Showing papers in "International Journal of Non-linear Mechanics in 2013"

Numerical and experimental investigation of a highly effective single-sided vibro-impact non-linear energy sink for shock mitigation

Abstract: In this paper a highly asymmetric, lightweight, vibro-impact non-linear energy sink (NES) leading to very efficient passive non-linear targeted energy transfer (TET) is investigated. To this end, a two degree-of-freedom linear system (the primary structure) is coupled to a single-sided vibro-impact (VI) attachment with highly asymmetric impact non-linearity (the VI NES). The proposed NES passively absorbs and rapidly dissipates a considerable amount of the impulse energy induced into the linear structure, leading to very effective shock mitigation compared to a double-sided (symmetric) VI NES. We find that appropriate selection of the weak linear stiffness that couples the non-linear VI attachment to the linear structure plays a significant role in the proposed design. Moreover, in contrast to the double-sided VI NES which has optimal performance for a narrow range of input energies, the proposed single-sided asymmetric VI NES maintains a high level of performance over a broad range of high input energies. Hence, the proposed design is especially suitable for severe shock mitigation in infrastructure. To quantify the enhanced shock mitigation performance of the asymmetric VI NES we employ measures of effective damping and stiffness developed in previous works to demonstrate that the primary structure with attached NES possesses drastically increased effective damping and stiffness compared to its nominal properties when no NES is attached. A series of experimental results fully validates the theoretical predictions.

New convex yield functions for orthotropic metal plasticity

Abstract: Two new yield functions for orthotropic sheet metals are proposed. The first one, called Yld2011-18p, provides 18 parameters that may be calibrated to experimental data. The second one, called Yld2011-27p, is a straightforward extension and provides 27 parameters. Both yield functions are unconditionally convex. Their formulations are based on the established concept of multiple linear transformations of the stress deviator. Furthermore, they are able to account for planar as well as for three-dimensional stress states. The proposed yield functions are applied to describe complex plastic anisotropies of different alloys. The ability of accurately predicting earing in cup-drawing is demonstrated by means of a non-linear finite element analysis.

Deterministic quantities characterizing noise driven Hopf bifurcations in gas turbine combustors

Abstract: Lean premix gas turbine combustors are prone to high amplitude pressure oscillations driven by non-linear thermoacoustic coupling. These pulsations are unwanted because they can affect the lifetime of the combustor parts. The standard strategy to get rid of these oscillations is to implement acoustic damping devices. Knowing the deterministic components characterizing the acoustic-flame coupling, the linear growth rates in particular, is necessary to properly design the dampers. However, the time scale associated with the variations of the engine operating conditions is much larger than the one of the acoustic pressure amplitude dynamics. Therefore, the linearly unstable system cannot be observed during the exponential amplitude growth of one of the acoustic eigenmodes and it is not possible to directly determine the linear growth rates. They can only be estimated from signals recorded when the system is operating on limit cycling states. Fortunately, these states are driven by a strong stochastic forcing produced by the highly turbulent reactive flow. It is shown in this article that the deterministic quantities can be extracted from the noise perturbed limit cycles data by making use of the stochastic differential equations describing the combustion instabilities. A straightforward experimental set-up allowing to reproduce the main features of the thermoacoustic coupling observed in gas turbines is used to validate the proposed identification methods. In a second step, these latter methods are applied to engine data.

A hybrid contact force model of revolute joint with clearance for planar mechanical systems

Abstract: The contact force model during the contact process of revolute joints with clearance is one of the most important contents. This paper presents a new contact force model of revolute joint with clearance for planar mechanical systems, which is a hybrid model of the Lankarani–Nikravesh model and the improved elastic foundation model. The framework of the Lankarani–Nikravesh model is used with the nonlinear stiffness coefficient derived using the improved elastic foundation model and the damping applied in introducing the ratio of the nonlinear stiffness coefficient of the improved elastic foundation model and contact stiffness of Lankarani–Nikravesh model. Furthermore, the hybrid contact force model is analyzed and compared with Lankarani–Nikravesh model as well as other existing contact models. The tangential contact is represented by using modified Coulomb friction model. And then, the dynamic characteristics of mechanical system with revolute clearance joint are analyzed based on the hybrid contact force model. The correctness and validity of the hybrid contact force model of the revolute joint clearance is verified through the demonstrative application example. Finally, the numerical simulation results show that the presented hybrid contact force model is an effective and new method to predict the dynamic characteristics of planar mechanical systems with clearance in revolute joints.

