# Transverse isotropy

Illustrative explanation of the transversal isotropy.
The material is rotationally symmetrical with respect to the 1 axis, which is perpendicular to the isotropic 2-3 plane.
A round bar made of this material oriented in this way can be rotated around its longitudinal axis without changing its properties.

The transversal isotropy (from Latin transversus "across" as well as ancient Greek ἴσος isos "equal" and τρόπος tropos "rotation, direction") is a special type of directional dependence of a material. Transversely isotropic materials have three properties:

1. There is a preferred direction, the 1-direction in the picture, in which the force-deformation behavior of the material is different than perpendicular to it.
2. At right angles to the preferred direction, in 2 and 3 directions, the material properties are independent of the direction (isotropic plane) and
3. In a reference system parallel to the preferred direction, there is no coupling between normal strains and shear distortions.

In planes that are not perpendicular to the preferred direction, the force-deformation behavior of the material is direction-dependent.

The special case that a material (on a particle) shows the same force-deformation behavior regardless of the direction of load, is called isotropy . The general case that the force-deformation behavior depends on the direction of load, on the other hand, is called anisotropy . The transversal isotropy is a special case of orthotropy and anisotropy and contains isotropy as a special case.

A linear elastic transversely isotropic material has a maximum of five material parameters.

## Significance in construction

Unidirectionally reinforced plastics are, to a good approximation, transversely isotropic when undamaged. They have high strength in the direction of the fibers and are more flexible perpendicular to them. In construction, transversely isotropic materials are often used because they allow the material properties to be adapted to the load. Among other things, the low density and high strength in the direction of load have led to a sharp increase in the use of fiber-reinforced plastics. These materials generally lose their transverse isotropy as a result of damage.

## Symmetry group

The directional dependence of a material is characterized by the fact that the force-deformation behavior is independent (invariant) compared to only certain rotations of the material: With transverse isotropy, these are any rotations around the preferred direction or 180-degree rotations perpendicular to the preferred direction. These rotations form the symmetry group of the transversely isotropic 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 initially rotated parallel to the preferred direction or by 180 degrees perpendicular to it. Then you apply the same force as in the first experiment and measure the deformation again. In the case of transversely isotropic 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 perpendicular to the preferred direction. If the special case of isotropy is not present, one will now always measure a different deformation than in the first experiment.

## Transversely isotropic elasticity

A transversely isotropic, linearly elastic material is characterized by the fact that the coupling terms in its stiffness or flexibility matrix are not occupied. Shear stresses in planes parallel or perpendicular to the preferred direction do not lead to normal strains. In such a material there is an orthonormal basis in which the stress-strain relationship exists${\ displaystyle {\ hat {e}} _ {1,2,3}}$

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

with the compliance matrix shown between the stresses and the strains . The dimensions of the modulus of elasticity and modulus of shear are force per area, while the Poisson's contraction numbers are dimensionless. The indices of the Poisson's contraction numbers are carefully defined by the negative ratio of the normal elongation in the j direction (effect) to that in the i direction when pulling in the i direction (cause): ${\ displaystyle S}$${\ displaystyle \ sigma _ {ij}}$${\ displaystyle \ varepsilon _ {ij}}$ ${\ displaystyle E_ {1}, E_ {2}, E_ {3}}$ ${\ displaystyle G_ {12}, G_ {23}, G_ {13}}$ ${\ displaystyle \ nu _ {ij}}$${\ displaystyle \ varepsilon _ {yy}}$${\ displaystyle \ varepsilon _ {ii}}$

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

### Material parameters

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

Formula symbol meaning
${\ displaystyle E _ {\ |}}$ Modulus of elasticity in the preferred direction
${\ displaystyle E _ {\ bot}}$ Modulus of elasticity perpendicular to the preferred direction
${\ displaystyle \ nu}$ Poisson's ratio when pulling in the preferred direction
${\ displaystyle G _ {\ |}}$ Shear modulus in planes parallel to the preferred direction
${\ displaystyle G _ {\ bot}}$ Shear modulus in the isotropic plane

