Study of a transition in the qualitative behavior of a simple oscillator with Coulomb friction

A simple mass-spring system is submitted to a constant force in addition to a periodic perturbation of rectangular wave shape. It has been obtained in a previous study that the range of the period-amplitude plane of this perturbation, where the trajectories are sliding with no loss of contact, is divided into two parts, one in which there exist infinitely many equilibrium states and no periodic solutions, and another one where there exist periodic solutions and no equilibrium states. The present work focuses on the transition between these two parts. All along the transition line, there exists a single equilibrium state. Initial data out of equilibrium lead either to a periodic trajectory, or to a trajectory, which tends to the equilibrium or to a periodic solution, either in finite time or at infinity.     

The Sub-Harmonic Bifurcation of Stokes Waves

Steady periodic water waves on the free surface of an infinitely deep irrotational flow under gravity without surface tension (Stokes waves) can be described in terms of solutions of a quasi-linear equation which involves the Hilbert transform and which is the Euler-Lagrange equation of a simple functional. The unknowns are a 2π-periodic function w which gives the wave profile and the Froude number, a dimensionless parameter reflecting the wavelength when the wave speed is fixed (and vice versa). Although this equation is exact, it is quadratic (with no higher order terms) and the global structure of its solution set can be studied using elements of the theory of real analytic varieties and variational techniques. In this paper it is shown that there bifurcates from the first eigenvalue of the linearised problem a uniquely defined arc-wise connected set of solutions with prescribed minimal period which, although it is not necessarily maximal as a connected set of solutions and may possibly self-intersect, has a local real analytic parametrisation and contains a wave of greatest height in its closure (suitably defined). Moreover it contains infinitely many points which are either turning points or points where solutions with the prescribed minimal period bifurcate. (The numerical evidence is that only the former occurs, and this remains an open question.) It is also shown that there are infinitely many values of the Froude number at which Stokes waves, having a minimal wavelength that is an arbitrarily large integer multiple of the basic wavelength, bifurcate from the primary branch. These are the sub-harmonic bifurcations in the paper's title. (In 1925 Levi-Civita speculated that the minimal wavelength of a Stokes wave propagating with speed c did not exceed 2πc ²/g. This is disproved by our result on sub-harmonic bifurcation, since it shows that there are Stokes waves with bounded propagation speeds but arbitrarily large minimal wavelengths.) Although the work of Benjamin & Feir} and others [9, 10] has shown Stokes waves on deep water to be unstable, they retain a central place in theoretical hydrodynamics. The mathematical tools used to study them here are real analytic-function theory, spectral theory of periodic linear pseudo-differential operators and Morse theory, all combined with the deep influence of a paper by Plotnikov [36]. Accepted: December 6, 1999     

内压载荷作用下薄膜椭球的稳定性分析

超弹性橄榄状和南瓜状薄膜椭球在内压载荷作用下存在不同的分岔形式.对橄榄状薄膜椭球来说,细长比大于某一临界值时,在一定内压作用下会发生梨形分岔;小于该临界值时,薄膜椭球的分岔行为与圆管的局部起鼓现象相类似.对南瓜状薄膜椭球,无论圆扁,当内压达到某载荷值时都会发生梨形分岔.本文采用能量判据,分析了在压强控制和质量控制两种加载方式作用下,不同形状的薄膜椭球的均匀解及分岔解的稳定性.通过计算要考察的平衡状态及施加小扰动之后状态的能量差来判断当前状态是否稳定,结果表明,在压强控制下,P-V曲线下降段的均匀解和分岔解均为不稳定解.但在质量控制下,在P-V曲线下降段中只有均匀解出现时,均匀解为稳定解;而在均匀解和分岔解共存的区间内,均匀解为不稳定解,分岔解为稳定解.另外,P-V曲线两个上升段的均匀解则均为稳定解.     

