Thin-walled corrugated tubes that have a bending multistability, such as the bendy straw, allow for variable orientations over the tube length. Compared to the long history of corrugated tubes in practical applications, the mechanics of the bending stability and how it is affected by the cross sections and other geometric parameters remain unknown. To explore the geometry-driven bending stabilities, we used several tools, including a reduced-order simulation package, a simplified linkage model, and physical prototypes. We found the bending stability of a circular two-unit corrugated tube is dependent on the longitudinal geometry and the stiffness of the crease lines that connect separate frusta. Thinner shells, steeper cones, and weaker creases are required to achieve bending bi-stability. We then explored how the bending stability changes as the cross section becomes elongated or distorted with concavity. We found the bending bi-stability is favored by deep and convex cross sections, while wider cross sections with a large concavity remain mono-stable. The different geometries influence the amounts of stretching and bending energy associated with bending the tube. The stretching energy has a bi-stable profile and can allow for a stable bent configuration, but it is counteracted by the bending energy which increases monotonically. The findings from this work can enable informed design of corrugated tube systems with desired bending stability behavior.

Electrolyte in a rechargeable Li-ion battery plays a critical role in determining its capacity and efficiency. While the typically used electrolytes in Li-ion batteries are liquid, soft solid electrolytes are being increasingly explored as an alternative due to their advantages in terms of increased stability, safety and potential applications in the context of flexible and stretchable electronics. However, ionic conductivity of solid polymer electrolytes is significantly lower compared to liquid electrolytes. In a recent work, we developed a theoretical framework to model the coupled deformation, electrostatics and diffusion in heterogeneous electrolytes and also established a simple homogenization approach for the design of microstructures to enhance ionic conductivity of composite solid electrolytes. Guided by the insights from the theoretical framework, in this paper, we ex- amine specific microstructures that can potentially yield significant improvement in the effective ionic conductivity. We numerically implement our theory in the open source general purpose finite element package FEniCS to solve the governing equations and present numerical solutions and insights on the effect of microstructure on the enhancement of ionic conductivity. Specifically, we investigate the effect of shape by considering ellipsoidal inclusions. We also propose an easily manufacturable microstructure that increases the ionic conductivity of the composite electrolyte by forty times, simply by the addition of dielectric columns parallel to the solid electrolyte phase.

An automatic complex topology lightweight structure generation method (ACTLSGM) is presented to automatically generate 3D models of lightweight truss structures with a boundary surface of any shape. The core idea of the ACTLSGM is to use the PIMesh, a mesh generation algorithm developed by the authors, to generate node distributions inside the object representing the boundary surface of the target complex topology structures; raw lightweight truss structures are then generated based on the node distributions; the resulting lightweight truss structure is then created by adjusting the radius of the raw truss structures using an optimization algorithm based on finite element truss analysis. The finite element analysis-based optimization algorithm can ensure the resulting structures satisfy the design requirements on stress distributions or stiffness. Three demos, including a lightweight structure for a cantilever beam, a femur bone scaffold, and a 3D shoe sole model with adaptive stiffness that can be used to adjust foot pressure distributions for patients with diabetic foot problems, are generated to demonstrate the performance of the ACTLSGM. The ACTLSGM is not limited to generating 3D models of medical devices, but can be applied in many other fields, including 3D printing infills and other fields where customized lightweight structures are required.

Transferring completed electronic devices onto curvilinear surfaces is popular for fabricating three-dimensional curvilinear electronics with high performance, while the problems of conformality between the unstretchable devices and the surfaces need to be considered. Prior conformability design based on conformal mechanics model is a feasible way to reduce the non-conformal contact. Former studies mainly focused on stretchable film electronics conforming onto soft bio-tissue with a sinusoidal form microscopic morphology or unstretchable film conforming onto rigid sphere substrate, which limits its applicability in the aspect of shapes and modulus of the substrate. Here, a conformal mechanics model with general geometric shape and material is introduced by choosing a bicurvature surface as the target surface, and the conformal contact behavior of film electronics is analyzed. All eight fundamental local surface features are obtained by adjusting two principal curvatures of the bicurvature surface, and the conformal performance is simulated. A dimensionless conformal criterion is given by minimizing the total energy as a function of seven dimensionless parameters, including four in geometric and three in the material. Thickness–width ratio and length–width ratio of flexible electronics are two key geometric parameters that decide the conformal behavior, and smaller thickness–width ratio and length–width ratio are favorite for conformal contact. The model and analysis results are verified by the finite element analysis, and it can guide the prior conformability design of the curvilinear electronic devices during the planar manufacturing process.

