A methodology to obtain material design allowables from high-fidelity compression after impact simulations on composite laminates

Abstract

Aeronautical industries address the structural reliability of designs by defining design allowables that account for any uncertainties. The Composite Materials Handbook-17 proposes A/B-basis values as design allowables. In this study, a new methodology to estimate the design allowables of the Compression After Impact (CAI) strength is presented. The CAI strength is predicted with high-fidelity simulations using finite element models featuring in-house constitutive damage models. The uncertainty associated to parameters of the model is defined and propagated to obtain the CAI strength distribution. To efficiently estimate this distribution, a Monte Carlo simulation is carried out employing a response surface previously calibrated with a reduced number of high-fidelity simulations. The A/B-basis values for the CAI strength are estimated from the strength distribution obtained and then compared with experimental results. The methodology proposed allows to reduce the number of experimental tests associated with generating design allowables, thus leading to an optimised cost-effective design.

Introduction

The design allowables most widely used in the aeronautic sector are the A/B-basis values, which are defined as 95% of the lower one-sided confidence bound of the 1st and 10th percentiles of the population measured [1], respectively. Consequently, if the load or stress in the part is greater than the allowable design value, then the design criteria is not fulfilled. Current industry practice uses experimental tests that generally adhere to the Composites Materials Handbook-17 (CMH-17 approach) [1] to determine design allowables (as in Laurin et al. [2]). According to the CMH-17 approach, design allowables are approximated by using the localisation and scale parameters of the results (i.e. mean value and standard deviation, respectively, in a normal distribution) and a coefficient factor according to the distribution of the results and the number of samples. To obtain accurate design allowables, the population distribution of the parameter must be measured, which implies a large number of laboratory tests and, therefore, the associated increase in time and economic costs.

Alternatively, the basis values can be obtained using advanced models and an appropriate methodology to quantify and manage uncertainty. The uncertainty associated to the analysis method can be grouped into two types: (i) uncertainty associated to the repeatability of the model or (ii) uncertainty due to the intrinsic variability of the parameters of the model. The uncertainty associated to the repeatability of the model does not apply to this study because the prediction from the high-fidelity model will only change if the parameters of the model are changed. Hence, the variability in the output results is caused by the uncertainty associated to the input parameters. This uncertainty can derive from different sources such as the manufacturing process, batch-to-batch variability of raw materials, the test method used to characterize the material, or the intrinsic variability of the material [1]. In addition, it can also be uncertainty associated to the boundary conditions (e.g. a misalignment in the applied load) and to the geometry of the specimen.

Therefore, to determine the design allowable values from high-fidelity simulations, the uncertainty associated to the different input parameters must be quantified. This uncertainty is propagated into the model to obtain the population of the output results. Further, the design allowables are estimated using a proper statistical analysis.

One way to propagate the uncertainty is to use the Stochastic Finite Element Method (SFEM) [3], which is based on the random variation of the input parameters of the model according to their distribution. Meanwhile, the easy-to-implement Crude Monte Carlo Simulations (CMCS) makes it a widely-used method for uncertainty propagation analysis. Vallmajó et al. [4] proposed a new methodology for estimating the B-basis value of notched composite laminates by means of CMCS using an analytical framework. The authors demonstrated that, for large sample sizes ($>{10}^4$), the B-value can be estimated to the 10th percentile. However, as the CMCS requires a large number of results, this is not feasible for simulations that require high computational times (e.g. FE analysis), thus, more economic methods, in terms of computational time, need to be employed.

The First Order Reliability Method (FORM) and the Second Order Reliability Method (SORM) are used in some applications to approximate the probability of a function with random input parameters. They are based on a Taylor series assuming that the output results follow a normal distribution and, therefore, require a small number of simulations. Gosling et al. [5] presented a methodology to estimate the reliability of a complex shear-deformable composite laminate using FORM, while Delbariani-Nejad et al. [6] studied the reliability of the delamination growth under mode I, mode II and mixed mode in composite laminates applying the FORM and SORM methods. According to their results, FORM provides a good balance between accuracy and economic cost in terms of computational time. Hussein at el. [7] used the First Order Second Moment (FOSM) method to maximize plate stiffness using the minimum carbon reinforcement polymer volume fraction in a plate under uniform pressure loading. The authors concluded, however, that these methods are not suitable when the model has non-linearities.

Nowadays, a large number of numerical works addresses the simulation of Low-Velocity Impact (LVI) and Compression After Impact (CAI) tests on composite structures [8], [9], [10], [11], [12], [13], [14], [15], [16], [17]. The interest in simulating the CAI test has grown considerably, as CAI strength is a design-driver for some aeronautical components. The prediction of CAI strength is quite complex and challenging as it is based on the previous impact simulation and involves complex contact interactions and progressive material degradation and the interaction of several failure mechanisms. For airworthiness certification, the analysis must be supported by test. For this study, the analysis of the CAI strength is performed in sub-element-level and, hence, the input parameters that feed the model have been experimentally tested.

English et al. [18] used the SFEM approach to simulate an LVI test on a laminate. The authors used the Latin Hypercube Sampling (LHS) technique to define an input test matrix. Afterwards, the results from the FE simulations were compared with experimental data to adjust and validate the FE model. Patel et al. [19], [20] performed a probabilistic analysis using a Gaussian response surface method in an LVI test by SFEM. The authors estimated the probability of the failure criteria for the matrix cracking and the delamination with different impacted energies by taking into account the uncertainty of the material properties. A sensitivity analysis was performed to determine which input parameters had a greater influence on the probability of failure.

Despite the large amount of published works, there is hardly any work that addresses determining design allowables directly from damage tolerance simulations using advanced constitutive models. This work proposes a cost-effective methodology to estimate the A/B-basis values for the CAI strength of laminated composites using high-fidelity FE simulations and statistical analysis. The main objective of this study is to present the methodology in detail, followed by the accurate design allowables. Nevertheless, the authors would like to remark that the methodology presented in this study to obtain the design allowables of the LVI & CAI test can reduce the number of tests to be performed but, it will not ever completely replace the experimental tests. Details of the modelling approach used to simulate the LVI and CAI experimental tests performed by Airbus and the post-processing of the results are presented in Section 2. Section 3 details the methodology used to obtain the A/B-basis values, while the results and discussion are presented in Section 4. All the data shown in this study is normalized, due to confidentiality and data rights from Airbus. The paper ends with concluding remarks in Section 5.