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Abstract A Ritz approach for the analysis of buckling and post-buckling of plates with through-the-thickness cracks is presented. The plate behavior is described by the first order shear deformation theory and von Karman's geometric nonlinearity. The admissible functions used in the displacements approximation are series of regular orthogonal polynomial supplemented with special functions able to decribe the dicontinuity across the crack and the singularity at the crack tips; boundary functions are used to fullfill the homogeneous essential boundary conditions. Convergence studies and analysis results are presented for buckling and post-buckling of plates with a central through-the-thickness crack evidencing differences in the structural response between pre- and post-buckling regimes, which can substantially affect the plate residual strength. The performed analyses show the efficiency and potential of the method, which provides accurate results in conjunction with a reduced number of degrees of freedom and simplified data preparation. [ABSTRACT FROM AUTHOR]

Abstract The prebuckling deformation of structures is usually very small in conventional concepts, and is always neglected in the conventional buckling theory (CBT) and numerical method (CNM). In this paper, we find a class of structures from the emerging field of stretchable electronics, of which the prebuckling deformation becomes large and essential for determining the critical buckling load. Although great progress has been made for the buckling theory in the past hundred years, it is still challenging to analyze the buckling problems with finite prebuckling deformation (FPD buckling) straightforwardly. Here, the experimental stretch of a series of serpentine interconnects was firstly conducted as a representative example to show the FPD buckling behaviors and inapplicability of the CBT and CNM. The CNM can yield a huge error of 50% on the critical buckling load for the case with thickness-to-width ratio of the cross section h / b = 0.6. Most importantly, a systematic and straightforward theory (FPD buckling theory) is developed to analyze the FPD buckling behaviors of beams with the coupling of bending, twist and stretch/compression. As a comparison, various theoretical and numerical methods are applied to three classic problems, including lateral buckling of a three-point-bending beam, lateral buckling of a pure bending beam and Euler buckling. Our FPD buckling theory for beams is able to give a good prediction, while the CBT (by Timoshenko et al.) and CNM (by commercial program packages) yield unacceptable results (with 70% error for a three-point-bending beam with h / b = 0.8, for example). Discussion on the FPD buckling of bulk structures is deferred to a following paper. [ABSTRACT FROM AUTHOR]

Understanding the effects of water absorption on the geometry of a paper sheet is important for inkjet printing applications, since internal moisture content differences may cause unacceptable out-of-plane deformations. The present work focuses on moisture-induced deformations due to a moisture content that is uniform over the thickness of the sheet. Large enough in-plane differences will cause the sheet to buckle, leading to a wavy pattern at the edges of the sheet. Two approaches are utilized to study this for levels of moisture content up to and including the threshold for buckling. An analytical approach, based on geometrically nonlinear plate theory (von Kármán theory), and a numerical one, using commercially available finite element software, are presented. As a first reference problem an isotropic circular plate, wetted uniformly at its center, is solved both analytically and numerically for the in-plane stress distribution, the buckling threshold, and the resulting buckling mode. Secondly, the same is done for an orthotropic rectangular plate, wetted at its edges. Here, in the analytical approach a Rayleigh–Ritz analysis is employed to approximate the buckling threshold. The latter is also calculated by numerical means. The results show that the methods provide results consistent with each other. [ABSTRACT FROM AUTHOR]

Tape-springs – straight strips with transversely curved cross-sections – undergo a flattening of the cross-section upon bending, and accompanying torsional buckling in equal-sense bending which then "recedes" at greater deformation. The exact analytical solution for tape-spring bending was solved by Mansfield; however, the resulting expression for the twisting curvature at the onset and recession of torsional buckling cannot be solved in closed-form. Approximate closed-form expressions are therefore sought: firstly, by utilising Taylor series approximations of Mansfield's extensive functions; and secondly, by assuming a uniform transverse curvature, thereby rendering the problem algebraic in nature, and deriving new governing expressions. Further approximations assume the cross-section does not deform upon bending, and that the tape-spring remains a developable surface in the inextensible limit of zero thickness. All approximate solutions are shown to agree closely with Mansfield's solution at small curvatures, with the uniform curvature approximation also capturing the recession of torsional buckling to a reasonable degree of accuracy. The accuracy of the uniform curvature approximation is related to a pertinent dimensionless parameter, which governs the size of a boundary layer in Mansfield's exact solution: as this parameter tends to infinity, the uniform curvature approximation and Mansfield's exact solution are shown to converge; as it tends to zero, all solutions are shown to converge on the inextensible limit. • Approximate solutions for tape-spring bending assuming uniform curvature. • Closed form expressions for the onset and recession of torsional buckling. • Approximate solution in good agreement with exact Mansfield solution. • Exact and approximate solutions converge in the zero-thickness inextensible limit. [ABSTRACT FROM AUTHOR]

A general formulation for the two dimensional lateral buckling problem is developed employing classical rod theory. A nonlinear differential system with seven unknowns is derived. Governing equations include a number of previously unaccounted nonlinear terms expressing interaction of various variables representing pipeline geometry and external loads. A rigorous derivation of the transformation of surface loads due to internal and external pressures to line loads revealed some previously unknown limitations on its applicability. Several boundary value problems are formulated corresponding to common pipeline engineering problems. Nonlinear dynamics methods are employed to draw some preliminary results on the behavior of the solutions. It is shown that the trivial solution corresponding to a perfectly straight pipeline experiencing no initial deformation and initial force is unstable. [ABSTRACT FROM AUTHOR]

We present a novel technique to simulate the deformation of a cantilever elastic beam constrained in a curved solid channel subject to end forces. We pose this as the minimization of an energy functional and solve it by a variant of a dynamic programming approach called the Viterbi algorithm. The core idea of this approach is to discretize the variables describing the potential energy and to construct a set of admissible configurations of the beam. The Viterbi algorithm is then employed to search through the set of possible beam configurations and locate the one with the minimum potential energy in a very computationally efficient way. The new approach does not require any gradient computations and could be considered as a direct search method, and thus can be guaranteed to find the global minimum potential energy. Also the constraints can be automatically satisfied by constructing the proper set of all the possible configurations. The approach can also be used to find feasible starting configurations associated with conventional minimizing algorithms. [ABSTRACT FROM AUTHOR]

