Search

In this paper, a linear smoothing scheme over eight-node Reissner-Mindlin plate element under the framework of the CS-FEM is employed to buckling analysis of laminated composite plates based on the first-order shear deformation theory. The modified stain matrix is computed by the divergence theorem between the nodal shape functions and their derivatives using Taylor's expansion. Isoparametric mapping and the computation of interior derivatives of shape function is not required in the proposed method, furthermore, all the computation are based on the global Cartesian coordinates. Some numerical examples are given at the end to demonstrate that the present method has good performance to alleviate the shear-locking phenomenon and improve the quality of the solutions with distorted meshes. [ABSTRACT FROM AUTHOR]

• A GMLS-based numerical manifold method (NMM) framework is proposed for the fourth-order problems with rotational degrees of freedom. • The dual-cover system has been fitted to GMLS-based numerical manifold method (NMM) to treat plates of complex geometry and openings without resorting to those complicated criteria and tailored operations. • The proposed GMLS-NMM has been applied for analyzing the bending, free vibration and buckling behaviors of thin plates complicated shape or cutouts. • The feasibility and accuracy of the proposed method are verified with extensive numerical examples considering plates with various shapes and cutouts. We proposed a generalized moving least squares (GMLS) based numerical manifold method (NMM) and applied to investigate mechanical behaviors of thin plates with various shape and cutouts. The influence domains of nodes in generalized moving least squares (GMLS) interpolation are employed to construct the mathematical cover, which is specifically fitted for the fourth-order problems with rotational degrees of freedom. Moreover, the GMLS-based NMM can naturally treat plates with arbitrary complex geometry and openings without resorting to those complicated criteria and tailored operations while simplifying the physical cover generation process. In numerical study, a suitable mass lumping scheme is formulated and advised for transverse vibration analysis. Bending, vibration and buckling behaviors of thin plates with different shapes and cutouts are studied. Numerical results manifest that GMLS-based NMM can accurately model transverse mechanical behaviors. [ABSTRACT FROM AUTHOR]

This research studies the nonlinear buckling of two-dimensional functionally graded (2D-FG) nanotubes with porosity based on the Zhang-Fu theory of tubes, Timoshenko beam theory, and the nonlocal gradient strain theory (NSGT) as well as Von-Karmen nonlinear theory. In this paper, the formulation of the problem for various boundary conditions is generated according to the energy method. Then, the results are extracted by incorporating the generalized differential quadrature method (GDQM) coupled with the iteration method. The accuracy of the results is proven through comparative studies, and finally, the impact of different parameters, which influence the buckling of the nanotube, is investigated. [ABSTRACT FROM AUTHOR]

An efficient way to introduce elastic energy that can bias an origami structure toward desired shapes is to allow curved tiles between the creases. The bending of the tiles supplies the energy and the tiles themselves may have additional functionality. In this paper, we present a basic theorem and systematic design methods for quite general curved origami structures that can be folded from a flat sheet, and we present methods to accurately find the stored elastic energy. Here the tiles are allowed to undergo curved isometric mappings, and the associated creases necessarily undergo isometric mappings as curves. These assumptions are consistent with a variety of practical methods for crease design. The h 3 scaling of the energy of thin sheets (h = thickness) spans a broad energy range. Different tiles in an origami design can have different values of h , and individual tiles can also have varying h. Following developments for piecewise rigid origami Fan Feng et al. (2020), we develop further the Lagrangian approach and the group orbit procedure in this context. We notice that some of the simplest designs that arise from the group orbit procedure for certain circle groups provide better matches to the buckling patterns observed in compressed cylinders and cones than known patterns. [ABSTRACT FROM AUTHOR]

We develop a comprehensive, geometrically-exact theory for an end-loaded heavy rod constrained to deform on a cylindrical surface. The cylinder can have arbitrary orientation relative to the direction of gravity. By viewing the rod-cylinder system as a special case of an elastic braid, we are able to obtain all forces and moments imparted by the deforming rod to the cylinder as well as all contact reactions. This framework allows for the monitoring of stresses to ascertain whether the cylinder, along with its end supports, is able to sustain the rod deformations. As an application of the theory we study buckling of the constrained rod under compressive and torsional loads, as well as the tendency of the rod to lift off the cylinder under further loading. The cases of a horizontal and vertical cylinder, with gravity having only a lateral or axial component, are amenable to exact analysis, while numerical results map out the transition in buckling mechanism between the two extremes. Weight has a stabilising effect for near-horizontal cylinders, while for near-vertical cylinders it introduces the possibility of buckling purely due to self-weight. Our results are relevant for many engineering and medical applications in which a slender structure is inserted into a cylindrical cavity. [ABSTRACT FROM AUTHOR]

Basal dislocations and ripplocations both offer explanations for the deformation of layered crystalline solids in response to compressive strain. Existing work, however, presents no clear definition of a ripplocation and the distinction between these two mechanisms remains unclear. Molecular dynamics simulations in graphite and modified graphitic models reveal that equivalent additional half-planes of material can induce both dislocations and ripplocations. In this work, we find that ripplocations are essentially an elastic buckling phenomenon operating at the atomic scale, rather than a discrete crystallographic defect. Dislocation core confinement produces incommensurate planar elastic strains with sufficient energy to trigger a buckling instability that transfers strain energy to the surrounding lattice. The interplay between incommensurate strain and buckling is a quasi-continuum phenomenon where both mechanisms may accommodate arbitrary disregistry, not only the discrete values accommodated by perfect or partial basal dislocations. The mechanistic transition depends critically on the atomic-scale interfacial energies between layers and the continuum elastic behavior of the bulk material. Buckling is possible, though unfavorable, in the MAX phase Ti 3 AlC 2 and systems of multiple layers buckle at lower strains than only a single layer. These extensions of the ripplocation model in van der Waals layered solids suggest that nanoscale elastic buckling potentially plays a role in the deformation of a wide range of layered crystalline systems in response to common mechanical configurations. [ABSTRACT FROM AUTHOR]

• A meshfree Galerkin approach is presented for free vibration and buckling analysis of a strain gradient thin plate. • A unified consistent integration scheme with nodal first to third order smoothed gradients is employed. • Effects of internal scale parameter, classical and high order boundary conditions are investigated. In this paper, a meshfree Galerkin approach is presented for analysis of free vibration and buckling of a strain gradient thin plate. A cubic moving least square or reproducing kernel approximation with C 2 continuity is used to ensure convergence. Due to smoothness of the meshfree approximation, the only variable is the deflection of a thin plate. To improve the computational accuracy, a consistent integration scheme with gradient smoothing is introduced to construct stiffness and geometrical stiffness matrices. The effectiveness of such an approach is confirmed by numerical results, and it is superior to a standard Gauss integration based meshfree Galerkin method. Further, the influence of classical and high order boundary conditions is investigated on the natural frequencies and critical buckling loads of strain gradient thin plates. [ABSTRACT FROM AUTHOR]

