# Orthotropy

The coordinate system with the three orthotropic axes radial, transversal, longitudinal
Wood as a typical orthotropic material in engineering
Due to its internal structure, a material (here 2D) is rotationally symmetrical with respect to a rotation by 180 degrees around an axis perpendicular to the plane of the sheet. It is orthotropic, and the red and green lines are its orthotropic axes. In 3D it could also be symmetrical with respect to rotations of 180 degrees around the red and green axes.

The Orthotropie (from Greek ορθός orthos "correctly, vertical, standing straight" and τρόπος tropos "way fashion") is a specific type of directional dependence of a material. Orthotropic materials as in the picture have the following properties:

1. The force-deformation behavior does not change if the material is rotated 180 degrees around the orthotropic axes.
2. In the reference system parallel to the orthotropic axes, there is no coupling between normal strains and shear distortions .

A linear elastic orthotropic material has a maximum of nine material parameters.

A material is isotropic if it has the same force-deformation behavior regardless of direction. With anisotropic materials, the force-deformation behavior depends on the direction of the load. Orthotropy is a special case of anisotropy and in turn contains transversal isotropy and isotropy as special cases.

Many construction materials are orthotropic, e.g. B. technical wood, fabric, many fiber-plastic composites and rolled sheets with texture.

## Symmetry group

The directional dependency of a material is characterized by the fact that the force-deformation behavior is independent (invariant) compared to only certain rotations of the material: In orthotropy, these are all 180-degree rotations around the orthotropic axes. These rotations form the symmetry group of the orthotropic material.

The invariance of these rotations of the material is illustrated by two experiments on a particle: In the first experiment, a certain force is applied to the particle and the resulting deformation is measured. In the second experiment, the material is rotated one after the other around any orthotropic axes - by 180 degrees. Then you apply the same force as in the first experiment and measure the deformation again. In the case of orthotropic material, the same deformation will be measured in the second experiment as in the first. Even with non-linear elastic material behavior.

The dependence on the rotations of the material can be seen if one rotates in the second experiment by an angle other than 180 degrees. If there is no special case of transverse isotropy or isotropy, one will now always measure a different deformation than in the first experiment.

## Orthotropy in the linear theory of elasticity

Are given two tensors second stage and with 3 × 3 coefficients respectively . The most general linear relationship that exists between these coefficients is: ${\ displaystyle {\ boldsymbol {\ sigma}}}$${\ displaystyle {\ boldsymbol {\ varepsilon}}}$${\ displaystyle \ sigma _ {ij}}$${\ displaystyle \ varepsilon _ {ij}}$

${\ displaystyle f_ {C}: \ varepsilon _ {kl} \ rightarrow \ sigma _ {ij} = \ sum _ {k, l = 1} ^ {3} C_ {ijkl} \ varepsilon _ {kl}}$.

This contains 81 coefficients with which the nine components are mapped to nine components . In linear elasticity theory , which is the symmetric stress tensor and the symmetric strain tensor , the number of independent tensor components is reduced to six so that only 36 coefficients are independent. This relationship between stresses and distortions can now also be written in Voigt's notation as a matrix equation: ${\ displaystyle C_ {ijkl}}$${\ displaystyle \ varepsilon _ {ij}}$${\ displaystyle \ sigma _ {ij}}$${\ displaystyle {\ boldsymbol {\ sigma}}}$${\ displaystyle {\ boldsymbol {\ varepsilon}}}$${\ displaystyle C_ {ijkl}}$

${\ displaystyle \ sigma _ {i} ^ {\ text {v}} = \ sum _ {j = 1} ^ {3} C_ {ij} ^ {\ text {v}} \ varepsilon _ {j} ^ { \ text {v}} \ Leftrightarrow {\ begin {bmatrix} \ sigma _ {11} \\\ sigma _ {22} \\\ sigma _ {33} \\\ sigma _ {23} \\\ sigma _ { 13} \\\ sigma _ {12} \\\ end {bmatrix}} = {\ begin {bmatrix} C_ {1111} & C_ {1122} & C_ {1133} & C_ {1123} & C_ {1113} & C_ {1112} \ \ C_ {2211} & C_ {2222} & C_ {2233} & C_ {2223} & C_ {2213} & C_ {2212} \\ C_ {3311} & C_ {3322} & C_ {3333} & C_ {3323} & C_ {3313} & C_ {3312 } \\ C_ {2311} & C_ {2322} & C_ {2333} & C_ {2323} & C_ {2313} & C_ {2312} \\ C_ {1311} & C_ {1322} & C_ {1333} & C_ {1323} & C_ {1313} & C_ {1312} \\ C_ {1211} & C_ {1222} & C_ {1233} & C_ {1223} & C_ {1213} & C_ {1212} \ end {bmatrix}} {\ begin {bmatrix} \ varepsilon _ {11} \\\ varepsilon _ {22} \\\ varepsilon _ {33} \\ 2 \ varepsilon _ {23} \\ 2 \ varepsilon _ {13} \\ 2 \ varepsilon _ {12} \\\ end {bmatrix}}}$.

