Stress

Stress is the force-per-unit-area exerted at a point within a medium. Since the force is a vector (with 3 components), and the unit-area is specified by its normal vector (with 3 components), stress is a 3x3 tensor. Since the stress does not cause the medium to spin, it must be symmetric across the main diagonal, and so may be written as:

${\displaystyle \sigma _{ij}={\begin{pmatrix}\sigma _{11}&\sigma _{12}&\sigma _{13}\\\sigma _{12}&\sigma _{22}&\sigma _{23}\\\sigma _{13}&\sigma _{23}&\sigma _{33}\\\end{pmatrix}}}$

The indices refer to the directions of the coordinate system, and so, if the coordinate system is changed, all the stress components change. For example, if the coordinate system is rotated with a rotation matrix R, the resulting stress tensor is

σmn=ΣσijRmiRnj

where the sum is over the repeated indices. Since the stress tensor is symmetric, there is always a special coordinate system wherein the rotated stress tensor has the simple form:

${\displaystyle \{\sigma '_{ij}\}={\begin{pmatrix}\sigma _{1}&0&0\\0&\sigma _{2}&0\\0&0&\sigma _{3}\\\end{pmatrix}}}$

This special coordinate system is called the principal coordinate system, and since these principle directions are mutually perpendicular, the stress tensor is said to be orthogonal. The three non-zero stresses are called the principal stresses. In mountainous areas, the principal coordinate system may be oriented in any direction, depending on the tectonics which built the mountains. However, in lightly-deformed areas , it is common that the principal coordinate system has one axis vertical (parallel to gravity), and (deeper than a few tens of meters) the greatest principal stress sV is the vertical stress (from the weight of the overlying rock). Hence the stress tensor has the form:

${\displaystyle \{s'_{ij}\}={\begin{pmatrix}s_{h}&0&0\\0&s_{H}&0\\0&0&s_{V}\\\end{pmatrix}}}$

The horizontal stresses are commonly significantly smaller than the vertical stress, with the least horizontal stress sh only marginally smaller than the greatest horizontal sH.

Within the marine layer, of course the stress is isotropic:

${\displaystyle \{s_{ij}\}={\begin{pmatrix}-P&0&0\\0&-P&0\\0&0&-P\\\end{pmatrix}}}$

where P is pressure, and the minus sign is the common convention.

Stress is related to elastic strain by Hooke's law [1].