• A high-order model considering shear deformation and thickness stretching effect is established to investigate the bending and buckling responses of porous FG curved nanobeam. • The governing equation is derived based on the consistent two-phase local/non-local stress gradient theory, and the analytical solution of curved nanobeams is obtained by Navier's method. • The difference of buckling mode sensitivity between curved characteristics, scale effect and thickness stretching effect of the curved beam is analyzed. In this work, we provide an analysis of the bending and buckling properties of functionally graded (FG) nanobeams with trigonometric function-dependent porosity distribution. Herein, a high-order curved nanobeam model considering both shear deformations and thickness stretching effect is established. Two-phase local/nonlocal stress-driven gradient theory is utilized to capture small-scale effects. Hamilton's principle is adopted to obtain the governing equation and boundary conditions, and the analytical solutions, such as exact formulas for lateral displacement and critical buckling load are obtained by employing Navier's method for simply supported boundary conditions. Furthermore, the effects of local volume fraction, nonlocal parameter, material length scale parameter, shear deformation, material distribution, and porosity distribution pattern in stress-driven models are analyzed in detail. The parametric studies indicate that the thickness stretching effect has a different influence on bending and buckling. In addition, the thickness stretching effect and the opening angle show different coupling mechanisms during bending and buckling. On the other hand, the higher-order buckling mode leads to a more significant scale effect and thickness tensile effect, but reduces the influence of the opening angle. Finally, we find that due to the coupling effect between the pore distribution pattern and the orientation bias of the FG nanobeam, the variation of the material gradient index leads to a different order of the dimensionless critical buckling loads predicted by the three pore distributions. [ABSTRACT FROM AUTHOR]

An efficient numerical model is developed in this work for the bifurcation analysis within Föppl-von Karman plate theory. This procedure is based on the use of the method of fundamental solutions in a high order continuation method that relies on Taylor series expansion with respect to a path parameter. This numerical model has an adaptive step length, which is effective especially for solving nonlinear problems and detecting bifurcation points. Despite of the huge application field of nonlinear elasticity, the method of fundamental solution was very rarely applied in this field and never to a nonlinear elastic plate model. The governing equations are strongly formulated in terms of the two unknowns: the deflection and the stress function. The singular value decomposition regularization is used to overcome the difficulty of the resulting system ill-conditioning. The accuracy and efficiency of the numerical model are illustrated on buckling numerical examples. [ABSTRACT FROM AUTHOR]

Subjected to compressive stresses, soft polymers with stiffness gradients can display various buckling patterns. These compressive stresses can have different origins, like mechanical forces, temperature changes, or, for hydrogel materials, osmotic swelling. Here, we focus on the influence of the transient nature of osmotic swelling on the initiation of buckling in confined layered hydrogel structures. A constitutive model for transient hydrogel swelling is outlined and implemented as a user-subroutine for the commercial finite element software Abaqus. The finite element procedure is benchmarked against linear perturbation analysis results for equilibrium swelling showing excellent correspondence. Based on the finite element results we conclude that the initiation of buckling in a two-layered hydrogel structure is highly affected by transient swelling effects, with instability emerging at lower swelling ratios and later in time with a lower diffusion coefficient. In addition, for hard-on-soft systems the wavelength of the buckling pattern is found to decrease as the diffusivity of the material is reduced for gels with a relatively low stiffness gradient between the substrate and the upper film. This study highlights the difference between equilibrium and transient swelling when it comes to the onset of instability in hydrogels, which is believed to be of importance as a fundamental aspect of swelling as well as providing input to guiding principles in the design of specific hydrogel systems. [ABSTRACT FROM AUTHOR]

Abstract Under a variety of external stimuli, hydrogels can undergo coupled solid deformation and fluid diffusion and exhibit large volume changes. The numerical analysis of this process can be complicated by numerical instabilities when using mixed formulations due to the violation of the inf-sup condition. In addition, the large deformations produce complex instability patterns causing singularities in the underlying set of equations. For these reasons, the experimentally observed complex patterns remain elusive and poorly understood. Furthermore, a stability criterion suitable to detect critical conditions and predict post-instability patterns is lacking for hydrogel simulations. Here we investigate the stability criterion for coupled problems with a saddle point nature and propose a generic framework to study diffusion-driven swelling-induced instabilities of hydrogels. Adopting a numerically stable subdivision-based mixed isogeometric analysis, we show that the proposed framework for stability analysis accurately captures instability points during the transient swelling of hydrogels. The influence of geometrical and material parameters on the critical conditions are also presented in stability diagrams for two useful problems involving the buckling of hydrogel rods and the wrinkling on the surface of hydrogel bilayers. The results show that the short-time response of hydrogels immersed in water are highly unstable. We believe that this generic scheme provides a theoretical and computational foundation to study the morphogenesis in nature, and it also paves the way to create functional materials and design novel hydrogel devices through stability diagrams. [ABSTRACT FROM AUTHOR]

Abstract This paper presents a study of the mechanical response of hexachiral honeycombs with transversally curved ligaments under planar uniaxial tensile, compressive and shear loads. The impact of the chiral cell design parameters on the resulting macroscopic behaviour is assessed utilising finite elements calculations. It is shown that the presence of ligament curvature permits to attain mechanical responses which are not achievable through conventional chiral honeycomb designs. In addition, the resulting responses exhibit, for all considered load cases, significant tunability through the investigated geometrical design parameters. Two chiral lattices with identical geometries, only differing in their ligament curvature, were manufactured and experimentally tested to validate the finite elements predictions. A connection and assembly strategy is presented and utilised, offering a fast and robust approach to build larger finite lattice structures through 3D printed single basic cells. The hexachiral lattices were tested in tension, compression and in-plane shear, showing good agreement with the numerical predictions. [ABSTRACT FROM AUTHOR]

Abstract Interaction between two straight-sided buckles propagating in opposite directions has been investigated by atomic force microscopy. Below a critical separation distance, it is observed that the two buckles are attracted towards each other once their fronts have crossed. A mechanical analysis using finite element simulations has shown that the buckle interaction is strongly influenced by the substrate elasticity. The deviation of the two buckles from their initial straight propagation is discussed at the light of mode mixity mappings extracted from finite elements simulations along the crack path of the two interacting buckles. Finally, a phase diagram depending on both the separation distance and the film/substrate elastic mismatch is proposed, in good agreement with the experimental results. Graphical abstract Image, graphical abstract [ABSTRACT FROM AUTHOR]

