PERTURBATION ANALYSIS FOR WAVE EQUATION OF NONLINEAR ELASTIC ROD

The longitudinal oscillation of a nonlinear elastic rod with lateral inertia was studied. Based on the far field and simple wave theory, a nonlinear Schrodinger (NLS) equation was established under the assumption of small amplitude and long wavelength. It is found that there are NLS envelop solitons in this system. Finally the soliton solution of the NLS equation was presented.     

非线性弹性杆中纵波传播过程的数值模拟

基于Hamilton空间体系的多辛理论研究了包含材料非线性效应和几何弥散效应的非线性弹性杆中纵波传播问题。导出了其Bridges意义下的多辛形式及其多种守恒律,并构造了等价于Box多辛格式的隐式多辛格式对不考虑材料非线性效应和几何弥散效应、只考虑材料非线性效应、只考虑几何弥散效应、同时考虑材料非线性效应和几何弥散效应四种情况下不同截面参数的圆杆中的纵波传播过程进行数值模拟,模拟结果不仅全面地反映了非线性效应和几何弥散效应对纵波传播的影响,而且也反映了多辛方法的两大优点:精确的保持多种守恒律和良好的长时间数值行为。     

The strain solitary waves in a nonlinear elastic rod

Solitary strain waves in a nonlinear elastic rod are analysed in this paper; influence of the physical and geometrical parameters of the rod on the waves are discussed; some main properties of the solitary waves are pointed out.     

New explicit multi-symplectic scheme for nonlinear wave equation简

Based on splitting multi-symplectic structures, a new multi-symplectic scheme is proposed and applied to a nonlinear wave equation. The explicit multi-symplectic scheme of the nonlinear wave equation is obtained, and the corresponding multi-symplectic conservation property is proved. The backward error analysis shows that the explicit multi-symplectic scheme has good accuracy. The sine-Gordon equation and the Klein-Gordon equation are simulated by an explicit multi-symplectic scheme. The numerical results show that the new explicit multi-symplectic scheme can well simulate the solitary wave behaviors of the nonlinear wave equation and approximately preserve the relative energy error of the equation.     

A new auxiliary equation method for finding travelling wave solutions to KdV equation

In this paper, a new auxiliary equation method is used to find exact travelling wave solutions to the （1＋1）-dimensional KdV equation. Some exact travelling wave solu- tions with parameters have been obtained, which cover the existing solutions. Compared to other methods, the presented method is more direct, more concise, more effective, and easier for calculations. In addition, it can be used to solve other nonlinear evolution equations in mathematical physics.     

The parameter perturbation method on elastic wave equation in inhomogeneous medium

I.IntroductionTheelasticwaveininhomogeneousmediumiscomplicatedbecauseofthediffracting,scattering,andtransmutingofthewavetapes.Exceptforsomesimpleandregularmediummode1s,thesolutionofelasticwavehasnotbeengotyet.Nowadays,theresearchoftheelasticwavescattering…     

Elastic-plastic deformation of sandwich rod on elastic basis

Sandwich composite material possesses advantages of both light weight and high strength. Although the mechanical behaviors of sandwich composite material with the influence of single external environment have been intensively studied, little work has been done in the study of mechanical property, in view of the nonlinear behavior of sandwich composites in the complicated external environments. In this paper, the problem about the bending of the three-layer elastic-plastic rod located on the elastic base, with a compressibly physical nonlinear core, has been studied. The mechanical response of the designed three-layer elements consisting of two bearing layers and a core has been examined. The complicated problem about curving of the three-layer rod located on the elastic base has been solved. The convergence of the proposed method of elastic solutions is examined to convince that the solution is acceptable. The calculated results indicate that the plasticity and physical nonlinearity of materials have a great influence on the deformation of the sandwich rod on the elastic basis.     

Modeling cylindrical waves in nonlinear elastic composites

A procedure of deriving nonlinear wave equations that describe the propagation and interaction of hyperelastic cylindrical waves in composite materials modeled by a mixture with two elastic constituents is outlined. Nonlinearity is introduced by metric coefficients, Cauchy-Green strain tensor, and Murnaghan potential. It is the quadratic nonlinearity of all governing relations. For a configuration (state) dependent on the radial coordinate and independent of the angular and axial coordinates, quadratically nonlinear wave equations for stresses are derived and a relationship between the components of the stress tensor and partial strain gradient is established. Four combinations of physical and geometrical nonlinearities in systems of wave equations are examined. Nonlinear wave equations are explicitly written for three of the combinations __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 6, pp. 63–72, June 2007.     

螺旋细杆的均匀扭转振动

讨论螺旋细杆的特殊形式扭转振动，即均匀扭转振动．以非圆截面杆和有原始曲率的圆截面杆为研究对象．杆作均匀扭转振动时各截面有相同的扭角变化规律，且杆中心线的几何形状不受振动过程的影响．研究表明，扭振来源于杆截面的非对称性及杆的原始曲率．杆的扭振规律与单摆运动相似，其动力学方程存在精确解．圆环杆的均匀扭振为螺旋杆的倾角为零时的特例．     

Kink wave determined by parabola solution of a nonlinear ordinary differential equation

By finding a parabola solution connecting two equilibrium points of a planar dynamical system,the existence of the kink wave solution for 6 classes of nonlinear wave equations is shown.Some exact explicit parametric representations of kink wave solutions are given.Explicit parameter conditions to guarantee the existence of kink wave solutions are determined.     

