Low-frequency wave propagation in post-buckled structures

Nonlinear wave propagation in solids and material structures provides a physical basis to derive nonlinear canonical equations which govern disparate phenomena such as vortex filaments, plasma waves, and traveling loops. Nonlinear waves in solids however remain a challenging proposition since nonlinearity is often associated with irreversible processes, such as plastic deformations. Finite deformations, also a source of nonlinearity, may be reversible as for hyperelastic materials. In this work, we consider geometric bucking as a source of reversible nonlinear behavior. Namely, we investigate wave propagation in initially compressed and post-buckled structures with linear-elastic material behavior. Such structures present both intrinsic dispersion, due to buckling wavelengths, and nonlinear behavior. We find that dispersion is strongly dependent on pre-compression and we compute waves with a dispersive front or tail. In the case of post-buckled structures with large initial pre-compression, we find that wave propagation is well described by the KdV equation. We employ finite-element, difference-differential, and analytical models to support our conclusions.     

Numerical study on Fermi–Pasta–Ulam–Tsingou problem for 1D shallow-water waves

Beginning with the first mode as the initial condition, long-term evolutions of gravity waves in shallow water are simulated based on the full nonlinear Boussinesq model. Evident recurrence is observed in long basins with appropriate initial amplitudes. Equipartition can be obtained in the case of a long basin, large initial amplitude or a long evolution time. Well-defined solitary waves are present during the recurrence stage and completely lost at the equipartition stage. The transition from regular to chaotic motion is conjectured to be related to the ratio of the dispersion and nonlinearity of the initial condition. For short basins with small initial amplitudes, nonlinearity is much smaller than dispersion, energy transfer is weak, and no recurrence can be observed. If dispersion and nonlinearity are chosen to be the same order in the initial condition, recurrence clearly emerges. However, if nonlinearity is chosen to be larger than dispersion, recurrence is absent and the system reaches equipartition rapidly.     

Rayleigh wave in a quadratic nonlinear elastic half-space (Murnaghan model)

The nonlinear equations that underlie the analysis of classical Rayleigh waves are derived for the two-dimensional case of nonlinear elastic deformation described by the Murnaghan model. In addition to the case of presence of both geometrical and physical nonlinearities, two special cases are considered where one only type of nonlinearity is taken into account. It is shown that unlike the one-dimensional problems for plane waves where only three types of nonlinear interaction should be allowed for, the two-dimensional problems should include 24 types of nonlinear interaction. In the case of geometrical nonlinearity alone, a preliminary analysis of the nonlinear equations is carried out. Second-order equations are derived. The second approximation includes the second harmonics of the wave itself and its attenuating amplitude and is nonlinearly dependent on the initial amplitude of the Rayleigh wave and linearly increasing with the distance traveled by the wave     

Waves in a solid medium with liquid-filled pores

The dynamic nonlinear theory of deformation of a two-phase medium, a solid with pores filled with a liquid, is developed. The variational principle is used to derive nonlinear equations that take into account the motions of the solid and liquid phases and the porosity variations. All types of nonlinearity, including nonlinear friction, are also taken into account. Formulas for the velocities of the linear and nonlinear waves and the absorption coefficient are derived. The one- and three-dimensional cases are considered. In the three-dimensional case, an equation describing the wave profile evolution is obtained as well as a nonlinear Schrödinger equation. Their solutions are analyzed; soliton-type solutions and solutions for narrow beams are obtained.     

非圆截面杆中的非线性扭转波及其行波解

研究了非圆截面杆中非线性扭转波的传播特性.由于非圆截面杆的扭转运动会伴随有横截面的翘曲,这种翘曲运动将引起扭转波的弥散.如果同时考虑有限扭转变形和翘曲弥散的共同作用,将会得到非线性扭转波的方程.在相平面上,对非线性扭转波动方程进行定性分析,结果表明,在一定条件下方程存在同宿轨道或异宿轨道,分别相应于方程的孤波解或冲击波解.本文利用Jacobi椭圆函数展开法,对该非线性方程进行求解,得到了非线性波动方程的三类准确周期解及相应的孤波解和冲击波解,讨论了这些解存在的必要条件.这些条件与定性分析的结果相一致.     

