise direction, and negative shear stresses rotate the material in the clockwise direction. This way, the shear stress component is positive in the Mohr-circle space, and the shear stress component is negative in the Mohr-circle space.
Two options exist for drawing the Mohr-circle space, which produce a mathematically correct Mohr circle:

Positive shear stresses are plotted upward
Positive shear stresses are plotted downward, i.e., the -axis is inverted.

Plotting positive shear stresses upward makes the angle on the Mohr circle have a positive rotation clockwise, which is opposite to the physical space convention. That is why some authors prefer plotting positive shear stresses downward, which makes the angle on the Mohr circle have a positive rotation counterclockwise, similar to the physical space convention for shear stresses.
To overcome the "issue" of having the shear stress axis downward in the Mohr-circle space, there is an alternative sign convention where positive shear stresses are assumed to rotate the material element in the clockwise direction and negative shear stresses are assumed to rotate the material element in the counterclockwise direction. This way, positive shear stresses are plotted upward in the Mohr-circle space and the angle has a positive rotation counterclockwise in the Mohr-circle space. This alternative sign convention produces a circle that is identical to the sign convention #2 in Figure 5 because a positive shear stress is also a counterclockwise shear stress, and both are plotted downward. Also, a negative shear stress is a clockwise shear stress, and both are plotted upward.
'This article follows the engineering mechanics sign convention for the physical space and the alternative'' sign convention for the Mohr-circle space '''