Post-critical Behavior Research Articles

Study on the buckling behavior of aluminum alloy sheets - before and after repaired with composite patches

In this paper, the compressive buckling behavior of aluminum alloy plates with an elliptical hole of various sizes is investigated. In order to improve the stability of these thin plates, T700/QY8911 composite laminate is used as a patch to repair the hole. The study included an analysis of the critical and post-critical behaviour using experimental and numerical methods. Experiments focus on buckling loads, post-buckling behavior and the relationship between sizes of holes and buckling load. Meanwhile, the buckling load and buckling mode are determined by finite element analysis, using linear analysis of eigenvalue problems modes, and then, the nonlinear analysis of structures with initiated geometrically imperfection is carried out, studying its post-buckling behavior, damage behavior and transfer of load. The results show that the buckling load of the open-hole specimen is related to size of opening. The existence of patch has a significant influence on stress distribution, and the buckling capability of repaired specimens is noticeably improved to the plate without a hole. And the compression experimental results are consistent with the numerical results, revealing that the developed finite element model of the structure is correct.

Stability of tensegrity-inspired structures fabricated through additive manufacturing

This study encompasses experimental and theoretical research on the stability of tensegrity-inspired structures in view of additive manufacturing technology. The aim of the research is to produce a qualitative and quantitative estimation of the post-critical behavior of struts in 3D-printed tensegrity-inspired modules. Presented results of compression tests performed on modules and single struts show that the bars work in the post-critical state before they reach their load bearing capacity. Comparison of theoretical solutions with experimental tests allowed us to conclude that the struts can carry axial loads significantly higher than the limits resulting from the classical Euler theory.

Experimental investigations on mechanical propertiesof 3D-printed tensegrity-inspired metamaterialsbased on 4-strut simplex module

The present study is focused on experimental investigations of mechanical properties of 3D-printed tensegrity-inspired metamaterials. Tensegrity systems have many advantageous features, such as: light weight, high stiffness-to-mass ratio, controllability, inherent attributes of smart structures, and unique mechanical behaviour. They may be applied not only in macro-scale, but they can also be used to create cellular mechanical metamaterials and lattices in various scales. Metamaterials are understood here as human-designed artificial materials, which do not exist in nature, and whose mechanical properties result from the morphology of the inner structure rather than from chemical or phase composition. Experimental studies on tensegrity metamaterials manufactured using 3D printing techniques are hardly present in the literature. This paper presents results of uniaxial compression tests carried out on a number of 3D-printed tensegrity-based modules corresponding to the metamaterial cells, differing in the manufacturing technology, parent material, and size. The following observations were made during the tests: one of the most important parameters that has a direct impact on the results is the elongation at break of the parent material; any inaccuracies at the production stage greatly affect the mechanical behaviour of the structure; it is crucial to ensure a free deformation consistent with the infinitesimal mechanism mode of tensegrity; a post-critical behaviour of the struts was clearly visible in the performed tests.

Investigation of FGM membrane wrinkling using reduced models based on a multi-scale method

In this article, we show that the reduced model proposed in our recent article to study the wrinkling of homogeneous elastic membranes does not reproduce the wrinkling observed in the case of an FGM-type composite membrane. New reduced models are then proposed to investigate the wrinkling of membranes made from Functionally Graded Materials (FGM). We examine, by this way, the influence of material functional parameters on determining the critical load and post-critical behavior under different loads. These reduced models, created using a multi-scale method based on Fourier series with slowly varying coefficients, enrich the reference model. In addition, these reduced models also allow us to significantly decrease the computation time and the number of finite elements compared to the complete model. The robustness of these reduced models is demonstrated in the case of wrinkling of membranes modeled by very thin shells, which are discretized using DKT18-type finite elements as described in literature. The resulting nonlinear problems are solved using a High-Order Continuation Method (HOCM).

Computational study of a homogenized nonlinear generalization of Timoshenko beam proposed by Turco et al.

