Snap-through Load Research Articles

Experimental study and numerical simulation of morphing characteristics of bistable laminates embedded with 3D printed shape memory polymers

Bistable laminates have potential applications in areas such as buffer and soft robotics due to their two stable states and ability to deform through both snap-through and snap-back processes. However, conventional orthotropic bistable laminates with two stable state configurations and snap loads are similar in magnitude. There is a problem that the deformation cannot be easily driven when the laminate stiffness is high. To solve this problem, a clever design is needed to reduce the snap-back load and increase the snap-through load so that the driven deformation can be realized more easily. In this paper, we propose a design method to suppress and control the deformation of bistable laminates by utilizing the shape memory of 3D-printed shape memory polymers (SMP), which enables the laminates to increase the stiffness while reducing the driving force. First, a numerical model of viscoelasticity of the laminate is established to study its deformation characteristics, and the numerical results are compared with experimental results with good agreement. The effect of SMPs on the snap load and principal curvature of bistable laminates is also investigated. Finally, the interlayer interface bonding of the bistable laminates is examined in microscopic perspective. The results demonstrate that 3D printed SMPs can effectively enhance the snap-through load and reduce the snap-back load of bistable laminates, achieving deformation suppression and control while maintaining good interlaminar bonding with carbon fiber composites. This study provides new insights and practical significance for the deformation suppression and active control of bistable structures.

Bistable hybrid symmetric laminates with a clamped-clamped boundary condition: An experimental and numerical investigation

In this work, a clamped-clamped boundary condition is considered for a family of bistable hybrid symmetric laminates in which the direction of curvature does not change during the snap-through process between the two stable configurations. The laminates were modeled using finite element analysis to predict the characteristics of stable configurations, including the distributions of curvatures and out-of-plane displacement in two directions, and then the results were validated by experiments. The snap-through behaviors of the bistable laminates under the clamped-clamped boundary condition were investigated numerically and experimentally and then compared with the preloaded laminates with the same boundary condition. The results showed that the bistable laminates require a lower snap-through load than the preloaded laminates when they have a similar maximum out-of-plane displacement. The loading position, lay-up design, and length of the clamped region were employed as variables to discover the features of snap-through behavior. It was found that the unsymmetric deformation mode was present regardless of the load position during the snap-through behavior of these bistable laminates. At last, a variable cross-section tube consisting of two bistable laminates was proposed, and the area of the cross-section could be increased by 80% after the snap-through of the two bistable laminates

Fatigue performance evaluation of bistable composites at different combinations of loading and environmental conditions

Bistable composite laminates have large-scale applications in morphing and energy-harvesting structures, but their fatigue performance remains largely unexplored. This study investigates the stiffness and damage progression and evaluates bistable performance to develop protocols for long-term applications. We analyze the effects of displacement-controlled fully reversible high cycle fatigue-loading on stiffness, damage, curvature, and snap-through load in the out-of-plane loading direction at eight different combinations of parameters with frequency from 1 to 10 Hz, two boundary conditions, and temperature from 22°C to 150°C up to 3 to 10 million cycles. Stiffness and damage evolution analysis demonstrate the first two stages in out-of-plane fatigue loading. The study proposes a damage definition in terms of load adapting with two fatigue damage models: (1) Shiri Model and (2) Wu Model, while both models exhibit reasonable accuracy in predicting damage for the first two stages despite deviating at the final cycle due to assuming this cycle as the final failure cycle. Of the two models, the Shiri model provided a smaller range of model parameter values, 0.22 and 0.43, for parameters p and q, respectively, which reflects adjustability to different test conditions by maintaining a moderate range. Specimens encountered no final failure by fiber breakage and did not lose bistability for any combination. Curvature and snap-through load measurements have not substantially changed due to fatigue loading. These findings confirm application protocols with a broad range of parameters for which the laminates can operate without significant fatigue damage and maintain their bistable performance for an infinite lifetime.

A bistable pure carbon fiber composite symmetric laminate and its preparation, modeling, and experimental verification

A novel bistable symmetric laminate of pure carbon fiber composite is presented in this article. First, an analytical model was established based on the results of finite element analysis (FEA) using the segmented continuous displacement function. Then, the bistable pure carbon fiber composite symmetric laminate was prepared in the experiment. Both the analytical and FEA results are in considerable agreement with the experimental results. Finally, the effects of various laminate lengths, widths, ratios, thicknesses and materials were investigated. The results showed that the bistable pure carbon fiber composite symmetric laminate has larger deformation and lower snap-through load.

