Anticlastic Bending Research Articles

Phase Diagrams for Anticlastic and Synclastic Bending Curvatures of Hexagonal and Re-entrant Honeycombs

Abstract Transformation between saddle- and dome-like bending shapes of regular and re-entrant hexagonal honeycomb panels are explored. An analytical model is proposed to uncover the underlying mechanisms and identify the controlling parameter when the cell walls are slender beams. Then, 3D finite element simulations are performed to examine the architecture dependence of bending shape and construct the phase diagrams of anticlastic and synclastic curvatures when the cell walls have a general geometry. The results are believed helpful to the design and application of related honeycomb structures.

Derivation of an effective plate theory for parallelogram origami from bar and hinge elasticity

Periodic origami patterns made with repeating unit cells of creases and panels bend and twist in complex ways. In principle, such soft modes of deformation admit a simplified asymptotic description in the limit of a large number of cells. Starting from a bar and hinge model for the elastic energy of a generic four parallelogram panel origami pattern, we derive a complete set of geometric compatibility conditions identifying the pattern’s soft modes in this limit. The compatibility equations form a system of partial differential equations constraining the actuation of the origami’s creases (a scalar angle field) and the relative rotations of its unit cells (a pair of skew tensor fields). We show that every solution of the compatibility equations is the limit of a sequence of soft modes — origami deformations with finite bending energy and negligible stretching. Using these sequences, we derive a plate-like theory for parallelogram origami patterns with an explicit coarse-grained quadratic energy depending on the gradient of the crease-actuation and the relative rotations of the cells. Finally, we illustrate our theory in the context of two well-known origami designs: the Miura and Eggbox patterns. Though these patterns are distinguished in their anticlastic and synclastic bending responses, they show a universal twisting response. General soft modes captured by our theory involve a rich nonlinear interplay between actuation, bending and twisting, determined by the underlying crease geometry.

Mechanical Properties of Glass Plate During Anticlastic Cold Bending

Anticlastic glass surfaces play a significant role in free-form glass facades. For realizing anticlastic surface, cold bending by loading at the corner of the plate is more adaptive and more economic than traditional hot bending method. Previous research on anticlastic cold bending mainly focuses on the description of instability phenomenon and qualitative analysis of parameters. However, the failure mechanisms of glass plates during cold bending and the influence of lamination remains unclear. In this paper, the anticlastic cold bending test was conducted to explore the influence of various factors, including aspect ratio, scale and composition of the plates. Subsequently, an effective finite element model was established and validated by test results, which is used for further explore the cold bending controlling condition for better engineering practice. The failure modes are considered as instability and strength failure. For laminated glass, maximum stresses can be derived from monolithic glass based on equivalent thickness method. The instability is induced by the compression area in the middle surface of glass plate which is significantly influenced by the composition of the laminates. Consequently, thin glass laminates exhibit enhanced stability because a reduced glass to PVB thickness ratio changes the compression area from bi-directional to uni-directional.

Unexpected bending behavior of architected 2D lattice materials.

Architected two-dimensional (2D) lattice materials consisting of elastic beams are appealing because of their tunable sign of Poisson's ratio. A common belief is that such materials with positive and negative Poisson's ratios display anticlastic and synclastic curvatures, respectively, when bent in one direction. Here, we show theoretically and demonstrate experimentally that this is not true. For 2D lattices with star-shaped unit cells, we find a transition between anticlastic and synclastic bending curvatures controlled by the beam's cross-sectional aspect ratio even at a fixed Poisson's ratio. The mechanisms lay in the competitive interaction between axial torsion and out-of-plane bending of the beams and can be well captured by a Cosserat continuum model. Our result may provide unprecedented insights to the design of 2D lattice systems for shape-shifting applications.

