非线性应力波传播理论的发展及应用

应力波传播理论是分析结构和材料在爆炸/冲击载荷作用下的响应及破坏特性的基础,在国防和民用工程上有重大价值。本文对作者们近半个世纪来在非线性应力波传播理论的发展及其工程应用方面所开展的主要研究作一回顾和讨论,包括：几类非线性应力波相互作用及失效,非线性粘弹性波传播理论及应用,动态破坏和应力波相互作用,以及应力波理论在防护工程中的应用等。     

Spectral element technique for efficient parameter identification of layered media. Part III: viscoelastic aspects

This article addresses the issues of wave propagation in elastic–viscoelastic layered systems and viscous parameter identification from non-destructive dynamic tests. A methodology that combines the spectral element technique, for the simulation of wave propagation, with the differential operator technique, for stress–strain relationship in viscoelastic materials, is adopted. The compatibility between the two techniques stems from the fact that both can be treated in the frequency domain, which enables naturally the adoption of Fourier superposition. The mathematical formulation of spectral elements for Burger's viscoelastic material model is highlighted. Also, an inverse procedure for the identification of the material's Young's moduli and complex moduli for layer systems is described. It is shown that the proposed methodology enables the substitution of an expensive laboratory testing procedure for the determination of material complex moduli with non-destructive dynamic testing.     

Waves in a viscoelastic bar surrounded by soils under smooth loading

The results of numerical solution of the nonstationary wave problem of longitudinal wave propagation in the bar-soil system are given. The barmaterial and the soil are assumed to be linearly viscoelastic (standard linear body). A nonlinear interaction condition is satisfied on the bar-soil contact boundary. The laws of propagation of longitudinal waves and variations in the longitudinal stresses in the bar are determined depending on the mechanical characteristics of the soil and the bar and on the interaction parameters.     

Crack propagation in the power-law nonlinear viscoelastic material

I.IntroductionCrazingdamageisacommonphenon1enonoffractureofpolymericmaterials.Theformationofcrazezoneisamid-stateinthefractureprocessofthematerialsfromperfectstatetofaiIurc.Microscopically,inthisregionthereexistssomefibrilslinkingthetwocracksurfacesandres…     

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.     

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.     

Plane nonaxisymmetric deformation of a viscoelastic cylinder with consideration of its physical and geometric nonlinearity

The nonaxisymmetric plane problem of the nonlinear theory of viscoelasticity is solved for a cylinder reinforced by an elastic circular shell. The cylinder has an internal cut resembling a Maltese cross in shape. The identification of the nonlinear endochronous theory of aging viscoelastic materials is conducted by a genetic algorithm method on the basis a nonmonotonic experimental stress-strain dependence. Some numerical results obtained for the stress-strain state of this cylinder under the action of internal pressure are discussed with consideration of the above physical nonlinearity and the finite logarithmic strains.     

Laplace变换法研究SHPB实验中试件的黏弹性波传播问题

对分离式霍普金森压杆(split Hopkinson pressure bar, SHPB) 实验中试件的黏弹性波传播的控制方程组进行Laplace 变换,并结合恰当的初始-边界条件求解,得到变换域的应力、速度、应变等变量的像函数的精确表达式. 采用该方法处理SHPB 实验中涉及黏弹性试件内部应力非均匀性问题,并给出数值反变换解. 作为特例,对于弹性试件分别采用级数展开法和留数定理进行反Laplace 变换,从而给出弹性夹层介质中应力波传播问题的解析解.      

Rayleigh Surface Waves on a Kelvin-Voigt Viscoelastic Half-Space

In this paper we consider the propagation of Rayleigh surface waves in an exponentially graded half-space made of an isotropic Kelvin-Voigt viscoelastic material. Here we take into account the effect of the viscoelastic dissipation energy upon the corresponding wave solutions. As a consequence we introduce the damped in time wave solutions and then we treat the Rayleigh surface wave problem in terms of such solutions. The explicit form of the secular equation is obtained in terms of the wave speed and the viscoelastic inhomogeneous profile. Furthermore, we use numerical methods and computations to solve the secular equation for some special homogeneous materials. The results sustain the idea, existent in literature on the argument, that there is possible to have more than one surface wave for the Rayleigh wave problem.     