Quantifying the strength of stability of multi-stable structures: A new design perspective

In recent years the use of multi-stable components for designing adaptive structures and energy harvesting systems has been broadly investigated. Most of these studies have concentrated on analyzing the equilibrium shapes and the snap-through loads of the bi/multi-stable elements. However, there is minimal work on quantifying the strength of these stable equilibrium shapes. The strength of stable states is a measure of the ease or difficulty of the transition between the different stable states of any multi-stable system. Quantifying the strength of stability becomes paramount while designing adaptive structures which integrate multi-stable elements. Previous studies have argued that snap-through loads represent the goodness of stability; however, the magnitude of the snapthrough load highly depends on the boundary conditions of the structure and the loads’ application point. This work presents a mathematical framework that quantifies the strength of stability of the stable equilibrium shapes of multi-stable elements and identifies the minimum energy path (MEP) between two stable states. The method is based on finding, either with an analytical approach or finite element (FE) models, the stable and transition points of the energy landscape. Results prove that the method provides a quantitative measure of stability, a key parameter when designing adaptive structures. In addition, the minimum energy path between two stable states provides a tool to simplify the design of efficient actuation strategies for multi-stable systems. Four numerical examples are detailed to demonstrate the robustness of the approach.

Effect of snap-through force location on morphing of bistable variable stiffness laminates

Unsymmetric composite laminates exhibit two or more stable states due to the presence of residual thermal stresses induced during the curing process. The paper aims to exploit the possibility of assessing the morphing behaviour of bistable composite laminates using semi-analytical and numerical models. A Rayleigh-Ritz based semi-analytical model is used to investigate the variable stiffness (VS) bistable laminates generated using curvilinear fibre alignments. The obtained results are verified using a fully non-linear numerical model developed in a commercially available finite element package. In order to identify the influence of loading location on the snap-through behaviour, a parametric study is proposed in the present research. Snap-through is achieved through transverse point loads. The novelty of this work resides on the investigation of load application location on the snap-through action where the transverse load is applied at various locations on the lamina surface. Earlier research on the morphing of bistable square composites is mainly on the application of concentrated load at the centre of the laminate geometry, which is considered as an ideal location for snap-through action. The designer may also have to choose other locations for the application of snap-through loads since the lamina centre may not be appropriate always in structural applications. Four different loading locations are selected for the present analysis, and snap-through loads are calculated to check the effect of loading location. Parametric studies are performed using the proposed semi-analytical and numerical models to investigate the effect of selected loading point positions on the snap-through process. The analysis is extended to a selected VS laminate family to understand the influence of loading point positions on the curvilinear fibre alignments.

Nonlinear dynamic buckling and vibration of thermally post-buckled temperature-dependent FG porous nanobeams based on the nonlocal theory

In this paper, nonlinear dynamic snap-through buckling and vibration behavior of the thermally post-buckled functionally graded (FG) porous nanobeams subjected to static and sudden mechanical loads are investigated utilizing the nonlocal elasticity theory. The physical properties of the nanobeam are considered to be functions of temperature based on the Touloukian model. In addition, to describe the FG porous materials, two different patterns of porosity distribution are adopted using trigonometric functions through the thickness of the nanobeam. The equations of motion in conjunction with the von-Kármán nonlinear assumption are established in the framework of Hamilton’s principle. By employing the Chebyshev-Ritz procedure, the nonlinear equations are discretized for three types of edge supports. Following that, the cylindrical arc-length technique is employed to assess the vibrational responses of the post-buckled nanobeam during static snap-through buckling. To evaluate the nonlinear dynamic buckling of the graded nanobeam under a sudden dynamic load, the Newmark time integration scheme together with the Newton-Raphson iterative method are utilized. Next, by means of the Budiansky-Roth criterion and the phase-plane approach, the dynamic snap-through loads are identified. After validating the developed mathematical model, a comprehensive investigation is carried out to determine the role of various physical and geometrical parameters on the dynamic snap-through buckling and vibration characteristics of the post-buckled FG nanobeams.

Snap-through load improvement of bistable composite shell with initial curvature by an adjustive angle

Bistable composite shells with an initial curvature have been extensively studied owing to their light weight and morphing. The bistable behaviour is influenced by the thermal residual stress generated during the manufacturing of bistable composite shells with initial curvature. Hence, this study proposes a method based on an adjustive angle to improve the snap-through load and enhance the load-bearing capacity of bistable composite shells with an initial curvature. The critical snap-through and snap-back loads at different adjustive angles are obtained by experiments and the finite element method, while the adjustive angle for the maximum critical load is captured. Moreover, the effect of the initial radius on the adjustive angle is provided, and the load displacement after adjustment at different radii and cross-section angles are presented.

