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

Abstract

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 outof-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.

Multi-stable composite twisting structure for morphing applications

“…Indeed, there are three distinct stability characteristics depending on the value of this ratio. The reader is referred to the work by Bloomfield et al (2009) The strain energy for this particular lay-up is shown in figure 8a, where the edge of the surface describes a flat circle denoting the constant level of strain energy for any helix angle −180 • < 2q < 180 • . This is also observed for a = 2 in the contour plot shown in figure 8b and is highlighted by the dashed circle.…”

Composite stacking sequence optimization for aeroelastically tailored forward-swept wings

“…The feasible LP region is known to be convex [8,12,23], and a closed-form expression has been defined only for a limited number of combinations of the lamination parameters [41]. Expressions for a limited set of LPs have been derived by different authors.…”

“…A method for obtaining an approximate feasible region is presented in [41]. [12] developed an approach to define a feasible region for all 12 lamination parameters, which was further extended by [8]. In particular, [13] derived explicit expressions describing the feasible region for ply angles limited to {0 • , ±45 • , 90 • }.…”

“…A closed-form solution to equations (7)(8) or (10)(11) can only be obtained for specific wing properties [7,22]. Therefore, the Ritz method based on Hamilton's principle has been used.…”

Effects of Long-Term Stowage on the Deployment of Bistable Tape Springs

“…Àh=2 cos 2h; cos 4h; sin 2h; sin 4h ð Þ z 2 dz (8) where z is the distance to the midplane, and h is the total laminate thickness [17]. The material property matrices vector C is defined as…”