Snap-through Phenomenon Research Articles

Critical Points for Variable Length Elastica With a Fixed Point Constraint Under Displacement Control

Abstract This paper studies a variable length elastica with a fixed point constraint by an assembly method that regards the whole elastica as an assembly of two components, i.e., pinned-clamped elasticas. The pinned-clamped elastica is obtained based on the post-buckled deformed shape with one internal inflection point. Thus, multiple coexisting solutions can be located accurately, which reveals three distinct equilibrium paths for the complete load–displacement curves. Under displacement control, two critical points on two equilibrium paths are found at saddle-node bifurcations. Interestingly, a new critical point is located at the boundary point of one equilibrium path, where the shapes of two pinned-clamped elasticas are two different post-buckled deformed shapes. Changing the location of the fixed point constraint allows the position of the boundary point to be easily manipulated, and the associated snap-through phenomenon can occur on different equilibrium paths. This flexible generation of the snap-through phenomenon is useful for designing engineering systems that require controllable snap-through.

Nonlinear thermal buckling and postbuckling analysis of bidirectional functionally graded tapered microbeams based on Reddy beam theory

In the present study, the nonlinear thermal buckling, postbuckling, and snap-through phenomenon of higher-order shear deformable tapered bidirectional functionally graded (BDFG) microbeams are comprehensively investigated under different types of thermal loading, for the first time. The thermomechanical properties of the BDFG microbeam are assumed to be functions of temperature, axial, and thickness directions. Reddy’s parabolic shear deformation beam theory with the von Karman nonlinearity is employed to derive the variable coefficient governing nonlinear differential equations on the basis the physical neutral surface concept. Size-dependent effect is captured in the formulation employing the modified couple stress theory. The generalized differential quadrature method (GDQM) is used to discretize the motion equations considering different boundary conditions. The resulting system of nonlinear algebraic equations is solved iteratively using Newton’s method. Theoretical analysis and numerical results indicate that, depending on the shear deformation beam theory, boundary conditions, and the type of thermal load, the response of the BDFG microbeam may be of the bifurcation or snap-through buckling type of instability. Numerical parametric studies are conducted to explore the influences of thermal load type, material property gradient indexes, boundary conditions, material temperature dependency, taperness ratio, and microstructural length scale on critical thermal buckling load, thermal postbuckling, and equilibrium paths of the BDFG microbeam.

Snap-through of an elastica under bilateral displacement control at a material point

Snap-through phenomenon widely occurs for elastic systems, where the systems lose stability at critical points. Here snap-through of an elastica under bilateral displacement control at a material point is studied, by regarding the whole elastica as two components, i.e., pinned-clamped elasticas. Specifically, stiffness−curvature curves of two pinned-clamped elasticas are firstly efficiently located based on the second-order mode, which are used to determine the shapes of two components. Similar transformations are used to assemble two components together to form the whole elastica, which reveals four kinds of shapes. One advantage of this way compared with other methods such as the shooting method is that multiple coexisting solutions can be located accurately. On the load−deflection curves, four branches correspond to four kinds of shapes and first two branches are symmetrical to the last two branches relative to the original point. For the bilateral displacement control, the critical points can only appear at saddle-node bifurcations, which is different to that for the unilateral displacement control. Specifically, one critical point is found on the first branch and two critical points are found on the secondary branch. In addition, the snap-through among different branches can be well explained with these critical points.

Nonlinear thermal behavior of shear deformable FG porous nanobeams with geometrical imperfection: Snap-through and postbuckling analysis

