A new nonlocal bending model for Euler–Bernoulli nanobeams

This paper is concerned with the bending problem of nanobeams starting from a nonlocal thermodynamic approach. A new coupled nonlocal model, depending on two nonlocal parameters, is obtained by using a suitable definition of the free energy. Unlike previous approaches which directly substitute the expression of the nonlocal stress into the classical equilibrium equations, the proposed approach provides a methodology to recover nonlocal models starting from the free energy function. The coupled model can then be specialized to obtain a nanobeam formulation based on the Eringen nonlocal elasticity theory and on the gradient elastic model. The variational formulations are consistently provided and the differential equations with the related boundary conditions are thus derived. Nanocantilevers are solved in a closed-form and numerical results are presented to investigate the influence of the nonlocal parameters.     

Nonlinear vibration analysis of fractional viscoelastic Euler–Bernoulli nanobeams based on the surface stress theory

The nonlinear vibrations of viscoelastic Euler–Bernoulli nanobeams are studied using the fractional calculus and the Gurtin–Murdoch theory. Employing Hamilton's principle, the governing equation considering surface effects is derived. The fractional integro-partial differential governing equation is first converted into a fractional–ordinary differential equation in the time domain using the Galerkin scheme. Thereafter, the set of nonlinear fractional time-dependent equations expressed in a state-space form is solved using the predictor–corrector method. Finally, the effects of initial displacement, fractional derivative order, viscoelasticity coefficient, surface parameters and thickness-to-length ratio on the nonlinear time response of simply-supported and clamped-free silicon viscoelastic nanobeams are investigated.     

A simple approach to detect the nonlocal effects in the static analysis of Euler–Bernoulli and Timoshenko beams

In this paper, the well-known Mohr analogy is applied to the computation of displacements and rotations of carbon nanotubes, and some simple formula is derived which allows the direct generalization of the Mohr theory to the nonlocal Euler–Bernoulli and Timoshenko beam theories. Finally, some examples show the effectiveness and simplicity of the proposed approach.     

Correction of local elasticity for nonlocal residuals: application to Euler–Bernoulli beams

Complications exist when solving the field equation in the nonlocal field. This has been attributed to the complexity of deriving explicit forms of the nonlocal boundary conditions. Thus, the paradoxes in the existing solutions of the nonlocal field equation have been revealed in recent studies. In the present study, a new methodology is proposed to easily determine the elastic nonlocal fields from their local counterparts without solving the field equation. This methodology depends on the iterative-nonlocal residual approach in which the sum of the nonlocal fields is treaded as a residual field. Thus, in this study the corrections of the local linear and nonlinear elastic fields for the nonlocal residuals in materials are presented. These corrections are formed based on the general nonlocal theory. In the context of the general nonlocal theory, two distinct nonlocal parameters are introduced to form the constitutive equations of isotropic elastic continua. In this study, it is demonstrated that the general nonlocal theory outperforms Eringen’s nonlocal theory in accounting for the impacts of the material’s Poisson’s ratio on its mechanics. To demonstrate the effectiveness of the proposed approach, the corrections of the local static bending, vibration, and buckling characteristics of Euler–Bernoulli beams are derived. Via these corrections, bending, vibration, and buckling behaviors of simple-supported nonlocal Euler–Bernoulli beams are determined without solving the beam’s equation of motion.     

Exact solution of the multi-cracked Euler–Bernoulli column

The use of distributions (generalized functions) is a powerful tool to treat singularities in structural mechanics and, besides providing a mathematical modelling, their capability of leading to closed form exact solutions is shown in this paper. In particular, the problem of stability of the uniform Euler–Bernoulli column in presence of multiple concentrated cracks, subjected to an axial compression load, under general boundary conditions is tackled. Concentrated cracks are modelled by means of Dirac’s delta distributions. An integration procedure of the fourth order differential governing equation, which is not allowed by the classical distribution theory, is proposed. The exact buckling mode solution of the column, as functions of four integration constants, and the corresponding exact buckling load equation for any number, position and intensity of the cracks are presented. As an example a parametric study of the multi-cracked simply supported and clamped–clamped Euler–Bernoulli columns is presented.     

