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We develop a new numerical-analytic method for solving geometrically nonlinear problems of bending of complex-shaped plates. The statement of the problem is realized within the framework of a refined higher-order theory, which takes into account the quadratic law of distribution of the transverse tangential stresses over the thickness. For linearization of the nonlinear problem, we apply the method of continuous extension with respect to the parameter. For the variational statement of the linearized problem, we construct a functional in the Lagrange form given on the kinematically admissible rates of displacements and the rates of shear functions. To find the main unknowns in the problem of nonlinear bending of a plate, we formulate the Cauchy problem for a system of ordinary differential equations. This Cauchy problem is solved by the Runge–Kutta–Merson method with automatic choice of the step. The right-hand sides of the differential equations (corresponding to the Runge–Kutta–Merson scheme) are found for fixed values of the parameter of loading from the solution of the variational problem for the Lagrange functional. The variational problems are solved by the Ritz method in combination with the R-function method. The test problems are solved for the cases of a rigidly restrained plate and a simply supported plate subjected to the action of uniformly distributed loads of different intensities. The problem of bending of plates of complex shape is solved. The influence of the geometric shape on the stress-strain state is investigated. [ABSTRACT FROM AUTHOR]

Intelligent magneto-responsive structures are widely used due to their fast response speed and noncontact control. In this study, the peeling behavior of a slender magneto-responsive sheet under the magnetic field's action is investigated experimentally and theoretically. Firstly, two identical magneto-responsive sheets, with the liquid film adhesion and the absence of the liquid film respectively, are analyzed under the control of a cylindrical magnet. In addition, the peeling process is abstracted via the classical elastica model, and the expression of the potential energy functional is established. The approximate solutions of the deflection and the adhesion length of the magneto-responsive sheets during the peeling process are obtained based on the Rayleigh–Ritz method. The effects of the magnetic field generated by cylindrical magnets and the work of adhesion on the maximum deflection and adhesion length of magneto-responsive sheets are further predicted. The theoretically approximate solution agrees very well with the experimental data. These findings can provide new implications in a wide range of industrial areas, such as medical microsensors and intelligent structures for drug delivery. [ABSTRACT FROM AUTHOR]

High-aspect-ratio aeroelastic wings are essential for energy efficient greener aircraft with improved aerodynamic efficiency and performance. In this work, a computational procedure based on a coupled computational fluid dynamics (CFD)–computational structural dynamics (CSD) technique is established to study the static aeroelastic behavior of a high-aspect-ratio wing at various flow velocities. Since these wings are structurally very flexible and experience large deformations, the present fluid structure interaction (FSI) approach couples a Navier–Stokes (N–S) based CFD solver with a finite element method (FEM) based structural solver incorporating geometric nonlinearity. Here, the N-S solver is based on the finite volume method (FVM) and the nonlinear FE structural solver is based on the Newton-Raphson method. The capability of the present FSI technique is demonstrated by comparing tip displacements and twist angles of the wing with the available experimental and numerical data in the literature. The effect of wing deformations on the pressure coefficients (Cp) of the linear and geometrically nonlinear wing models is also discussed. It shows that the linear model overpredicts Cp as compared to the geometrically nonlinear wing model. Further, a detailed comparison of span-wise horizontal displacement, vertical displacement, twist distributions as well as lift and side forces is shown between the linear and nonlinear structural models at various flow velocities. Numerical results show significant difference between the linear and nonlinear structural models at higher flow velocities. It indicates that the design of high-aspect-ratio wings must consider FSI approach to capture their nonlinear static aeroelastic behaviour accurately. [ABSTRACT FROM AUTHOR]

Multi-stable systems, known for having complex nonlinear dynamics, have been shown in past studies to outperform their linear counterparts when designed for operation in vibration energy harvesters. This work investigates the dynamics and performance of a multi-stable electromagnetic energy harvesting device. The objectives are to enhance the levels of generated power and evaluate the useful bandwidths for the various modes of stability. Nonlinearity is introduced through a set of inclined springs and strongly influenced by the geometric parameters, as the shape of the potential energy curves can be adjusted by the selection of these properties. The effect of the geometric parameters' uncertainties on the potential energy as a result of imperfections in manufacturing and installation, and the performance of the proposed harvester, is examined. A nominal tri-stable configuration is considered, and the sensitivity of the system to input design uncertainties is explored. The system's response bifurcation diagrams and dynamical characteristics as well as the power output under up- and reverse-sweeping excitations are also reported. It is shown that the overall response and effectiveness of the nominal multi-stable harvester may drastically change in terms of resonance regions, periodicity of the motion, and levels of harvested power. A series of mono-stable and bi-stable harvesters arise due to the presence of geometrical uncertainties. [ABSTRACT FROM AUTHOR]

Although rotating blades are designed to operate away from structural frequencies, internal resonance can still occur at certain rotation speeds, significantly affecting the blades' stiffness. Therefore, a detailed investigation of the resonance mechanisms of rotating blades, especially those introduced by nonlinearities, is crucial for structural design and optimization. To address this, a dynamic model of a functionally graded graphene/titanium alloy composite trapezoid plate is established to investigate the nonlinear dynamics of a rotating blade with varying cross-sections. The ordinary differential equations of the plate are formulated based on the energy method and Lagrange equations. The occurrence of 1:3 internal resonance in the trapezoid plate is identified through the analysis of linear and nonlinear free vibrations. Subsequently, the steady-state responses of the rotating plate in the case of 1:3 internal resonance are numerically calculated and verified using the approximation solution of the multiple scales method. Parametric studies are conducted to investigate the effects of rotation speed, taper ratio, excitation amplitude, and excitation frequency on the amplitude–frequency and amplitude–force curves. [ABSTRACT FROM AUTHOR]

An extremely simplified transformation model of dynamic deformatio of a strip consisting of two sections along its length is constructed. It is based on the classical geometrically nonlinear Kirchhoff–Love rod model on the loose section, and the fixed section of finite length is considered to be attached to the rigid and fixed support element via elastic layers. On the fixed section, the deflections of the rod and interlayers are considered zero, and for displacements in the axial direction over the thicknesses of the rod and interlayers, approximations are adopted according to the Timoshenko shear model subject to the continuity conditions at the points of their junction with each other and immobility at the points of attachment of the interlayers to the support element. The conditions for the kinematic coupling of the unfixed and fixed sections of the rod are formulated, and, based on them, using the d'Alembert–Lagrange variational princinple, the corresponding equations of motion and boundary conditions, as well as the static conditions for coupling the sections, are derived for the sections under consideration. [ABSTRACT FROM AUTHOR]