This paper has revisited the roof deformation and collapse of stamps with isolated grooves based on a contact mechanics approach, with emphasis on establishing the nonadhesive and adhesive contact solutions for surfaces containing a shallow rectangular groove with the effects of applied load and interfacial adhesion taken into account. By solving singular integral equations and using the energy release rate approach, closed-form solutions are derived analytically for the deformed groove shapes, interfacial stress distributions, and equilibrium relations between load and contact size, which reduce to the previously proposed solutions without adhesion or without applied load. Finite element (FE) analysis is performed to validate the nonadhesion solutions, while experiment results of stamp collapse reported in the literature are adopted to examine the adhesion solutions. By introducing the Johnson parameter α to represent a competition between surface energy and elastic strain energy of the groove, four kinds of contact behaviors of the groove roof can be characterized appropriately: nonadhesion, weak adhesion, intermediate adhesion, and strong adhesion. Hysteresis loop and energy loss due to distinct load/unloading paths are revealed in the cases of intermediate and strong adhesion. We have also provided the critical applied pressure to achieve roof collapse and the corresponding equilibrium contact size for full range of α.

A method is presented for evaluating the axial critical loads and deformed shapes of a Vierendeel periodic girder. They are obtained by solving the large-deflections equilibrium problem of a micro-polar equivalent model. The elastic properties of this model have been derived from those of an ideal girder whose cells may deform only according to the inner forces transferring modes of the unit cell. In particular, the strain energy density of the equivalent medium is obtained by evaluating the limit of the girder elastic energy for the cell to girder size ratio tending to zero. To solve the Engesser/Haringx discord, the large-deflections equilibrium equations are deduced by the virtual work principle, without any a priori assumption on the shear force. For this aim, also the external work of the substitute medium is evaluated by the same procedure as the strain energy density. Actually, it is first written for the ideal girder under the assumption of negligibility of the axial effects at the cell scale and then specialized for the continuous medium by examining the limit value for cell to girder size ratio going to zero. Closed form solutions for the critical deformed shapes are given, together with an accurate formula for the girder buckling load.

Motivated by the observations of snap-through phenomena in pre-stressed strips and curved shells, we numerically investigate the snapping of a pre-buckled hemispherical gridshell under apex load indentation. Our experimentally validated numerical framework on elastic gridshell simulation combines two components: (i) discrete elastic rods method, for the geometrically nonlinear description of one-dimensional rods, and (ii) a naive penalty-based energy functional, to perform the non-deviation condition between two rods at joint. An initially planar grid of slender rods can be actuated into a three-dimensional hemispherical shape by loading its extremities through a prescribed path, known as buckling-induced assembly; next, this pre-buckled structure can suddenly change its bending direction at some threshold points when compressing its apex to the other side. We find that the hemispherical gridshell can undergo snap-through buckling through two different paths based on two different apex loading conditions. The structural rigidity increases as the number of rods in the gridshell structure becomes denser, which emphasizes the mechanically nonlocal property in hollow grids, in contrast to the local response of continuum shells. The findings may bridge the gap among rods, grids, knits, and shells, for a fundamental understanding of a group of thin elastic structures, and inspire the design of novel micro-electro-mechanical systems and functional metamaterials.

This article presents a novel algorithm to predict the shape and the support configurations of a magnetic robot acting like a supple beam, which deforms due to an imposed magnetic field. The soft magnet loosely rests on a flat horizontal surface, which provides the pivoting supports to the robot to attain its deformed shape. The key difficulty in analysis of the mechanics is the lack of any prior knowledge about the location of the contacts where the distorted beam finds support from the ground. This article outlines an algorithm in which different possibilities referred to as modes are checked systematically to locate the placement and the nature of such supports. Consequently, the 2D shape of the soft beam can be determined without any heuristic assumption about where the magnetic robot is touching the solid surface. This study focuses on single contact sections although the algorithm idea is valid for multiple contacts as well. The mathematical theory and the numerical scheme are validated by comparing the simulated results with existing experimentally obtained configurations. Also, the parametric space of system-defining nondimensional parameters is explored to determine when a transition happens from one mode to another and which magnetization methods are desirable for higher stability. Through theoretical simulation, the results show that a thin rectangular cross section provides higher deformation when compared to the other two tested shapes (circular and equilateral triangle cross section).