On the truth of nanoscale for nanobeams based on nonlocal elastic stress field theory： equilibrium, governing equation and static deflection

This paper has successfully addressed three critical but overlooked issues in nonlocal elastic stress field theory for nanobeams： （i） why does the presence of increasing nonlocal effects induce reduced nanostructural stiffness in many, but not consistently for all, cases of study, i.e., increasing static deflection, decreasing natural frequency and decreasing buckling load, although physical intuition according to the nonlocal elasticity field theory first established by Eringen tells otherwise？ （ii） the intriguing conclusion that nanoscale effects are missing in the solutions in many exemplary cases of study, e.g., bending deflection of a cantilever nanobeam with a point load at its tip; and （iii） the non-existence of additional higher-order boundary conditions for a higher-order governing differential equation. Applying the nonlocal elasticity field theory in nanomechanics and an exact variational principal approach, we derive the new equilibrium conditions, do- main governing differential equation and boundary conditions for bending of nanobeams. These equations and conditions involve essential higher-order differential terms which are opposite in sign with respect to the previously studies in the statics and dynamics of nonlocal nano-structures. The difference in higher-order terms results in reverse trends of nanoscale effects with respect to the conclusion of this paper. Effectively, this paper reports new equilibrium conditions, governing differential equation and boundary condi- tions and the true basic static responses for bending of nanobeams. It is also concluded that the widely accepted equilibrium conditions of nonlocal nanostructures are in fact not in equilibrium, but they can be made perfect should the nonlocal bending moment be replaced by an effective nonlocal bending moment. These conclusions are substantiated, in a general sense, by other approaches in nanostructural models such as strain gradient theory, modified couple stress models and experiments.     

On the response spectrum of Euler–Bernoulli beams with a moving mass and horizontal support excitation

In this study, the response spectrum of a time-varying system such as a beam subjected to moving masses under a harmonic and earthquake support excitations is explored. The excitations are supposed to act on the horizontal directions of the beam axis. The inertial effect of the moving masses on the natural frequencies of the beam for different cases of loading is investigated and a critical value of a so called parameter “mass staying time” is presented to avoid dynamic instability of the system. Finally, some 3D response spectra for different supports excitations as well as the beam natural frequencies are depicted.     

A new flux-limiting approach–based kinetic scheme for the Euler equations of gas dynamics

This paper proposes a new kinetic-theory-based high-resolution scheme for the Euler equations of gas dynamics. The scheme uses the well-known connection that the Euler equations are suitable moments of the collisionless Boltzmann equation of kinetic theory. The collisionless Boltzmann equation is discretized using Sweby's flux-limited method and the moment of this Boltzmann level formulation gives a Euler level scheme. It is demonstrated how conventional limiters and an extremum-preserving limiter can be adapted for use in the scheme to achieve a desired effect. A simple total variation diminishing criteria relaxing parameter results in improving the resolution of the discontinuities in a significant way. A 1D scheme is formulated first and an extension to 2D on Cartesian meshes is carried out next. Accuracy analysis suggests that the scheme achieves between first- and second-order accuracy as is expected for any second-order flux-limited method. The simplicity and the explicit form of the conservative numerical fluxes add to the efficiency of the scheme. Several standard 1D and 2D test problems are solved to demonstrate the robustness and accuracy.     

The influence of the axial force on the vibration of the Euler–Bernoulli beam with an arbitrary number of cracks

The exact closed-form solution for the vibration modes and the eigen-value equation of the Euler–Bernoulli beam-column in the presence of an arbitrary number of concentrated open cracks is proposed. The solution is provided explicitly as functions of four integration constants only, to be determined by the standard boundary conditions. The enforcement of the boundary conditions leads the exact evaluation of the vibration frequencies as well as the buckling load of the beam-column and the corresponding eigen-modes. Furthermore, the presented solution allows a comprehensive evaluation of the influence of the axial load on the modal parameters of the beam. The cracks, which are not subjected to the closing phenomenon, are modelled as a sequence of Dirac’s delta generalised functions in the flexural stiffness. The eigen-mode governing equation is formulated over the entire domain of the beam without enforcement of any further continuity condition. The influence of the axial load on the vibration modes of beam-columns with different number and position of cracks, under different boundary conditions, has been analysed by means of the proposed closed-form expressions. The presented parametric analysis highlights some abrupt changes of the eigen-modes and the corresponding frequencies.     