Abstract: The precisely controlled buckling of stiff thin films (e.g., Si or GaAs nano ribbons) on the patterned surface of elastomeric substrate (e.g., poly(dimethylsiloxane) (PDMS)) with periodic inactivated and activated regions was designed by Sun et al. [Sun, Y., Choi, W.M., Jiang, H., Huang, Y.Y., Rogers, J.A., 2006. Controlled buckling of semiconductor nanoribbons for stretchable electronics. Nature Nanotechnology 1, 201–207] for important applications of stretchable electronics. We have developed a post-buckling model based on the energy method for the precisely controlled buckling to study the system stretchability. The results agree with Sun et al.’s (2006) experiments without any parameter fitting, and the system can reach 120% stretchability. [Copyright &y& Elsevier]

Abstract: A novel re-defining of the orthotropic material properties in terms of a so-called associated geometric mean isotropic (GMI) material is used to develop a thorough buckling analysis of an axially-loaded orthotropic circular cylinder. A membrane prebuckling condition is assumed and an expression for the buckling stress is derived in terms of cylinder geometry, orthotropic material properties, and the number of waves in the buckling deformation pattern in the axial and circumferential directions. By assuming the number of waves in each direction are real-valued variables, as opposed to integers, conditions which result in stationary values of the buckling stress are sought, and once found, examined for their character as regards representing minima, maxima, or saddle points. Three quite different buckling characteristics are predicted, the particulars depending on the shear modulus of the orthotropic material relative to that of the associated GMI material. It is shown that if the shear modulus of the orthotropic material is greater than the shear modulus of the associated GMI material, the cylinder buckles into a unique axisymmetric deformation pattern. If the shear modulus of the orthotropic material is less than the shear modulus of the associated GMI material, the cylinder buckles into a unique nonaxisymmetric deformation pattern. If the shear modulus of the orthotropic material is exactly equal to the shear modulus of the associated GMI material (this is the situation for an isotropic cylinder), the cylinder can buckle into either axisymmetric or nonaxisymmetric deformation patterns. Moreover it is shown that, in this case, there exists a number of deformation patterns, all at essentially the same stress level. Closed-form lower-bound expressions for the buckling stress are developed using the adopted notation, the value of the shear modulus relative to the shear modulus of the GMI material determining which expression is applicable. The results of this analysis are applied to a circular cylinder constructed of a lattice structure consisting of helical and circumferential ribs, a so-called orthogrid lattice cylinder, where it is assumed that the ribs of the lattice structure are dense enough to be able to represent the elastic properties of the lattice with an equivalent homogenized orthotropic material. An isogrid cylinder, where the helical rib angle is 30° relative to the axial direction, is a special case. The orthotropic cylinder analysis is reformulated in terms of the material properties of the ribs and the angle of the helical ribs. For this situation the isogrid case is the GMI material, and the rib angle determines whether the shear modulus of the equivalent orthotropic material is greater than or less than the GMI material. This translates into the character of the buckling deformations depending directly on the rib angle. [Copyright &y& Elsevier]

Abstract: We study mechanical response of a uniaxially pre-stretched soft elastic film to spatially modulated electric fields. The film is adhered onto a rigid, conductive substrate, and the electric field is generated through a periodically patterned electrode over the film. Our work shows that at low applied voltages the deformation of the film follows the pattern of the electrode, allowing for the transfer of the surface morphology of the electrode onto the film surface in opposite sign. However, when the voltage reaches a threshold, the film buckles to form stripped pattern parallel to the tensile direction. This newly resulting deformation is superposed on that before onset of buckling, leading to complicated and ordered topography of the film. The phenomenon may provide a new clue for creating ordered surface structures with the aid of elastic buckling. [ABSTRACT FROM AUTHOR]

Abstract: A solution for buckling of a stiff strip of finite width bonded to a compliant elastic half-space and subjected to uniform axial compression is presented. Approximate semi-analytical and finite element solutions are obtained and compared with a two-dimensional case of a plate on elastic foundation. The comparison demonstrates that the two-dimensional solution can be applied to predict the buckling wavelength and critical compressive strain when the width of the strip is equal to or larger than the buckling wavelength. For narrow strips, the wavelength is smaller and critical strain is higher than that of a plate on foundation. [Copyright &y& Elsevier]

Abstract: In this paper, to investigate the buckling characteristics of carbon nanotubes, an equivalent beam model is first constructed. The molecular mechanics potentials in a C–C covalent bond are transformed into the form of equivalent strain energy stored in a three dimensional (3D) virtual beam element connecting two carbon atoms. Then, the equivalent stiffness parameters of the beam element can be estimated from the force field constants of the molecular mechanics theory. To evaluate the buckling loads of multi-walled carbon nanotubes, the effects of van-der Waals forces are further modeled using a newly proposed rod element. Then, the buckling characteristics of nanotubes can be easily obtained using a 3D beam and rod model of the traditional finite element method (FEM). The results of this numerical model are in good agreement with some previous results, such as those obtained from molecular dynamics computations. This method, designated as molecular structural mechanics approach, is thus proved to be an efficient means to predict the buckling characteristics of carbon nanotubes. Moreover, in the case of nanotubes with large length/diameter, the validity of Euler’s beam buckling theory and a shell model with the proper material properties defined from the results of present 3D FEM beam model is investigated to reduce the computational cost. The results of these simple theoretical models are found to agree well with the existing experimental results. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the application of the p-Ritz method for the plastic buckling analysis of thick plates. In order to allow for the effect of transverse shear deformation in thick plates, the Mindlin plate theory is adopted. The plastic buckling behaviour of the plate is captured by using the incremental and deformation theories of plasticity. The material property of the plate is assumed to obey the Ramberg–Osgood stress–strain relation. The p-Ritz method will be applied to obtain the governing eigenvalue equation for the plastic buckling analysis of uniformly stressed plates with edges defined by polynomial functions. In the p-Ritz method, the displacement functions of the plate are approximately represented by the product of mathematically complete two-dimensional polynomial functions and boundary equations raised to appropriate powers that ensure the satisfaction of the geometric boundary conditions. The validity, convergence and accuracy of the method were demonstrated for various plate shapes such as rectangular, triangular and elliptical shapes. A parametric study was also undertaken to study the plastic buckling behaviour and the effect of transverse shear deformation. [Copyright &y& Elsevier]