Approximate equations for torsional waves in a nonlinear elastic rod with weak viscoelasticity

Approximate equations are derived for nonlinear torsional waves propagating along a thin circular viscoelastic rod. Ignoring the thermal effect, ‘nearly elastic’ compressible viscoelastic solids are considered in which a weak dependence of stresses on a history of strain is assumed. With the assumption that the rod is subjected to a finite angle of torsion, but that the rod is thin, the displacement is sought in a power series of the radial coordinate. The effects of geometrical and material nonlinearity give rise to the normal stress effect, which introduces deformations in the cross sectional and longitudinal dimensions of rod. Taking account of both the effect of nonlinearity and that of viscoelasticity, one dimensional approximate equations are obtained for the angle of torsion coupled with the longitudinal deformation.     

Studies of nonlinear deformation and stability of oval and elliptic cylindrical shells under axial compression

The stability of noncircular shells, in contrast to that of circular ones, has not been studied sufficiently well yet. The publications about circular shells are counted by thousands, but there are only several dozens of papers dealing with noncircular shells. This can be explained on the one hand by the fact that such shells are less used in practice and on the other hand by the difficulties encountered when solving problems involving a nonconstant curvature radius, which results in the appearance of variable coefficients in the stability equations. The well-known solutions of stability problems were obtained by analytic methods and, as a rule, in the linear approximation without taking into account the moments and nonlinearity of the shell precritical state, i.e., in the classical approximation. Here we use the finite element method in displacements to solve the problem of geometrically nonlinear deformation and stability of cylindrical shells with noncircular contour of the transverse cross-section. We use quadrilateral finite elements of shells of natural curvature. In the approximations to the element displacements, we explicitly distinguish the displacements of elements as rigid bodies. We use the Lagrange variational principle to obtain a nonlinear system of algebraic equations for determining the unknown nodal finite elements. We solve the system by a step method with respect to the load using the Newton-Kantorovich linearization at each step. The linear systems are solved by the Kraut method. The critical loads are determined with the use of the Silvester stability criterion when solving the nonlinear problem. We develop an algorithm for solving the problem numerically on personal computers. We also study the nonlinear deformation and stability of shells with oval and elliptic transverse cross-section in a wide range of variations in the ovalization and ellipticity parameters. We find the critical loads and the shell buckling modes. We also examine how the critical loads are affected by the strain nonlinearity and the ovalization and ellipticity of shells.     

Inelastic Deformation of Flexible Spherical Shells with Two Circular Openings

Elastoplastic analysis of thin-walled spherical shells with two identical circular openings is carried out with allowance for finite deflections. The shells are made of an isotropic homogeneous material and subjected to internal pressure of known intensity. The distributions of stresses (strains or displacements) along the contours of the openings and in the zone of their concentration are studied by solving doubly nonlinear boundary-value problems. The solution obtained is compared with the solutions that account for only physical nonlinearity (plastic deformations) and only geometrical nonlinearity (finite deflections) and with a numerical solution of the linearly elastic problem. The stress–strain state near the two openings is analyzed depending on the distance between the openings and the nonlinear factors accounted for     

Wave propagation analysis in 2D nonlinear hexagonal periodic networks based on second order gradient nonlinear constitutive models

We analyze the propagation of nonlinear waves in homogenized periodic nonlinear hexagonal networks, considering successively 1D and 2D situations. Wave analysis is performed on the basis of the construction of the effective strain energy density of periodic hexagonal lattices in the nonlinear regime. The obtained second order gradient nonlinear continuum has two propagation modes: an evanescent subsonic mode that disappears after a certain wavenumber and a supersonic mode characterized by an increase of the frequency with the wavenumber. For a weak nonlinearity, a supersonic mode occurs and the dispersion curves lie above the linear dispersion curve (v_p =v_p0). For a higher nonlinearity, the wave changes from a supersonic to an evanescent subsonic mode at s=0.7 and the dispersion curves drops below the linear case and vanish for certain values of the wavenumber. An important decrease in the frequency occurs for both subsonic and supersonic modes when the lattice becomes auxetic, and the longitudinal and shear modes become very close to each other. The influence of the lattice geometrical parameters of the lattice on the dispersion relations is analyzed.     

A model technique for analysis of nonlinear structures

Analysis of structural systems which display geometrical nonlinearity may be carried out in some cases with the aid of a digital computer. Convergence of numerical computations is, however, not always assured, and the computer time required is usually a multiple of the time needed for linear analysis. An experimental method to determine stress resultants and displacements, by taking into account geometrical nonlinearity, is suggested. The method consists of loading a model in two stages. First, a fraction of the dead and live loads is applied such that the displacements of the system remain small and its behavior may be considered as linear. Then, the model is loaded with its full dead and live loads under which it displays its real, nonlinear behavior. Strains and displacements are recorded for both cases. By comparing the two readings, that is, by dividing the second readings by the first ones, the nonlinearity of the system is determined. The method is suggested for every project in which nonlinear behavior may be a significant factor.     