The paper extends the recent work (Wu, C., and Yin, H., 2021, “Elastic Solution of a Polygon-Shaped Inclusion With a Polynomial Eigenstrain,” ASME J. Appl. Mech., 88(6), p. 061002) of Eshelby’s tensors for polynomial eigenstrains from a two-dimensional (2D) to three-dimensional (3D) domain, which provides the solution to the elastic field with continuously distributed eigenstrain on a polyhedral inclusion approximated by the Taylor series of polynomials. Similarly, the polynomial eigenstrain is expanded at the centroid of the polyhedral inclusion with uniform, linear, and quadratic order terms, which provides tailorable accuracy of the elastic solutions of polyhedral inhomogeneity using Eshelby’s equivalent inclusion method. However, for both 2D and 3D cases, the stress distribution in the inhomogeneity exhibits a certain discrepancy from the finite element results at the neighborhood of the vertices due to the singularity of Eshelby’s tensors, which makes it inaccurate to use the Taylor series of polynomials at the centroid to catch the eigenstrain at the vertices. This paper formulates the domain discretization with tetrahedral elements to accurately solve for eigenstrain distribution and predict the stress field. With the eigenstrain determined at each node, the elastic field can be predicted with the closed-form domain integral of Green’s function. The parametric analysis shows the performance difference between the polynomial eigenstrain by the Taylor expansion at the centroid and the C 0 continuous eigenstrain by particle discretization. Because the stress singularity is evaluated by the analytical form of Eshelby’s tensor, the elastic analysis is robust, stable, and efficient.

Force chains have been regarded as an important hallmark of granular materials. Numerous studies have examined their evolution, properties, and statistics in highly idealized, often circular-shaped, granular assemblies. However, particles found in nature and handled in industries come in a wide variety of shapes. In this article, we experimentally investigate the robustness of force chains with respect to particle shape. We present a detailed analysis on the particle- to continuum-scale response of granular materials affected by particle shape, which includes the force transmission and mobilized shear strength. The effect of shape is studied by comparing experimental results collected from shear tests performed on 2D analog circular- and arbitrarily shaped granular assemblies. Particle shapes are directly discretized from X-ray CT images of a real sand sample. By inferring individual contact forces using the granular element method (GEM), we provide a direct visualization of the force network, a statistical characterization of the force transmission and a quantitative description of the shear strength in terms of rolling, sliding, and interlocking contact mechanisms. We report that force chains are less prevalent in assemblies of arbitrarily-shaped particles than in circular-shaped samples. Furthermore, interlocking is identified as the essential contact mechanism that (1) furnishes a stable structure for force chains to emerge and (2) explains the enhanced shear strength observed in the arbitrarily-shaped samples. These findings highlight the importance of accounting for particle shape to capture and predict the complex mechanical behavior of granular materials across scales.

The Kresling truss structure, derived from Kresling origami, has been widely studied for its bi-stability and various other properties that are useful for diverse engineering applications. The stable states of Kresling trusses are governed by their geometry and elastic response, which involves a limited design space that has been well explored in previous studies. In this work, we present a magneto-Kresling truss design that involves embedding nodal magnets in the structure, which results in a more complex energy landscape, and consequently, greater tunability under mechanical deformation. We explore this energy landscape first along the zero-torque folding path and then release the restraint on the path to explore the complete two-degree-of-freedom behavior for various structural geometries and magnet strengths. We show that the magnetic interaction could alter the potential energy landscape by either changing the stable configuration, adjusting the energy well depth, or both. Energy wells with different minima endow this magneto-elastic structure with an outstanding energy storage capacity. More interestingly, proper design of the magneto-Kresling truss system yields a tri-stable structure, which is not possible in the absence of magnets. We also demonstrate various loading paths that can induce desired conformational changes of the structure. The proposed magneto-Kresling truss design sets the stage for fabricating tunable, scalable magneto-elastic multi-stable systems that can be easily utilized for applications in energy harvesting, storage, vibration control, as well as active structures with shape-shifting capability.

