# Unidirectional layer

Unidirectional layer (short: UD layer ) is the name for a layer of a fiber-plastic composite in which all fibers are oriented in a single direction. The fibers are assumed to be ideally parallel and homogeneously distributed. The unidirectional layer is transversely isotropic . The UD layer is the basic element of layered fiber-plastic composites.

## application

The UD layer is the basic element of layered fiber-plastic composites . This concerns both the calculation of the elastic properties with the help of the classical laminate theory , the net theory and the strength calculation ( break criteria for fiber reinforced plastics ).

• Fiber-plastic laminates are made up of elementary UD layers. The respective UD layers are rotated by certain angles in order to achieve certain properties in the respective material directions.
• The unidirectional layer is used in wing spar straps . There they absorb the normal forces from the bending moments.
• Pipes or containers manufactured using winding technology are built up from balanced angle connections. A balanced angle composite consists of two unidirectional layers that have a different fiber angle in the sign of the winding angle, e.g. B. ± 30 °.

In terms of its rigidity and strength, the UD layer is the ideal of a fiber-reinforced layer. The rigidity and strength properties of UD layers are therefore often reduced. In particular, the assumption of the ideally stretched fiber often does not apply in reality. However, wavy layers have, in particular, a low parallel compressive strength . ${\ displaystyle R _ {\ |} ^ {+}}$

## Mechanical properties

Coordinate system of the UD layer. Physical system and 123 system

The UD-layer, regardless of the fiber or matrix type , transversely isotropic . UD layers that consist of an isotropic fiber (e.g. glass fiber ) also result in orthotropic elasticity. The transverse isotropy is a special form of the rhombic anisotropy or orthotropy . With transverse isotropy, a plane of symmetry of the material is invariant to rotation. If the UD layer is loaded outside of its orthotropic axes, it shows anisotropic behavior.

The transversal isotropy of the UD layer is characterized by the lack of the displacement-strain coupling, a direction-dependent modulus of elasticity and the invariance with respect to the rotation around the fiber axis.

### indexing

The description of the UD-layer is in a physical - coordinate system , or in the 1,2,3 system. 1 and corresponds to the direction parallel to the fibers and 2, 3 or the direction perpendicular to the fibers. The orthotropic axes coincide with the coordinate axes. ${\ displaystyle \ |, \ perp}$${\ displaystyle \ |}$${\ displaystyle \ perp}$

The 1,2,3 indexing is common internationally. In English-speaking countries, however, a different indexing of the cross-contraction numbers is often used. Since the Poisson's numbers and are not identical, errors can quickly occur here. The Poisson's ratio is always the smaller of the two. To avoid confusion, it is called the minor poisson ratio in English . ${\ displaystyle \ nu _ {\ | \ perp}}$${\ displaystyle \ nu _ {\ perp \ |}}$${\ displaystyle \ nu _ {\ perp \ |}}$

### Calculation of the basic elasticity quantities

Polar diagram of the modulus of elasticity of a UD layer.

The basic elasticity of a UD layer is calculated from the elasticity of the fiber and matrix on a micromechanical basis. The elasticity values are included in the overall stiffness according to their fiber volume . A distinction is made between series connection and parallel connection of the stiffnesses.

The elastic parameters of the orthotropic material can be reduced to the moduli of elasticity and , shear modulus and and Poisson's ratio and due to the symmetry in the 2-3 plane . These can be divided into independent and dependent elasticity quantities. ${\ displaystyle E _ {\ |}}$${\ displaystyle E _ {\ perp}}$ ${\ displaystyle G _ {\ | \ perp}}$${\ displaystyle G _ {\ perp \ perp}}$ ${\ displaystyle \ nu _ {\ | \ perp}}$${\ displaystyle \ nu _ {\ perp \ perp}}$

The independent elasticity quantities are specific to the UD layer and cannot be calculated in any other way. Alternatively, they can be determined in experiments. However, this proves difficult, especially in the case of Poisson's numbers. Therefore, the calculation method using micromechanics is predominantly chosen. A number of corrections are necessary in order to take effects such as the increase in strain into account, but these are taken into account in micromechanics. Calculation methods can be found in the literature.

