ELASTIC SH WAVE PROPAGATION IN PERIODIC LAYERED COMPOSITES WITH A PERIODIC ARRAY OF INTERFACE CRACKS

The interaction of anti-plane elastic SH waves with a periodic array of interface cracks in a multi-layered periodic medium is analyzed in this paper. A perfect periodic structure without interface cracks is first studied and the transmission displacement coefficient is obtained based on the transfer matrix method in conjunction with the Bloch-Floquet theorem. This is then generalized to a single and periodic distribution of cracks at the center interface and the result is compared with that of perfect periodic cases without interface cracks. The dependence of the transmission displacement coefficient on the frequency of the incident wave, the influences of material combination, crack configuration and incident angle are discussed in detail. Compared with the corresponding perfect periodic structure without interface cracks, a new phenomenon is found in the periodic layered system with a single and periodic array of interface cracks.     

A theory of the densification-induced fragmentation in glasses and ceramics under dynamic compression

A boundary of failure which follows a shock front is observed in glasses and ceramics above a critical compressive shock load. This boundary, called the failure wave leaves behind a damaged material with newly evolved properties. These include the Tresca yield behavior and the reduction in sound speed. The evolution of the Mescall zone during long-rod penetration of these materials is associated with this wave. But, the failure wave and this accompanying process of fragmentation under dynamic compression are still not understood. It is known that some brittle solids undergo an irreversible density increase when subjected to high compression. This phenomenon, called densification is linked to the formations of slip lines and cracks in intensely compressed regions of silica glass. It also corresponds directly to the losses in shock wave speeds. Once densified, a region tends to shrink, straining the interface between it and the original solid. Stressed interfaces are unstable and may roughen, causing local cracking. On this basis, the failure wave is idealized as a propagating fracture boundary layer where the solid is comminuted by a process of densification interface roughness. The kinetics for this process are established using the fluctuation dissipation theorem. Shear and tensile modes of fragmentation are studied in plane stress. The theory predicts the powder size in the Mescall zone of silica glass. Nevertheless, this theory still needs experimental verification.     

A continuous loading solution for spherical elastic-plastic waves of small amplitude

A solution is constructed for the wave motion induced by the application of a continuously increasing uniform pressure on the surface of a spherical cavity in an infinite elastic-plastic medium. The small amplitude linearization in conjunction with parabolic work-hardening leads to constant wave speeds in both elastically and plastically deforming regions, so that linear wave function representations can be used. The induced wave pattern is an elastic (continuous) front propagating into the medium followed by an expanding region of continuous elastic loading, in turn followed by a region of continuous plastic loading which persists to the cavity boundary. The motion of the continuous elastic-plastic interface is part of the solution to be determined, in contrast to discontinuous interfaces which move with a pre-determined wave speed. Validity of this wave pattern requires appropriate pressure increase on the boundary. Several examples are presented to illustrate different forms of interface motion.     

Periodic set of limited electrically permeable interface cracks with contact zones

Plane problem for an infinite space composed of two different piezoelectric or piezoelectric/dielectric semi-infinite spaces with a periodic set of limited electrically permeable interface cracks is considered. Uniformly distributed electromechanical loading is applied at infinity. The frictionless contact zones at the crack tips are taken into account. The problem is reduced to the combined Dirichlet–Riemann boundary value problem by means of the electromechanical factors presentation via sectionally analytic functions, assuming that the electric flux is uniformly distributed inside the cracks. An exact solution of the problem is proposed. It permits to find in a closed form all necessary electromechanical characteristics at the interface and to formulate the equation for the determination of the electric flux value. Analysis of this equation confirms the correctness of the assumption concerning the uniform distribution of the electric flux in the crack region.Formulae for stresses, electric displacement vector, elastic displacements and electric potential jump at the interface as well as the intensity factors at the crack tips are given. Equation for the contact zone length determination is presented. Calculations for certain material combinations are carried out. The influence of electric permeability of cracks on electromechanical fields and the fracture mechanical parameters is analyzed.     

双材料界面裂纹复应力强度因子的正则化边界元法

双材料界面裂纹渐近位移和应力场表现出剧烈的振荡特性,许多用于表征经典平方根(r1/2)和负平方根(r?1/2)渐近物理场的传统数值方法失效,给界面裂纹复应力强度因子(K1+iK2)的精确求解增加了难度.引入一种含有复振荡因子的新型"特殊裂尖单元",可精确表征裂纹尖端渐近位移和应力场的振荡特性,在避免裂尖区域高密度网格剖...     