Programming desired shapes of stimuli-responsive nematic sheets is a challenging yet crucial problem for morphing applications. Here, we propose a strategy for inverse design of axisymmetric shapes via circular bilayers composed of two glassy nematic layers with radially symmetric and orthogonal director fields. In this situation, an explicit link can be established between the director orientation and the bilayer deflection. The result provides a recipe for how to construct a director pattern which, upon stimulation, leads to a prescribed shape of revolution. For illustration, two examples concerning simply supported and loosely clamped bilayer discs are given.

In the present paper the author discusses the influence of such factors as temperature, width, and geometrical similarity in dimensions of specimens on the impact values as obtained by means of the notched-bar test. Various shapes of specimens for notched-bar tests are compared and their impact values determined.

The nonlinear bending of straight and circular-arc cantilevers under vertical and horizontal point loads is analyzed from a unified approach. Formulas for determining the deflected shape of the cantilevers are presented. Closed-form solutions are obtained for bending under two types of distributed loads. In particular, the problem of bending under a uniformly distributed normal load is solved.

Tests have been carried out on thin closed-ended tubes of alpha brass subjected to various combinations of torque and internal pressure. The effect of loading, unloading, and reloading along different loading paths has been investigated, including the effect on the shape of the yield surface, and the form of the curve of representative stress and representative strain. The behavior of the material for initial loading suggests that the material is isotropic in the plane of the surface of the tube but is anisotropic in a radial direction. A form of yield criterion and representative stress which correlates these results has been deduced, but the results of the tests for the second loadings of the specimens cannot be correlated in terms of a nested set of similar yield surfaces. The evidence for or against the existence of corners or pointed vertices on the yield surface is inconclusive.

Frequency equations and modal shapes are derived for the transverse vibrations of beam grillages consisting of two sets of equally spaced uniform beams, the mass of the beams and associated plating being taken as lumped at the nodes.

This paper contains an exact solution for the stress distribution in an elastic ellipsoid rotating about a free axis. The surface of the body is free from boundary stresses. The stresses are expressed with the use of 12 coefficients which have been tabulated for 209 ellipsoids of different shapes. General diagrams give the maximum values of stresses as functions of two shape ratios.

Limit loads are found for rigid-plastic rings of arbitrary cross section subjected to uniform twisting moments and radial forces distributed over the circumference. It is shown that the interaction curves for limiting values of twisting moment and radial force have the same shape as interaction curves for limiting bending and stretching of straight beams. For rings of small uniform thickness the collapse loads are found to agree very well with those predicted by thin-plate theory. The theory is applied to find the collapse load of thin-walted corrugated tubes under axial thrust.

A method is presented for calculating the development of momentum and thermal laminar boundary layers on a heated cylinder of arbitrary shape when the cylinder moves in an incompressible fluid at rest with an unsteady velocity. This analysis is based on solutions to the unsteady momentum and energy-integral equations in conjunction with a set of universal functions, derived from exact solutions to the boundary-layer equations for a specific unsteady problem. These universal functions are given in tabulated form. Those associated with the energy-integral equation are calculated with a Prandtl number of 0.7. The reliability and limitation of these functions are indicated and discussed in the light of several simple problems of which solutions are available. A detailed calculation procedure for the general unsteady problem is given and then followed by a numerical example.

The solution of the torsion problem for a slender rectangular section has been made previously by approximate methods based on the Prandtl membrane analogy. In this paper approximate methods are employed in the solution of both the torsion and flexural shear problem for slender sections having a variety of shapes, most of them being doubly symmetric. Solutions obtained in this manner are compared with exact solutions, when these are available, and otherwise with solutions obtained by relaxation. It is shown that approximate methods provide an adequate solution for elements such as compressor-turbine blades when pretwist and taper can be neglected. Some attention is given to the problem of elastic-plastic torsion and elastic-plastic flexural shear of slender sections.