Wave propagation and dispersion in elasto-plastic microstructured materials

A Mindlin continuum model that incorporates both a dependence upon the microstructure and inelastic (nonlinear) behavior is used to study dispersive effects in elasto-plastic microstructured materials. A one-dimensional equation of motion of such material systems is derived based on a combination of the Mindlin microcontinuum model and a hardening model both at the macroscopic and microscopic level. The dispersion relation of propagating waves is established and compared to the classical linear elastic and gradient-dependent solutions. It is shown that the observed wave dispersion is the result of introducing microstructural effects and material inelasticity. The introduction of an internal characteristic length scale regularizes the ill-posedness of the set of partial differential equations governing the wave propagation. The phase speed does not necessarily become imaginary at the onset of plastic softening, as it is the case in classical continuum models and the dispersive character of such models constrains strain softening regions to localize.     

Multi-soliton solutions for the coupled nonlinear Schr?dinger-type equations

Nonlinear Schr?dinger-type equations can model the nonlinear waves in fluids, plasmas, nonlinear optics and atmosphere. In this paper, integrable coupled nonlinear Schr?dinger-type equations are investigated. With the aid of symbolic computation, the equations are transformed into their bilinear forms, by virtue of which the multi-soliton solutions are derived. Soliton interactions are analyzed, the elastic interactions are seen, while the dark, anti-dark, M- and W-shape solitons are exhibited with some parameters selected. The propagating solitons can preserve their properties after the interaction, and the profiles of them depend on the corresponding dispersion relations. The amplitudes, velocities of the solitons are found to be influenced by the coefficient of the original equations, which is detailed in the paper.     

Existence of Weak Solutions for the Three-Dimensional Motion of an Elastic Structure in an Incompressible Fluid

We study here the three-dimensional motion of an elastic structure immersed in an incompressible viscous fluid. The structure and the fluid are contained in a fixed bounded connected set Ω. We show the existence of a weak solution for regularized elastic deformations as long as elastic deformations are not too important (in order to avoid interpenetration and preserve orientation on the structure) and no collisions between the structure and the boundary occur. As the structure moves freely in the fluid, it seems natural (and it corresponds to many physical applications) to consider that its rigid motion (translation and rotation) may be large. The existence result presented here has been announced in [4]. Some improvements have been provided on the model: the model considered in [4] is a simplified model where the structure motion is modelled by decoupled and linear equations for the translation, the rotation and the purely elastic displacement. In what follows, we consider on the structure a model which represents the motion of a structure with large rigid displacements and small elastic perturbations. This model, introduced by [15] for a structure alone, leads to coupled and nonlinear equations for the translation, the rotation and the elastic displacement.     

Fluid–structure interaction between a two-dimensional mat-type VLFS and solitary waves by the Green–Naghdi theory

The two-dimensional, nonlinear hydroelasticity of a mat-type very large floating structure (VLFS) is studied within the scope of linear beam theory for the structure and the nonlinear, Level I Green–Naghdi (GN) theory for the fluid. The beam equation and the GN equations are coupled through the kinematic and dynamic boundary conditions to obtain a new set of modified GN equations. These equations represent long-wave motion beneath an elastic plate. A set of jump conditions that are necessary for the continuity (or the matching) of the solutions in the open water region and that under the structure is derived through the use of the postulated conservation laws of mass, momentum, and mechanical energy. The resulting governing equations, subjected to the boundary and jump conditions, are solved by the finite-difference method in the time domain. The present model is applicable, for example, to the study of the hydroelastic response of a mat-type VLFS under the action of a solitary wave, or a frontal tsunami wave. Good agreement is observed between the model results and other published theoretical and numerical predictions, as well as experimental data. The results show that consideration of nonlinearity is important for accurate predictions of the bending moment of the floating elastic plate. It is found that the rigidity of the structure greatly affects the bending moment and displacement of the structure in this nonlinear theory.     

New solitons and periodic wave solutions for nonlinear physical models

In this paper, an extended tanh method with computerized symbolic computation is used for constructing the traveling wave solutions of coupled nonlinear equations arising in physics. The obtained solutions include solitons, kinks, and plane periodic solutions. The applied method will be used in further works to establish more entirely new solutions for other kinds of nonlinear evolution equations arising in physics.     

