# What are orthotropic materials 1

## Orthotropy

If a material shows the same force-deformation behavior regardless of the direction of load, it is called isotropy. The general case that the force-deformation behavior depends on the direction of load is called anisotropy. The **Orthotropy** (from Greek ορθός*orthos* “Right, correct, right” and τρόπος*tropos* "Path, direction, manner") is a special case of anisotropy. The material shows in *certain* Directions have the same force-deformation behavior. From the general to the specific one can get through anisotropy-orthotropy-isotropy. Every isotropic material is also orthotropic, but not every orthotropic material is isotropic.

This article only discusses orthotropy in elasticity theory.

The law of elasticity of an orthotropic material in relation to an orthonormal basis along the orthotropic axes in 3D in Voigt's notation reads:

- $ \ begin {bmatrix} \ varepsilon_ {11} \ \ varepsilon_ {22} \ \ varepsilon_ {33} \ 2 \ varepsilon_ {23} \ 2 \ varepsilon_ {13} \ 2 \ varepsilon_ {12} \ end {bmatrix} = \ begin {bmatrix} \ frac {1} {E_1} & - \ frac {\ nu_ {12}} {E_1} & - \ frac {\ nu_ {13}} {E_1} & 0 & 0 & 0 \ & \ frac {1} {E_2} & - \ frac {\ nu_ {23}} {E_2} & 0 & 0 & 0 \ & & \ frac {1} {E_3} & 0 & 0 & 0 \ & & & \ frac {1} {G_ {23}} & 0 & 0 \ & \ text {sym} & & & \ frac {1} {G_ {13}} & 0 \ & & & & & \ frac {1} {G_ {12}} \ end {bmatrix} \ begin {bmatrix} \ sigma_ {11} \ \ sigma_ {22} \ \ sigma_ {33} \ \ sigma_ {23} \ \ \ sigma_ {13} \ \ sigma_ {12} \ end {bmatrix} $

And in 2D:

- $ \ begin {bmatrix} \ varepsilon_ {11} \ \ varepsilon_ {22} \ 2 \ varepsilon_ {12} \ end {bmatrix} = \ begin {bmatrix} \ frac {1} {E_1} & - \ frac { \ nu_ {12}} {E_1} & 0 \ & \ frac {1} {E_2} & 0 \ \ text {sym} & & \ frac {1} {G_ {12}} \ end {bmatrix} \ begin {bmatrix} \ sigma_ {11} \ \ sigma_ {22} \ \ sigma_ {12} \ \ end {bmatrix} $

### Linear elasticity theory and Voigt notation

The relationship between stresses $ \ sigma $ and distortions $ \ varepsilon $ is linear according to:

- $ f_C: \ varepsilon_ {kl} \ rightarrow \ sigma_ {ij} = C_ {ijkl} \ varepsilon_ {kl} $

This applies regardless of whether the material is orthotropic or not. It is the most general linear relationship that exists between two second order tensors. C maps 3x3 components to 3x3 components. And thus has 81 = 3x3 x 3x3 components itself. In linear elasticity theory (symmetrical stress tensor, symmetrical strain tensor, potential, see Voigt's notation) one can also define a 6x6 matrix $ C ^ {\ text {v}} $ for the relationship between stresses and strains, so that

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

### Rigidity matrix for orthotropy

A material is called orthotropic if there is an orthonormal basis, so that the law of elasticity represented in relation to this basis takes the following form (with only 9 material parameters):

- $ \ begin {align} \ begin {bmatrix} \ sigma_ {11} \ \ sigma_ {22} \ \ sigma_ {33} \ \ sigma_ {23} \ \ sigma_ {13} \ \ sigma_ {12 } \ \ end {bmatrix} & = \ begin {bmatrix} C_ {1111} & C_ {1122} & C_ {1133} & & & & \ & C_ {2222} & C_ {2233} & & & & \ \ & & C_ {3333} & & & & \ & & & C_ {2323} & & & \ & \ text {sym} & & & C_ {1313} & & \ & & & & & C_ {1212 } & \ \ end {bmatrix} \ begin {bmatrix} \ varepsilon_ {11} \ \ varepsilon_ {22} \ \ varepsilon_ {33} \ 2 \ varepsilon_ {23} \ 2 \ varepsilon_ {13} \ \ 2 \ varepsilon_ {12} \ end {bmatrix} \ end {align} $

The inverse of the stiffness matrix (the compliance matrix) is also symmetrical and only has non-zero values in the same places as the stiffness matrix. For the representation of the material law with the compliance matrix, the representation used at the very top is common, in which $ E_1, E_2, E_3, \ nu_ {12}, \ nu_ {13}, \ nu_ {23}, G_ {12}, G_ { 13}, G_ {23} $ can be used as the 9 material parameters.

