Finite amplitude acceleration waves in incompressible BKZ viscoelastic fluids— The propagation condition

A general propagation condition is derived which permits the calculation of the speed of propagation of a second-order acceleration wave passing through a particular non-linear incompressible viscoelastic fluid. The viscoelastic fluid is taken to obey the Bernstein, Kearsley and Zapas single integral constitutive model. The analysis is valid for arbitrary finite amplitude waves propagating through a medium undergoing an arbitrary large deformation. Three examples, rest history, steady simple shearing flow and steady simple extensional flow, are given to demonstrate the utility of the propagation condition.     

On steady periodic waves on the surface of a fluid of finite depth

A solution of Nekrasov’s integral equation is obtained, and the range of its existence in the theory of steady nonlinear waves on the surface of a finite-depth fluid is determined. Relations are derived for calculating the wave profile and propagation velocity as functions of the ratio of the liquid depth to the wavelength. A comparison is made of the velocities obtained using the linear and nonlinear theories of wave propagation.     

On the approximation of acceleration waves in rods

This paper employs an approximate form of analysis based on the assumption of plane stress to find the transport equation and corresponding evolution law governing the intensity of acceleration wave propagation in an elastic rod of slowly varying area of cross-section. The result is then extended to include the case of slightly bent rods. In each of these cases it is shown that for a medium in which the strain energy function Σ(p) is such that d³Σ/dp³ ≠ 0, with p the displacement gradient, the acceleration wave intensity is governed by a Bernoulli equation. The work is concluded by showing that the analysis may also be applied to the case of a composite rod comprising an arbitrary number of homogeneous isotropic plane layers normal to the direction of acceleration wave propagation.     

Bounds for Arbitrary Steady Gravity Waves on Water of Finite Depth

Necessary conditions for the existence of arbitrary bounded steady waves are proved (earlier, these conditions, that have the form of bounds on the Bernoulli constant and other wave characteristics, were established only for Stokes waves). It is also shown that there exists an exact upper bound such that if the free-surface profile is less than this bound at infinity (positive, negative, or both), then the profile asymptotes the constant level corresponding to a unform stream (supercritical or subcritical). Finally, an integral property of arbitrary steady waves is obtained. A new technique is proposed for proving these results; it is based on modified Bernoulli’s equation that along with the free surface profile involves the difference between the potential and its vertical average.     

波流与结构物相互作用的数值模拟

采用高阶边界元法建立波流与任意形状结构物相互作用的时域数学模型。在小流速假设下,将速度势进行摄动展开,边界条件分解为双物体假设下的零阶波陡稳定问题和一阶波陡下的不稳定波浪问题。物面速度势及自由水面速度势的法向导数等未知量通过求解高阶边界元积分方程得到,而积分方程的求解则通过一个数值程序来实现。在每个时间步上采用四阶Runge-Kutta法更新下一时刻自由水面波面和速度势,自由水面上采用人工阻尼层消除散射波。通过波流与直立圆柱相互作用的数值计算研究了一阶激振力和二阶慢漂力随波数的变化关系以及圆柱周围波幅的分布曲线,并与已有频域结果和时域结果对比验证了所建模型的准确性,进而应用本文模型研究了波流与实际工程结构物相互作用的问题。     

On the Passage of a Shock Wave through a Layer of Charged Gas

A class of exact solutions of the ideal electrohydrodynamics equations is presented. These solutions describe the propagation of a plane shock wave along a static background with decreasing density in the presence of gravity and longitudinal electric fields. This class of solutions contains an arbitrary function of the Lagrangian variable which makes it possible to consider many physically different cases.     

Nonreflecting boundary conditions based on nonlinear multidimensional characteristics

Nonlinear characteristic boundary conditions based on nonlinear multidimensional characteristics are proposed for 2‐ and 3‐D compressible Navier–Stokes equations with/without scalar transport equations. This approach is consistent with the flow physics and transport properties. Based on the theory of characteristics, which is a rigorous mathematical technique, multidimensional flows can be decomposed into acoustic, entropy, and vorticity waves. Nonreflecting boundary conditions are derived by setting corresponding characteristic variables of incoming waves to zero and by partially damping the source terms of the incoming acoustic waves. In order to obtain the resulting optimal damping coefficient, analysis is performed for problems of pure acoustic plane wave propagation and arbitrary flows. The proposed boundary conditions are tested on two benchmark problems: cylindrical acoustic wave propagation and the wake flow behind a cylinder with strong periodic vortex convected out of the computational domain. This new approach substantially minimizes the spurious wave reflections of pressure, density, temperature, and velocity as well as vorticity from the artificial boundaries, where strong multidimensional flow effects exist. The numerical simulations yield accurate results, confirm the optimal damping coefficient obtained from analysis, and verify that the method substantially improves the 1‐D characteristics‐based nonreflecting boundary conditions for complex multidimensional flows. Copyright © 2009 John Wiley & Sons, Ltd.     

