Complex Parametric Vibrations of Flexible Rectangular Plates

In this paper we consider parametric oscillations of flexible plates within the model of von Kármán equations. First we propose the general iterational method to find solutions to even more general problem governed by the von Kármán–Vlasov–Mushtari equations. In the language of physics the found solutions define stress–strain state of flexible shallow shell with a bounded convex space R ² and with sufficiently smooth boundary . The new variational formulation of the problem has been proposed and his validity and application has been discussed using precise mathematical treatment. Then, using the earlier introduced theoretical results, an effective algorithm has been applied to convert problem of finding solutions to hybrid type partial differential equations of von Kármán form to that of the ordinary differential (ODEs) and algebraic (AEs) equations. Mechanisms of transition to chaos of deterministic systems with infinite number of degrees of freedom are presented. Comparison of mechanisms of transition to chaos with known ones is performed. The following cases of longitudinal loads of different sign are investigated: parametric load acting along X direction only, and parametric load acting in both directions X and Y with the same amplitude and frequency.     

Microstructure-dependent couple stress theories of functionally graded beams

A microstructure-dependent nonlinear Euler-Bernoulli and Timoshenko beam theories which account for through-thickness power-law variation of a two-constituent material are developed using the principle of virtual displacements. The formulation is based on a modified couple stress theory, power-law variation of the material, and the von Kármán geometric nonlinearity. The model contains a material length scale parameter that can capture the size effect in a functionally graded material, unlike the classical Euler-Bernoulli and Timoshenko beam theories. The influence of the parameter on static bending, vibration and buckling is investigated. The theoretical developments presented herein also serve to develop finite element models and determine the effect of the geometric nonlinearity and microstructure-dependent constitutive relations on post-buckling response.     

Flexural-torsional bifurcations of a cantilever beam under potential and circulatory forces II. Post-critical analysis

The post-critical behavior of a cantilever beam with rectangular cross-section, under simultaneous action of conservative and non-conservative loads, is analyzed. An internally constrained Cosserat rod model is adopted to describe the dynamics of the beam in finite displacement regime. The bifurcation equations for simple buckling (divergence), simple flutter (Hopf) and double-zero (Takens-Bogdanova-Arnold) bifurcations are derived by means of the multiple time scales method. Due to the nilpotent eigenvalue at the double-zero critical point, the evaluation of the generalized Keldysh's eigenfunctions is required. Finally, some numerical results are shown and the bifurcation scenario of the beam is discussed.     

Bifurcation analysis for spherically symmetric systems using invariant theory

We study a degenerate steady state bifurcation problem with spherical symmetry. This singularity, with the five dimensional irreducible action ofO(3), has been studied by several authors for codimensions up to 2. We look at the case where the topological codimension is 3, theC -codimension is 5. We find a tertiary Hopf bifurcation and a heteroclinic orbit. Our analysis does not use any specific properties of the five dimensional representation and can in principle be used for higher representations as well. The computations are based on invariant theory and orbit space reduction.     

Supercritical and subcritical buckling bifurcations for a compressible hyperelastic slab subjected to compression

We study the buckling bifurcation of a compressible hyperelastic slab under compression with sliding–sliding end conditions. The combined series-asymptotic expansions method is used to derive the simplified model equations. Linear bifurcation analysis yields the critical stress value of buckling, which gives a non-linear correction to the classical Euler buckling formula. The correction is due to the geometrical non-linearities coupled with the material non-linearities. Then through non-linear bifurcation analysis, the approximate analytical solutions for the post-buckling deformations are obtained. The amplitude of buckling is expressed explicitly in terms of the aspect ratio, the incremental dimensionless engineering stress, the mode of buckling and the material constants. Most importantly, we find that both supercritical and subcritical buckling could occur for compressible materials. The bifurcation type depends on the material constants, the geometry of the slab and the mode numbers.     

