Probabilistic Finite Element Simulation of Sheet Molding Compound Composites with Multimodal Weibull Distribution

Sheet Molding Compound (SMC) is a type of ready to mold composites material. The most common SMC consists of glass fiber bundles about one inch long distributed randomly in a B-stage polyester resin. SMC possesses good mechanical properties and manufacturing flexibility in forming complex shaped parts and is relatively low cost, making it one of the attractive choices to replace metallic parts in automotive industry. Nevertheless, SMC composites have not been utilized in critical automobile components owing to the lack of a satisfactory predictive model, especially for crashworthiness simulations. The main challenges in analysis of SMC structures are the large scatter in mechanical properties and the large difference in strengths under different stress distribution or loading conditions. For example, SMC demonstrates 1.5-2 times higher strength under 3-point (3-pt) bending in comparison to uniaxial tension strength. This phenomenon is known as the size effect on strength and can be explained by Weibull's statistical strength theory. For materials with large size effect such as SMC, simulations carried out with the mean mechanical properties (i.e., tension, compression, and shear data) would result in a significant underprediction of flexural responses of the structure. To improve the predictions, the statistical distributions of the mechanical properties need to be considered and the size effect should be examined. Although statistical analysis has long been considered in composite designs, probabilistic finite element (PFE) analysis based on statistical strength models has also been employed to consider uncertainties and design reliability at every scale in composites, little work has been done to examine the size effect of strength in FE simulations. This work aims to incorporate the size effect in probabilistic simulations of SMC composite structures. First, we extended the unimodal Weibull strength model into multimodal one by combining the tensile and flexural Weibull strength models. This approach was examined with a glass fiber SMC composite. A randomization algorithm was developed to incorporate the strength distribution in PFE models. The strength distribution model was discretized into a limited number of segments and the values of the average strength for each segment and their probabilities were determined. The strength values were then randomly assigned across the integration points in the PFE model according to their probabilities. This approach successfully reproduced the tensile and flexural responses with the mean peak load, post peak behavior, and energy absorption similar to experimental results within ten iterations. Next, in addition to the tensile strength, the statistical distributions of the elastic modulus and compressive strength were also considered. The tensile strength and compressive strength were modeled by bimodal Weibull strength distributions corresponding to the uniaxial and 3-pt bending experiments. To determine the mixture weight fraction of the bimodal models and some difficult to measure parameters in the damage mechanics based composite material model, model optimization was explored using two techniques: (1) Artificial Neural Network (ANN)-based machine learning (ML) and (2) Random Search. It was observed that although computationally inexpensive, ANN-ML was rather complicated for a general-purpose regression. On the other hand, RS is easy to implement. Its high computational cost is acceptable as the optimization has to be done only once for any specific material model. The PFE models optimized with RS were examined with four verification cases including tensile, compression, 3-pt and 4-pt bending, and three validation cases including open hole tension, disk bending with a fixed boundary and with a simply supported boundary conditions. The PFE predictions agreed well with the experimental results across these load cases.