A combined experimental/analytical effort is carried out to elucidate the post-buckling response of bilaterally constrained columns and plates under monotonically increasing edge displacement. Real-time observations of the outward deformation of the plate via the shadow moire technique facilitate deep insight into the behavior of this class of problems. The lateral constraints lead to a sequential plate snapping process that arise from secondary buckling of a contact zone(s) in the plate. The critical load for mode transition is evaluated systematically as a function of the confining gap, the plate aspect ratio and other system parameters. The results generally exhibit significant scatter that is attributed to the inherent asymmetry of the deformation pattern in such systems. A large-deformation finite element analysis incorporating frictional contact algorithm is used to model the problem. The deformation patterns as well as the critical loads predicted from this analysis generally agree well with the experimental results. A simple analytic expression, based on the linear theory of beams, is developed for the critical loads. The experimental data are found to be well contained by lower and upper bounds from this analysis once the plate surface interacts with the two confining planes. The results of this work may serve as a basis for multilaminate design possessing unique mechanical response and superior energy absorption capability. [Copyright &y& Elsevier]

This paper presents a theoretical model for coupled extension and flexure with shear deformations of an electroelastic plate under biasing fields. The governing equations of this model, defined in the middle plane of the plates, are derived from the full three-dimensional theory of electroelasticity for small fields superposed upon finite biasing fields, under the assumption that the stress component normal to the plate vanishes identically. As examples to illustrate the applications of this model, the authors include their analysis of buckling of three plates, one single-layered plate and two double-layered plates (i.e., bimorphs) of distinct poling configurations. This analysis indicates that the electromechanical coupling strengthens the plates against buckling. [Copyright &y& Elsevier]

Magneto-responsive slender bodies are used in a range of promising applications, such as artificial cilia, magnetic fiber networks and cellular actuators. To accurately describe the magneto-elastic deformations, both the demagnetization field as well as the resulting magnetic loads on the body should be properly accounted for. The calculation of the demagnetization field for a general sample shape is very challenging, which has hampered the experimental characterization of the intrinsic magnetic and magnetoelastic properties. Here, a methodology is developed to accurately calculate the demagnetization field for slender bodies, i.e., for long beams having a rectangular or circular cross-section. We propose two different expressions for the magnetic load on slender bodies. To validate the two expressions, we solve the magnetic buckling problem for cantilever beams using an analytical approach for small deflections. We compare the critical buckling fields with an energy approach and with experimental results from three different studies. The load and energy methods were found to be similar and to correspond very well with the experimental data. To also validate our slender body approach (i.e., demagnetization field calculation and magnetic load expression) for large deflections, we analytically solve for the large (postbuckling) rotational deformation of slender beams. To do so, we formulated a weak form of the governing equations using a variational approach, which can be readily solved using finite elements and does not require a discretization of the space surrounding the magnetic material. We use this generic 3D continuum formulation as a starting point to derive the governing equations for slender bodies, which can be solved in a weak sense to find an approximate analytical formulation for large deflections and non-linear ferromagnetic materials. We compared the analytical results with experimental data on the post-buckling deformation of long cantilever beams and found excellent agreement. We anticipate that our results will be valuable for magneto-elastic (soft) robotics, homogenization approaches of magneto-elastic constitutive relations and other applications where strong magneto-elastic coupling is important. [ABSTRACT FROM AUTHOR]

Abstract: Thermal buckling analysis of rectangular functionally graded plates (FGPs) with geometrical imperfections is presented in this paper. The equilibrium, stability, and compatibility equations of an imperfect functionally graded plate are derived using the classical plate theory. It is assumed that the nonhomogeneous mechanical properties of the plate, graded through thickness, are described by a power function of the thickness variable. The plate is assumed to be under three types of thermal loading as uniform temperature rise, nonlinear temperature rise through the thickness, and axial temperature rise. Resulting equations are employed to obtain the closed-form solutions for the critical buckling temperature change of an imperfect FGP. The results are reduced and compared with the results of perfect functionally graded and imperfect isotropic plates. [Copyright &y& Elsevier]

Buckling due to diffusion-induced compressive stress has been observed during the lithiation of various nanowires, and can affect the mechanical and electrochemical performance of nanowire-based electrodes. This study is focused on the diffusion-induced buckling of nanowires. Two diffusion paths are analyzed; one with radial diffusion only, and the other with axial diffusion. For the diffusion-induced buckling of nanowires with radial diffusion, the theory of large deformation is used in the description of the coupling between mass transport and deformation in large deformed solids. Using the linear theory, analytical solution of the critical length of a nanowire, below which there is no buckling, is obtained, which is dependent on the constraint of the ends of the nanowire and the volumetric strain of the nanowire at the fully lithiated state. The comparison between the linear analytical solution of the critical length and the numerical solution from the theory of large deformation shows that the linear analytical solution is valid for the influx less than 1 mol · m −2 · s −1 and configurations considered in the work. Numerical analysis shows that the critical buckling time decreases with the increase of nanowire length and current density, and the nanowire with two fixed ends has a larger critical buckling time than that for the same nanowire with a fixed end and a pinned end. The nanowire length plays a bigger role in determining the critical state of charge for the onset of the buckling than that of the diffusion flux. The state of charge ( SOC ) at the state of critical buckling decreases with the increase of the nanowire length and increases with the increase of current density. For the diffusion-induced buckling of nanowires with axial diffusion, numerical analysis is performed. The numerical result reveals similar trend, i.e., the critical buckling time decreases with the increase of nanowire length and current density. The comparison of the critical buckling times for the conditions with axial diffusion, radial diffusion and diffusion from all sides, respectively, reveals that the contribution from axial diffusion is negligible even for small aspect ratio, and it does not change significantly with the increase of the aspect ratio. [ABSTRACT FROM AUTHOR]

