Spherical tensor

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Linear mapping of a vector by a tensor .
Illustration of a vector by a spherical tensor .

Spherical tensors , axiators or spherical tensors are in continuum mechanics tensors that are proportional to the unit tensor of the second order. Second level tensors are used here as linear mappings from geometric vectors to geometric vectors, which are generally rotated and stretched in the process, see the figure above on the right. Spherical tensors are the special images that cause a pure, homogeneous stretching in all three spatial directions without rotation, as in the lower picture on the right. The spherical or spherical part of a tensor is the spherical tensor, which has the same trace as the tensor.

Spherical tensors occur in continuum mechanics with all-round, hydrostatic pressure or with uniform expansion or compression of a body in all three spatial directions. They are therefore used to model the material behavior under these conditions.

definition

Spherical tensors are second order tensors that are times the unit tensor :

.

The spherical part of a tensor is denoted with a superscript "K" or "sph":

.

The trace "Sp" of the unit tensor is equal to the dimension of the underlying space, here and in the following three.

Expansion and compression

Expansion of a sphere

As mentioned at the beginning, spherical tensors occur with uniform expansion or compression of a body in all three spatial directions, which can be described as follows. Be

the position vector of a particle of a body in the undeformed starting position. The numbers are called the material coordinates of the particle and are related to the standard basis of the three-dimensional Euclidean vector space . In the current configuration, the particle currently has the position

with spatial coordinates after deformation as a result of movement . With pure expansion or compression without rotation there is a center of expansion and a stretching factor such that

applies to all particles, see figure on the right. Formation of the gradient according to the material coordinates provides the deformation gradient

,

which is a spherical tensor here. The arithmetic symbol is the dyadic product and denotes the Kronecker delta . The determinant of the deformation gradient is the volume ratio before and after the expansion:

.

Incompressibility

The volume-changing deformation described in the previous section is impossible for an incompressible material, because incompressibility is characterized by a constant volume ratio of one. Mathematically, this is made possible by the secondary condition

expressed in terms of the motion function. Such a secondary condition is ensured with a Lagrangian multiplier , which here corresponds to the pressure in the material. The associated reaction voltage is the pressure tensor

,

who is a spherical tensor. Examples of this description can be found in hyperelasticity .

Place in the eigenvalue space

Hydrostatic axis in the eigenvalue space

As a multiple of the unit tensor, each spherical tensor has three identical eigenvalues that lie in the eigenvalue space on the hydrostatic axis , see figure on the right. If only symmetrical tensors are considered, this axis is formed by the spherical tensors.

Invariants of spherical tensors

The three main invariants of a spherical tensor are

The amount is the Frobenius norm , which deals with the Frobenius scalar product " " to

calculated.

See also

literature