Nonlocal nonlinear static/dynamic snap-through buckling and vibration of thermally post-buckled imperfect functionally graded circular nanoplates

The bi-stability characteristics of the post-buckled plates have demonstrated extensive potential applications for energy harvesting and vibration isolation. In this article, at first, thermal post-buckling, nonlinear bending, and vibration behavior of the functionally graded (FG) circular nanoplates in bifurcation buckling are investigated according to nonlocal elasticity theory. Then, the nonlinear static/dynamic snapping phenomena and free vibration responses of the thermally post-buckled nanoplate subjected to static and sudden types of mechanical load are presented. For this aim, the equations of motion in conjunction with the von-Kármán nonlinearity and geometrical imperfection are established in the framework of Hamilton's principle. In addition, two distinct cases of temperature distribution as well as two types of edge conditions are taken into consideration. The equations of motion are discretized using the Chebyshev-Ritz procedure along with three different numerical algorithms. To evaluate the nonlinear dynamic snap-through buckling, the Newmark time integration scheme is adopted. Next, by means of the Budiansky-Roth criterion and the phase-plane approach, the dynamic snap-through loads are identified. A set of parametric studies is presented to provide an insight into influences of the nonlocal parameter, geometrical imperfection, gradient index, and edge supports on the static and dynamic characteristics of the nanosystem.

Magnetic actuation of switchable bistable structures: a numerical study

Bistable laminates have posed a significant research interest in the recent past owing to their potential applications in morphing and energy harvesting. An important aspect of research has been their snapthrough analysis, wherein the use of solid state actuators like shape memory alloys (SMAs) and macro-fiber composites (MFCs) has been prevalent. These actuators however, interfere with the structural response of the laminate altering its bistability and snapthrough characteristics. Recently, reversible snapthrough was demonstrated in 3D printed thermoplastic filaments composed of PLA with finely ground iron particles embedded. Remote actuation was achieved by means of permanent magnets providing the necessary snapthrough loads. In this manuscript, a numerical model has been developed based on a Rayleigh-Ritz minimization scheme, to predict the equilibrium states and the snapthrough loads. The means of snapthrough is the interaction between the ferromagnetic layers and the nonlinear magnetic field generated by a solenoid, which has been modeled using Biot-Savart’s law. The developed numerical model has been validated against finite element simulations in ABAQUS®, wherein the magnetoelastic interactions have been modeled as body forces using the DLOAD subroutine. This manuscript details one of the preliminary efforts in modeling of the remote snapthrough of switchable multistable structures and can provide valuable insights into the design of such non-contact actuation systems.

Snap-through of bistable variable stiffness laminates using MFC actuators

Unsymmetric composite laminates exhibit two or more stable equilibrium shapes with opposite curvature as a result of residual thermal stresses induced during the curing process. Surface bonded Macro Fibre Composite (MFC) actuators are generally employed to trigger snap-through between one stable shape to another by applying external voltages. However, these actuators require high actuation voltage input to change from one stable shape to another, especially when used in morphing applications. If the snap-through voltage is too high, morphing becomes infeasible or may require several actuators, which is contrary to the weight requirements. Variable Stiffness (VS) laminates provide the possibility for an enlarged design space with the possibility to tailor stiffness parameters, leading to lower snap-through loads and consequently reducing the size of the actuators.On the prediction of snap-through voltages, the existing semi-analytical models have shown reasonably high discrepancies between numerical and experimental results. An improved analytical model is proposed in this study to predict the snap-through of bistable VS laminates with MFC actuators. In the improved model, the equations resulting from the compatibility and the in-plane equilibrium are described equivalent to a standard plane elasticity problem which can be solved using a standard finite element (FE) approach. In addition, the total potential energy is written in terms of curvatures. The improved semi-analytical results are compared with a full geometrically nonlinear FE calculation.

