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Abstract In this study, a comprehensive analysis of visco-hyper-elastic thick soft arches under an external time-independent as well as time-dependent loads is presented from bending and internal resonance phenomenon perspectives. Axial, transverse and rotation motions are considered for modelling the thick and soft arch in the framework of the Mooney–Rivlin and Kelvin–Voigt visco-hyper-elastic schemes and third-order shear deformable models. The arch is assumed to be incompressible and is modelled using von Kármán geometric nonlinearity in the strain–displacement relationship. Using a virtual work method, the bending equations are derived. For the vibration analysis, three, coupled, highly nonlinear equations of motions are obtained using force-moment balance method. The Newton–Raphson method together with the dynamic equilibrium technique is used for the bending and vibration analyses. A detailed study on the influence of having visco-hyper-elasticity and arch curvature in the frequency response of the system is given in detail, and the bending deformation due to the applied static load is presented. The influence of having thick, soft arches with different slenderness ratios is shown, and the forced vibration response is discussed. Moreover, internal resonance in the system is studied showing that the curvature term in the structure can lead to three-to-one internal resonances, showing a rich nonlinear frequency response. The results of this study are a step forward in studying the visco-hyper-elastic behaviour of biological structures and soft tissues. Graphic abstract

The impact of roughness skewness (ksk) on turbulent boundary layer (TBL) flow with a zero pressure gradient over three-dimensional (3D) sinusoidal rough surfaces was experimentally investigated using a single hotwire anemometer. Nine 3D sinusoidal profiles were manufactured with positive, negative, and zero roughness skewness values. Measurements were taken at three different freestream velocities for each surface and compared with smooth wall TBL results. This study covered a range of friction Reynolds numbers (Reτ) from approximately 1000 to 4000, with δ/k≈20 ± 2, where δ represents the local boundary layer thickness, and k is the maximum height of the roughness, measured from the valley to peak. The results indicate that the wall-unit normalized streamwise mean velocity profiles for all rough surfaces exhibit a downward shift compared to the smooth wall profiles. Surfaces with positive roughness skewness produced the highest drag, leading to the largest downward shift. The friction coefficient (Cf) decreased as ksk decreased. The percentage increase in Cf and ΔU+ (the roughness function) was much larger when moving from negative to zero roughness skewness than when moving from zero to positive roughness skewness. The small differences in turbulence intensity profiles and higher-order turbulence statistics in the outer region of the TBL support the outer layer similarity hypothesis for the roughness considered in this study. The autocorrelation study revealed that surfaces with positive roughness skewness tend to shorten the average length of turbulence structures in the near-wall region.

In this study, the time-dependent mechanics of multilayered thick hyperelastic beams are investigated for the first time using five different types of shear deformation models for modelling the beam (i.e. the Euler–Bernoulli, Timoshenko, third-order, trigonometric and exponential shear deformable models), together with the von Kármán geometrical nonlinearity and Mooney–Rivlin hyperelastic strain energy density. The laminated hyperelastic beam is assumed to be resting on a nonlinear foundation and undergoing a time-dependent external force. The coupled highly nonlinear hyperelastic equations of motion are obtained by considering the longitudinal, transverse and rotation motions and are solved using a dynamic equilibrium technique. Both the linear and nonlinear time-dependent mechanics of the structure are analysed for clamped–clamped and pinned–pinned boundaries, and the impact of considering the shear effect using different shear deformation theories is discussed in detail. The influence of layering, each layer’s thickness, hyperelastic material positioning and many other parameters on the nonlinear frequency response is analysed, and it is shown that the resonance position, maximum amplitude, coupled motion and natural frequencies vary significantly for various hyperelastic and layer properties. The results of this study should be useful when studying layered soft structures, such as multilayer plastic packaging and laminated tubes, as well as modelling layered soft tissues.

A developing zero pressure gradient (ZPG) turbulent boundary layer (TBL) over different three-dimensional (3D) sinewave roughnesses is investigated experimentally using single hot-wire anemometry. Seven different sinewave profiles are fabricated with the same amplitude and with different wavelengths in the streamwise (sx) and spanwise (sz) directions. The effects of varying sx and sz on turbulence statistics and the drag coefficient (Cf) are assessed. The wall-unit normalized streamwise mean velocity profile is shifted downward compared with the smooth wall profile for all roughnesses. The streamwise spacing to height ratio sx/k has a more significant effect on the roughness function ΔU+ and Cf compared with the spanwise spacing to height ratio sz/k. However, sz/k has a large impact on the streamwise turbulence intensities in the log and outer layer. An excellent collapse is observed among the mean streamwise velocity profiles plotted in defect form in the outer region. However, a lack of similarity between TBLs over different rough surfaces is observed in the outer region for the turbulence intensities profiles. For isotropic 3D sinusoidal roughness (equal streamwise and spanwise spacing to height ratios), the contours of premultiplied streamwise turbulent energy spectrograms show an increase in energy in the outer layer with increasing spacing to height ratios. For anisotropic 3D sinusoidal roughness (unequal streamwise and spanwise spacing to height ratios), the contours of premultiplied streamwise turbulent energy spectrograms show an increase in energy in the outer layer with increasing sz/sx from half to two in this study.

Parallel robots are preferred over serial robots owing to their enhanced accuracy and rigidity which comes from their higher stiffness. However, there are applications both in industry and in healt...

