Geometric Nonlinear Effects Research Articles

On the time-dependent mechanics of membranes via the nonlinear finite element method

In this work, the problem of finite generalized and viscoelastic deformation of thin membranes with different geometries, made of incompressible hyperelastic materials, is formulated. The multiplicative decomposition of the deformation gradient tensor into elastic and viscous parts, and making use of dissipation inequality, nonlinear evolution equations for the internal variables of the models are obtained. The mechanical behavior of the dampers is assumed to be linearly viscous. Therefore, the Cauchy-like stress in the dampers is similar to that in Newtonian fluids and includes terms for the hydrostatic pressure and viscosity. The implicit and second-order accurate trapezoidal method is employed for the time integration of the evolution equations. Due to the highly nonlinear governing differential equations including the effects of geometric nonlinearity and viscoelasticity, a nonlinear finite element formulation based on isoparametric elements is developed. The accuracy and performance of the developed formulation and time-dependent solutions are verified by studying several numerical examples. The obtained results are compared with theoretical and experimental data available in the literature. The proposed formulation can appropriately predict the experimental results of viscoelastic membranes for both in-plane and out-of-plane deformations.

Assessment of interfacial energy effect on finite fracture behavior of composite adhesively bonded joints under four-point bending

Due to their remarkable advantages, composite adhesively bonded joints are intensively used in many engineering fields. There are many investigations performed by researchers to study the fracture behavior of the adhesively bonded joints. However, there exist discrepancies between the numerical predictions and the experimental results in some cases. This paper reports an investigation on the finite fracture behavior of composite adhesively bonded joints subjected to four-point bending. Experiments are performed for composite adhesively bonded joints with four different configurations. The fracture behavior is observed experimentally by utilizing a high-speed camera. The failure analysis employing the energy-stress coupled criterion is performed. The experimental results are compared with the numerical predictions by using an experimental-based approach where both geometric nonlinearity and interfacial energy are considered. Numerical illustrations are carried out to assess the effect of interface energy, geometric nonlinearity, joint type and dimensions on the fracture behavior of the considered bonded joints.

A finite element study on the effects of thickness and material nonlinearity on the equilibrium path of axially loaded circular cylindrical shells

Research has been carried out in recent years dealing with Lighter than Air Structures (LTAS), in which the innermost volume is taken to be a vacuum. Variations of a thin membrane over a frame were considered in order for such structures to have buoyancy potential while being capable of sustaining atmospheric pressures. The need for extremely thin membranes became evident as a result of the weight constraint The potential for the development of a viscoplastic stress field and instability led to the study of a classical problem: the instability of circular cylindrical shells under compressive loading. Thus, in order to obtain information related to the nonlinear geometric and material effects that can eventually be used in the LTAS, this study focused on what increasingly smaller membrane thicknesses and changes in the modulus can have on the equilibrium path of circular cylindrical shells. Considering shell thicknesses from 1E−3 to 1E−6m and using the Johnson-Cook constitutive relationship, results show that (1) the equilibrium path is highly dependent on shell thickness, and (2) inelastic strains drive path deviations and distinct post-collapse shapes. Furthermore, a reduction in the yield stress can result in plasticity before collapse, effectively changing the path of equilibrium and the post-collapse shape. Solutions were found using SIMULIA® Abaqus finite element (FE) software. At the present time these membrane thicknesses and material properties are not available. It is felt that, with the advancement of material science, they will become available in the immediate future.

Effects of geometric nonlinearities on the vibration reduction performance of tuned mass dampers on top of flexible structures

A tuned mass damper (TMD) on top flexible structures may sustain considerable rotation because of the significant flexural deformation of the supporting structure. The effects of such rotation on the vibration reduction performance are generally neglected in the analysis and design of TMD-controlled systems, by assuming that the mass moves only laterally in a fixed global coordinate. This technical note critically reviews this assumption by comparing the dynamic responses of a lumped-mass Euler beam with a TMD on top in either fixed or corotational coordinates to harmonic ground excitations. The results show that the corotational local coordinate of TMDs, which introduces geometric nonlinearity into the system, tends to bias the peaks of the frequency response curve toward lower excitation frequency, but such an effect is less profound than that induced by the P- Δ effect, another source of geometric nonlinearity in the system. The relative error in the peak frequency response of neglecting this effect increases for larger lateral drift of the supporting structure and smaller mass ratio of the TMD. An empirical equation for estimating this error is proposed to assist the decision-making of whether this assumption is appropriate for specific applications of TMDs.