Abstract We study the dynamics of snap-through when viscoelastic effects are present. To gain analytical insight we analyse a modified form of the Mises truss, a single-degree-of-freedom structure, which features an 'inverted' shape that snaps to a 'natural' shape. Motivated by the anomalously slow snap-through exhibited by spherical elastic caps, we consider a thought experiment in which the truss is first indented to an inverted state and allowed to relax while a specified displacement is maintained; the constraint of an imposed displacement is then removed. Focussing on the dynamics for the limit in which the timescale of viscous relaxation is much larger than the characteristic elastic timescale, we show that two types of snap-through are possible: the truss either immediately snaps back over the elastic timescale or it displays 'pseudo-bistability', in which it undergoes a slow creeping motion before rapidly accelerating. In particular, we demonstrate that accurately determining when pseudo-bistability occurs requires the consideration of inertial effects immediately after the indentation force is removed. Our analysis also explains many basic features of pseudo-bistability that have been observed previously in experiments and numerical simulations; for example, we show that pseudo-bistability occurs in a narrow parameter range at the bifurcation between bistability and monostability, so that the dynamics is naturally susceptible to critical slowing down. We then study an analogous thought experiment performed on a continuous arch, showing that the qualitative features of the snap-through dynamics are well captured by the truss model. In addition, we analyse experimental and numerical data of viscoelastic snap-through times reported previously in the literature. Combining these approaches suggests that our conclusions may also extend to more complex viscoelastic structures used in morphing applications. [ABSTRACT FROM AUTHOR]

Highlights • Poroelastic chemomechanical theory for analyzing vascular collapse in solid tumors. • Effect of biochemmechanical coupling mechanism and fluid transport on vessel buckling. • Biochemomechanical finite element method for buckling and postbuckling analysis. Abstract The collapse of blood vessels are widely observed in solid tumors, but the mechanisms underpinning this abnormal behavior remain unclear. In this paper, we investigate the stability of blood vessels embedded in a growing solid tumor by using a chemomechanical poroelastic theory. Linear stability analysis is first made to give the critical condition of vascular buckling. Thereby, pattern diagrams are provided to predict the collapsed shape of a blood vessel from the mechanical and geometric parameters of the system. Due to the mechano-chemo-biological coupling effect in tumors which involve both fluid transport and tissue growth, the buckling of blood vessels exhibits distinctly different features from such previously reported tubular tissues as airways and esophagi. We show that interstitial fluid pressure tends to drive blood vessels to buckle with a lower buckling mode, while perivascular cell proliferation favors a higher mode. Furthermore, a nonlinear biochemomechanical finite element method is formulated to track the morphological evolution of blood vessels during postbuckling. It is found that the blood vessels with the second mode of buckling may readily evolve into a crack-like shape. This method can reproduce the essential features of vascular collapse observed in in vivo tumors and provide clues for vascular normalization and stress alleviation in cancer treatments. Graphical abstract Image, graphical abstract [ABSTRACT FROM AUTHOR]

Abstract Buckling behavior of flat and cylindrical shells including through-the-thickness crack (through crack) is examined employing an effective reproducing kernel (RK) meshfree method. The concept of convected coordinate system is adopted to deal with general curvilinear surfaces. Both field variables and shell geometry are approximated by RKs, which is conceptually same procedure with isoparametric Finite Element Method (FEM). Each node has five degrees of freedom (DOFs). The numerical integration of stiffness matrices is conducted by strain smoothing approaches. In the present study, a crack modeling is introduced into the curved shell geometry for analyzing cracked cylinder buckling problems effectively. The presented approach has an attractive feature, i.e., five DOFs cracked flat shell model is only required for analyzing three-dimensional (3D) cracked curved shell problems. The accuracy and effectiveness of the present method are critically examined through several numerical examples in which the obtained results are compared with reference solutions as well as with the results of commercial FEM package (ANSYS). Effects of the element types in the FEM computations are also examined by comparison of the results by linear and quadratic shell elements. The results shed light on the significant effects of considered configurations on buckling coefficients and mode shapes. [ABSTRACT FROM AUTHOR]

This paper studies the buckling behavior and imperfection sensitivity of geodesic and stellated shells subject to external pressure. It is shown that these structures can completely eliminate the severe imperfection sensitivity of spherical shells and can achieve buckling pressure and mass efficiency higher than the perfect sphere. Key results of this paper are as follows. First, a shell with the shape of an icosahedron can carry external pressure significantly higher than a spherical shell, when the effects of geometric imperfections are considered. Second, stellated shells are generally insensitive to imperfections. For pyramids with height-to-radius ratios greater than 35% the buckling pressure is greater than for a perfect sphere. The specific ratio 45% gives the highest buckling pressure, 28% higher than the perfect sphere. Third, stellated icosahedra with concave pyramids have higher mass efficiency than the perfect sphere. Fourth, in terms of volume efficiency, geodesic shells are comparable to spherical shells with a knockdown factor of 0.2 and convex stellated shells are comparable to spherical shells with a knockdown factor of 0.65. [ABSTRACT FROM AUTHOR]

The global buckling and wrinkling behavior of sandwich plates with anisotropic facesheets is investigated by means of a linearized stability analysis. High-order plate models are formulated referring to a sublaminate variable-kinematic approach: an axiomatic through-thickness description of the displacement field is introduced, wherein the selected model is employed for an arbitrarily defined group of plies, i.e., the sublaminate. The two-dimensional governing equations of the plate model are solved in weak form by means of the Ritz method. The modeling approach is applied to sandwich plates with anisotropic laminated facesheets and orthotropic cores. A wide set of configurations is analyzed: critical loads and wrinkling patterns are determined for panels with foam and honeycomb cores, subjected to uniaxial as well as multiaxial loads. The proposed approach is shown to provide accurate, quasi-3D predictions for both long and short wavelength buckling with a reduced computational effort. [ABSTRACT FROM AUTHOR]

We investigate pattern transformations of periodic hydrogel systems that are triggered by swelling-induced structural instabilities. The types of microstructures considered in the present work include single-phase and two-phase voided hydrogel structures as well as reinforced hydrogel thin films. While the observed transformations of the single-phase structures show good agreement with experimental findings, the two-phase materials provide novel patterns associated with wrinkling of internal surfaces. Furthermore, an extensive parametric study on the reinforced hydrogel thin films reveals new opportunities for the design of complex out-of-plane surface modes caused by swelling-induced instabilities. Next to the mentioned buckling-type instabilities, we encountered the development of micro-creases at the internal surfaces of periodic media. [ABSTRACT FROM AUTHOR]

