Stress tensor

A stress tensor is a second order, which describes the mechanical stress at a particular point within the material tensor. It is a significant amount of continuum mechanics.

Used this tensor especially in physics ( solid state physics, voltage and Classical Mechanics, partially Geophysics ) and in electrodynamics.

Definition

In an imaginary cut surface through matter the cut-away in thought matter exerts on the remaining matter consists of a voltage (acting perpendicular to the cut surface ) and two shear stress components composed ( in the cut surface acting) posing as voltage vector from a normal stress component.

At the sites intersect three such imaginary cut surfaces, and the three voltage vectors in them are grouped together for the stress tensor:

The basis vectors of the coordinate system and the dyadic product call ( tensor product of two vectors).

In matrix notation, the stress tensor is generally given in the following form:

As the stress tensor is symmetric ( etc ), it does not consist of nine independent variables but only six, and can be written in the Voigt 's notation as a vector, so that the notation is simplified:

Conjunction with other sizes

For gases, the stress tensor reduces just to the pressure p times the identity matrix:

And is called the pressure tensor.

The Maxwell stress tensor of electrodynamics is a sub- matrix of the energy-momentum tensor.

The stress tensor in the volume element

The following figure illustrates this with the example of a cut out very small volume element for Cartesian coordinates; the considerations apply to the upper right rear corner:

In a concrete -sectional area, their orientation will be described by its outward normal vector, results in the voltage vector by multiplying the stress tensor with the normal vector:

Or for a symmetric stress tensor: