Micromechanical analysis of the behavior of stiff clay

Cementations formed in geological timescale are observed in various stiff clays.A micromechanical stress strain model is developed for modeling the effect of cementation on the deformation behavior of stiff clay.The proposed approach considers explicitly cementations at intercluster contacts,which is different from conventional model.The concept of inter-cluster bonding is introduced to account for an additional cohesion in shear sliding and a higher yield stress in normal compression.A damage law for inter-cluster bonding is proposed at cluster contacts for the debonding process during mechanical loading.The model is used to simulate numerous stress-path tests on Vallericca stiff clay.The applicability of the present model is evaluated through comparisons between the predicted and the measured results.In order to explain the stress-induced anisotropy arising from externally applied load,the evolution of local stresses and local strains at inter-cluster planes are discussed.     

Finite element analysis of 2D SMA beam bending

A thermomechanical model of a shape memory alloy beam bending under tip force loading is implemented in finite element codes.The constitutive model is a one dimensional model which is based on free energy and motivated by statistical thermodynamics.The particular focus of this paper is on the aspects of finite element modeling and simulation of the inhomogeneous beam bending problem.This paper extends previous work which is based on the small deformation Euler-Bernoulli beam theory and by treating an SMA beam as consisting of multi-layers in a twodimensional model.The flux terms are involved in the heat transfer equation.The simulations can represent both shape memory effect and super-elastic behavior.Different thermal boundary condition effect and load rate effect can also be captured.     

The non-linear dynamic response of the Euler-Bernoulli beam with an arbitrary number of switching cracks

In this study the non-linear dynamic response of the Euler-Bernoulli beam in presence of multiple concentrated switching cracks (i.e. cracks that are either fully open or fully closed) is addressed. The overall behaviour of such a beam is non-linear due to the opening and closing of the cracks during the dynamic response; however, it can be regarded as a sequence of linear phases each of them characterised by different number and positions of the cracks in open state. In the paper the non-linear response of the beam with switching cracks is evaluated by determining the exact modal properties of the beam in each linear phase and evaluating the corresponding time history linear response through modal superposition analysis. Appropriate initial conditions at the instant of transition between two successive linear phases have been considered and an energy control has been enforced with the aim of establishing the minimum number of linear modes that must be taken into account in order to obtain accurate results. Some numerical applications are presented in order to illustrate the efficiency of the proposed approach for the evaluation of the non-linear dynamic response of beams with multiple switching cracks. In particular, the behaviour under different boundary conditions both for harmonic loading and free vibrations has been investigated.     

Nonlinear dynamic response of beam and its application in nanomechanical resonator

Nonlinear dynamic response of nanomechanical resonator is of very important characteristics in its application. Two categories of the tension-dominant and curvature-dominant nonlinearities are analyzed. The dynamic nonlinearity of four beam structures of nanomechanical resonator is quantitatively studied via a dimensional analysis approach. The dimensional analysis shows that for the nanomechanical resonator of tension-dominant nonlinearity, its dynamic nonlinearity decreases monotonically with increasing axial loading and increases monotonically with the increasing aspect ratio of length to thickness; the dynamic nonlinearity can only result in the hardening effects. However, for the nanomechanical resonator of the curvature-dominant nonlinearity, its dynamic nonlinearity is only dependent on axial loading. Compared with the tension-dominant nonlinearity, the curvature-dominant nonlinearity increases monotonically with increasing axial loading; its dynamic nonlinearity can result in both hardening and softening effects. The analysis on the dynamic nonlinearity can be very helpful to the tuning application of the nanomechanical resonator.     

On the elastoplastic cyclic analysis of plane beam structures using a flexibility-based finite element approach

This paper presents the extension of a flexibility-based large increment method (LIM) for the case of cyclic loading. In the last few years, LIM has been successfully tested for solving a range of non-linear structural problems involving elastoplastic material models under monotonic loading. In these analyses, the force-based LIM algorithm provided robust solutions and significant computational savings compared to the displacement-based finite element approach by using fewer elements and integration points. Although in cyclic analysis a step-by-step solution procedure has to be adopted to account for the plastic history, LIM will still have many advantages over the traditional finite element method. Before going into the basic idea of this extension, a brief discussion regarding LIM governing equations is presented followed by the proposed solution procedure. Next, the formulation is specified for the treatment of the elastic perfectly plastic beam element. The local stage for the beam behavior is discussed in detail and the required improvement for the LIM methodology is described. Illustrative truss and beam examples are presented for different non-linear material models. The results are compared with those obtained from a standard displacement method and again highlight the potential benefits of the proposed flexibility-based approach.     

The SPH approach to the process of container filling based on non-linear constitutive models

In this work,the transient free surface of container filling with non-linear constitutive equation’s fluids is numerically investigated by the smoothed particle hydrodynamics(SPH) method.Specifically,the filling process of a square container is considered for non-linear polymer fluids based on the Cross model.The validity of the presented SPH is first verified by solving the Newtonian fluid and OldroydB fluid jet.Various phenomena in the filling process are shown,including the jet buckling,jet thinning,splashing or spluttering,steady filling.Moreover,a new phenomenon of vortex whirling is more evidently observed for the Cross model fluid compared with the Newtonian fluid case.     

Geometric nonlinear dynamic analysis of curved beams using curved beam element

Instead of using the previous straight beam element to approximate the curved beam,in this paper,a curvilinear coordinate is employed to describe the deformations,and a new curved beam element is proposed to model the curved beam.Based on exact nonlinear strain-displacement relation,virtual work principle is used to derive dynamic equations for a rotating curved beam,with the effects of axial extensibility,shear deformation and rotary inertia taken into account.The constant matrices are solved numerically utilizing the Gauss quadrature integration method.Newmark and Newton-Raphson iteration methods are adopted to solve the differential equations of the rigid-flexible coupling system.The present results are compared with those obtained by commercial programs to validate the present finite method.In order to further illustrate the convergence and efficiency characteristics of the present modeling and computation formulation,comparison of the results of the present formulation with those of the ADAMS software are made.Furthermore,the present results obtained from linear formulation are compared with those from nonlinear formulation,and the special dynamic characteristics of the curved beam are concluded by comparison with those of the straight beam.     

Force-displacement characteristics of simply supported beam laminated with shape memory alloys

As a preliminary step in the nonlinear design of shape memory alloy(SMA) composite structures,the force-displacement characteristics of the SMA layer are studied.The bilinear hysteretic model is adopted to describe the constitutive relationship of SMA material.Under the assumption that there is no point of SMA layer finishing martensitic phase transformation during the loading and unloading process,the generalized restoring force generated by SMA layer is deduced for the case that the simply supported beam vibrates in its first mode.The generalized force is expressed as piecewise-nonlinear hysteretic function of the beam transverse displacement.Furthermore the energy dissipated by SMA layer during one period is obtained by integration,then its dependencies are discussed on the vibration amplitude and the SMA’s strain(Ms-Strain) value at the beginning of martensitic phase transformation.It is shown that SMA’s energy dissipating capacity is proportional to the stiffness difference of bilinear model and nonlinearly dependent on Ms-Strain.The increasing rate of the dissipating capacity gradually reduces with the amplitude increasing.The condition corresponding to the maximum dissipating capacity is deduced for given value of the vibration amplitude.The obtained results are helpful for designing beams laminated with shape memory alloys.     

Non-linear response of a beam to periodic loading

The steady state response of a non-linear beam under periodic excitation is investigated. The non-linearity is attributed to the membrane tension effect which is induced in the beam when the deflection is not small in comparison to its thickness. The effects of multimode participation are investigated for simply supported and clamped boundary conditions. The finite element technique is used to formulate the non-linear differential equations of the straight beam and the method of averaging is used to obtain an approximate solution to the non-linear equations under harmonic loading. An analog computer was used to simulate the non-linear beam equation which was subjected to harmonic excitation. The agreement between theoretical and experimental values is reasonably good.     