Boundary control of an Euler–Bernoulli beam with input and output restrictions

Nonlinear Dynamics - In this study, boundary control is considered for an Euler–Bernoulli beam subject to bounded input, bounded output, and external disturbances. Through utilizing the...     

Analysis of unsteady gas–liquid flows in a rectangular tank: Comparison of Euler–Eulerian and Euler–Lagrangian simulations

We study the dynamics of gas–liquid flows experimentally and computationally in a rectangular bubble column where the gas source is introduced at the corner. The flow in this reactor is complex and inherently unsteady in nature. The two-dimensional liquid phase velocity field is calculated by an Eulerian approach solving the unsteady Reynolds Averaged Navier Stokes equations. The conservation equations are closed using a two parameter turbulence model. The two-way coupling was accounted for by adding source terms in the conservation equations of the continuous phase to take into account the interaction with the dispersed phase. Bubble tracking is achieved through a Lagrangian approach. Here the equations of motion are solved taking into account the drag, pressure, buoyancy and gravity forces. The time-averaged flows along with the variables which characterize turbulence are analyzed for a wide range of gas flow-rates using Euler–Lagrangian simulations. These simulation predictions are validated with Euler–Eulerian simulations where the gas-phase distribution is captured as a void fraction and PIV experiments. The motion of bubbles induces turbulence in the flow. The applicability of two parameter models for turbulence like the standard k–ε model on time-averaged flow properties is addressed. From the results of the time averaged velocity field, turbulence intensity, turbulent viscosity and gas hold-up profiles, it is concluded that the Euler–Lagrangian model is applicable at lower gas flow-rates. The Euler–Eulerian approach was found to be valid at lower as well as higher gas flow-rates.     

Physically-based Dirac’s delta functions in the static analysis of multi-cracked Euler–Bernoulli and Timoshenko beams

Dirac’s delta functions enable simple and effective representations of point loads and singularities in a variety of structural problems, leading very often to elegant and otherwise unworkable closed-form solutions. This is the case of cracked beams under static loads, whose theoretical and practical significance has attracted in recent years the interest of many researchers. Nevertheless, analytical formulations currently available for this problem are not completely satisfactory, either in terms of computational efficiency, when the continuity conditions must be enforced with auxiliary equations, or in terms of physical consistency, when the singularities in the beam’s flexural rigidity are represented with Dirac’s delta functions having a questionable negative sign. These considerations motivate the present study, which offers a novel and physically-based modelling of slender Euler–Bernoulli beams and short Timoshenko beams with any number and severity of cracks, conducing in both cases to exact closed-form solutions. For validation purposes, a standard finite element code is used, along with two nascent deltas (uniform and Gaussian density functions) to describe a smeared increase in the bending flexibility around the abscissa of the crack.     

Identification methodology and comparison of phenomenological ductile damage models via hybrid numerical–experimental analysis of fracture experiments conducted on a zirconium alloy

The present study details the methodology of the identification process of uncoupled damage formulations proposed by Bai and Wierzbicki (B&W model – Bai and Wierzbicki (2008) and Modified Mohr–Coulomb – Bai and Wierzbicki (2010)) as well as the Lemaitre and enhanced Lemaitre coupled damage models. These uncoupled models were first implemented in the Finite Element (FE) software Forge2009®, then their parameters identifications were carried out. These identifications involve two steps: identification of hardening law parameters and identification of damage parameters. In the first step, the method to obtain the coefficient of a non-linear friction law in compression test is also presented. The second step differs between the above-mentioned models: the identification of uncoupled models is carried out through the experimental fracture strains of different loading paths, while the identification of Lemaitre’s model is based on the softening effect of damage. The latter model is enhanced by accounting for the influence of the Lode parameter to improve its ability to predict fracture in shear loading. The results show that, among the studied models, the proposed enhanced Lemaitre model gives overall best results in terms of fracture prediction for all the tests. The proportionality of studied loading paths is also discussed. It is shown that the compression is not suitable to identify the parameters of the studied uncoupled damage models.     