Propagation of steady shock waves in non-linear thermoviscoelastic solids

This paper is concerned with the dynamic response of a class of thermoviscoelastic solids. In particular a specific one-dimensional model, consistent with the laws of continuum thermodynamics, is proposed and applied to the problem of the propagation of steady shock waves. The governing equations are written in terms of material response functions which can be determined from shock wave, thermophysical, and bulk response data. The results of the analysis are compared with experimental steady wave studies involving the solid polymer, polymethyl methacrylate.     

Lagrangian description of transport equations for shock waves in three-dimensional elastic solids

A set of transport equations for the growth or decay of theamplitudes of shock waves along an arbitrary propagation directionin three-dimensional nonlinear elastic solids is derived using theLagrangian coordinates.The transport equations obtained showthat the time derivative of the amplitude of a shock wave alongany propagation ray depends on (i) an unknown quantity immediatelybehind the shock wave,(ii) the two principal curvatures of theshock surface,(iii) the gradient taken on the shock surface ofthe normal shock wave speed and (iv) the inhomogeneous term.whichis related to the motion ahead of the shock surface.vanisheswhen the motion ahead of the shock surface is uniform.Severalchoices of the propagation vector are given for which the tran-sport equations can be simplified.Some universal relations,which relate the time derivatives of various jump quantities toeach other but which do not depend on the constitutive equationsof the material,are also presented.     

Dispersion and polarization of surface Rayleigh waves for the Cosserat continuum

In the framework of the nonsymmetric theory of elasticity (the Cosserat contimum), we consider the problem of propagation of a surface acoustic Rayleigh wave in the half-space. The wave is represented as a wave packet of arbitrary form bounded both in the time space and the Fourier space. We assume that the material strain is described by not only the displacement vector but also an independent rotation vector. The general analytic solution of this problem is obtained in displacements. We perform comparative analysis of the obtained solution and the corresponding solution for the classical elastic medium. We introduce and analyze macroparameters characterizing the difference between the stress-strain state and the state predicted by the classical theory of elasticity: the elasticity coefficient, the wave number, and the phase and group velocities. It should be noted that these parameters can be measured experimentally.     

Emergent scale behaviour in the effective equations for an elastic metamaterial

Asymptotic homogenisation of an elastic metamaterial consisting of a series of plates interspaced by a fluid is considered. It is shown that the usual method must be extended by the inclusion of a second macroscale, the “emergent scale”, in order to correctly capture the behaviour of this metamaterial at low frequency. The leading order solutions for plane wave propagation are found and the effective constitutive equations derived. At leading order, the dispersion curves for plane wave propagation depend on the direction of propagation and the branch associated with the emergent scale is dispersive. The effective constitutive equations contain some terms, associated with the shear normal to the plates, for which stress is not proportional to strain but rather depends on the third derivative of the normal displacement with respect to transverse position. These terms are not present in the usual form of the Willis equations and the effective fields are non-analytic functions of frequency as the frequency tends to zero.     

Stochastic solution to a time-fractional attenuated wave equation

The power law wave equation uses two different fractional derivative terms to model wave propagation with power law attenuation. This equation averages complex nonlinear dynamics into a convenient, tractable form with an explicit analytical solution. This paper develops a random walk model to explain the appearance and meaning of the fractional derivative terms in that equation, and discusses an application to medical ultrasound. In the process, a new strictly causal solution to this fractional wave equation is developed.     

Modeling micro-inertia in heterogeneous materials under dynamic loading

A continuum model including micro-inertia for heterogeneous materials under dynamic loading is proposed using a micro-mechanics method. The macro strain and stress are defined as the volume averages of the strain and stress fields in the representative volume element (RVE). The macro equations of motion are derived by using Hamilton’s principle together with the strain energy density and kinetic energy density involving the micro-inertia terms. The new macro equations of motion are used to study harmonic and transient wave propagation in layered media. Using a simple linear displacement field for the RVE, the dispersion curves obtained from the present model agree with the exact solutions very well for a range of wavelengths. The present model is also applied to analyze the transient response of layered media subjected to a triangular pulse loading. Comparison is made between the results of the present model and a finite element analysis.     