Surface and non-local effects for non-linear analysis of Timoshenko beams

In this paper, we present a non-local non-linear finite element formulation for the Timoshenko beam theory. The proposed formulation also takes into consideration the surface stress effects. Eringen׳s non-local differential model has been used to rewrite the non-local stress resultants in terms of non-local displacements. Geometric non-linearities are taken into account by using the Green–Lagrange strain tensor. A C⁰ beam element with three degrees of freedom has been developed. Numerical solutions are obtained by performing a non-linear analysis for bending and free vibration cases. Simply supported and clamped boundary conditions have been considered in the numerical examples. A parametric study has been performed to understand the effect of non-local parameter and surface stresses on deflection and vibration characteristics of the beam. The solutions are compared with the analytical solutions available in the literature. It has been shown that non-local effect does not exist in the nano-cantilever beam (Euler–Bernoulli beam) subjected to concentrated load at the end. However, there is a significant effect of non-local parameter on deflections for other load cases such as uniformly distributed load and sinusoidally distributed load (Cheng et al. (2015) [10]). In this work it has been shown that for a cantilever beam with concentrated load at free end, there is definitely a dependency on non-local parameter when Timoshenko beam theory is used. Also the effect of local and non-local boundary conditions has been demonstrated in this example. The example has also been worked out for other loading cases such as uniformly distributed force and sinusoidally varying force. The effect of the local or non-local boundary conditions on the end deflection in all these cases has also been brought out.     

Asymptotic analysis of outwardly propagating spherical flames

Asymptotic analysis is conducted for outwardly propagating spherical flames with large activation energy.The spherical flame structure consists of the preheat zone,reaction zone,and equilibrium zone.Analytical solutions are separately obtained in these three zones and then asymptotically matched.In the asymptotic analysis,we derive a correlation describing the spherical flame temperature and propagation speed changing with the flame radius.This correlation is compared with previous results derived in the limit of infinite value of activation energy.Based on this correlation,the properties of spherical flame propagation are investigated and the effects of Lewis number on spherical flame propagation speed and extinction stretch rate are assessed.Moreover,the accuracy and performance of different models used in the spherical flame method are examined.It is found that in order to get accurate laminar flame speed and Markstein length,non-linear models should be used.     

Precracking and interfacial delamination in a bi-material structure: Static and dynamic loadings

The behavior of a precracked bi-material structure interface under given static and dynamic axial loading is an interest object in the present paper.Firstly,it is shown that the shear-lag model is a proper tool to analyze a delamination process in a precracked bi-material structure undergoing static loading.Secondly,the"shear-lag model"is applied to the structure under dynamic loading.To solve the problem for an interface delamination of the structure and to determine the debond length along the interface,our own 2D boundary element method(BEM)code is proposed in the case of static loading,and the shear-lag model together with the Laplace transforms and half-analytical calculations are used in the case of dynamic loading.The interface layer is assumed as a very thin plate compared with the other two.The parametric(geometric and elastic)analysis of the debond length and interface shear stress is done. The results from the 2D BEM code proved the validity of analytical solutions to the shear-lag model.In the dynamic case,the influence of loading characteristics,i.e.,frequencies and amplitude fluctuations on the shear stress and the value of debond length for an interval of time,is discussed. The analysis of the obtained results is illustrated by an example of the modern ceramic-metal composite,namely cermet, and depicted in figures.     