Reflection and Refraction of Anti-Plane Shear Waves from a Moving Phase Boundary

The reflection and refraction of anti-plane shear waves from an interface separating half-spaces with different moduli is well understood in the linear theory of elasticity. Namely, an oblique incident wave gives rise to a reflected wave that departs at the same angle and to a refracted wave that, after transmission through the interface, departs at a possibly different angle. Here we study similar issues for a material that admits mobile elastic phase boundaries in anti-plane shear. We consider an energy minimal equilibrium state in anti-plane shear involving a planar phase boundary that is perturbed due to an incident wave of small magnitude. The phase boundary is allowed to move under this perturbation. As in the linear theory, the perturbation gives rise to a reflected and a refracted wave. The orientation of these waves is independent of the phase boundary motion and determined as in the linear theory. However, the phase boundary motion affects the amplitudes of the departing waves. Perturbation analysis gives these amplitudes for general small phase boundary motion, and also permits the specification of the phase boundary motion on the basis of additional criteria such as a kinetic relation. A standard kinetic relation is studied to quantify the subsequent energy partitioning and dissipation on the basis of the properties of the incident wave.     

Scattering of an incident wave from an interface separating two fluids

We consider the scattering of an incident pulse from an interface separating two fluids. The interface can be either an elastic membrane or a two-fluid interface with surface tension. By considering the limit where the ratio of acoustic wavelength to the surface wavelength is small, we systematically derived a boundary condition relating the scattered wave and the surface deformation. This condition is local and can be used to derive a partial differential equation for the deformation of the interface. This equation includes the contribution of the acoustic waves induced by the motion of the interface and once it is solved it can be used to determine the scattered field. At leading order in our analysis we find the plane wave approximation. The addition of the next order terms results in an on surface condition equivalent to that of Kriegsmann and Scandrett. We present numerical calculations to show that our results are in good agreement with the exact numerical solution as well as that of Kriegsmann and Scandrett. Physical situations where the conditions of our analysis are valid are presented.     

Elastic Energies in Circular Inhomogeneities: Imperfect Versus Perfect Interfaces

The change in the total potential energy in a stressed elastic plane system, consisting of an unbounded matrix containing a cylindrical inhomogeneity of circular cross-section, is studied, when an imperfect bonding is formed across the interface. The imperfect bonding is simulated by linearly elastic springs distributed over the interface. Two loading cases are examined: an equilibrium system of fixed uniform tractions acting in the remote boundary of the matrix, and a phase transformation in the inhomogeneity prescribed by stress free uniform eigenstrains distributed in the inhomogeneity region. For both loadings, the fully elastic fields in explicit forms are derived involving the spring compliances and three new two-phase parameters depending on the elastic properties of the two materials. The elastic energies stored in the whole system and in its constituents are determined in simple and compact forms. It is shown that, in both loading cases, the total potential energy of the system is reduced. It is found that, in nanoscale, the ratio of the elastic energy stored in interface to the elastic energy stored in inhomogeneity increases rapidly for small values of the circular radius and tends to zero for large values. Also, this ratio increases as the matrix becomes softer compared to the inhomogeneity.     

Dual boundary element analysis of wave scattering from singularities

The dual boundary element method is used to obtain an efficient solution of the Helmholtz equation in the presence of geometric singularities. In particular, time-harmonic waves in a membrane which contains one or more fixed edge stringers (or cracks) are investigated. The hypersingular integral equation is used in the procedure to ensure a unique solution for the problem with a degenerate boundary. The method yields a solution for the entire membrane as well as the dynamic stress intensity factor. Numerical results are presented for a circular membrane containing a single edge stringer, two edge stringers and an internal stringer. Also, the first three critical wave numbers of the membrane with the homogeneous boundary condition are determined, and the dynamic stress intensity factors are found for problems with the nonhomogeneous boundary condition. Good agreement is found after comparing the results with exact solutions, and with results obtained using DtN-FEM and ABAQUS.     

Why continuum damage is nonlocal: Justification by quasiperiodic microcrack array

Strain-softening damage due to distributed cracking is modeled by an elastic continuum with a quasiperiodic array of cracks of regular spacing but varying sizes. As a model for the initial stage, the cracks are penny-shaped and small compared to their spacing, and as a model for the terminal stage the uncracked ligaments between the cracks are circular and small compared to their spacing. The strain due to cracks and the compliance per crack are calculated. The cracked material is homogenized in such a manner that the macroscopic continuum strains satisfy exactly the condition of compatibility with the actual strains due to cracks, and the macroscopic continuum stress satisfies exactly the condition of work equivalence with the actual stresses in the cracked material. The results show that, contrary to the existing theories, the damage variable used in continuum damage mechanics should be nonlocal, while the elastic part of the response should be local. In particular, the nonlocal continuum damage should be considered as a function of the spatial average of the cracking strain rather than its local value. The size of the averaging region is determined by the crack spacing.     