The effects of different geometries of two-dimensional (2-D) roughness elements in a zero pressure gradient (ZPG) turbulent boundary layer (TBL) on turbulence statistics and drag coefficient are assessed using single hot-wire anemometry. Three kinds of 2-D roughness are used: (i) circular rods with two different heights, $k= 1.6$ and 2.4 mm, and five different streamwise spacing of $s_{x}= 6k$ to $24k$ , (ii) three-dimensional (3-D) printed triangular ribs with heights of $k= 1.6$ mm and spacing of $s_{x}= 8k$ and (iii) computerized numerical control (CNC) machined sinewave surfaces with two different heights, $k= 1.6$ and 2.4 mm, and spacing of $s_{x}= 8k$ . These roughnesses cover a wide range of ratios of the boundary layer thickness to the roughness height ( $23 < \delta /k < 41$ ), where $\delta$ is the boundary layer thickness. All roughnesses cause a downward shift on the wall-unit normalised streamwise mean velocity profile when compared with the smooth wall profiles agreeing with the literature, with a maximum downward shift observed for $s_{x}= 8k$ . In the fully rough regime, the drag coefficient becomes independent of the Reynolds number. Changing the roughness height while maintaining the same spacing ratio $s_{x}/k$ exhibits little influence on the drag coefficient in the fully rough regime. On the other hand, the effective slope $(ES)$ and the height skewness $(k_{sk})$ appear to be major surface roughness parameters that affect the drag coefficient. These parameters are used in a new expression for $k_{s}$ , the equivalent sand grain roughness, developed for 2-D uniformly distributed roughness in the fully rough regime.

An attempt is made in this article to analyse the large-amplitude local dynamics of nanofluid-conveying nanotubes with geometrical imperfections. Each element of the nanotube can have displacements along both longitudinal and transverse directions. A nonlinear damping model is also taken into account utilising the Kelvin–Voigt approach. The stress nonlocality and strain gradient influences are modelled using an advanced scale-dependent theory. Moreover, the Beskok–Karniadakis approach is applied for relative motions at the nanotube wall. To present the coupled motion equations of the coupled nanotube, Hamilton’s approach is used. Moreover, to develop a reliable solution procedure, Galerkin’s method along with continuation technique is utilised. The effects of nonlinear damping, geometrical imperfection, being at nanoscales, fluid velocity and relative motion at the wall on the large-amplitude local dynamics are investigated.

This article reviews, for the first time, the mechanical behaviour of functionally graded structures at small-scale levels. Functionally graded nanoscale and microscale structures are an advanced class of small-scale structures with promising applications in nanotechnology and microtechnology. Recent advancements in fabrication techniques such as the advent of powder metallurgy, made it possible to tailor the mechanical properties of structures at small-scale levels by fabricating them out of functionally layerwise mixture of two or more materials; this class of structures, called functionally graded (FG), can be used to improve the performance of many microelectromechanical and nanoelectromechanical systems due to their spatially varying mechanical and electrical properties. The increasing number of published papers on the mechanical behaviours of FG nanoscale and microscale structures, such as their buckling, vibration and static deformation, employing scale-dependent continuum-based models, has proved their importance in academia and industry. Generally, the nonlocal elasticity-based models have been used for FG nanostructures whereas modified versions of couple stress and strain gradient theories have been utilised for FG microstructures. In this review paper, first, various scale-dependent theories of elasticity for FG small-scale structures are explained. Then, available studies on the mechanical behaviours of FG nanostructures such as FG nanobeams and nanoplates are described. Moreover, available investigations on the mechanics of microstructures made of FG materials are reviewed. In addition, in each case, the most important findings of available studies are reviewed. Finally, further possible applications of advanced continuum mechanics to FG small-scale structures are inspired.

A viscoelastic axially functionally graded (FG) imperfect microbeam is considered and the complex nonlinear dynamics is analysed for the first time. The distribution of the material properties from beam's left end to the right one is of an exponential form. The small nature of the system is incorporated using the couple stress theory (CST) in a modified form. The viscosity between microbeam's infinitesimal elements is incorporated through the Kelvin-Voigt scheme. The elastic part of the energy is formulated via the elastic component of the stress and its couple; the viscous parts are incorporated via a negative work. The energy of kinetic type is also formulated. Without ignoring the axial displacement/inertia, Hamilton's principle of energy is used to formulate the viscoelastic mechanical model of the microsystem. A high-dimensional truncation/discretisation is conducted and numerical simulations are performed based upon it. The complex nonlinear dynamics of the axially FG imperfect beam is analysed in the presence/absence of viscosity.

In the present article, an effort is made to analyse the coupled global dynamics of nanoscale fluid-conveying tubes. The influences of geometric nonlinearity are captured through the nonlinear Euler–Bernoulli strain relation of beams. Moreover, the size influences related to the nanoscale tube are captured via developing a nonlocal strain gradient model of beams. The Beskok–Karniadakis theory is also used for capturing the size influences related to the nanofluid. In addition to size influences, Coriolis acceleration effects together with the influences of the centrifugal acceleration are taken into account. Hamilton's principle gives two coupled equations of motions, which are discretised utilising Galerkin's technique. A time integration scheme is used for extracting the global dynamic characteristics of the nanotube containing nanofluid flow. The non-dimensional critical speed associated with buckling is also determined. It is found that the nanofluid speed plays a crucial role in the global dynamics in both the subcritical and supercritical regimes.