Abstract: The equilibrium and buckling equations are derived for the lateral buckling of a prismatic straight beam. A consistent finite strain constitutive law is used, which is based on a hyperelastic model for an isotropic material. The kinematics of the cross-sectional deformations are based on a Timoshenko type beam displacement of the cross-sectional plane using Euler angles and two shear finite rotations coupled with warping taken normal to the displaced plane. Also derived are the second order approximations to the displacements, curvatures, twist and internal actions. The constitutive relationships for the internal actions reveal new coupling terms between the bending moments, torsion and bimoment, which are functions of the cross-sectional warping and shear deformations. New Wagner type nonlinear torsion terms are derived which are functions of the warping of the cross-sectional plane, and are coupled to the twisting and shear deformations of the cross-section. Solutions are determined for the lateral buckling of a prismatic monosymmetric beam under pure bending and the flexural–torsional buckling under axial compression. For the flexural–torsional buckling problem it is found that the Euler type column buckling formula is consistent with Haringx’s column buckling formula while the torsional buckling formula is different to conventional equations. The second variation of the total potential is also derived. The effects of shear deformations are explored by examining the non-dimensional lateral buckling equation for a simply supported beam. [ABSTRACT FROM AUTHOR]

Abstract: Here, the nonlinear thermo-elastic buckling/post-buckling characteristics of laminated circular conical–cylindrical/conical–cylindrical–conical joined shells subjected to uniform temperature rise are studied employing semi-analytical finite element approach. The nonlinear governing equations, considering geometric nonlinearity based on von Karman’s assumption for moderately large deformation, are solved using Newton–Raphson iteration procedure coupled with displacement control method to trace the pre-buckling/post-buckling equilibrium path. The presence of asymmetric perturbation in the form of small magnitude load spatially proportional to the linear buckling mode shape is assumed to initiate the bifurcation of the shell deformation. The study is carried out to highlight the influences of semi-cone angle, material properties and number of circumferential waves on the nonlinear thermo-elastic response of the different joined shell systems. [Copyright &y& Elsevier]

Highlights: [•] Timoshenko column in presence of an arbitrary number of concentrated cracks is considered. [•] Cracks have been modeled as concentrated reduction of the flexural stiffness. [•] Explicit expressions of buckling shapes and the buckling load equations are proposed. [•] Tensile buckling phenomenon is captured by the proposed explicit solutions. [•] The influence of multiple cracks on the tensile and the compressive buckling is assessed. [Copyright &y& Elsevier]

This paper presents the stability analysis of simply supported and cantilever, thin walled, open section, orthotropic composite beams subjected to concentrated end moments, concentrated forces, or uniformly distributed load. In the analysis, both the transverse shear and the restrained warping induced shear deformations are taken into account. An explicit expression is derived for the lateral-torsional buckling load of composite beams.A simple expression is also presented, which shows the approximate reduction in the buckling load due to shear deformation. It enables us to decide whether the effect of shear deformation is negligible. [Copyright &y& Elsevier]

Abstract: The analogy between the governing equations for the analysis of buckling in elastic structures and the elastodynamic equations of motion for wave propagation is presented. By employing this analogy, the exact and approximate buckling stresses of periodic layered materials and continuous fiber composites, respectively, are established. This is performed by utilizing micromechanically based dispersion relations for elastic wave propagating in the composite materials, which provide for a given wave length the corresponding phase velocity. By a specific change of variables in these dispersion relations, the corresponding buckling stresses can be determined. Results are presented and compared with solutions based on the mechanics of materials approach as well as with the well known Rosen’s fiber buckling predictions. [Copyright &y& Elsevier]

Abstract: This paper describes an investigation into elastic buckling of an embedded multi-walled carbon nanotube under combined torsion and axial loading, which takes account of the radial constraint from the surrounding elastic medium and van der Waals force between two adjacent tube walls. Depending on the ratio of radius to thickness, the multi-walled carbon nanotubes discussed here are classified as thin, thick, and nearly solid. Critical buckling load with the corresponding mode is obtained for multi-walled carbon nanotubes under combined torsion and axial loading, with various values of the radius to thickness ratio and surrounded with different elastic media. The study indicates that the buckling mode (m, n) of an embedded multi-walled carbon nanotube under combined torsion and axial loading is unique and it is different from that with axial compression only. New features for the buckling of an embedded multi-walled carbon nanotube under combined torsion and axial loading and the meaningful numerical results are useful in the design of nanodrive device, nanotorsional oscillator and rotational actuators, where multi-walled carbon nanotubes act as basic elements. [Copyright &y& Elsevier]

Abstract: This paper studies the pure bending and bending-induced local buckling of a nanocomposite beam reinforced by a single-walled carbon nanotube (SWNT). The Airy stress-function method was employed to analyse the deformation of the matrix, and the cross-sectional change of the SWNT in bending was taken into account. A particular consideration was given to the effect of the SWNT’s radial flexibility on the strain/stress states and buckling. It was found that in thicker matrix layers the SWNT buckles locally at smaller bending angles and greater flattening ratios. This causes higher strains/stresses in the surrounding matrix and in turn degrades the strength of the nanocomposite structure. [Copyright &y& Elsevier]

The third-order shear deformation plate theory (TPT) is employed to solve the axisymmetric bending and buckling problems of functionally graded circular plates. Relationships between the TPT solutions of axisymmetric bending and buckling of functionally graded circular plates and those of isotropic circular plates based on the classical plate theory (CPT) are presented, from which one can easily obtain the TPT solutions for the axisymmetric bending and buckling of functionally graded plates. It is assumed in analysis that the mechanical properties of the functionally graded plates vary continuously through the thickness of the plate and obey a power law distribution of the volume fraction of the constituents. Effects of material gradient property and shear deformation on the bending and buckling of functionally graded plates are discussed in the frameworks of the first-order plate theory (FPT) and third-order plate theories. Also, comparisons of the TPT solutions to the FPT and CPT solutions are presented, which show that the first-order shear deformation plate theory is enough to consider the effect of shear deformation on the axisymmetric bending and buckling of functionally graded circular plate, a much higher order and more complex plate theory (say TPT) is not necessary for such a kind of problem. [Copyright &y& Elsevier]

Abstract: A finite-deformation theory is developed to study the mechanics of thin buckled films on compliant substrates. Perturbation analysis is performed for this highly nonlinear system to obtain the analytical solution. The results agree well with experiments and finite element analysis in wavelength and amplitude. In particular, it is found that the wavelength depends on the strain. Based on the accurate wavelength and amplitude, the membrane and peak strains in thin films, and stretchability and compressibility of the system are also obtained analytically. [Copyright &y& Elsevier]