A finite element based parametric study to assess the suitability of factors used in EJMA for estimation of the meridional stresses in bellows

Bellows are highly engineered components. In general, the design of bellows is carried out as per the standards of the expansion joints manufacturer’s association (EJMA). In EJMA, the basic form of the expressions for estimation of stresses was derived by considering the half convolution of the bellows as a strip beam. Since axisymmetric shell theory leads to a more realistic response of the bellows, EJMA had introduced factors Cp, Cd and Cf for estimation of the meridional stress in the bellows due to pressure and deflection. These factors in EJMA are plotted as a function of two parameters viz. QW and QDT, which are dependent on bellows geometry and ply thickness. This paper discusses the suitability of the above-mentioned factors considering various configurations of the bellows. A finite element based parametric study was carried out and meridional stresses in the bellows were calculated due to pressure and deflection loading for various values of the QW and QDT. Around 36 bellows geometries with different ranges of QW and QDT were analysed numerically. Based on FE analysis, the EJMA factors Cp, Cd and Cf values were estimated and compared with the corresponding values given in EJMA. It was observed that for both shallow (QW approaching 1) and deeper convolutions (QW approaching 0.2), the EJMA solutions lead to an under prediction of membrane stress due to deflection loading and over-prediction of meridional bending stress due to deflection and pressure loading. The effect of geometric nonlinearities on elastically calculated stresses bellows was quantified for various geometries.

Nonlinear dynamic analysis of wrinkled membrane structure

PurposeThis paper studies the nonlinear dynamics of membrane structure considering wrinkling effect. The coupling between wrinkles and vibration is investigated elaborately, and new insight on the dynamics of wrinkled membrane is unveiled.Design/methodology/approachBased on the stability theory of plates and shells, the wrinkling model of the membrane structure is established. Considering the effects of wrinkling and nonlinearity, the dynamic response is calculated with NewMark method.FindingsWrinkling will impact the dynamics of the membrane structure significantly for asymmetrical tension loading cases, dynamic response of the wrinkled membrane structure can be classified into three categories: when the vibration is small, the dynamics of the wrinkled membrane structure will behave linearly, and the wrinkles will only affect the dynamic properties as initial conditions; when the vibration is relatively large, the wrinkles will interact with the vibration during the dynamic process, and the dynamics of the structure shows very complex features; when the vibration is large enough, the dynamics will be dominated by the geometric nonlinearity of large-amplitude vibration.Originality/valueIn the previous works on dynamics of wrinkled membrane structure, only the vibration modes have been studied, which means all those investigations are confined with linear vibration; little research has been conducted on the nonlinear dynamics of wrinkled membrane structure. In view of this, this paper presents an investigation of dynamic properties of membrane structure considering the wrinkling and geometric nonlinear effects. This research work presents some novel discoveries on the nonlinear dynamics of wrinkled membrane.

Numerical modeling and nonlinear analysis of semi-rigid jointed steel frames

This paper investigates the effect of semi-rigid connection, geometric nonlinearity and material nonlinearity in steel frame analysis. To study the independent and combined effects of different nonlinearities, elastic and inelastic analyses are carried out using general purpose finite element software SAP2000. Connections are modeled as rotational springs and frame as one dimensional beam element. Verification and application of the simplified numerical model developed in the present study is demonstrated by considering different examples and comparing the results of present study with those available in the published literature. Numerical alternatives, key features about modeling and necessary input parameters are discussed in detail. Rotational springs are characterized by linear, bilinear, multi-linear or nonlinear moment-rotation relationships of the connection. Results in terms of beam moment and nodal displacement are presented for both elastic and inelastic analyses. With the increase in connection flexibility, beam mid-span moments and displacements increase, but beam-end moments decrease. It is observed that the influence of connection nonlinearity dominates over material and geometric nonlinearities.

Analytical analysis of nonlinear internal resonance bandgap of pendulum-type metamaterial

In this paper, a pendulum-type metamaterial (PTM) is designed with a pendulum bob hinged at the primary mass. Considering the effect of geometric nonlinearity, the nonlinear dynamic equations of PTM unit cell are presented with the aid of the Bloch theorem. The analytical formula of dispersion equation is deduced to discuss the nonlinear internal resonance bandgap using the harmonic balance method. The obtained bandgap of the metamaterial is in good agreement with the numerical simulation result. The nonlinear geometric influence of pendulum on PTM bandwidth is significant. The bandgaps under different internal resonances are substantially different from each other due to the nonlinear effects. The upper boundaries of the frequency bandgap under 1:1/2 and 1:1/3 internal resonance rise nonlinearly to higher than those under linear and 1:1 internal resonance conditions. It shows that the impact of 1:1/2 and 1:1/3 internal resonance on the bandgap could be more obvious as the strong nonlinearity is taken into consideration.