Because of the transverse isotropy, the following expressions are identical in the 1-2-3 system:

${\ displaystyle {\ begin {array} {lcl} E_ {1} & = & E _ {\ |} \\ E_ {2} = E_ {3} & = & E _ {\ bot} \\ G_ {12} = G_ { 13} & = & G _ {\ |} \\ G_ {23} & = & G _ {\ bot} \\\ nu _ {12} = \ nu _ {13} & = & \ nu \\\ nu _ {21} = \ nu _ {31} && \\\ nu _ {23} = \ nu _ {32} && \ end {array}}}$

For thermodynamic reasons (compare Cauchy elasticity and hyperelasticity ) the compliance matrix is ​​symmetrical and lays so

${\ displaystyle {\ frac {\ nu _ {21}} {E_ {2}}} = {\ frac {\ nu _ {12}} {E_ {1}}} \ quad \ leftrightarrow \ quad \ nu _ { 21} = \ nu _ {31} = {\ frac {E _ {\ bot}} {E _ {\ |}}} \ nu}$

firmly. The Poisson's ratio in the plane perpendicular to the preferred direction is ultimately bound by the assumption of isotropy:

${\ displaystyle G _ {\ bot} = {\ frac {E _ {\ bot}} {2 (1+ \ nu _ {23})}} \ quad \ rightarrow \ quad \ nu _ {23} = \ nu _ { 32} = {\ frac {E _ {\ bot}} {2G _ {\ bot}}} - 1 \ ,.}$

All twelve parameters are based on the five material parameters. Isotropy arises with

${\ displaystyle {\ begin {array} {lcl} E_ {1} & = & E_ {2} = E \\ G_ {12} & = & G_ {23} = G = {\ dfrac {E} {2 (1+ \ nu)}} \ end {array}}}$

as a special case.

### Stress-strain relationship

The law of elasticity for transversally isotropic, linear elasticity is:

${\ displaystyle {\ begin {bmatrix} \ varepsilon _ {11} \\\ varepsilon _ {22} \\\ varepsilon _ {33} \\ 2 \ varepsilon _ {23} \\ 2 \ varepsilon _ {31} \ \ 2 \ varepsilon _ {12} \ end {bmatrix}} = {\ begin {bmatrix} {\ frac {1} {E_ {1}}} & - {\ frac {\ nu _ {12}} {E_ { 1}}} & - {\ frac {\ nu _ {12}} {E_ {1}}} & 0 & 0 & 0 \\ & {\ frac {1} {E_ {2}}} & - {\ frac {\ nu _ {23}} {E_ {2}}} & 0 & 0 & 0 \\ && {\ frac {1} {E_ {2}}} & 0 & 0 & 0 \\ &&& {\ frac {1} {G_ {23}}} & 0 & 0 \\ & \ mathrm {sym} &&& {\ frac {1} {G_ {12}}} & 0 \\ &&&&& {\ frac {1} {G_ {12}}} \ end {bmatrix}} {\ begin {bmatrix} \ sigma _ {11} \\\ sigma _ {22} \\\ sigma _ {33} \\\ sigma _ {23} \\\ sigma _ {31} \\\ sigma _ {12} \ end {bmatrix}}}$

between the stresses and the strains . By inverting the compliance matrix, the stiffness matrix is ​​obtained: ${\ displaystyle \ sigma _ {ij}}$${\ displaystyle \ varepsilon _ {ij}}$

${\ displaystyle \ left [{\ begin {array} {c} \ sigma _ {11} \\\ sigma _ {22} \\\ sigma _ {33} \\\ sigma _ {23} \\\ sigma _ {13} \\\ sigma _ {12} \ end {array}} \ right] = \ left [{\ begin {array} {cccccc} C_ {1111} & 2 \ nu _ {12} (\ lambda + G_ { 23}) & 2 \ nu _ {12} (\ lambda + G_ {23}) & 0 & 0 & 0 \\ & \ lambda + 2G_ {23} & \ lambda & 0 & 0 & 0 \\ && \ lambda + 2G_ {23} & 0 & 0 & 0 \\ &&& G_ {23 } & 0 & 0 \\ & \ mathrm {sym} &&& G_ {12} & 0 \\ &&&&& G_ {12} \ end {array}} \ right] \ left [{\ begin {array} {c} \ varepsilon _ {11} \\ \ varepsilon _ {22} \\\ varepsilon _ {33} \\ 2 \ varepsilon _ {23} \\ 2 \ varepsilon _ {13} \\ 2 \ varepsilon _ {12} \ end {array}} \ right] }$