The matrix with the 36 independent components represents the elasticity tensor of the material. In the case of hyperelasticity , this matrix is symmetrical , so that only 21 entries are then independent. ${\ displaystyle C ^ {\ text {v}}}$${\ displaystyle C_ {ijkl}}$

### Law of elasticity for 3D

A material is linearly elastic orthotropic if an orthonormal basis exists, so that the law of elasticity represented in relation to this basis takes the following form (with only twelve independent entries):

${\ displaystyle {\ begin {bmatrix} \ varepsilon _ {11} \\\ varepsilon _ {22} \\\ varepsilon _ {33} \\ 2 \ varepsilon _ {23} \\ 2 \ varepsilon _ {13} \ \ 2 \ varepsilon _ {12} \ end {bmatrix}} = \ underbrace {\ begin {bmatrix} {\ frac {1} {E_ {1}}} & - {\ frac {\ nu _ {21}} { E_ {2}}} & - {\ frac {\ nu _ {31}} {E_ {3}}} &&& \\ - {\ frac {\ nu _ {12}} {E_ {1}}} & { \ frac {1} {E_ {2}}} & - {\ frac {\ nu _ {32}} {E_ {3}}} &&& \\ - {\ frac {\ nu _ {13}} {E_ { 1}}} & - {\ frac {\ nu _ {23}} {E_ {2}}} & {\ frac {1} {E_ {3}}} &&& \\ &&& {\ frac {1} {G_ {23}}} && \\ &&&& {\ frac {1} {G_ {13}}} & \\ &&&&& {\ frac {1} {G_ {12}}} \ end {bmatrix}} _ {=: S } {\ begin {bmatrix} \ sigma _ {11} \\\ sigma _ {22} \\\ sigma _ {33} \\\ sigma _ {23} \\\ sigma _ {13} \\\ sigma _ {12} \ end {bmatrix}}}$

The dimension of the modulus of elasticity and the modulus of shear is force per area while the Poisson's contraction numbers are dimensionless. Pointer contraction numbers describe how one moves along a direction - e.g. B. the 1-direction - drawn material sample across it - z. B. in 2-direction - contracted. The corresponding Poisson's ratio would then be . The normal expansion in the i direction is denoted by. Then, for any material, the Poisson 's ratio is the negative ratio of normal elongation in the j direction (effect) to that in the i direction when pulling in the i direction (cause): ${\ displaystyle E_ {1}, E_ {2}, E_ {3}}$ ${\ displaystyle G_ {12}, G_ {23}, G_ {13}}$ ${\ displaystyle \ nu _ {ij}}$${\ displaystyle \ nu _ {12}}$${\ displaystyle \ varepsilon _ {ii}}$${\ displaystyle \ nu _ {ij}}$

${\ displaystyle \ nu _ {ij} = {\ frac {- \ varepsilon _ {jj}} {\ varepsilon _ {ii}}}}$

The matrix S is the compliance matrix of the material.

### Material parameters

The twelve characteristic values ​​occurring in the above flexibility matrix result from orthotropic linear elasticity from only nine material parameters, which can be determined in tests on macroscopic samples :

Formula symbol meaning
${\ displaystyle E_ {1}, E_ {2}, E_ {3}}$ Modulus of elasticity in the orthotropic axes
${\ displaystyle G_ {12}, G_ {13}, G_ {23}}$ Shear modulus in planes perpendicular to the orthotropic axes
${\ displaystyle \ nu _ {12}, \ nu _ {13}, \ nu _ {23}}$ Pointer contraction numbers when pulling in the direction of an orthotropic axis

The symmetry of the compliance matrix determines the three remaining Poisson's ratio:

${\ displaystyle \ nu _ {21} = {\ frac {E_ {2}} {E_ {1}}} \ nu _ {12} \ ,, \ quad \ nu _ {31} = {\ frac {E_ { 3}} {E_ {1}}} \ nu _ {13} \ ,, \ quad \ nu _ {32} = {\ frac {E_ {3}} {E_ {2}}} \ nu _ {23} \ ,.}$

The law of elasticity for orthotropic, linear elasticity is:

${\ displaystyle {\ begin {bmatrix} \ varepsilon _ {11} \\\ varepsilon _ {22} \\\ varepsilon _ {33} \\ 2 \ varepsilon _ {23} \\ 2 \ varepsilon _ {13} \ \ 2 \ varepsilon _ {12} \ end {bmatrix}} = {\ begin {bmatrix} {\ frac {1} {E_ {1}}} & - {\ frac {\ nu _ {12}} {E_ { 1}}} & - {\ frac {\ nu _ {13}} {E_ {1}}} & 0 & 0 & 0 \\ & {\ frac {1} {E_ {2}}} & - {\ frac {\ nu _ {23}} {E_ {2}}} & 0 & 0 & 0 \\ && {\ frac {1} {E_ {3}}} & 0 & 0 & 0 \\ &&& {\ frac {1} {G_ {23}}} & 0 & 0 \\ & { \ textsf {sym}} &&& {\ frac {1} {G_ {13}}} & 0 \\ &&&&& {\ frac {1} {G_ {12}}} \ end {bmatrix}} {\ begin {bmatrix} \ sigma _ {11} \\\ sigma _ {22} \\\ sigma _ {33} \\\ sigma _ {23} \\\ sigma _ {13} \\\ sigma _ {12} \ end {bmatrix} }}$

