18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS Design of fiber reinforced composite laminates and discussion on feasible region of lamination parameters S.T. Liu*, Y.P. Hou Department of Engineering Mechanics, Dalian University of Technology, Dalian, China *stliu@dlut.edu.cn Abstract : The relationship of parameters in material property need to be known in simultaneous design optimi- zation of material and structure, thus the obtained materials can be explained in physics. The relations of stiffness properties in fiber reinforced composite laminates can be defined by 12 lamination parameters of in-plane, coupl- ing and out-of-plane stiffness. At present, the obtained feasible region of 12 lamination parameters is necessary condition but not sufficient for laminate materials which can be realized in physics. In this paper, the change of feasible region of 12 lamination parameters with the increase of the number of plies was studied firstly under the assumption that the laminate thickness is fixed, the angles and thicknesses of ply can be changed arbitrary. We found that the feasible region is essentially unchanged when the number of plies is more than 6 plies. Then by considering least(6 plies) ply angles and ply thickness and fiber volume fraction in micro configuration as design variables, a topology optimization based method is proposed to minimize compliance of laminates under the con- dition of given fiber volume ratio. Finally, numerical examples are presented to demonstrate the validity of the proposed approach. Keywords : lamination parameters, laminates optimization, feasible region corresponding to laminate lay-up. The feasible re- 1 Introduction gion for in-plane or out-of-plane lamination parame- One of the basic purposes of structural engi- ters has been given in analytical form by [6]. How- neering optimization is to find a stiffest structure ever, feasible region for 12 lamination parameters with a given volume which is able to bear more has not analytical expressions because of their com- loadings under the conditions of a given set of boun- plexity. Literature [7-9] provided methods to obtain dary conditions and loads. Simultaneous optimiza- approximate feasible region, and expressions for tion of structures and materials is known as the op- describe the region were given by [9], but the litera- timal design to achieve this purpose. It not only ture also indicated that these expressions are neces- could optimize material distribution in space but also sary but not sufficient conditions for realizing ma- could design elastic property of local materials. The terial in physics. property of local materials is not limited except en- Literature [6] concluded that an optimal design suring material can be realized in physics [1-4]. The can be realized with at most two plies for pure difficulty of simultaneous optimization of structures membrane problems under the condition of laminate and materials is the optimal results can’t be realized thickness is fixed, but ply angles and ply thickness easily in physics. On one hand, continuous variabili- can be changed arbitrary. Inspired by this idea, in ty of material property in design domain makes ma- this paper, the change of feasible region with 12 la- terial manufacture inconvenience; on the other hand, mination parameters was discussed by increasing the parameters in material property relate to each other number of plies, and solved the least plies for feasi- and it is difficult to determine the range of each pa- ble region when it is essentially unchanged. Then rameter. Therefore, the obtained materials can’t be ply angles and ply thickness are treated as variables ensured to realize by physical materials. in optimization problems for design laminate lay-up For fiber reinforced laminates, the elastic stiff- with the given plies. In this case, the optimal materi- ness can be denoted by 12 lamination parameters al can be obtained and also be ensured to realize in and invariants [5], lamination parameters represent physics. The relationship of parameters in material lay-up of laminates (ply angles, ply thickness and property is not needed, so the difficulty of feasible number of plies). As parameters of elastic stiffness region of lamination parameters is avoided. must have some relations, then the 12 lamination This paper is organized as follows. Firstly, the parameters also have the same case. In other word, least plies for satisfying the feasible region of 12 the feasible region of lamination parameters must be lamination parameters was determined. Then, a to- determined in order to ensure lamination parameters pology optimization formulation for simultaneously
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