In this paper, a scale-dependent coupled nonlinear continuum-based model is developed for the mechanical behaviour of imperfect nanoscale tubes incorporating both the effect of the stress nonlocality and strain gradient effects. The scale effects on the nonlinear mechanics are taken into consideration employing a modified elasticity theory on the basis of a refined combination of Eringen's elasticity and the strain gradient theory. According to the Euler–Bernoulli theory of beams, the nonlocal strain gradient theory (NSGT) and Hamilton's principle, the potential energy, kinetic energy and the work performed by harmonic loads are formulated, and then the coupled scale-dependent equations of the imperfect nanotube are derived. Finally, Galerkin's scheme, as a discretisation technique, and the continuation method, as a solution procedure for ordinary differential equations, are used. The effects of geometrical imperfections in conjunction with other nanosystem parameters such as the nonlocal coefficient as well as the strain gradient coefficient on the coupled large-amplitude mechanical behaviour are explored and discussed.

An elastically supported viscoelastic functionally graded (FG) microcantilever is considered and its nonlinear mechanics is analysed for the first time. Moreover, for the first time, energy transfer via internal resonances and motion complexity are analysed. A nonlinear spring model is incorporated as an elastic support which is representative of elasticity induced from neighbouring devices. Size effects are incorporated using the modified couple stress theory (MCST). Mori-Tanaka formula is utilised for FG-material-property variations. Kinematics/kinetics for an infinitesimal beam elements in conjunction with Hamilton's method are used for large curvatures. Galerkin's technique is used for reductions and truncations of the dynamic model of elastically supported viscoelastic FG microsystem. Both base-excitation/frequency continuations are performed and the dynamics is investigated.

This paper presents complete nonlinear electromechanical models for energy harvesting devices consisting of multiple piezoelectric bimorphs (PBs) connected in parallel and series, for the first time. The proposed model is verified against available experimental results for a specific case. The piezoelectric and beam constitutive equations and different circuit equations are utilized to derive the complete nonlinear models for series and parallel connections of the PBs as well as those of piezoelectric layers in each bimorph, i.e., four nonlinear models in total. A multi-modal Galerkin approach is used to discretize these nonlinear electromechanical models. The resultant high-dimensional set of equations is solved utilizing a highly optimized and efficient numerical continuation code. Examining the system behavior shows that the optimum load resistance for an energy harvester array of 4 PBs connected in parallel is almost 4% of that for the case with PBs connected in series. It is shown an energy harvesting array of 8 PBs could reach a bandwidth of 14 Hz in low frequency range, i.e., 20–34 Hz. Compared with an energy harvester with 1 PB, it is shown that the bandwidth can be increased by more than 300% using 4 PBs and by more than 500% using 8 PBs. Additionally, the drawbacks of a multi-PB energy harvesting device are identified and design enhancements are proposed to improve the efficiency of the device.

This paper examines the extremely large nonlinear vibrations of a cantilever subject to base excitation in primary and secondary resonance regions for the first time. To predict extremely large vibration amplitudes accurately, a geometrically exact continuous model of the cantilever is developed for the centreline rotation of the cantilever; the proposed model's accuracy is verified for extremely large deformations through comparison to a nonlinear finite element model. The theory of Euler-Bernoulli, along with inextensibility assumption, and the Kelvin-Voigt material damping model are utilised to develop the geometrically exact model. The main feature of the geometrically exact model is that all nonlinear trigonometric terms in the model are kept intact before and after the discretisation process, which itself is performed utilising the Galerkin scheme. The numerical results show that the cantilever undergoes extremely large oscillations even at relatively small base excitation amplitudes. It is shown that for some cases the amplitude of the tip of the cantilever grows so large that it “bends backward”; a behaviour which can only be captured using the proposed geometrically exact model.

Broadband nonlinear energy harvesting capabilities of a parametrically excited bimorph piezoelectric energy harvester is investigated for the first time. The performance of the energy harvester is significantly enhanced via use of stoppers and an added tip mass in conjunction with parametric excitation. A fully nonlinear electromechanical model of the energy harvester was developed using beam theory of Euler-Bernoulli and the coupled constitutive equations for piezoelectric materials, with the motion constraints modelled as nonlinear springs. A multi-modal discretisation was conducted utilising the Galerkin scheme; the resultant set of equations was examined numerically through use of continuation technique. It is shown that a resonance bandwidth of 46% (normalised with respect to parametric resonance frequency) is achieved which is almost 10 times the resonance bandwidth of the system without any constraints.

This research paper investigates the in-plane static, stability and dynamic characteristics of tapered Timoshenko bi-directional functionally graded (BDFG) sandwich curved elastic arches (symmetric and asymmetric) in thermo-mechanical environments using the differential quadrature element method (DQEM). Physical properties such as Young's modulus , the mass density, Poisson's ratio and the thermal expansion coefficient are defined to vary both with respect to temperature and position while shear deformation , axial deformation , and rotary inertia effects are incorporated in the analysis. Different types of arches namely circular, parabolic, catenary, elliptic and sinusoidal models are modelled by having non-uniform cross-section and the BDFG properties. General functions are used to vary the physical properties through the thickness and length of the curved. Incorporating the influence of having a thermal environment for simplified models, the current methodology for studying such structures is verified by comparing the results with previously published literature and finite element software simulations. The effect of the temperature rise, physical properties variation and geometrical parameters (such as the ratio of thickness to length, the opening angle and non-uniform cross-section) on the static deformation, buckling load and free vibration characteristics of BDFG arches with metal/ceramics in the inner surface and ceramics/metal in the outer surface of the arch is investigated. In addition, the mechanical behaviour of sandwich arches for various combinations such as homogeneous or FG for outer skins and the inner core is presented under thermo-mechanical loadings.