The geometrically nonlinear problem for a long rectangular sandwich plate under a uniform surface load has been formulated and analytically solved. The basic equations have been derived using the geometrically nonlinear theory in the quadratic approximation, the curved-normal hypothesis, and Hooke's law for transversely isotropic materials. The exact expressions for the components of the stress-strain state of the sandwich plate with clamped longitudinal edges have been obtained. Numerical results for both the geometrically nonlinear and the linear boundary value problems have been presented for three options of mechanical parameters of the core material. [ABSTRACT FROM AUTHOR]

This paper reviews the recent advances on the combination of component mode synthesis and model order reduction for geometrically nonlinear analysis, focusing on fixed-interface substructures and non-intrusive reduced-order models. These approaches offer significant potential for engineering applications by enabling accurate mechanical behavior prediction without detailed geometric and material information. Additionally, they facilitate safe sharing and collaboration between companies, reducing design times and production costs while also offering computational efficiency. However, challenges remain, indicating the need for future improvements in this area. [ABSTRACT FROM AUTHOR]

This work is concerned with the dynamic thermo-electro-mechanical response of smart composite laminates (SCLs) integrated with homogeneous piezoelectric layers. It extends our earlier quasistatic model and accounts for the coupled dynamic behaviour of SCL; with the piezoelectric layers working as smart sensors or active actuators. In the current nonlinear dynamic model, the geometric nonlinearity is accounted for using von Kármán's formulations and the displacement field is described using a third-order shear deformation hypothesis. The fully coupled thermo-electro-mechanical constitutive laws are applied using temperature dependent material properties. The resulting nonlinear system of partial differential equations is converted into a robust and efficient finite element (FE) model. The FE model is used to examine the thermal stress states, the natural frequencies and frequency responses of the SCL plate under different temperatures. The results of our extensive analysis show a decrease in the natural frequencies and increase in vibration amplitudes of the SCL plate with increased temperatures. This is due to the development of thermal stresses as well as the thermal sensitivity of the material properties. Additionally, our results reveal the thermal effects on the electric field, the voltage output of the piezoelectric sensors and the control efficiency of actuators. [ABSTRACT FROM AUTHOR]

We present a kinetodynamic quarter-car model for passenger cars with McPherson suspension, which includes geometric nonlinearity due to the constraints imposed by the interconnection of elements. We retain flexibility in the strut as well as the tire, and use the simplest linear constitutive relationship for these flexible members. Accordingly, our tire is modeled using only two springs with linear stiffnesses; one is aligned along the road surface while the other is perpendicular to it. The strut is modeled using a linear traditional Kelvin–Voigt element, while other suspension elements are assumed to be rigid. The equations of motion for the resulting two-degrees-of-freedom system are derived using the Lagrangian formulation, including the potential energy associated with gravity. The gravitational forces result not only in constant external force but also appear as parameters in the coefficient of various terms involving the generalized coordinates. We validate our analytical model by comparing the vibration characteristics of the linearized model as well as the time-domain results of the fully nonlinear model against comparable models in the multibody dynamics software MSC Adams/View. We study the importance of geometric nonlinearity in capturing the true response of the vehicle body by comparing results from the fully nonlinear model against those from the linearized model both for harmonic as well as standard road inputs. Finally, we highlight the importance of including gravity in our formulation by pointing out the inadequacy of the models without gravity in capturing the true vibration characteristics (both linear and nonlinear). [ABSTRACT FROM AUTHOR]

The problem of forced bending vibrations of a plane rod with a finite-length fastening section under the action of an external transverse force at its free end was solved. The classical Kirchhoff–Love model in the classical geometrically nonlinear approximation was used to describe the deformation process of the free part of the rod. The deformation of its fixed part was described by the Timoshenko refined shear model that takes into account transverse strains. The conditions of kinematic conjugation of the free and fixed parts of the rod were formulated. The equations of motion, the corresponding boundary conditions, and the force conditions of conjugation of the rod parts were obtained using the Hamilton–Ostrogradsky variational principle. An exact analytical solution of the problem of forced vibrations of a rod under the action of a harmonic transverse force at the free end of the unfastened part of the rod was deduced. Numerical experiments were carried out to study the resonant vibrations of rods made of unidirectional fiber composite. The effect of a noticeable increase of the amplitudes of transverse vibrations of the ends of the cantilever parts of the rods studied due to transverse contraction of the fixed section was revealed. Taking into account the transverse contraction caused an almost twofold reduction of the maximum transverse shear stresses in the fixed part of the duralumin rod. [ABSTRACT FROM AUTHOR]

Simultaneously determining the random vibration responses and dynamic reliabilities of thin shell coupling geometric nonlinearity and multisource uncertainties is an intractable task. A novel non-intrusive framework based on direct probability integral method (DPIM) is proposed in the present study, which offers an efficient and competitive solution tool to tackle this challenging issue. New framework incorporating DPIM with adaptive schemes can address efficiently stochastic dynamic responses and reliability determination of geometrically nonlinear thin shells in a unified way. Adaptive choosing strategy of the smoothing parameter of Dirac function and the number of representative points is adopted. Importantly, a judgment criterion is established to adaptively perform nonlinear theory and linear theory of large deflection for thin shell, which breaks the limitation of using a single nonlinear or linear theory and results in more accurate responses. Finally, several numerical examples demonstrate that the proposed framework possesses high accuracy and efficiency when compared to Monte Carlo simulation (MCS) and quasi-MCS for computing stochastic deflection and stress responses and dynamic reliabilities. The transform of probability density function of deflection responses from unimodal to bimodal distribution implies that the stochastic P-bifurcation occurs in random vibration of nonlinear thin shell. The remarkable effects of sound pressure level of noise excitation, power spectral density of random excitation, random parameter variability and boundary conditions on uncertainty quantification of thin shells are revealed. [ABSTRACT FROM AUTHOR]