Failure behavior of laminated composite plates under anticlastic bending

In this study, the failure behavior of laminated composite plates subjected to anticlastic bending is investigated. Anticlastic bending is a special case of bi-axial out-of-plane loading that produces predominantly twisting moment. A test fixture is designed to actualize anticlastic loading condition. Carbon-fiber-reinforced epoxy composite is selected as the material of the specimens. Totally nine different layup configurations are chosen and at least four specimens are tested for each configuration. Acoustic emission monitoring (AEM) is utilized to detect the first-ply-failure load and the accumulation of damage in the laminates. In this method, elastic waves resulting from initiation and progression of damage are detected, and then AE signals are processed to identify the failure modes and determine the first-ply-failure load. A finite element model is developed to simulate the anticlastic bending test. A code is developed using ANSYS parametric design language to predict the first-ply-failure loads using various failure criteria like Tsai–Wu, Hashin, and Puck. The experimental and numerical results are then compared. Relative strengths and weaknesses of the failure criteria in estimating failure of laminated composite plates under anticlastic bending are discussed.

Beam steering with quasi-mosaic bent silicon single crystals: computer simulations for 855 MeV and 6.3 GeV electrons and comparison with experiments

Monte Carlo simulations have been performed for 855 MeV and 6.3 GeV electrons channeling in silicon single crystals at circular bent (111) planes. The aim was to identify critical experimental parameters which affect the volume-deflection and volume-capture characteristics. To these belongs the angular alignment of the crystal with respect to the nominal beam direction. The continuum potential picture has been utilized. The simulation results were compared with experiments. It turns out that the assumption of an anticlastic bending of the crystal, bent on the principle of the quasi-mosaic effect, is not required to reproduce the gross features of the experimental observations for two examined examples.Graphical abstract

Bending of hard-magnetic soft beams: A finite elasticity approach with anticlastic bending

Soft-materials that respond to external stimuli are very useful for application in soft robotics, stretchable electronics, biomedical applications etc. Recent times have seen a surge in research of magnetically activated polymers due to their wide range of material properties, applications and important features such as non-contact, fast and non-invasive actuation. Furthermore, these applications ask for Hard-magnetic particles, which can retain their magnetization even after the applied magnetic field has been removed. In this work, we focus on developing a thermodynamically-consistent analytical solution to the large bending deformation of hard-magnetic soft hyperelastic beams under the influence of an applied uniform magnetic field. The principal stress in the cross-sections, which arise due to anticlastic bending were also calculated. Lastly, a Prony series approximation was used to encapsulate the time dependent response of the material properties of the soft beam. The model was verified by comparing the results to previously developed experimental, numerical and analytical results.

Study on the mechanical response of anticlastic cold bending insulating glass and its coupling effect with uniform load.

Cold bending is a characteristic of significance for the beautiful curved glass curtain walls, because it affects them in terms of energy-efficiency and cost-efficiency. The increasing engineering projects call for more special studies on the mechanical properties of cold-bent glass panels, especially when the walls are built by insulating glass that is currently widely used while its relevant research is very scarce. This paper is devoted to studying the mechanical properties of anticlastic cold-bent insulating glass while taking different factors into consideration, including glass thickness, cold-bent torsion rate and cavity thickness. 9 pieces of insulating glass were manufactured for anticlastic cold-bending test and their coupled effect with identical load is also studied, and numerical finite element analysis sessions were carried out to simulate the experimental results for each one of them. Further, we analyzed the stress distribution performance of the sample pieces under cold bending and a uniform load, followed by discussions about stress transfer controls in glass plates. The results showed that the cold-bent control stress is on the surface with direct loads from cold bending and close to the cold-bent corner on the short edge, and it is transferred from the parts around the corner to the center when the uniform load plays a leading role in generating stress. This transfer could occur under a relatively small load with a small cold-bent torsion rate. A higher cold-bent torsion rate in cold bending contributed mostly to greater center stress in the glass, and as the glass thickness grows, stress and deflection at the plate center would significantly drop. However, the effect of cavity thickness on the anticlastic mechanical response of insulating glass was found to be trivial.