The stiffness matrix is ​​obtained by inverting the compliance matrix

${\ displaystyle {\ begin {bmatrix} \ sigma _ {11} \\\ sigma _ {22} \\\ sigma _ {33} \\\ sigma _ {23} \\\ sigma _ {13} \\\ sigma _ {12} \\\ end {bmatrix}} = {\ begin {bmatrix} {\ frac {1- \ nu _ {23} \ nu _ {32}} {D}} E_ {1} & {\ frac {\ nu _ {13} \ nu _ {32} + \ nu _ {12}} {D}} E_ {2} & {\ frac {\ nu _ {12} \ nu _ {23} + \ nu _ {13}} {D}} E_ {3} & 0 & 0 & 0 \\ & {\ frac {1- \ nu _ {13} \ nu _ {31}} {D}} E_ {2} & {\ frac {\ nu _ {21} \ nu _ {13} + \ nu _ {23}} {D}} E_ {3} & 0 & 0 & 0 \\ && {\ frac {1- \ nu _ {12} \ nu _ {21}} {D}} E_ {3} & 0 & 0 & 0 \\ &&& G_ {23} & 0 & 0 \\ & {\ text {sym}} &&& G_ {13} & 0 \\ &&&&& G_ {12} \ end {bmatrix}} {\ begin {bmatrix} \ varepsilon _ {11} \\\ varepsilon _ {22} \\\ varepsilon _ {33} \\ 2 \ varepsilon _ {23} \\ 2 \ varepsilon _ {13} \\ 2 \ varepsilon _ {12} \ end {bmatrix}}}$

With

${\ displaystyle D = 1- \ nu _ {12} \ nu _ {21} - \ nu _ {13} \ nu _ {31} - \ nu _ {23} \ nu _ {32} -2 \ nu _ {12} \ nu _ {23} \ nu _ {31} \ ,.}$

The compliance matrix and the stiffness matrix are symmetrical and have values ​​other than zero at the same places.

Transverse isotropy occurs with:

${\ displaystyle {\ begin {array} {lcl} E_ {2} & = & E_ {3} \\ G_ {12} & = & G_ {13} \\\ nu _ {12} & = & \ nu _ {13 } \\ G_ {23} & = & {\ frac {E_ {2}} {2 (1+ \ nu _ {23})}} \ end {array}}}$

### Level state of tension

For the plane state of stress (σ₃₃ = 0, σ₁₃ = 0, σ₂₃ = 0) the linear orthotropic elasticity law given above reads:

${\ displaystyle {\ begin {bmatrix} \ varepsilon _ {11} \\\ varepsilon _ {22} \\\ varepsilon _ {33} \\\ varepsilon _ {12} \ end {bmatrix}} = {\ begin { bmatrix} {\ frac {1} {E_ {1}}} & - {\ frac {\ nu _ {12}} {E_ {1}}} & 0 \\ - {\ frac {\ nu _ {21}} {E_ {2}}} & {\ frac {1} {E_ {2}}} & 0 \\ - {\ frac {\ nu _ {31}} {E_ {3}}} & - {\ frac {\ nu _ {32}} {E_ {3}}} & 0 \\ 0 & 0 & {\ frac {1} {2 \ cdot G_ {12}}} \ end {bmatrix}} {\ begin {bmatrix} \ sigma _ {11 } \\\ sigma _ {22} \\\ sigma _ {12} \\\ end {bmatrix}}}$

where:

${\ displaystyle {\ frac {\ nu _ {12}} {E_ {1}}} = {\ frac {\ nu _ {21}} {E_ {2}}}}$

so that in the linear orthotropic elasticity there are six material parameters for the plane stress state, if only the distortions in the plane are interested, there are only four material parameters.

### Flat state of distortion

Here the distortions are exclusively in the plane, but there are also tensions from the plane.

${\ displaystyle {\ begin {bmatrix} \ sigma _ {11} \\\ sigma _ {22} \\\ sigma _ {33} \\\ sigma _ {12} \\\ end {bmatrix}} = {\ begin {bmatrix} {\ frac {1- \ nu _ {23} \ nu _ {32}} {D}} E_ {1} & {\ frac {\ nu _ {13} \ nu _ {32} + \ nu _ {12}} {D}} E_ {2} & 0 \\ {\ frac {\ nu _ {13} \ nu _ {32} + \ nu _ {12}} {D}} E_ {2} & {\ frac {1- \ nu _ {13} \ nu _ {31}} {D}} E_ {2} & 0 \\ {\ frac {\ nu _ {12} \ nu _ {23} + \ nu _ {13}} {D}} E_ {3} & {\ frac {\ nu _ {21} \ nu _ {13} + \ nu _ {23}} {D}} E_ {3} & 0 \\ 0 & 0 & 2 \ cdot G_ {12} \ end {bmatrix}} {\ begin {bmatrix} \ varepsilon _ {11} \\\ varepsilon _ {22} \\\ varepsilon _ {12} \ end {bmatrix}}}$

With

${\ displaystyle D = 1- \ nu _ {12} \ nu _ {21} - \ nu _ {13} \ nu _ {31} - \ nu _ {23} \ nu _ {32} -2 \ nu _ {12} \ nu _ {23} \ nu _ {31} \ ,.}$

### Stability criteria

The material parameters cannot be chosen arbitrarily, but must meet certain stability criteria. These follow from the requirement that the stiffness and compliance matrices must be positive and definite . This leads to the conditions:

• All diagonal elements of the stiffness and compliance matrix must be positive (so that the material stretches in the tensile direction when you pull on it and not compresses) and
• the determinant of the stiffness and compliance matrix must be positive (so that it compresses under pressure and does not expand).