Interaction of subway LIM vehicle with ballasted track in polygonal wheel wear development

This paper develops a coupled dynamics model for a linear induction motor (LIM) vehicle and a subway track to investigate the influence of polygonal wheels of the vehicle on the dynamic behavior of the system. In the model, the vehicle is modeled as a multi-body system with 35 degrees of freedom. A Timoshenko beam is used to model the rails which are discretely supported by sleepers. The sleepers are modeled as rigid bodies with their vertical, lateral, and rolling motions being considered. In order to simulate the vehicle running along the track, a moving sleeper support model is introduced to simulate the excitation by the discrete sleeper supporters, in which the sleepers are assumed to move backward at a constant speed that is the same as the train speed. The Hertzian contact theory and the Shen- Hedrick-Elkins’ model are utilized to deal with the normal dynamic forces and the tangential forces between wheels and rails, respectively. In order to better characterize the linear metro system (LMS), Euler beam theory based on modal superposition method is used to model LIM and RP. The vertical electric magnetic force and the lateral restoring force between the LIM and RP are also taken into consideration. The former has gap-varying nonlinear characteristics, whilst the latter is considered as a constant restoring force of 1 kN. The numerical analysis considers the effect of the excitation due to polygonal wheels on the dynamic behavior of the system at different wear stages, in which the used data regarding the polygonal wear on the wheel tread are directly measured at the subway site.     

Free vibration of size-dependent magneto-electro-elastic nanoplates based on the nonlocal theory

In this paper, the free vibration of magneto- electro-elastic （MEE） nanoplates is investigated based on the nonlocal theory and Kirchhoff plate theory. The MEE nanoplate is assumed as all edges simply supported rectan gular plate subjected to the biaxial force, external electric potential, external magnetic potential, and temperature rise. By using the Hamilton＇s principle, the governing equations and boundary conditions are derived and then solved analytically to obtain the natural frequencies of MEE nanoplates. A parametric study is presented to examine the effect of the nonlocal parameter, thermo-magneto-electro-mechanical loadings and aspect ratio on the vibration characteristics of MEE nanoplates. It is found that the natural frequency is quite sensitive to the mechanical loading, electric loading and magnetic loading, while it is insensitive to the thermal loading.     

Stochastic response of an axially moving viscoelastic beam with fractional order constitutive relation and random excitations

A convenient and universal residue calculus method is proposed to study the stochastic response behaviors of an axially moving viscoelastic beam with random noise excitations and fractional order constitutive relationship, where the random excitation can be decomposed as a nonstationary stochastic process, Mittag-Leffler internal noise, and external stationary noise excitation. Then, based on the Laplace transform approach, we derived the mean value function, variance function and covariance function through the Green's function technique and the residue calculus method, and obtained theoretical results. In some special case of fractional order derivative α , the Monte Carlo approach and error function results were applied to check the effectiveness of the analytical results, and good agreement was found. Finally in a general-purpose case, we also confirmed the analytical conclusion via the direct Monte Carlo simulation.     

Non-linear dynamic response of a rotating radial Timoshenko beam with periodic pulse loading at the free-end

Consideration is given to the dynamic response of a Timoshenko beam under repeated pulse loading. Starting with the basic dynamical equations for a rotating radial cantilever Timoshenko beam clamped at the hub in a centrifugal force field, a system of equations are derived for coupled axial and lateral motions which includes the transverse shear and rotary inertia effects, as well. The hyperbolic wave equation governing the axial motion is coupled with the flexural wave equation governing the lateral motion of the beam through the velocity-dependent skew-symmetric Coriolis force terms. In the analytical formulation, Rayleigh-Ritz method with a set of sinusoidal displacement shape functions is used to determine stiffness, mass and gyroscopic matrices of the system. The tip of the rotating beam is subjected to a periodic pulse load due to local rubbing against the outer case introducing Coulomb friction in the system. Transient response of the beam with the tip deforming due to rub is discussed in terms of the frequency shift and non-linear dynamic response of the rotating beam. Numerical results are presented for this vibro-impact problem of hard rub with varying coefficients of friction and the contact-load time. The effects of beam tip rub forces transmitted through the system are considered to analyze the conditions for dynamic stability of a rotating blade with intermittent rub.     