The wedge subjected to tractions: a paradox re-examined

The classical two-dimensional solution for the stress distribution in an elastic wedge loaded by a uniform pressure on one side of the wedge becomes infinite when the wedge angle 2 satisfies the equation tan 235-1. This paradox was resolved recently by Dempsey who obtained a solution which is bounded at 235-2. However, for not equal but very close to 235-3, the classical solution can still be very large as approaches 235-4. In this paper we re-examine the paradox. We obtain a solution which remains bounded as approaches 235-5 and reproduces Dempsey's solution in the limit 235-6. At 235-7 the stress distribution contains a (ln r) term for general loadings. The (ln r) term disappears under a special loading and the stresses are bounded for all r. Moreover, the solution is not unique. In other words, for the wedge angle 235-8 under a special loading, infinitely many solutions exist for which the stresses are bounded for all r. We also obtain solutions which are bounded and approach Dempsey's solutions when 2= and 2. Again, under a special loading infinitely many solutions exist for which the stresses are bounded for all r. Care has been exercised in this paper to present the solutions in a form in which the terms r ^- and ln r are replaced by R ^-gl and ln R where R=r/r ₀is the dimensionless radial distance and r ₀ is an arbitrary constant having the dimension of length.     

Criteria for Floating I

Conditions are determined under which solid bodies will float on a liquid surface in stable equilibrium, under the influence of gravity and of surface tension. These include configurations in which the density of the body exceeds the density of the ambient liquid, so that for an infinitely deep liquid in a downward gravity field there is no absolute energy minimum. Of notable interest are the results (a) that if a smooth body is held rigidly and translated downward into an infinite fluid bath through a family of fluid equilibrium configurations in a downward gravity field, the transition is necessarily discontinuous, and (b) a formal proof that there can be a free-floating locally energy minimizing configuration that does not globally minimize, even if the density of the body exceeds that of the liquid. The present work is limited to the two dimensional case corresponding to a long cylinder that is floating horizontally. The more physical three-dimensional case can be studied in a similar way, although details of behavior can change significantly. That work will appear in an independent study written jointly with T. I. Vogel.     

Nonlinear steady states of hyperelastic membrane tubes conveying a viscous non-Newtonian fluid

We study possible steady states of an infinitely long tube made of a hyperelastic membrane and conveying either an inviscid, or a viscous fluid with power-law rheology. The tube model is geometrically and physically nonlinear; the fluid model is limited to smooth changes in the tube’s radius. For the inviscid case, we analyse the tube’s stretch and flow velocity range at which standing solitary waves of both the swelling and the necking type exist. For the viscous case, we first analyse the tube’s upstream and downstream limit states that are balanced by infinitely growing upstream (and decreasing downstream) fluid pressure and axial stress caused by fluid viscosity. Then we investigate conditions that can connect these limit states by a single solution. We show that such a solution exists only for sufficiently small flow speeds and that it has a form of a kink wave; solitary waves do not exist. For the case of a semi-infinite tube (infinite either upstream or downstream), there exist both kink and solitary wave solutions. For finite-length tubes, there exist solutions of any kind, i.e. in the form of pieces of kink waves, solitary waves, and periodic waves.     

Solitons in Several Space Dimensions:¶Derrick's Problem and¶Infinitely Many Solutions

In this paper we study a class of Lorentz invariant nonlinear field equations in several space dimensions. The main purpose is to obtain soliton-like solutions. These equations were essentially proposed by C. H. Derrick in a celebrated paper in 1964 as a model for elementary particles. However, an existence theory was not developed. The fields are characterized by a topological invariant, the charge. We prove the existence of a static solution which minimizes the energy among the configurations with nontrivial charge. Moreover, under some symmetry assumptions, we prove the existence of infinitely many solutions, which are constrained minima of the energy. More precisely, for every n∈:N there exists a solution of charge n. Accepted March 13, 2000?Published online September 12, 2000     

Displacement of oil by solutions of an active additive from a uniform stratum with allowance for gravity

Self-similar solutions describing the displacement of oil by solutions of an adsorbed active additive have been obtained and investigated [1–3] in the framework of a one-dimensional flow model with neglect of diffusion, capillary, and gravity effects. In the present paper, a self-similar solution is constructed for the problem of oil displacement by an aqueous solution of an active additive from a thin horizontal stratum with allowance for gravity under the assumption that there is instantaneous vertical separation of the phases. This makes it possible to estimate the effectiveness of flooding a stratum by solutions of surfactants and polymers in the cases when gravitational segregation of the phases cannot be ignored.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 87–92, January–February, 1984.     