This paper investigates the nonlinear forced dynamical behaviour of a geometrically imperfect viscoelastic shear-deformable microplate. The third-order shear deformation plate theory and the Kelvin–Voigt viscoelastic model are utilised in the framework of the modified version of the couple-stress theory to develop a model for the microsystem. The developed model is in the form of partial differential equations (PDEs) and accounts for geometric nonlinearities, damping nonlinearities, micro-scale size effects, and initial imperfection. Five coupled PDEs are derived for the five independent displacements and rotations. These PDEs are truncated to a set of nonlinearly coupled ordinary differential equations via application of a two-dimensional modal decomposition based on the Galerkin technique. The final set of equations consists of quadratic and cubic nonlinear terms for both damping and stiffness. An efficient numerical algorithm based on a continuation scheme is utilised to analyse the nonlinear forced vibration characteristics of such complicated system. The effects imperfection amplitude, damping nonlinearities, and micro-scale size on forced resonant vibration response are highlighted.

One of the most high-risk locations of plaque growth and rupture initiation (and hence occurrence of heart attack) is the first main bifurcation of the left main coronary artery; the aim of this investigation is to analyse the nonlinear three-dimensional biomechanics of bifurcated atherosclerotic left coronary artery. In order to examine the influence of different system parameters, a biomechanical model of a bifurcated coronary artery is developed. Three plaques of varying geometry and material properties inside the three branches of the left main (LM), the left anterior descending (LAD), and the left circumflex (LCx) are modelled incorporating three-dimensionality, nonlinear geometric and material properties, asymmetry, viscosity, and hyperelasticity, and fluid-solid interaction. A finite element method (FEM) is employed to incorporate all of the above-mentioned important features in addition to physiological blood pulsation, heart motion, active media layer contraction, lipid plaque, calcium deposition, three different artery layers, micro-calcification, and non-Newtonian model for blood. Moreover, the effects of different system features such as stenosis, curved shape of the artery, plaque location, and fibrous cap thickness on the stress field (shear and structural) are examined. The developed biomechanical model could be utilised to estimate the risk of the initiation of plaque rupture inside the human coronary artery and the occurrence of heart attack.

Understanding the mechanical behaviour of nanostructures is of great importance due to their applications in nanodevices such as in nanomechanical resonators, nanoscale mass sensors, electromechanical nanoactuators and nanogenerators. Due to the difficulties of performing accurate experimental measurements at nanoscales and the high computational costs associated with the molecular dynamics simulations, the continuum modelling of nanostructures has attracted a considerable amount of attention. Since size influences have a crucial role in the mechanics of structures at nanoscale levels, classical continuum-based theories have been modified to incorporate these effects. Among various modified continuum-based theories, the nonlocal elasticity and the nonlocal strain gradient elasticity have been employed to estimate the mechanical behaviour of nanostructures. In this review paper, first these two modified elasticity theories are briefly explained. Then, the nonlocal motion equations for different nanostructures including nanorods, nanorings, nanobeams, nanoplates and nanoshells are derived. Several papers which reported on the size-dependent mechanical behaviour of nanostructures using modified continuum models are reviewed. Furthermore, important results reported on the vibration, bending and buckling of nanostructures as well as the results of size-dependent wave propagation analyses are discussed.

A nonlocal strain gradient elasticity approach is proposed for the mechanical behaviour of fluid-conveying nanotubes; a nonlinear analysis, incorporating stretching, is conducted for a model based on both a nonlocal theory along with a strain gradient one. A clamped–clamped nanotube conveying fluid, as a conservative gyroscopic nanosystem, is considered and the motion energy and size-dependent potential energy are developed via use of constitutive and strain–displacement relations. An energy minimisation is conducted via Hamilton's method for an oscillating nanotube subject to external forces. This gives the nonlinear equation of the motion which is reduced to a high DOF system via Galerkin's technique. As many nanodevices operate near resonance, the resonant motions are obtained using a frequency-continuation method. The effect of different nanosystem/fluid parameters, including fluid/solid interface and the flow speed, on the nonlinear resonance is analysed.

The focus of this paper is to model and simulate the nonlinear dynamics of atherosclerotic coronary arteries as a tool to predict the initiation of heart attack. A dynamic three-dimensional visco/hyperelastic fluid–structure interaction model of an atherosclerotic coronary artery is developed by means of the finite element method (FEM) using ANSYS. Simulations are undertaken using the model to examine the risk of plaque rupture with the following parameters taken into account with varying levels of stenosis: physiological pulsatile blood flow; tapered shape of the artery; viscoelasticity and hyperelasticity of the artery wall; effect of the motion of the heart; active artery muscle contraction; the lipid core inside the plaque; three layers of the artery wall; non-Newtonian characteristics of the blood flow; and micro-calcification; this paper is the first to incorporate all these effects. The generated model can potentially be used as a predictive tool for plaque rupture to identify the conditions that are high risk for atherosclerosis plaque rupture.

This paper addresses the nonlinear resonant dynamics of axially functionally graded (AFG) tapered beams subjected to initial geometric imperfections, based on the Timoshenko beam theory. A rigorous coupled axial–transverse–rotational nonlinear model is developed taking into account the geometric nonlinearities due to the large deformations coupled with an initial imperfection along the length of the beam as well as nonlinear expressions accounting for nonuniform tapered geometry and mechanical properties. The Hamilton’s energy principle is used to balance the kinetic and potential energies of the AFG imperfect tapered Timoshenko beam with the work done by damping and the external excitation load. This results in a set of strongly nonlinear partial differential equations (PDEs). The Galerkin decomposition scheme involving an adequate number of both symmetric and asymmetric modes is utilized to reduce the PDEs to a set of nonlinear ordinary differential equations. A well-optimized numerical scheme is developed to handle the high-dimensional discretized model.