<p>Mechanical metamaterials are most often assemblies of stocky beam elements connected through rigid connections, hinges, or flexural joints. The description of these materials through classical beam theories is challenging because of the wide variety of complex phenomena observed in the severe deformation regime mechanical metamaterials must undergo and because most classical beam theories can only be applied to elements with sufficiently high slenderness. In the spirit of Hencky, Turco et al. (2020) has recently formulated an intrinsically discrete nonlinear elastic model suitable for the design of mechanical metamaterials. The objective of this contribution was to present a numerical study of the nonlinear generalization of the Timoshenko beam that results from the asymptotic homogenization of the discrete model introduced by Turco et al. The present numerical study took into account several loading cases and elucidated the sensitivity of the homogenized continuum with respect to axial, bending, and shear stiffness parameters, as well as to load imperfections, in terms of mechanical behavior, including buckling onset and post-critical behavior. It was found that the predictions obtained with the homogenized model in the large deformation regime matched excellently with those of the discrete model proposed by Turco et al.</p>

Gravity-coupled flutter and contact of a flag near a wall

The stability and postcritical behaviour of a horizontal flag undergoing gravity-induced deformation and periodic contact with a nearby horizontal rigid wall are experimentally investigated. The results elucidate the combined effects of gravity and contact on flutter, and reveal design principles for application to triboelectric energy harvesting. By varying the free-stream velocity, flag thickness and distance between the flagpole and the wall, the dynamics of the flag are classified into quasistatic equilibrium, flutter, partial contact and saturated contact modes. Considering the significance of gravitational effects, a new dimensionless flow velocity is proposed to identify the distribution of the dynamic modes, and its definition varies according to whether the wall is placed above or below the flag. The critical conditions for transitions between the dynamic modes are determined from the balance of fluid dynamic and gravitational effects. The distance from the flagpole to the wall is found to be more critical for transitions in the lower-wall configuration than in the upper-wall configuration. The peak contact force as well as the oscillation amplitude and frequency at postequilibrium exhibits remarkably different trends depending on the location of the wall. The peak contact force imposed on the wall by the fluttering flag weakens as the distance to the wall increases in the case of an upper wall, whereas it becomes stronger in the case of a lower wall.

Postcritical Imperfection Sensitivity of Functionally Graded Piezoelectric Cylindrical Nanoshells Using Boundary Layer Solution

The nonlocal (NL) and the strain gradient (SG) based equivalent continuum theories were proposed for nanomechanics by accommodating the long-range molecular interactions and to bypass the computationally expensive atomistic simulations. Nanostructures are often made of smart materials (e.g., piezoelectric, piezoceramic, and flexoelectric) for multifunctional properties. Applications of nanostructures in critical systems deserve accurate analysis. The buckling and postcritical behavior of an axially loaded, piezoelectric, nonlocal strain gradient (NLSG) thin cylindrical shells with functionally graded elastic properties were presented in earlier studies following Donnell’s approach, leading to a set of stiff, nonlinear, partial differential equations, accommodating the prebuckling nonlinearity and large postcritical deflection. This study extends the previous work by accommodating geometric imperfections in the formulation. Furthermore, a consistent thickness-wise distribution for the electric potential is also adopted following the Maxwell equation. The boundary layer (BL) concept is employed to solve the resulting nonlinear equations via asymptotic expansions of the regular and the BL fields. The solution is numerically illustrated on moderately short shells, illustrating the influence of the geometric imperfections. The critical load remains imperfection sensitive, yet the imperfections change an unstable postcritical path into a stable one. The influence of the functional gradation and the external electric fields on the imperfection sensitivity behavior are illustrated.

