Post-buckling Deformation Research Articles

特発性側弯症発症メカニズムの非線形座屈解析による検討

The present paper describes the numerical results by the finite-element method of unstable phenomena of the spine induced by the growth of vertebral bodies in order to investigate the cause of the idiopathic scoliosis. In the previous study, through the linear buckling analysis using simple plate models, we confirmed the existence of the buckling phenomena, and made clear the region of geometrical parameters in which the buckling phenomena occur. These results support the buckling hypothesis as a cause of initiation of the idiopathic scoliosis. However, this theory is valid for infinitesimal deformation and is not applicable to estimate the stability for post-buckling behavior. In this study, we conduct post-buckling simulations caused by the growth of vertebral bodies considering the geometrical nonlinearity using the simple plate model and spine models with full width and half width. From the results, the existence of the non-linear buckling phenomena are confirmed in the simple plate model and the spine model with half width. Moreover, after the non-linear buckling phenomena, stable deformations are obtained excepting each one case in the simple plate model and the spine model with half width. However, the magnitudes of the post-buckling deformations are too small to explain the pathogenesis of the severe deformity observed in patients of idiopathic scoliosis.

Mode interaction in thin-walled I-section struts with semi-rigid flange–web joints

A recently developed variational model that describes the non-linear interaction between the global and local buckling of a thin-walled I-section strut under pure compression is extended to include semi-rigid flange–web connections. A formulation combining the Rayleigh–Ritz method and continuous displacement functions is used to derive a system of differential and integral equilibrium equations for the structural component. Cellular buckling, observed in the case where the web is assumed to provide no more than a simple support to the flanges, is found to be rapidly eroded by increasing the connection rigidity, although the local buckling wavelength still reduces as the post-buckling deformation is increased. The model is validated using finite element analysis; the results compare very well particularly when a high rigidity between the section web and flanges is simulated.

Quasi-static buckling simulation of single-layer graphene sheets by the molecular mechanics method

This paper presents a quasi-static nonlinear buckling analysis of compressed single-layer graphene sheets (SLGSs) using the molecular mechanics method. Bonded interactions between carbon atoms are simulated using a modified parameter set of the DREIDING force field that leads to better agreement between simulated mechanical properties of graphene and reference literature data than the standard parameter set of this force field (see Mayo et al., J Phys Chem 1990; 94: 8897–8909). Identification of constraints of atoms of the SLGS edges with the boundary conditions of clamped and simply supported thin plates is made. The buckling loads and modes obtained by linear and nonlinear buckling analysis of a compressed quadratic SLGS with a side length of 6 nm are shown to be close to each other. In addition, it has been found by nonlinear buckling analysis that only equilibrium configurations with modes of initial post-buckling deformed configurations correlated with the one-half-wave column-like buckling mode have stable equilibrium configurations for clamped and simply supported SLGSs. As the edges of a simply supported SLGS approach each other, the geometry of this mode of post-buckling deformation with inclusion of the non-bonded van der Waals (vdW) interactions between carbon atoms becomes closer to the geometry of a single-walled carbon nanotube, and without inclusion of the vdW interactions, this mode has the geometry of a cylinder with a drop-shaped cross-section.

Experimental and numerical analysis of post-buckling deformation states of integrally stiffened thin-walled components of load-bearing aircraft structures

The study presents results of experiments and numerical analyses concerning thin-walled shells used as components of aircraft structures. The solution, which is proposed here, consists in stiffening such elements by means of integral ribs. A comparative analysis has been carried out between the suggested design solution and the reference structure. In the experimental part of the study, an optical scanner with digital image correlation has been used. Nonlinear numerical analyses have been carried out with the use of software based on the finite element method.