The coupled nonlinear mechanical behavior of nonlocal strain gradient nanotubes subject to distributed excitation forcing is investigated for the first time. Both longitudinal displacements and tra...

This paper is the first to analyse the coupled fluid-structure viscoelastic dynamical characteristics of a fluid-conveying viscoelastic microtube resting on a nonlinear elastic bed subject to large rotations. None of the axial and transverse motions/accelerations is neglected in the modelling and simulations. The dissipation is modelled using the Kelvin–Voigt scheme for the deviatoric segment of the symmetric couple stress tensor and the stress tensor. Based on the Euler–Bernoulli theory, in which the microtube cross-section remains perpendicular to the centreline, and the modified couple stress theory (MCST), the energies and the work of external load and damping are formulated. Through use of Hamilton's principle, the coupled transverse-longitudinal equations governing the motion of the fluid-conveying viscoelastic microtube are developed. A weighted-residual-based discretisation method is applied to the continuous vibration model and the resultant reduced model is simulated via a continuation technique. The coupled fluid-structure dynamical characteristics of the fluid-conveying viscoelastic microtube are analysed by constructing the frequency-amplitude diagrams. It is shown that slight changes in the flow speed significantly affects the resonant response and modal interactions.

This study examines the nonlinear large-amplitude static and dynamic responses of a doubly curved shallow microshell in the framework of the modified couple stress (MCS) theory. To this end, the expressions for the classical and higher-order stresses and strains are consistently derived in an orthogonal curvilinear coordinate system employing the Novozhilov shell theory. The strain energy of the system is then consistently derived utilising the Novozhilov shell formulations in the framework of the MCS theory. The kinetic energy of the microshell is obtained while accounting for all out-of-plane and in-plane displacements. Furthermore, the work of the distributed out-of-plane load is accounted for and the energy dissipation is taken into account via the Rayleigh energy dissipation function. An assumed-mode technique is utilised to expand the out-of-plane and in-plane displacements via series expansions. The Lagrange equations are then utilised to derive the discretised equations of motion in the form of a set of nonlinearly coupled ordinary differential equations (ODEs). This set of nonlinear ODEs is solved making use of a continuation technique (for the nonlinear static and dynamic analyses) as well as an eigenvalue extraction method (for the linear natural frequency analysis). Extensive numerical simulations are carried out for both static and dynamic cases and the effects of different parameters, such as the radius of curvature, the magnitude and direction of the applied distributed load, and the small-scale parameter are investigated. The numerical results are constructed in the form of nonlinear static deflection curves, nonlinear dynamic frequency-amplitude diagrams, time traces, and phase-plane portraits.

Vibration energy harvesters are emerging as a promising solution for powering small-scale electronics, such as sensors and monitoring devices, especially in applications where batteries are costly or difficult to replace. However, current vibration energy harvesters are only effective within a limited frequency bandwidth, whereas most ambient vibrations occur randomly over a wide frequency range. Many techniques, such as tuning, coupling between modes, multimodal arrays and hybrid transduction methods, can be used for performance enhancement of vibration-based energy harvesters. Among these techniques is the introduction of nonlinearities to the energy harvesting system. In most cases, using nonlinear techniques for energy harvesting results in a larger frequency bandwidth when compared to a linear system. In certain systems, the introduction of nonlinearities can also result in a higher amplitude response. The aim of this paper is to conduct a critical review of nonlinear techniques which have been investigated for performance enhancement of energy harvesters in the past decade and present state of the art of energy harvesters which utilise this technique. This includes discussions of several techniques that have been employed for enhancing energy harvesting, such as stochastic loading, internal resonances, being multi-degree-of-freedom, mechanical stoppers and parametric excitations, which all lead to nonlinear behaviour and enhancement of the system. These techniques are capable of significantly extending the frequency bandwidth and, in some cases, increasing the amplitude response. The enhancement in performance results in devices that can harvest energy more efficiently from ambient vibrations.

As the first endeavour, the coupled nonlinear mechanical behaviour of extensible functionally graded microbeams, when both viscoelasticity and imperfection are present, is investigated. The imperfect viscoelastic microbeam is subject to a transverse harmonic excitation load of a constant amplitude. The Kelvin–Voigt viscoelastic model and Mori–Tanaka homogenisation method are used together in order to describe the internal energy loss and the variation of the material properties of the microsystem along the transverse direction, respectively. The geometric imperfection is modelled by imposing an initial curvature in the transverse deformation of the viscoelastic microscale beam. Using the Euler–Bernoulli strain–displacement relations, the geometric nonlinearity is taken into account. The non-classical nonlinear equations of motion are derived on the basis of Hamilton's principle and the modified couple stress theory. The resulting equations are found to be coupled between transverse and longitudinal oscillations. Galerkin's technique and the method of pseudo-arclength continuation as well as direct time-integration approach are finally employed to solve the governing differential equations for oscillation frequencies and nonlinear dynamic response. It is found that the nonlinear forced oscillations of extensible functionally graded microbeams are greatly affected by the internal energy loss together with geometric imperfection. It is shown that the simultaneous presence of viscoelasticity and geometric imperfections governs both the amplitude and softness/hardness of the dynamical behaviour.

