Visualizing the stress tensor: Stress ellipse and Mohr circle

Adopting the principal stress axes as our reference system we know from Equation 16 that the traction vector on any plane with normal ${n}$ is

$$-\ensuremath{\mbox{\boldmath${t}$}}= (\ensuremath{\sigma_{1}}n_1,\ensuremath{\sigma_{2}}n_2, \ensuremath{\sigma_{3}}n_3)$$

Since all normals are assumed to have unit length, i.e. $n_1^2 + n_2^2 + n_3^2 = 1$, it follows that the components of the traction vector lie on an ellipsoid with semi-axes $\sigma_{1}$, $\sigma_{2}$ and $\sigma_{3}$

$$\frac{t_x^2}{\ensuremath{\sigma_{1}}^2} + \frac{t_y^2}{\ensuremath{\sigma_{2}}^2} + \frac{t_z^2}{\ensuremath{\sigma_{3}}^2} = 1$$ (37)

The stress ellipsoid, see Figure 5 left column, is a convenient tool to easily visualize the orientations and magnitudes of the principal stresses and hence the prevalent faulting regime and the magnitude of the shearing stresses.

Figure: Stress ellipsoids and Mohr circles for two stress tensors with equal value of $R = 0.6$. The upper tensor has $\sigma_{1}$= 1.3, $\sigma_{2}$ = 1 and $\sigma_{3}$= 0.8 and the lower tensor has $\sigma_{1}$= 1.9, $\sigma_{2}$ = 1 and $\sigma_{3}$= 0.4. Observe the difference in the shapes of the stress ellipsoids but the identical shapes (except for a scale factor) of the Mohr circle diagrams.

The stress ellipsoid is, however, less convenient for showing the relationship between the orientation of a plane in the stress field and the resulting magnitudes of the normal and shear stresses upon it. To this respect we use the Mohr circle diagram which, as we shall see, is very useful also in the discussion of rock failure. I will not derive the angular relations for the Mohr circle here, see e.g. Jaeger and Cook [1979], instead the relationship between the normal to a fault plane and the normal, shear and principal stresses are illustrated in Figure 6.

Figure: Figure modified after Jaeger and Cook [1979]. A) One octant of the stress ellipsoid showing the definitions of the angles $\alpha$ and $\theta$ and the traces of two ellipses for fixed values of $\alpha$ (GEH) and $\theta$ (DEF). B) The Mohr diagram corresponding to the stress ellipsoid in A.

The angle between the $\sigma_{1}$ axis and the normal is denoted $\alpha$ and the angle between the $\sigma_{3}$ axis and the normal is $\theta$. Using the equations for the magnitudes of the normal and shear stresses, Equations 17 and 20, we find that if one angle is kept fixed, varying the other angle traces out a circle in $\sigma_n$, $\tau$ space. If we fix one angle in the plane of two principal stresses, the three main circles in Figure 6B will be traced out, e.g. for a fault normal in the $\sigma_{1}$ - $\sigma_{2}$ plane ( $\theta = \ensuremath{{90}^{\circ}}$), Figure 6A, we have a circle centered on A (normal stress $(\ensuremath{\sigma_{1}}+ \ensuremath{\sigma_{2}})/2$) in Figure 6B, with radius $(\ensuremath{\sigma_{1}}- \ensuremath{\sigma_{2}})/2$. We see from Figure 6B that normal stress for the circle varies from $\sigma_{1}$ to $\sigma_{2}$ and shear stress from 0 to $(\ensuremath{\sigma_{1}}- \ensuremath{\sigma_{2}})/2$, as we expect. Setting $\theta$ to any other angle will trace out other circles centered on A, such as the DEF circle segment. The largest of the three main circles traces out the stress magnitudes on normals in the $\sigma_{1}$ - $\sigma_{3}$ plane. If desired, the Mohr circles can be mirrored in the $\sigma_n$ axis to allow for negative shear stress magnitudes. The space enclosed between the two smaller circles and the larger circle shows the possible shear and normal stresses on any plane in the stress field. Using the Mohr diagram we can hence read off the normal and shear stress magnitudes for any value of $\alpha$ and $\theta$, i.e. any direction of the fault plane normal.

Comparing the stress ellipsoids and the Mohr diagrams in Figure 5 we see that although the ratio $R$ is often referred to as the shape factor it does not describe the shape of the stress ellipsoid but instead the position of the $\sigma_{2}$ point in the Mohr diagram. $R$ reflects how close $\sigma_{2}$ is to $\sigma_{1}$ or $\sigma_{3}$, but it does not give any information on the magnitude of ( $\sigma_{1}$ - $\sigma_{3}$), i.e. the size of the Mohr diagram. An unscaled Mohr diagram can be constructed using only the directions of the principal stresses and $R$. Although without information on the absolute magnitude of the shear stress, such a diagram is valuable for the analysis of the relative stability of fault planes, since the location of the fault plane normal in the Mohr diagram only depends on the relative orientation of the fault plane normal with respect to the principal stress axes.