Hierarchical optimization of laminated fiber reinforced composites

1. Abstract The aim of this work is to perform hierarchical optimization [1] in laminated composite structures, considering simultaneously macroscopic and microscopic levels in the design of the structure and its material. In the macroscopic level, the optimization algorithm cares with orientations and fiber volume fractions of unidirectionally reinforced composite material layers. In the microscopic level, the goal is to define the microstructure of the layers by determining the cross-sectional size and shape of the reinforcement fibers. Both macro/micro scales are coupled by a resource constraint and interchange derivative information. The objective is to minimize compliance under a total fiber volume fraction constraint. A previous work in this line [2] optimized a laminated composite representing it by a 3D finite element model, where each layer was treated as a group of such elements. In each of those layers, a unit cell of microstructure was defined by topology optimization in accordance with a hierarchical optimization approach. However, in the present case, this unit cell is considered simply as a portion of matrix reinforced by a piece of fiber whose cross-sectional dimensions will be defined by optimization. These fibers’ cross sections are constrained to be elliptical, but with variable size and semi-axes aspect ratio. The present approach is more restrictive, since the general topology layout is predefined, but interesting in face of the cross-sectional shape of fibers more commonly available for composites fabrication. Moreover, the layered composites are here treated by finite elements based on laminated plate/shell theory. The variation of the size and shape of the fibers is here considered by means of response surfaces for the constitutive parameters of an unidirectional composite lamina in terms of the fiber dimensions. Such surfaces are built upon function and derivative information [3] of constitutive parameters, evaluated from material microstructural models using asymptotic homogenization techniques [4]. The layers’ orientations are chosen using the discrete material optimization (DMO) approach [5], where lists of candidate materials are interpolated by weighting functions, whose values are to be determined by optimization. Results in laminated plates show the influence of the reinforcement fibers’ shape and volume fraction in the global behavior of the test structures. It is shown that the present optimization procedure permits, in many cases, to improve the global behavior of structures when elliptical fibers are used to reinforce layers of a laminate. The optimal microstructures obtained are strongly influenced by the global loading conditions considered. In a final part, microstructural stresses are calculated in a laminated plate whose microstructure was already optimized, using results from asymptotic homogenization, in order to assess stress concentrations induced in the matrix by the fibers of distinct shapes. 2.