Post critical characteristics of a side-box steel concrete composite girder: Experimental investigation and mechanism analysis

The present study is aimed to investigate the post-critical behavior of a side-box steel-concrete composite girder under varying wind angles of attack, by conducting a series of section model testing with single-degree and two-degrees of freedom. The essence of flutter, as well as the adverse impact of vortex shedding lock-in on exacerbating post-critical flutter responses, was revealed in detail. The results indicate that the coupled motion of flutter for the prototype composite girder can be characterized as a supercritical Hopf bifurcation with significant phase shifting between motions. The circular character of the limit cycle oscillation (LCO) expressed in the phase diagram is altered to a non-circular form due to the apparent aerodynamic stiffness nonlinearity within a wide range of wind speeds. Interestingly, the coupling vibration between vortex-induced vibrations (VIV) and flutter, i.e., VIV-flutter coupling vibration is observed for the prototype girder, due to the evolution of total damping ratio concave shape with wind speed. Two mitigation countermeasures, i.e., a L-shaped skirt plate and a sharpened wind fairing, were proposed to improve the aeroelastic stability, caused by a significant increase in uncoupled positive aerodynamic damping and alternation of VIV-flutter coupling. In addition, the specific effects of structural damping on the VIV-flutter coupling were discussed in detail.

A cost-effective algorithm for the co-simulation of unsteady and non-linear aeroelastic phenomena

Increasingly flexible structures force engineers and scientists to improve simulation capabilities incorporating an adequate description of non-linear phenomena. Furthermore, many critical operating conditions of aeronautical and civil structures are associated with unsteady phenomena, which makes it necessary to tackle this kind of problems. A critical aspect to be analyzed is the stability of the system: the determination of, not only the critical flutter speed, but also the post-critical behavior, is fundamental to guarantee the safety of designs and determine mitigation strategies. In this work, a numerical technique that allows simulating the non-linear and unsteady behavior of highly flexible structures immersed in a fluid at high Reynolds numbers is presented. This problem is addressed using a co-simulation approach, coupling the Unsteady Vortex Lattice Method (as the aerodynamic model) with the Finite Element Method using non-linear beams and rigid bodies (as the structural dynamics model). The coupling is achieved through a weak interaction algorithm. The resulting method has a very good compromise between versatility and computational cost, and is not restricted to periodic motions or linear equations of motion. Nonlinearity is included through the possibility to represent large displacements and rotations of the structure, a visco-elastic material behavior, and a nonlinear relation between the aerodynamic loads and the incidence of the incoming flow. The domains considered in the submodels and the temporal and spatial discretizations used for each one are essentially different; the interaction process must take into account these aspects. The computational implementation is carried out by combining a general-purpose structural dynamics code with an aerodynamic tool oriented to the analysis of three-bladed horizontal axis wind turbines. Examples to validate the code and show the potential of the proposed method are presented in the Results section.

Postcritical Behavior of Nonlocal Strain Gradient Arches: Formulation and Differential Quadrature Solution

Arches are important components in nanostructures and systems. Because of the remarkable strength of nanomaterials, nanoarches are slender, thus enhancing their vulnerability to geometric instability, such as buckling. Although molecular simulations are often employed to analyze nanostructures, their use in routine analysis/design is formidable due to prohibitively exhaustive computation. Equivalent continuum models were developed as alternatives. The buckling and postcritical phenomena of the classical arch also form an important benchmark problem in nonlinear mechanics. This study investigates the buckling and the postcritical behavior of nanoarch subjected to inward pressure using nonlocal (NL), strain gradient (SG) continuum theory. The governing equations are derived in the form of a sixth-order nonlinear integrodifferential equation, unlike the fourth-order for a classical arch. The equation is then solved numerically using a differential quadrature (DQ) with appropriate boundary conditions. An incremental-iterative arc-length continuation is employed for the solution of the resulting algebraic system of equations. The equilibrium paths are traced for the possible instability modes, such as symmetric/antisymmetric bifurcations, snap-through and limit point instability, ascertained with the internal-external force diagrams. A particular instability mode is triggered at a certain threshold/range of the slenderness ratio of the arch, which is significantly influenced by the NL and SG interactions. These interactions not only cause quantitative changes in the instability behavior but also lead to qualitative changes, such as cessation, shift, and conversion of modes, more prominently for SG arches. Similar to classical arches, the prebuckling nonlinearity is shown to be significant.

