Octahedral stresses

Octahedral stresses we call the normal and shear stresses that are acting on some specific planes inside the stressed body, the octahedral planes. If we consider the principal directions as the coordinate axes (see also the article: Principal stresses and stress invariants), then the plane whose normal vector forms equal angles with the coordinate system is called octahedral plane. There are eight such planes forming a regular octahedron as it is illustrated in Fig. 1.

Octahedral stresses and their corresponding planes
Figure 1: A plane whose normal vector makes equal angles to the axes of the principal stresses is called octahedral plane. There are eight such planes that form a regular octahedron. The normal and shear stresses that act on these planes are called octahedral stresses.

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Deviatoric stress and invariants

The stress tensor can be expressed as the sum of two stress tensors, namely: the hydrostatic stress tensor and the deviatoric stress tensor. In this article we will define the hydrostatic and the deviatoric part of the stress tensor and we will calculate the invariants of the stress deviator tensor. The invariants of the deviatoric stress are used frequently in failure criteria.

Consider a stress tensor \( \sigma_{ij} \) acting on a body. The stressed body tends to change both its volume and its shape. The part of the stress tensor that tends to change the volume of the body is called mean hydrostatic stress tensor or volumetric stress tensor. The part that tends to distort the body is called stress deviator tensor. Hence, the stress tensor may expressed as:

\[ \sigma_{ij}=s_{ij}+p\delta_{ij} \]
(1)

where \( \delta_{ij} \) is the Kronecker delta (with \( \delta_{ij}=1 \) if \( i=j \) and \( \delta_{ij}=0 \) if \( i\neq j \) ), \( p \) is the mean stress given by:

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Principal stresses and stress invariants

In this article we will discuss the derivation of the principal stresses and the stress invariants from the Cauchy stress tensor. The principal stresses and the stress invariants are important parameters that are used in failure criteria, plasticity, Mohr’s circle etc.

For every point inside a body under static equilibrium there are three planes, called the principal planes, where the stress vector is normal to the plane and there is no shear component (see also: Calculation of normal and shear stress on a plane). These normal stress vectors are called principal stresses. From the mathematical point of view, the derivation of the principal stresses and their direction is known as a problem of determining the eigenvalues and their corresponding eigenvectors from a square matrix. Fig. 1 illustrates the principal stresses and their direction for a point inside the body compared to the initial system of coordinates and the stress tensor.

Principal stresses
Figure 1: Principal stresses and their direction with respect to the initial coordinate system.

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Calculation of normal and shear stress on a plane

Frequently it is necessary to calculate the normal and the shear stress on an arbitrary plane (with unit normal vector \( n \)) that crosses a rigid body in equilibrium. Faults or cracks that cross the rock mass and may lead to rock failure are cases that require the estimation of the stress components on a plane. Another example may be the calculation of the normal and shear stress on the failure surface of a specimen during a typical rock mechanics experiment.

The components of the stress vector \( T \) acting on any plane crossing an arbitrary point inside a rigid body may be calculated as follows:

\[ T_{i}^{(n)}=\sigma_{ij}n_{j} \]
(1)

where \( \sigma_{ij} \) is the stress tensor describing the stress state at that point and \( n_{j} \) are the components of the unit normal vector of the plane. In the above Eq. (1), the summation convention has been used.

The stress vector can be broken down into two components, the normal stress and the shear stress as shown in Fig. 1.

Normal and shear component of the stress vector on a plane
Figure 1: Normal and shear component of the stress vector on a plane

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Transformation of a tensor to a new coordinate system

The transformation (rotation) of a tensor into a new coordinate system is a common problem in rock mechanics and in continuum mechanics in general. In this article we will present the necessary equations and an example case. We will use the stress tensor as example.

Consider a rigid body in equilibrium and a coordinate system. The stress state of any internal point of this body is given by the stress tensor (cf. Fig. 1):

\[ \sigma_{ij}=\left[\begin{array}{ccc}\sigma_{11} & \sigma_{12} & \sigma_{13}\\ \sigma_{21} & \sigma_{22} & \sigma_{23}\\ \sigma_{31} & \sigma_{32} & \sigma_{33}\end{array}\right] \]
(1)
Geometrical representation of the Cauchy stress tensor
Figure 1: Elementary volume representing a point in a body under static equilibrium and geometrical representation of the components of the Cauchy stress tensor.

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