Present paper deals with the nonlinear thermal bending response, thermal postbuckling behavior, and snap-through phenomenon due to lateral mechanical load in a thermally preloaded functionally graded (FG) porous perfect/imperfect nanobeam subjected to two types of thermal loading, including heat conduction across the thickness and uniform temperature rise. Heterogenous material properties of the porous nanobeams are assumed to be position/temperature-dependent, where dependency is obtained according to the modified rule of mixture and Touloukian formulation. Two types of porosity distribution and also geometrical imperfection of the nanobeam are considered. Assuming the Timoshenko beam model and von-Karman nonlinearity, the nonlinear equilibrium equations are extracted on the basis of the nonlocal theory of elasticity. With the establishment of the principle of virtual displacement and Chebyshev polynomial of the first kind as the basic functions, the Ritz method is utilized to obtain the matrix form of the nonlocal governing equations. Two different strategies, including the Newton-Raphson technique and direct displacement control scheme, are first proposed to extract the nonlinear bending and postbuckling trajectories of the graded porous nanobeam. Afterwards, to trace the snapping behavior beyond limit loads of the graded porous nanobeam with thermal preloading, the cylindrical arch-length scheme as a path-following method is adopted. Numerical parametric investigations are given to discuss the influences of the power-law index, two types of porosity distribution and thermal loads, size-dependent nonlocal parameter, imperfection amplitude, and boundary conditions on the snap-through behavior and the nonlinear thermal stability of the FG nanobeam.

Second-order inelastic analysis of shallow and non-shallow steel arches

This work presents a second-order inelastic analysis of steel arches. The analysis of shallow and non-shallow arches with several cross sections and boundary and loads conditions are discussed. The computational platform used is the homemade CS-ASA, which performs advanced nonlinear static and dynamic analysis of structures. The nonlinear geometric effects are considered using a co-rotational finite element formulation; the material inelasticity is simulated by coupling the Refined Plastic Hinge Method (RPHM) with the Strain Compatibility Method (SCM), and the static nonlinear solution is based on an incremental-iterative strategy including continuation techniques. In the simulated nonlinear steel arch models, special attention is given to the equilibrium paths, the influence of rise-to-span ratio, support and loading conditions and full yield curves among other factors. The numerical results obtained show good agreement with those from literature and highlight that the arch rise-to-span ratio has great influence on the structure resistance and that the shallow arches can lose stability through the snap-through phenomenon.

Nonlinear aeroelastic behavior of an airfoil with free-play in transonic flow

An investigation has been made into the nonlinear aeroelastic behavior of an airfoil system with free-play nonlinear stiffness in transonic flow. Computational Fluid Dynamics (CFD) and Reduced Order Model (ROM) based on Euler and Navier-Stokes equations are implemented to calculate unsteady aerodynamic forces. Results show that the nonlinear aeroelastic system experiences various bifurcations with increasing Mach number. Regular subcritical bifurcations are observed in low Mach number region. Subsequently, complex Limit Cycle Oscillations (LCOs) and even non-periodic motions appear at specific airspeed regions. When the Mach number is increased above the freeze Mach number, regular subcritical bifurcations occur again. Comparisons with inviscid solutions are used to identify and elaborate the effect of viscosity with the help of aeroelastic analysis techniques, including root locus, Single Degree of Freedom (SDOF) flutter and aerodynamic influence coefficient (AIC). For low Mach numbers in the transonic regime, the viscosity has little effect on the linear flutter characteristic because of limited influence on AIC, but a remarkable impact on the nonlinear dynamic behavior due to the sensitivity of the nonlinear structure. As the Mach number increases, the viscosity becomes significantly important due to the existence of shock-boundary layer interaction. It affects the unstable mechanism of linear flutter, impacts the aerodynamic center and hence the snap-through phenomenon, influences the AIC and consequently the nonlinear aeroelastic response. When the Mach number is increased further, the shock wave dominates the air flow and the viscosity is of minor importance.

Dynamic snap-through of shallow spherical shells subjected to thermal shock

Present study deals with the dynamic snap-through phenomenon of an isotropic shallow spherical cap under transient type of thermal loading. The inner surface of the shell is subjected to sudden temperature elevation, whereas the outer surface is kept at reference temperature. Transient thermal shock is applied uniformly and shell thickness is assumed to be thin enough. Therefore, transient heat conduction equation is analytically solved across the thickness direction (one dimensional). Immovable simply-supported boundary conditions are assumed for the shell. Since the boundary conditions and the applied thermal shock are axisymmetric, the governing motion equations of the shell are restricted to the case of axisymmetric. First order shear deformation theory of shells is utilized to approximate the displacement field. The von Kármán type of geometrical non-linearity is used in strain-displacement relations. With the establishment of the associated Hamilton principle, differential equations of motion are extracted. Motion equations are discretized within the shell domain by means of the harmonic differential quadrature method (HDQM). The Newmark time marching scheme based on the constant average acceleration method is applied to turn the motion differential equations into a system of algebraic equations at each time step. Highly coupled equations are solved implementing the well-known Newton-Raphson iterative technique. By means of the Budiansky criterion, critical thermal shock parameters are distinguished. Critical dynamic snap-through temperatures are also verified using the phase-plane presentation. Comparison study is performed to validate the formulation and solution method of the present research with simple cases. Also, parametric investigations are performed to demonstrate the geometrical effects on the dynamic critical buckling temperatures of the shallow spherical shells subjected to thermal shock.