Mechanically-guided three-dimensional (3D) assembly of mesostructures through controlled compressive buckling represents a promising approach, because of the versatile applicability to a broad set of advanced materials, over length scales from sub-micrometers to centimeters. Based on this approach, a spatial variation of thickness in the initial 2D structures was demonstrated as an effective strategy to produce engineered folding deformations at regions with lower bending stiffness, thereby with an ability to enable autonomous origami assembly. The reliability of this strategy requires the development of a theoretical model as a design reference, such that targeted folding deformations can be achieved without inducing any structural failure. This work presents a finite-deformation model of controlled buckling in straight ribbons with engineered thickness reductions at two or three sites. A comparison of predicted maximum strains and deformed configurations to the finite element analyses (FEA) and experimental results elucidates the validity of the developed model. The results uncover the coupled effect of different geometric parameters on the maximum strain. For relative flexible creases that lead to evident folding deformations, the theoretical model gives approximate analytic solutions to the deformed configurations and maximum strain. Furthermore, the theoretical model was exploited in a design optimization to achieve highly sensitive rotatable micro-mirrors with a desired mechanical tunability of the optical transmittance. This study can be useful in the future designs of 2D precursor structures for the origami-inspired assembly of different 3D mesostructures and micro-devices. [ABSTRACT FROM AUTHOR]

Abstract: The physical parameters of optimal discrete link-spring models, which maximize the buckling load, are reconstructed. The cases where the system is pinned or clamped at one end, and supported by linear spring at the other end, are analyzed. It is shown that the optimal system can be determined recursively by using a one parameter iterative loop. For the pinned-spring-supported boundary conditions there is a region where the optimal solution is not unique. This case is characterized in the paper for the continuous system and its discrete model representation. This study enlightens the results obtained by Tadjbakhsh and Keller in 1962. In particular, it allows evaluation of the strongest column in the transition from the clamped–free to the pinned–pinned boundary condition; and from the (unstable) pinned–free configuration to that associated with the pinned–pinned. We have restricted our study to the regular cases, where there are no internal vanishing cross-sectional areas in the column. It has been shown that for certain range of spring constant the optimal columns are regular in the sense that their cross-section area is positive. [Copyright &y& Elsevier]

Abstract: Long, open-ended, hollow sandwich cylinders with ultralightweight cellular cores are optimized under uniform internal pressure for minimum weight design. Five different core topologies are considered: Kagomé truss, single-layered pyramidal truss, double-layered pyramidal truss, single-layered corrugated core and double-layered corrugated core. The highly porous cellular materials are homogenized to obtain effective constitutive relations. Close-formed solutions are presented for the forces and stresses in individual structural members of the sandwich, which are then validated by finite element calculations. Optimization of the sandwich-walled hollow cylinder is achieved using a quadratic optimizer, subjected to the constraints that none of the following failure modes occurs: facesheet yielding; facesheet punch shearing (active only for truss-cored sandwiches); core member buckling; core member yielding. In comparison with hollow cylinders having solid walls, truss-core sandwich cylinders and single-layer corrugated core sandwich cylinders are found to have superior weight advantages, especially for more heavily loaded cases. With the consideration of both weight efficiency and failure modes, sandwich-walled hollow cylinders having Kagomé truss core with pyramidal sub-geometry have the best overall performance in comparison with other core topologies. [Copyright &y& Elsevier]

A multiobjective optimization problem is presented to determine the optimal layer thickness and optimal closed loop control function for a symmetric cross-ply laminate subjected to thermomechanical loadings. The optimization procedure aims to maximize the critical combination of the applied edges load and temperature levels and to minimize the laminate dynamic response subject to constraints on the thickness and control energy. The objective of the optimization problem is formulated based on a consistent first-order shear deformation theory without introducing a shear correction factor. The dynamic response is measured as the sum of the total elastic energy of the laminate and a penalty term involving a closed loop control force. Laipunov–Bellman theory is used to obtain solutions for the controlled deflections and optimal control force. The layer thickness is taken as a design variable, and is presented as a function of the number of layers. A numerical study is made for simply supported symmetric laminates with an odd number of layers to show the advantages of the present control optimization. The study indicates that the present control optimization is active with most laminates cases for all values of aspect ratio, number of layers, orthotropy ratio, thermal expansions ratio and thickness ratio. [Copyright &y& Elsevier]

Abstract: An experimental and computational study of the bending response of steel sandwich panels with corrugated cores in both transverse and longitudinal loading orientations has been performed. Panel designs were chosen on the basis of failure mechanism maps, constructed using analytic models for failure initiation. The assessment affirms that the analytic models provide accurate predictions when failure initiation is controlled by yielding. However, discrepancies arise when failure initiation is governed by other mechanisms. One difficulty is related to the sensitivity of the buckling loads to the rotational constraints of the nodes, as well as to fabrication imperfections. The second relates to the compressive stresses beneath the loading platen. To address these deficiencies, existing models for core failure have been expanded. The new results have been validated by experimental measurements and finite element simulations. Limit loads have also been examined and found to be sensitive to the failure mechanism. When face yielding predominates, appreciable hardening follows the initial non-linearity, rendering robustness. Conversely, for designs controlled by buckling (either elastic or plastic) failure initiation is immediately followed by softening. The implication is that, when robustness is a key requirement, designs within the face failure domain are preferred. [Copyright &y& Elsevier]

Energy absorption structures have been pursued to protect personnel and infrastructures over the last few decades. In this study, a novel hybrid foam-core/solid-shell spherical (FSS) structure is presented and investigated. The internal foam core inherits merits of large deformation from conventional foam structures and the introduction of the external thin solid shell is to reach high strength and delay deformation further, thereby achieving high energy absorption efficiency in FSS structures. Theoretical models are developed to characterize elastic modulus and buckling behavior of FSS structures under a compressive loading, and are verified through extensive finite element analysis (FEA). Typical deformation mechanisms are revealed by addressing competition of buckling deformation of the foam core and solid shell, and are identified through the proposed theoretical models. Further, the energy absorption efficiency is proposed to optimize the specific energy absorption density and critical triggering force of activating energy absorption, and is correlated with deformation mechanism and geometric parameters of FSS structures. Both numerical and theoretical analyses show that the employment of a thin solid shell surrounding the foam structures will enhance the energy absorption efficiency with high capability and safe comfort. The present study is expected to provide a useful guideline for a hybrid design of future energy absorption structures with unprecedented performance. [ABSTRACT FROM AUTHOR]