### Reasons for the occupancy of the stiffness matrix

This section clarifies the question, **Why** the stiffness matrix is only occupied in the corresponding places. In general, 21 independent material constants appear in a linear material law (see Voigt's notation). In the case of orthotropy, however, the number of constants is reduced to 9. Why this is so is shown below.

### Rotary dies for 180 degree rotations

The (linear) images that describe 180-degree rotations around the orthotropic axes can be described with matrices. If a base is selected as a reference, the base vectors of which coincide with the axes of rotation that are perpendicular to one another, then these orthogonal matrices have the following form

- $ \ begin {align} A_x & = \ begin {bmatrix} 1 & 0 & 0 \ 0 & -1 & 0 \ 0 & 0 & -1 \ end {bmatrix}, A_y = \ begin {bmatrix} -1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & -1 \ end {bmatrix}, A_z = \ begin {bmatrix} -1 & 0 & 0 \ 0 & -1 & 0 \ 0 & 0 & 1 \ end {bmatrix} \ end {align} $

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

### Symmetry condition in index notation and Voigt notation

Thought experiment: A particle and its surroundings are subjected to a certain deformation and thus a certain distortion tensor $ \ varepsilon $. In the simplest case (which is not sufficiently general to define orthotropy) the particle could only be stretched in a certain direction. Now you actively change the stretching direction. This means that the material point is left as it is (i.e. the material is not rotated) and the point is subjected to (the same) stretching in a different direction. This leads to another strain tensor $ \ varepsilon '$.

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

- $ \ varepsilon '= A \, \ varepsilon \, A ^ {- 1} $

With the aid of a linear material law $ f_C $, the associated stress tensor can be determined for a given strain tensor. Be it

- $ \ begin {align} \ sigma &: = f_C (\ varepsilon) \ \ sigma '&: = f_C (\ varepsilon') \ end {align} $

In the general case of anisotropy, it is not true

- $ \ sigma '= A \, \ sigma \, A ^ {- 1} $

But this is exactly what is required for a subset of SO (3) described above in the case of orthotropy: A material is called orthotropic if for the function $ f_C $ the following symmetry transformation for each of the above-mentioned (orthogonal) rotary matrices and holds for any distortion

- $ \ begin {align} A f_C (\ varepsilon) A ^ {- 1} & = f_C (A \ varepsilon A ^ {- 1}) \ Leftrightarrow A f_C (\ varepsilon) A ^ T = f_C (A \ varepsilon A ^ T) \ end {align} $

In index notation

- $ \ begin {align} \ sigma_ {mn} '= A_ {mo} C_ {opjk} \ varepsilon_ {jk} A_ {np} & = C_ {mnil} \ varepsilon' _ {il} = C_ {mnil} A_ { ij} \ varepsilon_ {jk} A_ {lk} \ end {align} $

Now the same condition in Voigtsch notation: With the definition

applies

- $ \ begin {align} \ begin {bmatrix} \ sigma '_ {11} \ \ sigma' _ {22} \ \ sigma '_ {33} \ \ sigma' _ {23} \ \ sigma ' _ {13} \ \ sigma '_ {12} \ end {bmatrix} = A ^ {\ text {v}} _ \ sigma \ begin {bmatrix} \ sigma_ {11} \ \ sigma_ {22} \ \ sigma_ {33} \ \ sigma_ {23} \ \ sigma_ {13} \ \ sigma_ {12} \ end {bmatrix} \ Leftrightarrow {\ sigma '} ^ {\ text {v}} = A ^ { \ text {v}} _ \ sigma {\ sigma} ^ {\ text {v}}, \ qquad \ qquad \ begin {bmatrix} \ varepsilon '_ {11} \ \ varepsilon' _ {22} \ \ varepsilon '_ {33} \ \ varepsilon' _ {23} \ \ varepsilon '_ {13} \ \ varepsilon' _ {12} \ end {bmatrix} = A ^ {\ text {v}} _ \ sigma \ begin {bmatrix} \ varepsilon_ {11} \ \ varepsilon_ {22} \ \ varepsilon_ {33} \ \ varepsilon_ {23} \ \ varepsilon_ {13} \ \ varepsilon_ {12} \ end {bmatrix } \ end {align} $

With the new definition

surrendered

- $ \ begin {align} \ begin {bmatrix} \ varepsilon '_ {11} \ \ varepsilon' _ {22} \ \ varepsilon '_ {33} \ 2 \ varepsilon' _ {23} \ 2 \ varepsilon '_ {13} \ 2 \ varepsilon' _ {12} \ end {bmatrix} = A ^ {\ text {v}} _ \ varepsilon \ begin {bmatrix} \ varepsilon_ {11} \ \ varepsilon_ {22 } \ \ varepsilon_ {33} \ 2 \ varepsilon_ {23} \ 2 \ varepsilon_ {13} \ 2 \ varepsilon_ {12} \ end {bmatrix} \ Leftrightarrow {\ varepsilon '} ^ {\ text {v }} = A ^ {\ text {v}} _ \ varepsilon {\ varepsilon} ^ {\ text {v}} \ end {align} $