Large displacement analysis of shear deformable composite beams with interlayer slips

This paper presents a novel geometric non-linear finite element formulation for the analysis of shear deformable two-layer beams with interlayer slips. We adopt the co-rotational approach where the motion of the element is decomposed into two parts: a rigid body motion which defines a local coordinate system and a small deformational motion of the element relative to this local coordinate system. The main advantage of this approach is that the transformation matrices relating local and global quantities are independent to the choice of the geometrical linear local element. The effect of transverse shear deformation of the layers is taken into account by assuming that each layer behaves as a Timoshenko beam element. The layers are assumed to be continuously connected and partial interaction is considered by considering a continuous relationship between the interface shear flow and the corresponding slip. In order to avoid curvature and shear locking phenomena, the local linear element is formulated using “exact” displacement shape functions derived from the closed-form solution of the governing equations of a two-layer beam element. Finally, three numerical applications are presented in order to assess the performance of the proposed formulation.     

Stability and Hopf bifurcation of a delayed ratio-dependent predator-prey system

Since the ratio-dependent theory reflects the fact that predators must share and compete for food, it is suitable for describing the relationship between predators and their preys and has recently become a very important theory put forward by biologists. In order to investigate the dynamical relationship between predators and their preys, a so-called Michaelis-Menten ratio-dependent predator-prey model is studied in this paper with gestation time delays of predators and preys taken into consideration. The stability of the positive equilibrium is investigated by the Nyquist criteria, and the existence of the local Hopf bifurcation is analyzed by employing the theory of Hopf bifurcation. By means of the center manifold and the normal form theories, explicit formulae are derived to determine the stability, direction and other properties of bifurcating periodic solutions. The above theoretical results are validated by numerical simulations with the help of dynamical software WinPP. The results show that if both the gestation delays are small enough, their sizes will keep stable in the long run, but if the gestation delays of predators are big enough, their sizes will periodically fluc-tuate in the long term. In order to reveal the effects of time delays on the ratio-dependent predator-prey model, a ratiodependent predator-prey model without time delays is considered. By Hurwitz criteria, the local stability of positive equilibrium of this model is investigated. The conditions under which the positive equilibrium is locally asymptotically stable are obtained. By comparing the results with those of the model with time delays, it shows that the dynamical behaviors of ratio-dependent predator-prey model with time delays are more complicated. Under the same conditions, namely, with the same parameters, the stability of positive equilibrium of ratio-dependent predator-prey model would change due to the introduction of gestation time delays for predators and preys. Moreover, with the variation of time delays, the positive equilibrium of the ratio-dependent predator-prey model subjects to Hopf bifurcation.     

Three-dimensional finite element modeling of a vibrating beam with a breathing crack

This paper develops a full three-dimensional finite element model in order to study the vibrational behavior of a beam with a non-propagating surface crack. In this model, the breathing crack behavior is simulated as a full frictional contact problem between the crack surfaces, while the region around the crack is discretized into three-dimensional solid finite elements. The governing equations of this non-linear dynamic problem are solved by employing an incremental iterative procedure. The extracted response is analyzed utilizing either Fourier or continuous wavelet transforms to reveal the breathing crack effects. This study is applied to a cracked cantilever beam subjected to dynamic loading. The crack has an either uniform or non-uniform depth across the beam cross-section. For both crack cases, the vertical, horizontal, and axial beam vibrations are studied for various values of crack depth and position. Coupling between these beam vibration components is observed. Conclusions are extracted for the influence of crack characteristics such as geometry, depth, and position on the coupling of these beam vibration components. The accuracy of the results is verified through comparisons with results available from the literature.     

A new model for wrinkling and collapse analysis of membrane inflated beam

A new model is proposed to accurately predict the wrinkling and collapse loads of a membrane inflated beam. In this model, the pressure effects are considered and a modified factor is introduced to obtain an accurate prediction. The former is achieved by modifying the pressure-related structural parameters based on elastic small strain considerations, and the modified factor is determined by our test data. Compared with previous models and our test data, the present model, named as shell-membrane model, can accurately predict the wrinkling and collapse loads of membrane inflated beams.