On feasible regions of lamination parameters for lay-up optimization of laminated composites

The stiffness tensors of a laminated composite may be expressed as a linear function of material invariants and lamination parameters. Owing to the nature of orienting unidirectional laminae ply by ply, lamination parameters, which are trigonometric functions of the ply orientation, are interrelated. In optimization studies, lamination parameters are often treated as independent design variables constrained by inequality relationships to feasible regions that depend on their values. The relationships between parameters enclose a convex feasible region of lamination parameters which is generally unknown. The convexity properties allow the efficient optimization of laminated composite structures where lamination parameters are used as design variables. Herein, a two-level method is presented to determine the feasible regions of lamination parameters where potential ply orientations are a predefined finite set. At the first level, the feasible region of the in-plane, coupling and out-of-plane lamination parameters is determined separately using convex hulls. At the second level, a nonlinear algebraic identity is used to relate the in-plane, coupling and out-of-plane lamination parameters to each other. This general approach yields all constraints on the feasible regions of lamination parameters for a predefined set of ply orientations.