Morphing shell structures:A generalised modelling approach

AbstractMorphing shells are nonlinear structures that have the ability to change shape and adopt multiple stable states. By exploiting the concept of morphing, designers may devise adaptable structures, capable of accommodating a wide range of service conditions, minimising design complexity and cost. At present, models predicting shell multistability are often characterised by a compromise between computational efficiency and result accuracy. This paper addresses the main challenges of describing the multistable behaviour of thin composite shells, such as bifurcation points and snap-through loads, through the development of an accurate and computationally efficient energy-based method. The membrane and the bending components of the total strain energy are decoupled by using the semi-inverse formulation of the constitutive equations. Transverse displacements are approximated by using Legendre polynomials and the membrane problem is solved in isolation by combining compatibility conditions and equilibrium equations. This approach provides the strain energy as a function of curvature only, which is of particular interest, as this decoupled representation facilitates efficient solution. The minima of the energy with respect to the curvature components give the multiple stable configurations of the shell. The accurate evaluation of the membrane energy is a key step in order to correctly capture the multiple configurations of the structure. Here, the membrane problem is solved by adopting the Differential Quadrature Method (DQM), which provides accurate results at a relatively small computational cost. The model is benchmarked against three exemplar case studies taken from the literature

Bespoke extensional elasticity through helical lattice systems

© 2019 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited. Nonlinear structural behaviour offers a richness of response that cannot be replicated within a traditional linear design paradigm. However, designing robust and reliable nonlinearity remains a challenge, in part, due to the difficulty in describing the behaviour of nonlinear systems in an intuitive manner. Here, we present an approach that overcomes this difficulty by constructing an effectively one-dimensional system that can be tuned to produce bespoke nonlinear responses in a systematic and understandable manner. Specifically, given a continuous energy function E and a tolerance ℇ > 0, we construct a system whose energy is approximately E up to an additive constant, with L∞-error no more that ℇ. The system is composed of helical lattices that act as one-dimensional nonlinear springs in parallel. We demonstrate that the energy of the system can approximate any polynomial and, thus, by Weierstrass approximation theorem, any continuous function. We implement an algorithm to tune the geometry, stiffness and pre-strain of each lattice to obtain the desired system behaviour systematically. Examples are provided to show the richness of the design space and highlight how the system can exhibit increasingly complex behaviours including tailored deformation-dependent stiffness, snap-through buckling and multi-stability

Adaptive stiffness in lattice metastructures through tensile-buckling inspired topology morphing

This paper explores the use of simultaneous tensile buckling of unit cells to induce a transformation in lattice topology. Under tension, unit cells undergo passive transformation from a rectangle-like to a triangle-/pentagon-like topology, with an associated change in the effective stiffness properties. This behaviour is investigated through finite element analysis and experiments, with analytical results providing insights into the observed behaviour. The analysis identifies (i) that the initial unit cell topology (rectangular) is dominated by membrane effects, (ii) the transformation phase is associated with negative stiffness, and (iii) once formed, the new topology (triangular/pentagonal) exhibits increased stiffness in both compression and tension. Finite element analysis confirms that the unit cell behaviour is also preserved in lattices. Under tension, the lattice undergoes a seven-fold increase in stiffness as it transitions from its initial to the new topology, with a regime of negative stiffness during this transformation accounting for approximately 82% of its total elastic deformation. This new approach to elastically tailor the nonlinear response of (meta-)materials/structures has the potential to contribute to the development of novel tensile energy absorbers

Infinite-contrast periodic composites with strongly nonlinear behavior: Effective-medium theory versus full-field simulations

This paper presents a combined numerical-theoretical study of the macroscopic behavior and local field distributions in a special class of two-dimensional periodic composites with viscoplastic phases. The emphasis is on strongly nonlinear materials containing pores or rigid inclusions. Full-field numerical simulations are carried out using a Fast-Fourier Transform algorithm [H. Moulinec, P. Suquet, C. R. Acad. Sci. Paris II 318, 1417 (1994)] Moulinec, P. Suquet, C. R. Acad. Sci. Paris II 318, 1417 (1994), while the theoretical results are obtained by means of the `second-order' nonlinear homogenization method [P. Ponte Castaneda, J. Mech. Phys. Solids 50, 737 (2002)]. The effect of nonlinearity and inclusion concentration is investigated in the context of power-law (with strain-rate sensitivity m) behavior for the matrix phase under in-plane shear loadings. Overall, the `second-order' estimates are found to be in good agreement with the numerical simulations, with the best agreement for the rigidly reinforced materials. For the porous systems, as the nonlinearity increases (m decreases), the strain field is found to localize along shear bands passing through the voids (the strain fluctuations becoming unbounded) and the effective stress exhibits a singular behavior in the dilute limit. More specifically, for small porosities and fixed nonlinearity m>0, the effective stress decreases linearly with increasing porosity. However, for ideally plastic behavior (m = 0), the dependence on porosity becomes non-analytic. On the other hand, for rigidly-reinforced composites, the strain field adopts a tile pattern with bounded strain fluctuations, and no singular behavior is observed (to leading order) in the dilute limit.Comment: 28 pages, 28 B&W figures, 2 tables of color maps, to be published in International Journal of Solids and Structures (in press

