Non-probabilistic interval model-based system reliability assessment for composite laminates

Abstract

Composite materials have been widely applied to engineering fields by virtue of its superior mechanical properties. Therefore, the problem of composite laminates safety assessment has also been discussed. This paper develops the non-probabilistic interval model-based system reliability evaluation strategy based on the last ply failure criterion (LPF), which improves the rough reliability solution based on first ply failure criterion (FPF) for composite laminates, and overcomes the shortage of traditional LPF method based on the probabilistic reliability theory. Obviously, the probability density function (PDF) is difficult to be calculated due to the experimental cost and time. In the proposed method, considering the fiber failure and matrix failure (limit state functions) for composite laminates, the failure possibility of every single ply is evaluated by the non-probabilistic interval model method. Subsequently, the progressive damage method is combined with the branch-bound method (B&B) to search the significant failure paths of composite laminates, and then the whole system analysis process is completed based on LPF criterion. Finally, the system reliability model of composite laminates can be constructed by introducing non-probabilistic interval model. Furthermore, the correlation description for the non-probabilistic interval model among is applied to the failure modes, and the Cornell first-order bound theory is applied to achieve the system reliability of composite laminates. After the detailed analysis steps, the numerical example of laminated plate is presented to demonstrate the validity and reasonability of the developed methodology.

Appendix

In order to validate the accuracy and reasonability of the proposed method for system failure possibility of composite laminates, the numerical approach for system failure possibility based upon Monte Carlo simulation is discussed below.

Step 1:

Initialize the numbers of generated samples and failed samples as $N_{generated} = 1e6$$N_{failed}^{fiber}$ and $N_{failed}^{matrix}$, the $N_{failed}^{fiber}$ and $N_{failed}^{matrix}$ respectively represents fiber failure and matrix failure mode;

Step 2:

Substitute the sample data ${\text{x}}_{k} (N)$ into the two state functions Eq. 3 and conduct the response analysis, record sensitivity information of failure modes and restore the failed samples as follow rule:

• (i) if fiber failure mode happens, then $N_{failed}^{fiber} = N_{failed}^{fiber} + 1$, the fiber failure possibility is calculated as $P_{fiber} = \frac{{N_{failed}^{fiber} }}{{N_{generated} }}$;

• (ii) if matrix failure mode happens, then $N_{failed}^{matrix} = N_{failed}^{matrix} + 1$, the matrix failure possibility is calculated as $P_{matrix} = \frac{{N_{failed}^{matrix} }}{{N_{generated} }}$;

Step 3:

Compare the size of two failure modes and update the stiffness matrix based on the materials degradation criteria, further complete the search of main failure sequences with help of B&B method:

• (i) the max failure possibility is obtained $P_{\hbox{max} }$, therefore the $\max$th layer can be considered as destroyed firstly and the stiffness is deteriorated correspondingly, then go to step 1 and 2 for structure reanalysis;

• (ii) $P_{k}$ stands for each ply failure possibility, if the relation $P_{k} /P_{\hbox{max} } \ge 0.3$, the $k$th layer is record as branching point, which represents the $k$th layer can also be destroyed firstly, and then the stiffness is deteriorated correspondingly, then go back to step 1 and 2;

• (iii) above analysis procedure is end until all layers are destroyed or total stiffness matrix becomes singular $|\det {\text{K}}_{T} /\det {\text{K}}_{0} | \le \lambda \, (\lambda \to 0)$, and the main failure sequences are found.

Step 4:

According with sensitivity information obtained in step 2, calculate the correlation factor matrix by virtue of Eq. 16, the failure possibility of single main sequence $P_{f}^{k}$ is computed, and the system failure possibility $P_{f}$ is acquired with help of the weakest sequence