Out-of-plane laminate shear stiffness

II.1.6.6 Out-of-plane laminate shear stiffness

A linear relation between out-of-plane shear components of the strain tensor and the corresponding components of the Cauchy stress tensor is assumed:

{\begin{matrix} γ_{x z} (z) \\ γ_{y z} (z) \end{matrix}}_{lam} = {[g]}_{lam}^{- 1} (z) \cdot {\begin{matrix} τ_{x z} (z) \\ τ_{y z} (z) \end{matrix}}_{lam} .

To this relation should correspond a relation between the average out-of-plane shear strains and the out-of-plane shear force:

{\begin{matrix} Γ_{x z} \\ Γ_{y z} \end{matrix}}_{lam} = {[G]}_{lam}^{- 1} \cdot {\begin{matrix} Q_{x z} \\ Q_{y z} \end{matrix}}_{lam} .

(II.1.53)

This section attempts to justify the calculation of the ${[G]}_{lam}^{- 1}$ matrix in the previous expression.

II.1.6.6.1 Limitations of shear stiffness calculation

By writing equation (II.1.53), one implicitly makes the assumption that there is a linear relation between the average out-of-plane shear strain tensor components and the out-of-plane shear forces tensor components. We have seen however in section II.1.6.5 that the calculation of laminate out-of-plane shear stress tensor in out-of-plane shear forces loading axes leads to a non-linear dependence of shear stresses on shear forces. A consequence of this non-linear dependence is that expression (II.1.53) is not valid if laminate out-of-plane shear equilibrium equations are solved in shear loading axes.

The developments in current section II.1.6.6 assume that the $(μ_{x}, μ_{y})$ approach of section II.1.6.5 is adopted for stiffness calculations. This does not prevent us from using the “out-of-plane shear forces loading axes” approach for the calculation of ply out-of-plane shear stresses however:

We have seen in section II.1.6.5 that the $(μ_{x}, μ_{y})$ approach as well as “out-of-plane shear forces loading axes” approach are both inaccurate. Calculations mixing the two approaches loses some internal consistency, but does not lead to an increase of this inaccuracy.
Finite element solvers have adopted the $(μ_{x}, μ_{y})$ approach for the calculation of laminate out-of-plane shear stiffness properties. It is no big surprise, because the calculation of laminate stiffnesses is done in solver’s code to estimate shell element properties before any information on the loading is available.
An advantage of the $(μ_{x}, μ_{y})$ approach for shear stiffness calculation is that it simplifies considerably governing equations.
Also, the $(μ_{x}, μ_{y})$ approach allows calculation of the laminate shear stiffness properties only once before load response calculations. This leads to a considerable reduction of CPU usage. The out-of-plane shear stiffness tensor is calculated with respect to the laminate coordinate system. Its components can then easily be obtained in any other coordinate systems by performing simple rotations.

As the purpose of FeResPost and of its composite classes is to post-process finite element results, it makes sense to adopt approaches similar to those of other finite element solvers. Note, however, that other composite calculation software, like ESAComp, sometimes adopt different approaches.

II.1.6.6.2 Introduction of new notations

In the definition of loadings, the out-of-plane components of shear force $Q$ can be replaced by average out-of-plane shear stress $\overset{�}{τ}$ . Then the conversion between these two types of components is done simply by multiplication or division by laminate total thickness $t$ :

\overset{�}{τ} = \frac{Q}{t} .

Notations are introduced that simplify the writing of equations:

{Γ_{s}}_{*} = {\begin{matrix} Γ_{x z} \\ Γ_{y z} \end{matrix}}_{*},

{Q_{s}}_{*} = {\begin{matrix} Q_{x z} \\ Q_{y z} \end{matrix}}_{*},

{\overset{�}{τ_{s}}}_{*} = {\begin{matrix} {\overset{�}{τ}}_{x z} \\ {\overset{�}{τ}}_{y z} \end{matrix}}_{*} .

In these expressions the $*$ subscripts can be replaced by a symbol specific to the coordinate system in which the components of the vector are expressed (for example "load", "ply", "lam"...).

II.1.6.6.3 Energetic approach

The components of matrix ${[g]}_{lam} (z)$ are easily obtained from the orientation and material of plies. The components of ${[G]}_{lam}$ are obtained by a calculation of out-of-plane shear strain surface energy. An estimate of this surface energy is first calculated using the local expression of shear strains:

\begin{align} W_{shear}^{local} & = \frac{1}{2} \int_{z_{0}}^{z_{N}} {γ_{s} (z)} T_{lam} \cdot {τ_{s} (z)}_{lam} d z, \\ = \frac{1}{2} \int_{z_{0}}^{z_{N}} {τ_{s} (z)} T_{lam} \cdot {[g]}_{lam}^{- 1} (z) \cdot {τ_{s} (z)}_{lam} d z, \\ = \frac{1}{2} \int_{z_{0}}^{z_{N}} {Q_{s}} T_{lam} \cdot [Y] T_{lam} (z) \cdot {[g]}_{lam}^{- 1} (z) \cdot {[Y]}_{lam} (z) \cdot {Q_{s}}_{lam} d z, \\ = \frac{1}{2} {Q_{s}} T_{lam} \cdot {[\int_{z_{0}}^{z_{N}} [Y] T_{lam} (z) \cdot {[g]}_{lam}^{- 1} (z) \cdot {[Y]}_{lam} (z) d z]}_{lam} \cdot {Q_{s}}_{lam} . & (II.1.54) \end{align}

Note that we have used the ${[Y]}_{lam} (z)$ introduced by equation (II.1.48) in section II.1.6.5. This is possible only because we use the $(μ_{x}, μ_{y})$ approach.

