Experimental characterization of the instability of the vortex ring. Part I: Linear phase

The results of experiments performed to study the linear phase of the instability of vortex rings are presented. The experiments were performed in water. The vortex rings are generated by pushing water through the cylindrical nozzle of a pipe submerged in an aquarium. The experiments were made with the help of planar laser induced fluorescence as well as 2D2C and 2D3C particle image velocimetry. They show the straining field causing the instability, and for the first time experimentally the growth of a band of linear unstable modes. They also confirm previous studies concerning the shape of the instability and theories predicting the number of waves and the bandwidth of unstable modes. However, the measurement of the growth rate shows the influence of viscous damping, and consequently, the limit of the theories based on the hypothesis of an ideal fluid.     

Two-dimensional piezoelectricity. Part I: eigensolutions of nondegenerate and degenerate materials

General solutions of two-dimensional piezoelectricity, which yield all solutions of 2-D boundary values problems, are obtained by combining four complex conjugate pairs of independent eigensolutions, each containing an arbitrary analytic function. The forms of representation are fundamentally different for 14 different classes of nondegenerate and degenerate piezoelectric materials, as determined by the multiplicity and types of eigenvalues. Degenerate materials possess high-order eigensolutions, in which the eigenvectors of equal and lower orders are intrinsically coupled. Such coupling is nonexistent in nondegenerate cases including the well-known and analytically simple case with no multiple eigenvalues. The present analysis is drastically simplified by using the compliance-based formalism, instead of the stiffness-based, extended Eshelby–Stroh formalism. Explicit expressions are obtained for the eigensolutions, the pseudometrics, and the intrinsic tensors characterizing piezoelectric materials of every type.     

Experimental characterization of the instability of the vortex rings. Part II: Non-linear phase

The first stage of the instability of a vortex ring is linear and characterized by the growth of an azimuthal stationary wave which develops around the ring. Theoretical works predict its origin, shape, number of waves and growth rate. Apart for the growth rate, experimental and numerical results in viscous fluids fit well with the predictions based on an ideal fluid hypothesis. On the other hand, the next stages of the development of the instability (which are non-linear) are not well known. Only few phenomena are described, in an isolated way, in various partial contributions. The aim of this paper is to report on a complete experimental investigation of the non-linear phase of the instability of the vortex ring. The vortices were produced in water and their Reynolds number Re _p was varied from 2,650 to 6,100. Visualizations were performed using planar laser induced fluorescence and measurements with 2D2C and 2D3C particle image velocimetry. Based on a Fourier analysis of the results, it appears that the non-linear phase begins with the development of harmonics of the linear modes (first unstable modes). But the growth of those harmonics is rapidly stopped by the development of low order modes. Then appears an m=0 mode, which corresponds to a mean azimuthal velocity around the vortex. Simultaneously, secondary vortical structures develop all around the vortex in its peripheral zone. These vortical structures are linked with the ejection of vorticity in the wake of the ring and they appear just before the transition towards turbulence. A tentative is made here to place all these phenomena chronologically, in order to propose a scenario for the transition from the linear phase to turbulence.     

Fundamental solution for bonded materials with a free surface parallel to the interface. Part I: Solution of concentrated forces acting at the inside of the material with a free surface

The problem studied in this paper is that of a coated semi-infinite plane subjected to a concentrated force in the upper thin layer (or film). The elastic properties of the coating material are different from those of the substrate, and a perfect bond is assumed between the two materials. The exact solutions of stress functions in a series form are obtained by the method of image. The terms in series form of the stress functions correspond to the image points from the lower order to the higher. The recurrence relations of the stress functions are given, i.e., the stress functions corresponding to the higher order image points are determined by the lower ones. Hence, from the original stress functions for an infinite plane subjected to a concentrated force, the explicit formulas of all terms of the stress function series can be derived. Also, through comparisons between the theoretical results and the numerical results by FEM, it is verified that the convergence rate of the solutions is very rapid. In most practical cases only the first several image points are sufficient to ensure the accuracy of the solutions.     