RAMIFICATION OF EQUILIBRIA OF A COMPRESSED ELASTIC ORTHOTROPIC PLATE WITH INTERNAL STRESSES

The problem of loss of stability and post-critical behavior of a compressed orthotropic rectangular plate on a nonlinearly elastic base containing continuously distributed fields of dislocations and disclinations and under the influence of a small normal load is considered. The compressive force components are evenly distributed along the edges and act parallel to the main directions of elasticity. The problem is formulated as an analogue of a system of nonlinear Karman equations for an orthotropic plate containing a function called the incompatibility measure, which is expressed in terms of the densities of edge dislocations and wedge disclinations. The system of equations takes into account the small transverse pressure and the reaction of the elastic base in the form of a polynomial of the third degree of deflection. The following boundary conditions are considered: the edges of the plate are freely pinched or movably pivotally supported; two opposite edges of the plate are freely pinched or movably pivotally supported, and the other two are free from loads. The stress function is sought in the form of two components: the stress function caused by the presence of internal sources, determined from the linear boundary value problem, and the stress function caused by the external impact of compressive loads and a nonlinear elastic base, which is determined from the nonlinear boundary value problem. The nonlinear boundary value problem is investigated by the Lyapunov – Schmidt method. To solve the linearized equation from which the critical value of the compressive load is determined, the variational method is used in combination with the difference method. A system of branching equations of the Lyapunov – Schmidt method is constructed, which is investigated by numerical methods. The post-critical behavior of the plate is investigated and asymptotic formulas for new equilibria in the vicinity of the critical load are derived. For various values of compressive loads, the orthotropy parameters of the plate and the intensity parameter of internal stresses, the relations between the values of the base parameters are established, at which its bearing capacity is preserved in the vicinity of the classical value of the critical load.

ON BRANCHING OF EQUILIBRIUMS OF A COMPRESSED ELASTIC ROD ON A NONLINEAR ELASTIC BASE

The problem of buckling and postcritical behavior of a compressed elastic rod on a non-linearly elastic foundation under the action of a small transverse pressure is considered. The study is carried out on the basis of a non-linear equilibrium equation obtained taking into account the exact formula for the curvature of the axial line of the rod, while the curvature is approximated by a third-order asymptotic formula with respect to deflection. The boundary conditions correspond to free pinching or a movable hinged support of the ends of the rod. The influence of a small transverse load and the reaction parameters of a nonlinearly elastic foundation on the critical buckling loads is studied. The critical load is determined from the eigenvalue problem obtained by linearization of the equilibrium equation. The problem of eigenvalues for the case of free pinching of the ends of the rod is solved by the variational-difference method. To study the post-critical behavior of a compressed rod, the Lyapunov – Schmidt method is used in combination with numerical methods for calculating the coefficients of the system of branching equations. The cases of branching of equilibria of a compressed rod along one eigenmode are considered. Asymptotic formulas for new equilibria are constructed in the vicinity of the bifurcation point, taking into account a small normal load. The base reaction is considered as a cubic function of deflection. Conditions are established for the parameters of the nonlinearity of the elastic foundation, under which the compressed rod becomes sensitive to imperfections in the form of a small transverse load. The obtained results are compared for the classical case of the linear formula for expressing the curvature of the axial line of the rod through the second derivative of the deflection and the case when the curvature is approximated by a third-order asymptotic formula with respect to the deflection.

