Shankar C. Venkataramani
This is an invited commentary on "Geometrically incompatible confinement of solids", B. Davidovitch, Y. Sun and G. M. Grason (PNAS, doi:10.1073/pnas.1815507116 , arxiv:1809.06919).
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Buckling dynamics of a solvent-stimulated stretched elastomeric sheet
当单轴拉伸时，薄弹性片可表现出屈曲。屈曲的发生取决于板的几何性质和施加的应变的大小。在这里，我们表明，弹性体片材最初稳定的单轴拉伸下，可以不稳定时，暴露于溶剂，溶胀弹性体。我们通过实验和计算证明了屈曲模式的特征取决于拉伸的大小，这一观察提供了一种新的方法来控制膨胀均匀薄片的形状。
Effects of Buckling on Stress and Strain in Thin Randomly Disordered Tension-Loaded Sheets
研究了随机无序薄板中裂纹屈曲对应力应变的影响。片材被建模为梁的弹性晶格，其中每个梁具有单独的断裂阈值。一个统计分布的指数尾巴对弱或强的光束被用来产生的阈值和无序的大小可以在零和无穷大之间任意变化。沿晶格的顶行和底行施加单轴力偶，断裂根据应力场与局部强度的比率最强的地方进行。由于破裂是从第一个裂纹随机出现的完整板开始的，因此屈曲的开始和模式根据裂纹的生长位置和方式而变化。对于各种各样的疾病的屈曲板的应力-应变关系进行了比较与非屈曲板。的屈曲的最大外力的非屈曲值的系统可以容忍的破坏前的比率被发现，随着增加的混乱，是相应的位移的比率。
Thin sheets that are forced at their boundaries develop a variety of shapes aimed at minimising elastic energy by curving spontaneously in ways that break the symmetry of the sheet and the forcing. Characterising such buckling generally requires detailed analysis. Here, we follow a complementary approach, applicable to axisymmetric systems. It aims to predict the qualitative nature of any buckling using only the simple axisymmetric geometry of the initial system. We view an axisymmetric deformation according to the displacement it imposes on the radial lines making up the sheet. Thus, e.g., in…
Steady progress in the miniaturization of structures and devices has reached a scale where thermal fluctuations become relevant and it is thus important to understand how such fluctuations affect their mechanical stability. Here, we investigate the buckling of thermalized sheets and we demonstrate that thermal fluctuations increase the critical buckling load due to the enhanced scale-dependent bending rigidity for sheets that are much larger than a characteristic thermal length scale. The presented results are universal and apply to a wide range of microscopic sheets. These results are especia…
A growing or compressed thin elastic sheet adhered to a rigid substrate can exhibit a buckling instability, forming an inward hump. Our study shows that the strip morphology depends on the delicate balance between the compression energy and the bending energy. We find that this instability is a first order phase transition between the adhered solution and the buckled solution whose main control parameter is related to the sheet stretchability. In the nearly- unstretchable regime we provide an analytic expression for the critical threshold. Compressibility is the key assumption which allows us …
We consider the axial compression of a thin sheet wrapped around a rigid cylindrical substrate. In contrast to the wrinkling-to-fold transitions exhibited in similar systems, we find that the sheet always buckles into a single symmetric fold, while periodic solutions are unstable. Upon further compression, the solution breaks symmetry and stabilizes into a recumbent fold. Using linear analysis and numerics, we theoretically predict the buckling force and energy as a function of the compressive displacement. We compare our theory to experiments employing cylindrical neoprene sheets and find rem…
Programmed buckling by controlled lateral swelling in a thin elastic sheet
最近的实验已经在薄聚合物膜上施加了受控的溶胀模式，该模式随后弯曲成三维形状。我们开发了一个解决方案的设计问题所建议的这样的系统，即，如果以及如何可以产生特定的三维形状从薄的弹性片仅仅强加的二维图案的局部各向同性的增长。不是所有的形状都是可能的。可能会出现几种类型的阻塞，其中一些取决于板材厚度。我们提供了一些例子，使用轴对称形式的问题，这是分析听话。
Stochastic Model for the Interaction of Buckling and Fracture in Thin Tension-Loaded Sheets
我们介绍了一个模型的断裂，其中包括平面外的自由度来描述在薄板材料的屈曲。该模型是一个规则的正方形网格的弹性梁，刚性连接的节点，以保持旋转不变性。通过位移控制，在I型加载模式下在晶格的顶行和底行施加单轴力偶，引发断裂。这种方法自然地适合于无序的列入，并使各种各样的断裂行为进行研究，从一个简单的几何不连续性的系统，更复杂的裂纹几何形状和随机开裂。破裂可以从一个预裂纹板或从一个完整的板，其中第一个损伤随机出现，和屈曲开始时，位移矢量包含错位分量变得积极有利的一个没有。在本文中，我们只考虑无无序的中心裂纹板，并包括有关的力场和位移场，以及屈曲响应比的一些结果。目前的重点不是对这种系统进行全面研究，而是发展模型本身。
The edges of torn plastic sheets and growing leaves often display hierarchical buckling patterns. We show that this complex morphology (i) emerges even in zero strain configurations, and (ii) is driven by a competition between the two principal curvatures, rather than between bending and stretching. We identify the key role of branch-point (or "monkey-saddle") singularities in generating complex wrinkling patterns in isometric immersions, and show how they arise naturally from minimizing the elastic energy.
