Elsevier

Composites Part B: Engineering

Volume 123, 15 August 2017, Pages 136-147
Composites Part B: Engineering

Effect of matrix voids, fibre misalignment and thickness variation on multi-objective robust optimization of carbon/glass fibre-reinforced hybrid composites under flexural loading

https://doi.org/10.1016/j.compositesb.2017.05.022Get rights and content

Abstract

The robust and optimal design of carbon/glass fibre-reinforced epoxy hybrid composite laminates under the constraint of a specified minimum flexural strength was investigated in this study. Two conflicting objectives, minimizing material cost and weight, were considered with the design variables being fibre type, fibre orientation angle and fibre volume fraction of the laminas. Three sources of uncertainties, namely, fibre misalignment, lamina thickness variation and the presence of matrix voids were incorporated into the model. This multi-objective robust optimization problem was solved by combining a modified version of the non-dominated sorting genetic algorithm (NSGA-II) with a simple genetic algorithm (GA) as an anti-optimizer. Pareto optimal and robust solutions were found for different levels of minimum flexural strength and the significance of each uncertainty source on the optimal cost and weight of the optimal designs were investigated by conducting analysis of variance (ANOVA) tests. The results indicated that, in general, all three uncertainties affected the cost and weight of the optimal designs with the effect of voids being more critical for void contents of greater than 2%.

Introduction

Advanced composite materials have successfully replaced metals in many applications that require a combination of high strength and low density, particularly in the fields of aerospace, marine and automotive engineering [1]. Among the different types of composite materials available, hybrid composites such as those based on carbon/glass fibre-reinforced epoxy, have become increasingly popular due to their ability to exploit the properties of multiple fibre types. However, complexities in the design and manufacture of such hybrid composite materials have thus far limited their widespread application.

Difficulties related with the manufacture of carbon/glass fibre-reinforced epoxy laminated composites are mainly associated with the existence of defects such as voids, resin rich regions and fibre misalignment. These defects have a detrimental effect on the performance of hybrid composites and lead to an undesired variation in the resulting mechanical properties [2], [3], [4], [5]. Whilst several researchers have attempted to optimize the manufacturing process of composites through the minimization of defects [6], [7], [8], in practice, it is impossible to eliminate all defects and produce a “perfect” component.

The presence of voids is undesirable in most applications of composite materials with the void content typically being restricted below 5% [4]. However, in some aerospace applications, even a void content of 1% is considered unacceptable [9], [10]. Thus far, several studies have investigated the relationship between void content and strength of composite materials with it being concluded that matrix-dominated properties, e.g., flexural and compressive strength, are more influenced by voids when compared to fibre-dominated properties, e.g., longitudinal tensile strength [11]. In particular, flexural strength and modulus are extremely sensitive to void content [3], [10], [11], [12], [13]; therefore, consideration of the effect of matrix voids on mechanical properties is an important parameter in the design of composite materials. Dong [11] recently employed a finite element model based on representative volume elements (RVE) and proposed a simple regression model for predicting the strength of composite laminates with voids. This model was used in the present study to incorporate the degrading effect of void content within the optimization problem.

Other manufacturing related uncertainties and defects, such as resin rich regions and fibre misalignment, also affect the performance of composite products. Inspection of fibre-reinforced polymer (FRP) composite laminates produced by conventional manufacturing methods has shown that ±10% variation in the lamina thickness and ±3° deviation in fibre orientation angle from nominal values to be common [14], [15].

Such manufacturing related uncertainties of the resulting composite properties are known to degrade the performance of optimum designs. Therefore, following manufacture, highly optimized composites designed on the basis of nominal values of the design variables may exhibit lower performance than that originally predicted. Robust design optimization (RDO), which was first proposed by Taguchi [16], aims to find optimal solutions that are less sensitive to uncertainties in design variables and manufacturing methods. RDO methods are classified into probabilistic and non-probabilistic approaches. Probabilistic approaches use the probability distribution of the uncertain variables whereas non-probabilistic approaches only use information about the range of uncertain variables. Since in most cases the distribution of the design variables is not available at the early stage of the design process, non-probabilistic methods are generally considered more practical. One of the most popular non-probabilistic approaches is to consider uncertain inputs as bounded variables, i.e., uncertain-but-bounded, and find the worst combination of the bounds which results in the lowest performance of the design, i.e., the worst case [17].

