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Metamaterials are composed of structural elements and derive their properties mainly from the inner structure of the elements, rather than the properties of their constituent material. By designing an unstable structural element as the building block of a metamaterial, many interesting effective material properties can be obtained. The deformation and dissipation mechanisms of such a material built from unstable structural elements is studied in detail. To do so a combination of analytical, semi-analytical, and numerical models are applied to a single buckling element, a periodic cell, and finite size combinations of buckling elements including gradients in the properties of the building blocks. This not only provides insight into the micromechanics and the resulting effective behavior of such metamaterials, but also makes them accessible on the different relevant length scales. A metamaterial built from these building blocks shows programmable or switchable properties and can display energy dissipation with fully reversible deformation, distinguishing it from plastic materials, and timescale independent behavior, distinguishing it from viscoelastic materials. [ABSTRACT FROM AUTHOR]

The stability of thin viscous sheets has been studied so far in the special case where the base flow possesses a direction of invariance: the linear stability is then governed by an ordinary differential equation. We propose a mathematical formulation and a numerical method of solution that are applicable to the linear stability analysis of viscous sheets possessing no particular symmetry. The linear stability problem is formulated as a non-Hermitian eigenvalue problem in a 2D domain and is solved numerically using the finite-element method. Specifically, we consider the case of a viscous sheet in an open flow, which falls in a bath of fluid; the sheet is mildly stretched by gravity and the flow can become unstable by ‘curtain’ modes. The growth rates of these modes are calculated as a function of the fluid parameters and of the geometry, and a phase diagram is obtained. A transition is reported between a buckling mode (static bifurcation) and an oscillatory mode (Hopf bifurcation). The effect of surface tension is discussed. [ABSTRACT FROM AUTHOR]

Abstract: We consider the equilibrium shapes of a thin, annular strip cut out in an elastic sheet. When a central fold is formed by creasing beyond the elastic limit, the strip has been observed to buckle out-of-plane. Starting from the theory of elastic plates, we derive a Kirchhoff rod model for the folded strip. A non-linear effective constitutive law incorporating the underlying geometrical constraints is derived, in which the angle the ridge appears as an internal degree of freedom. By contrast with traditional thin-walled beam models, this constitutive law captures large, non-rigid deformations of the cross-sections, including finite variations of the dihedral angle at the ridge. Using this effective rod theory, we identify a buckling instability that produces the out-of-plane configurations of the folded strip, and show that the strip behaves as an elastic ring having one frozen mode of curvature. In addition, we point out two novel buckling patterns: one where the centerline remains planar and the ridge angle is modulated; another one where the bending deformation is localized. These patterns are observed experimentally, explained based on stability analyses, and reproduced in simulations of the post-buckled configurations. [Copyright &y& Elsevier]

Abstract: We present a combined theoretical and experimental study of the buckling of a thin film wrapped around a sphere under the action of capillary forces. A rigid sphere is coated with a wetting liquid, and then wrapped by a thin film having an approximately cylindrical shape. The equilibrium of the film is governed by the competing effects of elasticity and capillarity: elasticity tends to keep the film developable while capillarity tends to curve it in both directions so as to maximize the area of contact with the sphere. In the experiments, the region of contact between the film and the sphere has cylindrical symmetry when the sphere radius is small, but destabilizes to a non-symmetric, wrinkled configuration when the radius is larger than a critical value. We combine the Donnell equations for near-cylindrical shells to include a unilateral constraint with the impenetrable sphere, and the capillary forces acting along a moving edge. A non-linear solution describing the axisymmetric configuration of the film is derived. A linear stability analysis is then presented, which successfully captures the wrinkling instability, the symmetry of the unstable mode, the instability threshold and the critical wavelength. The motion of the free boundary at the edge of the region of contact, which has an effect on the instability, is treated without any approximation. [Copyright &y& Elsevier]

Abstract: In an earlier work, Elliott et al. [2006a, Stability of crystalline solids—II: application to temperature-induced martensitic phase transformations in bi-atomic crystals. Journal of the Mechanics and Physics of Solids 54(1), 193–232], the authors used temperature-dependent atomic potentials and path-following bifurcation techniques to solve the nonlinear equilibrium equations and find the temperature-induced martensitic phase transformations in stress-free, perfect, equi-atomic binary B2 crystals. Using the same theoretical framework, the current work adds the influence of stress to study the model''s stress-induced martensitic phase transformations. The imposition of a uniaxial Biot stress on the austenite (B2) crystal, lowers the symmetry of the problem, compared to the stress-free case, and leads to a large number of stable equilibrium paths. To determine which ones are possible reversible martensitic transformations, we use the (kinematic) concept of the maximal Ericksen–Pitteri neighborhood (max EPN) to select those equilibrium paths with lattice deformations that are closest, with respect to lattice-invariant shear, to the austenite phase and thus capable of a reversible transformation. It turns out that for our chosen parameters only one stable structure (distorted ) is found within the max EPN of the austenite in an appropriate stress window. The energy density of the corresponding configurations shows features of a stress-induced phase transformation between the higher symmetry austenite and lower symmetry martensite paths and suggests the existence of hysteretic stress–strain loops under isothermal load–unload conditions. Although the perfect crystal model developed in this work over-predicts many key material properties, such as the transformation stress and the Clausious–Clapeyron slope, when compared to real experimental values (based on actual polycrystalline specimens with defects), it is—to the authors'' knowledge—the first atomistic model that has been demonstrated to capture all essential trends and behavior observed in shape memory alloys. [Copyright &y& Elsevier]