Theory of nonlinear waves in a plasma

Stationary nonlinear waves propagating in a cold rarefied plasma composed of electrons and two types of ions are considered. The structure of isolated waves and shock waves is found. In recent years an intensive study has been made of finite-amplitude waves and collisionless shock waves in a rarefied plasma, in connection with laboratory experiments [1] and astrophysical applications (the problem of the interaction of the solar wind with the Earth's magnetosphere [2]). When allowance is made for dispersion effects associated with the departure of the dispersion law =(k) from the linear, and for the compensating nonlinear twisting of the wave profile, we are able to obtain the profile of stationary nonlinear waves of finite amplitude, and when allowance is made for damping we can also obtain the structure of a collisionless shock wave [3]. Such waves have been studied fairly fully for the case of a two-component plasma. The present paper examines stationary nonlinear waves propagating across a magnetic field in a cold rarefied quasi-neutral plasma composed of electrons and two types of ions.     

Free torsional vibrations of an elastic cylinder with laminated periodic structure

The theory of torsional vibrations of a circular cylinder, with a periodic variation of elastic constants and density normal to the axis of the cylinder, is developed in terms of Floquet waves. Floquet waves are quasi-periodic waves, whose amplitude profile has the same periodicity as that of the material and repeats with the periodicity of the cell. Using Floquet's theory, the dispersion spectrum is obtained for time-harmonic waves propagating in a laminated cylinder with periodic structure. It is shown that the dispersion spectrum has a band structure, consisting of passing bands and stopping bands. Motion in the case of grazing incidence, and motion at the end of the zones is discussed. It is also shown that as the radius of the cylinder tends to infinity, the torsional waves in a circular cylinder degenerate to SH-waves in laminated plates.     

Plane Waves in Cubically Nonlinear Elastic Media

The theory of plane waves in nonlinear materials described by the Murnaghan potential is proposed. The theory takes into account both the classical quadratic nonlinearity and the cubic nonlinearity of the basic wave equations. Some new opportunities for the wave interaction analysis are commented on: in addition to the second harmonics, a longitudinal plane wave generates the third one, a transverse plane wave generates the third harmonics, and horizontally and vertically polarized transverse plane waves jointly generate new waves     

具有精确色散性的非线性波浪数学模型

以完全非线性的自由表面边界条件为基础,以波面升高\eta和自由表面速度势\phi _\eta为待求变量,建立了新的波浪方程．方程在色散性上是完全精确的,非线性近似至三阶．与缓坡方程相比较,两者都具有精确的色散性,但该方程属于非线性模型,可模拟波浪的非线性效应,且适用于不规则波．方程的特点是属于微分-积分方程,对如何处理方程中积分项进行了讨论,并数值模拟了不同周期的线性波和二阶Stokes波,也模拟了波群的非线性演化,以对模型进行验证．      

Physical mechanisms of the rogue wave phenomenon

A review of physical mechanisms of the rogue wave phenomenon is given. The data of marine observations as well as laboratory experiments are briefly discussed. They demonstrate that freak waves may appear in deep and shallow waters. Simple statistical analysis of the rogue wave probability based on the assumption of a Gaussian wave field is reproduced. In the context of water wave theories the probabilistic approach shows that numerical simulations of freak waves should be made for very long times on large spatial domains and large number of realizations. As linear models of freak waves the following mechanisms are considered: dispersion enhancement of transient wave groups, geometrical focusing in basins of variable depth, and wave-current interaction. Taking into account nonlinearity of the water waves, these mechanisms remain valid but should be modified. Also, the influence of the nonlinear modulational instability (Benjamin–Feir instability) on the rogue wave occurence is discussed. Specific numerical simulations were performed in the framework of classical nonlinear evolution equations: the nonlinear Schrödinger equation, the Davey–Stewartson system, the Korteweg–de Vries equation, the Kadomtsev–Petviashvili equation, the Zakharov equation, and the fully nonlinear potential equations. Their results show the main features of the physical mechanisms of rogue wave phenomenon.     