Finite anticlastic bending of hyperelastic laminated beams with a rubberlike core

A novel analytical approach to investigate the finite bending of hyperelastic laminated beams is presented. Two different nonlinear material models are taken into account, which are the compressible Mooney-Rivlin for rubberlike mediums and the Saint Venant-Kirchhoff for less deformable materials. The anticlastic bending is included in the formulation and the analytical expression of the transverse radius of curvature is presented. The stress analysis is performed in each layer separately, by considering the actual stored energy function of the constituents, in both Lagrangian and Eulerian frameworks. The finite bending of a sandwich beam is investigated in terms of stresses and stretches.

Competing anticlastic and piezoelectric deformation at large deflections

In bending of a purely elastic beam or plate, it is well established that the cross-sectional shape changes character with decreasing bending radius-of-curvature and that the transition can be characterized by the Searle parameter. In a piezoelectric structure, the cross-sectional deformation is affected by the opposite anticlastic and electromechanical bending curvatures. The behavior is consequently more complicated and it is an open question how the cross-sectional shape develops with increasing bending. In this paper, analytical solutions are used to study the cross-sectional deformation of piezoelectric cantilever-actuators taking both anticlastic and electromechanical bending effects into account. We consider unimorph and bimorph actuators. In the case of electrical actuation, as for the purely mechanical case, we find that the Searle parameter is an important parameter characterizing the shape of the cross-section. A load scaling rule gives a criterion for fixed cross-section-deflection for different actuator widths. Using this scaling rule, the Searle parameter is kept unchanged. The analytical results are verified by non-linear finite element analysis using electric potential and mechanical moment as applied loads.

Closed-form solution and analysis of the plate twist test in sandwich and laminated composites

A new anticlastic bending solution for the deflection and rotations of laminated composite plates and sandwich panels applicable to plates under torsion or bending with opposite curvatures is developed using first-order shear deformation theory and the theorem of minimum potential energy. The novelty of the proposed solution relies on being algebraic, taking into account the overhang, employing variable shear correction factors and twisting moment (Mxy), and considering the influence of the twist (D66), bending (D11,D12,D22) and shear stiffnesses (A44,A55). A finite element model is also developed for comparison. Both approaches are validated with compliance measurements previously reported, observing good agreement (less than 10% difference with respect to experiments) for most orthotropic and isotropic plates. Because Mxy is the main driving-load of anticlastic bending, its distribution is analyzed on several plates. According to finite element modeling, Mxy attains a quarter of the applied load (P/4) within the mid-plane of a glass/epoxy 0/906s laminated composite, but it has a parabolic behavior for sandwich panels with compliant cores. Good match is obtained between compliance predictions by the first-order shear deformation theory and the finite element model for square laminate plates with aspect ratios between (8.4<width/thickness<101) and square sandwich panels with (3.9<width/thickness<59) and shear modulus ratios (from 1 to 1800) between core and face sheets.

Anticlastic bending impact on webs transiting rollers

Webs are subjected to large out-of-plane deformations when transiting rollers in process machinery. Webs are often treated as membranes in analysis but become subject to significant bending strains when transiting rollers. Anticlastic bending of thick plates is a known phenomenon. The anticlastic effect is often ignored for webs which are thin. The objective of this paper is to demonstrate that the large bending deformations webs are subjected to on rollers influence the internal membrane stresses and deformations in the web. The results will show that the concept of normal entry of a web to a roller has complexity that has previously not been considered. It will be demonstrated that a cross direction tensile membrane stress results from the large deformations that acts to stabilize the web and inhibit wrinkle formation.

