Last ply failure analysis

Background

The last ply failure analysis allows to perform a degradation analysis by successive Classical Laminate Theory calculations. Target is to evaluate the potential residual strength of a laminate after failure of individual layers.

Analysis process

The general procedure is shown in the flowchart below.

Based on this, the iterative analysis process is explained in the following. Hereby, the flowchart is referred by roman numerals given in green dots.

I - Input data

Aim of the last ply failure analysis is to determine the strength of a laminate after sequential failure of individual plies. Therefore, a laminate and a load case is required as input data. Additionally, optional settings can be defined, as explained in the following.

Required input

In order to perform a last ply failure analysis, a laminate has to be defined. For full functionality, a failure criterion allowing to distinguish between failure modes shall be used. A suggested option is the Failure Mode Concept (FMC) criterion.

Furthermore, a load case has to be defined by section forces and moments:

$$\left[\begin{array}{c} n_x \\\ n_y \\\ n_{xy} \\\ m_x \\\ m_y \\\ m_{xy} \end{array} \right] \ ,$$

which is assumed to define an ultimate load (UL).

Optional input

As optional inputs, the following parameters and settings can be defined:

• Ultimate load factor $j_{A}$ (default: 1.0),
• Degradation factor $f_{deg}$ (default: 1.0E-6),
• Maximum allowable strain $\varepsilon_{Allow}$ (default: 0.003),
• Option, if all stiffness values are degraded on fiber failure (default: true).

II - Iterative CLT analysis

Based on the laminate and load case defined, an iterative CLT analysis is peformed. For the first itertation ($i = 0$), the pristine laminate is assumed. For subsequent iterations ($i \geq 1$), the laminate may be degraded due to preceeding failure, as explained below.

III - Result of CLT analysis

From each CLT analysis $i$, the following results are used for the failure evaluation and the determination of resulting reserve factors:

• Minimum reserve factor $RF_{min,i}$ at the critical layer $j_{crit}$, determined as the minimum over all layers $j$ $\left(RF_{min,i} = RF_{j_{crit}} = \min\limits_{j} \left(RF_{j}\right)\right)$.
• Maximum absolute strain $\varepsilon_{Abs,i}$ determined as the maximum strain in fiber direction over all layers $j$, taking the top and bottom of the layer into account: $\varepsilon_{Abs,i} = \max\left(\left\vert \max\limits_{j} \left\vert \varepsilon_{\parallel,j}^{top} \right\vert \right\vert, \left\vert \max\limits_{j} \left\vert \varepsilon_{\parallel,j}^{bottom} \right\vert \right\vert\right)$.
• The failure type related to the minimum reserve factor $RF_{min,i}$.

Thereby, the minimum reserve factor is determined by looping over the layers starting at the first layer $j = 1$. Therefore, in case of several layers with identical reserve factors, the layer with the smaller layer number will usually be detected as the critical layer $j_{crit}$. This might be the case if a symmetric laminate is loaded in-plane only, unless numerical accuracy does not lead to slightly different values in terms of machine precision.

IV - Stopping criterion

The iterative CLT analysis is stopped on basis of the following stopping criteria.

1. Failure type at the critical layer $j_{crit}$ is fiber failure and the critical layer $j_{crit}$ has already been degraded due to fiber failure in previous iterations.
2. Failure type at the critical layer $j_{crit}$ is inter-fiber failure and the critical layer $j_{crit}$ has already been degraded due to inter-fiber failure in previous iterations.
3. Failure type at the critical layer $j_{crit}$ is general material failure and the critical layer $j_{crit}$ has already been degraded due to either fiber failure or inter-fiber failure in previous iterations.
4. The number of iterations is two times the number of layers.

If one of the stopping criteria is fulfilled, the last ply failure analysis is finished.