Abstract: This paper concerns the design of tensegrity structures with optimal mass-to-stiffness ratio. Starting from an initial layout that defines the largest set of allowed element connections, the procedure seeks the topology, geometry and prestress of the structure that yields optimal designs for different loading scenarios. The design constraints include strength constraints for all elements of the structure, buckling constraints for bars, and shape constraints. The problem formulation accommodates different symmetry constraints for structure parameters and shape. The static response of the structure is computed by using the nonlinear large displacement model. The problem is cast in the form of a nonlinear program. Examples show layouts of 2D and 3D asymmetric and symmetric structures. The influence of the material parameters on the optimal shape of the structure is investigated. [Copyright &y& Elsevier]

Abstract: A postbuckling analysis is presented for a simply supported, shear deformable functionally graded plate with piezoelectric actuators subjected to the combined action of mechanical, electrical and thermal loads. The temperature field considered is assumed to be of uniform distribution over the plate surface and through the plate thickness and the electric field considered only has non-zero-valued component E Z . The material properties of functionally graded materials (FGMs) are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents, and the material properties of both FGM and piezoelectric layers are assumed to be temperature-dependent. The governing equations are based on a higher order shear deformation plate theory that includes thermo-piezoelectric effects. The initial geometric imperfection of the plate is taken into account. Two cases of the in-plane boundary conditions are considered. A two step perturbation technique is employed to determine buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of perfect and imperfect, geometrically mid-plane symmetric FGM plates with fully covered or embedded piezoelectric actuators under different sets of thermal and electric loading conditions. The effects played by temperature rise, volume fraction distribution, applied voltage, the character of in-plane boundary conditions, as well as initial geometric imperfections are studied. [Copyright &y& Elsevier]

Abstract: This paper presents the development and illustrates the application of semi-analytical solutions for plastic buckling (bifurcation) problems involving perfectly straight and uniformly compressed thin-walled metal members. These solutions are derived on the basis of a non-linear Generalised Beam Theory (GBT) formulation, recently proposed by two of the authors, that resorts to a linearised buckling analysis that adopts Hill’s hypo-elastic comparison solid method to obtain the plastic bifurcation loads and associated buckling mode shapes. Moreover, both small-strain incremental and deformation plasticity theories are employed. Several numerical illustrative examples are presented and discussed throughout the paper and closed-form analytical formulae are also derived. The accuracy of the GBT-based semi-analytical plastic bifurcation predictions is assessed through the comparison with (i) available theoretical solutions and/or (ii) results yielded by a special shell finite element model developed by one of the authors. [Copyright &y& Elsevier]

This paper develops a static and dynamic large deformation model to investigate the post-buckling response of slender beams constrained by movable and flexible bilateral confinements. A novel discretization algorithm is proposed to convert the irregular constraints into gap vectors. Governing equations are formulated based on the Euler-Bernoulli beam theory. An energy method is presented to solve the equations using a modified Nelder–Mead algorithm. A constrained minimization of the total energy is carried out with respect to the gap vectors. The theoretical results of the proposed model, i.e., the deformed beam shape configuration and force-displacement relationship, are compared with existing studies and experiments. The regularly and movably constrained, static model is compared with the existing small and large deformation models, respectively. The regularly and flexibly confined, dynamic model is compared with experiments. Satisfactory agreements are observed. Parametric studies are conducted to investigate the post-buckling response in terms of loading and constraints conditions. In particular, the loading condition is first examined by changing the loading frequency. The highest achievable buckling mode is then studied with respect to the Young's moduli ratios of movable constraints-to-beam and flexible constraints-to-beam, respectively, and the ratio of walls gap-to-beam length. The proposed theoretical models are effective in understanding and predicting the static and dynamic post-buckling response of beams constrained by movable and flexible confinements. [ABSTRACT FROM AUTHOR]

A general framework to study the mechanical behaviour of a cylindrical silicon anode particle in a lithium ion battery as it undergoes lithiation is presented. The two-way coupling between stress and concentration of lithium in silicon, including the possibility of plastic deformation, is taken into account and two particular cases are considered. First, the cylindrical particle is assumed to be free of surface traction and second, the axial deformation of the cylinder is prevented. In both cases plastic stretches develop through the entire cylinder and not just near the surface as is commonly found in spherical anode particles. It is shown that the stress evolution depends both on the lithiation rate and the external constraints. Furthermore, as the cylinder expands during lithiation it can develop a compressive axial stress large enough to induce buckling, which in turn may lead to mechanical failure. An explicit criterion for swelling-induced buckling obtained as a modification of the classical Euler buckling criterion shows the competition between the stabilising effect of radius increase and the destabilising effect of axial stress. [ABSTRACT FROM AUTHOR]

Abstract: Part I presented a set of experiments in which pressurized tubes were cycled axially under stress control about a compressive mean stress. This loading history causes biaxial ratcheting involving compressive axial strain and expansion of the diameter of the tube. The compressive strain in turn induces the initiation and growth of axisymmetric wrinkles. Persistent cycling resulted in localization of the wrinkles and collapse. In Part II the problem is first modeled as a shell with initial axisymmetric imperfections while the biaxial ratcheting of the material is modeled using the Dafalias–Popov two-surface nonlinear kinematic hardening model. It is demonstrated that when suitably calibrated this modeling framework reproduces the prevalent ratcheting deformations and the evolution of wrinkling including the conditions at collapse accurately for all experiments. The calibrated model is then used to evaluate the ratcheting behavior of pipes under thermal-pressure cyclic loading histories experienced by axially restrained pipelines. [Copyright &y& Elsevier]

Abstract: The sinusoidal shear deformation plate theory, presented in the first part of this paper, is used to study the buckling and free vibration of the simply supported functionally graded sandwich plate. Effects of rotatory inertia are considered. The critical buckling load and the vibration natural frequency are investigated. Some available results for sandwich plates non-symmetric about the mid-plane can be retrieved from the present analysis. The influences of the transverse shear deformation, plate aspect ratio, side-to-thickness ratio and volume fraction distributions are studied. In addition, the effect of the core thickness, relative to the total thickness of the plate, on the critical buckling load and the eigenfrequencies is investigated. [Copyright &y& Elsevier]