Post-Buckling Behaviour of Steel Structures with Different Types of Imperfections

In this paper, the stability of steel members with a complex initial geometrical imperfection pattern are analysed. This issue is extremely important in the case of slender structures, characterised by multiple close critical loads and modal interactions, which can lead to unstable post-critical paths and imperfection sensitivity. Despite the fact that the loss of stability, as a result of complex geometrical imperfections, is a very common mechanism for the destruction of slender steel structures, there is still no unambiguous and adequate research in the literature and in scientific research taking into account multimodal buckling. Therefore, in this study, special attention was focused on the analysis of the equilibrium path of the structure in the pre- and post-buckling range. This was studied by introducing a model of a structure composed of four rigid bars connected by elastic nodes. For this model, as well as for the structure with and without initial geometrical imperfections, a set of nonlinear algebraic equations of equilibrium was developed. A complex pattern of imperfections was taken into account using a linear superposition of buckling modes obtained from a linear eigenvalue problem. In order to investigate the nature of bifurcation points, the concept of minimum of potential energy was adopted. By means of numerous examples, the influence of imperfections on the structural behaviour was discussed. It was found that, for special imperfection patterns, an increase in the amplitude of initial geometrical imperfection can result in an increase in the value of the critical load defining the bifurcation point. In these cases, initial geometrical imperfections can play a positive role, resulting in stable post-buckling behaviour. This phenomenon corresponds with the so-called “modal nudging” which aims to improve the buckling response of slender elastic structures by introducing a small disturbance in the primary geometry of the structure, which results in equilibrium paths of greater load-carrying capacity. Among other observations, a snap-through phenomenon caused by transition from the local to the global minimum of potential energy was also noted. The observed snap-through was caused by the specific configuration of initial geometrical imperfections, which in this case played quite a dangerous role. It should be emphasised that the proposed model structure allows for a full description of the post-critical behaviour and a trace of the influence of complex imperfection configurations in a simple and clear manner.

Stability of Strained Stanene Compared to That of Graphene.

Stanene, composed of tin atoms, is a member of 2D-Xenes, two-dimensional single element materials. The properties of the stanene can be changed and improved by applying deformation, and it is important to know the range of in-plane deformation that the stanene can withstand. Using the Tersoff interatomic potential for calculation of phonon frequencies, the range of stability of planar stanene under uniform in-plane deformation is analyzed and compared with the known data for graphene. Unlike atomically flat graphene, stanene has a certain thickness (buckling height). It is shown that as the tensile strain increases, the thickness of the buckled stanene decreases, and when a certain tensile strain is reached, the stanene becomes absolutely flat, like graphene. Postcritical behaviour of stanene depends on the type of applied strain: critical tensile strain leads to breaking of interatomic bonds and critical in-plane compressive strain leads to rippling of stanene. It is demonstrated that application of shear strain reduces the range of stability of stanene. The existence of two energetically equivalent states of stanene is shown, and consequently, the possibility of the formation of domains separated by domain walls in the stanene is predicted.

Limit state according to the condition of loss of stability of equilibrium forms

Objective. The purpose of the study is to determine the group of the limiting state according to the condition of loss of stability of the equilibrium form of structures. Method. The study is based on the provisions of the theory of stability of equilibrium states of building structures; branching theory of solutions of nonlinear equations; perturbation method; methods of catastrophe theory.Result. The results of the analysis of the post-critical behavior of structures based on the solution of the problem in higher approximations and from the fundamental provisions of the theory of catastrophes are generalized. It is proved that the study of the stability of equilibrium forms of structures using algebraic means and geometric images of the theory of catastrophes makes it possible to unambiguously determine the type of critical bifurcation points, predict the nature of the behavior of the structure, and determine the limit state group to which the state reached by the structure should be attributed. Conclusion. It seems necessary to rename the ordinal numbers of the types of critical points of bifurcations so that they coincide with the numbers of the groups of limit states corresponding to them.

Запредельные напряженно-деформированные состояния в поперечных сечениях элементов стальных конструкций

The article proposes using a technique of calculation the post-limit (residual) bearing capacity of steel frame elements relating to the strength check, in general case of loading, taking into consideration the post-critical behavior of plates making up the frame element and large plastic strain overflowing into the self-hardening stage. The solution is made by the “reverse” method in dimensionless parameters with increments of the strain of the most stressed fiber, which allows reducing by orders of magnitude the calculation time compared to existing methods. Herewith, the results of post-critical resistance of plates were used, which were obtained considering geometric imperfections and initial stresses under deformations in the form of limit relative slenderness of working parts of plates, that were approximated in form of functions for convenience and extrapolated to the values . The graphs showing how the bearing capacity reduction coefficient depends on the value of the normalized strain of the most stressed fiber for some of the cross-sections most widely used in construction practice are presented.