We develop a comprehensive geometrically-exact theory for an end-loaded elastic rod constrained to deform on a cylindrical surface. By viewing the rod-cylinder system as a special case of an elastic braid, we are able to obtain all forces and moments imparted by the deforming rod to the cylinder as well as all contact reactions. This framework allows us to give a complete treatment of static friction consistent with force and moment balance. In addition to the commonly considered model of hard frictionless contact we analyse in detail two friction models in which the rod, possibly with intrinsic curvature, experiences either lateral or tangential friction. As applications of the theory we study buckling of the constrained rod under compressive and torsional loads, finding critical loads to depend on Coulomb-like friction parameters, as well as the tendency of the rod to lift off the cylinder under further loading. Our results are relevant for many engineering and medical applications in which a slender structure winds inside or outside a cylindrical boundary. [ABSTRACT FROM AUTHOR]

• Wrinkling-assisted strategy for controlled interface delamination. • Analytical model of interface delamination in a bilayer film/substrate system. • Characteristic number K that dominates the process of interface delamination. • Assembly of highly-flexible 3D structures and those with large-area suspended features. Mechanically-guided three-dimensional (3D) assembly is a recently established method for the fabrication of 3D structures and devices at micro/nanoscale. Such assembly methods typically involve a process of compressive buckling in a patterned high-modulus thin film integrated with a low-modulus elastomer substrate at selective regions. A successful assembly requires, simultaneously, a sufficiently strong interface in selective regions to stay undelaminated, and a sufficiently weak interface in remaining regions to be fully separated. However, such requirements are very challenging to meet for highly flexible 3D mesostructures or those with large-area suspended features. Here, we propose a wrinkling-assisted strategy to substantially facilitate the delamination at desired regions of the film/substrate system. An additional assisting layer is introduced such that a weaker film/assisting-layer interface replaces the original film/substrate interface, and the wrinkles formed in the assisting layer induce additional driving forces to separate the film/assisting-layer interface. An analytical model is developed to capture the delamination of the thin film from the wrinkled assisting layer, which is validated by both experiments and finite element analyses (FEA). The model shows that a combined material/geometry parameter governs the process of delamination in the bilayer film/substrate system, offering a useful design reference for the mechanically-guided 3D assembly. Furthermore, our experiments and FEA demonstrate that the wrinkling-assisted strategy can enable the assembly of highly flexible 3D mesostructures and those with large areas attached to the substrate in the initial configuration, which are particularly challenging to achieve by using previous strategies. [ABSTRACT FROM AUTHOR]

Nonlocal strain gradient models are being used more and more extensively in examining the size-dependent effects on the statical and dynamical behaviors of micro/nano-structures. However, for the fake of simplification, the size-dependent effects are often assumed to be neglected in the thickness direction of beams and plates. This simplification is originally aiming at reducing the complexity of the studied beam and plate problems while retaining all the most important mechanical features. Based on the nonlocal strain gradient model without the thickness effect, the size-dependent effect may reveal stiffness-softening or stiffness-hardening effect, which only depends on the values of strain-gradient and stress-gradient parameters but is independent on the geometric feature. In this paper, a nonlocal strain gradient beam model incorporating the thickness effect is developed for the size-dependent buckling analysis of nanobeams, and closed-form solutions are derived for post-buckling configuration and critical buckling force. When incorporating the size-dependent effect of thickness, it is found that the stiffness-softening and stiffness-hardening effects depend not only on the ratio of stress-gradient parameter to strain-gradient parameter, but also on the geometric feature (slenderness ratio). Interestingly, the stiffness-softening effect can be only found for the longitudinal dispersion relation of Silicon, while the stiffness-hardening effect may be observed for the buckling behaviors of nanobeams made of Silicon due to the size-dependent effect of thickness. [ABSTRACT FROM AUTHOR]

In this paper, a comprehensive study on mechanical behavior of non-uniform small scale beams in the framework of nonlocal strain gradient theory is presented. The governing equations and boundary conditions have been developed using Hamilton's principle and solved with the aid of finite element method for introducing the bending, buckling and free vibration response of nano-beams. The current formulation could be used for all types of non-uniformities by varying the width and thickness of nano-beams along the length. The accuracy of the current model and formulation is verified by comparing the results with previous literature and those obtained by analytically solving simplified problems. In order to understand the influence of having non-uniform cross section on static and dynamic behavior of small scale beams, parametric study is presented and the effects of different parameters such as non-uniformity, non-local and strain gradient terms on natural frequency, buckling load and deformation are observed and discussed. It is seen that having non-uniform cross section in nonlocal strain gradient beams could lead to significant changes in mechanical behavior of such structures. [ABSTRACT FROM AUTHOR]

Protein microtubules (MTs) are one of the important intercellular components and have a vital role in the stability and strength of the cells. Due to applied external loads, protein microtubules may be involved buckling phenomenon. Due to impact of protein microtubules in cell reactions, it is important to determine their critical buckling load. Considering nature of protein microtubules, various parameters are effective on microtubules buckling. The small size of microtubules and also lack of uniformity of MTs properties in different directions caused the necessity of accuracy in the analysis of these bio-structure. In fact, microtubules must be considered as a size dependent cylinder, which behave as an orthotropic material. Hence, in the present work using first-order shear deformation model (FSDT), the buckling equations of anisotropic MTs are derived based on new modified couple stress theory (NMCST). After solving the stability equations, the influences of various parameters are measured on the MTs critical buckling load. [ABSTRACT FROM AUTHOR]

In present paper, a unified size-dependent high-order beam model which contains various higher-order shear deformation beam models as well as Euler–Bernoulli and Timoshenko beam models is developed to study the simultaneous effects of nonlocal stress and strain gradient on the bending and buckling behaviors of nanobeams by using the nonlocal strain gradient theory. For this objective, a nonlocal parameter is introduced to capture the nonlocal effect and a material length scale parameter is involved to evaluate the strain gradient effect. The governing equations and the associated boundary conditions are formulated by using Hamilton's principle. Navier's method is utilized to obtain the analytical solutions for bending and buckling of a simply supported nanobeam. The present model is validated by comparing the obtained results with those available in literature. The influences of nonlocal parameter, material length scale parameter, slenderness ratio and shear deformation on the bending and buckling behaviors of the nanobeam are examined in detail. Results reveal that within the framework of nonlocal strain gradient theory, results predicted by Timoshenko beam model and various higher-order beam models are almost same with some negligible differences. Moreover, it is found that the nanobeam could exhibit either stiffness-softening effect or stiffness-hardening effect, which depends on the relative magnitude of the nonlocal parameter and the material length scale parameter. [ABSTRACT FROM AUTHOR]