In this paper, a size-dependent continuum-based model is presented for the coupled nonlinear dynamics of extensible functionally graded (FG) microbeams with viscoelastic properties. The length-scale effect is incorporated based on the modified couple stress theory (MCST). Moreover, employing the Kelvin–Voigt viscoelastic model, viscous components are taken into consideration in both stress and deviatoric segments of the symmetric couple stress tensors. The variation of the material properties of the FG viscoelastic microbeam along the thickness is approximated with the use of the Mori–Tanaka homogenisation method. Both the transverse and longitudinal motion as well as inertial terms are included in the size-dependent continuum model and numerical calculations. The elastic potential energy, kinetic energy and the viscos work are obtained with the consideration of size effects. Using von Karman's strain-displacement relations together with Hamilton's principle, the coupled differential equations of motion are derived. Then, Galerkin's approach and a continuation technique are used in order to obtain the fundamental frequency and dynamic response of the FG viscoelastic microbeam. The effects of parameters such as the gradient index, excitation frequency, the amplitude of the harmonic load and viscoelastic parameters on the nonlinear frequency- and force-responses of the FG viscoelastic microbeams are investigated in details.

The three-dimensional nonlinear mechanics and pull-in characteristics of a microplate-based microelectromechanical system (MEMS) are investigated via a multi-degree freedom energy-based technique where the in-plane and out-of-plane motions are retained in the modelling and simulations; the deformable microplate is modelled using the Kirchhoff's plate theory in conjunction with von Karman nonlinear strains, and it is assumed to be fully clamped at all the edges; an electrical field in the form of a combination of DC and AC voltages is applied to the deformable electrode of microplate-type. The modified couple stress theory is employed to model the small-size effects. The potential energy with size-dependent characteristics, together with the deformable microplate's kinetic energy, is formulated as functions of the displacements and mechanical and geometric parameters of the system. These energy terms, along with the Rayleigh energy dissipation and the electrical potential energy, are inserted into Lagrange's equations to derive the discretised model of the microplate-based MEMS consisting of three sets of second-order coupled reduced-order models for the in-plane and out-of-plane motions. Numerical simulations are conducted for both static and dynamic responses of the MEMS device. The numerical simulations have been performed via use of the pseudo-arc-length continuation technique in conjunction with backward-differentiation-formula (BDF) (for the nonlinear analysis); the Floquet theory is used for stability analysis. An eigenvalue extraction is employed for the linear analysis. Results are shown through DC voltage-deformation and AC frequency-motion diagrams in order to highlight the motion characteristics as well as pull-in instability of the microplate-based MEMS device.

In this paper, the porous-hyperelastic properties of soft materials are obtained experimentally and a general model for a combination of porosity (of functional type) and hyperelasticity using the Mooney-Rivlin strain energy density is obtained. Porous-hyperelastic samples are fabricated using thermoplastics with different porosities by varying the infill rate of 3D-printing. Following the available standards, the stress-strain behaviour for different samples are obtained and a general model for hyperelastic closed-cell porosity is presented. After obtaining model's characteristics from the experimental testings, a general beam formulation is presented for hyperelastic beams with functional porosity through the length. Both the axial and transverse motions are considered in the model of hyperelastic beams in the framework of the Mooney-Rivlin material model and Hamilton's principle. A geometrical imperfection of the beam is also considered in the formulation. The nonlinear forced vibrations of the imperfect porous-hyperelastic beam are studied by simultaneously solving the axial and transverse nonlinear coupled equations using a dynamic equilibrium technique. It is shown that having a uniform and functional porosity has a significant effect in changing the nonlinear frequency response of the system. Geometrical imperfection leads to a significant coupling between the axial and transverse coordinates when the porosity varies functionally through the length which shows the importance of considering both motions while analysing such structures. The results are useful for better understanding the effects of imperfections in studying the mechanics of soft structures and can be useful in designing soft robotics and artificial organs.

The size-dependent nonlinear oscillation characteristics of a functionally graded microplate is investigated numerically, in which all the displacements, i.e. in-plane as well as out-of-plane, and their inertia are accounted for. The potential energy of the functionally graded microsystem is obtained based on a modified version of the couple stress theory, so as to account for size effects, together with the Mori-Tanaka homogenisation mixture model for the graded material property. The kinetic and size-dependent potential energies of the microsystem are dynamically balanced by the work of an external force via the Lagrange equations and truncated employing an assumed-mode discretization scheme. Extensive numerical simulations are conducted upon the discretised model of the microsystem through use of a continuation technique as well as an eigenvalue extraction method (for the nonlinear and linear studies, respectively). The effect of several functionally graded microsystem parameters, namely the material gradient index, the material length-scale parameter, and the frequency and amplitude of an exciting external force on the response is investigated.

The nonlinear mechanics of three-layered microcantilevers under base excitation is investigated numerically. Employing the modified version of the couple stress-based theory, together with the Bernoulli-Euler beam theory, the potential energy of the three-layered microsystem is derived, while accounting for size effects. Obtaining the kinetic energy, the equations of motion in the longitudinal and transverse directions are derived via Hamilton's principle. Application of the inextensibility condition reduces the two equations of motion to one nonlinear integro-partial differential equation for the transverse oscillation, consisting of geometrical and inertial nonlinearities. The nonlinear equation of partial-differential type is reduced to set of equations of ordinary-differential type through use of a weighted-residual method. Solving the resultant set of discretised equations via a continuation technique gives the frequency-amplitude and force-amplitude responses of the microsystem. The nonlinear response is investigated for different layer composition and different layer thicknesses. The effect of small-scale parameter, as well as base excitation amplitude, is also examined.