Abstract: Of interest here is the influence of loading rate on the stability of structures where inertia is taken into account, with particular attention to the comparison between static and dynamic buckling. This work shows the importance of studying stability via perturbations of the initial conditions, since a finite velocity governs the propagation of disturbances. The method of modal analysis that determines the fastest growing wavelength, currently used in the literature to analyze dynamic stability problems, is meaningful only for cases where the velocity of the perfect structure is significantly lower than the associated wave propagation speeds. As a model structure to illustrate this point we select an elastic ring subjected to external hydrostatic pressure which is applied at different rates (appropriately non-dimensionalized with respect to elastic axial wave speed). The ring’s stability is studied by following the evolution of a localized small perturbation. It is shown that for small values of the applied loading rate, the structure fails through a global mode, while for large values of the applied loading rate the structure fails by a localized mode of deformation. An analytically obtained localization time is found to be a very good estimate of the onset of instability time at high loading rates. [Copyright &y& Elsevier]

Abstract: In this paper, inhomogeneous deformation of a temperature-sensitive hydrogel has been studied and analyzed under arbitrary geometric and boundary conditions. We present the governing equations and equilibrium conditions of an isothermal process based on the monophase gel field theory of hydrogel via a variational approach. We have adopted and implemented an explicit form of energy for temperature-sensitive hydrogel in a three-dimensional finite element method (FEM) using a user-supply subroutine in ABAQUS. For verification purpose, a few numerical results obtained by the proposed approach are compared with existing experimental data and analytical solutions. They are all in good agreement. We also provide several examples to show the possible applications of the proposed method to explain various complex phenomena, including the bifurcation, buckling of membrane, buckling of thin film on compliant substrate and the opening and closure of flowers. [Copyright &y& Elsevier]

Abstract: The elastica behavior of an extensional sandwich panel with a “soft” core when subjected to in-plane compressive loads is presented and it is compared with the response of its extensional equivalent single layer (ESL) with shear deformations model. The field equations along with the appropriate boundary conditions for the sandwich and the ESL panels have been derived through a variational approach following the High-order SAndwich Panel Theory (HSAPT) approach that takes into account the vertical flexibility of the core. The governing equations include the effects of the extension of the mid-surfaces of the face sheets of the sandwich panel or the mid-plane of the ESL model which the classical elastica approach misses. The results of the elastica response of a clamped-simply-supported sandwich panel and its ESL counterpart are presented and compared. They include the response along the panel, deformed shapes and equilibrium curves of in-plane loads versus structural quantities such as displacements and internal stress resultants and stresses. These results reveal that the predicted buckling load of the ESL panel is larger than that of the sandwich panel and that deep in the non-linear range the upper face sheet wrinkles with increasing overall and edge displacements and a release of the load. Hence, the use of an equivalent single layer panel especially when a sandwich panel with a compliant core is considered may lead to unsafe and unreliable predictions when large displacements and large rotations are considered. [Copyright &y& Elsevier]

Highlights: [•] Director theory is used to formulate the equations of motion. [•] Shooting method is adopted in the numerical analysis. [•] Focus is placed on the effects of extensibility on the vibration characteristics. [•] The results from extensible elastica, inextensible elastica, and small-deformation theory are compared. [Copyright &y& Elsevier]

In the problem of the synthesis of metamaterials, the pantographic architecture revealed remarkable potentialities. Indeed it allowed for the synthesis of second gradient 2D (nonlinear) continua: i.e. 2D shells whose deformation energy depends also on the second derivatives of displacements in the tangent directions to the reference configuration. Moreover, pantographic architecture seems to be able to produce metamaterials whose macroscopic elongations are large, albeit remaining in the elastic regime. The theoretically shown potentialities have started to become of ≫practical≪ interest thanks to a series of experiments, which were made possible by the recent 3D additive manufacturing. The actual construction of pantographic architecture has been based on the design of two arrays of beams interconnected by small cylinders, whose behavior can be modeled in different ways: if they are very short they can be regarded as clamps, while if they are short enough as elastic (or inelastic for large rotations) cylindrical hinges connecting the beams of different arrays. Otherwise, they must be modeled as elastic (or inelastic) elements allowing for relative rotations and displacements. In this paper, we focus on this particular case and we introduce, after a homogenization based on heuristic arguments, a 2D generalized continuum model whose kinematics is characterized by two placement and rotation fields (one for each array of beams) and whose deformation energy depends on relative displacements and rotations. The offset between the two beams arrays is proven to be an essential tool for defining effective invariant kinematical deformation measures. In facts, one wants to postulate a deformation energy for the introduced 2D generalized continuum which gives predictions in agreement with those given by the more refined 3D model where the pantographic architecture is described with its maximum geometric complexity and where the constituting material is assumed to be modelable as a standard 3D first gradient continuum. In the present paper, in order to arrive at the correct conjecture for the postulated energy, we consider the concept of averages of rotations in SO(3) Lie group. The used enriched kinematics is seen to be a possible alternative to the adoption of second gradient 2D models. Some rather surprising deformation processes are studied, where interesting non-symmetric post-buckling phenomena are observed in both the models used. Mentioned post-buckling has been observed experimentally. [ABSTRACT FROM AUTHOR]

A micromorphic computational homogenization framework has recently been developed to deal with materials showing long-range correlated interactions, i.e. displaying patterning modes. Typical examples of such materials are elastomeric mechanical metamaterials, in which patterning emerges from local buckling of the underlying microstructure. Because pattern transformations significantly influence the resulting effective behaviour, it is vital to distinguish them from the overall deformation. To this end, the following kinematic decomposition into three parts was introduced in the micromorphic scheme: (i) a smooth mean displacement field, corresponding to the slowly varying deformation at the macro-scale, (ii) a long-range correlated fluctuation field, related to the buckling pattern at the meso-scale, and (iii) the remaining uncorrelated local microfluctuation field at the micro-scale. The micromorphic framework has proven to be capable of predicting relevant mechanical behaviour, including size effects and spatial as well as temporal mixing of patterns in elastomeric metamaterials, making it a powerful tool to design metamaterials for engineering applications. The long-range correlated fluctuation fields need to be, however, provided a priori as input parameters. The main goal of this study is experimental identification of the decomposed kinematics in cellular metamaterials based on the three-part ansatz. To this end, a full-field micromorphic Integrated Digital Image Correlation (IDIC) technique has been developed. The methodology is formulated for finite-size cellular elastomeric metamaterial specimens deformed in (i) virtually generated images and (ii) experimental images attained during in-situ compression of specimens with millimetre sized microstructure using optical microscopy. The proposed IDIC method identifies the different kinematic fields, both before and after the microstructural buckling, and without any prior knowledge determines correctly the relevant patterning modes required by the homogenization scheme. It is further argued that patterning modes are independent of the unit cell size, the hole diameter to cell size ratio, as well as local material properties, allowing for modelling and design of (finite- and infinite-size) metamaterials and specimens with graded microstructures in terms of geometry and/or material properties. It is shown that the proposed methodology is also applicable to cellular metamaterials and structures with different microstructural designs. • Decomposition of cellular metamaterials kinematics by an ansatz from literature. • Development of a novel Micromorphic Integrated Digital Image Correlation scheme. • Unravelling long-range correlated fluctuation modes & their spatial distributions. • Identification of micromorphic fields achieved directly from in-situ test images. [ABSTRACT FROM AUTHOR]