A class of finite difference schemes for interface problems with an HOC approach

In this paper, we propose a new methodology for numerically solving elliptic and parabolic equations with discontinuous coefficients and singular source terms. This new scheme is obtained by clubbing a recently developed higher‐order compact methodology with special interface treatment for the points just next to the points of discontinuity. The overall order of accuracy of the scheme is at least second. We first formulate the scheme for one‐dimensional (1D) problems, and then extend it directly to two‐dimensional (2D) problems in polar coordinates. In the process, we also perform convergence and related analysis for both the cases. Finally, we show a new direction of implementing the methodology to 2D problems in cartesian coordinates. We then conduct numerous numerical studies on a number of problems, both for 1D and 2D cases, including the flow past circular cylinder governed by the incompressible Navier–Stokes equations. We compare our results with existing numerical and experimental results. In all the cases, our formulation is found to produce better results on coarser grids. For the circular cylinder problem, the scheme used is seen to capture all the flow characteristics including the famous von Kármán vortex street. Copyright © 2016 John Wiley & Sons, Ltd.     

Linear and non-linear interactions between static and dynamic bifurcations of damped planar beams

The critical and post-critical behavior of a non-conservative non-linear structure, undergoing statical and dynamical bifurcations, is analyzed. The system consists of a purely flexible planar beam, equipped with a lumped visco-elastic device, loaded by a follower force. A unique integro-differential equation of motion in the transversal displacement, with relevant boundary conditions, is derived. Then, the linear stability diagram of the trivial rectilinear configuration is built-up in the parameter space. Particular emphasis is given to the role of the damping on the critical scenario. The occurrence of different mechanisms of instability is highlighted, namely, of divergence, Hopf, double zero, resonant and non-resonant double Hopf, and divergence-Hopf bifurcation. Attention is then focused on the two latter (codimension-two) bifurcations. A multiple scale analysis is carried-out directly on the continuous model, and the relevant non-linear bifurcation equations in the amplitudes of the two interactive modes are derived. The fixed-points of these equations are numerically evaluated as functions of two bifurcation parameters and some equilibrium paths illustrated. Finally, the bifurcation diagrams, illustrating the system behavior around the critical points of the parameter space, are obtained.     

A streamfunction–velocity approach for 2D transient incompressible viscous flows

Electrodeposition is a widely used technique for the fabrication of high aspect ratio microstructures. In recent years, much research has been focused within this area aiming to understand the physics behind the filling of high aspect ratio vias and trenches on substrates and in particular how they can be made without the formation of voids in the deposited material. This paper reports on the fundamental work towards the advancement of numerical algorithms that can predict the electrodeposition process in micron scaled features. Two different numerical approaches have been developed, which capture the motion of the deposition interface and 2‐D simulations are presented for both methods under two deposition regimes: those where surface kinetics is governed by Ohm's law and the Butler–Volmer equation, respectively. In the last part of this paper the modelling of acoustic forces and their subsequent impact on the deposition profile through convection is examined. Copyright © 2009 John Wiley & Sons, Ltd.     

Bifurcations and the penetrating rate analysis of a model for percussive drilling

In this paper, we investigate a low dimensional model of percussive drilling with vibro-impact to mimic the nonlinear dynamics of the bounded progression. Non- holonomity which arises in the stick-slip caused by the impact during drilling fails to be correctly identified via the classical techniques. A reduced model without non-holono- mity is derived by the introduction of a new state variable, of which averaging technique is employed successfully to detect the periodic motions. Local bifurcations are presented directly by using C-L method. Numerical simulations and the penetrating rate analysis along different choices of parame- ters have been carried out to probe the nonlinear behaviour and the optimal penetrating rate of the drilling system.     

A unified constitutive theory for polymeric liquids I. Basic considerations and a simplified model

By combining a continuum mechanical approach with considerations of network theory and thermodynamics of irreversible processes, a set of differentialtype constitutive equations for polymeric liquids are obtained which provide expressions for the stress tensor, evolution equations of the effective Finger strain and Cauchy strain for the network deformation, and a first order differential equation governing the rigidity modulus. Unlike Giesekus' recent unified approach that starts from the bead-spring model, the theory lends itself more readily to a better understanding of most of the current theories based on continuum mechanics and molecular network concepts. Different recent models such as those due to Leonov, Dashner—Van Arsdale, Phan Thien—Tanner, and Acierno et al. (or Marrucci) can be unambiguously interpreted as resulting from specific approximations or additional assumptions.     