Experimental Investigation of the Nonlinear Response of a Hanging Cable. Part I: Local Analysis

An experimental model of an elastic cable carrying eight concentrated masses and hanging at in-phase or out-of-phase vertically moving supports is considered. The system parameters are adjusted to approximately realize multiple 1:1 and 2:1 internal resonance conditions involving planar and nonplanar, symmetric and antisymmetric modes. Response measurements are made in various frequency ranges including meaningful external resonance conditions. A local analysis of the system response is made on the basis of numerous amplitude-frequency and amplitude-forcing plots obtained in different ranges of the control parameter space. Attention is mainly devoted to the detection of the main features of the regular motions exhibited by the system, and to the analysis of the relevant phenomena of nonlinear modal interaction, competition, and local bifurcation between planar and nonplanar regular responses. The resulting picture appears very rich and varied.     

A theory of strain-gradient plasticity for isotropic, plastically irrotational materials. Part I: Small deformations

This study develops a small-deformation theory of strain-gradient plasticity for isotropic materials in the absence of plastic rotation. The theory is based on a system of microstresses consistent with a microforce balance; a mechanical version of the second law that includes, via microstresses, work performed during viscoplastic flow; a constitutive theory that allows:

•

the microstresses to depend on , the gradient of the plastic strain-rate, and

•

the free energy ψ to depend on the Burgers tensor .

The microforce balance when augmented by constitutive relations for the microstresses results in a nonlocal flow rule in the form of a tensorial second-order partial differential equation for the plastic strain. The microstresses are strictly dissipative when ψ is independent of the Burgers tensor, but when ψ depends on G the gradient microstress is partially energetic, and this, in turn, leads to a back stress and (hence) to Bauschinger-effects in the flow rule. It is further shown that dependencies of the microstresses on lead to strengthening and weakening effects in the flow rule.Typical macroscopic boundary conditions are supplemented by nonstandard microscopic boundary conditions associated with flow, and, as an aid to numerical solutions, a weak (virtual power) formulation of the nonlocal flow rule is derived.     

Nonlinear flexural-flexural-torsional interactions in beams including the effect of torsional dynamics. I: Primary resonance

Nonlinear coupling between torsional and both in-plane and out-of-plane flexural motion is examined for inextensional beams (or beam-like structures) whose torsional and flexural eigenfrequencies are of the same order. The analysis presented here is based on a consistent set of nonlinear differential equations which contain both curvature and inertia nonlinearities, and account for torsional dynamics. Response characteristics, including stability, are determined for cantilever beams subjected to a lateral periodic excitation. The beam's response in the presence of a one-to-one internal resonance involving a torsional frequency and an in-plane bending frequency is investigated in detail.     

Analytical modeling of synthetic fiber ropes subjected to axial loads. Part I: A new continuum model for multilayered fibrous structures

Synthetic fiber ropes are characterized by a very complex architecture and a hierarchical structure. Considering the fiber rope architecture, to pass from fiber to rope structure behavior, two scale transition models are necessary, used in sequence: one is devoted to an assembly of a large number of twisted components (multilayered), whereas the second is suitable for a structure with a central straight core and six helical wires (1 + 6). The part I of this paper first describes the development of a model for the static behavior of a fibrous structure with a large number of twisted components. Tests were then performed on two different structures subjected to axial loads. Using the model presented here the axial stiffness of the structures has been predicted and good agreement with measured values is obtained. A companion paper (Ghoreishi, S.R. et al., in press. Analytical modeling of synthetic fiber ropes, part II: A linear elastic model for 1 + 6 fibrous structures, International Journal of Solids and Structures, doi:10.1016/j.ijsolstr.2006.08.032) presents the second model to predict the mechanical behavior of a 1 + 6 fibrous structure.     

Nonlinear Dynamics of a Rigid Unbalanced Rotor in Journal Bearings. Part I: Theoretical Analysis