If material parameters are identified in a real material that contradict these stability criteria, caution is required. The stability criteria are:

${\ displaystyle {\ begin {array} {l} E_ {1}, E_ {2}, E_ {3}, G_ {12}, G_ {13}, G_ {23}> 0 \\ | \ nu _ { 12} | <{\ sqrt {\ dfrac {E_ {1}} {E_ {2}}}} \ quad \ rightarrow \ quad 1- \ nu _ {12} \ nu _ {21}> 0 \\ | \ nu _ {13} | <{\ sqrt {\ dfrac {E_ {1}} {E_ {3}}}} \ quad \ rightarrow \ quad 1- \ nu _ {13} \ nu _ {31}> 0 \ \ | \ nu _ {23} | <{\ sqrt {\ dfrac {E_ {2}} {E_ {3}}}} \ quad \ rightarrow \ quad 1- \ nu _ {23} \ nu _ {32} > 0 \\ 1- \ nu _ {12} \ nu _ {21} - \ nu _ {13} \ nu _ {31} - \ nu _ {23} \ nu _ {32} -2 \ nu _ { 12} \ nu _ {23} \ nu _ {31}> 0 \ end {array}}}$

As the left hand side of the last inequality approaches zero, the material is increasingly resisting hydrostatic compression. From the symmetry of the compliance matrix, it also follows:

${\ displaystyle {\ begin {array} {l} | \ nu _ {21} | <{\ sqrt {\ dfrac {E_ {2}} {E_ {1}}}} \\ | \ nu _ {31} | <{\ sqrt {\ dfrac {E_ {3}} {E_ {1}}}} \\ | \ nu _ {32} | <{\ sqrt {\ dfrac {E_ {3}} {E_ {2} }}} \ end {array}}}$

## Reasons for the occupancy of the stiffness matrix

In this section, the question is clarified why the stiffness matrix is ​​only occupied at the appropriate places. In general, 21 independent material constants appear in a linear material law. 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 chosen as a reference, the base vectors of which coincide with the mutually perpendicular axes of rotation, then these orthogonal matrices have the following form

{\ displaystyle {\ begin {aligned} 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 {aligned}}}

These 3 matrices (and also 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 . In the simplest case (which, however, 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 you leave the material point as it is (i.e. do not rotate the material) and subject the point to (the same) stretching in a different direction. This leads to a different strain tensor . ${\ displaystyle \ varepsilon}$${\ displaystyle \ varepsilon '}$

The change in the direction of distortion can be described with a rotation matrix . It applies ${\ displaystyle A}$

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

With the aid of a linear material law , the associated stress tensor can be determined for a given strain tensor . Be it ${\ displaystyle f_ {C}}$

{\ displaystyle {\ begin {aligned} \ sigma &: = f_ {C} (\ varepsilon) \\\ sigma '&: = f_ {C} (\ varepsilon') \ end {aligned}}}

In the general case of anisotropy, it is not true

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

But this is exactly what is required for the subset of SO (3) described above in the case of orthotropy: A material is called orthotropic if the following symmetry transformation applies to the function for each of the above-mentioned (orthogonal) rotation matrices and for any distortions ${\ displaystyle f_ {C}}$

{\ displaystyle {\ begin {aligned} Af_ {C} (\ varepsilon) A ^ {- 1} & = f_ {C} (A \ varepsilon A ^ {- 1}) \ Leftrightarrow Af_ {C} (\ varepsilon) A ^ {T} = f_ {C} (A \ varepsilon A ^ {T}) \ end {aligned}}}

In index notation

{\ displaystyle {\ begin {aligned} \ sigma _ {mn} '= A_ {mo} C_ {opjk} \ varepsilon _ {jk} A_ {np} & = C_ {mnil} \ varepsilon' _ {il} = C_ {mnil} A_ {ij} \ varepsilon _ {jk} A_ {lk} \ end {aligned}}}