V - Strain reserve factor

The strain reserve factor $RF_{\varepsilon}$ indicates the reserve factor on basis of the maximum allowable strain and the results of the CLT analysis at the current iteration $i$. It is set at the first time $RF_{min,i} \geq 1$ is fulfilled and, once set, not overwritten in subsequent iterations:

$$RF_{\varepsilon} = \frac{\varepsilon_{Allow}}{\varepsilon_{Abs,i}} \ .$$

VI - Determination of failure type

Depending on the failure type, the stiffnesses of the critical layer $j_{crit}$ is degraded and the respective reserve factor is set.

If an overall failure criterion which does not allow to differentiate various failure types (e.g., Tsai-Wu) is used, general material failure is assumed as the failure type.

VII - Reserve factor on first fiber failure

At the first occurence of fiber failure, the reserve factor on first fiber failure is set:

$$RF_{first\_FF} = RF_{min,i} \ .$$

Once set, $RF_{first\_FF}$ is not updated in subsequent iterations.

VIII - Reserve factor on first inter-fiber failure

At the first occurence of inter-fiber failure, the reserve factor on first inter-fiber failure is set

$$RF_{first\_IFF} = RF_{min,i} \cdot j_{A} \ ,$$

taking the ultimate load factor $j_{A}$ into account. Assuming that the last ply failure analysis is performed for ultimate load (compare input data), the reserve factor is increased by the factor $j_{A}$ under the assumption, that inter-fiber failure may occure above limit load.

Once set, $RF_{first\_IFF}$ is not updated in subsequent iterations.

IX - Degradation of stiffnesses due to fiber failure

In case of fiber failure, the respective stiffness value of the layer in fiber direction is reduced by the degradation factor $f_{deg}$:

• Elastic modulus in fiber direction: $E_{\parallel}^{deg} = E_{\parallel} \cdot f_{deg}$.

If the option to degrade all stiffness values on fiber failure (see input data) is active, the stiffness values perpendicular to fiber direction also degraded:

• Elastic modulus perpendicular to fiber direction: $E_{\perp}^{deg} = E_{\perp} \cdot f_{deg}$,
• Shear modulus: $G_{\parallel \perp}^{deg} = G_{\parallel \perp} \cdot f_{deg}$.

The reduced stiffness value(s), marked by the superscript $^{deg}$ are then used for the respective layer in subsequent iterations.

X - Degradation of stiffnesses due to inter-fiber failure

In case of inter-fiber failure, the respective stiffness values of the layer perpendicular to the fiber direction are reduced by the degradation factor $f_{deg}$:

• Elastic modulus perpendicular to fiber direction: $E_{\perp}^{deg} = E_{\perp} \cdot f_{deg}$,
• Shear modulus: $G_{\parallel \perp}^{deg} = G_{\parallel \perp} \cdot f_{deg}$.

The reduced stiffness value(s), marked by the superscript $^{deg}$ are then used for the respective layer in subsequent iterations.

XI - Degradation of stiffnesses due to general material failure

If an overall failure criterion which does not allow to differentiate various failure types (e.g., Tsai-Wu) is used, general material failure is assumed as the failure type. In this case, all respective stiffness values of the layer are reduced by the degradation factor $f_{deg}$:

• Elastic modulus in fiber direction: $E_{\parallel}^{deg} = E_{\parallel} \cdot f_{deg}$,
• Elastic modulus perpendicular to fiber direction: $E_{\perp}^{deg} = E_{\perp} \cdot f_{deg}$,
• Shear modulus: $G_{\parallel \perp}^{deg} = G_{\parallel \perp} \cdot f_{deg}$.

The reduced stiffness value(s), marked by the superscript $^{deg}$ are then used for the respective layer in subsequent iterations.

XII - Exceedance factor

The exceedance factor is determined as the maximum value of all minimum reserve factors $RF_{min,i}$ over all iterations:

$$EF_{LPF} = \max\limits_{i} \left( RF_{min,i} \right) \ .$$

XIII - End of last ply failure analysis

After reaching one of the stopping critera, the last ply analysis is finished.