The present study aims to investigate the influence of geometric nonlinearity on post-flutter responses by developing a full-mode coupled nonlinear flutter analysis method (frequency-domain method) and a time-dependent nonlinear analysis scheme (time-domain method). This approach integrates the three-dimensional (3D) nonlinear finite element model and nonlinear self-excited force described by amplitude-dependent rational functions (RFs). By comparing post-flutter responses obtained from frequency-domain and time-domain methods, not only the influence of geometric nonlinearity on post-flutter responses is quantified, but also the underlying physical mechanism is revealed. The results show that the geometric nonlinear effect will become more significant with the increase of the amplitude and thus will induce a super-harmonic resonance behavior. The behavior is mainly characterized by the higher harmonic frequencies vibrations with higher-order mode shapes involved in the vertical and torsional displacement responses. Meanwhile, the larger the vibration amplitude, the more significant the super-harmonic resonance behavior. Besides, the geometric nonlinear effect will also cause a significant uplifting of the bridge deck in the vertical direction during 3D nonlinear flutter process. The main physical mechanism for the reduction in the amplitude of post-flutter response (dominated by the vibration with fundamental harmonic frequency) after considering the geometric nonlinear behavior is that the vibrations with higher harmonic frequencies play a role of absorbing energy and reducing vibration (similar to tuned mass damper effect) for the vibration with fundamental harmonic frequency. For the long-span suspension bridge with a main span of 1650 m studied in this study, the geometric nonlinear effect may need to be considered when the torsional amplitude at mid-span is only greater than 1.5°. [ABSTRACT FROM AUTHOR]

The direct parametrisation method for invariant manifold is a model-order reduction technique that can be applied to nonlinear systems described by PDEs and discretised, for example, with a finite element procedure in order to derive efficient reduced-order models (ROMs). In nonlinear vibrations, it has already been applied to autonomous and non-autonomous problems to propose ROMs that can compute backbone and frequency–response curves of structures with geometric nonlinearity. While previous developments used a first-order expansion to cope with the non-autonomous term, this assumption is here relaxed by proposing a different treatment. The key idea is to enlarge the dimension of the parametrising coordinates with additional entries related to the forcing. A new algorithm is derived with this starting assumption, and as a key consequence, the resonance relationships appearing through the homological equations involve multiple occurrences of the forcing frequency, showing that with this new development, ROMs for systems exhibiting a superharmonic resonance can be derived. The method is implemented and validated on academic test cases involving beams and arches. It is numerically demonstrated that the method generates efficient ROMs for problems involving 3:1 and 2:1 superharmonic resonances, as well as converged results for systems where the first-order truncation on the non-autonomous term showed a clear limitation. [ABSTRACT FROM AUTHOR]

A geometrically exact Kirchhoff beam formulation (GEKBF) established on the Lie group SO(3) for simulating the dynamics of a slender beam is proposed. The kinematic description, dynamic equilibrium equations and their linearization are derived in this framework. Then, a second-order interpolation function for the torsion angle is introduced to improve the convergence of the element. Next, the rotation vector parameterization is developed to reduce the geometric nonlinearity caused by the rotation of the rigid body. In addition, the influence of the reference frame is considered, and the derivation of the elastic force and Jacobian matrix is simplified using the spatial-parallel transport. The semi-discrete equations of motion are in the form of a second-order ordinary differential equation on Lie group, which are solved using the Lie group generalized α-method. Finally, the accuracy of the proposed formulation is verified using several numerical examples. [ABSTRACT FROM AUTHOR]

Support loss, as one of the main energy dissipation sources in micro/nano mechanical resonators, is generally evaluated by linear theories without considering mid-plane stretching. Predictions of support loss based on linear elastic theory cannot be applied to resonators working in nonlinear regime with large vibration amplitude. In this regard, we investigate the support loss of a doubly clamped micro/nano beam resonator in in-plane flexural vibration in this study, with both geometric nonlinearity and axial pre-tension considered, aiming to accurately model the support loss for high-performance resonators under large-deflection vibration. The expression of support loss is derived, with both the normal stress induced by the mid-plane stretching and shear stress distributed in the attachment region of the support taken into account. The analytical model proposed in this work is validated by finite element simulations and the experimental data of previous studies. Furthermore, the influence of the axial pre-tension on nonlinear vibration properties and that of the vibration amplitude on the support loss are probed. The effects of the axial pre-tension and the aspect ratio on the support loss are also analyzed. The analytical results demonstrate that the mid-plane stretching effect due to geometric nonlinearity leads to larger support loss, and the support loss increases with vibration amplitude. Hopefully, the nonlinear model of support loss developed in this work may provide guidelines to evaluate the support loss and elucidate the phenomenon of the support loss increasing with vibration amplitude. [ABSTRACT FROM AUTHOR]

Satellite borne flexible structure is a multi-degree-of-freedom system, which contains complex dynamic characteristics such as time-varying parameters, geometric nonlinearity, gap nonlinearity, and so on. Flexible structure suspension typically results in geometric nonlinearity. The oscillation equation with nonlinear term is established according to the law of motion of a nonlinear pendulum and considering the influence of medium swing angle and lateral force. The perturbation approach is used to get the relationship between vibration frequency and the nonlinear term, and the impact of factors on vibration characteristics is investigated. The satellite borne flexible structure's active vibration control (AVC) system is then established. Considering proportional differential (PD) or fuzzy control adjustment, variable step size least mean square (VSS-LMS) adaptive filtering algorithm is used to calculate the control signal, and considering the influence of geometric nonlinearity, the actuator is used to suppress the vibration of the satellite borne flexible structure. Finally, the vibration response's amplitude under steady-state excitation significantly decreases as an outcome of the vibration control simulation. [ABSTRACT FROM AUTHOR]

Purpose: The objective of the present work is to carry out simulation studies for prediction of nonlinear dynamic characteristics of functionally graded uniform beams. Methods: The material properties are considered to be varying simultaneously along thickness and length direction. Euler–Bernoulli beam theory is used to generate the constitutive relations and two distinct methodologies are employed, viz., whole domain method and finite element method. Geometric nonlinearity is incorporated in the methodology using Von Karman's strain–displacement relations. Results: The results are presented as backbone curve and mode shape in free vibration and frequency–response curve and operational deflection shape in forced vibration. The effect of material gradation on dynamic behaviour is investigated using different values of material gradient parameter. The nonlinear system is seen to display hard spring behaviour. A comparative study between WDM and FEM also shows the correctness of present study. Conclusion: In free vibration, a similar trend has been found in all cases that natural frequency increases with response amplitude. This behaviour is due to the stretching effect associated with large deflection resulting in additional stiffening of the system. Increase in vibration frequencies with response amplitude is formally known as hardening type nonlinearity. The same is also observed in forced vibration analysis where the frequency–response curve bends towards right. [ABSTRACT FROM AUTHOR]