Closed form solutions for predicting static and dynamic buckling behaviors of a drillstring in a horizontal well

The nonlinear static and dynamic buckling (snaking motion) analysis of a rotating drilling string laterally constrained in a horizontal well is presented, through a group of fourth-order nonlinear partial differential equations . The analytical approximate solutions to the static and dynamic buckling are obtained via combining Newton linearization with Harmonic Balance Method, and Galerkin's method, respectively. On the basis of the analytical approximate solutions, static post-buckling deformation, critical dynamic buckling load, and two different kinds of snaking motions (i.e. the pipe moves up and down around its static buckling configuration; the pipe moves from one side of the wellbore to the other side) are investigated. Accuracy of the approximate solutions is verified by comparing with numerical solutions. Theoretical results are useful for practical design applications related to calculation of buckling loads and selection of bottom hole-assembly (BHA) elements and pipe rotational speeds . What's more, the solving procedures of accurate analytical approximate solutions to the snaking motions yield rapid convergence with respect to exact numeric solutions. The present results are valid for small as well as large oscillation amplitudes. By combining Newton linearization with Harmonic Balance Method and Galerkin's method, the analytical approximate solutions are explicitly established for static and dynamic buckling of a horizontal drilling string, respectively. • Nonlinear static and dynamic buckling of drillpipe in a horizontal well is studied. • Analytical approximate solutions are obtained by Improved HB and Galerkin method. • The present solution procedures are brief. • Accuracy of approximate solution is verified by comparing with numerical solution.

Thermal Post-Buckling of Functionally Graded Material Circular Plates Subjected to Transverse Point-Space Constraints

Axisymmetrical thermal post-buckling of functionally graded material (FGM) circular plates with immovably clamped boundary and a transversely central point-space constraint was studied. The material properties of the plate were assumed to vary as power law functions in the thickness direction and the temperature rise field to change only in the thickness direction. Based on von Karman's non-linear plate theory, governing equations in terms of the displacements of the middle plane were established. Temperature rise field was obtained by solving the one-dimensional heat conduction equation associated with specified boundary conditions at the top and bottom surface of the plate. By using the shooting method, thermal post-buckling deformation of the FGM circular plate was obtained before and after the plate contacting the point-space constraint. The changes in the characteristics of the deformation and the internal forces of FGM plates were discussed. The effects of gradients of material properties and non-uniform temperature rise parameters on the thermal post-buckling behaviors of FGM circular plates were also examined.

MACRO-MICRO ANALYSIS FOR ANTI-PENETRATING POSTBUCKLING OF FIBER REINFORCED, DEBONED LAMINATES WITH CONTACT EFFECTS

This paper presents a close-form solution for the anti-penetrating postbuckling problem of fiber reinforced, rectangular laminates. The laminate has a through-width debonded region buried at the center position and is simply supported at its four edges subjected to the in-plane compressive load. Based on the previous analysis of the contact kinematics from micro mechanics of composite material, the nonlinear contact effects at the interfaces of the debonded region are considered and characterized by a contact factor. Governing equations are established and are solved using the perturbation approach. Asymptotic solutions from perturbation analysis are presented. It is found that for fiber reinforced, debonded laminates the postbuckling deformation involves global deflection and local buckling with amplitude of the global one 104 times larger than the local one. The global deflection is induced by the postbuckling load and the local buckling is caused by the anti-penetrating contact effects. Due to the contact effects, the load carry capacity of the delaminated laminate is tremendously enhanced. Further examination indicates that both the buckling load and the contact performances are intrinsic properties of the laminates. Parametric analyses show that (i) the buckling load decreases with the increasing of the delamination length and the delamination depth and is very sensitive to the changing of the aspect ratio, (ii) the contact performances, including the location and size of the contact areas, rely only on the geometries of the debonded region and the material properties but are hardly affected by the aspect ratio. The analysis is in effect by comparing with the well established results.

Study on the Influence of Thickness and Width on the Post-Buckling Deformation of Cold-Rolled Strip

As a consequence of post-buckling due to high residual stresses caused by the cold-rolling process, long free thin strips frequently show excessive wavy surface. In this paper, a new analytical approach by extending a classic post-buckling analysis method based on the minimum potential energy principle is used to study on the influence of strip thickness and width on the post-buckling deformation for the center and edge wave. It is concluded that the thickness of the strip has significant effect on the flatness defects, while the width does not.