Laplace变换法及其在粘弹性波中传播的应用

针对波传播分析理论的发展历程进行了简要的综述，详细介绍了几种处理粘弹性波传播问题的分析方法，重点讲解Laplace变换法以及Laplace变换在粘弹性波中的应用，对比分析几种方法在各自应用上的优劣，由于Laplace变换法能准确地描述应力波在任意时刻、任意点的波动情况，在处理大尺寸混凝土类构件中应力波传播问题时具有其独特的优势。     

USING LAPLACE TRANSFORM TO SOLVE THE VISCOELASTIC WAVE PROBLEMS IN THE SHPB EXPERIMENTS

对分离式霍普金森压杆(split Hopkinson pressure bar, SHPB) 实验中试件的黏弹性波传播的控制方程组进行Laplace 变换,并结合恰当的初始-边界条件求解,得到变换域的应力、速度、应变等变量的像函数的精确表达式. 采用该方法处理SHPB 实验中涉及黏弹性试件内部应力非均匀性问题,并给出数值反变换解. 作为特例,对于弹性试件分别采用级数展开法和留数定理进行反Laplace 变换,从而给出弹性夹层介质中应力波传播问题的解析解.     

Nonlinear flutter of viscoelastic rectangular plates and cylindrical panels of a composite with a concentrated masses

The problem of flutter of viscoelastic rectangular plates and cylindrical panels with concentrated masses is studied in a geometrically nonlinear formulation. In the equation of motion of the plate and panel, the effect of concentrated masses is accounted for using the δ-Dirac function. The problem is reduced to a system of nonlinear ordinary integrodifferential equations by using the Bubnov-Galerkin method. The resulting system with a weakly singular Koltunov-Rzhanitsyn kernel is solved by employing a numerical method based on quadrature formulas. The behavior of viscoelastic rectangular plates and cylindrical panels is studied and the critical flow velocities are determined for real composite materials over wide ranges of physicomechanical and geometrical parameters.     

SHPB试验中粘弹性材料的应力均匀性分析

采用特征线解法，对满足ZWT方程的粘弹性材料在高应变率SHPB试验中的应力均匀性进行了数值研究。着重分析了不同的材料本构粘性(松弛时间)、瞬态波阻抗比和入射波升时对于试样中应力均匀性、应变均匀性和平均应变率等的影响。为今后动态试验的试样设计提供了一定的理论依据。     

Uniaxial wave propagation in a nonlinear thermoviscoelastic medium with temperature dependent properties

The problem of finite wave propagation in a nonlinearly thermoviscoelastic thin rod whose viscoelastic properties are temperature dependent is considered. The rod is subjected to mechanical or thermal time-dependent loading. The coupled equations of motion and heat conduction are based on a constitutive theory of nonisothermal nonlinear viscoelasticity which is described by single-integral terms only. This theory is reformulated here for the uniaxial motion of a compressible rubbery material. The solution of the field equations is obtained by a numerical procedure which is developed for the present case and is able to handle successfully shock waves whenever they built up in the nonlinear material.     

Identification of parameters of models of nonlinear deformation of isotropic and composite materials on the basis of calculations and experiments aimed at analyzing the dynamic behavior of cylindrical metal–plastic shells

A method for identification of material parameters of the constitutive relations of elastoplastic and viscoelastic deformation of isotropic and composite materials is developed. The method is based on minimizing the functional of the residue of results of numerical and experimental analysis of unsteady deformation of structural elements made of examined materials. The method is tested, and prospects of its application for determining material parameters of viscoelastic and elastoplastic models of nonlinear deformation of cylindrical metal–plastic shells under explosive loading are demonstrated.     