Numerous powerful methods exist for developing reduced-order models (ROMs) from finite element (FE) models. Ensuring the accuracy of these ROMs is essential; however, the validation using dynamic responses is expensive. In this work, we propose a method to ensure the accuracy of ROMs without extra dynamic FE simulations. It has been shown that the well-established implicit condensation and expansion (ICE) method can produce an accurate ROM when the FE model's static behaviour are captured accurately. However, this is achieved via a fitting procedure, which may be sensitive to the selection of load cases and ROM's order, especially in the multi-mode case. To alleviate this difficulty, we define an error metric that can evaluate the ROM's fitting error efficiently within the displacement range, specified by a given energy level. Based on the fitting result, the proposed method provides a strategy to enrich the static dataset, i.e. additional load cases are found until the ROM's accuracy reaches the required level. Extending this to the higher-order and multi-mode cases, some extra constraints are incorporated into the standard fitting procedure to make the proposed method more robust. A curved beam is utilised to validate the proposed method, and the results show that the method can robustly ensure the accuracy of the static fitting of ROMs. [ABSTRACT FROM AUTHOR]

A transformation model of the dynamic deformation of an elongated orthotropic composite rod-type plate, consisting of two sections (fastened and free) along its length, is proposed. In the free section, the orthotropic axes of the material do not coincide with the axes of the Cartesian coordinate system chosen for the plate, and in the fastened section, the displacements of points of the contact's boundary surface (clamping) with the elastic support element are considered to be known. The constructed model is based on the use for the free section of the relations of the refined Timoshenko shear model, compiled for rods in a geometrically nonlinear approximation without taking into account lateral contraction. For the section fastened on the elastic support element, a one-dimensional shear deformation model is built taking into account lateral contraction, which is transformed into another model by satisfying the conditions of kinematic coupling with the elastic support element with given displacements of the interface points with the plate. The conditions for the kinematic coupling of the free and fastened sections of the plate are formulated. Based on the Hamilton–Ostrogradsky variational principle, the corresponding equations of motion and boundary conditions, as well as the static conditions for the matching of sections, are derived. The constructed model is intended to simulate natural processes and structures when solving applied engineering problems aimed at developing innovative oscillatory biomimetic propulsors. [ABSTRACT FROM AUTHOR]

This research investigates the nonlinear dynamics and performance enhancement of a suspension system using a diamond-shaped linkage with inerter (D-inerter). The proposed suspension system consists of two inerters embedded in a four-bar linkage mechanism connected with a spring and a damper. Both sinusoidal and random road profiles are considered as external excitation sources. The evaluation of vibration isolation and riding comfort performance is based on displacement transmissibility, acceleration amplitude, car body acceleration, suspension stroke, and dynamic tyre load. The results show that compared with a linear suspension system, the D-inerter has a broader bandwidth of enhanced isolation and lower resonant peak. It is found that a larger inertance value and initial length between the ends of inerter can effectively improve the suppression performance of the nonlinear suspension. The root mean square of vehicle body acceleration with the D-inerter is decreased by 21.5% at the speed of 30 m/s. Additionally, design guidance is provided to select optimal inertance values for improved suspension performance. The results demonstrate that the D-inerter is beneficial for enhancing the suspension structural stability, riding comfort, and vibration suppression, which can potentially be employed for vibration isolation in suspension systems. [ABSTRACT FROM AUTHOR]

The crank train inerter (CTI) is a recently developed vibration inerter consisting of a crank, a linking rod, a flywheel, several gears and pinions. The inertance of CTI is nonlinear when CTI undergoes large displacements. This study explores the nonlinear effect of inertance of a CTI for vibration isolation. The accurate expression of the inertial force in CTI is first derived, and its accuracy is verified by experiment. The equation of motion for the coupled structure–CTI system is formulated, in which a simplified expression of inertial force is adopted. The analytical solutions in forms of the frequency response, peak response and stability boundary are obtained via the harmonic balance method. The transmissibility of the CTI system is derived, and the vibration isolation performance of the CTI is evaluated. Finally, the control effect of CTI on a multistory isolated building for harmonic and seismic excitations is studied. The results show that the nonlinearity of CTI leads to stiffness softening and an unstable response. For the most unfavorable response, the CTI has better vibration isolation control performance than the linear inerter for harmonic and seismic excitations under the same inertance/mass ratio. [ABSTRACT FROM AUTHOR]

Purpose: To suppress the low-frequency vibration of dynamic systems such as underwater vehicles, this research proposes a novel geometrically nonlinear vibration isolator using hybrid nonlinear inertial and negative stiffness element. Methods: A spring and an inerter are integrated together into a 4-rod linkage structure to form geometric nonlinearity. The performance of isolator under force or base-motion excitation is analysed. The performance of the proposed isolator in a flexible base structure simulating vibration isolation in ships is also considered. The transmissibilities and vibrational energy transfer are used to evaluate the effectiveness of isolation. Results: The results demonstrate better performance in low-frequency vibration isolation comparing to conventional linear isolator. The combined use of spring and inerter in the linkage mechanism can create a frequency band of ultra-low transmissibility and energy flow at low frequencies. Conclusion: Structural parameters of the proposed hybrid nonlinear element can be designed to alter the dynamic characteristic of the nonlinear isolator to attenuate low-frequency vibration transmission. The proposed nonlinear isolator demonstrates a strong potential for application in naval architecture. [ABSTRACT FROM AUTHOR]

This paper compares four methods for formulating stability constraints in topology optimization with geometric nonlinearity. The methods are: a direct approach to compute the critical load factor, an approximation using an eigenvalue analysis at a load factor of 1, a new method based on an eigenvalue analysis at the constraint limit load factor, and an implicit method based on stiffness reduction, which has not previously been investigated for stability constraint formulation. These four methods are described in detail and then compared qualitatively and quantitatively (including optimization examples) in terms of accuracy, robustness, and computational efficiency. The results show that formulating the constraint using an eigenvalue analysis at a load factor of 1 is the most robust approach, as it is least likely to experience mode switching or mode skipping during optimization, which leads to poor convergence for the other three methods. It is also the most efficient, as it only requires a single eigenvalue solve, whereas other methods require additional linear solves to compute the constraint value. However, an eigenvalue analysis at a load factor of 1 only approximates the critical load factor, which may be over, or under-estimated. Therefore, none of the methods fully satisfy the criteria of accuracy, robustness, and efficiency, highlighting the need for further research, e.g., by improving the accuracy of the method based on an eigenvalue analysis at a load factor of 1, or by improving the robustness and efficiency of the direct approach. [ABSTRACT FROM AUTHOR]