A novel computational framework for the analysis of bistable composite beam structures

Over the last few years, several studies have shown the potential of multi-stability in engineering applications such as shape-adaptive structures and energy harvesting devices. Due to their highly non-linear mechanics, multi-stable structures are inherently more complex to model than conventional engineering structures. Especially when composite materials are involved, two-dimensional or three-dimensional finite elements solutions are often required in order to have a thorough understanding of the stress field evolution within the structure throughout the snap-through phenomenon. Nevertheless, such modelling strategies yield time-consuming design and optimisation procedures. In order to exploit multi-stability in industrial applications, an enhancement of the existing modelling capabilities is needed. Starting from the state-of-the-art methodologies available in the literature, a novel computational framework is proposed in this study for the analysis of bistable composite beam structures. An advanced structural modelling approach based on Carrera’s Unified Formulation (CUF) is adopted. By harnessing the potential of CUF, a full in-plane stress field (including axial, transverse shear and transverse normal stress) can be accurately yet efficiently predicted with no need to resort to two-dimensional or three-dimensional FEM solutions. Furthermore, the equilibrium paths of slender as well as thicker bistable beams are investigated, showing that refined predictions of the stable geometries and snap-through loads in bistability analyses can be obtained via the proposed advanced one-dimensional formulation. Validation of the proposed modelling approach towards reference solutions and commercial software finite elements is provided.

An efficient semi-analytical framework to tailor snap-through loads in bistable variable stiffness laminates

Multistable laminates are potential candidates for adaptive structures due to the existence of multiple stable states. Commonly, such bistable shapes are generated from the cool-down process of the unsymmetric laminates from the curing temperature. In this work, we exploit unsymmetric variable stiffness laminates with curvilinear fiber paths to generate similar bistable shapes as unsymmetric cross-ply laminates, but with the possibility to tailor the snap-through loads. Snap-through is a complex phenomenon in that is difficult to characterize using simple analytical models. An accurate yet computationally efficient semi-analytical model is proposed to compute the snap-through forces of bistable variable stiffness (VS) laminates. The differential equations resulting from the compatibility and the in-plane equilibrium equations are solved with negligible numerical error using the Differential Quadrature Method (DQM). As a result, the in-plane stress resultants and the total potential energy is written in terms of curvatures. The out-of-plane displacements are expressed in the form of Legendre polynomials where the unknown coefficients of the displacement function are found using the Rayleigh-Ritz formulation. The calculated snap-through loads are then compared with the Finite Element (FE) results. A parametric study is conducted to explore the tailoring capabilities of VS laminates for snap-through loads.

Changing the critical snap-through loads of post-buckled beams using piezoelectric actuation

Post-buckled and curved structures experience snap-through instabilities when external loads from mechanical, fluid, or thermal environments result in a loss of local stability and a dramatic jump to a remote stable equilibrium. Fatigue caused by snap-through is a concern in many engineered systems because of the large stress reversals involved. This paper studies the extent to which the strategic placement and actuation of piezoelectric materials bonded to clamped-clamped post-buckled beams can influence the loads at which snap-through occurs. The electromechanical system is modeled using elastica theory with extensions to account for the influence of piezoelectric actuation on the structure. Static equilibrium positions and their stability are computed across a large configuration space using numerical integration and a shooting method. The results indicate that the effect of piezoelectric actuation on critical snap-through load depends on the degree to which the beam is buckled, the location of the external load, the placement of the piezoelectric material, and the applied actuation voltage. Experiments are performed to validate the numerical results and provide a physical demonstration of changing snap-through loads with piezoelectric actuation. Experimental results demonstrate that critical snap-through loads can be altered by factors ranging from 0.4 to 2.0, and numerical results indicate that even larger changes to snap-through loads are physically realizable.

Static and dynamic symmetric snap-through of non-uniform shallow arch under a pair of end moments considering critical slowing-down effect

This paper presents analytical analysis of static and dynamic symmetric snap-through of non-uniform shallow circular shallow arch under a pair of end moments with the same magnitude. The non-uniformity is characterized by dividing the arch into three piecewise constant-stiffness segments. Hamilton's principle is used to derive the governing differential equations by assuming negligible axial inertia. The snap-through load and snap-through criterion are analyzed in detail and by an index plot, the stiffer center case is shown to behave distinctly when stiffer end case is compared. The dynamic snap-through when the moment is slightly higher than the snap-through moment is analyzed analytically by a perturbation method, and a critical slowing effect is observed when the moment is approaching to the snap-through moment. Comparison with dynamic FEA shows a good agreement with the analytical result and analysis on theoretical finite time blow-up phenomenon reveals that when geometric parameters are corresponding to the critical snap-through condition, the initial quadratic phase's motion is slow and relatively blow-up phase's motion is fast. The analytical formulations have been extended to include two limiting cases including rigid end case and rigid center case by using the constrained Hamilton's principle by Lagrangian multipliers. Snap-through criteria analysis reveals closed-form criterion for rigid center case and an asymptotic result for rigid end case. This paper serves to enhance the knowledge on snap-through and critical slowing down for shallow arches with non-uniformity under end moments.