Optimization of hybrid composite laminates is not a simple problem due to the contribution of several variables such as fibre type, orientation angle and volume fraction of individual laminas together with the existence of the hybrid effect [18]. This hybrid effect is defined as the deviation of a property from that calculated based on the rule of mixtures (RoM) [19] and has recently been explained in terms of the stress profile through the thickness of the composite [18]. Therefore, efficient optimization tools will be required to achieve the optimal and robust design of such composite materials. Several studies have investigated the problem of optimizing composite materials with different objectives and constraints such as maximizing strength, stiffness and buckling load capacity, and minimizing weight and cost [20], [21], [22], [23], [24], [25], [26], [27].

In most actual applications of optimization problems, more than one objective is normally incorporated and hence the optimization problem is multi-objective. Unlike single-objective optimization, the result of a multi-objective problem is not unique and a set of optimum results would be expected. Such a set of optimal results obtained from multi-objective optimization is called a Pareto optimal set. Existing methods for finding the Pareto optimal set for multi-objective optimization problems are classified into “preference-based classical methods” and “multi-objective evolutionary algorithms (MOEA)” [28]. Whilst preference-based classical methods are easier to solve compared to MOEAs, a disadvantage is that they require the preference of objectives as an input, which is not always clear to the designer without prior knowledge of the optimal results. In contrast to this, MOEAs start with initial random data and converge to a Pareto optimal set through a number of evolutions without using the preference of objectives. Once a Pareto optimal set has been obtained, different options can be compared and the final choice, i.e, optimum solution, can be determined. Several MOEAs have been proposed thus far, e.g., strength Pareto evolutionary algorithm (SPEA-II) [29], Pareto archived evolutionary strategy (PAES) [30] and the non-dominated sorting genetic algorithm (NSGA) [31]. A modified version of NSGA, NSGA-II, is one of the most popular MOEAs and has been used by several researchers in the field of composite optimization.

The present study is a continuation of previous research by the current authors [18], [32], [33], [34] and aims to investigate the multi-objective optimization and robust design of multi-directional carbon/glass fibre-reinforced hybrid composites when different sources of uncertainties are considered. The effects of three sources of uncertainties, namely, existence of matrix voids, fibre misalignment and thickness variation, on optimal and robust designs were studied with the main focus being the influence of matrix voids. For this purpose, a modified version of the non-dominated sorting genetic algorithm (NSGA-II) was employed for multi-objective optimization and the worst case of the uncertainties was found through an anti-optimization method [35] using a simple genetic algorithm (GA). The density and material cost were chosen as objectives and hybrid composites with different minimum required values of flexural strength were studied.

Section snippets

Material properties

Fibre-reinforced epoxy hybrid composites containing six laminas under three-point bending load were investigated as shown in Fig. 1 where L, w and h are the span, width and total thickness of the composite, respectively. High strength T700S carbon and S-2 glass fibres were chosen for the laminas in order to ensure the possibility of achieving a positive hybrid effect as noted previously [18], [32], [33]. The properties of the fibres and matrix used in this study are presented in Table 1.

Stiffness and strength when matrix voids are present

The

Effect of voids

Given the aforementioned minimum required flexural strengths, the Pareto optimal fronts for hybrid composites with different levels of void content are presented in Fig. 4.

It can be seen from Fig. 4 that, in general, the presence of voids up to 2% does not significantly affect the performance of the optimal results with the Pareto optimal fronts for hybrid composites with Vv = 0%, 1% and 2% being identical irrespective of the void content. However, for void contents of greater than 2%, i.e., Vv

Conclusions

The problem of multi-objective robust optimization for multi-directional carbon/glass fibre-reinforced epoxy composites has been investigated in this paper with manufacture related uncertainties in lamina thickness and fibre orientation angle, together with the presence of matrix voids, being considered. Material cost and density were considered as conflicting objectives with the minimum required flexural strength being chosen as a constraint. Multi-objective optimization was combined with an

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