Compression-controlled dynamic buckling in thin soft sheets
该研究通过实验和理论分析，揭示了橡胶薄片在高压缩速度下进入亚稳态并发生高阶模态增长与粗化的动态屈曲机制，阐明了材料横截面和压缩速度对屈曲动力学的影响。
As 2D materials such as graphene, transition metal dichalcogenides, and 2D polymers become more prevalent, solution processing and colloidal-state properties are being exploited to create advanced and functional materials. However, our understanding of the fundamental behavior of 2D sheets and membranes in fluid flow is still lacking. In this work, we perform numerical simulations of athermal semiflexible sheets with hydrodynamic interactions in shear flow. For sheets initially oriented in the flow-gradient plane, we find buckling instabilities of different mode numbers that vary with bending …
We investigate the mechanical response of thin sheets perforated with a square array of mutually orthogonal cuts, which leaves a network of squares connected by small ligaments. Our combined analytical, experimental and numerical results indicate that under uniaxial tension the ligaments buckle out-of-plane, inducing the formation of 3D patterns whose morphology is controlled by the load direction. We also find that by largely stretching the buckled perforated sheets, plastic strains develop in the ligaments. This gives rise to the formation of kirigami sheets comprising periodic distribution …
A rectangular thin elastic sheet is deformed by forcing a contact between two points at the middle of its length. A transition to buckling with stress focusing is reported for the sheets sufficiently narrow with a critical width proportional to the sheet length with an exponent 2/3 in the small thickness limit. Additionally, a spring network model is solved to explore the thick sheet limit and to validate the scaling behaviour of the transition in the thin sheet limit. The numerical results reveal that buckling does not exist for the thickest sheets and a stability criterion is established for…
The buckling of thin elastic sheets is a classic mechanical instability that occurs over a wide range of scales. In the extreme limit of atomically thin membranes like graphene, thermal fluctuations can dramatically modify such mechanical instabilities. We investigate here the delicate interplay of boundary conditions, nonlinear mechanics and thermal fluctuations in controlling buckling of confined thin sheets under isotropic compression. We identify two inequivalent mechanical ensembles based on the boundaries at constant strain (isometric) or at constant stress (isotensional) conditions.Rema…
Numerical simulations of thin sheets undergoing large deformations are computationally challenging. Depending on the scenario, they may spontaneously buckle, wrinkle, fold, or crumple. Nature's thin tissues often experience significant anisotropic growth, which can act as the driving force for such instabilities. We use a recently developed finite element model to simulate the rich variety of nonlinear responses of Kirchhoff-Love sheets. The model uses subdivision surface shape functions in order to guarantee convergence of the method, and to allow a finite element description of anisotropical…
We study the stress developed during the drying of a colloidal drop of silica nanoparticles. In particular, we use the wrinkling instability of a thin floating sheet to measure the net stress applied by the deposit on the substrate and we focus on the effect of the particle polydispersity. In the case of a bidisperse suspension, we show that a small number of large particles substantially decreases the expected stress, which we interpret as the formation of lower hydrodynamic resistance paths in the porous material. As colloidal suspensions are usually polydisperse, we show for different avera…
The edge of torn elastic sheets and growing leaves often form a hierarchical buckling pattern. Within non-Euclidean plate theory this complex morphology can be understood as low bending energy isometric immersions of hyperbolic Riemannian metrics. With this motivation we study the isometric immersion problem in strip and disk geometries. By finding explicit piecewise smooth solutions of hyperbolic Monge-Ampere equations on a strip, we show there exist periodic isometric immersions of hyperbolic surfaces in the small slope regime. We extend these solutions to exact isometric immersions through …