Abstract: A rod-based model for plant growth and branching is developed in this paper. Specifically, Euler''s theory of the elastica is modified to accommodate growth and remodeling. In addition, branching is characterized using a configuration force and evolution equations are postulated for the flexural stiffness and intrinsic curvature. The theory is illustrated with examples of multiple static equilibria of a branched plant and the remodeling and tip growth of a plant stem under gravitational loading. [Copyright &y& Elsevier]

Abstract: If a body with a stiffer surface layer is loaded in compression, a surface wrinkling instability may be developed. A bifurcation analysis is presented for determining the critical load for the onset of wrinkling and the associated wavelength for materials in which the elastic modulus is an arbitrary function of depth. The analysis leads to an eigenvalue problem involving a pair of linear ordinary differential equations with variable coefficients which are discretized and solved using the finite element method. The method is validated by comparison with classical results for a uniform layer on a dissimilar substrate. Results are then given for materials with exponential and error-function gradation of elastic modulus and for a homogeneous body with thermoelastically induced compressive stresses. [Copyright &y& Elsevier]

Abstract: When a thin elastic structure comes in contact with a liquid interface, capillary forces can be large enough to induce elastic deformations. This effect becomes particularly relevant at small scales where capillary forces are predominant, for example in microsystems (micro-electro-mechanical systems or microfluidic devices) under humid environments. In order to explore the interaction between capillarity and elasticity, we have developed a macroscopic model system in which an initially immersed vertical elastic rod is raised through a horizontal liquid surface. We follow a combined approach of experiments, theory and numerical simulations to study this system. In spite of its apparent simplicity, our experiment reveals a complex phase diagram, involving large hysteretic behaviour. We employ Kirchhoff equations for thin elastic rods and use path-following methods from which we obtain a variety of equilibrium states and associated transitions that are in excellent qualitative and quantitative agreement with those observed experimentally. [Copyright &y& Elsevier]

Abstract: The Kagome Double-Layer Grid (KDLG) is a sandwich-like structure, based on the planar Kagome pattern, which has properties that make it attractive for application as a morphing material. In order to understand the passive and active properties of the KDLG with rigid joints, an analysis is made of the determinacy of the pin-jointed version. The number of internal mechanisms and states of self-stress of the finite pin-jointed structure are calculated as a function of the size of the structure. A statically and kinematically determinate version is obtained by relocating the internal nodes and by prescribing a set of patch bars around the periphery. The actuation performance of the rigid-jointed version is then explored theoretically by replacing a single bar in the structure by an actuator. The resistance to actuation is determined in terms of the stiffness and the allowable actuation strain as dictated by yield and buckling. The paper concludes with the optimal design of a double-layer grid to maximise actuation performance. [Copyright &y& Elsevier]

We examine the problem of an elastic film bonded to an elastic substrate and subjected to compressive stress (induced by growth of the film or lateral compression of the whole system). Following the linear analysis previously carried out to determine the critical growth/compression ratio required to induce wrinkling in the film, we carry out a weakly-nonlinear analysis to derive an amplitude equation that describes the evolution of the wrinkling amplitude beyond the bifurcation point. We carry out a comprehensive numerical bifurcation analysis of the problem using the finite element method and show excellent agreements between the weakly-nonlinear analysis and the numerical experiments. We are also able to solve directly for the bifurcation point in our discretized system and characterize the effect of implementation details such as the aspect ratio of the computational domain on the observed bifurcation point. Finally, we explore solutions of the amplitude equation in the case that the wrinkling amplitude is allowed to vary over long spatial and/or temporal scales. [ABSTRACT FROM AUTHOR]

• Insight into the relative performance of various lattice structures. • Load bearing capacity analysis of lattices using nonlinear continuum mechanics. • General boundary and loading conditions for periodic cells at arbitrary orientations. • Stiffness based failure criterion covering both buckling and progressive softening. • Nonlinear analysis of hierarchical lattice optimized for buckling resistance. A framework is introduced for benchmarking periodic microstructures in terms of their ability to maintain their stiffness under large deformations, accounting in a unified manner both for buckling and softening due to geometric and material nonlinearities. The proposed framework is applied to three classical 2D lattice microstructures at different volume fractions as well as to an optimized hierarchical microstructure from the literature. The high slenderness of the structure members, often assumed in analyses, is demonstrated not to be valid at volume fractions of 10% and above, with the infinitesimal volume fraction solutions underestimating the actual buckling resistance considerably. The performed analyses provide useful and quantitative insight regarding the compressive load carrying capacity of materials with a moderately dense periodic microstructure, in a rather universal and practical form. [ABSTRACT FROM AUTHOR]

This paper addresses localization of the deformation due to buckling that occurs immediately following the onset of bifurcation in the axisymmetric buckling of a perfect spherical elastic shell subject to external pressure. The localization process is so abrupt that the buckling mode of the classical eigenvalue analysis, which undulates over the entire shell, becomes modified immediately after bifurcation transitioning to an isolated dimple surrounded by an unbuckled expanse of the shell. The paper begins by revisiting earlier attempts to analyze the initial post-buckling behavior of the spherical shell, illustrating their severely limited range of validity. The unsuccessful attempts are followed by an approximate Rayleigh-Ritz solution which captures the essence of the localization process. The approximate solution reveals the pathway that begins at bifurcation from the classical mode shape to the localized dimple buckle. The second part of the paper presents an exact asymptotic expansion of the initial post-buckling behavior which accounts for localization and which further exposes the analytic details of the abruptness of the transition. [ABSTRACT FROM AUTHOR]