Self-propulsion of slender micro-swimmers by curvature control: N-link swimmers

Abstract: We discuss a reduced model to compute the motion of slender swimmers which propel themselves by propagating a bending wave along their body. Our approach is based on the use of resistive force theory for the evaluation of the viscous forces and torques exerted by the surrounding fluid, and on discretizing the kinematics of the swimmer by representing its body through an articulated chain of N rigid links capable of planar deformations. The resulting system of ODEs governing the motion of the swimmer is easy to assemble and to solve, making our reduced model a valuable tool in the design and optimization of bio-inspired engineered microdevices. We test the accuracy and robustness of our approach on three benchmark examples: Purcell's 3-link swimmer, Taylor's swimming sheet and some recent quantitative observations of circular motion of a sperm cell. An explicit formula for the displacement of Purcell's 3-link swimmer generated by a square stroke of small amplitude is also discussed.

Coupled global dynamics of an axially moving viscoelastic beam

Abstract: The nonlinear global forced dynamics of an axially moving viscoelastic beam, while both longitudinal and transverse displacements are taken into account, is examined employing a numerical technique. The equations of motion are derived using Newton′s second law of motion, resulting in two partial differential equations for the longitudinal and transverse motions. A two-parameter rheological Kelvin–Voigt energy dissipation mechanism is employed for the viscoelastic structural model, in which the material, not partial, time derivative is used in the viscoelastic constitutive relations; this gives additional terms due to the simultaneous presence of the material damping and the axial speed. The equations of motion for both longitudinal and transverse motions are then discretized via Galerkin’s method, in which the eigenfunctions for the transverse motion of a hinged-hinged linear stationary beam are chosen as the basis functions. The subsequent set of nonlinear ordinary equations is solved numerically by means of the direct time integration via modified Rosenbrock method, resulting in the bifurcation diagrams of Poincare maps. The results are also presented in the form of time histories, phase-plane portraits, and fast Fourier transform (FFTs) for specific sets of parameters.

Steady-state transverse response of an axially moving beam with time-dependent axial speed

Abstract: This paper investigates the non-linear dynamics of an axially moving beam with time-dependent axial speed, including numerical results for the non-linear resonant response of the system in the sub-critical speed regime and global dynamical behavior. Using Galerkin's technique, the non-linear partial differential equation of motion is discretized and reduced to a set of ordinary differential equations (ODEs) by choosing the basis functions to be eigenfunctions of a stationary beam. The set of ODEs is solved by the pseudo-arclength continuation technique, for the system in the sub-critical axial speed regime, and by direct time integration to investigate the global dynamics. Results are shown through frequency–response curves as well as bifurcation diagrams of the Poincare maps. Points of interest in the parameter space in the form of time traces, phase-plane portraits, Poincare maps, and fast Fourier transforms (FFTs) are also highlighted. Numerical results indicate that the system displays a wide variety of rich and interesting dynamical behavior.

Nonlinear reduced order modeling of isotropic and functionally graded plates

Abstract: The focus of this investigation is on the development and validation of non-linear structural dynamic reduced order models of structures undergoing large deformations, with particular emphasis on aircraft panels. Significant efforts are devoted to the formulation and assessment of “dual modes” which when combined with the linear transverse modes form an excellent basis for the representation of the displacement and stress fields in the reduced order model. This task is first successfully achieved for isotropic/symmetric composite structures and then extended to asymmetric and functionally graded ones. Examples of application are presented that demonstrate the high accuracy of the proposed reduced order models as compared to full finite element preditions, even with a small number of modes.

The influence of residual stress on finite deformation elastic response

Abstract: This paper is concerned with the effect of residual stress on the elastic behaviour of materials undergoing finite elastic deformations. The theory is based on a general constitutive framework for hyperelastic materials with residual stress. Several simple problems, whose solutions are known in the situation where there is no residual stress, are analyzed in order to elucidate the influence of residual stress, and the results are illustrated for two prototype constitutive laws.