Design of periodic arched structures integrating the structural nonlinearity and band gap effect for vibration isolation

In this study, a Roman-bridge-inspired metamaterial for vibration isolation is proposed that simultaneously realizes a geometrical nonlinear structure and band gap effect, through the arrangement of periodic materials in a semicircular arched structure. Finite element analysis and band diagram analysis are performed to verify the effect of the periodic arch-structured metamaterial by calculating the transmissibility and band gap frequency range of cuboid, arched and periodic arched models. Based on the calculation results, the relationship between the geometrical parameters of the periodic arch structure and band gap frequency are analyzed, and an empirical correlation is formulated for the generation of a training database for a machine learning optimization model. The geometrical parameters of the metamaterial were optimized using the machine learning model based on an autoencoder architecture with supervised learning. An optimized metamaterial with a band gap area at three target frequencies is generated by the trained neural network, and the effectiveness of the optimized structure is validated by both numerical analysis and a frequency response test.

Prey localization in spider orb webs using modal vibration analysis

Spider webs are finely tuned multifunctional structures, widely studied for their prey capture functionalities such as impact strength and stickiness. However, they are also sophisticated sensing tools that enable the spider to precisely determine the location of impact and capture the prey before it escapes. In this paper, we suggest a new mechanism for this detection process, based on potential modal analysis capabilities of the spider, using its legs as distinct distributed point sensors. To do this, we consider a numerical model of the web structure, including asymmetry in the design, prestress, and geometrical nonlinearity effects. We show how vibration signals deriving from impacts can be decomposed into web eigenmode components, through which the spider can efficiently trace the source location. Based on this numerical analysis, we discuss the role of the web structure, asymmetry, and prestress in the imaging mechanism, confirming the role of the latter in tuning the web response to achieve an efficient prey detection instrument. The results can be relevant for efficient distributed impact sensing applications.

Nonlinear thermo-elastic stability of variable stiffness curvilinear fibres based layered composite beams by shear deformable trigonometric beam model coupled with modified constitutive equations

Nonlinear thermo-elastic buckling characteristics of composite variable stiffness beam with layers making use of curvilinear fibres under thermal environment is attempted here. The model is based on a shear deformable theory introducing trigonometric function, and considering von Kármán’s assumptions based geometrical nonlinear effect. The beam constitutive equation is modified according to the stress-free situation in the width direction of beam-Poisson’s effect in the formulation for predicting the behaviour of general lay-up composite beams. By the principle of minimum total potential energy, the governing equations in terms of incremental stiffness matrices are formed introducing general beam finite element. The global equilibrium equations formulated are solved for envisaging the post-buckling path through eigenvalue analysis iteratively, thus establishing the relationship of thermal temperature against moderate amplitude level of beam deflection. A systematic parametric analysis considering different lamina properties such as curvilinear fibre path angles and including lay-up sequences, thermal expansion coefficient, mixed laminate combining straight and curvilinear fibres-based layers is carried out on thermo-structural stability of curvilinear fibre-based beams. Also, the influence of geometric factors, flexible beam end support, and variation in thermal profile, etc. over the stability behaviour of beam is examined.

Efficient dynamics modeling and analysis for probabilistic uncertain beam systems with geometric nonlinearity and thermal coupling effect

Efficient dynamics modeling and analysis for probabilistic uncertain beam systems with geometric nonlinearity and thermal coupling effect

Frequency domain analysis of pre-stressed elastomeric vibration isolators

Two types of elastomeric vibration isolators used for equipment vibration isolation in aerospace vehicles are considered for the present study. These isolators are constructed using elastomers mounted in steel encasings. These isolators are initially deformed statically and dynamic loads are applied on the deformed configuration. To capture the static deformation, equivalent static load corresponding to its load rating and specified displacements are created. Static deformation is computed using Finite Element methods with four node axi-symmetric element which include the geometric non-linear effect for steel and with standard Yeoh hyper-elastic material model for elastomers (Muhammed and Zu, 2012) [1]. Yeoh material constants are derived from uni-axial tension test data of the elastomer specimen. These isolators are subjected to harmonic and random excitations in the pre-deformed state. For numerical analysis, elastomeric constants at dynamic conditions are obtained as complex function of frequency using Dynamic Mechanical Analyzer (DMA) for a range of frequencies. The standard material model of Yeoh is modified incorporating frequency dependant material characteristics and damping in the range of frequencies of interest. A multiplicative non-separable variables law is derived for Yeoh material model to include the effect of static pre-stress, based on the methodology given in literature (Nashif et al., 1985; Beda et al., 2014) [2,3]. The modifications of Yeoh model suitable for frequency domain analysis is the novelty in the present study. In the analysis, while dynamic loads are applied, the configuration is updated considering initial static loading. The frequency response of the isolators is computed using material properties evaluated at progressive dynamic strains until a match in natural frequency is observed. Appropriate damping corrections are then incorporated to match the test observed transmissibility. Then updated material properties are used to compute the random response which showed good agreement with results of experiments, validating the approach taken for the development of this model.

