Design Method Developed for Automated Fiber Placement

Summary

Automated Fiber Placement (AFP) is employed as an innovative fiber-reinforced composite manufacturing method in which fibers are laid directionally within various production constraints. This method allows fiber angles to be varied within a lamina layer during laying, providing flexibility in determining stacking sequences as with constant-stiffness composites, and enabling optimum solutions that deliver better performance compared to constant-stiffness structures. In this work, an innovative finite element method was developed to enable design in accordance with the AFP method and its production constraints. The proposed method was tested on plates with different boundary conditions and fiber distribution was determined in accordance with production constraints. The optimization results were found to be consistent with studies in the literature and provided better results in some cases.

Introduction

The "Automated Fiber Placement" (AFP) method is used to manufacture fiber-reinforced composite materials. Since fiber-reinforced composites have higher stiffness-to-weight ratios compared to metals, fiber-reinforced composite materials have become an alternative to metallic materials [1]. The AFP method is used in the aerospace industry for manufacturing critical aerospace components. The AFP method is a more attractive manufacturing method compared to conventional composite manufacturing methods because it provides manufacturing automation that reduces labor and increases the repeatability of the composite production process.

Figure 1. Fiber-reinforced composite laying with the Automated Fiber Placement (AFP) method [2]

Figure 1 shows a robotic manipulator performing the AFP process [2]. Tow pre-preg material is heated with an infrared heater and subsequently pressed onto the mold with a compression cylinder to adhere to it. For this method, fiber directions can be laid onto the mold in any desired direction as long as they exceed a limited radius of curvature [3, 4].  

Figure 2. Example stacking sequence of different lamina layers in a composite laminate

Figure 2 shows an example stacking sequence in which lamina layers are stacked on top of each other. Constant stiffness means stiffness that remains the same throughout a composite part. For example, if the stacking sequence and layer thickness of lamina layers remain the same throughout the part, the stiffness of that section remains constant. Many previous studies have focused on the design of constant-stiffness composite laminates by determining the laying sequence of lamina layers. Considerable research has been conducted on the design of constant-stiffness composites. In these studies, different objective functions were used such as stiffness, first buckling frequency, and failure load.         Figure 3. Linearly varying spline curve defined with two independent parameters T0 and T1, used by Gürdal and Olmedo [5] More recent studies focused on variable stiffness design after the AFP manufacturing method became commercially available. The problem here is to find the best fiber directions and their stacking sequences for given performance requirements. Gürdal and Olmedo proposed a linearly varying spline for defining fiber angles [5]. Similar studies used B-splines and NURBS functions following the same conceptual definition of splines [6, 7]. There are manufacturing constraints such as fiber continuity and the minimum radius of curvature that the fiber can be directed. Recent studies aim to find optimal fiber directions while taking manufacturing constraints into account [8].

Design Method

The design method was developed from a finite element model, an optimization method, and a new design method to find manufacturable fiber angles. This method uses finite element analyses within the optimization method. It employs lamination parameters. Classical laminate theory is implemented in MATLAB®. This is necessary to perform optimization in the same environment as the finite element environment for numerical efficiency. Lamination Parameters are used to convert the optimization into a convex problem. For example, in Equation (1), the in-plane lamination parameter vector, {ξ^A}, is shown as a function of the thickness coordinates of laminated layers, z_k, z_k+1, for the sum of N lamina layers with total laminate thickness, t. The fiber angles of "k" lamina layer are expressed as θk. Out-of-plane lamination parameters, {ξ^D}, are proportional to the cube of the thickness coordinate. The optimization aims to find maximum stiffness, i.e., minimum strain energy, as shown in Equation (3). The "fmincon" function of MATLAB® optimization toolbox was used to find the optimal fiber angles. Strain energy, R, is calculated using global finite element displacements, {d}, and global stiffness matrix, [K]. These two properties are functions of fiber distribution and change during optimization. After optimization, lamination parameters are determined at the center of each finite element and are converted directly to fiber angles and layer sequences using the direct search method. Manufacturability requires that fiber angles be continuous and curvatures remain within a maximum curvature limit. To achieve this, we also treated fiber angles as field variables. We then searched for fiber angles closest to the optimum solution and satisfying these constraints. Details of the method can be found in reference [9].

Results and Conclusions

The developed method was tested for three different boundary conditions and loading cases. Figure 4 shows three different boundary conditions. In this study, the length Ly is taken as a unit.

Figure 4. Boundary conditions used in method testing

Figure 5 shows the distribution of lamination parameters obtained after optimization for a simply supported plate. Since the lamination parameter consists of four components, their distributions on the plate are shown in four separate figures.   Figure 5. Optimal lamination parameter distribution for a simply supported and pressure-loaded plate The optimal lamination parameters found are converted to angles and sequences using the direct search method. Figure 6 shows the results of fiber distributions for a simply supported plate with out-of-plane pressure.

Figure 6a./6b. Angles obtained after optimization

Figure 6.a shows the angles obtained after optimization. Figure 6.b shows the angles obtained in accordance with manufacturing constraints. Upon careful examination, fiber angles show continuous distribution. At the same time, curvature values were limited with a penalty parameter.

Figure 7a./7b. Elemental curvature for penalty factors

The penalty value was calculated using the defined maximum elemental curvature value (max. κ_el). In this study, max. κ_el was taken as 5 m⁻¹. As shown in Figure 7.a, elemental curvature was calculated for various penalty factors and the penalty factor that met the desired value was found to be 0.0176. When applied to fiber angles, the maximum curvature value shown in Figure 7.b is 5 m⁻¹. Table 1 presents overall results and improvements in accordance with different geometries. The proposed solution is consistent with literature findings [10] for the same problem and in many cases yields even better results. Table 1. Optimization results for three different boundary condition types and comparison of results with reference [10]. For a cantilever plate, the normalized compliance, R, is given by the following relationship: R= E₂₂ t L³ _y R /(p² L⁵ _x), for simply supported and fixed plates, R=E₂₂ t³ R/ (p² L⁵ )x100, was used for cantilever beam boundary conditions. The proposed method provides the following advantages: • Finding optimal fiber angles and lamina sequences using a very simple and effective calculation method, • Consideration of manufacturing constraints related to fiber continuity and radius of curvature, • Finding better optimum solutions than literature in many cases. In future work, conversion of discontinuous angles to spline functions where tool paths will be determined with continuous functions, experimental applications and production, and modeling of actual part geometry with shell-type elements will be undertaken. Dr. Associate Professor Eralp Demir Sabancı University Faculty of Engineering and Natural Sciences Department of Mechatronics      

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