It is well known that an annular sheet could wrinkle as a result of axisymmetric tensile loads applied at the edges. In this system, regions under compression appear due to Poisson effect in the azimuthal direction yielding an incompatible excess in length, so that the membrane has to buckle out of the plane. Then, radial wrinkles emerge following the direction of the tensile principal stress. This so called Lamé configuration has been widely used as theoretical and experimental paradigms for wrinkling instabilities. In this work, we explore the consequences of changing the geometry of this mo…
The wrinkle pattern exhibited upon stretching a rectangular sheet has attracted considerable interest in the "extreme mechanics" community. Nevertheless, key aspects of this notable phenomenon remain elusive. Specifically -- what is the origin of the compressive stress underlying the instability of the planar state? what is the nature of the ensuing bifurcation? how does the shape evolve from a critical, near-threshold regime to a fully-developed pattern of parallel wrinkles that permeate most of the sheet? In this paper we address some of these questions through numerical simulations and anal…
Anisotropic surface tension of buckled fluid membrane
固体薄板和流体薄膜在侧向压力作用下表现出屈曲。在这里，它揭示了流体膜具有各向异性屈曲表面张力相反，固体片。令人惊讶的是，表面张力垂直于屈曲方向表现出更强的依赖性比平行于它。我们的理论预测支持无网格膜模型的数值模拟。该各向异性张力可用于测量膜的弯曲刚度。研究还发现多层屈曲膜之间存在相同步现象。
Wrinkling composite sheets
本文通过求解线性化Föppl-von Kármán方程，研究了弯曲刚度空间变化的受限复合薄片在液体基底上的屈曲行为，发现空间依赖弹性系数可消除均匀情形下的褶皱简并性，并使大尺寸薄片的弯曲能量局域化到柔软区域。
We address the fully-developed wrinkle pattern formed upon stretching a Hookean, rectangular-shaped sheet, when the longitudinal tensile load induces transverse compression that far exceeds the stability threshold of a purely planar deformation. At this "far from threshold" parameter regime, which has been the subject of the celebrated Cerda-Mahadevan (CM) model, the wrinkle pattern expands throughout the length of the sheet and the characteristic wavelength of undulations is much smaller than its width. Employing Surface Evolver simulations over a range of sheet thicknesses and tensile loads …
Buckling mediated by mobile localized elastic excitations
研究发现薄板结构转变由可移动的局部弹性激发介导，揭示了褶皱的动态行为及尺寸标度关系。
The preference of thin flat sheets to bend rather than stretch, combined with results from Geometry, mean that changes in a thin sheet's Gaussian curvature are prohibitively expensive. As a result, an imposed curvature in one principal direction inhibits bending in the other: so-called curvature-induced rigidity. Here, we study the buckling behaviour of a rectangular strip of finite thickness held horizontally in a gravitational field, but with a transverse curvature imposed at one end. The finite thickness of the sheet limits the efficacy of curvature-induced rigidity in two ways: (i) finite …
Everyday experience confirms the tendency of adhesive films to detach from spheroidal regions of rigid substrates -- what is a petty frustration when placing a sticky bandage onto an elbow or knee is a more serious matter in the coating and painting industries. Irrespective of their resistance to bending, a key driver of such phenomena is Gauss' \textit{Theorema Egregium}, which implies that naturally flat sheets cannot conform to doubly-curved surfaces without developing a strain whose magnitude grows sharply with the curved area. Previous attempts to characterize the onset of curvature-induc…
We study the periodic buckling patterns that emerge when elastic shells are subjected to geometric confinement. Residual swelling provides access to range of shapes (saddles, rolled sheets, cylinders, and spherical sections) which vary in their extrinsic and intrinsic curvatures. Our experimental and numerical data show that when these structures are radially confined, a single geometric parameter -- the ratio of the total shell radius to the amount of unconstrained material -- predicts the number of lobes formed. We then generalize our model to account for a relaxation of the in-plane constra…
Self-Ordering of Buckling, Bending, Bumping Beams
本文通过实验、模拟和统计力学理论，研究了平行生长弹性梁系统中由屈曲、弯曲和碰撞导致的自有序化现象，预测了保证全局秩序所需的生长量，并发现该超材料的压缩刚度和储存弯曲能与几何受挫梁的数量成正比，为设计可调抗压力学超材料提供了依据。
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