The inconsistency of Eringen’s nonlocal differential model, as applied to investigate nanostructures, has recently triggered the study of nonlocal integral models. In this paper we adopt Eringen’s two-phase nonlocal integral model to carry out an analytical study on the buckling problem of Euler-Bernoulli beams. By using a reduction method rigorously proved in the previous work, the resulting integro-differential equation for the problem is firstly reduced to a fourth order differential equation with mixed boundary conditions. Exact characteristic equations are then obtained for four types of boundary conditions. Further, after some detailed asymptotic analysis, asymptotic solutions for the critical buckling loads are obtained, which are shown to have a good agreement with the numerical solutions. The analytical solutions show clearly that the nonlocal effect reduces the buckling loads. It is also found that the effect could be first-order or second order depending on the boundary conditions. [ABSTRACT FROM AUTHOR]

Metamaterials are composed of structural elements and derive their properties mainly from the inner structure of the elements, rather than the properties of their constituent material. By designing an unstable structural element as the building block of a metamaterial, many interesting effective material properties can be obtained. The deformation and dissipation mechanisms of such a material built from unstable structural elements is studied in detail. To do so a combination of analytical, semi-analytical, and numerical models are applied to a single buckling element, a periodic cell, and finite size combinations of buckling elements including gradients in the properties of the building blocks. This not only provides insight into the micromechanics and the resulting effective behavior of such metamaterials, but also makes them accessible on the different relevant length scales. A metamaterial built from these building blocks shows programmable or switchable properties and can display energy dissipation with fully reversible deformation, distinguishing it from plastic materials, and timescale independent behavior, distinguishing it from viscoelastic materials. [ABSTRACT FROM AUTHOR]

The buckling behavior of nanocomposite plates of polygonal planform under in-plane loads is examined. The plate under consideration is reinforced by single-walled carbon nanotubes (CNTs). The governing eigenvalue equation to this problem is derived based on the first-order shear deformation plate theory (FSDT) with a set of element-free shape functions in approximating the two-dimensional displacement fields. To solve this eigenvalue equation, the element-free IMLS-Ritz method is employed to furnish the buckling solution. The convergence of the solution for the CNT-reinforced composite plates is examined. Comparison study is further carried out to validate the accuracy of the solution. A parametric study is performed by varying the CNT volume fraction, CNT distribution, plate thickness-to-apothem ratio and boundary conditions. This first known buckling solution may serve as benchmarks for future research. [ABSTRACT FROM AUTHOR]

A central question in developmental biology is how size and position are determined. The genetic code carries instructions on how to control these properties in order to regulate the pattern and morphology of structures in the developing organism. Transcription and protein translation mechanisms implement these instructions. However, this cannot happen without some manner of sampling of epigenetic information on the current patterns and morphological forms of structures in the organism. Any rigorous description of space- and time-varying patterns and morphological forms reduces to one among various classes of spatio-temporal partial differential equations. Reaction-transport equations represent one such class. Starting from simple Fickian diffusion, the incorporation of reaction, phase segregation and advection terms can represent many of the patterns seen in the animal and plant kingdoms. Morphological form, requiring the development of three-dimensional structure, also can be represented by these equations of mass transport, albeit to a limited degree. The recognition that physical forces play controlling roles in shaping tissues leads to the conclusion that (nonlinear) elasticity governs the development of morphological form. In this setting, inhomogeneous growth drives the elasticity problem. The combination of reaction-transport equations with those of elasto-growth makes accessible a potentially unlimited spectrum of patterning and morphogenetic phenomena in developmental biology. This perspective communication is a survey of the partial differential equations of mathematical physics that have been proposed to govern patterning and morphogenesis in developmental biology. Several numerical examples are included to illustrate these equations and the corresponding physics, with the intention of providing physical insight wherever possible. [ABSTRACT FROM AUTHOR]

The stability of thin viscous sheets has been studied so far in the special case where the base flow possesses a direction of invariance: the linear stability is then governed by an ordinary differential equation. We propose a mathematical formulation and a numerical method of solution that are applicable to the linear stability analysis of viscous sheets possessing no particular symmetry. The linear stability problem is formulated as a non-Hermitian eigenvalue problem in a 2D domain and is solved numerically using the finite-element method. Specifically, we consider the case of a viscous sheet in an open flow, which falls in a bath of fluid; the sheet is mildly stretched by gravity and the flow can become unstable by ‘curtain’ modes. The growth rates of these modes are calculated as a function of the fluid parameters and of the geometry, and a phase diagram is obtained. A transition is reported between a buckling mode (static bifurcation) and an oscillatory mode (Hopf bifurcation). The effect of surface tension is discussed. [ABSTRACT FROM AUTHOR]

Graphene finds application in nanoscale electronic devices, nano sensors and nanocomposites in which graphene sheets usually rest on an elastic medium. Buckling of graphene sheets can occur in such structures, which significantly influences its performance. Therefore, it is essential to investigate the buckling behavior of graphene for better engineering design and manufacture. The present work provides an element-free kp-Ritz framework to analyze the buckling behavior of graphene. The element-free kp-Ritz method is first implemented to investigate the buckling behavior of single-layered graphene sheets (SLGSs) embedded in elastic foundations, based on the classical plate theory (CLP), incorporating the non-local elasticity theory, which takes small effects into account when dealing with nanostructures. A Winkler-type model is adopted to simulate the functioning of the elastic medium. Numerical solutions for critical buckling loads of SLGSs are obtained by solving the governing differential equations derived from the principle of minimum potential energy using the element-free kp-Ritz method. The concepts of non-local parameter effects and non-local effects are distinguished. The influence of non-local parameters for both side length and aspect ratio on critical buckling loads of SLGSs is analyzed. Additionally, the effects of non-local parameters and the Winkler modulus parameter on the buckling patterns are discussed. [ABSTRACT FROM AUTHOR]

We develop a reduced model for hard-magnetic, thin, linear-elastic shells that can be actuated through an external magnetic field, with geometrically exact strain measures. Assuming a reduced kinematics based on the Kirchhoff–Love assumption, we derive a reduced two-dimensional magneto-elastic energy that can be minimized through numerical analysis. In parallel, we simplify the reduced energy by expanding it up to the second order in the displacement field and provide a physical interpretation. Our theoretical analysis allows us to identify and interpret the two primary mechanisms dictating the magneto-elastic response: a combination of equivalent magnetic pressure and forces at the first order, and distributed magnetic torques at the second order. We contrast our reduced framework against a three-dimensional nonlinear model by investigating three test cases involving the indentation and the pressure buckling of shells under magnetic loading. We find excellent agreement between the two approaches, thereby verifying our reduced model for shells undergoing nonlinear and non-axisymmetric deformations. We believe that our model for magneto-elastic shells will serve as a valuable tool for the rational design of magnetic structures, enriching the set of reduced magnetic models. • We derive a geometrically exact model for magneto-active shells undergoing 3D deformation. • The reduced magnetic energy is interpreted as the energy potential of magnetic forces and torques. • Our analysis provides an understanding of the magneto-elastic coupling for curved geometries. • We contrast the results from our theory with 3D FEM to test the model. • The reduced 2D model is ten times faster than its 3D counterpart. [ABSTRACT FROM AUTHOR]