Abstract: Molecular mechanics (MM) simulations have been carried out to determine energetically favorable double-walled carbon nanotube (DWNT) structures, and analyze their infinitesimal extensional, torsional, radial expansion/contraction, and bending deformations. Loads are applied either to one wall or simultaneously to both walls of an open-ended DWNT. These results are compared against single-walled carbon nanotube (SWNT) results to determine differences and similarities between responses of SWNTs and DWNTs, and the validity of using SWNT results to predict the response of a DWNT. It is found that for small deformations such as simple tension and torsion, results for a DWNT can be derived from those for its constituent SWNTs within 3% error. Results of radial expansion/contraction of a SWNT are used to deduce an expression for the van der Waals force. Based on these results, a continuum model is proposed for a MWNT whose response to mechanical deformations computed using engineering theories is the same as that of the MWNT obtained via MM simulations. The continuum structure is comprised of concentric cylindrical tubes interconnected by truss elements. Young’s modulus, Poisson’s ratio, the thickness of each concentric tube, and the stiffness of the truss elements are given. The proposed continuum model is validated by studying bending and the onset of global buckling deformations of a DWNT and its proposed equivalent continuum structure. Carbon nanotubes can be replaced by their equivalent continuum structures when deriving mechanical properties of nanotube reinforced polymeric composites. [Copyright &y& Elsevier]

In this paper a semi-analytic solution for the post-critical behavior of compressed thin walled members with generic cross sections is presented. It is based on the Koiter approach and the method of separation of variables. The buckling solution is exactly evaluated using a single sinusoidal function and the initial post-critical behavior is obtained as a decoupled sinusoidal series solution along the beam axis. A specialized integration scheme allows obtaining the solution with only a few terms and a very low cost, with respect to standard finite element analyses. This tool is then used to highlight the reason of the poor behavior of beam models enriched with cross section deformation modes in reconstructing the post-critical solution. Successful strategies are proposed in order to overcome these limitations. A series of numerical tests are reported. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a study on stretch-induced wrinkling of thin polyethylene sheets when subjected to uniaxial stretch with two clamped ends. Three-dimensional digital image correlation was used to measure the wrinkling deformation. It was observed that the wrinkle amplitude increased as the nominal strain increased up to around 10%, but then decreased at larger strain levels. This behavior is consistent with results of finite element simulations for a hyperelastic thin sheet reported previously (Nayyar et al., 2011). However, wrinkles in the polyethylene sheet were not fully flattened out at large strains (>30%) as predicted for the hyperelastic sheet, but exhibited a residual wrinkle whose amplitude depended on the loading rate. This is attributed to the viscoelastic response of the material. Two different viscoelastic models were adopted in finite element simulations to study the effects of viscoelasticity on wrinkling and to improve the agreement with the experiments, including residual wrinkles and rate dependence. It is found that a parallel network model of nonlinear viscoelasticity is suitable for simulating the constitutive behavior and stretch-induced wrinkling of the polyethylene sheets. [Copyright &y& Elsevier]

Abstract: Light-activated polymers can undergo complex deformation in response to the combination of mechanical and optical stimuli. These materials are attractive for remote actuation and sensing applications. However, the behavior of such materials subjected to photomechanical patterning is not well understood. In this paper we consider a polymer that operates by photoactivated stress relaxation; at the molecular level, photoinitiation of residual initiator molecules generate free radicals that break and then reform in-chain functionalities of stretched chains in an elastomeric network, which results in macroscopic stress relaxation. We carry out experiments and finite element calculations that demonstrate the sequence of deformation events culminating in the formation of a buckled spot as a result of biaxially stretching the elastomeric film then irradiating a circular region followed by releasing the mechanical constraint. In order to better understand the photomechanics, we analyze a simpler model problem wherein a linear elastic, stress relaxing disk is subjected to (i) radial extension, (ii) irradiation of a concentric circular region, and (iii) release of the applied displacements in (i), which results in deformation and stress redistribution. In the final step, the deformation may transition from planar to buckling out of the plane depending on system parameters. Companion finite element calculations are performed against which our analytical results are in good agreement. Although not directly comparable, the analytic model qualitatively agrees with the experiments. The results of this work provide a useful foundation from which to explore more interesting behavior of periodically photo-mechanically patterned films and other more challenging actuation problems. [Copyright &y& Elsevier]

Abstract: The energy absorption and load-bearing capacity under axial compression of some model cellular structures are studied with an eye toward optimization based on structural mass or volume available for deformation. Three configurations are considered: multilayer, multi-cell and multi-tube, all of a rectangular-cell topology. Loading is applied either parallel or normal to the cell axis. The cell’s aspect ratio and the relative density of the material ρ are systematically varied. The specimens are laterally confined by rigid walls to stabilize the deformation, but the effect of confinement diminishes for sufficiently large number of cells. A square-cell topology seems to be optimal. Together with an appropriate value for ρ, this provides an optimal constraint on the wavelength of the characteristic buckle and consequently extensive energy dissipation throughout the material body. When considering mean stress, crush energy and stroke or densification strain on the basis of minimum mass and volume simultaneously, ρ ≈0.5 seem to be a viable compromise among conflicting trends. The mechanical performance in this case is considerably improved over common cellular structures, for which ρ is typically <0.1. [Copyright &y& Elsevier]