Bi-stable hybrid composite laminates containing metallic strips: an experimental and numerical investigation

This study scrutinizes the static characteristics of a new category of bi-stable hybrid composite laminates (BHCLs) that contain multiple metallic strips (aluminum) distributed along the middle layer of a carbon fiber reinforced epoxy laminate. With this new class of BHCLs, the direction of curvature of the laminates does not change during the snap-through between the two stable states, unlike the conventional bi-stable cross-ply laminates. The laminates were modeled using finite element analysis and were experimentally validated. BHCLs with two to five metallic strips were fabricated for this purpose. The effect of the number, width, and thickness of the metallic strips on the static characteristics of the laminate were numerically investigated using ABAQUS. Further, the curvatures, out-of-plane displacement, and static snap-through load of the laminates were determined experimentally. The results showed a strong relationship between the residual curvature and the load-carrying capability of the laminate and the number, width, and thickness of the metallic strips, as well as the laminate geometry. Knowledge of this characteristic will allow designers to tune the parameters for a given application and to achieve the desired performance. Good qualitative and quantitative agreement was observed between the numerical and the experimental results.

Characterization of bi-stable pure and hybrid composite laminates – An experimental investigation of the static and dynamic responses

In this study, the static and dynamic responses of bi-stable hybrid composite laminates [0/AL/90]T and [02/AL/902]T were scrutinized, and their behavior was compared to bi-stable pure composite laminates including [0/90]T and [02/902]T. The work consisted of an analytical study that was validated experimentally. An analytical method based on Hamilton’s principle was developed to investigate the static and vibration characteristics of the laminates. Experimentally, curvatures and out-of plane-displacement, and snap-through load were measured using a quasi-static loading on a universal testing machine. Further experimental analysis was performed to characterize the damping viscous ratio, natural frequency, and critical base excitation that cause the snapping between the two different stable shapes. The results show that the hybridization of bi-stable pure composite laminates has the potential to increase the stable curvatures and enhance the static load-carrying capability up to five times when compared to a pure bi-stable composite laminate of the same thickness. It was also observed that the hybridization of bi-stable pure composite laminates may result in a dramatic change in the dynamic response. The natural frequencies of bi-stable hybrid composite laminates are increased in comparison with bi-stable pure composite laminates. The critical base excitation required for snapping has increased significantly for the hybrid composite laminate. The qualitative and quantitative comparisons between the analytical and experimental results were very promising and they agreed well.

Design of a multistable composite laminate by variable cross-section method and applying the displacement constraint

This paper presents a multistable variable cross-section laminate (MVCL), which is designed by applying the displacement constraint at the two opposing ends of the laminate. A theoretical model describing the MVCL as rigid bars connected with compliant hinges is built to predict the equilibrium configurations and snap-through loads of the MVCL. The snap-through process of the multistable structure is investigated via experiment and finite element analysis (FEA). Subsequently, the theoretical model is improved by simplifying the middle segment of the MVCL as an elastic beam. Good agreement is obtained between the experimental test, FEA, and improved theoretical model results. It is shown that the current approach can be successfully applied to predict the equilibrium configurations and snap-through loads of an MVCL. The snap-through loads can be adjusted by changing the middle segment length of an MVCL.

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.

Vibration analysis of bi-stable composite cross-ply laminates using refined shape functions

In this article, static and dynamic responses of cross-ply bi-stable composite plates were studied. To accurately predict the natural frequencies and snap-through load, a set of higher order shape functions were proposed. In static analysis, the stable configurations, the deflection of corners, and the midpoint of the plate were calculated. For dynamic analysis, Hamilton’s principle is used to provide approximate solutions to the vibration problem under study. The responses of the plate under ramp and harmonic applied forces were determined, the effect of shape functions on the prediction of the first natural frequency of the plate and the required force for snap-through were investigated. A finite element model is also developed to study the static and vibration characteristics of bi-stable composite plate. The qualitative and quantitative comparisons between the finite element method results and those obtained from the present analysis are generally good and satisfactory. The developed analytical model can also be used for parametric study and further design modification.