有限厚度砂床对波浪载荷的响应

本文采用水-土两相介质混合物的连续介质力学理论分析了有限厚度砂床对波浪载荷的响应。土床看作饱和孔隙弹性介质,用Biot理论来描述流动和变形的耦合效应,波浪用线性水波描述。在上述假设下得到了描述土床对波浪载荷响应的基本方程。用解沂-数值方法给出了土床中位移、应力和孔隙水压力的分布并将理沦计算结果与波浪-砂床模型试验的实验资料进行了比较。文章还对影响土床对波浪响应的参量特性进行了分析和讨论。根据参量研究、理论计算与实验资料比较的结果,可以得出结论:本文所提供的理论模型和计算方法能用来估算海床对波浪载荷的影响。     

Finite amplitude, free vibrations of an axisymmetric load supported by a highly elastic tubular shear spring

The finite amplitude, free vibrational characteristics of a simple mechanical system consisting of an axisymmetric rigid body supported by a highly elastic tubular shear spring subjected to axial, rotational, and coupled shearing motions are studied. Two classes of elastic tube materials are considered: a compressible material whose shear response is constant, and an incompressible material whose shear response is a quadratic function of the total amount of shear. The class of materials with constant shear response includes the incompressible Mooney-Rivlin material and certain compressible Blatz-Ko, Hadamard, and other general kinds of models. For each material class, the quasi-static elasticity problem is solved to determine the telescopic and gyratory shearing deformation functions needed to evaluate the elastic tube restoring force and torque exerted on the body. For all materials with constant shear response, the differential equations of motion are uncoupled equations typical of simple harmonic oscillators. Hence, exact solutions for the forced vibration of the system can be readily obtained; and for this class, engineering design formulae for the load-deflection relations are discussed and compared with experimental results of others'. For the quadratic material, however, the general motion of the body is characterized by a formidable, coupled system of nonlinear equations. The free, coupled shearing motion for which either the axial or the azimuthal shear deformation may be small is governed by a pair of equations of the Duffing and Hill types. On the other hand, the finite amplitude, pure axial and pure rotational motions of the load are described by the classical, nonlinear Duffing equation alone. A variety of problems are solved exactly for these separate free vibrational modes, and a number of physical results are presented throughout.     

Nonlinear Oscillations of Elastic Solar Panels of a Spacecraft at Finite Turn by Roll

Non-stationary rotational motion and nonlinear oscillations in the roll plane of a spacecraft with two elastic panels of solar batteries are considered. The spacecraft and sections of solar panels are considered to be non-deformable; they are interconnected by elastic-hinged joints that allow large angles of rotation. The spacecraft makes a turn about its axis as a solid body. Motion of elastic system under consideration is described in a connected coordinate system. Equations of motion are obtained on the basis of principle of possible displacements. Examples of numerical integration of a system of nonlinear differential equations with the analysis of convergence are given.     

Bright–dark vector soliton solutions for the coupled complex short pulse equations in nonlinear optics

Under investigation in this paper are the coupled complex short pulse equations, which describe the propagation of ultra-short optical pulses in cubic nonlinear media. Through the Hirota method, bright–dark one- and two-soliton solutions are obtained. Interactions between two bright or two dark solitons are verified to be elastic through the asymptotic analysis. With different parameter conditions, the oblique interactions, bound states of solitons and parallel solitons are analyzed.     

Nonlinear wave propagation in damaged hysteretic materials using a frequency domain-based PM space formulation

In this work, a new and simple numerical approach to simulate nonlinear wave propagation in purely hysteretic elastic solids is presented. Conversely to classical time discretization method, which fully integrates the nonlinear equation of motion, this method utilizes a first-order approximation of the nonlinear strain in order to separate linear and nonlinear contributions. The problem for the nonlinear displacements is then posed as a linear one in which the solid is enforced with nonlinear forces derived from the linear strain. In this manner, a frequency analysis can be easily conducted, leading directly to a well-known frequency spectrum for the nonlinear strain. A mesoscale approach known as Preisach–Mayergoyz space (PM space) is used for the chacterization of the nonlinear elastic region of the solid. A meshless element free Galerkin method is implemented for the discretized equations of motion. Nevertheless, a mesh-based method can be still used as well without loss of generality. Results are presented for bidimensional isotropic plates both in plane stress and in plane strain subjected to harmonic monotone excitation.     

