Geometric Nonlinear Effects Research Articles

Heuristic molecular modelling of quasi-isotropic auxetic metamaterials under large deformations

A two-dimensional molecular model is presented for the elastic non-linear modelling and design of meta-materials. The fundamental unit-cell, based on a heuristic molecule (HM) approach, is composed of atoms that interact through centred and non-centred spring-based bonds. The kinematics formulation allows to consider large displacements and finite strains while the specific topology of the HM can be parametrized to modify the shape of the rigid atoms and the size of the bonds. The HM is frame indifferent and provides a remarkable quasi-isotropic elastic response for both deviatoric and volumetric large deformation modes. At a macro-scale, the relationship with different continuum materials is given through a standard isotropic Cauchy up to an isotropic Cosserat solid. Evidence on the interest of the model as a calculation tool is provided by studying the elastic response of standard and auxetic materials subjected to a non-homogeneous deformation field, as well as the response of auxetic foams under large deformations. Aside the numerical agreement, it is highlighted how the tailoring of the HM topology can be effective to approximate the non-linear geometric effects that occur at finer scales of auxetic foams. In perspective, we address how the exotic mechanical properties provided by the HM, together with the assumed physical-driven framework, can foster the engineering application and the design of new meta-materials.

Viscoelastic Soil–Structure Interaction Procedure for Building on Footing Foundations Considering Consolidation Settlements

This paper presents a numerical methodology to analyze frame structures supported on footing foundations subjected to slow strains caused by consolidation settlements. A building project on a subsurface layer of soft soil has been analyzed. The Boundary Element Method with the Mindlin fundamental solution has been applied to compute the displacement resulting from the interference between pressure bulbs on the foundation. The rheological Kelvin–Voigt model has also been used for soil–structure interactions. Terzaghi’s Theory of Consolidation was used to fit the displacement–time curves. Finally, the rheological model was coupled through an iterative procedure, employing structural non-linear geometric effects. The results are consistent with settlement predicted effects and revealed that the slow distribution of efforts can cause relevant increases in some regions in the structure of the building.

Nanoscale flow-induced nonlinear vibration of multilayer graphene based-resonators: Slip and transition flow regimes

This paper presents a nanoscale fluid-structure interaction (FSI) model for flow-induced nonlinear vibration of multilayer graphene based-resonators in the slip and transition regimes. Based on the multibeam interlayer shear model with von Kármán geometric nonlinearity and nonlocal modified couple stress theory, the size-dependent equations of motion are obtained for beam resonators laminated of NG-graphene layer and (NG−1) interlayer crosslinks. The small-scale effect on effective viscosity and slip boundary conditions of nanoscale flow is formulated by introducing the average velocity correction factor. For unbounded and bounded (by a rigid wall) nanoflows, the closed-form expressions of velocity correction factor in terms of Knudsen number, as a dimensionless discriminant parameter, are developed to correct the pressure distribution predicted by the potential theory. The present FSI model considers the size effects through the length scale parameter for microstructure local rotation, the nonlocal parameter for long-range interatomic interactions, and Knudsen number for slip condition of nanoscale flow. The Galerkin's method in conjunction with the multiple-scale perturbation technique is used to obtain closed-form solutions of frequency ratio and time-history response. The natural frequencies of cantilever multilayer graphene nanostrips are validated with the ones existing in the literature. Parametric studies are carried out to investigate the influences of small-scale, number of layers, interlayer shear modulus, and fluid characteristics on the linear and nonlinear natural frequencies. Findings show that the critical upstream speed for dynamic instability of the coupled system can change considerably by taking into account the size effects of nanoflow and nanostructures. Therefore, the classical FSI model is inadequate for analyzing nanoscale flow-induced vibration of nanoresonators. Also, it is seen that the effect of geometric nonlinearity becomes very significant for multilayer graphene beams with small interlayer shear modulus.