Surface acoustic wave measurements of small fatigue cracks initiated from a surface cavity

A model for the low frequency scattering of a surface acoustic wave by a surface cylindrical cavity with two corner cracks is presented. It is applied to determine the depth of the small fatigue cracks initiated from a pit-type surface flaw. The general scattering formalism based on the elastodynamic reciprocity principle is employed. The effect of the cylindrical cavity on the surface wave reflection from cracks is considered using an approximate stress intensity factor for the corner cracks. In situ surface acoustic wave measurements have been performed during fatigue tests for an Al 2024-T3 sample. The surface wave signal was acquired continuously at different cyclic load levels. The model is verified by comparing calculated reflection signals and spectra with those from experiments. The depths of fully and partially open cracks are determined from the predicted and the measured surface wave reflections. The surface wave reflection is observed to be sensitive to crack closure.     

Nonlinear behaviour of interacting dielectric cracks in piezoelectric materials

Existing studies on the fracture of cracked piezoelectric materials have been limited mostly to the electrically impermeable and permeable crack models, which represent the limiting cases of the physical boundary condition along the crack surfaces. This paper presents a study on the electromechanical behaviour of interacting dielectric cracks in piezoelectric materials. The cracks are filled with dielectric media and, as the result, the electric boundary condition along the crack surfaces is governed by the opening displacement of the cracks. The formulation of this nonlinear problem is based on simulating the cracks using distributed dislocations and solving the resulting nonlinear singular integral equations. Multiple deformation modes are observed. A solution technique is developed to determine the desired deformation mode of the interacting cracks. Numerical results are given to show the effect of the interaction between parallel cracks. Attention is paid to the transition between permeable and impermeable models with increasing crack opening.     

Wave induced sliding at the interface between a layered elastic medium and a half-space

The sliding interface between an unrestrained elastic half-space and a grounded layered half-space excited by an incident harmonic wave is investigated. The problem is formulated considering various possible boundary conditions and boundary inequalities at the sliding interface. The Coulomb friction model without distinction between the static and kinetic coefficients of friction is considered to govern the sliding condition. Three possible bands at the interface, namely slip, stick, and separation, are considered. The interface is assumed to be preloaded under normal and shear stresses. The solution is developed by modifying the problem of welded interface, which then is reduced to a set of algebraic equations. The effects of the incident angle, layer thickness, friction coefficient and externally applied stresses on the drift velocity of the unrestrained half-space are studied numerically for a pair of materials. It is shown that the sliding interface, and hence the drift velocity of unrestrained half-space is noticeably influenced by the layered medium. These results are expected to be useful for the development of a new kind of ultrasonic drive in future.     

考虑床面边界层效应的波浪海床相互作用分析

在考虑海床表面附近黏性边界层的影响下, 考察了波浪与弹性多孔海床之间的相互作用．波浪场包括势流部分和边界层部分,海床域控制方程由比奥固结理论给出．波浪场和海床域通过交界面处应力连续和速度连续条件进行耦合．在简谐波和小变形的前提下,通过联立求解势流方程、波浪边界层方程和海床准静态比奥固结方程得到了波浪运动及相应的海床动力响应的解析解．通过与以往文献的实验结果进行对比,解析解的合理性得到了验证．通过参数分析讨论了实际问题中需要考虑波浪和海床相互作用机制的海床土质条件,以及流体黏性对波能衰减的影响规律．      

General boundary conditions for the wave equation around non-homogenous scatterers

To solve the wave equation inside a region that contains an inhomogenous dielectric material of arbitrary shape under the influence of an incoming wave, we establish a generalized boundary condition. The solutions inside a finite region resulting form a given incoming wave from the outside, are determined by a linear relation between the normal gradient and the function values on the boundary. This boundary condition is non-local and we show how it can be used in conjunction with the variational principle applied to an open system.     