A geometric nonlinear modeling approach for strong rigid–flexible–thermal coupling dynamics of a hub and multiplate system considering frictional contact is proposed. Based on the absolute nodal coordinate formulation (ANCF), an ANCF thin-plate element with thermoelasticity is developed, where the temperature field is expressed with Taylor polynomials to yield heat-conduction equations. In contrast to the traditional coupling formulations, the influences of the attitude motion and structural deformation on the intensity of the solar radiation, the geometric nonlinearity of the plate as well as the frictional contact are taken into account. The frictional-contact formulations for a thin plate and a rigid body are presented, which can capture the stick–slip transition and address the multiple-point contact scenarios. To solve the strong rigid–flexible–thermal coupling equations, a novel numerical approach combining the generalized- α method and the modified central-difference method is proposed. Two validations are performed to verify the proposed model, which proves the importance of considering the geometric nonlinearity and reveals the phenomena of thermally induced vibrations. Then, the thermal–dynamic coupling analysis for the satellite and solar-array multibody system in a thermal environment is carried out. The dynamic characteristics of the thermally induced vibration can be successfully revealed by the rigid–flexible–thermal coupling model. Furthermore, it is indicated that the influence of contact and thermal load on the nonlinear behavior of the solar-array deployment is essential, which demonstrates the feasibility of the proposed approach. [ABSTRACT FROM AUTHOR]

The statement and the analytical and numerical methods for solving a geometrically nonlinear problem for a long noncircular cylindrical panel compliant to transverse shear under a normal surface load are described. The basic equations are based on the geometrically nonlinear theory of shallow shells in quadratic approximation, the Timoshenko hypothesis, and Hooke's law for transversally isotropic materials. Analytical expressions for the components of the stress–strain state are obtained for a shell with hinged longitudinal edges, and the limiting values of the generalized geometric parameter are determined. Numerical results are obtained for long open cylindrical shells of elliptical and oval cross-sections under uniform normal pressure. [ABSTRACT FROM AUTHOR]

A new numerical-analytical method for solving geometrically nonlinear bending problems for thin shallow shells of complex shape in plan made of functionally graded materials is developed. The method is based on the R-functions method and the parameter continuation method. Test problems for clamped homogeneous and hinged functionally graded plates under uniformly distributed load of different intensity are solved. The results for deflections and stresses obtained with the developed method are compared with those obtained with other methods. The bending problem for a functionally graded shell of complex shape is solved. The influence of the gradient properties of the material and geometric shape on the stress–strain state is studied. [ABSTRACT FROM AUTHOR]

Finite element models (FEM) of geometrically nonlinear structures are increasingly computationally expensive as physical systems become more complex. Nonlinear reduced order models (NLROM) are an efficient alternative to simulate the dynamic behavior of practical engineering structures. However, the computation complexity of ROM increases exponentially with the number of selected modes, which makes the calculation effort very expensive. This study proposes a fusion method that combines reduced-order and predictive modeling, referred to as the multi-fidelity surrogate-based nonlinear reduced-order model (MFS-NLROM). The MFS-NLROM approach defines the nonlinear solution as the high-fidelity response and the linear solution as the low-fidelity response, enabling faster calculation of nonlinear FEM. By incorporating a discrepancy function to represent the nonlinear force, the proposed method increases the complexity of the fitting function while reducing the number of nonlinear terms. This simplifies nonlinear problems through the predicted model of the nonlinear force. The MFS-NLROM method is applicable to static, dynamic, and nonlinear normal mode analyses. In dynamic analysis, error compensation is introduced through the fusion of the MFS-NLROM and Newmark methods. Additionally, the first derivative of the multi-fidelity surrogate is utilized to obtain nonlinear normal modes (NNMs) using the multi-harmonic balance (MHB) methods, which enables the evaluation of the accuracy of the proposed method. The feasibility and effectiveness of the proposed framework are verified using examples involving a flat beam and a wing cantilever beam, in which the Implicit Condensation and Expansion (ICE) model and MFS-NLROM are fully compared. The evaluation results show that MFS-NLROM has better computation efficiency under the premise of ensuring accuracy. In short, the proposed framework can leverage the advantages of computational accuracy and efficiency, which can serve as a means to promote the development of digital twins in industrial fields (dynamic system damage monitoring, safety assessment, and other aspects). [ABSTRACT FROM AUTHOR]

Double cable system (DCS) with a movable guide pulley (MGP) is a common component in many mechanical equipment. It is difficult to calculate the unstressed length of cable under a preload, and the form-finding analysis after obtaining the unstressed length. To solve this problem, a method of establishing double-layer nonlinear equations for calculating the unstressed length of cable under a given preload is proposed. For the inner single cable, a new cable element is presented based on Hermite interpolation. Later, initial cable form is obtained by approximating catenary as a three-point parabola, and the initial values for the nonlinear equations are derived. For the outer double cable, unstressed length is a descriptive variable of the DCS. In view of the constraints on a single cable posed by the guide pulleys, corresponding constraint equations are obtained by introducing the angle of rotation of the MGP, which is a process variable. Under a given preload at an end of single cable, equations for the DCS with a MGP are constructed. Moreover, the tangent stiffness matrixes are derived for quick solution, and unstressed length for the DCS is obtained. On this basis, by changing the partial constraint equations of the DCS, the form-finding analysis of the cable can be achieved. A few numerical examples and comparisons demonstrate the reliability of the proposed new approach for geometrically nonlinear analysis of cables. [ABSTRACT FROM AUTHOR]

This paper presents an efficient mesh updating scheme (MUS) for the arbitrary Lagrangian–Eulerian (ALE) formulation of an arbitrarily curved beam based on the corotational method. By discretizing the beam using both Lagrangian elements and ALE elements, the proposed MUS can take full advantage of the simple expression form of the Lagrangian formulation and the accurate moving-load description of the ALE node. The deleting-node and adding-node procedures of the MUS can avoid the negative influence of the variation of the ALE element length on the element accuracy and stiffness matrix singularity. In contrast to the adding-node procedure for Lagrangian elements, interpolation cannot be used directly. Inserting a Lagrangian node in an ALE element is investigated, and the displacement, velocity, and acceleration of the newly added node are evaluated accurately based on the corotational method. Three examples are investigated to verify the validity, computational accuracy and computational efficiency of the proposed MUS by comparing the results of the MUS with those from literature that utilized traditional ALE formulation. These examples show that the proposed MUS has significant advantages in terms of computational time and computer memory. [ABSTRACT FROM AUTHOR]