Dynamic analysis on generalized linear elastic body subjected to large scale rigid rotations

The dynamic analysis of a generalized linear elastic body undergoing large rigid rotations is investigated. The generalized linear elastic body is described in kine- matics through translational and rotational deformations, and a modified constitutive relation for the rotational deformation is proposed between the couple stress and the curvature tensor. Thus, the balance equations of momentum and moment are used for the motion equations of the body. The floating frame of reference formulation is applied to the elastic body that conducts rotations about a fixed axis. The motion-deformation coupled model is developed in which three types of inertia forces along with their incre- ments are elucidated. The finite element governing equations for the dynamic analysis of the elastic body under large rotations are subsequently formulated with the aid of the constrained variational principle. A penalty parameter is introduced, and the rotational angles at element nodes are treated as independent variables to meet the requirement of C1 continuity. The elastic body is discretized through the isoparametric element with 8 nodes and 48 degrees-of-freedom. As an example with an application of the motion- deformation coupled model, the dynamic analysis on a rotating cantilever with two spatial layouts relative to the rotational axis is numerically implemented. Dynamic frequencies of the rotating cantilever are presented at prescribed constant spin velocities. The maximal rigid rotational velocity is extended for ensuring the applicability of the linear model. A complete set of dynamical response of the rotating cantilever in the case of spin-up maneuver is examined, it is shown that, under the ultimate rigid rotational velocities less than the maximal rigid rotational velocity, the stress strength may exceed the material strength tolerance even though the displacement and rotational angle responses are both convergent. The influence of the cantilever layouts on their responses and the multiple displacement trajectories observed in the floating frame is simultaneously investigated. The motion-deformation coupled model is surely expected to be applicable for a broad range of practical applications.     

Nonlinear waves in diatomic crystals

A possible improvement of a continuum model for diatomic crystals is examined using continuum limit of a discrete diatomic model. For this purpose, various discrete models of diatomic lattice are compared at the linearized and weakly nonlinear levels. The suitable numbering of the atoms in the lattice is found which is better adopted for continualization than the familiar pair numbering introducing two sub-lattices. The coupled governing partial nonlinear differential equations for longitudinal strain and relative distance between the atoms are obtained in the continuum limit that allows us to describe localization of the strains due to the presence of the atoms of two kinds. It is found, that the equations obtained possess two kinds of localized wave solutions, one related to the acoustical branch and the other one related to the optical branch.     

Constructing an analytical solution for lamb waves using the cosserat continuum approach

The problem of propagation of a Lamb elastic wave in a thin plate is considered using the Cosserat continuum model. The deformed state is characterized by independent displacement and rotation vectors. Solutions of the equations of motion are sought in the form of wave packets specified by a Fourier spectrum of an arbitrary shape for three components of the displacement vector and three components of the rotation vector which depend on time, depth, and the longitudinal coordinate. The initial system of equations is split into two systems, one of which describes a Lamb wave and the second corresponds to a transverse wave whose amplitude depends on depth. Analytical solutions in displacements are obtained for the waves of both types. Unlike the solution for Lamb waves, the solution obtained for the transverse wave has no analogs in classical elasticity theory. The solution for the transverse wave is compared with the solution for the Lamb wave. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 1, pp. 143–150, January–February, 2007. An erratum to this article is available at .     

Nonlinear analysis of the dynamics of articulated composite rotor blades

The general nonlinear intrinsic equations of motion of an elastic composite beam are solved in order to obtain the elasto-dynamic response of a rotating articulated blade. The solution utilizes the linear Variational-Asymptotic Method (VAM) cross-sectional analysis, together with an improved damped nonlinear model for the rigid-body motion analysis of helicopter blades in coupled flap and lead-lag motions. The explicit (direct) integration algorithm implements the perturbation method in order to solve the transient form of the nonlinear intrinsic differential equations of motion and obtain the elasto-dynamic behavior of an accelerating composite blade. The specific problem considered is an accelerating articulated helicopter blade of which its motion is analyzed since it starts rotating from rest until it reaches the steady-state condition. It is observed that the steady-state solution obtained by this method compares very well with other available solutions. The resulting simulation code is a powerful tool for analyzing the nonlinear response of composite rotor blades; and for serving the ultimate aim of efficient noise and vibration control in helicopters.     