Loss of Pressure Boundary through Buckling-Induced Fracture in the Ciudad Nezahualcóyotl Pipeline

AbstractA study is presented to investigate the fatigue and local buckling-induced fracture incident of a buried continuous pipeline during the 1985 Mexico City earthquake. Assuming pipeline wrinkling is caused by longitudinal wave propagation, peak ground strains are estimated from recorded velocity time histories near the site of the fractured pipeline and critical buckling strains are calculated from experimentally based empirical relationships. A physics-based model of the postbuckling deformation of the pipeline provides estimated deformation demands on the pipe, whereas a micromechanical fatigue-fracture model is used to determine the potential for fracture considering the estimated axial deformations from the recorded ground motion, plastic strain demands, and stress state at the critical fracture location. The ground motion at the Ciudad Nezahualcoyotl site in Mexico is assumed to have approximately seven cycles of ground deformation large enough to exceed the critical buckling strain of the membe...

Cyclic axial response and energy dissipation of cold-formed steel framing members

This paper summarizes results from an experimental program that investigated the cyclic axial behavior and energy dissipation of cold-formed steel C—sections structural framing members. Fully characterized cyclic axial load–deformation response of individual members is necessary to facilitate performance-based design of cold-formed steel building systems. Specimen cross-section dimensions and lengths were selected to isolate specific buckling modes (i.e., local, distortional or global buckling). The cyclic loading protocol was adapted from FEMA 461 with target displacements based on elastic buckling properties. Cyclic response showed large post-buckling deformations, pinching, strength and stiffness degradation. Damage accumulated within one half-wave after buckling. The total hysteretic energy dissipated within the damaged half-wave decreased with increasing cross-section slenderness. More energy dissipation comes at the cost of less cumulative axial deformation before tensile rupture.

Asymptotic solutions for buckling delamination induced crack propagation in the thin film-compliant substrate system

In a thin film-substrate system in-plane compressive stress is commonly generated in the film due to thermal mismatch in operation or fabrication process. If the stress exceeds a critical value, part of the film may buckle out of plane along the defective interface. After buckling delamination, the interface crack at the ends may propagate. In the whole process, the compliance of the substrate compared with the film plays an important role. In this work, we study a circular film subject to compressive stress on an infinitely thick substrate. We study the effects of compliance of the substrate by modeling the system as a plate on an elastic foundation. The critical buckling condition is formulated. The asymptotic solutions of post-buckling deformation and the corresponding energy release rate of the interface crack are obtained with perturbation methods. The results show that the more compliant the substrate is, the easier for the film to buckle and easier for the interface crack to propagate after buckling.

Vibration Analysis of a Postbuckled Microscale FG Beam Based on Modified Couple Stress Theory

On the basis of modified couple stress theory, the postbuckling behavior of the Euler-Bernoulli microscale FG beams is investigated by means of an exact solution method. The modified couple stress theory as a nonclassical continuum theory is capable of interpreting the size dependencies which become more significant at micro/nanoscales. The Von-Karman type nonlinear strain-displacement relationships are employed. The thermal effects are also incorporated into formulation. The governing equation of motion and the corresponding boundary conditions are derived using Hamilton’s principle. The material properties are assumed to be graded in the thickness direction according to the power-law distribution. A closed-form solution is obtained for the postbuckling deformation which is beyond the critical buckling load. To study the vibrations taking place in the vicinity of a buckled equilibrium position, the linear vibration problem is exactly solved around the first three buckled configurations. The natural frequencies of the lowest vibration modes around each of the first three buckled configurations are obtained. The influences of power-law exponent, boundary condition, length scale parameter, and thermal environment changes on the static deflection and free vibration frequencies are studied. A comparison is also made between the present results and those obtained via the classical beam theories.

