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Register a new userBuckling and wrinkling from geometric and energetic viewpoints
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We discuss shape profiles emerging in inhomogeneous growth of squeezed tissues. Two approaches are used simultaneously: i) conformal embedding of two-dimensional domain with hyperbolic metrics into the plane, and ii) a pure energetic consideration based on the minimization of the total energy functional. In the latter case the non-uniformly pre-stressed plate, which models the inhomogeneous two-dimensional growth, is analyzed in linear regime under small stochastic perturbations. It is explicitly demonstrated that both approaches give consistent results for buckling profiles and reveal self-similar behavior. We speculate that fractal-like organization of growing squeezed structure has a far-reaching impact on understanding cell proliferation in various biological tissues.
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Periodic wrinkling of a rigid capping layer on a deformable substrate provides a useful method for templating surface topography for a variety of novel applications. Many experiments have studied wrinkle formation during the compression of a rigid film on a relatively soft pre-strained elastic substrate, and most have focused on the regime where the substrate thickness can be considered semi-infinite relative to that of the film. As the relative thickness of the substrate is decreased, the bending stiffness of the film dominates, causing the bilayer to transition to either local wrinkling or a global buckling instability. In this work optical microscopy was used to study the critical parameters that determine the emergence of local wrinkling or global buckling of freestanding bilayer films consisting of a thin rigid polymer capping layer on a pre-strained elastomeric substrate. The thickness ratio of the film and substrate as well as the pre-strain were controlled and used to create a buckling phase diagram which describes the behaviour of the system as the ratio of the thickness of the substrate is decreased. A simple force balance model was developed to understand the thickness and strain dependences of the wrinkling and buckling modes, with excellent quantitative agreement being obtained with experiments using only independently measured material parameters.
Thin dielectric elastomers with compliant electrodes exhibit various types of instability under the action of electromechanical loading. Guided by the thermodynamically-based formulation of Fosdick and Tang (J. Elasticity 88, 255-297, 2007), here we provide an energetic perspective on the stability of dielectric elastomers and we highlight the fundamental energetic divide between voltage control and charge control. By using the concept of energy relaxation, we describe wrinkling for neo-Hookean ideal elastomers, and we show that in voltage control wrinkling is stable as long as the tension-extension inequality holds, whereas wrinkling is always stable in charge control. We finally illustrate some examples involving both homogeneous and inhomogeneous deformations, showing that the type and hierarchy of instabilities taking place in dielectric membranes can be tuned by suitable choices of the boundary conditions.
Motivated by recent experiments showing the buckling of microtubules in cells, we study theoretically the mechanical response of, and force propagation along elastic filaments embedded in a non-linear elastic medium. We find that, although embedded microtubules still buckle when their compressive load exceeds the critical value $f_c$ found earlier, the resulting deformation is restricted to a penetration depth that depends on both the non-linear material properties of the surrounding cytoskeleton, as well as the direct coupling of the microtubule to the cytoskeleton. The deformation amplitude depends on the applied load $f$ as $(f- f_c)^{1/2}$. This work shows how the range of compressive force transmission by microtubules can be as large as tens of microns and is governed by the mechanical coupling to the surrounding cytoskeleton.
This article investigates the large deflection and post-buckling of composite plates by employing the Carrera Unified Formulation (CUF). As a consequence, the geometrically nonlinear governing equations and the relevant incremental equations are derived in terms of fundamental nuclei, which are invariant of the theory approximation order. By using the Lagrange expansion functions across the laminate thickness and the classical finite element (FE) approximation, layer-wise (LW) refined plate models are implemented. The Newton-Raphson linearization scheme with the path-following method based on the arc-length constraint is employed to solve geometrically non-linear composite plate problems. In this study, different composite plates subjected to large deflections/rotations and post-buckling are analyzed, and the corresponding equilibrium curves are compared with the results in the available literature or the traditional FEM-based solutions. The effects of various parameters, such as stacking sequence, number of layers, loading conditions, and edge conditions are demonstrated. The accuracy and reliability of the proposed method for solving the composite plates geometrically nonlinear problems are verified.
We investigate wrinkling patterns in a tri-layer torus consisting of an expanding thin outer layer, an intermediate soft layer and an inner core with a tunable shear modulus, inspired by pattern formation in developmental biologies, such as follicle pattern formation during the development of chicken embryos. We show from large-scale finite element simulations that hexagonal wrinkling patterns form for stiff cores whereas stripe wrinkling patterns develop for soft cores. Hexagons and stripes co-exist to form hybrid patterns for cores with intermediate stiffness. The governing mechanism for the pattern transition is that the stiffness of the inner core controls the degree to which the major radius of the torus expands this has a greater effect on deformation in the long direction as compared to the short direction of the torus. This anisotropic deformation alters stress states in the outer layer which change from biaxial (preferred hexagons) to uniaxial (preferred stripes) compression as the core stiffness is reduced. As the outer layer continues to expand, stripe and hexagon patterns will evolve into Zigzag and segmented labyrinth, respectively. Stripe wrinkles are observed to initiate at the inner surface of the torus while hexagon wrinkles start from the outer surface as a result of curvature-dependent stresses in the torus. We further discuss the effects of elasticities and geometries of the torus on the wrinkling patterns.
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