Theory of water waves in an elastic vessel

Recent experiments related to the Dragon Wash phenomena showed that axisymmetric capillary waves appear first from excitation, and circumferential capillary waves appear after increase of the excitation strength. Based on this new finding, a theory of parametric resonance is developed in detail to explain the on-set of the prominent circumferential capillary waves. Numerical computation is also carried out and the results agree generally with the experiments. Analysis and numerical computation are also presented to explain the generation of axisymmetric low-frequency gravity waves by the high-frequency external excitation.     

Scattering of elastic waves in an elastic matrix containing an inclusion with interfaces

Based on the theory of elastic dynamics, the scattering of elastic waves and dynamic stress concentration in fiber-reinforced composite with interfaces are studied. Analytical expressions of elastic waves in different medium areas are presented and an analytic method of solving this problem is established. The mode coefficients are determined by means of the continuous conditions of displacement and stress on the boundary of the interfaces. The influence of material properties and structural size on the dynamic stress concentration factors near the interfaces is analyzed. It indicates that they have a great influence on the dynamic properties of fiber-reinforced composite. As examples, numerical results of dynamic stress concentration factors near the interfaces are presented and discussed. This paper provides reliable theoretical evidence for the study of dynamic properties in fiber-reinforced composite. Project supported by the National Natural Science Foundation of China (No. 19972018).     

Nonlinear elastic shear waves

Summary The dynamics of one-dimensional two-component shear motion in elastic-isotropic homogeneous media is studied assuming isentropic finite displacements. Wave breaking of initially continuous waves on the infinite interval is discussed for weakly nonlinear waves.The problem of a resonating finite-thickness shear layer in primary resonance for single-component motion exhibits jump discontinuities of particle velocity, shear strain and stress in a finite frequency band near primary resonance.Under certain conditions two-component motion can be reduced to a quasi-single-component motion.     

Solitary waves in initially stressed thin elastic tubes

In the present work, we study the propagation of non-linear waves in an initially stressed thin elastic tube filled with an inviscid fluid. Considering the physiological conditions of the arteries, in the analysis, the tube is assumed to be subjected to a uniform inner pressure P₀ and an axial stretch ratio λ_z. It is assumed that due to blood flow, a finite dynamical displacement field is superimposed on this static field and, then, the non-linear governing equations of the elastic tube are obtained. Using the reductive perturbation technique, the propagation of weakly non-linear waves in the longwave approximation is investigated. It is shown that the governing equations reduce to the Korteweg-deVries equation which admits a solitary wave solution. It is observed that the present model equations give two solitary wave solutions. The results are also discussed for some elastic materials existing in the literature.     

Propagation of guided waves in pressure vessel

Pressure vessels usually operate under extremes of high/low temperatures and high pressures. Defect, such as crack and corrosion, can result in leakage or rupture failures, even catastrophic incidents. Guided wave-based structural health monitoring (SHM) technology is one of the most prominent options in non-destructive evaluation and testing (NDE/NDT) techniques. Propagation of guided waves in a typical pressure vessel is systematically investigated in this study for the application of guided wave-based SHM. Shape of the pressure vessel is a cylinder with two end caps. Because of geometric similarity, theory of guided wave propagation in the cylinderical structure is analyzed to study dispersive features of guided waves in the pressure vessel. Dispersion curves of three different types of guided wave modes, viz. the longitudinal, torsional and flexural modes, are calculated using theoretical method. Based on the analyses and experimental wave signals, central frequency and wave parameters of incident wave are optimized. Effect of contained liquid on propagation of guided waves, especially the L(0, 2) mode, in the pressure vessel is further investigated to minimize energy leakage of the wave to the contained liquid. The analytical method, finite element analysis (FEA) and experiments are applied to study propagation characteristics of guided waves in the pressure vessel, so as to demonstrate the feasibility of guided wave-based non-destructive evaluation and testing (NDE/NDT) for such kind of complex structures.     

Rayleigh waves in an isotropic elastic half-space coated by a thin isotropic elastic layer with smooth contact

In the present paper, we are interested in the propagation of Rayleigh waves in an isotropic elastic half-space coated with a thin isotropic elastic layer. The contact between the layer and the half space is assumed to be smooth. The main purpose of the paper is to establish an approximate secular equation of the wave. By using the effective boundary condition method, an approximate, yet highly accurate secular equation of fourth-order in terms of the dimensionless thickness of the layer is derived. From the secular equation obtained, an approximate formula of third-order for the velocity of Rayleigh waves is established. The approximate secular equation and the formula for the velocity obtained in this paper are potentially useful in many practical applications.     