Thermal postbuckling behavior of size-dependent functionally graded Timoshenko microbeams

Abstract: The thermal postbuckling characteristics of microbeams made of functionally graded materials (FGMs) undergoing thermal loads are investigated based on the modified strain gradient theory (MSGT). The volume fraction of the ceramic and metal phases of FGM microbeams is expressed by using a power low function. The non-classical beam model presented herein is capable of interpreting size effects through introducing material length scale parameters and encompasses the modified couple stress theory (MCST) and classical theory (CT). Based on the non-linear Timoshenko beam theory and the principle of virtual work, the stability equations and associated boundary conditions are derived and are then solved through the generalized differential quadrature (GDQ) method in conjunction with a direct approach without linearization. The influences of the material gradient index, length scale parameter, and boundary conditions on the thermal postbuckling behavior of FGM microbeams are comprehensively investigated. Also, this study compares the results obtained from the MSGT with those from CT. The effect of geometrical imperfection on the buckling deformation of microbeams in prebuckled and postbuckled states is discussed.

Stationary response of Duffing oscillator with hardening stiffness and fractional derivative

Abstract: The stationary response of Duffing oscillator with hardening stiffness and fractional derivative under Gaussian white noise excitation is studied. First, the term associated with fractional derivative is separated into the equivalent quasi-linear dissipative force and quasi-linear restoring force by using the generalized harmonic balance technique, and the original system is replaced by an equivalent nonlinear stochastic system without fractional derivative. Then, the stochastic averaging method of energy envelope is applied to the equivalent nonlinear stochastic system to yield the averaged Ito equation of energy envelope, from which the corresponding Fokker–Planck–Kolmogorov (FPK) equation is established and solved to obtain the stationary probability densities of the energy envelope and the amplitude envelope. The accuracy of the analytical results is validated by those from the Monte Carlo simulation of original system.

Solution of the model of beam-type micro- and nano-scale electrostatic actuators by a new modified Adomian decomposition method for nonlinear boundary value problems

Abstract: In this paper we solve the common nonlinear boundary value problems (BVPs) of cantilever-type micro-electromechanical system (MEMS) and nano-electromechanical system (NEMS) using the distributed parameter model by the Duan–Rach modified Adomian decomposition method (ADM). The nonlinear BVPs that are investigated include the cases of the single and double cantilever-type geometries under the influence of the intermolecular van der Waals force and the quantum Casimir force for appropriate distances of separation. The new Duan–Rach modified ADM transforms the nonlinear BVP consisting of a nonlinear differential equation subject to appropriate boundary conditions into an equivalent nonlinear Fredholm–Volterra integral equation before designing an efficient recursion scheme to compute approximate analytic solutions without resort to any undetermined coefficients. The new approach facilitates parametric analyses for such designs and the pull-in parameters can be estimated by combining with the Pade approximant. We also consider the accuracy and the rate of convergence for the solution approximants of the resulting Adomian decomposition series, which demonstrates an approximate exponential rate of convergence. Furthermore we show how to easily achieve an accelerated rate of convergence in the sequence of the Adomian approximate solutions by applying Duan's parametrized recursion scheme in computing the solution components. Finally we compare the Duan–Rach modified recursion scheme in the ADM with the method of undetermined coefficients in the ADM for solution of nonlinear BVPs to illustrate the advantages of our new approach over prior art.

Modeling and measuring visco-elastic properties: From collagen molecules to collagen fibrils

Abstract: Collagen is the main structural protein in vertebrate biology, determining the mechanical behavior of connective tissues such as tendon, bone and skin. Although extensive efforts in the study of the origin of collagen exceptional mechanical properties, a deep knowledge of the relationship between molecular structure and mechanical properties remains elusive, hindered by the complex hierarchical structure of collagen-based tissues. Understanding the viscoelastic behavior of collagenous tissues requires knowledge of the properties at each structural level. Whole tissues have been studied extensively, but less is known about the mechanical behavior at the submicron, fibrillar and molecular level. Hence, we investigate the viscoelastic properties at the molecular level by using an atomistic modeling approach, performing in silico creep tests of a collagen-like peptide. The results are compared with creep and relaxation tests at the level of isolated collagen fibrils performed previously using a micro-electro-mechanical systems platform. Individual collagen molecules present a non-linear viscoelastic behavior, with a Young's modulus increasing from 6 to 16 GPa (for strains up to 20%), a viscosity of 3.84±0.38 Pa s, and a relaxation time in the range of 0.24–0.64 ns. At the fibrils level, stress–strain–time data indicate that isolated fibrils exhibit viscoelastic behavior that could be fitted using the Maxwell–Weichert model. The fibrils showed an elastic modulus of 123±46 MPa. The time-dependent behavior was well fit using the two-time-constant Maxwell–Weichert model with a fast time response of 7±2 s and a slow time response of 102±5 s.