Запредельная несущая способность стержневых элементов стальных конструкций после потери общей устойчивости

The «reverse» numerical-analytical method for definition the residual bearing capacity of steel frame elements after buckling is proposed. The method allows to take in consideration the post-critical behavior of plates making up the frame element and large plastic strain overflowing into the self-hardening stage.The solution is carried out by the reverse sequence: it is offered to define values of deformational and non-deformational internal forces by numerical unitless procedure according to a given post-limit deformed state of the most loaded cross-section of an element under compression and bending in one or two planes. Wherein deformational forces are assumed as actually acting forces, taking in consideration geometrical nonlinearity, and non-deformational forces are assumed as fictitious forces, compensating nonlinear material behavior and local buckling. The latter ones are considered to be the result of sequential symmetrical acting of two additional axial fictitious forces with one or two eccentricities on the elastic element. The first fictitious force compensates the initial stage of increase of plastic strains before element buckling, and the second one compensates local buckling and further increase of plastic strains in some part of element span after element buckling. The effect of the geometrical nonlinearity on internal forces is defined by «straight» unitless analytical solution of the problem, considering deformed state, using the elastic model of element compressed by actually acting and fictitious forces with their eccentricities. Evaluated internal forces are used in the reverse solution to define eccentricities of actually acting force on the ends of element. The proposed method has a sufficient accuracy for practical application and allows to reduce by orders the magnitude of computational time compared to numerical methods. The graphs showing how the post-buckling bearing capacity reduction coefficients depends on the value of the normalized strain of the most stressed fiber for some of the cross-sections most widely used in construction practice are presented.

Experimental parametric study on dynamic divergence instability and chaos of circular cylindrical shells conveying airflow

Experiments have shown that soft circular cylindrical shells supported at both ends and conveying airflow lose stability by so-called dynamic divergence. The present study investigates experimentally the effect of geometric parameters of a silicone rubber shell, namely length-to-radius (L/R) and thickness-to-radius (h/R) ratios, on the dynamic divergence instability. Bifurcation diagrams of the rms velocity of the shell vibration versus flow velocity are obtained for different shells, displaying a strongly subcritical nonlinear behaviour. Then, the onset of instability and post-critical behaviour of the shells are compared: (i) thinner and longer shells lose stability at lower flow velocities, (ii) thinner shells have higher rms vibration velocity, and (iii) by decreasing L/R, the subcritical behaviour is weakened for thin shells, while it is strengthened for thick shells. The existence of chaos and the influence of geometric parameters on the chaotic behaviour of the system are deeply examined by means of several measures.

Hopf bifurcations for an oversteer vehicle – the influence of wheel load changes

AbstractIn [1] the occurence of a Hopf bifurcation at steady‐state cornering for a two‐wheel oversteering vehicle with a brush model for the tyre forces was found and following the bifurcating branch of periodic solutions we observed a Canard phenomenon and relaxation oscillations, which are typical for singularly perturbed systems. In that article we assumed for simplicity, that the car's center of mass lies at street level. In this presentation we investigate the influence of an elevated height on the stability of the car's steady motion and the postcritical behaviour.

Composite plate analysis made in an unsymmetric configuartion

The work presents a thin-walled plate element with the central rectangular cut-out which can be use as an elastic or load-bearing element. Plates were made of carbon epoxy laminate and subjected to uniform compression. Plates were simply supported on shorter edges, and loaded axial load. The study included analysis of the critical and weakly post-critical behavior using experimental and numerical methods. Numerical analysis was performed with using linear analysis of eigenvalue problem to determination critical loads. The second step connected nonlinear analysis of structure with initiated geometrically imperfection corresponding to the flexural-torsional buckling mode of the plate. To the numerical calculations the commercial ABAQUS program was used.