An energy-based computational scheme for the analysis of reinforced concrete structures with geometric and material nonlinearities

ABSTRACT This paper proposes a new energy-based numerical technique for the nonlinear analysis of reinforced concrete beam elements. The mathematical concept and equations are presented in detail for a general nonlinear quasi-static problem. This method can create a physical sense of the types of energy in a system for an analyst to easily consider any material type and geometric nonlinearity effects from the analysis of reinforced concrete members. The governing equations of this computational scheme have a quadratic form, which leads to proving a complementary relationship for a stability check. In other words, a distinguishing feature of the proposed method is paving the path for selecting the optimum size of displacement increments during a nonlinear numerical analysis. In the proposed method, the steps have been made easier for analysts. Unlike several existing methods, the formulation has also considered the shear and normal deformations, making the analysis results closer to experimental realities. Moreover, the results obtained from solving, evaluating, and comparing several examples demonstrate the acceptable accuracy and convergence of this energy-based approach for reinforced concrete beams. In addition, the quadratic nature of its formulation provides an analyst with information about the accuracy and stability of the obtained structural responses.

Geometrically nonlinear model of piezoelectric wind energy harvesters based on vortex-induced vibration and galloping

The interaction between vortex-induced vibration (VIV) and galloping could enhance the performance of wind energy harvesters (WEHs). Though VIV-galloping interaction may cause large amplitude wind-induced vibrations, the effects of geometrical nonlinearity were not considered in the modeling of VIV-galloping interactive piezoelectric WEHs (PWEHs). In this work, based on the extended Hamilton’s principle, a geometrically nonlinear model (GNM) of cantilevered PWEHs with VIV-galloping interaction was derived. The model includes both the transverse and axial aerodynamic forces, and considers the effect of the rotation of the bluff body on the aerodynamic forces. The aerodynamic coefficients were extracted by a piecewise polynomial fitting in a relatively large range of angle of attack for the square cross-sectional bluff body. Two flexible PWEH prototypes were fabricated and tested in a small wind tunnel to verify the proposed model. After the mechanical damping ratio of the low-coupling piezoelectric energy harvester prototypes were identified based on purely electrical measurements, the steady-state root mean square voltages of the prototypes with increasing wind speed were worked out using geometrically linear model (GLM) and the proposed GNM, respectively, and then compared with experiments. Both models can accurately predict the VIV-galloping interaction, but GNM is much more accurate than GLM at a relatively high wind speed. The proposed GNM provides a powerful tool to develop VIV-galloping interactive PWEHs.

Geometric nonlinear transient analysis of mechanically and thermally shocked functionally graded shell panels

The present paper investigates the von-Karman geometrically nonlinear structural responses of ceramic-metal functionally graded shell panels under the effect of the cases of mechanical and thermal shocks on their ceramic-rich surface. The constituents in the FGM are temperature-dependent and are graded as per the recently proposed four-parameter power laws while their effective properties in the shells are obtained using the Mori-Tanaka homogenization scheme. The transient thermoelastic problem is solved by importing the temporally varying temperature distribution from the nonlinear 1D heat equation as thermal loads into the FE code developed for thin and moderately thick shells based on higher-order shear deformation theory. The nonlinearity in the heat equation is linearized using Picard method in each Crank-Nicholson time-step while the linearization step in the geometric nonlinear problem uses Newton-Raphson method in each Newmark time-step. The new results presented in the manuscript study the effect of geometric nonlinearity on the responses of FG shell panels and the effect of different parameters of four-parameter FG laws. It was observed that including geometric nonlinearity significantly affects the responses of the shell panel structures when the loading direction, either mechanical or thermal, along with the gradation direction is reversed.

