Nonlinear internal resonance of double-walled nanobeams under parametric excitation by nonlocal continuum theory

In the present work, the nonlinear internal resonance of double-walled nanobeams under the external parametric load is studied. The nonlocal continuum theory is applied to describe the nano scale effects and the nonlinear governing equations are derived by the multiple scale method. The parametric internal resonance is considered and the relation between the frequency and amplitude is discussed. From the numerical simulation, it can be observed that small scale effects are more obvious for short structures. Three different nonlinear cases can be found. The gap between the stable and instable regions is reduced by the van der Walls (vdW) interaction but enhanced by the excitation amplitude. Moreover, the dynamical motions of double-walled nanobeams are sensitive to the initial condition and excitation frequency.     

Linear and nonlinear vibrations of fractional viscoelastic Timoshenko nanobeams considering surface energy effects

In this paper, the linear and nonlinear vibrations of fractional viscoelastic Timoshenko nanobeams are studied based on the Gurtin–Murdoch surface stress theory. Firstly, the constitutive equations of fractional viscoelasticity theory are considered, and based on the Gurtin–Murdoch model, stress components on the surface of the nanobeam are incorporated into the axial stress tensor. Afterward, using Hamilton's principle, equations governing the two-dimensional vibrations of fractional viscoelastic nanobeams are derived. Finally, two solution procedures are utilized to describe the time responses of nanobeams. In the first method, which is fully numerical, the generalized differential quadrature and finite difference methods are used to discretize the linear part of the governing equations in spatial and time domains. In the second method, which is semi-analytical, the Galerkin approach is first used to discretize nonlinear partial differential governing equations in the spatial domain, and the obtained set of fractional-order ordinary differential equations are then solved by the predictor–corrector method. The accuracy of the results for the linear and nonlinear vibrations of fractional viscoelastic nanobeams with different boundary conditions is shown. Also, by comparing obtained results for different values of some parameters such as viscoelasticity coefficient, order of fractional derivative and parameters of surface stress model, their effects on the frequency and damping of vibrations of the fractional viscoelastic nanobeams are investigated.     

Flexoelectric and surface effects on the electromechanical behavior of graphene-based nanobeams

In this novel work, the electromechanical behavior of graphene-based nanocomposite (GNC) beams with flexoelectric and surface effects were investigated using size-dependent Euler-Bernoulli theory, linear piezoelectricity and Galerkin's weighted residual method along with modified strength of materials and finite element (FE) approaches. In addition, analytical and FE models were developed to study the static response of flexoelectric GNC nanobeams with various boundary conditions: cantilever, simply-supported and clamped-clamped. The developed models predict that the effective piezoelectric coefficients of GNC are responsible for the actuation capability of a graphene layer in the transverse direction due to the applied field in its axial direction and the predictions by both the models are found to be in good agreement. Results reveal that the flexoelectric and surface effects on the static response of GNC nanobeams are significant and should be taken into account. The electromechanical response of GNC nanobeams can be tailored to achieve the required coupled electromechanical characteristics of a vast range of NEMS using various boundary conditions and thickness of nanobeam as well as volume fraction of graphene. Our fundamental study sheds a light on the possibility of developing high-performance and lightweight graphene-based NEMS such as nanosensors, nanogenerators and nanoresonators using non-piezoelectric graphene.     

Surface effects on nonlinear free vibration of functionally graded nanobeams using nonlocal elasticity

In this paper, to consider all surface effects including surface elasticity, surface stress, and surface density, on the nonlinear free vibration analysis of simply-supported functionally graded Euler–Bernoulli nanobeams using nonlocal elasticity theory, the balance conditions between FG nanobeam bulk and its surfaces are considered to be satisfied assuming a cubic variation for the component of the normal stress through the FG nanobeam thickness. The nonlinear governing equation includes the von Kármán geometric nonlinearity and the material properties change continuously through the thickness of the FG nanobeam according to a power-law distribution of the volume fraction of the constituents. The multiple scale method is employed as an analytical solution for the nonlinear governing equation to obtain the nonlinear natural frequencies of FG nanobeams. The effect of the gradient index, the nanobeam length, thickness to length ratio, mode number, amplitude of deflection to radius of gyration ratio and nonlocal parameter on the frequency ratios of FG nanobeams is investigated.     

Nonlinear interaction of internal and surface gravity waves in a two-layer fluid with free surface

A new nonlinear model of the propagation of wave packets in the system “liquid layer with solid bottom–liquid layer with free surface” is considered. With the use of the method of multiple-scale expansions, the first three linear approximations of the nonlinear problem are obtained. Solutions of problem of the first approximation are constructed and analyzed in detail. It is shown that there exist internal and surface components of the wave field, and their interaction is analyzed.     

