Strain tensor

Strain tensors or deformation tensors describe the ratio of instantaneous configuration to the initial configuration in the deformation of continuous bodies and thus change the relative position relations of matter elements. This change ( = deformation ) of the inner array corresponds to a change in the external shape of the solid and thus for example as stretching, compression, shear, etc. visible.

In the literature, a variety of such definitions is known to be formed from the deformation gradient.

First, let so that the right Cauchy-Green tensor

With respect to the initial configuration and the left Cauchy-Green tensor

Form with respect to the current configuration.

Thus, these two tensors are forcibly symmetrical and in case of non - deformation equal to the identity - as a matrix that is expressed is the unit matrix with respect to the respective base system.

For engineering applications, it is usually but wishes sizes that represent a zero in non- deformation. This leads to definitions of the Green-Lagrange strain tensor -

Or the Euler Almansi strain tensor

In addition, however still exists a number of other similar definitions, each with their rights and benefits in different theories.

Principal axis representation of the strain tensors

The tensors and can also be presented by the principal axis transformation in its principal axis system. The resulting specific three eigenvalues ​​are identical for and and are the squares of the so-called principal stretches dar.

The linearized strain tensor

For the description of small strains, the linearized strain tensor is used in engineering mechanics usually. It is obtained by linearization of the sizes or. For this purpose is given to using the definition of the deformation gradient in the strain tensor,

For small displacements, the last term can be neglected. This yields the linearized strain tensor

With the components