Asymptotic Research of Nonlinear Wave Processes in Saturated Porous Media

The problem of nonlinear wave dynamics of a fluid-saturated porous medium is investigated. The mathematical model proposed is based on the classical Frenkel--Biot--Nikolaevskiy theory concerning elastic wave propagation and includes mass, momentum, energy conservation laws, as well as rheological and thermodynamic relations. The model describes nonlinear, dispersive, and dissipative medium. To solve the system of differential equations, an asymptotic modified two-scales method is developed and a Cauchy problem for initial equations system is transformed to a Cauchy problem for nonlinear generalized Korteweg--de Vries--Burgers equation for modulated quick wave amplitudes and an inhomogeneous set of equations for slow background motion. Stationary solutions of the derived evolutionary equation that have been constructed numerically reflect different regimes of elastic wave attenuation: diffusive, oscillating, and soliton-like.     

Types of coefficient constraints of coupled nonlinear Schrödinger equations for elastic and inelastic interactions between spatial solitons with symbolic computation

With symbolic computation and Hirota method, analytic two-soliton solutions for the coupled nonlinear Schrödinger (CNLS) equations, which describe the propagation of spatial solitons in an AlGaAs slab waveguide, are derived. Two types of coefficient constraints of the CNLS equations to distinguish the elastic and inelastic interactions between spatial solitons are obtained for the first time in this paper. Asymptotic analysis is made to investigate the spatial soliton interactions. The inelastic interactions are studied under the obtained coefficient constraints of the CNLS equations. The influences of parameters for the obtained soliton solutions are discussed. All-optical switching and soliton amplification are studied based on the dynamic properties of inelastic interactions between spatial solitons. Numerical simulations are in good agreement with the analytic results. The presented results have applications in the design of birefringence-managed switching architecture.     

Finite amplitude,free vibrations of an axisymmetric load supported by a highly elastic tubular shear spring

The finite amplitude, free vibrational characteristics of a simple mechanical system consisting of an axisymmetric rigid body supported by a highly elastic tubular shear spring subjected to axial, rotational, and coupled shearing motions are studied. Two classes of elastic tube materials are considered: a compressible material whose shear response is constant, and an incompressible material whose shear response is a quadratic function of the total amount of shear. The class of materials with constant shear response includes the incompressible Mooney-Rivlin material and certain compressible Blatz-Ko, Hadamard, and other general kinds of models. For each material class, the quasi-static elasticity problem is solved to determine the telescopic and gyratory shearing deformation functions needed to evaluate the elastic tube restoring force and torque exerted on the body. For all materials with constant shear response, the differential equations of motion are uncoupled equations typical of simple harmonic oscillators. Hence, exact solutions for the forced vibration of the system can be readily obtained; and for this class, engineering design formulae for the load-deflection relations are discussed and compared with experimental results of others'. For the quadratic material, however, the general motion of the body is characterized by a formidable, coupled system of nonlinear equations. The free, coupled shearing motion for which either the axial or the azimuthal shear deformation may be small is governed by a pair of equations of the Duffing and Hill types. On the other hand, the finite amplitude, pure axial and pure rotational motions of the load are described by the classical, nonlinear Duffing equation alone. A variety of problems are solved exactly for these separate free vibrational modes, and a number of physical results are presented throughout.     

Geometric nonlinear formulation for thermal-rigid-flexible coupling system

This paper develops geometric nonlinear hybrid formulation for flexible multibody system with large deformation considering thermal efect. Diferent from the conventional formulation, the heat flux is the function of the rotational angle and the elastic deformation, therefore, the coupling among the temperature, the large overall motion and the elastic deformation should be taken into account. Firstly,based on nonlinear strain–displacement relationship, variational dynamic equations and heat conduction equations for a flexible beam are derived by using virtual work approach,and then, Lagrange dynamics equations and heat conduction equations of the first kind of the flexible multibody system are obtained by leading into the vectors of Lagrange multiplier associated with kinematic and temperature constraint equations. This formulation is used to simulate the thermal included hub-beam system. Comparison of the response between the coupled system and the uncoupled system has revealed the thermal chattering phenomenon. Then, the key parameters for stability, including the moment of inertia of the central body, the incident angle, the damping ratio and the response time ratio, are analyzed. This formulation is also used to simulate a three-link system applied with heat flux. Comparison of the results obtained by the proposed formulation with those obtained by the approximate nonlinear model and the linear model shows the significance of considering all the nonlinear terms in the strain in case of large deformation. At last, applicability of the approximate nonlinear model and the linear model are clarified in detail.