Torsion and curvature effects on fluid flow in a helical annulus

Abstract: In this article incompressible viscous flow in a helical annulus is studied numerically. A second order finite difference method based on the projection algorithm is used to solve the governing equations written in the helical coordinate system. Considering the hydrodynamically fully developed flow, the effects of different physical parameters such as aspect ratio, torsion, curvature and Reynolds number on the flow field are investigated in detail. The numerical results obtained indicate that a decrease in the aspect ratio and torsion number leads to the increase of the friction factor at a given Dean number.

Non-linear response of buckled beams to 1:1 and 3:1 internal resonances

Abstract: The non-linear response of a buckled beam to a primary resonance of its first vibration mode in the presence of internal resonances is investigated. We consider a one-to-one internal resonance between the first and second vibration modes and a three-to-one internal resonance between the first and third vibration modes. The method of multiple scales is used to directly attack the governing integral–partial–differential equation and associated boundary conditions and obtain four first-order ordinary-differential equations (ODEs) governing modulation of the amplitudes and phase angles of the interacting modes involved via internal resonance. The modulation equations show that the interacting modes are non-linearly coupled. An approximate second-order solution for the response is obtained. The equilibrium solutions of the modulation equations are obtained and their stability is investigated. Frequency–response curves are presented when one of the interacting modes is directly excited by a primary excitation. To investigate the global dynamics of the system, we use the Galerkin procedure and develop a multi-mode reduced-order model that consists of temporal non-linearly coupled ODEs. The reduced-order model is then numerically integrated using long-time integration and a shooting method. Time history, fast Fourier transforms (FFT), and Poincare sections are presented. We show period doubling bifurcations leading to chaos and a chaotically amplitude-modulated response.

A new development and interpretation of the Navier–Stokes fluid which reveals why the “Stokes assumption” is inapt

Abstract: In this short paper I present an alternate way of describing fluids in general, and the Navier–Stokes fluid in particular, from a phenomenological point of view, that shows clearly that the putative assumption conjectured by Stokes is not a reasonable assumption. I also show that the procedure presented here is more suited for incorporating constraints such as that of incompressibility, as well as having other advantages. The approach also helps to pinpoint several serious errors in the justifications that are provided in classical texts for the development of the Navier–Stokes model. Finally, from the point of view of the role that causality plays in Newtonian mechanics, the approach suggested is the preferable approach.

Subharmonic resonance of a symmetric ball bearing–rotor system

Abstract: The performance of a ball bearing–rotor system is often limited by the occurrence of subharmonic resonance with considerable vibration and noise. In order to comprehend the inherent mechanism and the feature of the subharmonic resonance, a symmetrical rotor system supported by ball bearings is studied with numerical analysis and experiment in this paper. A 6DOF rotordynamic model which includes the non-linearity of ball bearings, Hertzian contact forces and bearing internal clearance, and the bending vibration of rotor is presented and an experimental rig is offered for the research of the subharmonic resonance of the ball bearing–rotor system. The dynamic response is investigated with the aid of orbit and amplitude spectrum, and the non-linear system stability is analyzed using the Floquet theory. All of the predicted results coincide well with the experimental data to validate the proposed model. Numerical and experimental results show that the resonance frequency is provoked when the speed is in the vicinity of twice synchroresonance frequency, while the rotor system loses stability through a period-doubling bifurcation and a period-2 motion i.e. subharmonic resonance occurs. It is found that the occurrence of subharmonic resonance is due to the together influence of the non-linear factors, Hertzian contact forces and internal clearance of ball bearings. The effect of unbalance load on subharmonic resonance of the rotor system is minor, which is different from that of the sliding bearing–rotor system. However, the moment of couple has an impact influence on the subharmonic resonances of the ball bearing–rotor system. The numerical and experimental results indicate that the subharmonic resonance caused by ball bearings is a noticeable issue in the optimum design and failure diagnosis of a high-speed rotary machinery.