The nonlinear dynamical characteristics of a doubly curved shallow microshell are investigated thoroughly. A consistent nonlinear model for the microshell is developed on the basis of the modified couple stress theory (MCST) in an orthogonal curvilinear coordinate system. In particular, based on Donnell’s nonlinear theory, the expressions for the strain and the symmetric rotation gradient tensors are obtained in the framework of MCST, which are then used to derive the potential energy of the microshell. The analytical geometrically nonlinear equations of motion of the doubly microshell are obtained for in-plane displacements as well as the out-of-plane one. These equations of partial differential type are reduced to a large set of ordinary differential equations making use of a two-dimensional Galerkin scheme. Extensive numerical simulations are conducted to obtain the nonlinear resonant response of the system for various principal radii of curvature and to examine the effect of modal interactions and the length-scale parameter.

In this paper, the nonlinear size-dependent forced vibrations of a functionally graded extensible microbeam is investigated based on the modified couple stress and Timoshenko beam theories; the transverse and longitudinal displacements and inertia as well as rotation and rotary inertia are included in the system modelling and simulations. The Mori-Tanaka homogenisation technique is employed to estimate the material properties of the extensible microbeam along the thickness. According to the modified couple stress theory, the elastic potential energy of the Timoshenko microbeam is obtained. The kinetic energy of the system as well as the work due to an external force is also developed. In order to derive the equations of motions, Hamilton's principle is employed retaining the transverse and longitudinal displacements and inertia as well as the rotation and rotary inertia; the Galerkin scheme is used to convert the nonlinear partial differential equations of motions into nonlinear ordinary differential equations. A numerical solution process, based on the pseudo-arclength continuation technique in conjunction with direct time-integration, is used; the nonlinear resonant dynamic responses of the system are obtained and frequency-response and force-response curves are plotted. Moreover, the effect of the length-scale parameter on the nonlinear forced dynamics of the system is analysed by comparing the results obtained using the modified couple stress theory versus the classical continuum theory. Finally, the influence of other system parameters, involving the material gradient index, on the frequency-responses and force-responses is investigated.

The nonlinear oscillations of viscoelastic microplates is addressed in this paper based on the modified couple stress theory (MCST). Employing the Kirchhoff plate theory, both out-of-plane and in-plane motions as well as the corresponding inertia are taken into account; an internal damping mechanism, based on Kelvin–Voigt model, is employed to model the behaviour of the material. The strain energy, the kinetic energy, the work done by the viscous parts of the classical and non-classical stresses, and the work of the external time-dependent force are obtained and implemented into Hamilton's framework so as to derive a set of fully coupled nonlinear partial differential equations (PDEs) for motions in the out-of-plane and in-plane directions. The Galerkin scheme is utilized to reduce the set of viscoelastically coupled nonlinear PDEs into a set of nonlinear ordinary differential equations (ODEs). Thereupon, this set of equations is transformed into a new set of size-dependent viscoelastically coupled nonlinear first-order ODEs and then is solved with the aid of a continuation scheme. The nonlinear oscillations is thoroughly investigated through conducting extensive numerical simulations and plotting force-response and frequency-response diagrams of the viscoelastic microsystem. The results reveal that the contributions of the nonlinear damping terms, arising due to employing a viscoelastic model, in the response of the viscoelastic microsystem increase substantially when the forcing amplitude is increased. Moreover, the concurrent presence of the nonlinear amplitude-dependent damping mechanism and the length-scale parameter affects the resonant response of the microplate significantly in both linear and nonlinear senses.

Modal interactions and internal energy transfers are investigated in the large-amplitude oscillations of a functionally graded microcantilever with an intermediate spring-support. Based on the Mori–Tanaka homogenization technique and the modified couple stress theory, the energy terms of the functionally graded microsystem (kinetic and size-dependent potential energies) are developed and dynamically balanced. Large-amplitude deformations, due to having one end free, are modeled taking into account curvature-related nonlinearities and assuming an inextensibility condition. The continuous model of the functionally graded microsystem is reduced, by means of the Galerkin method, yielding an inertial- and stiffness-wise nonlinear model. Numerical simulations on this highly nonlinear reduced-order model of the functionally graded microcantilever are performed using a continuation method; a possible case of modal interactions is determined by obtaining the natural frequencies of the microsystem. The nonlinear oscillations of the microcantilever are examined, and it is shown how the energy fed to the functionally graded microsystem (from the base excitation) is transferred between different modes of oscillation.

This paper, for the first time, analyses the size-dependent global dynamics of imperfect axially forced microbeams and shows that how a small initial imperfection (due to improper manufacturing of microbeams) can substantially change the size-dependent global dynamical behaviour of the microsystem; moreover, it investigates the effect of small size of the microbeam on the appearance and vanishing of different chaotic and quasiperiodic motions. More specifically, the continuous expressions for the size-dependent potential energy as well as kinetic energy of the microsystem are constructed and dynamically balanced via an energy method. A transformation to a reduced-order model is performed via a weighted-residual method. The bifurcation diagrams of Poincare maps are constructed by means of direct time-integrating the reduced-order model for the imperfect microsystem. Poincare sections, phase-plane diagrams, time histories, and fast Fourier transforms are also plotted for some cases in order to shed light on the microsystem size-dependent global dynamics.

