Orthotropic material

The special case that a material, regardless of the load in each direction is the same load-deformation behavior are called isotropic. The general case in that the force -strain behavior is dependent on the loading direction, is known as anisotropy. The orthotropy (from Greek orthos ορθός " accurate, correct, right," and τρόπος tropos " way, direction, way ") is a special case of anisotropy (orthogonal - anisotropic). And from general to specific leads therefore on anisotropy orthotropy isotropy. Each isotropic material is orthotropic so also ( and anisotropically ), but not any orthotropic is isotropic.

In this article, the orthotropy is discussed in the theory of elasticity only.

The elasticity law of a (linear) material is orthotropic with respect to an orthonormal basis along the Orthotropieachsen in 3D in Voigt shear notation:

And in 2D:

Linear elasticity theory and Voigt notation

The relationship between stresses and strains is linear according to:

This applies regardless of whether the material is orthotropic or not. It is the most general linear relationship that exists between two tensors of the second stage. C forms 3x3 components onto 3x3 components. And that itself has 81 = 3x3 x 3x3 components. In the linear theory of elasticity (balanced stress tensor, bal. Strain tensor, potential, see Voigt notation) there is the relationship between stresses and strains define a 6x6 matrix, so that

In the general case thus remain independent material parameters in the material law 21.

Stiffness matrix for orthotropic

An orthotropic material is called when an orthonormal basis exists, so that the material information presented with respect to this basis the following form accepts (with only nine material parameters ):

The inverse of the stiffness matrix (the compliance matrix ) is also symmetrical and filled with non-zero values ​​as the stiffness matrix even at the same locations. For the representation of the constitutive model to the flexibility matrix, the representation used at the top is common to be used in the 9 than the material parameters.

Reasons for Busy awareness of the stiffness matrix

In this section, the question is unclear why the stiffness matrix is only open in the appropriate places. In general, diving in a linear material law 21 independent material constants (see Voigt notation). In the case of orthotropy, however, reduced the number of constants to 9 Why this is so is shown below.

Rotation matrices at 180 degree rotations

The (linear) pictures that describe 180 degree rotations about the Orthotropieachsen, can be described with matrices. If we choose as a reference, a basis whose basis vectors correspond to the mutually perpendicular axes of rotation, then the orthogonal matrices have the following form

These three matrices (and in addition the identity matrix ) form a subgroup of the rotation group SO (3).

Symmetry condition in index notation and Voigt shear notation

Thought experiment: a particle and its surroundings is subjected to a certain deformation and thus a particular tensor. In the simplest case ( which, however, is generally not sufficient to define the orthotropia ) particles could be stretched only in a specific direction. Now you change the stretching direction active. That letting the material point as it is ( ie not rotating the material ) and undergoes the point but ( same ) stretching in the other direction. So you come to a different strain tensor.

The change in the direction of distortion can be described by a rotation matrix A. It is

Using a linear material law can be determined for given strain tensor, the associated stress tensor. It should be

Although the general case of anisotropy does not

But just this one calls for a for the above subset of SO (3) in the case of orthotropy: A material is called orthotropic if the function has the following symmetry transformation for each of the above ( orthogonal ) rotation matrices and for any distortions applies

In index notation

Now the same condition in Voigt shear notation: With the definition

Applies

The new definition

Results

In Voigt notation shear is thus obtained as a symmetry condition

And since this must hold for arbitrary strains, the symmetry condition

Special case 180 degree rotations

Since the 3x3 matrices matrices A are occupied only on the main diagonal in the special case of orthotropy, the definitions from above simplify to

Altogether, the three 3x3 matrices corresponding to the three 6x6 matrices

Evaluation of the symmetry conditions for the special case

The symmetry condition is evaluated for these matrices yields

In the last three equations we see that C can only have the following form

Since these Voigt stiffness matrix is also symmetric ( see Voigt notation), remains

Summary

• The orthotropy in the linear theory of elasticity can be defined as a special case of anisotropy, in which the stiffness or compliance matrix takes a particularly simple form (9 constants instead of 21 constants in the general case ).
• In addition to the orthotropy there are other special cases of anisotropy, eg Transversalisotropie, isotropy, etc.. This same symmetry conditions are specified. Only then other subgroups of the rotation group (ie other matrices A ) is considered.
• On the shape of the elastic law can be seen that the coupling between train and push for loading along the Orthotropierichtungen omitted.