The dependent elasticity quantities are calculated from the independent elasticity quantities.

### Independent basic elasticity quantities

The following basic elasticity parameters are necessary to fully describe a UD layer.

• 2-dimensional law of elasticity: ${\ displaystyle E _ {\ |}, E _ {\ perp}, G _ {\ | \ perp}, \ nu _ {\ | \ perp}}$
• 3-dimensional law of elasticity: ${\ displaystyle E _ {\ |}, E _ {\ perp}, G _ {\ | \ perp}, \ nu _ {\ | \ perp}, \ nu _ {\ perp \ perp}}$

In contrast to isotropic materials, the shear modulus cannot be calculated from the Poisson's ratio. This shear modulus is therefore an independent variable. ${\ displaystyle G _ {\ | \ perp}}$

The modulus of elasticity in the direction parallel to the fibers is significantly greater than the modulus in the direction perpendicular to the fibers . This is because the parallel direction is dominated by the stiff fibers, while the perpendicular direction is matrix-dominated. ${\ displaystyle E _ {\ |}}$${\ displaystyle E _ {\ perp}}$

### Dependent basic elasticity quantities

The missing dependent elasticity values ​​can be calculated from the independent elasticity values. The shear modulus in the transverse isotropic level is calculated as the isotropic of: . The second Poisson's ratio is calculated from the Maxwell-Betti relationship . ${\ displaystyle G _ {\ perp \ perp} = {\ frac {E _ {\ perp}} {2 (1+ \ nu _ {\ perp \ perp})}}}$${\ displaystyle \ nu _ {\ perp \ |}}$ ${\ displaystyle \ nu _ {\ perp \ |} = \ nu _ {\ | \ perp} {\ frac {E _ {\ perp}} {E _ {\ |}}}}$

### Stiffness matrix

With the help of the dependent and independent elasticity quantities, the stiffness matrix can be set up for the plane stress drop.

${\ displaystyle {\ begin {bmatrix} \ displaystyle {\ frac {E _ {\ |}} {1- \ nu _ {\ | \ perp} \ nu _ {\ perp \ |}}} & \ displaystyle {\ frac {\ nu _ {\ perp \ |} E _ {\ perp}} {1- \ nu _ {\ | \ perp} \ nu _ {\ perp \ |}}} & 0 \\ [3ex] \ displaystyle {\ frac {\ nu _ {\ | \ perp} E _ {\ |}} {1- \ nu _ {\ | \ perp} \ nu _ {\ perp \ |}}} & \ displaystyle {\ frac {E _ {\ perp }} {1- \ nu _ {\ | \ perp} \ nu _ {\ perp \ |}}} & 0 \\ [3ex] 0 & 0 & G _ {\ perp \ |} \ end {bmatrix}}}$

The matrix for the spatial voltage drop is the same.

## transformation

Schematic representation of the strain-shift coupling with loading outside the orthotropic axes

With the help of the polar transformation , the stiffness matrix of the UD layer can be rotated around the 3-axis (axis normal to the layer plane). So z. B. the construction of a balanced angle connection is possible. The polar transformation by an angle leads to anisotropy in the UD layer and thus to the strain-shift coupling. ${\ displaystyle \ alpha \ neq {\ frac {\ pi} {2}}, \ pi, {\ frac {3 \ pi} {2}} \ dots}$

## Application in the finite element method (FEM)

The basic elasticity values ​​can be used in all common FEM programs to describe the orthotropic or anisotropic behavior of a UD layer. However, a sufficiently fine discretization in the direction of the layer thickness must be ensured.

## literature

• Helmut Schürmann: Constructing with fiber-plastic composites. 2nd edition Springer, 2007.