The nonlinear size-dependent dynamics of a functionally graded micro-cantilever is investigated when subject to a base excitation resulting in large-amplitude oscillations. A geometric nonlinearities due to large changes in the curvature is taken into account. Employing the Mori–Tanaka homogenisation technique (for the material properties), the modified couple stress theory (MCST) is used to formulate the potential and kinetic energies of the system in terms of the transverse and axial motions. A dynamic energy balance is performed between the energy terms, yielding the continuous models for the axial and transverse displacements. The inextensibility condition results in the size-dependent model of the functionally graded micro-cantilever involving inertial and stiffness nonlinear terms. The resultant model is discretised based on a weighted-residual technique yielding a high-dimensional truncated model (required for accurate simulations). A parameter-continuation scheme together with a time integration method is introduced to the truncated model so as to determine the resonances with stable and unstable solution branches with special consideration to the effect of different system parameters, such as material gradient index and the length-scale effect on the nonlinear dynamics of the functionally graded micro-cantilever.

A viscoelastic model for the nonlinear analysis of the coupled transverse, longitudinal, and rotational oscillations of an imperfect shear deformable microbeam is developed, for the first time, based on the modified couple stress theory. An energy dissipation mechanism is developed via use of the Kelvin–Voigt internal energy dissipation mechanism. For the stress and deviatoric part of the symmetric couple stress tensors, the viscous components along with the corresponding work terms are obtained. The size-dependent elastic energy along with the kinetic energy of the viscoelastic microsystem is formulated in terms of the displacement field together with system geometric and physical parameters. The internal energy dissipation is developed via the work done by the viscous components of the stress and the deviatoric part of the symmetric couple stress tensors by means of the Kelvin–Voigt mechanism. These work and energy terms are inserted into Hamilton’s principle together with the work due to an external force in order to obtain three viscoelastically coupled equations governing the transverse, longitudinal, and rotational motions with cubic and quadratic nonlinear terms. A high-dimensional Galerkin approximation method is applied for all the three equations, yielding three sets of second-order coupled ordinary differential equations with cubic and quadratic nonlinearities. Upon application of a transformation, a continuation technique along with the backward differentiation formula (BDF) is employed in order to obtain the time-variant response of the system subject to a harmonic load. Special attention is paid to the effect of the Kelvin–Voigt type viscoelasticity on the system response in the presence of the length-scale parameter.

The nonlinear motion characteristics of a bilayered Timoshenko microbeam is analysed taking into account all the translational (i.e. longitudinal and transverse) and rotational motions; the effect of size is included through use of the modified couple stress theory. Considering a continuous variation through the thickness for the displacement field, the geometrically nonlinear strain terms are developed. Relating these strain terms to stress terms, via use of constitutive relations, leads to a stress tensor in terms of the displacement field; the same is applied to the symmetric curvature tensor and the deviatoric part of the symmetric couple stress tensor by means of a constitutive relation incorporating size effects. The potential energy of the bilayered microbeam is formulated via the modified couple stress theory. The kinetic energy along with the work of an external dynamic force is formulated in terms of the displacement field of the bilayered microbeam. A dynamic equilibrium state is obtained via balancing the energy and work. Three-dimensional reduced-order models for the transverse, longitudinal, and rotational motions are obtained via Galerkin's method. These coupled models are solved via a continuation method coupled with direct time-integration. The resonant responses are constructed with special consideration to the effects of the length-scale parameter and the material percentage and thickness of each layer; a comparison is also made between the motion characteristics of the bilayered and monolayered microbeam.

The vibration behaviour of a geometrically imperfect three-layered shear-deformable microbeam is analysed via model development and numerical simulations. Taking into account all the translational and rotational motions, considering continuous variations through the thickness for the displacement field, employing the modified couple stress theory for including small-size effects, and using constitutive relations for both stress and the deviatoric part of the couple stress tensors, the size-dependent elastic energy stored in the three-layered shear-deformable microbeam is obtained. The kinetic energy, work of an internal damping mechanism, and the work due to an external harmonic excitation force of the three-layered microsystem are also obtained and dynamically balanced by the size-dependent elastic energy by means of Hamilton's principle. The continuous expressions obtained for the axial, transverse, and rotational motions of the three-layered microsystem are truncated to high-dimensional reduced-order models with the help of a weighted-residual method. Numerical simulations are conducted by means of the backward differentiation formula (BDF) in conjunction with a continuation method. The nonlinear vibration behaviour of the microsystem is then analysed by plotting the coupled frequency-responses. The effect of the length-scale parameters as well as the material fraction and thickness of each layer on the vibration behaviour of the microsystem is highlighted.

In this paper, the complex viscoelastically coupled global mechanics of fluid-conveying microtubes is examined for the first time. The externally excited microtube is assumed to be embedded in a nonlinear elastic medium. A scale-dependent theoretical model is presented with consideration of curvature nonlinearity within the context of the modified version of the couple stress theory (CST). According to Hamilton's energy/work principle, the coupled nonlinear equations of fluid-conveying microscale tubes are presented. Both the transverse and longitudinal displacements and inertia are taken into account in the continuum-based model and numerical calculations. In order to discretise the governing nonlinear differential equations, Galerkin's weighted-residual procedure is employed. The bifurcation characteristics of the fluid-conveying microsystem with clamped-clamped boundary conditions are obtained within the framework of a direct time-integration procedure. It is found that the complex global dynamics of the fluid-conveying microsystem is very sensitive to the speed of the flowing fluid.