Investigation into Dynamic Postbuckling of Thin Rectangular Plate under Elastic Compression Wave

The initial buckling displacements are taken as initial conditions to solve the dynamic postbuckling problem. Non-dimensional dynamic postbuckling equilibrium equations denoted by the displacements are derived and solved by use of the finite difference method. The results reveal the regulation that the dynamic postbuckling deformation develops from the initial low step mode to the subsequent higher step modes under elastic compression wave.

On the free vibration of thermally pre/post-buckled shear deformable SMA hybrid composite beams

The objective of this article is to analytically investigate the free vibration of shape memory alloy hybrid composite (SMAHC) beams in thermally pre/post-buckled domains. Non-linear equations of motion of SMAHC beams are derived based on the first-order shear deformation theory and von Karman geometrical non-linearity via the extended Hamilton principle. The recovery stress generated by temperature-induced martensitic phase transformation of the pre-strained SMA fibers is computed by means of the Brinson model. Exact closed-form solutions are presented for the buckling temperature, post-buckling deformation and temperature-deformation equilibrium path of symmetric and asymmetric simply supported SMAHC beams under uniform temperature rise. The vibration of the symmetric SMAHC beam around the first buckled configuration is also investigated and an analytical solution for the fundamental frequency and its associated mode shape is obtained. Based on the developed closed-form solutions, extensive numerical results are presented to provide an insight into the influence of SMA fibers volume fraction, pre-strain in the SMA fibers, location of SMA fibers, temperature rise and geometrical parameters on the static and dynamic responses of the SMAHC beams in the thermally pre/post-buckling regimes. Due to the absence of similar results in the specialized literature, this paper is likely to fill a gap in the state of the art of this problem.

Large Deformation and Failure of Thin-Walled Film at the Post-Buckling Delamination

An accurate 1D model of post-buckling deformation of a thin sub-laminate of layered composite is develloped using nonlinead theory of slender plate. The strain energy realize rate at delamination propagation is obtained via the elliptical integrals. A model of fracture buckled sub-laminate is implemented and used for general analysis of progressive damage of composite

Buckling Behavior of Elliptical Shells of Changing Thickness under Uniform Axial Compression

Many elliptical shells are used in structural applications in which the dominant loading condition is axial compression. Due to the fact that the radius varies along the cross-section midline, the buckling behavior is more difficult to identify than those of cylindrical shells. The general concerned aspects in cylindrical shell buckling analyses such as the buckling mode, the pre-buckling deformation and post-buckling deformation are all quite different related to specific elliptical shell geometry. The buckling behavior of elliptical cylindrical shells with uniform thickness has been widely studied by many researchers. However, the thickness around the circumference may change for some specific structural forms, the femoral neck for example, which makes the buckling behavior more complex. It is known that the buckling strength of thin cylindrical shells is quite sensitive to imperfections, so it is natural to explore the imperfection sensitivity of elliptical shells. This paper explores the buckling behavior of imperfect elliptical shells under axial compression. It is hoped that the results will make a useful contribution in this field.

Analytical Solution for Initial Postbuckling Deformation of the Sandwich Beams Including Transverse Shear

This paper presents analytical approximate solutions for the initial postbuckling deformation of sandwich beams including transverse shear. The approximate procedure is based on the nonlinear beam equation (with transverse shear included) by coupling the well-known Maclaurin series expansion and orthogonal Chebyshev polynomials; the governing differential equation with sinusoidal nonlinearity can be reduced to form a cubic-nonlinear equation. Analytical approximations to the resulting boundary condition problem are established by combining the methods of Newton linearization and harmonic balance. Unlike the classical method of harmonic balance, the linearization is performed prior to proceeding with harmonic balancing thus resulting in a set of linear algebraic equations instead of one of nonlinear algebraic equations, thereby establishing analytical approximate solutions. Illustrative examples are presented for a few typical sandwich construction configurations, and it is shown that the proposed approximate solutions are in excellent agreement with the numerical solution obtained via the shooting method for both small and large angles of rotation.