Solution behavior of the falling sphere problem in viscoelastic flows

Thehp-finite element method and pseudo arc-length method are combined to track solution behavior of the steady flow of a sphere falling in a tube filled with viscoelastic fluids. The computations is proved to be convergent and stable by a posteriori error analyses; the solutions smooth and with extremely steep stress gradient are obtained by using appropriate high-order interpolation distributions. The commonly used drag coefficient is proved not a reliable indicator for the approximation error. The solution curves of the upper convected Maxwell (UCM) and the Oldroyd-B fluids clearly indicate that there exist limiting points, parameterized by the Deborah number, where the magnitude of the solutions and the axial normal stress gradients behind the rear stagnation rise up rapidly, maybe go to infinity. The limiting points can be simply removed by employing the modified Chilcott and Rallison (MCR) fluid, which replaces the infinitely extendable, linear Gaussian chain with the finitely extendable nonlinear elastic spring (the FENE chain). Therefore the limiting behavior is caused by the physical model; it is not a numerical artifact. The project supported by the National Natural Science Foundation of China (20274041) and the Cooperative Research Center for Polymers, Australia     

Standing Waves on an Infinitely Deep Perfect Fluid Under Gravity

The existence of two-dimensional standing waves on the surface of an infinitely deep perfect fluid under gravity is established. When formulated as a second-order equation for a real-valued function w on the 2-torus and a positive parameter μ, the problem is fully nonlinear (the highest order x-derivative appears in the nonlinear term but not in the linearization at 0) and completely resonant (there are infinitely many linearly independent eigenmodes of the linearization at 0 for all rational values of the parameter μ). Moreover, for any prescribed order of accuracy there exists an explicit approximate solution of the nonlinear problem in the form of a trigonometric polynomial. Using a Nash-Moser method to seek solutions of the nonlinear problem as perturbations of the approximate solutions, the existence question can be reduced to one of estimating the inverses of linearized operators at non-zero points. After changing coordinates these operators become first order and non-local in space and second order in time. After further changes of variables the main parts become diagonal with constant coefficients and the remainder is regularizing, or quasi-one-dimensional in the sense of [22]. The operator can then be inverted for two reasons. First, the explicit formula for the approximate solution means that, restricted to the infinite-dimensional kernel of the linearization at zero, the inverse exists and can be estimated. Second, the small-divisor problems that arise on the complement of this kernel can be overcome by considering only particular parameter values selected according to their Diophantine properties. A parameter-dependent version of the Nash-Moser implicit function theorem now yields the existence of a set of unimodal standing waves on flows of infinite depth, corresponding to a set of values of the parameter μ>1 which is dense at 1. Unimodal means that the term of smallest order in the amplitude is cos x cos t, which is one of many eigenfunctions of the completely resonant linearized problem.     

Multiple three-dimensional equilibrium solutions for cantilever beams loaded by dead tip and uniform distributed loads

The development of multiple solutions for orthotropic cantilever beams in a fully three-dimensional setting is investigated. The governing equations are solved using an iterative shooting procedure that converts the original boundary value problem into a sequence of initial value problems that converge to the desired solution. This method is well suited to finding multiple equilibrium solutions. Several classes of equilibrium configurations are described and illustrated including planar shapes, buckled planar shapes and fully three-dimensional configurations which appear far removed from the initial plane of loading. The solutions for the planar shapes and the buckled configurations compare favourably to previously published results. The development of the far-removed shapes is shown to be qualitatively similar to that of the planar shapes. The behaviour is shown to be highly dependant upon the aspect ratio of the cross-section. For certain aspect ratios it is shown, somewhat surprisingly, that out-of-plane equilibrium solutions can exist at loads below those required for multiple planar solutions.     