With

${\ displaystyle {\ begin {array} {lcl} \ lambda & = & {\ dfrac {\ nu _ {12} \ nu _ {21} + \ nu _ {23}} {(1- \ nu _ {23 } -2 \ nu _ {12} \ nu _ {21}) (1+ \ nu _ {23})}} E_ {2} \\ C_ {1111} & = & {\ dfrac {1- \ nu _ {23}} {1- \ nu _ {23} -2 \ nu _ {12} \ nu _ {21}}} E_ {1} \ end {array}}}$.

This linear matrix equation between stresses and strains, written in Voigt's notation for small strains, can be generalized with hyperelasticity to nonlinear elastic transversely isotropic behavior.

### 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:

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

As the left hand side of the last inequality approaches zero, the material is increasingly resisting hydrostatic compression. From the symmetry relationship it follows in addition:

${\ displaystyle | \ nu _ {21} | <{\ sqrt {\ dfrac {E_ {2}} {E_ {1}}}}}$.

## example

Pulling a transversely isotropic material sample with a force at an angle to the preferred direction${\ displaystyle \ mathrm {F}}$${\ displaystyle \ alpha}$
Example of the directional dependence of the modulus of elasticity and the Poisson's ratio for transverse isotropy

A transversely isotropic linear elastic material has the characteristics

${\ displaystyle {\ begin {array} {lcl} E_ {1} & = & 2000 \, \ mathrm {MPa} \\ E_ {2} & = & 1000 \, \ mathrm {MPa} \\ G_ {12} & = & 700 \, \ mathrm {MPa} \\ G_ {23} & = & 350 \, \ mathrm {MPa} \\\ nu _ {12} & = & 0 {,} 25 \ end {array}}}$

The stability criteria are met:

${\ displaystyle {\ begin {array} {l} E_ {1}, E_ {2}, G_ {12}, G_ {23}> 0 \\ | \ nu _ {23} | = \ nu _ {23} = {\ frac {E_ {2}} {2G_ {23}}} - 1 = 0 {,} 4285 \ ldots <1 \\ | \ nu _ {12} | = 0 {,} 25 <{\ sqrt { \ frac {E_ {1}} {E_ {2}}}} = 1 {,} 4142 \ ldots \\ | \ nu _ {21} | = \ nu _ {12} {\ frac {E_ {2}} {E_ {1}}} = 0 {,} 125 <{\ sqrt {\ frac {E_ {2}} {E_ {1}}}} = 0 {,} 7071 \ ldots \\ 1- \ nu _ { 12} \ nu _ {21} = 0 {,} 96875> 0 \\ 1- \ nu _ {23} ^ {2} = 0 {,} 8163 \ ldots> 0 \\ (1- \ nu _ {23 } -2 \ nu _ {12} \ nu _ {21}) (1+ \ nu _ {23}) = 0 {,} 7270 \ ldots> 0 \ end {array}}}$.

If a sample of this material is loaded uniaxially at an angle to the preferred direction as in the picture above, the modulus of elasticity and the Poisson's ratio as shown in the picture below would be measured. With isotropy, the curves would be concentric circles. ${\ displaystyle \ alpha}$

## Individual evidence

1. main (2000)
2. Helmut Schürmann: Constructing with fiber-plastic composites . 2nd Edition. Springer, 2008, p. 182 f .
3. ^ H. Altenbach (2012)

## literature

• H. Altenbach: Continuum Mechanics: Introduction to the material-independent and material-dependent equations . Springer, 2012, ISBN 3-642-24119-0 .
• 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 .