Contact zone assessment for a fast growing interface crack in an anisotropic bimaterial

Summary A plane strain problem for a crack with a frictionless contact zone at the leading crack tip expanding stationary along the interface of two anisotropic half-spaces with a subsonic speed under the action of various loadings is considered. The cases of finite and infinite-length interface cracks under the action of a moving concentrated loading at its faces are considered. A finite-length crack for a uniform mixed-mode loading at infinity is considered as well. The associated combined Dirichlet-Riemann boundary value problems are formulated and solved exactly for all above-mentioned cases. The expressions for stresses and the derivatives of the displacement jumps at the interface are presented in a closed analytical form for an arbitrary contact zone length. Transcendental equations are obtained for the determination of the real contact zone length, and the associated closed form asymptotic formulas are found for small values of this parameter. It is found that independently of the types of the crack and loading, an increase of the crack tip speed leads to an increase of the real contact zone length and the correspondent stress intensity factor. The latter increase significantly for an interface crack tip speed approaching the Ragleigh wave speed.     

多个共面任意分布表面裂纹的应力强度因子

采用线弹簧模型求解多个共面任意分布表面裂纹的应力强度因子。基于Ｒｅｉｓｓｎｅｒ板理论和连续分布位错思想,通过积分变换方法,将含有多个共面任意分布表面裂纹的无限平板问题归结为一组Ｃａｕｃｈｙ型奇异积分方程。利用Ｇａｕｓｓ－Ｇｈｅｂｙｓｈｅｖ笔法获得了奇异积分方程的数值解。为验证本文法的正确性,文中最后给出了有关应力强度因子或Ｐ－Ｖ曲线的数值结果并与现有的理论结果或实验结果进行了对比。结果表明了连续位     

An elastodynamic computational time-reversal method for shape reconstruction of traction-free scatterers

This study proposes an elastodynamic time-reversal imaging method for the shape reconstruction of flaws with the traction-free boundary. The proposed method is a generalization of the synthetic aperture focusing technique (SAFT) implemented with the aid of time-reversal back-propagation computation. The reconstruction formula is derived from the boundary condition applied on the flaw boundary without using a diffraction theory yielding an inherently normalized objective function. No restriction is imposed on the wave source or the material properties as long as the wave medium is linearly elastic and lossless. There are two major advantages with the proposed method. First, it is free from ray tracing. The method can thus work with an arbitrary source and the wave modes including inhomogeneous waves. Second, signal conditioning is not required to focus a scattered field to the correct location in the reconstructed image. Numerical examples are presented to demonstrate the performance and benefits of the proposed method. In the numerical examples, the shape of cavities and cracks in an elastic layer is reconstructed from synthetic data. The results show that the method works well with the reverberating incident fields in which inhomogeneous and body waves coexist. Also demonstrated is that the curvature of the cavities is resolved well at the scale of the incident wave length. Through the numerical examples, the method is shown to be a versatile imaging method potentially useful for ultrasonic nondestructive characterization of small flaws.     

On the derivation of an admissibility condition for phase boundary propagation in an SMA bar based on a 3-D formulation

There is some considerable difficulty in determining the solution uniquely for a propagating phase boundary in shape memory alloy (SMA) bar. In this paper, we establish an admissibility condition starting from a three-dimensional (3-D) internal-variable formulation to resolve this issue. We adopt a 3-D formulation in literature which is based on a constitutive model with specific forms of the Helmholtz free energy and dissipation rate. Then the 3-D dynamical equations are reduced to the 1-D rod equations for three phase regions (coupled with the radial effect and surface condition) by using two small parameters. Connection conditions at the phase interfaces are determined. By considering the traveling-wave solution for the rod system, we eventually derive three conditions across a sharp phase boundary corresponding to the 1-D sharp-interface model, including the two usual jump conditions and an additional condition. The third condition is then used to supplement the 1-D sharp-interface model to study an impact problem. The unique solution is constructed analytically for all possible impact velocity, including three kinds of wave patterns according to different levels of the impact velocity. The results are compared with those obtained by the maximal dissipation rate criterion.     

含单排周期微裂纹平板结构中的弹性波传播

受损伤固体中含有的微裂纹或微孔洞往往具有周期性，对含周期性缺陷结构中的弹性波分析是力学研究中的重要课题，它直接关系到结构的强度和使用寿命。目前对损伤固体中弹性波散射与透射研究结果主要是弹性动力学平面问题。1995年。Scarpetta和Sumbatyan采用解析法研究了平面波在双周期裂纹弹性介质中的传播问题。并推出显式分析结果。本文基于弹性动力学理论，分析研究了含有单排横向周期裂纹的平板中弯曲波的反射与透射问题。给出了含单排裂纹时反射波与透射波系数的数值结果。对于多排裂纹情况，可采用具有退化核第一类Fredholm积分方程方法分析求解，在求解中给出相应的无量纲数，例如无量纲波数、裂纹尺寸比等。本文分析结果可望能在工程振动控制中应用。