Theoretical Design and Analysis of Thermally Induced Bistable Plate


Student thesis: Doctoral Thesis

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Awarding Institution
Award date13 Aug 2020


Bistable composite laminates have attracted considerable attention in recent decades owing to their potential applications in the structure morphing, aerospace industry and energy harvesting. Bistability of composite laminate appears as a result of residual thermal stresses induced during the curing process due to the unsymmetric stacking sequence. A bistable plate can transform from one equilibrium position to another with small external forces. The shape transition characteristics make these materials good candidates for adaptive structures, such as morphing skins in aerofoils and wind turbine blades. Because large deformation occurs during shape transition process, bistable plates are also good broadband energy harvesters. After about 40 years of research, the prediction of cured shapes of bistable composite laminates has been well addressed. Current interests are directed toward innovative designs of bistable laminates to meet a desired need, develop new mathematical models that can accurately predict the complex nonlinear dynamics and actively control snap-through motion for morphing applications. A remaining challenge in developing a mathematical model is how to accurately address complex geometrical nonlinearity.

The analytical model based on the Rayleigh-Ritz method is widely used and provides valid cured shape predictions of unsymmetric composite laminates, but its accuracy is reduced when modelling large-amplitude nonlinear dynamics. Commercial finite element (FE) software, such as Abaqus, is a robust tool for performing nonlinear analysis. However, for bistable or multistable plates, hands-on user intervention is often required for the solver to converge to the desired equilibrium because they are not designed to solve such a specific problem. Additionally, most of the commercial FE software cannot be directly used to carry out the active feedback control and energy harvesting analysis. The lack of an effective mathematical model leads to some basic mechanical behaviours of bistable laminates remaining unknown which greatly limits their practical applications.

In this thesis, to solve above disadvantages, a nonlinear finite element model (FEM) is proposed, which can model the complex nonlinear dynamics and perform electro-mechanical analysis. This model is applied to investigate several research hotspots related to thermally induced bistable plates. All the simulations are carried out by own-developed MATLAB code and the results are validated by comparing with published experimental or numerical results. The main parts of this thesis are presented as follows.

The accuracy, convergence and computational efficiency of the FEM strongly depend on the selected element. In this thesis, to develop an accurate FEM that can rapidly simulate a thermally induced bistable plate, five plate elements are introduced and tested to find a suitable one. These plate elements include a 3-node Mindlin (MIN3) plate element with improved transverse shear, 4-node and 9-node Mindlin plate elements, a 4-node conforming plate element and a displacement-based 4-node quadrilateral element RDKQ-NL20. Using von Kármán large deflection theory and total Lagrangian (TL) approach, the nonlinear FE governing equations for composite laminates under external loads are derived based on the minimum potential energy principle. Convergence analysis and numerical testing for the five elements are conducted. The results show that the RDKQ-NL20 element has the best accuracy and convergence among the tested elements.

Traditional bistable plates made of carbon-fibre reinforced composite laminate possess low load-carrying capacity. In this thesis, a novel bistable plate using functionally graded carbon nanotube-reinforced composite (FG-CNTRC) is proposed. Single-walled carbon nanotubes (SWCNTs) in this nano-composite are assumed to have a functionally graded distribution along the thickness direction following a power law. A higher-order Rayleigh-Ritz model is presented to investigate the bistability and buckling behaviour of the proposed bistable plates. The arc-length method is used in the buckling analysis to trace the load-displacement path. The cured shapes and snap-through loads are predicted. The analytical results are compared with the nonlinear FEA for validation. Through a comprehensive parametric study, it is found that by varying the distribution type, volume fraction, and volume fraction exponent of the SWCNTs, bistable FG-CNTRC plates can be generated with multiple stable shapes and a wide range of load-carrying capacities.

The proposed bistable FG-CNTRC plate is bonded with two piezoelectric layers for broadband energy harvesting. Using the 4-node RDKQ-NL20 element and based on Hamilton’s principle, a nonlinear FE electro-mechanical model is developed to evaluate the energy harvesting performance of the proposed energy harvester. The complex cross-well dynamics are accurately captured by above developed FEM. The open-circuit voltages of the bistable energy harvester and their linear counterparts under different excitation levels are predicted and compared. Frequency response diagrams of the root mean square (RMS) voltage reveal that bistable FG-CNTRC plates can operate over a wide range of frequencies and deliver higher power than their linear counterparts.

The proposed FE electro-mechanical model is extended in this thesis to study active dynamic snap-through control of the bistable plate. A piezoelectric actuator and sensor made of macro-fibre composites (MFC) are attached to the bistable laminate surface to form a closed-loop control system. The displacement feedback control law is implemented in an active control system to realise shape transitions. Some critical displacement feedback gains required to induce dynamic snap-through are predicted. The results show that active full-state configuration control can be realised and the displacement feedback control algorithm is effective in suppressing undesired chaotic oscillations.

Bistable composite plates have the potential to be the morphing skins in aircraft, so understanding their supersonic aeroelastic flutter behaviour is necessary. Aeroelastic analysis and active flutter control of variable stiffness composite laminates (VSCLs) are first conducted to provide a basic understanding of panel flutter in supersonic flow. In the FEM, fibre-path angles in the VSCLs are assumed to vary linearly between the centre and edges of the panel. The first-order piston theory is used to model the aerodynamic load caused by the supersonic airflow. The critical buckling temperature and aerodynamic pressure for the VSCLs are predicted through a parametric study. The optimal location and shape of piezoelectric actuators for controlling the dynamic responses of VSCLs are determined by comparing the norms of the feedback control gain. Numerical simulations show that unstable panel flutter and thermal post-buckling deflection can be effectively suppressed by optimal design of the piezoelectric patches. A supersonic aeroelastic nonlinear flutter study of bistable plate is also carried out. For comparison purposes, the critical dynamic pressures of shallow shells that have the same geometry, material properties and stacking sequence with bistable laminate are calculated. The effects of changing flow yaw angle and curing temperature on the critical dynamic pressure for bistable laminates are investigated. The load-carrying capacity and pre- and post-flutter-onset responses for bistable laminates are studied in detail.

    Research areas

  • Active control, analytical model, bistable plate, broadband energy harvesting, CLPT, first-order shear deformation laminated plate theory (FSDT), FEM, functionally graded carbon nanotube reinforced composite, geometrical nonlinearity, piezoelectric, snap-through, supersonic flutter, VSCLs