Buckling is a long studied mechanical process that has been tackled from a variety of theoretical and numerical methods over the past two and a half centuries. Despite this, predicting buckling behavior of materials with complex structure and components, such as notched beams of non-homogeneous architected composites, remains non-trivial. Inspired by recent advancements in applying artificial intelligence methods to model physical relationships directly from observational data, here we use a Variational Autoencoder in concert with a Long Short-Term Memory network to model the buckling behavior of notched beams. Our model, DeepBuckle, qualitatively and quantitatively learns buckling behavior of homogeneous polymer beams, and has the capacity to predict novel designs that yield certain buckling behaviors with creative, out-of-the-box implementation. Importantly, we subsequently demonstrate that our approach directly generalizes to beams comprised of a far more complex composite foam material, without the increased computational resources or ancillary knowledge of material characteristics required by a more traditional finite element or other continuum approaches. Notably, the method reported here uses a simple table top experimental setup and can easily be transferred to other applications, for use in fundamental studies of mechanical phenomena, or in educational settings. [ABSTRACT FROM AUTHOR]

We present a combined analytical approach and numerical study on the stability of a ring bound to an annular elastic substrate, which contains a circular cavity. The system is loaded by depressurizing the inner cavity. The ring is modeled as an Euler–Bernoulli beam and its equilibrium equations are derived from the mechanical energy which takes into account both stretching and bending contributions. The curvature of the substrate is considered explicitly to model the work done by its reaction force on the ring. We distinguish two different instabilities: periodic wrinkling of the ring or global buckling of the structure. Our model provides an expression for the critical pressure, as well as a phase diagram that rationalizes the transition between instability modes. Towards assessing the role of curvature, we compare our results for the critical stress and the wrinkling wavelength to their planar counterparts. We show that the critical stress is insensitive to the curvature of the substrate, while the wavelength is only affected due to the permissible discrete values of the azimuthal wavenumber imposed by the geometry of the problem. Throughout, we contrast our analytical predictions against finite element simulations. [ABSTRACT FROM AUTHOR]

A size-dependent nonlinear Euler–Bernoulli beam is considered in the framework of the nonlocal strain gradient theory. The geometric nonlinearity due to the stretching effect of the mid-plane of the size–dependent beam is considered here. The governing equations and boundary conditions are derived by employing the Hamilton principle. The post-buckling deflections and critical buckling forces of simply supported size-dependent beams are analytically derived. The derived results are compared with those of strain gradient theory, nonlocal elasticity theory and classical elasticity theory. It is found that the post-buckling deflections can be increased by increasing the nonlocal parameter or decreasing the material characteristic parameter. The high-order buckling deflections are more sensitive to size-dependent parameters than the low-order buckling deflections. Furthermore, the critical buckling force can be increased by decreasing the nonlocal parameter when the nonlocal parameter is larger than the material characteristic parameter, or increasing the nonlocal parameter when the nonlocal parameter is smaller than the material characteristic parameter. [ABSTRACT FROM AUTHOR]

An element-free based improved moving least squares-Ritz (IMLS-Ritz) method is proposed to study the buckling behavior of functionally graded nanocomposite plates reinforced by single-walled carbon nanotubes (SWCNTs) resting on Winkler foundations. The first-order shear deformation theory (FSDT) is employed to account for the effect of shear deformation of plates. The IMLS is used for construction of the two-dimensional displacement field. We derive the energy functional for moderately thick plates. By minimizing the energy functional via the Ritz method, solutions for the critical buckling load of the functionally graded carbon nanotube (FG–CNT) reinforced composite plates on elastic matrix are obtained. Numerical experiments are carried out to examine the effect of the Winkler modulus parameter on the critical buckling loads. The influences of boundary condition, plate thickness-to-width ratio, plate aspect ratio on the critical buckling loads are also investigated. It is found that FG–CNT reinforced composite plates with top and bottom surfaces of CNT-rich have the highest critical buckling loads. [ABSTRACT FROM AUTHOR]

The popularity of the studies is getting further on the flexomagnetic (FM) response of nano-electro-magneto machines. In spite of this, there are a few incompatibilities with the available FM model. This study indicates that the accessible FM model is inappropriate when considering the converse magnetization effect that demonstrates the necessity and importance of deriving a new FM relation. Additionally, the literature has neglected the converse FM coefficient in the Lifshitz invariant inside the free energy constitutive relation. This fact inspires us to endeavor and conduct a new characteristic formulation for static analysis of axially compressed piezomagnetic nanobeams comprising the FM effect. This novel FM model is competent and suitable for various boundary conditions, encompassing analytical, semi-analytical, and numerical solving strategies. However, based on the previous FM equation established with respect to Euler-Bernoulli and Timoshenko beams, the governing equations are ill-posed due to the corresponding energy density. Despite that, this error will not remain in the finalized equations in the present model by conjecturing a gradient of the magnetic field and a different formulation. Moreover, the inverse FM parameter will appear in the magnetic field relation. As the literature reported, non-uniform deformed piezomagnetic structures are capable of presenting more outstanding flexomagneticity. In actuality, a non-uniform elastic strain appears as a response to the magnetic field gradient (converse effect) that causes this study to deduce the nanobeam with higher-order shear deformations. Furthermore, the local governing equations will be transferred into the nonlocal phase according to the nonlocal differential, particularly nonlocal integral elasticity which itself is a strong nonlocality. Through this theory, and in regard to the converse FM impact, an analytical expression is applied for computing critical buckling loads within several ends conditions of the nanobeam. Our present results and achievements will hopefully be an effective contribution to theoretical studies on the mechanics of intelligent nanostructures. [ABSTRACT FROM AUTHOR]