Nonlinear steady state vibrations of beams made of the fractional Zener material using an exponential version of the harmonic balance method

This paper presents the application of an exponential version of the harmonic balance method to the analysis of steady state vibration of geometrically nonlinear systems. A detailed description of the method and of the corresponding numerical procedure is provided. The von Karman theory is used to describe the effects of geometric nonlinearity. The material of the beams is modelled with the help of the Zener model using the fractional calculus. The problem is solved using an exponential version of the harmonic balance method. In the above-mentioned version, the complex calculus is used in contrast to the ordinary harmonic balance method, where the steady state vibrations are described with the help of the trigonometric functions. It significantly simplifies derivation of the amplitude equations. Moreover, the exponential version of the harmonic balance method is more elegant in comparison with the ordinary one. A detailed derivation of the amplitude equations is presented. The modified continuation method is proposed to solve the nonlinear amplitude equations and to determine the response curves. Moreover, the results of the exemplary calculation are presented and compared with known results in order to justify the efficiency and the correctness of the proposed approach.

Theoretical and experimental investigations on steady-state responses of rotor-blade systems with varying rotating speeds based on a new nonlinear dynamic model

Rotor-blade systems are major units in turbine structures, whose dynamic responses will affect structural safety, energy conversion efficiency, overall durability and service performance. During high-speed rotation, periodic disturbances near resonant regions may lead to a large deflection of flexible blades and an unstable rotation of rotor-blade systems. Due to the coupling effect of rotation and vibration, the distributions of mass/inertia, damping and stiffness are related with varying rotation speed and local deformation in rotor-blade systems, which has not been concerned in conventional harmonic balance methods. In this paper, a modified harmonic balance method is proposed to obtain dynamic responses of rotor-blade systems with a comprehensive model in incremental form, which considers geometric nonlinearity, centrifugal and gyroscopic effects of flexible blades and nonlinear constraints between rotor and multiple blades. Based on Reddy’s high order shear deformation plate theory, dynamic equations for a rotating composite blade with a pre-setting angle is established with consideration of von Kármán type geometrical nonlinearity, varying rotating speeds and equivalent aerodynamic loads. A coupling motion equation for rotor-blade systems is derived out by assembling corresponding elemental equilibrium and joint nonlinear constraints. Steady-state responses are obtained by using a modified harmonic balance method which accounts nonlinearities of mass, damping and stiffness. Numerical and experimental studies for rotor-blade systems are carried out to validate the effectiveness of the proposed model. Results show that a higher rotating speed leads to larger natural frequencies of rotor-blade systems, and a more evidently coupling effect between motion and vibration. Meanwhile, joint constraints, damping effect and load distributions have evident influences on vibration amplitudes and resonant points of system’s steady-state responses.

On the Effect of Geometric Nonlinearity on the Dynamics of Flared Folding Wingtips

Recent studies have considered the use of flared folding wingtips (FFWTs) to enable higher aspect ratios—reducing overall induced drag—while reducing gust loading and meeting airport operational requirements. The majority of these analyses have employed linear assumptions despite the presence of large wingtip deformations. In this paper the effect of geometric nonlinearities introduced by an FFWT on the static and dynamic aeroelastic response of a wing is assessed. A geometrically exact expression was formulated to describe the change in the local angle of attack in the chordwise direction across all fold angles. This expression highlighted that the aerodynamic stiffness of an FFWT and, therefore, quantities such as the linear flutter speed are a function of the fold angle and, hence, the attitude of the wing. This effect was verified using both a wind tunnel model of a flexible semispan wing and a numerical model utilizing MSC Nastran, which linearized the model about the equilibrium position of the wingtip. These experiments showed that the geometric nonlinearities introduced due to the large deformations of FFWTs can significantly affect the dynamics of the system, with flutter speeds varying by over 28%, simply by changing the angle of attack of the model.

Refined Beam Theory for Geometrically Nonlinear Pre-Twisted Structures

This paper proposes a novel fully nonlinear refined beam element for pre-twisted structures undergoing large deformation and finite untwisting. The present model is constructed in the twisted basis to account for the effects of geometrical nonlinearity and initial twist. Cross-sectional deformation is allowed by introducing Lagrange polynomials in the framework of a Carrera unified formulation. The principle of virtual work is applied to obtain the Green–Lagrange strain tensor and second Piola–Kirchhoff stress tensor. In the nonlinear governing formulation, expressions are given for secant and tangent matrices with linear, nonlinear, and geometrically stiffening contributions. The developed beam model could detect the coupled axial, torsional, and flexure deformations, as well as the local deformations around the point of application of the force. The maximum difference between the present deformation results and those of shell/solid finite element simulations is 6%. Compared to traditional beam theories and finite element models, the proposed method significantly reduces the computational complexity and cost by implementing constant beam elements in the twisted basis.