A polymeric split Hopkinson pressure bar instrumented with velocity gages

Polymeric split Hopkinson pressure bars are often used to test low-impedance materials at elevated strain rates. However, they tend to be viscoelastic, and a viscoelastic wave propagation model is required to analyze the data. This considerably complicates the analysis over the more common linear elastic split Hopkinson bar. In this research, a polymeric split Hopkinson bar is instrumented with electromagnetic velocity gages. The gages are placed at the interfaces between the bars and the specimen. By using this arrangement, viscoelastic effects in the bars are negligible and the need for a viscoelastic correction is eliminated. The method is applied by testing low-density foams.     

On the effective behavior of nonlinear inelastic composites: II: A second-order procedure

A new method for determining the overall behavior of composite materials comprising nonlinear viscoelastic and elasto-viscoplastic constituents is presented. Part I of this work showed that upon use of an implicit time-discretization scheme, the evolution equations describing the constitutive behavior of the phases can be reduced to the minimization of an incremental energy function. This minimization problem is rigorously equivalent to a nonlinear thermoelastic problem with a transformation strain which is a nonuniform field (not even uniform within the phases). In part I of this paper the nonlinearity was handled using a variational (or secant) technique. In this second part of the study, a proper modification of the second-order procedure of Ponte Castañeda is proposed and leads to replacing, at each time-step, the actual nonlinear viscoelastic composite by a linear viscoelastic one. The linearized problem is even further simplified by using an “effective internal variable” in each individual phase. The resulting predictions are in good agreement with exact results and improve on the predictions of the secant model proposed in part I of this paper.     

Propagation of nonlinear waves along a viscoelastic bar

Various types of nonlinear waves propagating along a viscoelastic bar are considered. The rheological equation of state has strong physical and geometric nonlinearities, and nonisothermal effects are included. Both weak (isentropic) and shock waves of loading and unloading are investigated. It is shown that, for certain rubber-like materials, stable shock waves of extension can exist along with the shock waves of compression at very large strains. We then consider the strike of a viscoelastic bar of finite length against a rigid obstacle. Numerical solutions to this problem illustrate the influence of stress relaxation on nonlinear wave processes. A model for sticking and bouncing off is formulated and the mass-averaged velocity of the bar at the moment when it bounces off the obstacle is calculated.     

A Numerical Method for Wave Propagation in Viscoelastic Stratified Porous Media

This paper presents a numerical method, a transmission matrix method, for the wave propagation in viscoelastic stratified saturated porous media. The wave propagation in saturated media, based on Biot theory, is a coupled problem. In this stratified three-dimensional model we do the Laplace transform for the time variable and the Fourier transform for the horizontal space coordinate. The original problem is transformed into ordinary differential equations with six independent unknown variables, which are only the function of the coordinate of depth. Thus, we get a transmission matrix of the wave problem for each layer. In the process of solution we use numerical method to calculate the eigenvalues and the eigenvectors of the transmission matrices. In the first step of the solution process we can obtain the wave field in the transformed space. The fast Fourier transform (FFT) method is used to do the inverse Laplace and the inverse Fourier transforms to get the solution in the time space. The detailed formulae are derived and some numerical examples are given.     

A Case Study on the Use of Fractional Derivatives: The Low-Frequency Viscoelastic Uni-Directional Behavior of Polyurethane Foam

A fractional derivative model of dissipative effects is combined with a nonlinear elastic model to model the response of polyurethane foam in quasi-static compression tests. A system identification method is developed based on a separation of the elastic and viscoelastic parts of the response, which is possible because of symmetries in the imposed deformation time-history. Simulations are used to evaluate the proposed identification method when noise is present in the response. The system identification technique is also applied with some success to experimental data taken from several compression experiments on two types of polyurethane foam blocks.