Propogation of waves in a linear elastic fluid

The non-classical symmetry method is used to determine particular forms of the arbitrary velocity and forcing terms in a linear wave equation used to model the propogation of waves in a linear elastic fluid. The behaviour of solutions derived using the non-classical symmetry method are discussed. Solutions satisfy a given initial profile and wave velocity. For some solutions the arbitrary forcing terms and wave velocity can be written in terms of the initial wave profile. Relationships between the arbitrary forcing, arbitrary velocity and the solution are derived.     

各向异性介质中弹性波的数值模拟

提出了一种非均匀各向异性介质中弹性波传播的数值模拟算法。该方法可以灵活地运用于具有任意地表形状、内部孔洞、固液边界和不规则内部交界面的介质情况,另外,该方法自然满足复杂几何边界的自由表面条件。这种基于三角形和四边形离散网格的算法使用的是围绕每个节点的积分平衡方程,而不是其它有限差分法中使用的各个节点满足的弹性动力学的微分方程。该文工作是非均匀各向同性介质中弹性波传播格子法研究的继续。除了研究各向异性介质中波的传播以外,还给出了一种能够省时的四边形网格的格子法。     

Exact Solitary Water Waves with Vorticity

The solitary water wave problem is to find steady free surface waves which approach a constant level of depth in the far field. The main result is the existence of a family of exact solitary waves of small amplitude for an arbitrary vorticity. Each solution has a supercritical parameter value and decays exponentially at infinity. The proof is based on a generalized implicit function theorem of the Nash–Moser type. The first approximation to the surface profile is given by the “KdV” equation. With a supercritical value of the surface tension coefficient, a family of small amplitude solitary waves of depression with subcritical parameter values is constructed for an arbitrary vorticity.     

Constitutive modeling of the dynamic fracture of smooth tensile bars

A recently developed viscoplastic-damage type of constitutive theory for high strain-rate flow processes and ductile fracture is used to model the deformation and fracture of dynamically loaded smooth cylindrical tensile bars. The analysis assumes polycrystalline materials which usually contain microvoids with an average density of the order of 10⁶ per cm³ that are dispersed homogeneously throughout. It is shown that for dynamically imposed loading that produce nominal strain rates ranging between 5 × 10² − 5 × 10³ sec ⁻¹, the inhomogeneous fields of stress and deformation caused by wave propagation and wave reflection induce necking at different locations along the gauge section, depending upon the strain-rate imposed. This occurs without imposition of any geometrical or material irregularity to preposition the location of the necking. The imposed rate of strain is also shown to affect the magnitude of the strain at which necking initiates, as well as the strain required for fracture.     

Wave propagation in nonlinear and hysteretic media––a numerical study

This paper considers the problem of one dimensional wave propagation in nonlinear, hysteretic media. The constitutive law in the media is prescribed by an integral relationship based on the Duhem model of hysteresis. It is found that the well known nonlinear elastic stress–strain relationship is a special case of this integral relationship. It is also shown that the stress–strain relationship from the McCall and Guyer model of hyesteretic materials can also be derived from this integral stress–strain relationship. The first part of this paper focuses on a material with a quadratic stress–strain relationship, where the initial value problem is formulated into a system of conservation laws. Analytical solutions to the Riemann problem are obtained by solving the corresponding eigenvalue problem and serve as reference for the verification and illustration of the accuracy obtained using the applied numerical scheme proposed by Kurganov and Tadmor. The second part of this research is devoted to wave propagation in hysteretic media. Several types of initial excitations are presented in order to determine special characteristics of the wave propagation due to material nonlinearity and hysteresis. The results of this paper demonstrate the accuracy and the robustness of this numerical scheme to analyze wave propagation in nonlinear materials.     

曲面曲率对Rayleigh波传播特性的影响

对任意形状的均匀各向同性线弹性曲面物体,用 WKB～（1）方法求解了沿曲面传播的Rayleigh表面波的运动微分方程,同时考虑了波传播方向及其垂直方向曲面曲率对波的穿透性的影, 所获波动方程的势函数解答表明,在一般情况下垂直波传播方向的曲面曲率对波的穿透深度的影响是不容忽视的.进而以同种介质平面表面情况下的Rayleigh面波的传播特性为基准,给出了曲面曲率引起波数或波速变化的解析表达式.通过理论分析和数值算例,描述了曲面上Rayleigh面波传播行为的一些基本特征.