Finite element simulation for material accumulation and welding processes including a free melt surface

A modeling and simulation approach for problems with solid-liquid-solid phase transitions and a free surface, feasible for material accumulation processes based on laser-based free form heading and welding processes for joining different metallic materials is presented. Both named processes are modeled within the framework of continuum mechanics by coupling the Stefan problem with the Navier-Stokes equations including a free capillary surface. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A nonlocal higher-order model including thickness stretching effect for bending and buckling of curved nanobeams

An analytical approach for static bending and buckling analyses of curved nanobeams using the differential constitutive law, consequent to Eringen’s strain-driven integral model coupled with a higher-order shear deformation accounting for through thickness stretching is presented. The formulation is general in the sense that it can be deduced to examine the influence of different structural theories, for static and dynamic analyses of curved nanobeams. The governing equations derived using Hamiltons principle are solved in conjunction with Naviers solutions. The formulation is validated considering problems for which solutions are available. A comparative study is made here by various theories obtained through the formulation. The effects various structural parameters such as thickness ratio, beam length, rise of the curved beam, and nonlocal scale parameter are brought out on bending and stability characteristics of curved nanobeams.     

Non-linear forced vibration analysis of nanobeams subjected to moving concentrated load resting on a viscoelastic foundation considering thermal and surface effects

Analytical solution for the steady-state response of an Euler–Bernoulli nanobeam subjected to moving concentrated load and resting on a viscoelastic foundation with surface effects consideration in a thermal environment is investigated in this article. At first, based on the Eringen's nonlocal theory, the governing equations of motion are derived using the Hamilton's principle. Then, in order to solve the equation, Galerkin method is applied to discretize the governing nonlinear partial differential equation to a nonlinear ordinary differential equation; solution is obtained employing the perturbation technique (multiple scales method). Results indicate that by increasing of various parameters such as foundation damping, linear stiffness, residual surface stress and the temperature change, the jump phenomenon is postponed and with increasing the amplitude of the moving force and the nonlocal parameter, the jump phenomenon occurs earlier and its frequency and the peak value of amplitude of vibration increases. In addition, it is seen that the non-linear stiffness and the critical velocity of the moving load are important factors in studying nanobeams subjected to moving concentrated load. Presence of the non-linear stiffness of Winkler foundation resulting nanobeam tends to instability and so, the jump phenomenon occurs. But, presence of the linear stiffness will lead to stability of the nanobeam. In the next sections of the paper, frequency responses of the nanobeam made of temperature-dependent material properties under multi-frequency excitations are investigated.     

Solvability of a supercritical free surface flow problem under gravity-capillarity

In this paper, we consider a problem of a supercritical free surface flow over an obstacle lying on the bottom of a channel in 2D. The flow is irrotational, stationary and the fluid is ideal and incompressible. We take into account both the gravity and the effects of the superficial tension. The problem is nonlinear, it is formulated by the Laplace operator and the dynamic condition defined on the free surface of the fluid domain (Bernoulli equation). Using the perturbation stream function, we linearize the problem and we give a priori properties of the solution. These a priori properties allow us to construct a space where we can use the Lax–Milgram’s theorem to prove the existence and the uniqueness of the solution of the problem.     

Nonlinear free surface condition due to second order diffraction by a pair of cylinders

An analysis of the non-homogeneous term involved in the free surface condition for second order wave diffraction on a pair of cylinders is presented. In the computations of the nonlinear loads on offshore structures the most challenging task is the computation of the free surface integral. The main contribution to this integrand is due to the nonhomogeneous term present in the free surface condition for second order scattered potential. In this paper, the free surface condition for the second order scattered potential is derived. Under the assumption of large spacing between the two cylinders, waves scattered by one cylinder may be replaced in the vicinity of the other cylinder by equivalent plane waves together with non-planner correction terms. Then solving a complex matrix equation, the first order scattered potential is derived and since the free surface term for second order scattered potential can be expressed in terms of the first order potentials, the free surface term can be obtained using the knowledge of first order potentials only.     

Streamwise oscillations of a cylinder beneath a free surface: Free surface effects on vortex formation modes

A computational study of a viscous incompressible two-fluid model with an oscillating cylinder is investigated at a Reynolds number of 200 and at a dimensionless displacement amplitude of A=0.13 and for the dimensionless forcing cylinder oscillation frequency-to-natural vortex shedding frequency ratios, f/f₀=1.5,2.5,3.5. Specifically, two-dimensional flow past a circular cylinder subject to forced in-line oscillations beneath a free surface is considered. The method is based on a finite volume discretization of the two-dimensional continuity and unsteady Navier-Stokes equations (when a solid body is present) on a fixed Cartesian grid. Two-fluid model based on improved volume-of-fluid method is used to discretize the free surface interface. The study focuses on the laminar asymmetric flow structure in the near wake region and lock-on phenomena at a Froude number of 0.2 and for the dimensionless cylinder submergence depths, h=0.25, 0.5 and 0.75. The equivorticity patterns and pressure distribution contours are used for the numerical flow visualization. The code validations in special cases show good comparisons with previous numerical results.     