Nonlinear thermal stability and snap-through buckling of temperature-dependent geometrically imperfect graded nanobeams on nonlinear elastic foundation

In this research, thermal postbuckling and nonlinear thermal bending of size-dependent functionally graded (FG) perfect/imperfect nanobeams in-contact with a nonlinear elastic foundation and exposed to thermal loading are first scrutinized. The second objective of this research is to investigate snap-through instability in a thermally postbuckled FG nanobeam due to uniform lateral load. The geometrical imperfection of the nanobeam is taken into account. Thermo-mechanical material properties are temperature-dependent and are assumed to vary continuously throughout the nanobeam thickness on the basis of the power-law model. The nonlinear equilibrium equations are derived according to the Euler-Bernoulli beam theory in conjunction with the nonlinear von-Karman assumptions based on the nonlocal elasticity theory. Using Chebyshev polynomial of the first kind, Ritz method is utilized into the principle of virtual displacement to form nonlocal governing equations. Three different methodologies, including direct displacement control approach, Newton-Raphson iterative method, and cylindrical arch-length scheme are utilized to investigate the nonlinear thermal stability curves and snap-through phenomenon through limit points of a thermally postbuckled FG nanobeam. The primary purpose of this research is to explore the influences of the nonlocal parameter, nonlinear elastic foundation coefficients, power-law index, imperfection amplitude as well as the different types of boundary conditions on the nonlinear thermal stability and snap-through phenomenon of the FG nanobeam.

Delayed electromechanical instability of a viscoelastic dielectric elastomer balloon.

When a dielectric elastomer (DE) balloon is subjected to electromechanical loading, instability may happen. In recent experiments, it has been shown that the instability configuration of a DE balloon under electromechanical loading can be very different from that only subjected to mechanical load. It has also been observed in the experiments that the electromechanical instability phenomena of a DE balloon can be highly time-dependent. In this article, we adopt a nonlinear viscoelastic model for the DE membrane to investigate the time-dependent electromechanical instability of a DE balloon. Using the model, we show that under a constant electromechanical loading, a DE balloon may gradually evolve from a convex shape to a non-convex shape with bulging out in the centre, and compressive hoop stress can also gradually develop the balloon, resulting in wrinkles as observed in the experiments. We have further shown that the snap-through instability phenomenon of the DE balloon also greatly depends on the ramping rate of the applied voltage.

Equilibrium Paths for von Mises Trusses in Finite Elasticity

This paper deals with the equilibrium problem of von Mises trusses in nonlinear elasticity. A general loading condition is considered and the rods are regarded as hyperelastic bodies composed of a homogeneous isotropic material. Under the hypothesis of homogeneous deformations, the finite displacement fields and deformation gradients are derived. Consequently, the Piola-Kirchhoff and Cauchy stress tensors are computed by formulating the boundary-value problem. The equilibrium in the deformed configuration is then written and the stability of the equilibrium paths is assessed through the energy criterion. An application assuming a compressible Mooney-Rivlin material is performed. The equilibrium solutions for the case of vertical load present primary and secondary branches. Although, the stability analysis reveals that the only form of instability is the snap-through phenomenon. Finally, the finite theory is linearized by introducing the hypotheses of small displacement and strain fields. By doing so, the classical solution of the two-bar truss in linear elasticity is recovered.

