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Lightweight thin-walled structures are crucial for many engineering applications. Advanced manufacturing methods are enabling the realization of composite materials with spatially varying material properties. Variable angle tow fibre composites are a representative example, but also nanocomposites are opening new interesting possibilities. Taking advantage of these tunable materials requires the development of computational design methods. The failure of such structures is often dominated by buckling and can be very sensitive to material configuration and geometrical imperfections. This work is a review of the recent computational developments concerning the optimisation of the response of composite thin-walled structures prone to buckling, showing how baseline products with unstable behaviour can be transformed in stable ones operating safely in the post-buckling range. Four main aspects are discussed: mechanical and discrete models for composite shells, material parametrization and objective function definition, solution methods for tracing the load-displacement path and assessing the imperfection sensitivity, structural optimisation algorithms. A numerical example of optimal material design for a curved panel is also illustrated.

Carbon nanotube/polymer nanocomposite plate- and shell-like structures will be the next generation lightweight structures in advanced applications due to the superior multifunctional properties combined with lightness. Here material optimization of carbon nanotube/polymer nanocomposite beams and shells is tackled via ad hoc nonlinear finite element schemes so as to control the loss of stability and overall nonlinear response. Three types of optimizations are considered: variable through-the-thickness volume fraction of random carbon nanotubes (CNTs) distributions, variable volume fraction of randomly oriented CNTs within the mid-surface, aligned CNTs with variable orientation with respect to the mid-surface. The collapse load, which includes both limit points and deformation thresholds, is chosen as the objective/cost function. An efficient computation of the cost function is carried out using the Koiter reduced order model obtained starting from an isogeometric solid-shell model to accurately describe the point-wise material distribution. The sensitivity to geometrical imperfections is also investigated. The optimization is carried out making use of the Global Convergent Method of Moving Asymptotes. The extensive numerical analyses show that varying the volume fraction distribution as well as the CNTs orientation can lead to significantly enhanced performances towards the loss of elastic stability making these lightweight structures more stable. The most striking result is that for curved shells, the unstable postbuckling response of the baseline material can be turned into a globally stable response maintaining the same amount of nanostructural reinforcement but simply tailoring strategically its distribution.

A study of the pre- and post-buckling state of square plates built from functionally graded materials (FGMs) and pure ceramics is presented. In contrast to the theoretical approach, the structure under consideration contains a finite number of layers with a step-variable change in mechanical properties across the thickness. An influence of ceramics content on a wall and a number of finite layers of the step-variable FGM on the buckling and post-critical state was scrutinized. The problem was solved using the finite element method and the asymptotic nonlinear Koiter’s theory. The investigations were conducted for several boundary conditions and material distributions to assess the behavior of the plate and to compare critical forces and post-critical equilibrium paths.

The Koiter's asymptotic method is combined with the assumed strain solid shell element formulation for postbuckling analysis of composite and sandwich structures. The assumed strain solid shell element is free of locking and the small angle assumption, and it allows multiple plies through the element thickness. While laminated composite structures are modeled with single element through the thickness, sandwich structures are modeled with three elements stacked through the thickness to model the face sheets and the core independently. The Koiter's method is used to trace initial postbuckling path. Subsequently, the Koiter's method is switched to the arc-length method to investigate postbuckling behavior involving large deflections. The transition point at which the switching occurs is determined using the postbuckling coefficients, obtained from the asymptotic analysis with the fourth order expansion. Numerical tests demonstrate the validity and effectiveness of the present approach.

In this work, the nonlinear forced vibrations of doubly curved shells are studied. For this, the Forced Resonance Curves of four different shells were determined: a shallow cylindrical panel, a shallow spherical panel, a non-shallow spherical panel, and a hyperbolic paraboloid. To model the shells, the Koiter’s nonlinear shell theory, for both shallow and non-shallow shells, was applied. The forced resonance curves were determined using an adaptive harmonic balance method and through a reduced-order model (ROM) via parameterization method for invariant manifolds. The findings of this study reveal the complex dynamic behavior exhibited by doubly curved shells, with various types of bifurcations such as Saddle–Node, Neimark–Sacker, and Period Doubling bifurcations. Thanks to the general treatment of the forcing term implemented in the parameterization method, the results highlight how complex high-order resonances can be retrieved by the ROM, up to a comfortable range of vibration and forcing amplitudes tested. Finally, it clearly demonstrates how the Nonlinear Normal Modes as invariant manifolds provide accurate and efficient ROMs for nonlinear vibrations of shells.

We study a mutually coupled mesoscopic-macroscopic-shell system of equations modeling a dilute incompressible polymer fluid which is evolving and interacting with a flexible shell of Koiter type. The polymer constitutes a solvent-solute mixture where the solvent is modelled on the macroscopic scale by the incompressible Navier–Stokes equation and the solute is modelled on the mesoscopic scale by a Fokker–Planck equation (Kolmogorov forward equation) for the probability density function of the bead-spring polymer chain configuration. This mixture interacts with a nonlinear elastic shell which serves as a moving boundary of the physical spatial domain of the polymer fluid. We use the classical model by Koiter to describe the shell movement which yields a fully nonlinear fourth order hyperbolic equation. Our main result is the existence of a weak solution to the underlying system which exists until the Koiter energy degenerates or the flexible shell approaches a self-intersection.