Thermoelastic and vibration response analysis of shape memory alloy reinforced active bimorph composites

In this paper, thermoelastic and free vibration responses of hybrid Shape Memory Alloy (SMA) and E-glass fiber-reinforced composites are investigated using finite element analysis. We consider four composites for analysis and comparison of performances, (1) single layer SMA fiber-reinforced hybrid composite (unimorph SMAHC), (2) a two-layer (cross-ply) SMA fiber-reinforced hybrid composite (bimorph SMAHC), (3) single layer SMA fiber-reinforced with a honeycomb core, and (4) single layer SMA fiber-reinforced with reentrant-honeycomb or auxetic core. We analyze the bending response of all the aforementioned composite structures, focusing on the bimorph anticlastic and synclastic bending with negative and positive Gaussian curvatures, respectively. The uniqueness of this work is utilizing the shape memory effect of the SMA fibers to achieve active shape control of low-stiffness and lightweight auxetic composite structures. We investigated the free vibration response and analyzed the effect of the fiber volume fraction and the fiber orientation angle on the natural frequency of the system. Further, the role of the reentrant angle and stiffness of honeycomb struts on the overall effective mechanical properties of the ply is also studied. Results confirm the non-linearity of the parametric combination and the importance of optimization of these parameters for improved Gaussian curvature. The findings are envisaged to be useful for the design of lightweight rigidizable space structures and tensairity structures.

Viscoelastic web curl due to storage in wound rolls

Winding is often the final operation in a roll-to-roll manufacturing process. Web materials, i.e., materials that are thin compared to their length, are wound into rolls because this form is the only practical means to store them. The resulting bending strains and associated stresses are large for thick webs and laminates. As many webs are viscoelastic on some time scale, bending stresses lead to creep and inhomogeneous changes in length. When the web material is unwound and cut into discrete samples, a residual curvature remains. This curvature, called curl, is the inability for the web to lie flat at no tension. Curl is an undesirable web defect that causes loss of productivity in a subsequent web process. This paper describes the development and implementation of modeling and experimental tools to explore and mitigate curl in homogenous webs. Two theoretical and numerical methods that allow the prediction of curl in a web are developed: a winding software based on bending recovery theory, and the implementation of dynamic simulations of winding. One experimental method is developed that directly measures the curl online by taking advantage of the anticlastic bending resulting from the curl. These methods are demonstrated for a low-density polyethylene web.

The Bending of Beams in Finite Elasticity

In this paper the analysis for the anticlastic bending under constant curvature of nonlinear solids and beams, presented by Lanzoni, Tarantino (J. Elast. 131:137–170, 2018), is extended and further developed for the class of slender beams. Following a semi-inverse approach, the problem is studied by a three-dimensional kinematic model for the longitudinal inflexion, which is based on the hypothesis that cross sections deform preserving their planarity. A compressible Mooney-Rivlin law is assumed for the stored energy function and from the equilibrium equations, the free parameter of the kinematic model is computed. Thus, taking into account the three-dimensionality of the beam, explicit formulae for the displacement field, the stretches and stresses in every point of the body, following both Lagrangian and Eulerian description, are derived. Subsequently, slender beams under variable curvature were examined, focusing on the local determination of the curvature and bending moment along the deformed beam axis. The governing equations take the form of a coupled system of three equations in integral form, which is solved numerically. The proposed analysis allows to study a very wide class of equilibrium problems for nonlinear beams under different restraint conditions and subject to generic external load systems. By way of example, the Euler beam and a cantilever beam loaded by a dead or live (follower) concentrated force applied at the free end have been considered, showing the shape assumed by the beam as the load multiplier increases.

Exploring photo-deformation of polydomain liquid crystal polymers by homogenization

ABSTRACTThis article explores photo-deformation of polydomain liquid crystal polymers. Photo-isomerization-driven stimuli differ from thermal ones in mechanical response in that domains deform in differing amounts according to their relative alignment with respect to light polarization, by contracting along their liquid crystal orientation by an amount maximized at parallel alignment, and therefore polydomains’ response to light is complicated. Theoretical analysis of polydomain deformation with random domain alignments can be greatly simplified by applying homogenization method to obtain effective overall deformation. Not only is polydomain photo deformation very different from monodomain, but domain distributions enable very different photo responses. Anticlastic, syn-clastic and cylindrical-shell bending modes will be predicted, dictated by order parameter and material thickness. This work serves as a guide to theoretical analysis of composite materials with microstructure and to controlling their morphing shapes experimentally.