퇴화 쉘요소를 사용한 원통형 복합재 패널의 비선형 좌굴후 현상해석

In this study, the post - buckling analysis of cylindrical panel of isotropic material and composites was investigated. The arc-length nonlinear analysis method has been used for the analysis of post-buckling phenomena where snap-through and snap-back phenomenon occur. In this study, a method to find bifurcation point in buckling analysis using arc-length method is presented. The bifurcation point of the buckling was found in the cylindrical shell structure of isotropic material to ensure the reliability of the current approach. The bifurcation buckling point was successfully calculated with composites having extremely high structural instability. The formulations, mathematical model and analysis procedures and boundary conditions proposed in this study can be used as important data for the unstable buckling analysis of other structures.

A nonlinear resonant mass sensor with enhanced sensitivity and resolution incorporating compressed bistable beam

A new piezoelectric actuated nonlinear mass sensor is proposed by using the snap-through phenomenon of a compressed bistable beam to enhance the sensitivity and resolution, which can be used to weigh or detect threshold mass by tracking the bifurcation frequency shift. According to the nonlinear finite element modeling technique, the nonlinear dynamic response of the combined nonlinear structure is numerically calculated, which shows that the bifurcation point can be accurately identified by the sharp and great amplitude change regardless of the damping effect, thus providing an effective way for tracking the bifurcation frequency. Hence, the nonlinear sensitivity depicted by the bifurcation frequency shift per unit mass can reach 3.3 times the sensitivity of linear mass sensors having the same size. Also, as a mass switch, the response amplitude jumps sharply when the added mass is greater than or equal to the threshold value, which is dependent on the excitation frequency. Meanwhile, the influences of the beam compression and excitation voltage on the sensitivity and minimum detectable mass were obtained for sensor optimization. For concept validation, a macro-sized nonlinear mass sensor was fabricated with the geometric size of 58.0 mm long and 4.0 mm wide, and the experimental results show that the sensitivity is around 575.0 Hz/g compared with the simulated sensitivity of 542.0 Hz/g. For a mass switch, the minimum threshold mass is 0.2 mg. The fair agreement between the simulation and experiments adequately validated the proposed nonlinear bistable mass sensor.

Voltage-Induced Snap-Through of an Asymmetrically Laminated, Piezoelectric, Thin-Film Diaphragm Micro-Actuator—Part 2: Numerical and Analytical Results

Abstract In Part 2 of the two-paper series, the asymmetrically laminated piezoelectric shell subjected to distributed bias voltage as modeled in Part 1 is analytically and numerically investigated. Three out-of-plane degrees-of-freedom (DOFs) and a number of in-plane DOFs are retained to study the shell's snap-through phenomenon. A convergence study first confirms that the number of the in-plane DOFs retained affects not only the number of predicted equilibrium states when the bias voltage is absent but also the prediction of the critical bias voltage for snap-through to occur and the types of snap-through mechanisms. Equilibrium states can be symmetric or asymmetric, involving only a symmetric out-of-plane DOF, and additional asymmetric out-of-plane DOFs, respectively. For symmetric equilibrium states, the snap-through mechanism can evolve from the classical bidirectional snap-through and latching to a new type of snap-through that only allows snap-through in one direction (i.e., unidirectional snap-through), depending on the distribution of the bias voltage. For asymmetric equilibrium states, degeneration can occur to the asymmetric bifurcation points when the radii of curvature are equal. Finally, the unidirectional snap-through renders an explanation to the experimental findings in Part 1.

Small scale effects on buckling and postbuckling behaviors of axially loaded FGM nanoshells based on nonlocal strain gradient elasticity theory

By means of a comprehensive theory of elasticity, namely, a nonlocal strain gradient continuum theory, size-dependent nonlinear axial instability characteristics of cylindrical nanoshells made of functionally graded material (FGM) are examined. To take small scale effects into consideration in a more accurate way, a nonlocal stress field parameter and an internal length scale parameter are incorporated simultaneously into an exponential shear deformation shell theory. The variation of material properties associated with FGM nanoshells is supposed along the shell thickness, and it is modeled based on the Mori-Tanaka homogenization scheme. With a boundary layer theory of shell buckling and a perturbation-based solving process, the nonlocal strain gradient load-deflection and load-shortening stability paths are derived explicitly. It is observed that the strain gradient size effect causes to the increases of both the critical axial buckling load and the width of snap-through phenomenon related to the postbuckling regime, while the nonlocal size dependency leads to the decreases of them. Moreover, the influence of the nonlocal type of small scale effect on the axial instability characteristics of FGM nanoshells is more than that of the strain gradient one.