New refined geometrically nonlinear equations of motion of composite elongated rod-type plates in plane stress-strain state are derived for the case when the axes of the selected coordinate system coincide with the axes of the orthotropy of the plate material. The equations are based on the previously proposed relations of a consistent version of the geometrically nonlinear theory of elasticity under small deformations and on the refined shear model of S.P. Timoshenko. Such equations describe the high-frequency torsional vibrations in elongated rod-type plates that can occur during low-frequency flexural vibrations. By limit transition to the classical model of the theory of rods, the derived equations simplified to a system of equations of a lower order. For the case of the same deformation models for a composite plate with inclined reinforcement, similar equations of motion in a linear approximation are obtained. It is shown that they describe the coupled flexural-torsional vibrations in the case of small displacements. [ABSTRACT FROM AUTHOR]

This study proposes a new numerical-and-analytical method for solving geometrically nonlinear problems of bending of complex-shaped plates made of functionally graded materials developed. The problem was formulated within the framework of a refined higher-order theory considering the quadratic law of distribution of transverse tangential stresses along the plate thickness. To linearize the nonlinear problem, we used the method of continuous continuation in the parameter associated with the external load. For the variational formulation of the linearized problem, a Lagrange functional was constructed, defined at kinematically possible displacement velocities. To find the main unknowns of the problem of nonlinear plate bending (displacements, strains, and stresses), the Cauchy problem for a system of ordinary differential equations is formulated. The Cauchy problem was solved by the Runge-Kutta–Merson method with automatic step selection. The initial conditions are found from the solution of the problem of geometrically linear deformation. The right-hand sides of the differential equations, at fixed values of the load parameter corresponding to the Runge-Kutta–Merson scheme, were obtained from the solution of the variational problem for the Lagrange functional. The variational problems were solved by the Ritz method in combination with the R-function method. The latter makes it possible to present an approximate solution as a formula. This solution structure exactly satisfies all (general structure) or part (partial structure) of the boundary conditions. Test problems are solved for a homogeneous rigidly fixed and functionally graded hinged square plate subjected to a uniformly distributed load of varying intensity. The results for deflections and stresses obtained by the developed method are compared with the solutions obtained by radial basis functions. The problem of bending of a functionally graded plate of complex shape is solved. The influence of the gradient properties of the material and geometric shape on the stress-strain state is investigated. [ABSTRACT FROM AUTHOR]

This work presents a topology optimization method for prestressed cable-truss structures considering geometric nonlinearity. Based on the large deformation kinematics theory, we establish a multi-node cable element model, which can fully take into account the geometric nonlinearity caused by external loads and prestress. Besides, nonlinear hyperelasticity principles are used to deal with cable materials’ unidirectional stress characteristics. To consider the impact of prestress on the global structural stiffness, we carried out the rebalancing of prestress and construct an objective function of prestress-modified compliance. Then, we equate the cable-truss structure as a particular two-phase material structure and realize the material interpolation based on a discrete material optimization-like method. Finally, we perform the adjoint sensitivity analysis of the objective function and solve the optimization problem through the gradient-based algorithm. Therefore, we establish a topology optimization method for prestressed cable-truss structures with stiffness as the objective, continuous density/size parameters as variables, and mass fraction as constraints. This method’s feasibility and reliability are demonstrated in 2D and 3D numerical examples. [ABSTRACT FROM AUTHOR]

This work blends the inexact Newton method with iterative combined approximations (ICA) for solving topology optimization problems under the assumption of geometric nonlinearity. The density-based problem formulation is solved using a sequential piecewise linear programming (SPLP) algorithm. Five distinct strategies have been proposed to control the frequency of the factorizations of the Jacobian matrices of the nonlinear equilibrium equations. Aiming at speeding up the overall iterative scheme while keeping the accuracy of the approximate solutions, three of the strategies also use an ICA scheme for the adjoint linear system associated with the sensitivity analysis. The robustness of the proposed reanalysis strategies is corroborated by means of numerical experiments with four benchmark problems—two structures and two compliant mechanisms. Besides assessing the performance of the strategies considering a fixed budget of iterations, the impact of a theoretically supported stopping criterion for the SPLP algorithm was analyzed as well. [ABSTRACT FROM AUTHOR]

The article considers the problems of the formation of a neck in a sample of complex shape and Lüders slip bands in a square plate with a central circular hole in a plane-deformed state in finite deformations, taking into account the perfectly plastic flow. The statement of tasks is presented. Due to the symmetry, quarters of the models were considered. Numerical results were obtained in the CAE Fidesys package using the finite element method for the first and second orders, as well as for the spectral element method: for seven orders for the necking problem and the third-order elements for the Lüders slip bands problem. Based on the results obtained, the analysis of the capabilities of the spectral element method relative to the finite element method is carried out. The results of the conducted research can be useful when deciding on the use of the spectral element method in industrial problems. [ABSTRACT FROM AUTHOR]

The stationary/nonstationary random vibration responses and dynamic reliability analyses of thin plate structures coupling geometrical nonlinearity and multisource randomness in both structural parameters and external excitations are the important issues to be attacked. In this study, the novel direct probability integral method (DPIM) is proposed to tackle them. Firstly, the differential equations of thin plate vibration with large deflection are revisited based on von Kármán nonlinear theory. Then, the non-intrusive DPIM decouples physical evolution and probability density propagation of structure, which is utilized to calculate nonlinear stochastic dynamic responses and reliability of thin plate in a unified way. Moreover, to determine the resultant large-amplitude nonlinear deformation of thin plate under random concentrated excitation and distributed excitation, an important criterion is devised, which can judge the applicability of geometrically nonlinear theory. Several numerical examples demonstrate the high accuracy and efficiency of proposed approach by comparing the results with those of Monte Carlo simulation. It is indicated that, the changes of structural parameters and random excitation cause the transition of probability density functions of nonlinear deflections from unimodal to bimodal distributions. Stochastic P-bifurcation occurs in nonlinear random vibration analysis of thin plate, which implies that the geometric nonlinearity should be rationally considered in stochastic dynamic analysis of structure. Finally, the remarkable effects of geometric nonlinearity, thickness, random parameter variability and boundary conditions on stochastic responses and reliabilities of thin plates are revealed. [ABSTRACT FROM AUTHOR]