The dynamic behaviour of a rigid rotor supported on plain journal bearings was studied, focusing particular attention on its nonlinear aspects. Under the hypothesis that the motion of the rotor mass center is plane, the rotor has five Lagrangian co-ordinates which are represented by the co-ordinates of the mass center and the three angular co-ordinates needed to express the rotor's rotation with respect to its center of mass. In such conditions, the system is characterised not only by the nonlinearity of the bearings but also by the nonlinearity due to the trigonometric functions of the three assigned angular co-ordinates. However, if two angular co-ordinates have values that are generally quite small because of the small radial clearances in the bearings, the system is de facto linear in these angular co-ordinates. Moreover, if the third angular co-ordinate is assumed to be cyclic [18], the number of degrees of freedom in the system is reduced to four and nonlinearity depends solely on the presence of the journal bearings, whose reactions were predicted with the -film, short bearing model. After writing the equations of motion in this way and determining a numerical routine for a Runge–Kutta integration the most significant aspects of the dynamics of a symmetrical rotor were studied, in the presence of either pure static or pure couple unbalance and also when both types of unbalance were present. Two categories of rotors, whose motion is prevailingly a cylindrical whirl or a conical whirl, were put under investigation.     

Free surface on a simple fluid between rotating eccentric cylinders. Part I: Analytical solution

The steady motion of a simple fluid between vertical cylinders which rotate about non-concentric axes is solved by means of domain perturbations. The theory is developed as a perturbation of the rest state in powers of the angular frequency ω of the inner cylinder, and the solution is carried out to O (ω²). The stress is expanded in a series of Rivlin-Ericksen tensors. At the second order only one material parameter, the climbing constant, enters the analysis. A procedure is developed for predicting the shape of the free surface on the fluid. Secondary motions generated by the eccentricity are shown to appear at the second order.     

Nonlinear dynamic analysis of Timoshenko beams by BEM. Part I: Theory and numerical implementation

In this two-part contribution, a boundary element method is developed for the nonlinear dynamic analysis of beams of arbitrary doubly symmetric simply or multiply connected constant cross section, undergoing moderate large displacements and small deformations under general boundary conditions, taking into account the effects of shear deformation and rotary inertia. Part I is devoted to the theoretical developments and their numerical implementation and Part II discusses analytical and numerical results obtained from both analytical or numerical research efforts from the literature and the proposed method. The beam is subjected to the combined action of arbitrarily distributed or concentrated transverse loading and bending moments in both directions as well as to axial loading. To account for shear deformations, the concept of shear deformation coefficients is used. Five boundary value problems are formulated with respect to the transverse displacements, to the axial displacement and to two stress functions and solved using the Analog Equation Method, a BEM based method. Application of the boundary element technique yields a nonlinear coupled system of equations of motion. The solution of this system is accomplished iteratively by employing the average acceleration method in combination with the modified Newton–Raphson method. The evaluation of the shear deformation coefficients is accomplished from the aforementioned stress functions using only boundary integration. The proposed model takes into account the coupling effects of bending and shear deformations along the member, as well as the shear forces along the span induced by the applied axial loading.

     

Inelastic impact dynamics of ships with one-sided barriers. Part I: analytical and numerical investigations

This two-part paper deals with impact interaction of ships with one-sided ice barrier during roll dynamics. The first part presents analytical and numerical studies for the case of inelastic impact. An analytical model of a ship roll motion interacting with ice is developed based on Zhuravlev and Ivanov non-smooth coordinate transformations. These transformations have the advantage of converting the vibro-impact oscillator into an oscillator without barriers such that the corresponding equation of motion does not contain any impact term. Such approaches, however, account for the energy loss at impact times in different ways. The present work, in particular, demonstrates that the impact dynamics may have qualitatively different response characteristics to different dissipation models. The difference between localized and distributed equivalent damping approaches is discussed. Extensive numerical simulations are carried out for all initial conditions covered by the ship grazing orbit for different values of excitation amplitude and frequency of external wave roll moment. The basins of attraction of safe operation are obtained and reveal the coexistence of different response regimes such as non-impact periodic oscillations, modulation impact motion, period added impact oscillations, chaotic impact motion and rotational motion. The second part will consider experimental validations of predicted results.     