Nonlinear Random Response Analyses of Panels Considering Transverse Shear Deformations under Combined Thermal and Acoustic Loads

The panel structures of flight vehicles at supersonic or hypersonic speeds are subjected to combined thermal, acoustic, and aerodynamic loads. Because of the combined thermal and acoustic loads, the panel structure may exhibit nonlinear random vibration responses, such as the snap‐through phenomenon and random vibrations. These unique dynamic behaviors of the panel structure under combined thermal and acoustic loads can result in serious damage or fatigue failure of the panel structures of high‐speed flight vehicles. This study investigates the nonlinear random responses of thin and thick panels under combined thermal and acoustic loads. The panels are modeled based on the first‐order shear deformation theory (FSDT) to account for transverse shear deformations. The von‐Karman nonlinear strain–displacement relationship is used for geometric nonlinearity in the out‐of‐plane direction of the panel. The thermal load distribution is assumed to be constant in the thickness direction of the panel. The random acoustic load is represented as stationary White–Gaussian random pressure with zero mean and uniform magnitude over the panels. Static and dynamic equations are derived using the principle of virtual work and the nonlinear finite element method. A thermal postbuckling analysis is conducted using the Newton–Raphson method, and the dynamic nonlinear equations are solved using the Newmark‐β time integration method. In the present numerical analyses, the snap‐through responses for both the thin and thick panels are investigated, and the results indicate that the loading conditions that cause snap‐through are different for thin and thick panels.

An accurate singularity-free geometrically exact beam formulation using Euler parameters

An accurate singularity-free geometrically exact formulation of a three-dimensional beam with large displacements, large deformations, and large rotations is developed. The undeformed configuration of the beam can be either straight or curved. Euler parameters are used to parametrize the rotation of a cross section of the beam to avoid singularity. While the position vector is interpolated using Hermite functions, Euler parameters are interpolated using a $$\hbox {C}^{1}$$ -continuous interpolation function. The stretch–shear strain vector of the centroid line of the beam can be obtained by independent interpolations of position and rotation fields. Governing equations of the beam are obtained using Lagrange’s equations for systems with constraints. An arc-length method is introduced to trace the force–displacement curve of equilibria of the beam. Several examples are simulated to show the performance of the current formulation. A static planar example with its exact solution is first simulated to show accuracy and convergency of the current formulation. In-plane buckling of a circular arch is then studied, and the snap-through phenomenon is found when the arch has clamped–clamped boundary conditions. The stiffness, natural frequencies, and mode shapes of a spring are calculated; the effect of the mass of the spring on natural frequencies and mode shapes of the spring–mass system is analyzed, and a curve veering phenomenon accompanied with mode shift and mode localization behaviors is found there. The current formulation is compared with the formulation in the commercial software ADAMS using a dynamic three-dimensional example, and their results are in excellent agreement. Another dynamic example of a moving force on a beam is also shown, and its dynamic behavior is analyzed.

Size Dependency in the Axial Postbuckling Behavior of Nanopanels Made of Functionally Graded Material Considering Surface Elasticity

Size-dependent modeling and analysis for the nonlinear postbuckling response of functionally graded (FG) cylindrical nanopanels under axial compressive load are presented based on the surface elasticity theory. The non-classical formulations are developed on the basis of the Gurtin–Murdoch elasticity theory within the framework of the classical shell theory. In order to capture the large deflections associated with the postbuckling regime, the geometrical nonlinearity is considered in von Karman sense. The material properties are supposed to be graded across the panel thickness in accordance with a simple power law function of the volume fractions of the silicon and aluminum constituents with considering the physical neutral plane position. To satisfy the balance conditions on the free surface layers, it is assumed that the bulk normal stress along the thickness direction varies linearly between those values of surface stress components at the inner and outer layers. A boundary layer theory of shell buckling is extended to the case of size-dependent FG nanopanels, and then, a singular perturbation technique is put to use to solve the nonlinear problem. It is displayed that due to the stiffening influence of surface effects, the depth of the postbuckling regime for FG nanopanels with higher material gradient index decreases, as for the metal-rich nanopanel, the snap-through phenomenon approximately diminishes.