Surface energy can also be estimated from the out-of-plane shear global equation:

\begin{align} W_{shear}^{global} & = \frac{1}{2} {Γ_{s} (z)} T_{lam} \cdot {Q_{s}}_{lam}, \\ = \frac{1}{2} {Q_{s}} T_{lam} \cdot {[G]}_{lam}^{- 1} \cdot {Q_{s}}_{lam} . & (II.1.55) \end{align}

Then, as there is only one surface energy, $W_{shear}^{global} = W_{shear}^{local}$ and one should have

\begin{align} {[G]}_{lam}^{- 1} & = \int_{z_{0}}^{z_{N}} [Y] T_{lam} (z) \cdot {[g]}_{lam}^{- 1} (z) \cdot {[Y]}_{lam} (z) d z, \\ = \int_{z_{0}}^{z_{N}} {[U]}_{lam} (z) d z . \end{align}

Here again, the integration can be calculated ply-by-ply. More precisely, the following is computed on ply $k$ :

\begin{align} {[U^{k}]}_{lam} (z) & = [Y^{k}] T_{lam} (z) \cdot {[g^{k}]}_{lam}^{- 1} \cdot {[Y^{k}]}_{lam} (z), \\ = {[U_{0}^{k}]}_{lam} + {[U_{1}^{k}]}_{lam} z + {[U_{2}^{k}]}_{lam} z^{2} + {[U_{3}^{k}]}_{lam} z^{3} + {[U_{4}^{k}]}_{lam} z^{4} . \end{align}

where

\begin{align} {[U_{0}^{k}]}_{lam} & = [Y_{0}^{k}] T_{lam} \cdot {[g^{k}]}_{lam}^{- 1} \cdot {[Y_{0}^{k}]}_{lam}, \\ {[U_{1}^{k}]}_{lam} & = [Y_{0}^{k}] T_{lam} \cdot {[g^{k}]}_{lam}^{- 1} \cdot {[Y_{1}^{k}]}_{lam} + [Y_{1}^{k}] T_{lam} \cdot {[g^{k}]}_{lam}^{- 1} \cdot {[Y_{0}^{k}]}_{lam}, \\ {[U_{2}^{k}]}_{lam} & = [Y_{0}^{k}] T_{lam} \cdot {[g^{k}]}_{lam}^{- 1} \cdot {[Y_{2}^{k}]}_{lam} + [Y_{1}^{k}] T_{lam} \cdot {[g^{k}]}_{lam}^{- 1} \cdot {[Y_{1}^{k}]}_{lam} + [Y_{2}^{k}] T_{lam} \cdot {[g^{k}]}_{lam}^{- 1} \cdot {[Y_{0}^{k}]}_{lam}, \\ {[U_{3}^{k}]}_{lam} & = [Y_{1}^{k}] T_{lam} \cdot {[g^{k}]}_{lam}^{- 1} \cdot {[Y_{2}^{k}]}_{lam} + [Y_{2}^{k}] T_{lam} \cdot {[g^{k}]}_{lam}^{- 1} \cdot {[Y_{1}^{k}]}_{lam}, \\ {[U_{4}^{k}]}_{lam} & = [Y_{2}^{k}] T_{lam} \cdot {[g^{k}]}_{lam}^{- 1} \cdot {[Y_{2}^{k}]}_{lam} . \end{align}

Then the integral above develops as follows:

\begin{align} {[G]}_{lam}^{- 1} & = \int_{z_{0}}^{z_{N}} {[U]}_{lam} (z) d z, \\ = \sum_{k = 1}^{N} \int_{z_{k - 1}}^{z_{k}} {[U^{k}]}_{lam} (z) d z, \\ = \sum_{k = 1}^{N} \int_{z_{k - 1}}^{z_{k}} (\sum_{i = 0}^{4} {[U_{i}^{k}]}_{lam} z^{i}) d z, \\ = \sum_{k = 1}^{N} (\sum_{i = 0}^{4} {[U_{i}^{k}]}_{lam} \int_{z_{k - 1}}^{z_{k}} z^{i} d z), \\ = \sum_{k = 1}^{N} \sum_{i = 0}^{4} ({[U_{i}^{k}]}_{lam} \frac{z_{k}^{i + 1} - z_{k - 1}^{i + 1}}{i + 1}) . \end{align}

The stiffness matrix ${[G]}_{lam}$ and the compliance matrix ${[G^{- 1}]}_{lam}$ are defined as above. Note that once the laminate out-of-plane shear stiffness and compliance matrices are known, the laminate out-of-plane shear equivalent moduli are calculated from the components of the compliance matrix with the following expressions:

G_{x z} = \frac{1}{t \cdot {[G^{- 1}]}_{11}},

G_{y z} = \frac{1}{t \cdot {[G^{- 1}]}_{22}},

in which the ${[G^{- 1}]}_{lam}$ matrix has first been rotated into the appropriate axes.

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