We consider a family of linearly viscoelastic shells with thickness 2ε, all having the same middle surfaceS=θ(ω¯)⊂IR3, where ω⊂IR2 is a bounded and connected open set with a Lipschitz-continuous boundary γ and θ∈C3(ω¯;IR3). The shells are clamped on a portion of their lateral face, whose middle line is θ(γ0), where γ0 is a non-empty portion of γ. The aim of this work is to show that the viscoelastic Koiter’s model is the most accurate two-dimensional approach in order to solve the displacements problem of a viscoelastic shell. Furthermore, the solution of the Koiter’s model, ξKε=(ξK,iε), is in H1(0,T;VK(ω)), with ξK,iε:[0,T]×ω¯→R the covariant components of the displacements field ξK,iεai of the points of the middle surface S and where VK(ω):={η=(ηi)∈H1(ω)×H1(ω)×H2(ω);ηi=∂νη3=0inγ0},with ∂ν denoting the outer normal derivative along γ. Under the same assumptions as for the viscoelastic elliptic membranes problem, we show that the displacement field, ξK,iεai, converges to ξiai (the solution of the two-dimensional problem for a viscoelastic elliptic membrane) in H1(0,T;H1(ω)) for the tangential components, and in H1(0,T;L2(ω)) for the normal component, as ε→0. Under the same assumptions as in the viscoelastic flexural shell problem, we show that the displacement field, ξK,iεai, converges to ξiai (the solution of the two-dimensional problem for a viscoelastic flexural shell) in H1(0,T;H1(ω)) for the tangential components, and in H1(0,T;H2(ω)) for the normal component, as ε→0. Also, we obtain analogous results assuming the same assumptions as in the viscoelastic generalized membranes problem. Therefore, we justify the two-dimensional viscoelastic model of Koiter for all kind of viscoelastic shells.

Bistable composite laminates provide an appealing platform for morphing applications. On the other hand they exhibit geometrically nonlinear behavior and they are sensitive to imperfections. Their curing behavior dictates bifurcation buckling analysis while their actuation requires snapthrough buckling analysis. This work proposes a generalized finite element analysis procedure applying Koiter’s asymptotic postbuckling theory to address their curing and actuation. Initially the postbuckling theory is discussed providing essential aspects required for its application into finite element analysis. A generalized scheme is established for the Koiter-based procedure to enable its incorporation into design optimization routines. To prove its generality, the procedure is implemented into three finite element commercial codes, namely, ABAQUS, ANSYS and LS-DYNA. Best practices for these implementations are provided, then their accuracies are assessed through multiple comparisons with published data. Moreover, Hyper-Elliptic Cambered Span (HECS) Wing design is developed utilizing bistable laminates. Stability characteristics of several design variations of the morphing HECS wing are assessed using the developed procedure. The Koiter-based finite element procedure is proven to be both general and suitable for implementation in different finite element codes to address designs with complex geometry. Therefore, this work provides a unique platform for novel designs employing bistable composites in various engineering applications. Furthermore, it presents a general framework to implement Koiter’s asymptotic postbuckling theory in finite element codes for bifurcation buckling and post-buckling studies of imperfection-sensitive structures.

The phenomena that occur during compression of hybrid thin-walled columns with open cross-sections in the elastic range are discussed. Nonlinear buckling problems were solved within Koiter’s approximation theory. A multimodal approach was assumed to investigate an effect of symmetrical and anti-symmetrical buckling modes on the ultimate load-carrying capacity. Detailed simulations were carried out for freely supported columns with a C-section and a top-hat type section of medium lengths. The columns under analysis were made of two layers of isotropic materials characterized by various mechanical properties. The results attained were verified with the finite element method (FEM). The boundary conditions applied in the FEM allowed us to confirm the eigensolutions obtained within Koiter’s theory with very high accuracy. Nonlinear solutions comply within these two approaches for low and medium overloads. To trace the correctness of the solutions, the Riks algorithm, which allows for investigating unsteady paths, was used in the FEM. The results for the ultimate load-carrying capacity obtained within the FEM are higher than those attained with Koiter’s approximation method, but the leap takes place on the identical equilibrium path as the one determined from Koiter’s theory.

The applicability and limitations of simplified models of thin elastic circular cylindrical shells for linear vibrations of double-walled carbon nanotubes (DWCNTs) are considered. The simplified models, which are based on the assumptions of membrane and moment approximate thin-shell theories, are compared with the extended Sanders–Koiter shell theory. Actual discrete DWCNTs are modelled by means of couples of concentric equivalent continuous thin, circular cylindrical shells. Van der Waals interaction forces between the layers are taken into account by adopting He’s model. Simply supported and free–free boundary conditions are applied. The Rayleigh–Ritz method is considered to obtain approximate natural frequencies and mode shapes. Different aspect and thickness ratios, and numbers of waves along longitudinal and circumferential directions, are analysed. In the cases of axisymmetric and beam-like modes, it is proven that membrane shell theory, differently from moment shell theory, provides results with excellent agreement with the extended Sanders–Koiter shell theory. On the other hand, in the case of shell-like modes, it is found that both membrane and moment shell theories provide results reporting acceptable agreement with the extended Sanders–Koiter shell theory only for very limited ranges of geometries and wavenumbers. Conversely, for shell-like modes it is found that a newly developed, simplified shell model, based on the combination of membrane and semi-moment theories, provides results in satisfactory agreement with the extended Sanders–Koiter shell theory in all ranges.