Now the same condition in Voigt's notation: With the definition

{\ displaystyle {\ begin {aligned} A _ {\ sigma} ^ {\ text {v}}: = {\ begin {bmatrix} A_ {11} A_ {11} & A_ {12} A_ {12} & A_ {13} A_ {13} & A_ {12} A_ {13} + A_ {13} A_ {12} & A_ {11} A_ {13} + A_ {13} A_ {11} & A_ {11} A_ {12} + A_ {12 } A_ {11} \\ A_ {21} A_ {21} & A_ {22} A_ {22} & A_ {23} A_ {23} & A_ {22} A_ {23} + A_ {23} A_ {22} & A_ { 21} A_ {23} + A_ {23} A_ {21} & A_ {21} A_ {22} + A_ {22} A_ {21} \\ A_ {31} A_ {31} & A_ {32} A_ {32} & A_ {33} A_ {33} & A_ {32} A_ {33} + A_ {33} A_ {32} & A_ {31} A_ {33} + A_ {33} A_ {31} & A_ {31} A_ {32} + A_ {32} A_ {31} \\ A_ {21} A_ {31} & A_ {22} A_ {32} & A_ {23} A_ {33} & A_ {22} A_ {33} + A_ {23} A_ { 32} & A_ {21} A_ {33} + A_ {23} A_ {31} & A_ {21} A_ {32} + A_ {22} A_ {31} \\ A_ {11} A_ {31} & A_ {12} A_ {32} & A_ {13} A_ {33} & A_ {12} A_ {33} + A_ {13} A_ {32} & A_ {11} A_ {33} + A_ {13} A_ {31} & A_ {11} A_ {32} + A_ {12} A_ {31} \\ A_ {11} A_ {21} & A_ {12} A_ {22} & A_ {13} A_ {23} & A_ {12} A_ {23} + A_ { 13} A_ {22} & A_ {11} A_ {23} + A_ {13} A_ {21} & A_ {11} A_ {22} + A_ {12} A_ {21} \\\ end {bmatrix}} \ end {aligned}}}

applies

{\ displaystyle {\ begin {aligned} {\ begin {bmatrix} \ sigma '_ {11} \\\ sigma' _ {22} \\\ sigma '_ {33} \\\ sigma' _ {23} \ \\ sigma '_ {13} \\\ sigma' _ {12} \ end {bmatrix}} = A _ {\ sigma} ^ {\ text {v}} {\ begin {bmatrix} \ sigma _ {11} \ \\ sigma _ {22} \\\ sigma _ {33} \\\ sigma _ {23} \\\ sigma _ {13} \\\ sigma _ {12} \ end {bmatrix}} \ Leftrightarrow {\ sigma '} ^ {\ text {v}} = A _ {\ sigma} ^ {\ text {v}} {\ sigma} ^ {\ text {v}}, \ qquad \ qquad {\ begin {bmatrix} \ varepsilon' _ {11} \\\ varepsilon '_ {22} \\\ varepsilon' _ {33} \\\ varepsilon '_ {23} \\\ varepsilon' _ {13} \\\ varepsilon '_ {12} \ end {bmatrix}} = A _ {\ sigma} ^ {\ text {v}} {\ begin {bmatrix} \ varepsilon _ {11} \\\ varepsilon _ {22} \\\ varepsilon _ {33} \\\ varepsilon _ {23} \\\ varepsilon _ {13} \\\ varepsilon _ {12} \ end {bmatrix}} \ end {aligned}}}

With the new definition

{\ displaystyle {\ begin {aligned} A _ {\ varepsilon} ^ {\ text {v}}: = {\ begin {bmatrix} A_ {11} A_ {11} & A_ {12} A_ {12} & A_ {13} A_ {13} & A_ {12} A_ {13} + A_ {13} A_ {12} & A_ {11} A_ {13} + A_ {13} A_ {11} & A_ {11} A_ {12} + A_ {12 } A_ {11} \\ A_ {21} A_ {21} & A_ {22} A_ {22} & A_ {23} A_ {23} & A_ {22} A_ {23} + A_ {23} A_ {22} & A_ { 21} A_ {23} + A_ {23} A_ {21} & A_ {21} A_ {22} + A_ {22} A_ {21} \\ A_ {31} A_ {31} & A_ {32} A_ {32} & A_ {33} A_ {33} & A_ {32} A_ {33} + A_ {33} A_ {32} & A_ {31} A_ {33} + A_ {33} A_ {31} & A_ {31} A_ {32} + A_ {32} A_ {31} \\ 2A_ {21} A_ {31} & 2A_ {22} A_ {32} & 2A_ {23} A_ {33} & A_ {22} A_ {33} + A_ {23} A_ { 32} & A_ {21} A_ {33} + A_ {23} A_ {31} & A_ {21} A_ {32} + A_ {22} A_ {31} \\ 2A_ {11} A_ {31} & 2A_ {12} A_ {32} & 2A_ {13} A_ {33} & A_ {12} A_ {33} + A_ {13} A_ {32} & A_ {11} A_ {33} + A_ {13} A_ {31} & A_ {11} A_ {32} + A_ {12} A_ {31} \\ 2A_ {11} A_ {21} & 2A_ {12} A_ {22} & 2A_ {13} A_ {23} & A_ {12} A_ {23} + A_ { 13} A_ {22} & A_ {11} A_ {23} + A_ {13} A_ {21} & A_ {11} A_ {22} + A_ {12} A_ {21} \\\ end {bmatrix}} \ end {aligned}}}

surrendered

{\ displaystyle {\ begin {aligned} {\ begin {bmatrix} \ varepsilon '_ {11} \\\ varepsilon' _ {22} \\\ varepsilon '_ {33} \\ 2 \ varepsilon' _ {23} \\ 2 \ varepsilon '_ {13} \\ 2 \ varepsilon' _ {12} \ end {bmatrix}} = A _ {\ varepsilon} ^ {\ text {v}} {\ begin {bmatrix} \ varepsilon _ { 11} \\\ varepsilon _ {22} \\\ varepsilon _ {33} \\ 2 \ varepsilon _ {23} \\ 2 \ varepsilon _ {13} \\ 2 \ varepsilon _ {12} \ end {bmatrix} } \ Leftrightarrow {\ varepsilon '} ^ {\ text {v}} = A _ {\ varepsilon} ^ {\ text {v}} {\ varepsilon} ^ {\ text {v}} \ end {aligned}}}

