Classic laminate theory

The classical laminate theory is a method for calculating the pane and plate stiffness as well as the stresses of a flat multilayer composite. The layers of the composite usually consist of orthotropic fiber-plastic composites or unidirectional layers . However, isotropic layers can also be treated. The calculation of the coupling between disk loads and disk deformations is of particular importance. If there is a coupling, a component z. B. bend under tension, one speaks colloquially of delay.

The classic laminate theory is the basis of a large number of calculation programs for fiber-reinforced plastics. For a strength calculation, it is essential to determine the layer stresses of a multi-layer composite according to the classical laminate theory.

The classical laminate theory is also abbreviated as CLT in German-speaking countries, based on the English term classical laminate theory . There is also the term multilayer theory .

Assumptions

The classical laminate theory is largely based on Kirchhoff's plate theory . It only applies to an infinitesimal undisturbed section. So no laminate edge effects or load application problems are taken into account. In detail, the following assumptions apply:

• The law of elasticity of the individual layers is ideally linearly elastic.
• The laminate is thin (thickness is small compared to the rest of the dimensions) ${\displaystyle t}$
• The laminate thickness is constant ${\displaystyle t}$
• The first order theory is valid (small deformations)
• The Bernoulli assumptions are valid (flat cross-sections, rigid in the direction of the thickness)
• The state of stress is flat due to the thin walls ( ). ${\displaystyle \sigma _{z}=\tau _{zx}=\tau _{zy}=0}$
• The layers are ideally glued together.
• The laminate lies in the plane. ${\displaystyle x,y}$

Calculation process

Before the calculation, the following sizes of the layers must be determined:

• Stiffness matrix of each UD layer (law of elasticity of the unidirectional layer ) ${\displaystyle Q_{ij}}$
• Layer angle of the unidirectional layers. In the case of isotropic layers, the layer angle is arbitrary. ${\displaystyle \alpha }$
• Layer thicknesses ${\displaystyle t}$
• Shift order

The designer tries to select the above sizes in such a way that the external loads on the laminate result in the most favorable possible stress on each UD layer. Furthermore, he must ensure that he meets the stiffness requirements. The following calculation process is therefore often run through iteratively.

1. Transformation of the disk stiffness matrices of the UD layers into the global system ${\displaystyle N}$ ${\displaystyle x,y}$ ${\displaystyle Q_{k}\rightarrow {\hat {Q}}_{k}}$
2. Calculation of the pane stiffness matrix , plate stiffness matrix and coupling stiffness matrix ${\displaystyle A}$ ${\displaystyle D}$ ${\displaystyle B}$
3. Assembling the disc-plate stiffness matrix
4. Invert the disk-plate stiffness matrix
5. Calculate the global strains and curvatures ${\displaystyle {\hat {\epsilon }}}$ ${\displaystyle {\hat {\kappa }}}$
6. Transforming the global strain into the strain of each UD layer in the layer coordinate system
7. Calculation of the stresses in each UD layer in the layer coordinate system with the help of the disk stiffness matrix of the UD layer ${\displaystyle Q_{ij}}$
8. This is usually followed by a strength analysis using fracture criteria for fiber-reinforced plastics .

As a by-product, the engineering constants of the layered composite are obtained from the inverse of the disk-plate stiffness matrix.

External loads

The classical laminate theory does not calculate with tensions as external loads, but with their flows. A force or moment flow is a size-related quantity.

In the case of isotropic panes and plates, pane loads only lead to pane deformations (expansion and displacement). In the case of layered laminates, pane loads can also lead to sheet deformation (curvatures and twists). Therefore, the following distinction between disk and plate loads is necessary.

Disk loads

Slice loads are normal and shear stresses in the laminate plane or its flows. corresponds to the coordinate direction normal to the laminate plane. ${\displaystyle z}$

${\displaystyle {\begin{bmatrix}n_{x}\\n_{y}\\n_{xy}\end{bmatrix}}=\int _{t}^{}{\begin{bmatrix}\sigma _{x}\\\sigma _{y}\\\tau _{xy}\end{bmatrix}}\,\mathrm {d} z}$

Disk loads

The plate loads consist of the bending moment flows and the torsional moment flow. The torsional moment can be interpreted as a torsional moment.

${\displaystyle {\begin{bmatrix}m_{x}\\m_{y}\\m_{xy}\end{bmatrix}}=-\int _{t}^{}{\begin{bmatrix}\sigma _{x}\\\sigma _{y}\\\tau _{xy}\end{bmatrix}}\cdot z\,\mathrm {d} z}$

Disc-plate stiffness matrix

The disk-plate stiffness matrix describes the elastic behavior of the entire laminate. It is composed of three sub-matrices , the disc stiffness matrix A , the plate stiffness matrix D and the coupling matrix B , which couples the first two matrices.