Absorbing boundary conditions for elastic waves in transversely isotropic media

In the numerical simulation of elastic wave propagation in the solid, it is essential to introduce absorbing boundary conditions to limit the large or unbounded domain of computation. In this paper, the absorbing boundaries for transversely isotropic media are composed of simple first-order partial differential operators, and each of the operators can perfectly absorb a plane wave outgoing at a certain angle. To test the absorbing ability, the reflection coefficient formulas for the quasi-P and quasi-S wave on the absorbing boundary are derived based on the potential functions theory of the elastic wave. Numerical examples show that the absorbing effect is good. The boundary conditions given here have a practical meaning.Supported by National Natural Science Foundation of China.     

Rayleigh lamb waves in micropolar isotropic elastic plate

The propagation of waves in a homogeneous isotropic micropolar elastic cylindrical plate subjected to stress free conditions is investigated. The secular equations for symmetric and skew symmetric wave mode propagation are derived. At short wave limit, the secular equations for symmetric and skew symmetric waves in a stress free circular plate reduces to Rayleigh surface wave frequency equation. Thin plate results are also obtained. The amplitudes of displacements and microrotation components are obtained and depicted graphically. Some special cases are also deduced from the present investigations. The secular equations for symmetric and skew symmetric modes are also presented graphically.     

Purely transverse waves in elastic anisotropic media

Formulas are obtained for decompositions of the third- and fourth-rank tensors symmetric in the last two and three indices, respectively, into irreducible parts invariant relative to the orthogonal group of coordinate transformation. The corresponding parts of the decompositions are orthogonal to each other. These decompositions are used to obtain a general representation of the displacement vectors of plane transverse waves in elastic isotropic and anisotropic solids. It is shown that the displacement vectors of transverse waves are second-, third-, and fourth-degree homogeneous polynomials of the wave normal. Special orthotropic materials are found that transmit purely transverse waves for any direction of the wave normal. The eigenmoduli, eigenstates, and engineering constants (bulk moduli, Youngs moduli, Poissons ratios, shear moduli, and Lame constants of the closest isotropic materials) are determined for these materials.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 1, pp. 160–172, January–February, 2005     

Scattering of elastic waves by a periodic array of cracks in 3-D space

The problem of scattering of normal incident time harmonic plane elastic waves by a co-planar periodic array of cracks in 3-D space is investigated. The scattered waves consist of a superposition of an infinite number of wave modes [M, N]^T and [M, N]^L,M. N=0, 1, 2, , but only a finite number of them are propagating wave modes. The numerical calculation has been made for rectangular cracks and P wave incidence. The reflection coefficient of [O, O] order,R ₀ ³ , has been studied in detail for various wave numbers and parameters of the geometry for the problem. The reliability of the numerical calculation has been checked by an application of the balance of rates of energies. For an elongated rectangular crack,R ₀ ³ in the corresponding 2-D problem in [2] is recovered. The dynamic stress intensity factors around the crack edge have been obtained. The results as the wave number goes to zero have been compared with those in the correspoding static case. Good agreement is observed.     

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.     

Problem of hydrodynamic pressure on suddenly starting vessel

In this paper,Lagrangian method is applied to discuss the problem of the hydrodynamic pressure on a suddenly starting vessel.The free surface profile and the coefficients of the hydrodynamic pressure on the vessel wall are obtained.And it is verified that the singularity of the pressure near the free surface is only logarithmic.     

Separation of elastic waves in split Hopkinson bars using one-point strain measurements

A method for the analysis of elastic waves in split Hopkinson bars for unlimited time durations is presented. This method allows the separation of component waves traveling in opposite directions in each bar using the strain history measured at one point on the bar and a known end condition for it. The method extends the time period for which valid experimental data can be extracted for a split Hopkinson bar apparatus and eliminates the need for a second independent measurement of the stress waves required in other methods for such extended analyses. Comparisons with the two-point method, which requires two independent strain measurements, show good agreement between the two methods. The accuracy and feasibility of the method are demonstrated through its application to impact experiments on composite laminates. The use of the current method in determining the response of a fiber-reinforced composite laminate under impact loading is described.     