Non-axisymmetric bending of thin annular plates due to circumferentially distributed moments

AbstractThe non-linear deformation of a thin annular plate subjected to circumferentially distributed bending moments is studied. A von Kármán plate model is adopted to formulate the equations of motion. Free–free boundary conditions have been applied at both inner and outer edges. The matrix formulation of the Differential Quadrature Method is used to discretise and solve the governing equations. Linear analysis predicts that the annular disk deforms axisymmetrically into a spherical dome. However, the proposed non-linear analysis shows that a symmetry breaking bifurcation may occur after which the linear solution becomes unstable and the plate transitions into a non-axisymmetric cylindrical deformation. This is the case when at least one of two parameters reaches a critical value. These parameters are the non-dimensionalised ratio between applied moment and bending stiffness and the ratio between inner and outer radius. Furthermore, it is noted that free-free boundary conditions and circumferentially distributed bending moments do not break the circular symmetry of the annular disk. Hence, the principal axes of curvature in the deformed configuration do not have a preferred orientation. Therefore, the present work describes a shell possessing infinite identical equilibria, having different yet no favoured direction and, hence, links to previous researches on neutrally stable structures

Bilinear stiffness and bimodular Poisson's ratio in cylindrical sinusoidal lattices through topology morphing

Bilinear elastic behaviour allows structural designs to respond in either a stiff or compliant manner depending on the load. Here a cylindrical sinusoidal lattice structure is described that stiffens beyond a certain load. When subjected to axial compression, the lattice can undergo a topological transformation by forming contact connections. This topology change involves a transition from rectangular-like unit cells to kagome-like unit cells, associated with an approximately fourfold increase in stiffness. The lattice exhibits negative Poisson's ratio with a step-change from ≈−0.66 to ≈−0.23 prior to and during contact formation, respectively. After contact formation, it displays a nonlinear Poisson's ratio behaviour. The mechanics underpinning these behaviours are analysed using a combination of experiments and numerical modelling. A comparison with similar planar lattices reveals the effect of the global topology of the lattice (e.g. planar, cylindrical) on the unit cell-level topology morphing. The proposed topology-morphing cylindrical sinusoidal lattice introduces new design possibilities in the application-rich context of tubular structures with nonlinear mechanical properties

Topology morphing lattice structures

Planar cellular lattice structures subject to axial compression may undergo elastic bending or buckling of the unit cells. If sufficient compression is applied, the columns of adjacent cells make contact. This changes the topology of the lattice by establishing new load paths. This topology change induces a corresponding shift in the effective stiffness characteristics of the lattice – in particular, the shear modulus undergoes a step-change. The ability to embed adaptive stiffness characteristics through a topology change allows structural reconfiguration to meet changing load/operational requirements efficiently. The concept, of topological reconfiguration, can be exploited across a range of length scales, from (meta-)materials to components. Here we focus on macroscopic behaviour presenting results obtained from finite element analysis that shows excellent correlation with the observed response of 3D-printed PLA lattices. Through a parametric study, we explore the role of key geometric and stiffness parameters and identify desirable regions of the design space. The non-linear responses demonstrated by this topology morphing lattice structure may offer designers a route to develop bespoke elastic systems. </p

Stiffness tailoring in sinusoidal lattice structures through passive topology morphing using contact connections

Structures with adaptive stiffness characteristics present an opportunity to meet competing design requirements, thus achieving greater efficiency by the reconfiguration of their topology. Here, the potential of using changes in the topology of planar lattice structures is explored to achieve this desired adaptivity and observe that lattice structures with rectangle-like unit-cells may undergo elastic buckling or bending of cell walls when subject to longitudinal compression. Under sufficient load intensity, cell walls can deform and contact neighbouring cells. This self-contact is harnessed to change the topology of the structure to that of a kagome-like lattice, thereby establishing new load paths, thus enabling enhancement, in a tailored manner, of the effective compressive and shear stiffness of the lattice. Whilst this phenomenon is independent of characteristic length scale, we focus on macroscopic behaviour (lattices of scale ≈ 200 mm). Experimentally observed responses of 3D-printed lattices correlate excellently with finite element analysis and analytical stiffness predictions for pre- and post-contact topologies. The role of key geometric and stiffness parameters in critical regions of the design space is explored through a parametric study. The non-linear responses demonstrated by this topology morphing lattice structure may offer designers a new route to tailor elastic characteristics.</p

Multiscale tailoring of helical lattice systems for bespoke thermoelasticity

© 2019 (Meta-)Materials, e.g. functional or architectured materials that change shape in response to external stimuli, often do so by exploiting solid–solid phase transitions or concerted elastic deformations. For the resulting system to be effective the (meta-)material needs to have desirable and tunable properties at length scales sufficiently small that desirable continuum behaviour of the resulting component is obtained. Developing such (meta-)materials has proven to be an endeavour which requires considerable expertise in science, engineering and mathematics. Here, we pursue an alternative approach where the design for functionality is integrated across multiple length scales in the system. We demonstrate this approach by designing and prototyping helical lattices that act as one-dimensional thermoelastic materials with unusual properties such as negative thermal expansivity—with magnitude far exceeding the most extreme values reported in the literature—and zero-hysteresis shape memory. Our strategy is independent of characteristic length scale, allowing us to design behaviour across a range of dimensions