Disc stiffness matrix A _ij

Schematic representation of the strain-displacement coupling through the or terms ${\displaystyle A_{16}}$ ${\displaystyle A_{26}}$

The pane stiffness matrix results from the parallel connection of the pane stiffnesses of all individual layers ( unidirectional layer ), weighting the layer thickness of the individual layer. ${\displaystyle {\hat {Q}}_{ij}}$ ${\displaystyle N}$

${\displaystyle A_{ij}=\sum _{k=1}^{N}{\hat {Q}}_{ij,k}\cdot t_{k}}$

Plate stiffness matrix D _ij

Schematic representation of the bending-drill coupling through the or terms ${\displaystyle D_{16}}$ ${\displaystyle D_{26}}$

The plate stiffness matrix results from the parallel connection of the bending stiffnesses of all individual layers ( unidirectional layer ) plus their Steiner component .

${\displaystyle D_{ij}=\sum _{k=1}^{N}{\hat {Q}}_{ij,k}\cdot t_{k}\cdot \left(\underbrace {\frac {t_{k}^{2}}{12}} _{\text{Biegesteifigkeit}}+\underbrace {\left(z_{k}-{\frac {t_{k}}{2}}\right)^{2}} _{\text{Steiner Anteil}}\right)}$

Coupling stiffness matrix B _ij

Schematic representation of the strain-curvature coupling through the or terms ${\displaystyle B_{11},B_{12}}$ ${\displaystyle B_{22}}$

The coupling stiffness matrix is ​​composed of the pane stiffnesses of the individual layers, weighted with their static moment. This means that for symmetrically layered laminates the coupling between pane and plate disappears.
The coupling between expansion and curvature is used in bimetal switches .

${\displaystyle B_{ij}=-\sum _{k=1}^{N}{\hat {Q}}_{ij,k}\cdot t_{k}\cdot \underbrace {\left(z_{k}-{\frac {t_{k}}{2}}\right)} _{\text{stat. Moment}}}$

interpretation

Put together, the law of elasticity of the disk-plate element, based on the neutral plane (index 0), is written as

${\displaystyle {\begin{bmatrix}n_{x}\\n_{y}\\n_{xy}\\m_{x}\\m_{y}\\m_{xy}\end{bmatrix}}={\begin{bmatrix}A_{11}&A_{12}&A_{16}&B_{11}&B_{12}&B_{16}\\A_{12}&A_{22}&A_{26}&B_{12}&B_{22}&B_{26}\\A_{16}&A_{26}&A_{66}&B_{16}&B_{26}&B_{66}\\B_{11}&B_{12}&B_{16}&D_{11}&D_{12}&D_{16}\\B_{12}&B_{22}&B_{16}&D_{12}&D_{22}&D_{26}\\B_{16}&B_{26}&B_{66}&D_{16}&D_{26}&D_{66}\\\end{bmatrix}}{\begin{bmatrix}\epsilon _{x}\\\epsilon _{y}\\\gamma _{xy}\\\kappa _{x}\\\kappa _{y}\\\kappa _{xy}\end{bmatrix}}_{0}}$

Selected laminates and their properties of the law of elasticity :

• UD layer : orthotropic as a disk (), orthotropic as a plate (), no disk-to-plate coupling () ${\displaystyle A_{16},A_{26}=0}$ ${\displaystyle D_{16}=D_{26}=0}$ ${\displaystyle B_{ij}=0}$
• UD layer outside the plane of symmetry : anisotropic as a disk ( ), anisotropic as a plate ( ), disk-to-plate coupling ( ) ${\displaystyle A_{16},A_{26}\neq 0}$ ${\displaystyle D_{16},D_{26}\neq 0}$ ${\displaystyle B_{ij}\neq 0}$
• balanced angle connection or cross connection : orthotropic as disc (), orthotropic as plate (), disc-plate coupling () ${\displaystyle A_{16},A_{26}=0}$ ${\displaystyle D_{16},D_{26}=0}$ ${\displaystyle B_{11},B_{22}\neq 0}$
• Balanced angle or cross connection , symmetrically layered : orthotropically as a disc (), anisotropic as a plate (), no disc-plate coupling () ${\displaystyle A_{16},A_{26}=0}$ ${\displaystyle D_{16},D_{26}\neq 0}$ ${\displaystyle B_{ij}=0}$

The plate orthotropy is also called arching orthotropy due to the lack of a torsion-twist coupling .