In Voigt's notation, one obtains the symmetry condition

{\ displaystyle {\ begin {aligned} {\ sigma '} ^ {\ text {v}} = A _ {\ sigma} ^ {\ text {v}} {\ sigma} ^ {\ text {v}} & = A _ {\ sigma} ^ {\ text {v}} C ^ {\ text {v}} {\ varepsilon} ^ {\ text {v}} = C ^ {\ text {v}} {\ varepsilon '} ^ {\ text {v}} = C ^ {\ text {v}} A _ {\ varepsilon} ^ {\ text {v}} {\ varepsilon} ^ {\ text {v}} \ end {aligned}}}

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

{\ displaystyle {\ begin {aligned} A _ {\ sigma} ^ {\ text {v}} C ^ {\ text {v}} = C ^ {\ text {v}} A _ {\ varepsilon} ^ {\ text {v}} \ end {aligned}}}

### Special case of 180 degree rotations

Since in the special case of orthotropy the 3 × 3 matrices are only occupied on the main diagonal , the definitions from above are simplified ${\ displaystyle A}$

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

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

{\ displaystyle {\ begin {aligned} A_ {x} ^ {\ text {v}} & = {\ begin {bmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 \ & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 \ & 0 & 0 & 0 & 0 \\ end {bmatrix}}, \ qquad A_ {y} ^ {\ text {v}} & = {\ begin {bmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 \ 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\\ end {bmatrix}}, \ qquad A_ {z} ^ {\ text {v}} & = {\ begin {bmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 1 \\\ end {bmatrix}} \ end {aligned}}}