Size dependency in axial postbuckling behavior of hybrid FGM exponential shear deformable nanoshells based on the nonlocal elasticity theory

The objective of this study is to examine the nonlocal nonlinear instability of functionally graded cylindrical shells at nanoscale integrated with piezoelectric nanolayers under combination of axial compressive load and lateral electric field. Eringen's nonlocal continuum elasticity is incorporated within the framework of the exponential shear deformation shell theory to consider the influence of transverse shear deformation in a refined form. Additionally, in order to eliminate the stretching-bending coupling terms, the change in the position of physical neutral plane corresponding to different volume fractions is taken into account. With the aid of the boundary layer theory of shell buckling and employing a perturbation-based solution methodology, explicit expressions for the size-dependent equilibrium paths before and after buckling point are proposed for functionally graded hybrid nanoshells with various nonlocal parameters, material property gradient indexes and subjected to different values of lateral electric field. It is indicated that the both width and depth of the snap-through phenomenon related to the axial postbuckling behavior of hybrid FGM nanoshells decrease due to the nonlocality influence.

Size-dependent nonlinear instability of shear deformable cylindrical nanopanels subjected to axial compression in thermal environments

In this work, modeling and analysis for size-dependent buckling and postbuckling behavior of cylindrical nanopanels under axial compression in thermal environments are presented. To this end, a non-classical panel model based upon the Gurtin–Murdoch elasticity theory in conjunction with the first-order shear deformation shell theory is developed which considers efficiently the effects of surface free energy. In order to capture the large deflections, the von Karman geometrical nonlinearity is taken into account. The size-dependent governing differential equations are deduced to a boundary layer type that incorporates the nonlinear prebuckling deformations and the large postbuckling deflections. Afterwards, by means of an efficient solution methodology using a perturbation solving process, the size-dependent critical buckling loads and associated postbuckling equilibrium paths are obtained corresponding to various geometrical parameters and thermal environments. It is seen that for the both classical and non-classical panel models, the wide of snap-through phenomenon is larger for nanopanels with higher ratio of length to width. Also, it is found that thermal environment has no influence on the value of minimum load of the postbuckling domain and associated maximum deflection.

Axisymmetric static and dynamics snap-through phenomena in a thermally postbuckled temperature-dependent FGM circular plate

Thermal postbuckling analysis and the axisymmetric static and dynamic snap-through phenomena due to static/sudden uniform lateral pressure in a thermally postbuckled functionally graded material circular plate are performed in this research. Plate is formulated using the first order shear deformation plate theory. Thermo-mechanical properties of the plate are assumed to be temperature dependent where dependency is described according to the higher order Touloukian representation. Two types of temperature loading are considered. Uniform temperature rise and heat conduction across the thickness direction. The one dimensional heat conduction equation in the thickness direction is obtained and discreted via the central finite difference method. The obtained system of equations is nonlinear since the thermal conductivity itself is a function of the unknown nodal temperatures. Using the von-Kármán assumptions, the governing equations of the plate are obtained in a matrix representation with the aid of the conventional Ritz method whose shape functions are developed using the Gram-Schmidt process. At first thermal postbuckling analysis is performed which is a nonlinear problem with respect to both temperature and displacements. Afterwards, response of the bulged thermally postbuckled plate is obtained under the static and dynamic uniform pressure. Snap-through phenomenon may be observed in both static and dynamic loading cases, due to the immovability of the edge of the plate and the initial deflection caused by postbuckling deflection. To capture the snapping phenomenon and trace the path beyond the limit loads, cylindrical arch-length technique is used. In dynamic snap-through analysis, the effect of structural damping is also included. Numerical results of this study reveal that the structure is sensitive to the initial deflection caused by thermal postbuckling load. Increasing the temperature prior to mechanical loads enhances the snap-through intensity and also increases both the upper and lower limit loads. As shown, dynamic snap-through loads are lower than the static ones, however dynamic snap-through intensity is more than the static snap-though intensity. Furthermore, structural damping enhances the dynamic buckling loads of the plate and decreases the dynamic postbuckling deflection of the plate.