Mohr’s circle

In two dimensional stress analysis, Mohr’s circle is a graphical representation of the stress state of a point in a body under static equilibrium. The two dimensional loading state can be either plane stress or plane strain loading. For three dimensional analysis we may use the extended Mohr circle for three dimensions (Mohr’s circle 3d).

Consider a body in equilibrium under two dimensional loading (cf. Fig. 1). The stress tensor for this case is given by:

\[ \sigma_{ij}=\left[\begin{array}{cc}\sigma_{11} & \sigma_{12}\\ \sigma_{21} & \sigma_{22}\end{array}\right] \]
(1)
Normal and shear stress on a plane in two dimensional analysis
Figure 1: Free body diagram of a point under static equilibrium in two dimensional analysis. Illustration of the normal and shear stresses acting on an arbitrary plane.

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Plane stress

Plane state of stress or simply plane stress we call a special case of loading which usually occurs to solid bodies where one dimension is very small compared to the other two. Consider a very thin solid body as shown in Fig. 1. The normal and shear stresses acting on the two opposite sides normal to \( x_{3} \) are all equal to zero. Due to the fact that the body is very thin, we may assume that \( \sigma_{33} \), \( \sigma_{31} \) and \( \sigma_{32} \) are approximately zero throughout the hole body:

\[ \sigma_{33}=\sigma_{31}=\sigma_{32}=0 \]
(1)
Plane stress loading conditions
Figure 1: Thin solid under plane stress loading conditions. The external forces in the direction of the third axis are zero.

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Equilibrium equations

A solid body is in static equilibrium when the resultant force and moment on each axis is equal to zero. This can be expressed by the equilibrium equations. In this article we will prove the equilibrium equations by calculating the resultant force and moment on each axis. A more elegant solution may be derived by using Gauss’s theorem and Cauchy’s formula. This approach may be found in international bibliography.

Consider a solid body in static equilibrium that neither moves nor rotates. Surface and body forces act on this body. We cut an infinitesimal parallelepiped inside the body and we analyze the forces that act on it as shown in Fig. 1. We will assume that the stress field is continuous and differentiable inside the whole body.

Infinitesimal parallelepiped in static equilibrium
Figure 1: Infinitesimal parallelepiped representing a point in a body under static equilibrium. The stresses acting on the opposite sides of the cube are slightly different.

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Index notation for tensors and vectors

Index notation is used extensively in literature when dealing with stresses, strains and constitutive equations. The reason is that it reduces drastically the number of terms in an equation and simplifies the expressions. We will use a right handed Cartesian coordinate system to describe the index notation (cf. Fig. 1). Moreover it is more convenient to name the axes \( x_{1} \), \( x_{2} \) and \( x_{3} \) instead of the more familiar notation \( x \), \( y \), \( z \).

Right handed Cartesian coordinate system
Figure 1: Illustration of a right handed Cartesian coordinate system and an arbitrary vector \( \vec{u} \).

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Mohr’s circle in 3 dimensions

Mohr’s diagram is a useful graphical representation of the stress state at a point. In this graphical representation the state of stress at a point is represented by the Mohr circle diagram, in which the abscissa \( \sigma \) and \( \tau \) give the normal and shear stress acting on a particular cut plane with a fixed normal direction. In the general 3 dimensional case, for a given state of stress at a point, the Mohr circle diagram has three circles as shown in Fig. 1. Mohr’s circle diagram is used frequently in conjunction with failure criteria like the Mohr-Coulomb failure criterion.

3D Mohr's circles
Figure 1: In the three dimensional case there are three Mohr’s circles. The diameters and the centers of the circles are calculated from the three principal stresses. The point representing the normal and shear stress on a plane does not necessarily lie on the perimeter of a circle.

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