Precise method to control elastic waves by conformal mapping

The transformation method to control waves has received widespread attention in electromagnetism and acoustics. However, this machinery is not directly applicable to the control of elastic waves, because it has been shown that the Navier's equation does not usually retain its form under coordinate transformation. In this letter, we prove the form invariance of the Navier's equation under the conformal mapping based on the Helmholtz decomposition method. The needed material parameters are provided to manipulate elastic waves. The validity of this approach is confirmed by an active stealth device which can disguise the signal source by changing its position. Experimental verifications and potential applications may be expected in nondestructive testing, structural seismic design and other fields.     

Critical states in the interaction of discontinuous waves with interfaces between elastic media

The ray method is used to develop a procedure for studying the kinematics and dynamics of the interaction of discontinuous waves with interfaces between elastic media. The critical states of such interactions are analyzed. For the cases of a free surface and two surfaces contacting with and without sliding, it is established that the kinematic and dynamic conditions for critical states to occur are satisfied simultaneously Translated from Prikladnaya Mekhanika, Vol. 45, No. 2, pp. 79–90, February 2009.     

Diffraction of discontinuous waves by ellipsoidal interfaces of transversely isotropic elastic media

Discontinuous waves penetrating the interfaces of transversely isotropic elastic media are studied using the ray-path method. The cases where such waves interact with curvilinear reflectors and with biconvex and biconcave lenses are considered. It is shown that discontinuous waves can focus or scatter depending on the physical properties and sequence of the media under consideration.Translated from Prikladnaya Mekhanika, Vol. 40, No. 10, pp. 98–106, October 2004.     

Scattering of harmonic elastic waves by a plane interface crack with linear adhesive tips in a layered half space

In this paper, the scattering of elastic waves by an interface crack with linear adhesive tips in a layered half space is considered. By use of integral transform and integral equation methods, the singular integral equations of this problem are derived, which are transformed into a set of algebraic equations by means of contour integration and Chebyshev polynomials expanding technique. The numerical results of the adhesive region and stress amplitudes are given in this paper.     

MULTIPLE SCATTERING AND DYNAMIC STRESS ANALYSIS OF ELASTIC WAVES IN A FIBER—REINFORCED COMPOSITE WITH INTERFACES

Based on the theory of elastic dynamics, multiple scattering of elastic waves and dynamic stress concentrations in fiber-reinforced composite are studied. The analytical expressions of elastic waves in different regions are presented. The mode coefficients of elastic waves are determined in accordance with the continuous conditions of displacement and stress on the boundary of the multi-interfaces. By using the addition theorem of Hankel functions, the formula of scattered wave fields in different local coordinates are transformed into those in one local coordinate to determine the unknown coefficients and dynamic stress concentration factors (DSCFs). The influences of the distance between two inclusions, material properties and structural size on the DSCFs near the interfaces are analyzed. As examples, the numerical results of DSCFs near the interfaces for two kinds of fiber-reinforced composites are presented and discussed. The project supported by the National Natural Science Foundation of China (19972018)     

基于积分方程方法的弹性波多重散射计算

运用积分方程方法计算了含多个随机分布椭圆柱型孔洞的随机非均匀介质中相干波的速度和衰减系数,分析了这种介质的频散特性。首先,建立了散射位移场满足的积分方程,推导了单个椭圆柱孔洞的散射截面计算公式。接着分析了在含多个随机分布椭圆柱型孔洞的随机非均匀介质中弹性波的多重散射,给出在统计平均意义下的相干波的波速和衰减系数计算公式。然后用Matlab进行了编程,给出了一个数值算例,并将计算结果与波函数展开法进行了比较,分析了随机空隙介质的频散特征及其孔洞椭圆偏心率和材料空隙率的影响。     

On the interaction of cubically nonlinear transverse plane waves in an elastic material

The paper presents theoretical results on the interaction of cubically nonlinear harmonic elastic plane waves in a nonlinear material described by the Murnaghan potential. The interaction of two harmonic transverse waves is studied using the method of slowly varying amplitude. Reduced and evolution equations and the Manley-Rowe relations are derived. An analysis is made of the mechanism of energy transfer from the strong pumping wave, which has frequency ω, to the weak signal wave, which has frequency 3ω because of this interaction. A switching mechanism for hypersonic waves in a nonlinear elastic material is described, which is similar to the switching mechanism observed in transistors __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 6, pp. 61–70, June 2006.