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

The symmetry condition evaluated for these matrices gives

{\ displaystyle {\ begin {aligned} {\ text {because of}} A_ {x} ^ {\ text {v}} C ^ {\ text {v}} = C ^ {\ text {v}} A_ {x } ^ {\ text {v}}: \ qquad {\ begin {bmatrix} C_ {11} ^ {\ text {v}} & C_ {12} ^ {\ text {v}} & C_ {13} ^ {\ text {v}} & C_ {14} ^ {\ text {v}} & C_ {15} ^ {\ text {v}} & C_ {16} ^ {\ text {v}} \\ C_ {21} ^ {\ text {v}} & C_ {22} ^ {\ text {v}} & C_ {23} ^ {\ text {v}} & C_ {24} ^ {\ text {v}} & C_ {25} ^ {\ text {v }} & C_ {26} ^ {\ text {v}} \\ C_ {31} ^ {\ text {v}} & C_ {32} ^ {\ text {v}} & C_ {33} ^ {\ text {v }} & C_ {34} ^ {\ text {v}} & C_ {35} ^ {\ text {v}} & C_ {36} ^ {\ text {v}} \\ - C_ {41} ^ {\ text { v}} & - C_ {42} ^ {\ text {v}} & - C_ {43} ^ {\ text {v}} & - C_ {44} ^ {\ text {v}} & - C_ {45 } ^ {\ text {v}} & - C_ {46} ^ {\ text {v}} \\ - C_ {51} ^ {\ text {v}} & - C_ {52} ^ {\ text {v }} & - C_ {53} ^ {\ text {v}} & - C_ {54} ^ {\ text {v}} & - C_ {55} ^ {\ text {v}} & - C_ {56} ^ {\ text {v}} \\ C_ {61} ^ {\ text {v}} & C_ {62} ^ {\ text {v}} & C_ {63} ^ {\ text {v}} & C_ {64} ^ {\ text {v}} & C_ {65} ^ {\ text {v}} & C_ {66} ^ {\ text {v}} \ end {bmatrix}} & = {\ begin {bmatrix} C_ {11} ^ {\ text {v}} & C_ {12} ^ {\ text {v}} & C_ {13} ^ {\ text {v}} & - C_ {14} ^ {\ text {v}} & - C_ { 15} ^ {\ text {v}} & C_ {16} ^ {\ text {v}} \\ C_ {21} ^ {\ text {v}} & C_ {22} ^ {\ text {v}} & C_ { 23} ^ {\ text {v}} & - C_ {24} ^ {\ te xt {v}} & - C_ {25} ^ {\ text {v}} & C_ {26} ^ {\ text {v}} \\ C_ {31} ^ {\ text {v}} & C_ {32} ^ {\ text {v}} & C_ {33} ^ {\ text {v}} & - C_ {34} ^ {\ text {v}} & - C_ {35} ^ {\ text {v}} & C_ {36 } ^ {\ text {v}} \\ C_ {41} ^ {\ text {v}} & C_ {42} ^ {\ text {v}} & C_ {43} ^ {\ text {v}} & - C_ {44} ^ {\ text {v}} & - C_ {45} ^ {\ text {v}} & C_ {46} ^ {\ text {v}} \\ C_ {51} ^ {\ text {v} } & C_ {52} ^ {\ text {v}} & C_ {53} ^ {\ text {v}} & - C_ {54} ^ {\ text {v}} & - C_ {55} ^ {\ text { v}} & C_ {56} ^ {\ text {v}} \\ C_ {61} ^ {\ text {v}} & C_ {62} ^ {\ text {v}} & C_ {63} ^ {\ text { v}} & - C_ {64} ^ {\ text {v}} & - C_ {65} ^ {\ text {v}} & C_ {66} ^ {\ text {v}} \ end {bmatrix}} \ \ {\ text {because of}} A_ {y} ^ {\ text {v}} C ^ {\ text {v}} = C ^ {\ text {v}} A_ {y} ^ {\ text {v} }: \ qquad {\ begin {bmatrix} C_ {11} ^ {\ text {v}} & C_ {12} ^ {\ text {v}} & C_ {13} ^ {\ text {v}} & C_ {14} ^ {\ text {v}} & C_ {15} ^ {\ text {v}} & C_ {16} ^ {\ text {v}} \\ C_ {21} ^ {\ text {v}} & C_ {22} ^ {\ text {v}} & C_ {23} ^ {\ text {v}} & C_ {24} ^ {\ text {v}} & C_ {25} ^ {\ text {v}} & C_ {26} ^ { \ text {v}} \\ C_ {31} ^ {\ text {v}} & C_ {32} ^ {\ text {v}} & C_ {33} ^ {\ text {v}} & C_ {34} ^ { \ text {v}} & C_ {35} ^ {\ text {v}} & C_ {36} ^ {\ text {v}} \\ - C_ {41} ^ {\ text {v}} & - C_ {42 } ^ {\ text {v}} & - C_ {43} ^ {\ text {v}} & - C_ {44} ^ {\ text {v} } & - C_ {45} ^ {\ text {v}} & - C_ {46} ^ {\ text {v}} \\ C_ {51} ^ {\ text {v}} & C_ {52} ^ {\ text {v}} & C_ {53} ^ {\ text {v}} & C_ {54} ^ {\ text {v}} & C_ {55} ^ {\ text {v}} & C_ {56} ^ {\ text { v}} \\ - C_ {61} ^ {\ text {v}} & - C_ {62} ^ {\ text {v}} & - C_ {63} ^ {\ text {v}} & - C_ { 64} ^ {\ text {v}} & - C_ {65} ^ {\ text {v}} & - C_ {66} ^ {\ text {v}} \ end {bmatrix}} & = {\ begin { bmatrix} C_ {11} ^ {\ text {v}} & C_ {12} ^ {\ text {v}} & C_ {13} ^ {\ text {v}} & - C_ {14} ^ {\ text {v }} & C_ {15} ^ {\ text {v}} & - C_ {16} ^ {\ text {v}} \\ C_ {21} ^ {\ text {v}} & C_ {22} ^ {\ text {v}} & C_ {23} ^ {\ text {v}} & - C_ {24} ^ {\ text {v}} & C_ {25} ^ {\ text {v}} & - C_ {26} ^ { \ text {v}} \\ C_ {31} ^ {\ text {v}} & C_ {32} ^ {\ text {v}} & C_ {33} ^ {\ text {v}} & - C_ {34} ^ {\ text {v}} & C_ {35} ^ {\ text {v}} & - C_ {36} ^ {\ text {v}} \\ C_ {41} ^ {\ text {v}} & C_ { 42} ^ {\ text {v}} & C_ {43} ^ {\ text {v}} & - C_ {44} ^ {\ text {v}} & C_ {45} ^ {\ text {v}} & - C_ {46} ^ {\ text {v}} \\ C_ {51} ^ {\ text {v}} & C_ {52} ^ {\ text {v}} & C_ {53} ^ {\ text {v}} & -C_ {54} ^ {\ text {v}} & C_ {55} ^ {\ text {v}} & - C_ {56} ^ {\ text {v}} \\ C_ {61} ^ {\ text {v}} & C_ {62} ^ {\ text {v}} & C_ {63} ^ {\ text {v}} & - C_ {64} ^ {\ text {v}} & C_ {65} ^ {\ text {v}} & - C_ {66} ^ {\ text {v}} \ end {bmatrix}} \\ {\ text {because of}} A_ {z} ^ {\ te xt {v}} C ^ {\ text {v}} = C ^ {\ text {v}} A_ {z} ^ {\ text {v}}: \ qquad {\ begin {bmatrix} C_ {11} ^ {\ text {v}} & C_ {12} ^ {\ text {v}} & C_ {13} ^ {\ text {v}} & C_ {14} ^ {\ text {v}} & C_ {15} ^ {\ text {v}} & C_ {16} ^ {\ text {v}} \\ C_ {21} ^ {\ text {v}} & C_ {22} ^ {\ text {v}} & C_ {23} ^ {\ text {v}} & C_ {24} ^ {\ text {v}} & C_ {25} ^ {\ text {v}} & C_ {26} ^ {\ text {v}} \\ C_ {31} ^ {\ text {v}} & C_ {32} ^ {\ text {v}} & C_ {33} ^ {\ text {v}} & C_ {34} ^ {\ text {v}} & C_ {35} ^ {\ text { v}} & C_ {36} ^ {\ text {v}} \\ C_ {41} ^ {\ text {v}} & C_ {42} ^ {\ text {v}} & C_ {43} ^ {\ text { v}} & C_ {44} ^ {\ text {v}} & C_ {45} ^ {\ text {v}} & C_ {46} ^ {\ text {v}} \\ - C_ {51} ^ {\ text {v}} & - C_ {52} ^ {\ text {v}} & - C_ {53} ^ {\ text {v}} & - C_ {54} ^ {\ text {v}} & - C_ { 55} ^ {\ text {v}} & - C_ {56} ^ {\ text {v}} \\ - C_ {61} ^ {\ text {v}} & - C_ {62} ^ {\ text { v}} & - C_ {63} ^ {\ text {v}} & - C_ {64} ^ {\ text {v}} & - C_ {65} ^ {\ text {v}} & - C_ {66 } ^ {\ text {v}} \ end {bmatrix}} & = {\ begin {bmatrix} C_ {11} ^ {\ text {v}} & C_ {12} ^ {\ text {v}} & C_ {13 } ^ {\ text {v}} & C_ {14} ^ {\ text {v}} & - C_ {15} ^ {\ text {v}} & - C_ {16} ^ {\ text {v}} \ \ C_ {21} ^ {\ text {v}} & C_ {22} ^ {\ text {v}} & C_ {23} ^ {\ text {v}} & C_ {24} ^ {\ text {v}} & -C_ {25} ^ {\ text {v}} & - C_ {26} ^ {\ text {v}} \\ C_ {31} ^ {\ te xt {v}} & C_ {32} ^ {\ text {v}} & C_ {33} ^ {\ text {v}} & C_ {34} ^ {\ text {v}} & - C_ {35} ^ {\ text {v}} & - C_ {36} ^ {\ text {v}} \\ C_ {41} ^ {\ text {v}} & C_ {42} ^ {\ text {v}} & C_ {43} ^ {\ text {v}} & C_ {44} ^ {\ text {v}} & - C_ {45} ^ {\ text {v}} & - C_ {46} ^ {\ text {v}} \\ C_ {51} ^ {\ text {v}} & C_ {52} ^ {\ text {v}} & C_ {53} ^ {\ text {v}} & C_ {54} ^ {\ text {v}} & - C_ {55} ^ {\ text {v}} & - C_ {56} ^ {\ text {v}} \\ C_ {61} ^ {\ text {v}} & C_ {62} ^ {\ text {v} } & C_ {63} ^ {\ text {v}} & C_ {64} ^ {\ text {v}} & - C_ {65} ^ {\ text {v}} & - C_ {66} ^ {\ text { v}} \ end {bmatrix}} \ end {aligned}}}