Numerical simulation of shear and the Poynting effects by the finite element method: An application of the generalised empirical inequalities in non-linear elasticity

Abstract: Finite element simulations of different shear deformations in non-linear elasticity are presented. We pay particular attention to the Poynting effects in hyperelastic materials, complementing recent theoretical findings by showing these effects manifested by specific models. As the finite element method computes uniform deformations exactly, for simple shear deformation and pure shear stress, the Poynting effect is represented exactly, while for the generalised shear and simple torsion, where the deformation is non-uniform, the solution is approximated efficiently and guaranteed computational bounds on the magnitude of the Poynting effect are obtained. The numerical results further indicate that, for a given elastic material, the same sign effect occurs under different shearing mechanisms, showing the genericity of the Poynting effect under a variety of shearing loads. In order to derive numerical models that exhibit either the positive or the negative Poynting effect, the so-called generalised empirical inequalities, which are less restrictive than the usual empirical inequalities involving material parameters, are assumed.

A comprehensive study of stability in an electro-statically actuated micro-beam

Abstract: This paper studies the mechanical and bifurcation behavior of a capacitive micro-beam suspended between two conductive stationary plates, which can be used as a micro-switch or as a RF resonator. The equation of dynamic motion of the micro-switch is obtained using Euler–Bernoulli beam theorem. The equilibrium positions or the fixed points of the micro-switch are obtained by solving the equation of the static deflection using the step-by-step linearization method (SSLM) and discretizing by Galerkin weighted residual method. In order to study the global stability of the obtained fixed points a modified non-linear mass–spring model is used. Non-linear motion trajectories in phase portraits are given and regions of bounded and unbounded solutions separated by a homoclinic or heteroclinic orbits and positions of the stationary conductive plates are illustrated. Critical values of the applied voltage leading to qualitative changes in the micro-beam behavior through a saddle node or pitch fork bifurcations for different values of the gap and voltage ratios are obtained. The effects of different gaps and voltage ratios also are investigated.

Flow of power-law fluid over a stretching surface: A Lie group analysis

Abstract: This paper investigates the boundary layer flow of power-law fluid over a permeable stretching surface. The use of Lie group analysis reveals all possible similarity transformations of the problem. The application of infinitesimal generator on the generalized surface stretching conditions leads to two possible surface conditions which leads to the possibility of two types of stretching velocities namely; the power-law and exponential stretching. The power-law stretching has already been discussed in the literature, however exponential stretching is investigated here for the first time. Interestingly, an exact analytical solution of the non-linear similarity equation for exponential stretching is developed for shear thinning fluid with power-law index n =1/2. This solution is further extended to a larger class of shear thinning fluids ( n ≈1/2) using perturbation method. In addition, the numerical solution for shear thinning fluid is also presented. The two solutions match excellently for shear thinning fluids. Analytical solution for shear thickening fluid is not tractable and the numerical solution is presented for completeness.

On the parametric large deflection study of Euler–Bernoulli cantilever beams subjected to combined tip point loading

Abstract: The problem of determining the parametric large deflection components of Euler–Bernoulli cantilever beams subjected to combined tip point loading is studied in this paper. We introduce the characteristic equation of the beam's deflection and, with employing the recently developed automatic Taylor expansion technique (ATET), present deflection solutions in terms of the loading parameters to the Euler–Bernoulli boundary value problem. The obtained ATET deflection solutions, verified by comparison with the numerical solutions, are valid for the entire beam length, and independently and efficiently adaptable for the very large loading conditions, and easily implementable for engineering analyses and syntheses. Exploiting these solutions as theoretical tools we study the beam's angular and axial deflections behavior for several tip point loading conditions. Besides the widely known beam's axial inflection points, we also recognize beam's angular inflection points for the mixed loading condition and show that the parametric solutions are intelligent in recognizing the right deflection branch for both inflection types.

Deformation dependent dielectric permittivity and its effect on actuator performance and stability

Abstract: We utilize a model for birefringence/permittivity based on the statistical mechanics of a Gaussian polymer chain to construct a relationship for the dependence of the dielectric permittivity of an elastomer on a general 3-dimensional state of deformation. The model, due to Kuhn and Grun (1942 [1] ), expresses the birefringence/permittivity of a Gaussian polymer chain elastomer as a function of the end-to-end distance of the chains, and assumes that the motions of the chains are affine to the overall deformation. The outcome is an expression for the permittivity tensor of the elastomer as a function of its stretch ratios. The permittivity is isotropic in the undeformed state and under pure dilatation, but otherwise becomes anisotropic during deformation. With this model, we use the free energy of the elastomer to compute the response of a neo-Hookean thin film in an actuator configuration subject to electric and mechanical loading for conditions where the permittivity in the through thickness direction is allowed to increase or decrease with the in-plane extension of the thin film. With such an approach, we study the deformation characteristics of the actuator and its stability under through thickness electric fields. Our calculations show that the deformation dependent permittivity can hasten or postpone an electromechanical instability that can cause a sudden thinning of the dielectric, accompanied by in-plane stretching, when the through thickness electric field is raised above a critical magnitude. Specifically, we consider the case of an actuator exhibiting a through thickness permittivity that decreases with in-plane extension. We observe that in such an actuator the instability is delayed to a higher electric field than would be the case if the dielectric permittivity were independent of strain. Furthermore, we establish that upon removal of the electric field the system follows a different path in terms of potential versus charge, and so develops a hysteresis loop, similar to that identified by Zhao et al. (2007 [2] ) for dielectric elastomers with constant isotropic permittivity, but that stiffen during straining.