轴对称横观各向同性热弹性圆柱的分解

为了得到精确的应力场、位移场、温度场,将扭转圆轴的精化理论研究方法推广到轴对称横观各向同性热弹性圆柱。利用Bessel函数以及轴对称横观各向同性热弹性圆柱的通解,给出了轴对称横观各向同性热弹性圆柱的分解定理。根据柱面齐次边界条件获得了精确的精化方程,精化方程可以分解为一阶方程、超越方程、温度方程,从而将横观各向同性热弹性圆柱的轴对称问题分解为轴向拉压问题、超越问题、热-应力耦合问题。超越部分对应端部自平衡情况,可以清晰地了解到端部应力分布对内部应力场的影响,热-应力耦合部分对应无外加应力场时圆柱内部因温度变化引起的热应力。     

Finite element analysis for steady-state hydride-induced fracture in metals by composite model

Delayed hydride cracking, which is observed in hydride-forming metals, due to the precipitation of hydrides near the crack tip, is investigated under conditions of constant temperature and crack velocity, plane strain and small-scale hydride-precipitation. The coupling of the operating physical processes of hydrogen-diffusion, hydride precipitation and material deformation is taken into account. The material is assumed to be an elastic composite made of hydrides and solid solution, with properties depending locally on the volume fraction of the hydrides. In the present analysis, the composite elastic properties have been derived by a generalized self consistent model for particulate composites. With respect to hydride-precipitation, two cases have been considered: (i) precipitation in a homogeneous medium with elastic properties, equal to the effective properties of the composite and (ii) precipitation in an inhomogeneous medium, where the expanding hydride has different elastic properties than those of the surrounding solid solution. The differences between the near-tip field distributions, produced by the two precipitation models, are relatively small. The effect of the hydrogen concentration far from the crack tip, on the near-tip field is also studied. It is shown that for small crack growth velocities, near the threshold stress intensity factor, the remote hydrogen concentration weakly affects the normalized stress distribution in the hydride-precipitation zone, which is controlled by the thermodynamically required hydrostatic stress, under hydrogen chemical equilibrium. However, for values of the applied stress intensity factor and the crack tip velocity, away from the threshold stress intensity factor and crack arrest, the effect of remote hydrogen concentration on the normalized near-tip stress field is strong. Reduction of the remote hydrogen concentration generally leads to reduction of the hydride-precipitation zone and increase of the near-tip stresses. Also reduction of the remote hydrogen concentration leads to distributions closer to those under hydrogen chemical equilibrium.     

Non-fourier heat conduction in a plane slab with prescribed boundary heat flux

The non-stationary heat conduction in an infinitely wide plane slab with a prescribed boundary heat flux is studied. An arbitrary time dependent boundary heat flux is considered and a non-vanishing thermal relaxation time is assumed. The temperature and the heat flux density distributions are determined analytically by employing Cattaneo-Vernotte's constitutive equation for the heat flux density. It is proved that the temperature and the heat flux density distributions can be incompatible with the hypothesis of local thermodynamic equilibrium. A comparison with the solution which would be obtained by means of Fourier's law is performed by considering the limit of a vanishing thermal relaxation time.     

General series solution for free convection past a non-isothermal vertical flat plate

Summary Free convection past a vertical plate is studied theoretically for a general class of nonlinear wall temperature distributions. Due to the broad approach it is possible in some cases to find two series solutions applying to the same wall temperature distribution which are valid in different parts of the x region. Through graphical joining of both solutions an overall valid solution is obtained. This is illustrated by examples.     

Nonlinear Spherical Caps and Associated Plate and Membrane Problems

This paper deals with the existence and multiplicity problem of the equilibrium solutions of an elastic spherical cap within nonlinear strain theory. We pose the problem in the form of a three parameter bifurcation problem, one parameter being related to the load, the others to the geometry. When the geometrical parameters are different from zero, the so-called generic case, we revisit the nonuniqueness results, and explore the solutions in the parameter space. Then we analyze the formal limits as the geometrical parameters tend to zero. When the curvature tends to zero, we obtain from the nonlinear shell a von Kármán plate, a new, although natural, result. When the thickness parameter tends to zero, we get a nonlinear membrane problem. A study of the latter gives infinitely many solutions, and we discuss the construction, shapes, and stability in detail. This revised version was published online in August 2006 with corrections to the Cover Date.     