Dynamics of a Rigid Unbalanced Rotor with Nonlinear Elastic Restoring Forces. Part I: Theoretical Analysis

In a previous paper, the dynamic behaviour of a Jeffcott rotor was studied in the presence of pure static unbalance and nonlinear elastic restoring forces. The present paper extends the analysis to a rigid rotor with an axial length such as to make the transverse moment of inertia greater than the axial one. As in the previous investigation, the elastic restoring forces are assumed to be nonlinear and the effects of couple unbalance are also included but, unlike the Jeffcott rotor, the system exhibits six degrees-of-freedom. The Lagrangian coordinates were fixed so as to coincide with the three coordinates of the centre of mass of the rotor and the three angular coordinates needed in order to express the rotor's rotations with respect to a reference frame having its origin in the centre of mass. The precession motions of such a rotor turn out to be cylindrical at low angular speeds and exhibit a conical aspect when operating at higher speeds. The motion equations of the rotor were written with reference to a system that was subsequently adopted for the experimental analysis. The particular feature of this system was the use of a steel wire (piano wire) for the rotor shaft, suitably constrained and with the possibility of regulating the tension of the wire itself, in order to increase or reduce the nonlinear character of the system. The numerical analysis performed with integration of the motion equations made it possible to point out that chaotic solutions were manifested only when the tension in the wire was given the lowest values – i.e. when the system was strongly nonlinear – in the presence of considerable damping and rotor unbalance values that were so high as to lose any practical significance. Under conditions commonly shared by analogous real systems characterised by poor damping, where the contribution to nonlinearity is almost entirely due to elastic restoring forces, the analysis pointed out that precession motions may be manifested with a periodic character, whether synchronous or not, or a quasi-periodic character, but in no case is the solution chaotic.     

Effect of surface layers on the constriction resistance of an isothermal spot. Part I: Reduction to an integral equation and numerical results

The problem of the constriction resistance of a circular spot on a half-space covered with a uniform layer of different material is considered. For the general case of any specified axisymmetric distributions of temperature over the spot and heat flux over the rest of the surface, the mixed boundary value problem governing the heat flow from the spot to the underlying layer-substrate composite is converted to a non-homogeneous Fredholm integral equation of the second kind. For the particular case of isothermal spot on otherwise insulated surface, the evaluation of the constriction resistance is reduced to the reciprocal of a simple integral with the solution of the relevant integral equation as integrand. The integral equation is solved numerically and very accurate results are obtained for the constriction resistance over four orders of magnitude variation of the ratio, , of layer thickness to spot radius and the ratio, k_r, of layer to substrate conductivities for both conducting (k_r > 1) and insulating (k_r < 1) layers. An extensive discussion of the numerical results is presented with particular emphasis on their implications for the contact resistance of practical joints in the presence of interfacial layers. Further, in the light of the numerical results, two widely used analytical approximations for the constriction resistance – the first of which results from replacing the isothermal condition over the spot by a special flux (herein called the Equivalent Isothermal Heat Flux, EIHF) condition which is believed to render the spot nearly isothermal and the second is a consequence of the assumption (herein termed the Thin Insulating Layer Approximation, TILA) that, for thin insulating layers like oxide films, the heat flow in the layer region right beneath the spot is purely axial – are assessed as to their levels of accuracy and ranges of applicability with respect to both and k_r.     

Multi-scale defect interactions in high-rate brittle material failure. Part I: Model formulation and application to ALON

Within this two part series we develop a new material model for ceramic protection materials to provide an interface between microstructural parameters and bulk continuum behavior to provide guidance for materials design activities. Part I of this series focuses on the model formulation that captures the strength variability and strain rate sensitivity of brittle materials and presents a statistical approach to assigning the local flaw distribution within a specimen. The material model incorporates a Mie–Grüneisen equation of state, micromechanics based damage growth, granular flow and dilatation of the highly damaged material, and pore compaction for the porosity introduced by granular flow. To provide initial qualitative validation and illustrate the usefulness of the model, we use the model to investigate Edge on Impact experiments (Strassburger, 2004) on Aluminum Oxynitride (AlON), and discuss the interactions of multiple mechanisms during such an impact event. Part II of this series is focused on additional qualitative validation and using the model to suggest material design directions for boron carbide.     

Nonlinear waves on the surface of a charged liquid. Instability,bifurcation, and nonequilibrium shapes of the charged surface

Plane nonlinear waves in shallow water are described by the Kortewegde Vries equation [1–3]. The present paper contains theoretical investigations of nonlinear waves and nonlinear equilibrium shapes on the surface of a charged liquid. The influence of the field on the velocity and shape of a hydrodynamic soliton is considered. The bifurcation of the equilibrium shapes is investigated. Problems of the equilibrium shapes of a charged liquid are solved in the nonlinear formulation of the dynamics of nonlinear solitary forms (lunes, trenches) on the surface.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 94–102, May–June, 1984.     