The last 3 equations show that it can only have the following form ${\ displaystyle C}$

{\ displaystyle {\ begin {aligned} C ^ {\ text {v}} & = {\ begin {bmatrix} C_ {11} ^ {\ text {v}} & C_ {12} ^ {\ text {v}} & C_ {13} ^ {\ text {v}} & 0 & 0 & 0 \\ C_ {21} ^ {\ text {v}} & C_ {22} ^ {\ text {v}} & C_ {23} ^ {\ text {v} } & 0 & 0 & 0 \\ C_ {31} ^ {\ text {v}} & C_ {32} ^ {\ text {v}} & C_ {33} ^ {\ text {v}} & 0 & 0 & 0 \\ 0 & 0 & 0 & C_ {44} ^ {\ text {v}} & 0 & 0 \\ 0 & 0 & 0 & 0 & C_ {55} ^ {\ text {v}} & 0 \\ 0 & 0 & 0 & 0 & 0 & C_ {66} ^ {\ text {v}} \ end {bmatrix}} \ end {aligned}}}

Since this Voigt stiffness matrix is ​​also symmetrical, remains

{\ displaystyle {\ begin {aligned} C ^ {\ text {v}} & = {\ begin {bmatrix} C_ {11} ^ {\ text {v}} & C_ {12} ^ {\ text {v}} & C_ {13} ^ {\ text {v}} & 0 & 0 & 0 \\ & C_ {22} ^ {\ text {v}} & C_ {23} ^ {\ text {v}} & 0 & 0 & 0 \\ && C_ {33} ^ {\ text {v}} & 0 & 0 & 0 \\ &&& C_ {44} ^ {\ text {v}} & 0 & 0 \\ & {\ text {sym}} &&& C_ {55} ^ {\ text {v}} & 0 \\ &&&&& C_ {66} ^ {\ text {v}} \ end {bmatrix}} \ end {aligned}}}

## Summary

• Orthotropy in the linear theory of elasticity 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. B. transverse isotropy, isotropy, etc. Here the same symmetry conditions are given. Only then other subgroups of the rotation group (i.e. other matrices ) are considered.${\ displaystyle A}$
• The shape of the elastic law shows that there is no coupling between tension and thrust for loading along the orthotropic directions.

Wiktionary: orthotropic  - explanations of meanings, word origins, synonyms, translations

## literature

• JF 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 . German publishing house for basic industry, Stuttgart 1996, ISBN 3-342-00681-1 .
• P. Haupt: Continuum Mechanics and Theory of Materials. Springer, 2000, ISBN 3-540-66114-X .
• H. Altenbach: Continuum Mechanics: Introduction to the material-independent and material-dependent equations. Springer, 2012. ISBN 3-642-24119-0 .

## Individual evidence

1. Haupt, 2000
2. H. Altenbach, 2012, p. 331.