Orbits Homoclinic to Exponentially Small Periodic Orbits for a Class of Reversible Systems. Application to Water Waves

In this paper a class of reversible analytic vector fields is investigated near an equilibrium. For these vector fields, the part of the spectrum of the differential at the equilibrium which lies near the imaginary axis comes from the perturbation of a double eigenvalue 0 and two simple eigenvalues , . In the first part of this paper, we study the 4-dimensional problem. The existence of a family of solutions homoclinic to periodic orbits of size less than μ^N for any fixed N, where μ is the bifurcation parameter, is known for vector fields. Using the analyticity of the vector field, we prove here the existence of solutions homoclinic to a periodic orbit the size of which is exponentially small ( of order . This result receives its significance from the still unsolved question of whether there exist solutions that are homoclinic to the equilibrium or whether the amplitudes of the oscillations at infinity have a positive infimum. In the second part of this paper we prove that the exponential estimates still hold in infinite dimensions. This result cannot be simply obtained from the study of the 4-dimensional analysis by a center-manifold reduction since this result is based on analyticity of the vector field. One example of such a vector field in infinite dimensions occurs when describing the irrotational flow of an inviscid fluid layer under the influence of gravity and small surface tension (Bond number ) for a Froude number F close to 1. In this context a homoclinic solution to a periodic orbit is called a generalized solitary wave. Our work shows that there exist generalized solitary waves with exponentially small oscillations at infinity. More precisely, we prove that for each F close enough to 1, there exist two reversible solutions homoclinic to a periodic orbit, the size of which is less than , l being any number between 0 and π and satisfying . (Accepted October 2, 1995)     

Internally Pressurized Radially Polarized Piezoelectric Cylinders

The piezoelectric phenomenon has been exploited in science and engineering for decades. Recent advances in smart structures technology have lead to a resurgence of interest in piezoelectricity, and in particular, in the solution of fundamental boundary-value problems. In this paper, we develop an analytic solution to the axisymmetric problem of an infinitely long, radially polarized, radially orthotropic piezoelectric hollow circular cylinder. The cylinder is subjected to uniform internal pressure, or a constant potential difference between its inner and outer surfaces, or both. An analytic solution to the governing equilibrium equations (a coupled system of second-order ordinary differential equations) is obtained. On application of the boundary conditions, the problem is reduced to solving a system of linear algebraic equations. The stress distributions in the cylinder are obtained numerically for two typical piezoceramics of technological interest, namely PZT-4 and BaTiO₃. It is shown that the hoop stresses in a cylinder composed of these materials can be made virtually uniform throughout the cross-section by applying an appropriate set of boundary conditions. This revised version was published online in June 2006 with corrections to the Cover Date.     

Geometrically non-linear transverse vibrations of C-S-S-S and C-S-C-S rectangular plates

This paper is concerned with a new improved formulation of the theoretical model previously developed by Benamar et al. based on Hamilton's principle and spectral analysis, for the geometrically non-linear vibrations of thin structures. The problem is reduced to a non-linear algebraic system, the solution of which leads to determination of the amplitude-dependent fundamental non-linear mode shapes, the frequency parameters, and the non-linear stress distributions. The cases of C-S-C-S and C-S-S-S rectangular plates are examined, and the results obtained are in a good qualitative and quantitative agreement with the previous available works, based on various methods. In order to obtain explicit analytical solutions for the first non-linear mode shapes of C-S-C-S RP² and C-S-S-S RP, which are expected to be very useful in engineering applications and in further analytical developments, the improved version of the semi-analytical model developed by El Kadiri et al. For beams and fully clamped rectangular plates, has been slightly modified, and adapted to the above cases, leading to explicit expressions for the higher basic function contributions, which are shown to be in a good agreement with the iterative solutions, for maximum non-dimensional vibration amplitude values up to 0.75 and 0.6 for the first non-linear mode shapes of C-S-C-S RP and C-S-S-S RP, respectively.