A new model for the study of rain-wind-induced vibrations of a simple oscillator

In this paper, a model equation is presented for the study of rain-wind-induced vibrations of a simple oscillator. As will be shown the presence of raindrops in the wind-field may have an essential influence on the dynamic stability of the oscillator. In this model equation the influence of the variation of the mass of the oscillator due to an incoming flow of raindrops hitting the oscillator and a mass flow which is blown and shaken off is investigated. The time-varying mass is modeled by a time harmonic function whereas simultaneously also time-varying lift and drag forces are considered.     

Flexural-torsional bifurcations of a cantilever beam under potential and circulatory forces I: Non-linear model and stability analysis

The stability of a cantilever elastic beam with rectangular cross-section under the action of a follower tangential force and a bending conservative couple at the free end is analyzed. The beam is herein modeled as a non-linear Cosserat rod model. Non-linear, partial integro-differential equations of motion are derived expanded up to cubic terms in the transversal displacement and torsional angle of the beam. The linear stability of the trivial equilibrium is studied, revealing the existence of buckling, flutter and double-zero critical points. Interaction between conservative and non-conservative loads with respect to the stability problem is discussed. The critical spectral properties are derived and the corresponding critical eigenspace is evaluated.     

A dynamic slip velocity model for molten polymers based on a network kinetic theory

In this paper the slip phenomenon is considered as a stochastic process where the polymer segments (taken as Hookean springs) break off the wall due to the excessive tension imposed by the bulk fluid motion. The convection equation arising in network theories is solved for the special case of a polymer/wall interface to determine the time evolution of the configuration distribution function (Q, t). The stress tensor and the slip velocity are calculated by averaging the proper relations over a large number of polymer segments. Due to the fact that the model is probabilistic and time dependent, dynamic slip velocity calculations become possible for the first time and therefore some new insight is gained on the slip phenomenon. Finally, the model predictions are found to match macroscopic experimental data satisfactorily.Nomenclature rate of creation of polymer segments - g(Q) constant of rate of creation of polymer segments - rate of loss of polymer segments - h(Q) constant of rate of loss of polymer segments - h(Q) constant of rate of loss of polymer segments due to destruction of its B-link - H Hookean spring constant - k Boltzmann's constant - n unit vector normal to the polymer/wall interface - n ₀ number density of polymer segments - n ₀ surface number density of polymer segments - Q vector defining the size and orientation of a polymer segment - Q ^* critical length of a segment beyond which the tension may overcome the W _adh - t time - t _h howering time of broken polymer segments - T absolute temperature - W _adh work of adhesion Greek Letters _n nominal strain - strain - _n nominal shear rate - shear rate - dimensionless constant in the expressions of h(Q), g(Q) - viscosity - ^T velocity gradient tensor - ₀ time constant - standard deviation of vectors Q at equilibrium - _w wall shear stress - stress tensor - ₀ equilibrium configuration distribution function of Q - configuration distribution function of Q     

On the modeling of granular traffic flow by the kinetic theory for active particles trend to equilibrium and macroscopic behavior

The modeling of vehicular traffic flow is developed by methods of the discrete mathematical kinetic theory for active particles. The discretization refers to the velocity variable in the case of spatially homogeneity. The discretization overcomes, at least in part, some technical difficulties related to the selection of the correct representation scale. Moreover, the modeling approach includes in the state equation of the vehicle an activity variable suitable to model the quality (low or high) of the vehicle-driver system. This paper aims to be the first of a project concerning traffic flow by active particles methods.     

Von Kármán rotating flow of Maxwell nanofluids featuring the Cattaneo-Christov theory with a Buongiorno model

This research paper analyzes the transport of thermal and solutal energy in the Maxwell nanofluid flow induced above the disk which is rotating with a constant angular velocity.The significant features of thermal and solutal relaxation times of fluids are studied with a Cattaneo-Christov double diffusion theory rather than the classical Fourier’s and Fick’s laws.A novel idea of a Buongiorno nanofluid model together with the Cattaneo-Christov theory is introduced for the first time for the Maxwel...     