On the postbuckling and free vibrations of FG Timoshenko beams

The postbuckling behavior of functionally graded beams is investigated by means of an exact solution method. The Von-Karman type nonlinear strain–displacement relationships are employed. The effects of the transverse shear deformation and rotary inertia are also included based upon the Timoshenko beam theory. After writing the kinetic and potential energy functionals, the governing equations of motion including the axial, transverse deflections and also the cross sectional rotation are derived using the Hamilton’s principle. The material properties are assumed to be graded in the thickness direction according to the power-law distribution. Neglecting the inplane inertia, the three equations of motion are reduced to two nonlinear partial–integral–differential equations in terms of the transverse mid-plane deflection and the cross sectional rotation. FG beams are considered to have fixed–fixed, fixed–hinged, and hinged–hinged end conditions. A closed-form solution is achieved for the postbuckling deformation as a function of the exerted axial load which is beyond the critical buckling load. In order to study the vibrations taking place in the vicinity of a buckled equilibrium position, the linear vibration problem is exactly solved around the first buckled configuration of a hinged–hinged FG beam. This leads to a characteristic equation whose eigenvalues are the natural frequencies and the corresponding eigenvectors also determine the mode shapes. The influences of power-law exponent, some commonly used boundary conditions and beam geometrical parameters on the static deflection and free vibration frequencies are studied. A comparison of the present results with those obtained via Euler–Bernoulli beam theory clarifies the overestimation of the frequencies by the later one.

Three-dimensional non-linear buckling of thick-walled elastic tubes under pressure

This paper is concerned with numerical simulations of three-dimensional finite deformation of a thick-walled circular elastic tube subject to internal or external pressure and zero displacement on its ends. We formulate the system of equations that can accommodate large strain and displacement for the incompressible isotropic neo-Hookean material. The fully non-linear governing equations are solved using the C++ based object-oriented finite element library libMesh. A Lagrangian mesh is used to discretize the governing equations, and a weighted residual Galerkin method and Newton iteration solver are used in the numerical scheme. To overcome the sensitivity of the fully non-linear system to small changes in the iterations, the analytical form of the Jacobian matrix is derived, which ensures a fast and better numerical convergence than using a numerically approximated Jacobian matrix.Results are presented for different parameters in terms of wall thickness/radius ratio, and length/radius ratio, as well as internal/external pressure. Validation of the model is achieved by the excellent agreement with the results obtained using the commercial package Abaqus. Comparison is also made with the previous work on axisymmetric version of the same system (Zhu et al., 2008 [34]; Zhu et al. 2010 [43]), and interesting fully three-dimensional post-buckling deformations are highlighted. The success of the current approach paves the way for fluid–structure interaction studies with potential application to collapsible tube flows and modeling of complex physiological systems.

An atomistic-based foliation model for multilayer graphene materials and nanotubes

We present a three-dimensional continuum model for layered crystalline materials made out of weakly interacting two-dimensional crystalline sheets. We specialize the model to multilayer graphene materials, including multi-walled carbon nanotubes (MWCNTs). We view the material as a foliation, partitioning of space into a continuous stack of leaves, thus loosing track of the location of the individual graphene layers. The constitutive model for the bulk is derived from the atomistic interactions by appropriate kinematic assumptions, adapted to the foliation structure and mechanics. In particular, the elastic energy along the leaves of the foliation results from the bonded interactions, while the interaction energy between the walls, resulting from van der Waals forces, is parametrized with a stretch transversal to the foliation. The resulting theory is distinct from conventional anisotropic models, and can be readily discretized with finite elements. The discretization is not tied to the individual walls and allows us to coarse-grain the system in all directions. Furthermore, the evaluation of the non-bonded interactions becomes local. We test the accuracy of the foliation model against a previously proposed atomistic-based continuum model that explicitly describes each and every wall. We find that the new model is very efficient and accurate. Furthermore, it allows us to rationalize the rippling deformation modes characteristic of thick MWCNTs, highlighting the role of the van der Waals forces and the sliding between the walls. By exercising the model with very large systems of hollow MWCNTs and suspended multilayer graphene, containing up to 109 atoms, we find new complex post-buckling deformation patterns.