Finite inflation analysis of a hyperelastic toroidal membrane of initially circular cross-section

Abstract: In this work, we have studied the finite inflation of a hyperelastic toroidal membrane with an initially circular cross-section under internal pressure. The membrane material is assumed to be a Mooney–Rivlin solid. The inflation problem is formulated as a variational problem for the total potential energy comprising the membrane strain energy and internal energy of the gas. The problem is then discretized and solved up to a high degree of accuracy through a sequence of approximations based on the Ritz expansion of the field variables combined with a potential energy density perturbation and Newton–Raphson method. The effects of the inflation pressure and material properties on the state of stretch and geometry of the inflated torus have been studied, and some interesting results have been obtained. The stability of the inflated configurations in terms of impending wrinkling of the membrane has been investigated on the principal stretch parameter plane for both isotropic and anisotropic (transversely isotropic) material cases. Certain shape factors quantifying the geometry of the membrane have been defined and calculated which characterize the cross-sectional shape and size of the torus during inflation.

A geometrically non-linear formulation of a three-dimensional beam element for solving large deflection multibody system problems

Abstract: A beam finite element formulation for large deflection problems in the analysis of flexible multibody systems has been proposed. In this formulation, a set of independent discrete deformation modes are defined for each element which are related to conventional small deflection beam theory in a co-rotational frame. The paper examines the applicability of this formulation for a shear-deformable three-dimensional Timoshenko beam model, in which geometric non-linearities due to large deflections, buckling loads and post-buckling are included. The geometric non-linearities are accounted for by additional second-order terms in the expressions for the deformation modes. Some numerical examples including large deflections are presented and discussed in order to illustrate the influence of these terms on the accuracy and rate of convergence. The influence of these terms on the displacements is small, except for bifurcation points where the load–deflection characteristics change drastically. It is demonstrated, by comparison with available results in the literature, that highly accurate solutions can be obtained with the present beam finite element formulation

On the contact problem of an inflated spherical hyperelastic membrane

Abstract: In this paper, the mechanics of contact of an inflated spherical non-linear hyperelastic membrane pressed between two rigid plates has been studied. We have considered the membrane material to be a homogeneous and isotropic Mooney–Rivlin hyperelastic solid. All three cases, namely frictionless, no-slip and stick–slip conditions have been considered separately in the plate-membrane contact region. The stretch of the membrane, and the surface traction (for no-slip contact) has been determined. For the stick–slip case, the sliding front is observed to be initiated at the contact periphery which moves towards the pole. The state at which the impending wrinkling condition occurs has been determined analytically. It is observed that the impending wrinkling state occurs at the periphery of the contact. Based on this, the minimum initial stretch (inflation) required to prevent wrinkling at any point in the membrane has been determined.

Influence of size effect and elastic boundary condition on the pull-in instability of nano-scale cantilever beams immersed in liquid electrolytes

Abstract: In this study, the static pull-in instability of nanocantilever beams immersed in a liquid electrolyte is theoretically investigated. In modeling the nanocantilever beam, the effects of van der Waals forces, elastic boundary condition and size dependency are considered. The modified couple stress theory, containing material length scale parameter, is used to interpret the size effect which appears in micro/nanoscale structures. The modified Adomian decomposition (MAD) method is used to gain an approximate analytical expression for the critical pull-in parameters which are essential for the design of micro/nanoactuators. The results show that the beam can deflect upward or downward, based on the values of the non-dimensional parameters. It is found that the size effect greatly influences the beam deflection and is more noticeable for small thicknesses. Neglecting size effect overestimates the deflection of the nanobeam. The findings reveal that the increase of ion concentration increases the pull-in voltage but decreases the pull-in deflection. Furthermore, an increase in ion concentration increases the influence of size-dependent effect on pull-in voltage.