Engineering constants

The engineering constants are obtained from the inverse disk-plate stiffness matrix. represents the layer thickness of the laminate. ${\displaystyle {\hat {E}}_{x},{\hat {E}}_{y},{\hat {G}}_{xy},{\hat {\nu }}_{xy}}$ ${\displaystyle t_{ges}}$ ${\displaystyle {\begin{bmatrix}{\bar {A}}&{\bar {B}}\\{\bar {B}}^{T}&{\bar {D}}\end{bmatrix}}={\begin{bmatrix}A&B\\B&D\end{bmatrix}}^{-1}}$

• ${\displaystyle {\hat {E}}_{x}={\frac {1}{{\bar {A}}_{11}t_{ges}}}}$

• ${\displaystyle {\hat {E}}_{y}={\frac {1}{{\bar {A}}_{22}t_{ges}}}}$

• ${\displaystyle {\hat {G}}_{xy}={\frac {1}{{\bar {A}}_{66}t_{ges}}}}$

• ${\displaystyle {\hat {\nu }}_{yx}=-{\frac {{\bar {A}}_{12}}{{\bar {A}}_{11}}}}$

Limits of Theory

The limits of validity of the CLT result mainly from the assumptions of Kirchhoff's plate theory (see: Assumptions). The quality of the solution decreases for thick and compact panels. Even very flexible layers under shear loads should not be calculated with the CLT. The reason is the increasing share of the shear reduction compared to the bending reduction of the plate.

When the pane is only loaded, the pane thickness does not affect the quality of the solution. In the case of very thick individual layers, there may be deviations, since the deformation caused by the interlaminar shear is not taken into account when the pane is loaded. In practice, however, this is of little importance, since laminates usually consist of thin individual layers.

Application in the finite element method

The engineering constants can be used directly in the finite element method in combination with volume elements in order to simulate the global behavior of a layered composite. If the layer stresses are of interest, they must be calculated from the global strains, as in the manual CLT calculation. The stresses calculated in the FEM represent the global stresses of the bond and do not correspond to the layer stresses. The global stress must not be transformed directly into the layer stress. However, modern FEM programs are also able to output the layer stresses with suitable modeling. Reference is made here to the modeling of a sandwich panel bending test using the FEM program Patran / Nastran. The sandwich panel consists of a total of 5 layers, 2 cover layers each ( prepreg layer 1 and layer 2), the core layer (honeycomb core) and then again two cover layers of prepreg (the prepreg layers can be viewed as a laminate). Depending on the type of modeling, the model can be designed as a laminate variant. For this purpose, the 2-d orthotropic element properties are simply assigned to the shell elements (shell elements) using the composite tool. In the result output, both the stress and the strain of the individual layer can then be displayed. When formulating the engineering constants using 3-dimensional volume elements, neither sliding-stretching couplings (anisotropy as a disk) nor bending-stretching couplings are taken into account. The anisotropy as a plate cannot be mapped either. There are special shell elements that can take these couplings into account.

In the case of symmetrically layered, balanced laminates ( orthotropy ), which is common in practice, modeling using volume elements and engineering constants is permitted.

Calculation programs

There are a number of CLT programs, some of which are available free of charge. They are mostly based on spreadsheet programs. A database with semi-finished fiber products and matrix systems is partially connected.

• Lami Cens (free of charge) [1]
• ESAComp (commercial)
• Compositor (commercial)
• AlfaLam (linear material law), AlfaLam.nl (non-linear material law), both programs are free of charge [2]
• eLamX 2.6 (Java, free of charge) [3]
• RF-LAMINATE 5.xx (commercial) [4]

...

literature

• J. Wiedemann: Lightweight Construction, Volume 1: Elements . Springer-Verlag, Berlin 1986. ISBN 3-5-40164049
• H. Altenbach, J. Altenbach, R. Rikards: Introduction to the mechanics of laminate and sandwich structures . German publishing house for basic industry, 1996. ISBN 3-3-42006811
• H. Altenbach, J. Altenbach, W. Kissing: Mechanics of Composite Structural Elements (2nd ed.). Springer, Singapore, 2018. ISBN 978-981-10-8934-3

Individual evidence

1. ↑ Manfred Flemming, Siegfried Roth: fiber composite construction properties . mechanical, constructive, thermal, electrical, ecological, economic aspects. Springer-Verlag, Berlin Heidelberg 2003, ISBN 978-3-642-55468-1 , Theory for the calculation of thin-walled laminates, p. 47 .