Geometrically non-linear bending of plates: Implications in curved building façades

Curved glazed façades of buildings are often realized by cold bending of flat plates. This paper is devoted to experimental tests under static loading on the geometrically non-linear anticlastic bending of aluminium plates, whose constituent material is used as a phantom material to mimic the elastic mechanical behaviour of glass. The 3D displacement field of the plates is reconstructed through the use of the DIC (Digital Image Correlation) technique and several numerical FE analyses are performed to simulate the experimental results. A critical value, corresponding to a configuration of the plate where a change in its curvature sign occurs, of the applied displacement at the plate corner is determined. The results are discussed in terms of critical applied displacement against the plate thickness for different plate sizes. It is shown that the critical displacements are almost independent on the plate size, but linearly depend on the plate thickness.

Interlaminar Stresses in Composite Laminates Subjected to Twisting Deformation

The interlaminar stress in angle-ply and cross-ply composite laminates subjected to twisting deformation are investigated. Two mechanisms of interlaminar load transfer have been developed by studying the angle-ply laminate and the cross-ply laminate subjected to uniform axial extension, thermoelastic deformation and anticlastic bending deformation. In the present, these mechanisms are investigated in laminates subjected to twisting deformation. It is shown that the mechanisms of interlaminar load transfer in twisting deformation are identical to those previously investigated, though they arise from different causes. Furthermore, a unified treatment of the mechanisms of interlaminar load transfer is presented for the angle-ply laminate and the cross-ply laminate subjected to the four aforementioned modes of deformation.

Bent Bragg–Laue monochromator for high-energy X-rays

A bent Bragg–Laue monochromator (BLM) is proposed for high-energy X-ray (∼25–60 keV) beamlines. The BLM has the unique feature of bi-directional focusing. A sagittally bent Laue crystal can focus the large horizontal fan of a bending magnet or wiggler source. A meridionally bent Bragg crystal focuses the beam vertically and corrects for the anticlastic bending effects of the Laue crystal. This monochromator geometry relies on the crystal orientations being optimized. It is shown that the focusing condition and Rowland condition can be simultaneously satisfied at a given energy. Detailed ray tracings indicate that a BLM can provide similar energy resolution and higher flux density compared to a sagittally bent double-Laue monochromator configuration. A prototype BLM with a symmetric Bragg crystal and an asymmetric Laue crystal was tested. Matching of the bend radii of the two crystals in the meridional direction was demonstrated. In general, the horizontal acceptance of the sagittally bent Laue crystal is limited by the large curvature. This horizontal BLM acceptance could be increased by translating the Laue crystal along its sagittal bending axis.

Finite Anticlastic Bending of Hyperelastic Solids and Beams

This paper deals with the equilibrium problem in nonlinear elasticity of hyperelastic solids under anticlastic bending. A three-dimensional kinematic model, where the longitudinal bending is accompanied by the transversal deformation of cross sections, is formulated. Following a semi-inverse approach, the displacement field prescribed by the above kinematic model contains three unknown parameters. A Lagrangian analysis is performed and the compressible Mooney-Rivlin law is assumed for the stored energy function. Once evaluated the Piola-Kirchhoff stresses, the free parameters of the kinematic model are determined by using the equilibrium equations and the boundary conditions. An Eulerian analysis is then accomplished to evaluating stretches and stresses in the deformed configuration. Cauchy stress distributions are investigated and it is shown how, for wide ranges of constitutive parameters, the obtained solution is quite accurate. The whole formulation proposed for the finite anticlastic bending of hyperelastic solids is linearized by introducing the hypothesis of smallness of the displacement and strain fields. With this linearization procedure, the classical solution for the infinitesimal bending of beams is fully recovered.