The post-buckling behavior of a beam subjected to lateral constraints is of practical importance in a variety of applications, such as stent procedures, filopodia growth in living cells, endoscopic examination of internal organs, and deep drilling. Even though in reality the constraining surfaces are often deformable, the literature has focused mainly on rigid and fixed constraints. In this paper, we make a first step to bridge this gap through a theoretical and experimental examination of the post-buckling behavior of a beam constrained by a fixed wall and a springy wall, i.e. one that moves laterally against a spring. The response exhibited by the proposed system is much richer compared to that of the fixed-wall system, and can be tuned by choosing the spring stiffness. Based on small-deformation analysis, we obtained closed-form analytical solutions and quantitative insights. The accuracy of these results was examined by comparison to large-deformation analysis. We concluded that the closed-form solution of the small-deformation analysis provides an excellent approximation, except in the highest attainable mode. There, the system exhibits non-intuitive behavior and non-monotonous force–displacement relations that can only be captured by large-deformation theories. Although closed-form solutions cannot be derived for the large-deformation analysis, we were able to reveal general properties of the solution. In the last part of the paper, we present experimental results that demonstrate various features obtained from the theoretical analysis. [ABSTRACT FROM AUTHOR]

Thin films with low adhesion and large residual stresses may buckle. The resulting morphologies are varied, but one of the most commonly observed is an intriguing oscillating pattern – the so-called “telephone cord” – which has been extensively investigated in the recent years. We have studied the kinematics of formation of telephone cords using a geometrically non-linear plate model and mode dependent interfacial toughness, captured via a cohesive zone. Through extensive Finite Element Simulations, we have demonstrated a simple, non-trivial relation between telephone cord wavelength and interfacial toughness. To validate this prediction, highly stressed Mo thin films were deposited on Si wafers, with a well defined interface and very reproducible adhesion. Studying the morphology of the resulting buckles for different film thicknesses and stresses, we observed a trend which was fully consistent with our simulation results. From the data fit, an adhesion energy of 0.58±0.04 J m −2 for the SiO 2 /Ag interface was inferred, which compares well with literature estimates. [ABSTRACT FROM AUTHOR]

A compression-induced buckling delamination test is employed to quantitatively characterize the interfacial adhesion of Ni thin film on steel substrate. It is shown that buckles initiate from edge flaws and surface morphologies exhibit symmetric, half-penny shapes. Taking the elastoplasticity of film and substrate into account, a three-dimensional finite element model for an edge flaw with the finite size is established to simulate the evolution of energy release rates and phase angles in the process of interfacial buckling-driven delamination. The results show that delamination propagates along both the straight side and curved front. The mode II delamination plays a dominant role in the process with a straight side whilst the curved front experiences almost the pure mode I. Based on the results of finite element analysis, a numerical model is developed to evaluate the interfacial energy release rate, which is in the range of 250–315 J/m 2 with the corresponding phase angle from −41° to −66°. These results are in agreement with the available values determined by other testing methods, which confirms the effectiveness of the numerical model. [ABSTRACT FROM AUTHOR]

Wrinkle networks are ubiquitous buckle-induced delaminations in supported graphene, which locally modify the electronic structure and degrade device performance. Although the strong property–deformation coupling of graphene can be potentially harnessed by strain engineering, it has not been possible to precisely control the geometry of wrinkle networks. Through numerical simulations based on an atomistically informed continuum theory, we understand how strain anisotropy, adhesion and friction govern spontaneous wrinkling. We then propose a strategy to control the location of wrinkles through patterns of weaker adhesion. This strategy is deceptively simple, and can in fact fail in several ways, particularly under biaxial compression. However, within bounds set by the physics of wrinkling, it is possible to robustly create by strain a variety of wrinkle network geometries and junction configurations. Graphene is nearly unstrained in the planar regions bounded by wrinkles, highly curved along wrinkles, and highly stretched and curved at junctions, which can either locally attenuate or amplify the applied strain depending on their configuration. These mechanically self-assembled networks are stable under the pressure produced by an enclosed fluid and form continuous channels, opening the door to nano-fluidic applications. [ABSTRACT FROM AUTHOR]

Thermo-mechanical size-dependent buckling analysis of embedded functionally graded (FG) microbeams is performed based on sinusoidal shear deformation beam and modified couple stress theories. It is assumed that material properties vary smoothly and continuously throughout the thickness. Winkler elastic foundation model is used to simulate the interaction between FG microbeam and elastic medium. The governing equations and corresponding boundary conditions are obtained with the aid of minimum total potential energy principle. The buckling characteristics of simply supported embedded FG microbeams in thermal environment are investigated. The obtained results are compared with the results of simple beam theory with no shear deformation effects and classical theory. Influences of thickness-to-material length scale parameter ratio, material property gradient index, slenderness ratio, temperature change and Winkler parameter on critical buckling loads of embedded FG microbeams are discussed in detail. [ABSTRACT FROM AUTHOR]

Magneto-rheological elastomers (MREs) are functional materials that can be actuated by applying an external magnetic field. MREs comprise a composite of hard magnetic particles dispersed into a nonmagnetic elastomeric (soft) matrix. Hard MREs have been receiving particular attention because the programmed magnetization remains unchanged upon actuation. Motivated by a new realm of applications, there have been significant theoretical developments in the continuum (three-dimensional) description of hard MREs. In this paper, we derive an effective theory for MRE rods (slender, mono-dimensional structures) under geometrically nonlinear deformation in three dimensions. Our theory is based on reducing the three-dimensional magneto-elastic energy functional for the hard MREs into an one-dimensional Kirchhoff-like description (centerline-based). Restricting the theory to two dimensions, we reproduce previous works on planar deformations. For further validation in the general case of three-dimensional deformation, we perform precision experiments with both naturally straight and curved rods under either constant or constant-gradient magnetic fields. Our theoretical predictions are in excellent agreement with both discrete simulations and precision-model experiments. Finally, we discuss some limitations of our framework, as highlighted by the experiments, where long-range dipole–dipole interactions, which are neglected in the theory, can play a role. [Display omitted] • We propose a reduced-order theory for the 3D deformation of a magneto-elastic rod. • The theory is validated experimentally in different magnetic-field conditions. • The theory is in excellent agreement with discrete simulations and experiments. • Dipole–dipole interactions are not included in the theory but studied experimentally. [ABSTRACT FROM AUTHOR]

• 2nd buckling mode diagram depends on Young's modulus ratio & imperfection magnitude. • Rectangular checkerboard & stripe modes give a family of herringbone patterns. • Superposition of identical modes in symmetric directions generates cognate modes. • Cognate modes reveal existence of diverse patterns cognate with herringbone pattern. • Robustness of herringbone patterns & diversity of cognate patterns are elucidated. In this study, we investigate the diversity of the bifurcations and deformations during evolution of periodic patterns on compressed films bonded to compliant substrates. Three-dimensional finite element analysis is performed assuming that the first bifurcation has either the hexagonal or square (i.e., checkerboard) dimple mode. Step-by-step eigenvalue buckling analysis is performed to explore sequential bifurcations on the bifurcated paths. It is found that at the second bifurcation, a rectangular checkerboard or stripe mode occurs depending not only on the Young's modulus ratio of the film and substrate, but also on the magnitude of the imperfection prescribed by the first bifurcation mode. Different bifurcation modes give a family of herringbone deformation patterns with different dimensions, i.e., the evolutional process is multiple and robust. Further, superposition of the identical modes in symmetric directions elucidates the existence of distinctive patterns cognate with the herringbone pattern, including a variety of experimentally observed patterns. [ABSTRACT FROM AUTHOR]