Nonlinear waves in two-dimensions generated by variable pressure acting on the free surface of a heavy liquid

The development of nonlinear waves on the free surface of a heavy liquid initially at rest is treated analytically in cases where the external pressure force of limited power is distributed over a large area in the free surface but is otherwise arbitrary. In [1] approximate (up to small terms of higher order) solution of the problem is obtained in the form of functional series. In the present article the convergence theorems for the series are proved. When the pressure varies with time sinusoidally, the sums of the series are found in closed form. By passing to the limit in the solution as time goes to infinity, the form of the nonlinear steady-state wave is found. According to the solution, when the steady-state wave gets away from the variable pressure zone, a long chain of structures develops similar to so called Kelvin-Helmholtz billows. The existence of nonlinear standing waves is discovered, which have a finite number of nodes in the free surface infinite in extent, and the frequency spectrum and the form of these waves are found explicitly.     

Simulation of particle ejecta effects from the surface under impulse action

This paper presents research results of ejecta due to plane shock wave arrival at a free smooth surface of a homogeneous pattern in the cluster dynamics (CD) simulation. Provided are 3D simulation results illustrating that ejecta effect is not only due to the physical reasons but also due to the side computation effects. One-dimensional model is developed to study the problems of cluster motion. This model was used to analyze the cluster behavior when a shock wave arrives at the boundary of the pattern. The oscillation character of the near-boundary clusters was analyzed as well as the impact of the interaction potential anharmonicity. It is shown that the most high-frequency mode of oscillations of the cluster lattice defined by potential anharmonicity (with neighbors moving in opposite phase) plays an important role in side ejecta effects. Hence, the criteria was developed to define the threshold loading velocity associated with ejecta in 3D problems.The method is suggested to eliminate the ejecta computation effects using the modification of cluster motion equations. The efficiency of this method is verified in a number of 3D simulations. It is shown that the suggested approach eliminates the side ejecta effects and keeps the fundamental physical regularities of loading and further motion of the pattern.     

Single-mode control and chaos of cantilever beam under primary and principal parametric excitations

A non-linear control law is proposed to suppress the vibrations of the first mode of a cantilever beam when subjected to primary and principal parametric excitations. The dynamics of the beam are modeled with a second-order non-linear ordinary-differential equation. The model accounts for viscous damping air drag, and inertia and geometric non-linearities. A control law based on quantic velocity feedback is proposed. The method of multiple scales method is used to derive two-first ordinary differential equations that govern the evolution of the amplitude and phase of the response. These equations are used to determine the steady state responses and their stability. Amplitude and phase modulation equations as well as external force–response and frequency–response curves are obtained. Numerical simulations confirm this scenario and detect chaos and unbounded motions in the instability regions of the periodic solutions.     

Nonlinear finite element reliability analysis of Concrete-Faced Rockfill (CFR) dams under static effects

The response of concrete slab on Concrete-Faced Rockfill (CFR) dams is very important. This study investigates the reliability of the concrete slab on a CFR dam by the improved Rackwitz–Fiessler method under static loads. For this purpose, ANSYS finite element analysis software and FERUM reliability analysis program are combined with direct coupled method and response surface method. Reliability index and probability of failure of the concrete are computed in the all critical points of the concrete slab by dam height. This study is also expanded for the reliability of CFR dams including different concrete slab thickness. In addition to the linear behavior, geometrically and materially non-linear responses of the dam are considered in the finite element analysis which is performed with reliability analysis. The Drucker–Prager method and the multi linear kinematic hardening method are, respectively, used for concrete slab and for rockfill and foundation rock. Finite element model used in the analyses includes dam–reservoir–foundation interaction. Reservoir water is modeled by the Lagrangian approach. Welded and friction contact based on the Coulomb’s friction law are considered in the joints of the dam. One-dimensional two noded contact elements are used to define friction. The self-weight of the dam and the hydrostatic pressure of the reservoir water are considered in the numerical solutions. According to this study, hydrostatic pressure, nonlinear response of the rockfill and the decrease in the concrete slab thickness reduce the reliability of the concrete slab of the CFR dam. Besides, the CFR dam models including friction are safer than the models including welded contact in the joints.     

Stability and local bifurcation analysis of functionally graded material plate under transversal and in-plane excitations

In this paper, stability and local bifurcation behaviors for a simply supported functionally graded material (FGM) rectangular plate subjected to the transversal and in-plane excitations in the uniform thermal environment are investigated using both analytical and numerical methods. Three kinds of degenerated equilibrium points of the bifurcation response equations are considered, which are characterized by a double zero eigenvalues, a simple zero and a pair of pure imaginary eigenvalues as well as two pairs of pure imaginary eigenvalues in nonresonant case, respectively. With the aid of Maple and normal form theory, the explicit expressions of transition curves are obtained, which may lead to static bifurcation, Hopf bifurcation and 2-D torus bifurcation. Finally, the numerical solutions obtained by using fourth-order Runge–Kutta method agree with the analytic predictions.     

On the competition of elastic energy and surface energy in discrete numerical schemes

The Γ-limit of certain discrete free energy functionals related to the numerical approximation of Ginzburg–Landau models is analysed when the distance h between neighbouring points tends to zero. The main focus lies on cases where there is competition between surface energy and elastic energy. Two discrete approximation schemes are compared, one of them shows a surface energy in the Γ-limit. Finally, numerical solutions for the sharp interface Cahn–Hilliard model with linear elasticity are investigated. It is demonstrated how the viscosity of the numerical scheme introduces an artificial surface energy that leads to unphysical solutions.