Exact solutions of linear and non-linear differential-difference heat and diffusion equations with finite relaxation time

Abstract: We consider heat and diffusion equations with finite relaxation time which ensure a finite speed of propagation of disturbances. We use the Cattaneo–Vernotte model for the heat flux and obtain a number of exact solutions to the corresponding linear differential-difference heat equation. We also give exact solutions to two one-dimensional Stokes problem for a differential-difference mass/heat transfer equation (without a source and with a linear source) with a periodic boundary condition. We describe a number of exact solutions to non-linear differential-difference heat equations of the form T ¯ t = div [ f ( T ) ∇ T ] + g ( T ¯ ) , T ¯ = T ( x , t + τ ) , where τ is the relaxation time. In addition, we obtain some exact solutions to non-linear systems of two coupled reaction–diffusion equations with finite relaxation time and present several exact solutions of non-linear reaction–diffusion equations with time-varying delay of the form u t = ku xx + F ( u , w ) , w = u ( x , t − τ ) , where τ = τ ( t ) . All equations in question contain arbitrary functions or free parameters. Their solutions can be used to solve certain problems and test numerical methods for non-linear partial differential-difference equations (delay partial differential equations).

Non-linear dynamic instability of functionally graded plates in thermal environments

Abstract: Geometrically non-linear parametric instability of functionally graded rectangular plates in thermal environments is investigated via a multi-degree-of-freedom energy approach. Non-linear higher-order shear deformation theory is used and the non-linear response to in-plane static and harmonic excitation in the frequency neighborhood of twice the fundamental mode is investigated. The boundary conditions are assumed to be simply supported movable. Numerical analyses are conducted by means of pseudo arc-length continuation and collocation technique to obtain force–amplitude relations in the presence of temperature variation in the thickness direction. The effect of volume fraction exponent and temperature variation on the onset of instability for both static and periodic in-plane excitation are fully discussed and the post-critical non-linear responses are obtained. Moreover, direct time integration of equations of motion is carried out and bifurcation diagrams, phase-space plots, Poincare maps and time histories are obtained showing complex non-linear dynamics through period-doubling and Neimark–Sacker bifurcations.

Double zero bifurcation of non-linear viscoelastic beams under conservative and non-conservative loads

Abstract: The combined effects of conservative and non-conservative loads on the mechanical behavior of an unshearable and inextensional visco-elastic beam, close to bifurcation, are investigated. The equations of motion and boundary conditions are derived via a constrained variational principle, and the Lagrange multiplier successively condensed, to get integro-differential equations. These latter, with the mechanical boundary conditions appended, are put in an operator-form, amenable to perturbation analysis. A linear stability analysis is carried out in the space of the two loading parameter, displaying the existence of codimension-1 and codimension-2 bifurcations. The influence of both internal and external damping on this scenario is thoroughly investigated. A post-critical analysis is carried out around a double-zero bifurcation, by using an adapted version of the multiple scale method, based on fractional series expansions in the perturbation parameter. The integro-differential problem is directly attacked, so that any a priori discretization is avoided. Emphasis is given to the interaction between the two damping coefficients. This reveals the existence, also in the non-linear range, of a phenomenon of destabilization, so far known only in the linear range.

Analysis of large deflections of a curved cantilever subjected to a tip-concentrated follower force

Abstract: The paper addresses the issue of effectively using the direct numerical method for static analysis of the flexible curved uniform cantilever beam under a tip-concentrated follower force. The angle of inclination of the follower force with respect to the deformed axis of the beam remains unchanged during deformation. After changing the variables, the original non-linear boundary value problem transforms into the initial-value problem for pendulum equation. The resulting initial value problem is solved numerically using a modified Numerov's method. In contrast to the usually used iteration methods (e.g. shooting technique), the problem is solved without iterations by direct numerical method. Some qualitative conclusions were made using Kirchhoff's kinetic analogy. It is shown that there are no critical loads in the Euler sense (divergence) for any values of the initial curvature and angle of inclination of the follower force. An extension of direct numerical method to curved spring-hinged cantilever subjected to follower force is also proposed. The paper presents some equilibrium configurations of the uniform curved fixed and spring-hinged cantilevers under normal and tangential follower force obtained by direct method.