A direct approach to fiber and membrane reinforced bodies. Part I. Stress concentrated on curves for modelling fiber reinforced materials

An approach is outlined to the equilibrium in fiber-reinforced materials in which the fibers are modeled as curves or lines with concentrated material properties. The system of forces representing the interaction of the fibers with the bulk matter is analyzed, and equilibrium of forces is derived from global laws. The displacements of the bulk matter are assumed to have continuous extension to the fibers. This forces the set of admissible deformations superquadratically integrable. This in turn forces the energy of the bulk of superquadratic growth. The material of the bulk matrix therefore cannot be linearly elastic. The energy of fibers can have a slower growth and can be quadratic. A formal set of assumptions is given under which an equilibrium state of minimum energy exists in the given external conditions. A weak form of equilibrium equations is derived for this equilibrium state. An explicitly calculable axisymmetric example is presented with an isotropic and quadratic energy of the matrix (linear elasticity) and linearly stretchable fiber. Since the superquadratic growth assumption is not satisfied, some peculiar features of the solution arise, such as the infinite limit of the radial displacement near the fiber. Nevertheless, from the obtained solution, we can compute the normal force in the fiber and the shear stress at the interface.     

Improved averaging method for turbulent flow simulation. Part I: Theoretical development and application to Burgers' transport equation

This is the first of two articles intended to develop, apply and verify a new method for averaging the momentum and mass transport equations for turbulence. The new method is based on Gaussian filtering in both the spatial and temporal domains. Application is made to the problem of momentum and scalar transport in a one-dimensional transient Burgers' flow field. No actual calculations, with the averaged equations, are presented in this paper. However, an ‘exact’ solution of the one-dimensional flow situation is presented as an economical tool for verifying the performance of the different turbulence models. In the second paper calculations are performed with the averaged one-dimensional equations on coarse grids, and the results are compared to the exact or fully simulated data with a statistical verification procedure.     

Nonlinear behaviour of different flexible size-dependent beams models based on the modified couple stress theory. Part 2. Chaotic dynamics of flexible beams

In the present part of the paper various problems of non-linear dynamics of nano-beams within the modified couple stress theory as well as the Bernoulli-Euler, Timoshenko, and Sheremetev-Pelekh-Reddy-Levinson models are studied taking into account the geometric non-linearity. Different characteristics of the vibrational process, including Fourier spectra, wavelet spectra, phase portraits, Poincaré maps as well as the largest Lyapunov exponents, are studied for the same physical-geometric parameter with and without consideration of the size-dependent behaviour. Vibration graphs are constructed and analysed, and scenarios of transition from regular to chaotic vibrations are illustrated and discussed.     

Two-to-one resonant multi-modal dynamics of horizontal/inclined cables. Part I: Theoretical formulation and model validation

This paper is first of the two papers dealing with analytical investigation of resonant multi-modal dynamics due to 2:1 internal resonances in the finite-amplitude free vibrations of horizontal/inclined cables. Part I deals with theoretical formulation and validation of the general cable model. Approximate nonlinear partial differential equations of 3-D coupled motion of small sagged cables – which account for both spatio-temporal variation of nonlinear dynamic tension and system asymmetry due to inclined sagged configurations – are presented. A multi-dimensional Galerkin expansion of the solution of nonplanar/planar motion is performed, yielding a complete set of system quadratic/cubic coefficients. With the aim of parametrically studying the behavior of horizontal/inclined cables in Part II [25], a second-order asymptotic analysis under planar 2:1 resonance is accomplished by the method of multiple scales. On accounting for higher-order effects of quadratic/cubic nonlinearities, approximate closed-form solutions of nonlinear amplitudes, frequencies and dynamic configurations of resonant nonlinear normal modes reveal the dependence of cable response on resonant/nonresonant modal contributions. Depending on simplifying kinematic modeling and assigned system parameters, approximate horizontal/inclined cable models are thoroughly validated by numerically evaluating statics and non-planar/planar linear/non-linear dynamics against those of the exact model. Moreover, the modal coupling role and contribution of system longitudinal dynamics are discussed for horizontal cables, showing some meaningful effects due to kinematic condensation.