Nonlinear vibration of shallow arch-rigid body (elastically supported) coupled system is asymptotically investigated, for refined modeling of dynamic interactions involved with a complex arch structure built in weak soils (implying non-ideal foundation). Through simplifying the weakly moving arch boundary (in weak soils) as a rigid body supported by elastic springs, an arch-rigid body coupled model is proposed in a direct perturbation formulation. The arch-rigid body two-way coupling coefficients are asymptotically derived. Two important factors turn out to be quantitative measures of coupling intensity, i.e., the moment of inertia and height of rigid body, which play key roles in arch-rigid body coupled dynamics. Furthermore, nonlinear arch-rigid body primarily resonant responses are fully investigated with stability/bifurcation characteristics determined, including both hardening and softening cases. A comparative study between the coupled model and ideal (i.e., uncoupled) arch dynamics demonstrates effectiveness of the newly proposed coupled model. Besides, an arch-foundation (boundary oscillator) coupled model, as a degenerate case of the arch-rigid body coupled model, is also investigated for comparison, partially unveiling limits of the previous arch-foundation (oscillator) model. [ABSTRACT FROM AUTHOR]

The problem of electrothermomechanical behavior and durability of inelastic shells of revolution with piezoelectric sensors and actuators under axisymmetric resonance vibrations is stated and a problem-solving method is presented. Numerical experiments for a round plate with a piezoelectric sensor were conducted to study the effect of geometric nonlinearity on the frequency dependences of deflection amplitudes, self-heating temperature, and electrical sensor index. The dependence of the service time of the system on extreme amplitudes of mechanical load and the thermal-transfer conditions on its surface is studied using a criterion of assessing local durability from the maximum self-heating temperature. [ABSTRACT FROM AUTHOR]

Despite its rapid development, Physics-Informed Neural Network (PINN)-based computational solid mechanics is still in its infancy. In PINN, the loss function plays a critical role that significantly influences the performance of the predictions. In this paper, by using the Least Squares Weighted Residual (LSWR) method, we proposed a modified loss function, namely the LSWR loss function, which is tailored to a dimensionless form with only one manually determined parameter. Based on the LSWR loss function, an advanced PINN technique is developed for computational 2D and 3D solid mechanics. The performance of the proposed PINN technique with the LSWR loss function is tested through 2D and 3D (geometrically nonlinear) problems. Thoroughly studies and comparisons are conducted between the two existing loss functions, the energy-based loss function and the collocation loss function, and the proposed LSWR loss function. Through numerical experiments, we show that the PINN based on the LSWR loss function is effective, robust, and accurate for predicting both the displacement and stress fields. The source codes for the numerical examples in this work are available at https://github.com/JinshuaiBai/LSWR_loss_function_PINN/. [ABSTRACT FROM AUTHOR]

The nonlinear characteristics of a general multiple degree-of-freedom suspension bridge under harmonic excitation are studied with an innovative approach consisting of the harmonic modes, the finite element modeling and the incremental harmonic balance method. The geometric nonlinearity induced by large-amplitude vibration is considered. The harmonic mode and mean kinetic energy of the system are proposed to describe the nonlinear responses. The harmonic mode can be expressed as a linear combination of different linearized system modes. The resonance responses of the structure are characterized with the participation ratio of these system modes. Its composition under harmonic excitations can easily be described in terms of the different harmonic mode shapes and then the system modes. The proposed method is further applied to analyze, in detail, the properties of the harmonic modes and the evolution of the steady-state nonlinear responses with variations in the excitation frequency. [ABSTRACT FROM AUTHOR]

Mechanical metamaterials with low-frequency and broadband bandgaps have great potential for elastic wave control. Inspired by the ancient window mullions, a novel plate-type metamaterial with a two-dimensional bandgap is designed. Based on the local resonance mechanism, the broadband low-frequency in-plane and out-of-plane bandgaps on the designed structure are realized. The bandgaps can be adjusted by the mass re-distribution of the main-slave resonators, the stiffness design of the support beam, and the adjustment of the excitation amplitude. A semi-analytical method is proposed to calculate the in-plane and out-of-plane bandgaps and the corresponding wave attenuation characteristics of the infinite periodic metamaterial. We explored how mass re-distribution, stiffness changes, and geometric nonlinearity influence the bandgap. Then, to verify the conclusions, we fabricated a finite periodic structure and obtained its wave transmission characteristics both numerically and experimentally. Finally, the designed metamaterial is applied to the waveguide control, elastic wave imaging, and vibration isolation. This study may provide new ideas for structural design and engineering applications of mechanical metamaterials. [ABSTRACT FROM AUTHOR]

Structural shape monitoring plays a vital role in the structural health monitoring systems. The inverse finite element method (iFEM) has been demonstrated to be a practical method of deformation reconstruction owing to its unique advantages. Current iFEM formulations have been applied to small deformation of structures based on the small-displacement assumption of linear theory. However, this assumption may be inapplicable to some structures with large displacements in practical applications. Therefore, geometric nonlinearity needs to be considered. In this study, to expand the practical utility of iFEM for large displacement monitoring, we propose a nonlinear iFEM algorithm based on a four-node inverse quadrilateral shell element iQS4. Taking the advantage of an iterative iFEM algorithm, a nonlinear response is linearized to compute the geometrically nonlinear deformation reconstruction, like the basic concept of nonlinear FE analysis. Several examples are solved to verify the proposed approach. It is demonstrated that large displacements can be accurately estimated even if the in-situ sensor data includes different levels of randomly generated noise. It is proven that the nonlinear iFEM algorithm provides a more accurate displacement response as compared to the linear iFEM methodology for structures undergoing large displacement. Hence, the proposed approach can be utilized as a viable tool to effectively characterize geometrically nonlinear deformations of structures in real-time applications. [ABSTRACT FROM AUTHOR]

A geometrically nonlinear topology optimization (GNTO) method with thermal–mechanical coupling is investigated. Firstly, the new expression of element coupling stress due to superimposed mechanical and thermal loading is obtained based on the geometrically nonlinear finite element analysis. The lightweight topology optimization (TO) model under stress constraints is established to satisfy the strength requirement. Secondly, the distortion energy theory is introduced to transform the model into structural strain energy constraints in order to solve the implicit relationship between stress constraints and design variables. Thirdly, the sensitivity analysis of the optimization model is derived, and the model is solved by the method of moving asymptotes (MMA). Numerical examples show that temperature has a significant effect on the optimal configuration, and the TO method considering temperature load is closer to engineering design requirements. The proposed method can be extended to the GNTO design with multiple physical field coupling. [ABSTRACT FROM AUTHOR]