In Voigt's notation, one obtains the symmetry condition

- $ \ begin {align} {\ sigma '} ^ {\ text {v}} = A ^ {\ text {v}} _ \ sigma {\ sigma} ^ {\ text {v}} & = A ^ {\ text {v}} _ \ sigma C ^ {\ text {v}} {\ varepsilon} ^ {\ text {v}} = C ^ {\ text {v}} {\ varepsilon '} ^ {\ text {v }} = C ^ {\ text {v}} A ^ {\ text {v}} _ \ varepsilon {\ varepsilon} ^ {\ text {v}} \ end {align} $

And since this must apply to any elongation, the symmetry condition is

- $ \ begin {align} A ^ {\ text {v}} _ \ sigma C ^ {\ text {v}} = C ^ {\ text {v}} A ^ {\ text {v}} _ \ varepsilon \ end {align} $

### Special case of 180 degree rotations

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

- $ \ begin {align} A ^ {\ text {v}} _ \ sigma = A ^ {\ text {v}} _ \ varepsilon & = \ begin {bmatrix} A_ {11} A_ {11} & 0 & 0 & 0 & 0 & 0 \ 0 & A_ {22} A_ {22} & 0 & 0 & 0 & 0 \ 0 & 0 & A_ {33} A_ {33} & 0 & 0 & 0 \ 0 & 0 & 0 & A_ {22} A_ {33} & 0 & 0 \ 0 & 0 & 0 & 0 & A_ {11} A_ {33} & 0 \ 0 & 0 & 0 & 0 & 0 & A_ { 11} A_ {22} \ \ end {bmatrix} \ \ end {align} $

So the three 3x3 matrices correspond to the three 6x6 matrices

### Evaluation of the symmetry conditions for the special case

The symmetry condition evaluated for these matrices gives

The last 3 equations show that C can only have the following form

- $ \ begin {align} C ^ {\ text {v}} & = \ begin {bmatrix} C ^ {\ text {v}} _ {11} & C ^ {\ text {v}} _ {12} & C ^ {\ text {v}} _ {13} & 0 & 0 & 0 \ C ^ {\ text {v}} _ {21} & C ^ {\ text {v}} _ {22} & C ^ {\ text {v}} _ {23} & 0 & 0 & 0 \ C ^ {\ text {v}} _ {31} & C ^ {\ text {v}} _ {32} & C ^ {\ text {v}} _ {33} & 0 & 0 & 0 \ 0 & 0 & 0 & C ^ {\ text {v}} _ {44} & 0 & 0 \ 0 & 0 & 0 & 0 & C ^ {\ text {v}} _ {55} & 0 \ 0 & 0 & 0 & 0 & 0 & C ^ {\ text {v}} _ {66} \ end {bmatrix} \ end { align} $

Since this Voigt stiffness matrix is also symmetrical (see Voigt notation), remains

- $ \ begin {align} C ^ {\ text {v}} & = \ begin {bmatrix} C ^ {\ text {v}} _ {11} & C ^ {\ text {v}} _ {12} & C ^ {\ text {v}} _ {13} & 0 & 0 & 0 \ & C ^ {\ text {v}} _ {22} & C ^ {\ text {v}} _ {23} & 0 & 0 & 0 \ & & C ^ {\ text {v}} _ {33} & 0 & 0 & 0 \ & & & C ^ {\ text {v}} _ {44} & 0 & 0 \ & \ text {sym} & & & C ^ {\ text {v}} _ {55} & 0 \ & & & & & C ^ {\ text {v}} _ {66} \ end {bmatrix } \ end {align} $

### Summary

- Orthotropy in linear elasticity theory can be defined as a special case of anisotropy in which the stiffness or compliance matrix takes on a particularly simple form (9 constants instead of 21 constants in the general case).
- In addition to orthotropy, there are other special cases of anisotropy, e.g. transversalisotropy, isotropy, etc. The same symmetry conditions are given here. Only then other subgroups of the rotation group (i.e. other matrices A) are considered.
- The form of the elastic law shows that there is no coupling between tension and thrust for loading along the orthotropic directions.

### literature

- J.F. Nye:
*Physical Properties of Crystals: Their Representation by Tensors and Matrices*. Oxford University Press. 1985. ISBN 978-0-19-851165-6. - H. Altenbach, J. Altenbach, R. Rikards:
*Introduction to the mechanics of laminate and sandwich structures*. Stuttgart: German publishing house for basic industry, 1996. ISBN 3-342-00681-1

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