Computer algebra-perturbation solution to a nonlinear wave equation

In this paper, the higher-order asymptotic solution to the Cauchy problem of a nonlinear wave equation is found by using a computer algebra-perturbation method. The secular terms in the solution from straightforward expansions are eliminated with the straining of characteristic, coordinates and the use of the renormalization technique, and the four-term uniformly valid solution is obtained with the symbolic computation by using a computer algebra system. The comparison of the derived asymptotic solution and the numerical solution shows that they coincide with each other for smaller ε and agree quite well for larger ε (e. g., ε=0.25) Project supported by the National Natural Science Foundation of China and Shanghai Municiple Natural Science Foundation     

Symmetry solutions of a nonlinear elastic wave equation with third-order anharmonic corrections

Lie symmetry method is applied to analyze a nonlinear elastic wave equation for longitudinal deformations with third-order anharmonic corrections to the elastic energy. Symmetry algebra is found and reductions to second-order ordinary differential equations （ODEs） are obtained through invariance under different symmetries. The reduced ODEs are further analyzed to obtain several exact solutions in an explicit form. It was observed in the literature that anharmonic corrections generally lead to solutions with time-dependent singularities in finite times singularities, we also obtain solutions which Along with solutions with time-dependent do not exhibit time-dependent singularities.     

Superfluous order and the proper solution of the Maxwell equation

In this paper it is proved that Maxwell equation is equivalent to a fourth order equation. Under a certain condition, its general solution is given by     

Klein-Gordon波动方程多波传播的非线性特性

基于双波初值问题,讨论非线性对多波传播的影响。通过选取合适的多重尺度,对Klein-Gordon波动方程进行变形,得到方程的解的多尺度展式首项近似和三波传播时速度相互影响的定量关系,揭示了多波传播的非线性特性;最后,应用Mathematica对波动方程进行数值仿真。研究结果表明,另外多个波的存在会使波的传播速度（相速）超过独自传播时的速度（相速）。     

Soliton solution to generalized nonlinear disturbed Klein-Gordon equation

A generalized nonlinear disturbed Klein-Gordon equation is studied. Using the homotopic mapping method, the corresponding homotopic mapping is constructed. A suitable initial approximation is selected, and an arbitrary-order approximate solution to the soliton is calculated： A weakly disturbed equation is also studied.     

Analysis of sloping elastic pile under arbitrary loads by line-loaded integral equation method

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非线性弹性大变形材料梁的抗爆特性

从理论上探讨了非线性弹性大变形材料应用于抗爆结构的可行性,为此,基于等效结构体系的分析原理,将两端固定铰支梁的横向和纵向位移表示为三角级数形式,应用第二类Lagrange方程建立了非线性大变形材料梁的非线性分析方法,并且用ABAQUS有限元软件中的超弹性材料模型验证了所提出的方法的有效性。对典型的爆炸荷载作用下非线性弹性大变形材料梁的抗爆特性进行了分析,讨论了动力放大系数和材料性质及动荷载之间的关系。结果表明:与线弹性小变形材料相比,非线性弹性大变形材料具有优良的抗爆特性,结构的抗爆能力随结构变形的增大而显著提高。     

Tunable three-dimensional nonreciprocal transmission in a layered nonlinear elastic wave metamaterial by initial stresses

In this work, the three-dimensional(3 D) propagation behaviors in the nonlinear phononic crystal and elastic wave metamaterial with initial stresses are investigated.The analytical solutions of the fundamental wave and second harmonic with the quasilongitudinal(qP) and quasi-shear(qS1 and qS2) modes are derived. Based on the transfer and stiffness matrices, band gaps with initial stresses are obtained by the Bloch theorem.The transmission coefficients are calculated to support the band gap prope...     

Classical and nonclassical symmetry classifications of nonlinear wave equation with dissipation

A complete classical symmetry classification and a nonclassical symmetry classification of a class of nonlinear wave equations are given with three arbitrary parameter functions. The obtained results show that such nonlinear wave equations admit richer classical and nonclassical symmetries, leading to the conservation laws and the reduction of the wave equations. Some exact solutions of the considered wave equations for particular cases are derived.     

Symplectic analysis for wave propagation in one-dimensional nonlinear periodic structures

The wave propagation problem in the nonlinear periodic mass-spring structure chain is analyzed using the symplectic mathematical method. The energy method is used to construct the dynamic equation, and the nonlinear dynamic equation is linearized using the small parameter perturbation method. Eigen-solutions of the symplectic matrix are used to analyze the wave propagation problem in nonlinear periodic lattices. Nonlinearity in the mass-spring chain, arising from the nonlinear spring stiffness effect, has profound effects on the overall transmission of the chain. The wave propagation characteristics are altered due to nonlinearity, and related to the incident wave intensity, which is a genuine nonlinear effect not present in the corresponding linear model. Numerical results show how the increase of nonlinearity or incident wave amplitude leads to closing of transmitting gaps. Comparison with the normal recursive approach shows effectiveness and superiority of the symplectic method for the wave propagation problem in nonlinear periodic structures.