An adaptive sparse polynomial chaos (PC) transfer matrix method (TMM) is proposed to study the dynamics modeling and analysis of probabilistic uncertain beam systems with geometric nonlinearity and thermal coupling effect. Based on the floating reference frame and the nonlinear elasticity theory, the dynamics equations of flexible beam with geometric nonlinearity and thermal coupling effect are deduced firstly, and then, its transfer equations and transfer matrices and that of the involved hinge elements are established, which can be used to assemble the overall transfer equation of beam system easily according to the system's topological structure. For a deterministic system, the dynamics of system can be obtained by solving the coupled transfer equation and transient heat conduction equation easily. The proposed TMM doesn't need the global differential–algebraic equation of system and greatly reduces the involved matrix order and calculation cost, so it is very useful for improving the valuation efficiency of the design of experiment in the subsequent uncertain analysis. When considering the stochastic parameters, above equations are all uncertain equations. Based on the hyperbolic basis truncation scheme, the least angle regression algorithm, and the posteriori cross-validation error estimation method, a nonintrusive adaptive sparse PC expansion method is adopted for uncertain analysis, which only needs a small number of model evaluations and could further reduce the calculation cost of uncertain analysis. The numerical simulation is presented to validate the correctness, feasibility, and computational efficiency of this method. By deducing new transfer equations of related body/hinge, this research can be extended to the dynamics modeling and analysis for general rigid–flexible–thermal coupling multibody systems with probabilistic uncertainties. [ABSTRACT FROM AUTHOR]

With onset of the twenty-first century, the earth is converted into a smart world in no time due to advancements in technology like the internet of things, wireless sensor network, and microelectromechanical system. Mobile energy source is highly essential to power these devices. However, the vibration energy harvester extracts the electrical energy from the mechanical vibratory system to fulfill the need for power requirements. The present study assesses the effect of geometrical nonlinearity of beam due to the occurrence of large deformation under the excitation of moving load. As an endeavor, the nonlinear analysis of the cantilever beam bonded with piezo-patch in the field of energy harvesting under the desired condition is examined. The desired basic formulation is based on the finite element approach. Hamilton's principle is incorporated to derive the equation of motion of the system. Further, Newmark's integration method is adopted to study the behavior of moving load. To solve the nonlinear set of equations Newton–Raphson method is successfully implemented. It is concluded from the analysis that with low speed and high moving force the power output is more. A decreasing trend is observed in displacement due to geometric nonlinearity which results in low voltage and power generation as compared to linear energy harvester. Later real coded genetic algorithm is successfully implemented to obtain the desired design parameter values for optimal power output. [ABSTRACT FROM AUTHOR]

A geometrically nonlinear problem for a shear-compliant long oval cylindrical shell undergoing uniform surface loading is stated and solved analytically. The basic equations are derived using the second-order geometrically nonlinear theory of shallow shells, Timoshenko's hypothesis, and Hooke's law. The exact expressions for the stress–strain state of a longitudinally hinged shell are obtained, the characteristic (limit) values of the generalized curvature parameter are determined, and the system of equations for finding the critical load is set up. Main partial and limiting cases are considered. [ABSTRACT FROM AUTHOR]

Ring structures can generate significant vibration due to the internal and external excitations. A mathematical model of a thin ring subjected to time-varying springs and forces is established by using Hamilton's principle, where the stiffnesses and geometric nonlinearity caused by external excitations are introduced. The external excitations, which usually act as a forcing term for forced vibrations, can be found in the stiffness coefficients. Thus, the external excitation is confirmed to be an primary factor in changing vibration characteristics. The natural frequencies of the ring subjected to identical equally-spaced external excitations split when the external excitations count N and wavenumber n satisfy 2n/N = integer. The parametric vibrations induced by time-varying internal and external excitations are analyzed and classified into seven cases in terms of the parameter combinations. The instability areas of typical cases with the multi-frequency excitations are obtained based on Floquét theory. The influences of primary parameters such as the time-varying stiffness and force, orientation angle, and damping on the instabilities are studied. The relationships between natural frequency splitting and parametric instability are revealed and compared with the existing results in the open literature. The results in this study lay a foundation for predicting the parametric instabilities of similar structures. [ABSTRACT FROM AUTHOR]

The diagonal inerter is integrated into a suspension vibration reduction system (SVRS). The dynamic model of the SVRS with diagonal inerter and damping is established. The dynamic model is of strong geometric nonlinearity. The retaining non-linearity up to cubic terms is validated under impact excitation. The conditions omitting the static deformation are determined. The effects of the diagonal inerter on the vibration reduction performance of the SVRS are explored under impact and random excitations. The vibration reduction performance of the proposed SVRS with both diagonal inerter and damping is better than that of either the SVRS without them or the SVRS with the diagonal damping only. [ABSTRACT FROM AUTHOR]

Topology optimization for three-dimensional (3D) thin plate structure is an attractive methodology for versatile industrial and biomedical applications. For this perspective, the topology optimization requires an appropriate treatment of the 3D geometric nonlinearity of thin structures that avoids numerical instabilities, which is a well-known challenge in topology optimization. This paper develops a density-based topology optimization for thin plates and considers geometric nonlinearity using a 3D corotational triangle element formulation. The corotational formulation is an approach for expressing a finite deformation by dividing small strains and finite rotations into local element coordinates. Thus, the mechanical behavior in local coordinates can be assumed to be linearly elastic behavior that follows the small strain theorem. This technique is expected to be effective and stable for topology optimization with geometric nonlinearity. Complementary work minimization with volume constraints was applied for density-based topology optimization of the plate structure by a solid isotropic material with penalization method. Numerical examples of two benchmarks demonstrated consistencies with existing related works. We conducted topology optimization of an ankle-foot orthosis (AFO) as a biomedical application and showed the capabilities of the proposed methodology and the minimum increases of the complementary work with an optimum design with a volume reduction ratio. These achievements highlight the capabilities of the developed topology optimization as an efficient framework and feasibilities for a new orthosis design. [ABSTRACT FROM AUTHOR]

In this investigation, a discrete element model of single-layer arranged particles, which is suitable for solving problems of plate structures, is proposed through the discrete element method (DEM) according to stress characteristics of thin plate structures. Additionally, mass of the particles and rotary inertia of the contact element corresponding to the discrete model are modified accordingly. Then, expressions of stiffness coefficient of the contact springs among the discrete particles of the plate are derived based on the energy equivalence principle. Afterward, the proposed discrete model is applied to solve elastic problems of plate structure, such as integrated plate and plate with an aperture under in-plane tension, cantilever plate under bending and large deformation plate subjected to transverse loads, the results of which are well consistent with results obtained through finite element method. The key of the discrete element model is interaction among the particles, which is obtained according to motion equations and contact relationships. Furthermore, the stiffness matrix and iteration are not required during the calculation process by adopting DEM. The model can automatically meet the requirements of large deformation. Therefore, it is suitable for solving highly nonlinear problems of plates with large deformation or large rotation. [ABSTRACT FROM AUTHOR]