Higher order model for soft and hard elastic interfaces

The present paper deals with the derivation of a higher order theory of interface models. In particular, it is studied the problem of two bodies joined by an adhesive interphase for which “soft” and “hard” linear elastic constitutive laws are considered. For the adhesive, interface models are determined by using two different methods. The first method is based on the matched asymptotic expansion technique, which adopts the strong formulation of classical continuum mechanics equations (compatibility, constitutive and equilibrium equations). The second method adopts a suitable variational (weak) formulation, based on the minimization of the potential energy. First and higher order interface models are derived for soft and hard adhesives. In particular, it is shown that the two approaches, strong and weak formulations, lead to the same asymptotic equations governing the limit behavior of the adhesive as its thickness vanishes. The governing equations derived at zero order are then put in comparison with the ones accounting for the first order of the asymptotic expansion, thus remarking the influence of the higher order terms and of the higher order derivatives on the interface response. Moreover, it is shown how the elastic properties of the adhesive enter the higher order terms. The effects taken into account by the latter ones could play an important role in the nonlinear response of the interface, herein not investigated. Finally, two simple applications are developed in order to illustrate the differences among the interface theories at the different orders.     

Weight function theory and variational formulations for three-dimensional plane elastic cracks advancing

The weight function theory for three-dimensional elastic crack analysis received great attention after the work of Rice (1985, 1989). Several applications have been considered since then, particularly in the context of configurational stability, crack path prediction, stress intensity factor expansions, perturbation approaches. In all cases, a specific hypothesis has been made on the variation of crack shape, in order to formulate the problem in terms of Cauchy principal value. In the present note, such hypothesis is further investigated and consequences discussed. A variational statement given in Salvadori and Fantoni (2013a) is thus rephrased in terms of weight functions. Its discrete formulation shows the potential to accurate approximation of crack front propagation.     

Non-linearities in rotation and thickness deformation in a new third-order thickness deformation theory for static and dynamic analysis of isotropic and laminated doubly curved shells

A geometrically non-linear theory is developed for shells of generic shape allowing for third-order thickness and shear deformation and rotary inertia by using eight parameters; geometric imperfections are also taken into account. The geometrically non-linear strain–displacement relationships are derived retaining full non-linear terms in all the 8 parameters, i.e. in-plane and transverse displacements, rotations of the normal and thickness deformation parameters; these relationships are presented in curvilinear coordinates, ready to be implemented in computer codes. Higher order terms in the transverse coordinate are retained in the derivation so that the theory is suitable also for thick laminated shells. Three-dimensional constitutive equations are used for linear elasticity. The theory is applied to circular cylindrical shells complete around the circumference and simply supported at both ends to study initially static finite deformation. Both radially distributed forces and displacement-dependent pressure are used as load and results for different shell theories are compared. Results show that a 6 parameter non-linear shell theory is quite accurate for isotropic shells. Finally, large-amplitude forced vibrations under harmonic excitation are investigated by using the new theory and results are compared to other available theories. The new theory with non-linearity in all the 8 parameters is the only one to predict correctly the thickness deformation; it works accurately for both static and dynamics loads.     

Correlation analysis of three influencing factors and the dust production rate for a free-falling particle stream

The effects of three factors (i.e., drop height h, hopper outlet diameter d₀, and material temperature T) on the dust generation rate derived from a free falling particle stream were investigated via full factorial experiments. The correlation between the three factors and dust generation rate was also analysed. Results show that T and h affect the first fugitive dust rate largely, whereas the second fugitive dust rate is mainly dominated by h and d₀. Through analysing the first fugitive dust percentage data, it is found that h and T should be considered first for higher temperatures and lower flow rates, whereas h and d₀ can be considered under contrasting conditions, and h should be controlled in the remaining two sets of conditions. Relationships between the influencing factors and total and first fugitive dust rates were developed via multiple regression to quantify the dust emission rates for different contact surfaces (rigid or water).