Experimental study and numerical simulation of the characteristics of a percussive gas–solid separator

The presence of solid particles in the flow of hypersonic wind tunnels damages the appearance of the experiment models in the wind tunnel and influences the accuracy of experimental results. The design of a highly efficient gas–solid separator was therefore undertaken. Particle trajectory imaging methods were used to measure trajectories under different conditions. The flow field and particle movement characteristics for different head angles (HAs) and separation tooth angles (STAs), inlet velocities, and the exhaust gas outlet pressures in the separator, were calculated using simulations based on the discrete phase model. The particle separation efficiency, pressure loss, and flow loss resulting from different structural parameters were also studied. In line with experimental observations, the characteristic angle of particle movements in the separator and the separation efficiency of the separator were found to increase with decreasing HA and with increasing STA. Separation efficiency improves with increasing inlet velocity and with increasing negative pressure of the exhaust gas outlet; however, the corresponding pressure loss and the flow rate of the waste gas also increased.     

Development and elaboration of numerical method for simulating gas–liquid–solid three-phase flows based on particle method

A numerical method for simulating gas–liquid–solid three-phase flows based on the moving particle semi-implicit (MPS) approach was developed in this study. Computational instability often occurs in multiphase flow simulations if the deformations of the free surfaces between different phases are large, among other reasons. To avoid this instability, this paper proposes an improved coupling procedure between different phases in which the physical quantities of particles in different phases are calculated independently. We performed numerical tests on two illustrative problems: a dam-break problem and a solid-sphere impingement problem. The former problem is a gas–liquid two-phase problem, and the latter is a gas–liquid–solid three-phase problem. The computational results agree reasonably well with the experimental results. Thus, we confirmed that the proposed MPS method reproduces the interaction between different phases without inducing numerical instability.     

Numerical simulation of a dense solid particle flow inside a cyclone separator using the hybrid Euler–Lagrange approach

This paper presents a numerical simulation of the flow inside a cyclone separator at high particle loads. The gas and gas–particle flows were analyzed using a commercial computational fluid dynamics code. The turbulence effects inside the separator were modeled using the Reynolds stress model. The two phase gas–solid particles flow was modeled using a hybrid Euler–Lagrange approach, which accounts for the four-way coupling between phases. The simulations were performed for three inlet velocities of the gaseous phase and several cyclone mass particle loadings. Moreover, the influences of several submodel parameters on the calculated results were investigated. The obtained results were compared against experimental data collected at the in-house experimental rig. The cyclone pressure drop evaluated numerically underpredicts the measured values. The possible reason of this discrepancies was disused.     

Constrained parameter-splitting perturbation method for the improved solutions to the nonlinear vibrations of Euler–Bernoulli cantilevers

In this paper, a new method named constrained parameter-splitting perturbation method for improving the solutions obtained from the parameter-splitting perturbation method is proposed for solving the problems in some extremal cases, such as the strongly nonlinear vibration of an Euler–Bernoulli cantilever. The proposed method takes the advantages of both the perturbation method and the harmonic balance method. The idea is that the solution obtained by the parameter-splitting perturbation method is substituted into the equation of motion and then the accumulative error of the equation is minimized for determining the unknown splitting parameters under the constraints constructed under the frame of harmonic balance method. The forced vibration of an oscillator with cubic geometric nonlinearity and inertia nonlinearity and the forced vibration of a planar microcantilever beam with a lumped tip mass are studied as examples to reveal the efficacy of the proposed method. The inspection of the steady-state response including its stability is conducted by means of comparing the frequency-response curves obtained by the proposed method with those obtained by the numerical continuation method and harmonic balance method, respectively, to show the efficacy and the advantages of the proposed method. Meanwhile, the nonlinear ordering effect on the solutions of the proposed method is also studied by comparing the results obtained by using different nonlinear orderings in the systems. In the last, we found through convergence examinations that it is necessary to have corrections to the erroneous solution which are obtained by harmonic balance method and Floquet theory in stability analysis.