Functional Description for Thick Bistable Carbon Fiber Laminates With Rayleigh-Ritz, Abaqus, and Experiments
Authors: Knippenberg Christopher H. , Myers Oliver J. , Nelon Christopher 
Published: November 2020

Abstract
Composite laminates constructed in an asymmetric layup orientation of [0i, 90i], i &gt; 0, exhibit two stable equilibrium positions and may be actuated to snap from a primary cure shape to an inversely related secondary stable shape. This study aims to aid in developing a comprehensive description of thick bistable laminates, whose increased thickness risks the loss of bistability, through previously established analytical approaches and verification via experimentation. The principle of minimum potential energy is applied to two materials and analyzed using the Rayleigh-Ritz minimization technique to determine the cure shapes of carbon fiber reinforced polymer laminates composed of AS4/8552 and TR50S-12k carbon fibers. These materials were modeled to act as square thick bistable laminated composites with sidelengths up to 0.914m. Visualizations of the out-of-plane displacements are shown with a description of the Rayleigh-Ritz analysis. Additionally, a finite element model (FEM) created in Abaqus CAE 6.14 and experiments using DA409/G35 and TR50S-12K/NP301 prepreg were used to further describe and develop the fundamental description for thick bistable laminates in terms of loss of bistability, actuation load, and principle shape.
The analytical model is an extension of Hyer’s (2002) and Mattioni’s (2009) work applied to thick bistable laminates where the primary assumption was the x-axis curvature equaled the negative y-axis curvature for the primary and secondary stable positions, respectively. This assumption leads to the already cemented conclusion that bistable laminates, once cured, take on one of two inversely related paraboloid shapes. FEA simulations contradicted this by showing an average 11% difference in curvature magnitude for the aforementioned shapes. Furthermore, fourth order polynomials were used to describe the curvature along the axes, differing from the previously used Menger curvatures, (three-point approximation). Bifurcation plots using peak deflections and average curvature generated from FEA simulations clearly showed bistability existed to approximately 50 plies; however, the energy landscape plots indicated a significant degradation of bistability starting at 36 plies. Experimentation was performed on a test stand mimicking the same boundary conditions used in FEA while applying a central out-of-plane load. Experimental observations showed decreased peak displacements of stable cure shapes. Observations also indicated that the x-axis curvature had a significant difference in magnitude compared to the negative y-axis curvature. However, the existence of bistability agreed with FEA energy landscape plots, with clear “snaps” ending at thicknesses of 28–36 plies. Moreover, actuation force was found to correlate well with FEA simulations. Differences in the critical point can be attributed to the combination of material property differences for DA409 and TR50S-12K, failure to capture polymer relaxation, limitations of the experimental setup, and hand layup fabrication errors. Lastly, this paper adds viability of thicker laminates for use in macroscale applications where shape morphing or shape-retention attributes are a necessary constraint, although only where low loads are expected.