Studies of nonlinear deformation and stability of stiffened oval cylindrical shells under combined loading by bending moment and boundary transverse force

The finite-element statement of stability problems for stiffened oval cylindrical shells is presented with the moments and the nonlinearity of their subcritical stress-strain state taken into account. Explicit expressions for the displacements of elements of noncircular cylindrical shells as solids are obtained by integration of the equations derived by equating the linear deformation components with zero. These expressions are used to construct the shape functions of the effective quadrangular finite element of natural curvature, and an efficient algorithm for studying the shell nonlinear deformation and stability is developed. The stability of stiffened oval cylindrical shells is studied in the case of combined loading by a boundary transverse force and a bending moment. The influence of the shell ovality and the deformation nonlinearity on the shell stability is investigated.     

Inelastic deformation of flexible cylindrical shells with a curvilinear hole

The elastoplastic state of thin cylindrical shells weakened by a curvilinear (circular) hole is analyzed considering finite deflections. The shells are made of an isotropic homogeneous material. The load is internal pressure of given intensity. The distributions of stresses (strains, displacements) along the hole boundary and in the zone of their concentration are studied. The results obtained are compared with solutions that account for physical (plastic strains) or geometrical (finite deflections) nonlinearity alone and with a numerical linear elastic solution. The stress-strain state around a circular hole is analyzed for different geometries in the case where both nonlinearities are taken into account __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 12, pp. 115–123, December, 2006.     

Physically and Geometrically Nonlinear Deformation of Spherical Shells with an Elliptic Hole

The elastoplastic state of thin spherical shells with an elliptic hole is analyzed considering that deflections are finite. The shells are made of an isotropic homogeneous material and subjected to internal pressure of given intensity. Problems are formulated and a numerical method for their solution with regard for physical and geometrical nonlinearities is proposed. The distribution of stresses (strains or displacements) along the hole boundary and in the zone of their concentration is studied. The results obtained are compared with the solutions of problems where only physical nonlinearity (plastic deformations) or geometrical nonlinearity (finite deflections) is taken into account and with the numerical solution of the linearly elastic problem. The stress—strain state in the neighborhood of an elliptic hole in a shell is analyzed with allowance for nonlinear factors __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 6, pp. 95–104, June 2005.     

Study of nonlinear deformation and stability of reinforced shells under combined loading by a bending moment and a transverse boundary force

We present a finite-element statement for the solution of stability problems for reinforced elliptic cylindrical shells with moment properties and nonlinearity in their precritical stressstrain state taken into account. Integrating the equations obtained by equating the linear strain components with zero, we find explicit expressions for the displacements of elements of noncircular cylindrical shells as rigid bodies. Using these expressions, we construct the shape functions of a fourangle finite element of natural curvature and develop an effective algorithm for studying nonlinear deformation and stability of shells. We study the stability of reinforced elliptic cylindrical shells under combined loading by a transverse boundary force and a bending moment and investigate how the ellipticity of the shells and the nonlinearity of deformation at the precritical stage affect the shell stability.     

Shear deformable bars of doubly symmetrical cross section under nonlinear nonuniform torsional vibrations—application to torsional postbuckling configurations and primary resonance excitations

In this paper a boundary element method is developed for the nonuniform torsional vibration problem of bars of arbitrary doubly symmetric constant cross section, taking into account the effects of geometrical nonlinearity (finite displacement—small strain theory) and secondary twisting moment deformation. The bar is subjected to arbitrarily distributed or concentrated conservative dynamic twisting and warping moments along its length, while its edges are subjected to the most general axial and torsional (twisting and warping) boundary conditions. The resulting coupling effect between twisting and axial displacement components is also considered and a constant along the bar compressive axial load is induced so as to investigate the dynamic response at the (torsional) postbuckled state. The bar is assumed to be adequately laterally supported so that it does not exhibit any flexural or flexural–torsional behavior. A coupled nonlinear initial boundary value problem with respect to the variable along the bar angle of twist and to an independent warping parameter is formulated. The resulting equations are further combined to yield a single partial differential equation with respect to the angle of twist. The problem is numerically solved employing the Analog Equation Method (AEM), a BEM based method, leading to a system of nonlinear Differential–Algebraic Equations (DAE). The main purpose of the present contribution is twofold: (i) comparison of both the governing differential equations and the numerical results of linear or nonlinear free or forced vibrations of bars ignoring or taking into account the secondary twisting moment deformation effect (STMDE) and (ii) numerical investigation of linear or nonlinear free vibrations of bars at torsional postbuckling configurations. Numerical results are worked out to illustrate the method, demonstrate its efficiency and wherever possible its accuracy.