| Summary: | This thesis sets out to develop an accurate and computationally efficient model to describe the multistability of geometrically non-linear shells with arbitrary curvature, suitable for modelling morphing applications. Governing equations are obtained for a system of curvilinear, non-orthogonal coordinates, based on the reference surface metric of a shell of arbitrary curvature. A novel, complete set of symmetric equilibrium equations and general boundary conditions is provided, derived from the principle of virtual work. The strain tensor reduces to the Sanders-Koiter-Leonard relations when recast for orthogonal coordinates and vanishes for rigid body motion. A novel expression of Gauss's compatibility conditions is obtained by consistently imposing assumptions of small strains and moderate rotations to the Riemann tensor. The model is reduced to the solution of von Karman plates with initial curvature when the shell is sufficiently shallow. In this case, the membrane and bending effects are decoupled via the semi-inverse formulation of constitutive equations. This beneficial effect reflects the through-thickness distribution of strains, assumed to be linear in a thin-walled structure, where the neutral axis does not necessarily lie at the mid plane. Strain energy can therefore be expressed as a function of curvature only. This decoupled representation facilitates efficient solution. The solution method recasts the governing equations by introducing suitable sets of differential operators, discretised using the pseudo-spectral Differential Quadrature Method (DQM), due to its accuracy and efficiency. The system of partial differential equations is consequently solved with simple matrix algebra. Physical insights are discussed, including the ability to devise neutrally stable structures independently from conventional stiffness tailoring techniques. Comparisons with state-of-the-art examples in literature are outlined in a series of case studies, that highlight advantages of the proposed model. First, snap-through loads for bistable plates are computed with greater accuracy and fewer degrees of freedom than prior state-of-the-art. Then, tristable shells with bending boundary layers on their perimeters are represented with no assumption on the symmetry of the displacement field and no constraining relations. Finally, a temperature-driven, multi-mode morphing plate, shows that the combination of DQM and orthogonal polynomial approximation allows an accurate description of membrane and bending components of the total strain energy, with relatively low computational cost.
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