Influence of Single External Store on Geometrical Nonlinear Flutter of High-aspect-ratio Wings

High-altitude and long-endurance UAVs, distributed electric propulsion aircraft and large passenger aircraft often use high-aspect-ratio wings, which have obvious geometric nonlinear large deformation effect, and are usually arranged with a certain number of external stores, which have an impact on their geometric nonlinear aeroelastic characteristics. Based on the “quasi-linear” hypothesis, the updated Lagrange incremental finite element method and doublet-lattice method are used to perform the geometrically nonlinear static aeroelastic analysis, and the mass, stiffness and aerodynamic matrices of large deformation are updated, and the geometrically nonlinear flutter characteristics of high-aspect-ratio wings are calculated. The influence of the position and mass of the external store on geometrical nonlinear flutter characteristics of high-aspect-ratio wing structure is investigated. Results show that the change of the position and mass of the external store causes the relative position between the elastic axis and the center of gravity of the wing, and the wing’s large deformation equilibrium state to change, resulting in differences in the stiffness characteristics of the wing and the natural vibration frequency of the equilibrium state, thus affecting the wing flutter characteristics. When a single external store is installed near the wing root, changing the mass and direction of the store has no significant effect on the geometric nonlinear flutter characteristics of the wing. The improvement of wing flutter speed is most obvious when the hanging point is moved to the leading edge of the wing near 0.65 times the wingspan.

Nonlinear dynamic thermal-mechanical behaviors of framed structures under temperature variations

An abnormal change in dynamic characteristics is considered an indicator of structural damage and can help the optimal design of structures suffering from dynamic loads, but these characteristics are also affected by ambient temperature. Therefore, a novel advanced computational method based on the fiber-based approach is proposed for the first time to predict the nonlinear dynamic behaviors of frames under environmental temperature changes with linear and nonlinear profiles by using Fortran programming language. The cross-section will be divided into a matrix of fibers, and by assigning the different temperatures to each fiber, all linear and nonlinear temperature gradients in both vertical and horizontal directions will be operated. The material nonlinearity is traced by the distributed plasticity model using the same mesh as the fiber-based approach whereas the effects of geometric nonlinearity due to member curvature and member chord rotation are considered by using the stability functions and a geometric matrix, respectively. A sequential nonlinear thermal incremental-iterative solution scheme based on the Newton-Raphson algorithm and Newmark-β method is also developed to address the nonlinear dynamic problems due to temperature variations. The accuracy and computational performance of the proposed method are validated by comparing the obtained results with those from experiments and the Abaqus. Results obtained show that the proposed method is exact for the nonlinear inelastic dynamic thermo-mechanical analysis of framed structures. It would offer a tool for thermal-structural design practice instead of using time and cost-consuming commercial software.

Experimental and numerical analysis of hyperelastic plates using Mooney-Rivlin strain energy function and meshless collocation method

In this research, the flexural behavior of a hyperelastic plate made of silicone under uniformly distributed loading was investigated. The displacements and strains were formulated using the first-order shear deformation plate theory (FSDPT). The right Cauchy-Green tensor and Mooney-Rivlin strain energy function were employed to formulate hyperelastic silicone plate governing equations. The strong-form of the governing equations of the silicone plate using Mooney-Rivlin strain energy function and Euler-Lagrange relations were derived for the first time. In the governing equations, the effects of geometrical and material nonlinearity effects were considered simultaneously. The meshless collocation method (MCM) and thin-plate spline radial basis function were utilized to discretize the governing equations. The arc-length continuation algorithm was also used for analyzing the system of non-linear equations. MCM results were compared to those of the experimental test to validate the results of the meshless method. For this purpose, a square silicone plate with simply supported boundary conditions under uniformly distributed loading was investigated via the digital image correlation (DIC) technique. According to the obtained results, the MCM based on radial basis function is an adequate method for non-linear analysis of hyperelastic plates with Mooney-Rivlin strain energy function under uniformly distributed loading.

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.