Highlights: [•] Nonlinear dynamics of a nanobeam with piezoelectricity viscoelasticity coupling. [•] Model establishment for a nanobeam subjected to van der Waals forces. [•] Numerical simulations of the nanobeam with incremental harmonic balance method. [•] Effects of system parameters on nanobeam buckling, post-buckling and instability. [Copyright &y& Elsevier]

Abstract: Application of a dead compressive load at the free end of an elastic rod (the ‘blade’) induces its penetration into a sliding sleeve ending with a linear elastic spring. Bifurcation and stability analysis of this simple elastic system shows a variety of unexpected behaviors: (i) an increase of buckling load at decreasing of elastic stiffness; (ii) a finite number of buckling loads for a system with infinite degrees of freedom (leading to a non-standard Sturm–Liouville problem); (iii) more than one bifurcation load associated to each bifurcation mode; (iv) a restabilization of the straight configuration after the second bifurcation load associated to the first instability mode; (v) the presence of an Eshelby-like (or configurational) force, deeply influencing stability. Only the first of these behaviors was previously known, the second and third ones disprove common beliefs, the fourth highlights a sort of ‘island of instability’, and the last one shows surprising phenomena and effects on stability. [Copyright &y& Elsevier]

Abstract: This paper, the second of two parts, presents three novel finite element case studies to demonstrate the importance of normal-tangential coupling in cohesive zone models (CZMs) for the prediction of mixed-mode interface debonding. Specifically, four new CZMs proposed in Part I of this study are implemented, namely the potential-based MP model and the non-potential-based NP1, NP2 and SMC models. For comparison, simulations are also performed for the well established potential-based Xu–Needleman (XN) model and the non-potential-based model of van den Bosch, Schreurs and Geers (BSG model). Case study 1: Debonding and rebonding of a biological cell from a cyclically deforming silicone substrate is simulated when the mode II work of separation is higher than the mode I work of separation at the cell-substrate interface. An active formulation for the contractility and remodelling of the cell cytoskeleton is implemented. It is demonstrated that when the XN potential function is used at the cell-substrate interface repulsive normal tractions are computed, preventing rebonding of significant regions of the cell to the substrate. In contrast, the proposed MP potential function at the cell-substrate interface results in negligible repulsive normal tractions, allowing for the prediction of experimentally observed patterns of cell cytoskeletal remodelling. Case study 2: Buckling of a coating from the compressive surface of a stent is simulated. It is demonstrated that during expansion of the stent the coating is initially compressed into the stent surface, while simultaneously undergoing tangential (shear) tractions at the coating-stent interface. It is demonstrated that when either the proposed NP1 or NP2 model is implemented at the stent-coating interface mixed-mode over-closure is correctly penalised. Further expansion of the stent results in the prediction of significant buckling of the coating from the stent surface, as observed experimentally. In contrast, the BSG model does not correctly penalise mixed-mode over-closure at the stent-coating interface, significantly altering the stress state in the coating and preventing the prediction of buckling. Case study 3: Application of a displacement to the base of a bi-layered composite arch results in a symmetric sinusoidal distribution of normal and tangential traction at the arch interface. The traction defined mode mixity at the interface ranges from pure mode II at the base of the arch to pure mode I at the top of the arch. It is demonstrated that predicted debonding patterns are highly sensitive to normal-tangential coupling terms in a CZM. The NP2, XN, and BSG models exhibit a strong bias towards mode I separation at the top of the arch, while the NP1 model exhibits a bias towards mode II debonding at the base of the arch. Only the SMC model provides mode-independent behaviour in the early stages of debonding. This case study provides a practical example of the importance of the behaviour of CZMs under conditions of traction controlled mode mixity, following from the theoretical analysis presented in Part I of this study. [Copyright &y& Elsevier]

Abstract: We consider the equilibrium shapes of a thin, annular strip cut out in an elastic sheet. When a central fold is formed by creasing beyond the elastic limit, the strip has been observed to buckle out-of-plane. Starting from the theory of elastic plates, we derive a Kirchhoff rod model for the folded strip. A non-linear effective constitutive law incorporating the underlying geometrical constraints is derived, in which the angle the ridge appears as an internal degree of freedom. By contrast with traditional thin-walled beam models, this constitutive law captures large, non-rigid deformations of the cross-sections, including finite variations of the dihedral angle at the ridge. Using this effective rod theory, we identify a buckling instability that produces the out-of-plane configurations of the folded strip, and show that the strip behaves as an elastic ring having one frozen mode of curvature. In addition, we point out two novel buckling patterns: one where the centerline remains planar and the ridge angle is modulated; another one where the bending deformation is localized. These patterns are observed experimentally, explained based on stability analyses, and reproduced in simulations of the post-buckled configurations. [Copyright &y& Elsevier]

Abstract: In this research, buckling analysis of three microbeam models are investigated based on modified couple stress theory. Using Euler–Bernoulli beam theory (EBT), Timoshenko beam theory (TBT) and Reddy beam theory (RBT), the effect of shear deformation is presented. To examine the effect of boundary condition, three kinds of boundary conditions i.e. hinged–hinged, clamped–hinged and clamped–clamped boundary conditions, are considered. These nonclassical microbeam models incorporated with Poisson effect, contain a material length scale parameter and can capture the size effect. These models can degenerate into the Classical models if the material length scale parameter and Poisson’s ratio are both taken to be zero. Governing equations and boundary conditions are derived by using principle of minimum potential energy. Generalized differential quadrature (GDQ) method is employed to solve the governing differential equations. Also an analytical solution is applied to determine the critical buckling load of microbeams with hinged–hinged boundary condition. Comparison between the results of GDQ and analytical methods reveals the accuracy of GDQ method. Some numerical results are exhibited to indicate the influences of beam thickness, material length scale parameter and Poisson’s ratio on the critical buckling load of these microbeams. [Copyright &y& Elsevier]

Highlights: [•] We model the growth of gut epithelial cells cultured upon a deformable substrate. [•] Growth generates buckling instabilities, contributing to crypt formation in vivo. [•] Variations in mechanical properties have little effect on resulting configurations. [•] Configurations are controlled by growth patterns & interactions with strata below. [Copyright &y& Elsevier]