Local instability of a plate with a circular nanohole under uniaxial tension

The problem of the influence of surface tension on the local instability of a plate under tension is investigated by the example of an infinite plate with a circular nanohole. The method for solving the problem under consideration is based on a previously developed approach for solving problems on local buckling of a thin uniaxially stretched elastic sheet with variously shaped macroholes. The critical (Euler) load, at which the instability occurs, is sought by the Ritz method in framework of the linearized Karman set of equations based on the principle of virtual motions as the smallest positive load giving the minimum of the potential energy of the plate deformation. The generalized plane stress state of the plate with a nanohole is found for its application taking into account surface stresses. The linearized Gurtin-Murdoch relationships for the surface elasticity, the generalized Young-Laplace law, and the Goursat-Kolosov complex potentials are used when deriving analytical dependences for the stress in this plane problem. The deflection of the plate is represented by a double row satisfying the damping condition at infinity and ensuring freedom of motion of hole edges. Numerical calculations are performed for the aluminum plate. Two variants of the elastic properties of a circular hole are considered. Zones of compressing circumferential tensions near the edges of the circular hole with radii 1, 2, and 4 nm under uniaxial tension are constructed. It is shown that the account of the surface stresses decreases the critical load, which has a tendency to decrease in both cases with a decrease in the cut radius in a nanometer variation range.     

Scattering of elastic waves by multiple elastic circular cylinders with imperfect interface

In the current paper a general method is presented for the rigorous solution for the scattering of elastic waves by a cluster of elastic circular cylinders of infinite length. The interface separating the cylinder from the surrounding media is considered to be homogeneous imperfect. Specifically, the tractions are continuous but the displacements are discontinuous and proportional in terms of interface stiffness parameters to their respective traction components. Using the exact theory of multipole expansion, analytic solutions for the scattered and internal fields excited by an incident plane P-wave, an incident cylindrical P-wave and an incident plane SV-wave are derived.

Numerical results for directivity patterns and scattering cross-sections are presented for a finite hexagonal array of elastic circular inclusions with imperfect interface. The results show that the sequence of maxima and minima in the curves of scattered cross-sections becomes more undistinguishable as the interface becomes more imperfect. Also, the results reveal that large low-frequency peaks of the scattered cross-sections, which correspond to resonance scattering, can be observed for both the low-velocity and high-velocity elastic cylinders with extremely imperfect interface while the small high-frequency peaks of the scattered cross-sections can appear for low-velocity elastic cylinders with relatively perfect interface. Furthermore, the results clearly show that the interaction effects between cylinders cannot be ignored for an incident plane SV-wave as compared to an incident plane P-wave. More importantly is the fact that the reciprocity relations, which hold for elastic wave scattering by a single cylinder, no longer apply for elastic wave scattering by multiple cylinders.     

Scattering of elastic waves by multiple elastic circular cylinders with imperfect interface

In the current paper a general method is presented for the rigorous solution for the scattering of elastic waves by a cluster of elastic circular cylinders of infinite length. The interface separating the cylinder from the surrounding media is considered to be homogeneous imperfect. Specifically, the tractions are continuous but the displacements are discontinuous and proportional in terms of interface stiffness parameters to their respective traction components. Using the exact theory of multipole expansion, analytic solutions for the scattered and internal fields excited by an incident plane P-wave, an incident cylindrical P-wave and an incident plane SV-wave are derived.

Numerical results for directivity patterns and scattering cross-sections are presented for a finite hexagonal array of elastic circular inclusions with imperfect interface. The results show that the sequence of maxima and minima in the curves of scattered cross-sections becomes more undistinguishable as the interface becomes more imperfect. Also, the results reveal that large low-frequency peaks of the scattered cross-sections, which correspond to resonance scattering, can be observed for both the low-velocity and high-velocity elastic cylinders with extremely imperfect interface while the small high-frequency peaks of the scattered cross-sections can appear for low-velocity elastic cylinders with relatively perfect interface. Furthermore, the results clearly show that the interaction effects between cylinders cannot be ignored for an incident plane SV-wave as compared to an incident plane P-wave. More importantly is the fact that the reciprocity relations, which hold for elastic wave scattering by a single cylinder, no longer apply for elastic wave scattering by multiple cylinders.     

Reconstruction of a complex-shaped defect from the known arrival time of the reflected ultrasonic wave

The problem of reconstructing the boundary of an arbitrarily shaped defect formed inside an elastic body from the measured time of arrival of the reflected ultrasonic wave in the echo method is considered. The characteristic size of the defect is assumed to be greater than the wavelength, and the defect is irradiated from the far-field zone, which means that the incident wave can be considered as plane. An algorithm is developed for reconstructing the convex envelope of a nonconvex defect from the arrival times of echo signals measured at different angles with the use of circular scanning.     

Symmetric and Anti-Symmetric Lamb Waves in a Two-Dimensional Phononic Crystal Plate

It is weft known that Lamb waves in a plate with a mirror plane can be separated into two uncoupled sets： symmetric and anti-symmetric modes. Based on this property, we present a revised plane wave expansion method （PWE） to calculate the band structure of a phononie crystal （PC） plate with a mirror plane. The developed PWE method can be used to calculate the band structure of symmetric and anti-symmetric modes separately, by which the depending relationship between the partial acoustic band gap （PABG）, which belongs to the symmetric and anti-symmetric modes alternatively, and the position of the scatterers can be determined. As an example of its application, the band structure of the Lamb modes in a two-dimensional PC plate with two layers of void circular inclusions is investigated. The results show that the band structure for the symmetric and anti-symmetric modes can be changed by the position of the scatterers drastically, and larger PABGs will be opened when the scatterers are inserted into the area of the plate, where the elastic potential energy is concentrated.     

Transverse vibrations of thin,elastic plates with concentrated masses and internal elastic supports

The fundamental frequency of vibration of a plate carrying concentrated masses and with internal elastic supports is determined. The case of an orthotropic, rectangular plate elastically restrained against rotation along the four edges is tackled first by using simple polynomial approximations and the Galerkin method. Then, vibrations of clamped and simply supported isotropic plates of regular polygonal shape are studied by using the conformal mapping technique coupled with the variational method. Finally the case of a circular plate elastically restrained against translation and rotation is considered.     

Holographic determination of the residual strength of arbitrarily clamped, centrally thinned circular plates

In holographic inspection, the perturbed fringe patterns which reveal internal defects in components are generally distinct. However, in the case of a uniformly loaded circular flat plate containing a central circular defect, distinct fringe perturbation is not visually obvious except through quantitative analysis of the fringe pattern. During the analysis, ambiguities frequently arise in correctly assigning the fringe order, which is often fractional, at the centre of the plate. With the use of carrier fringes, accurate fringe orders at the central region of the plate can be determined. These, together with an iterative procedure, enable accurate estimation of both the size and depth of the defect. As the present method also determines the actual clamping condition along the plate boundary, the use of the von Mises yield criterion enables estimation of the residual strength, expressed in terms of the proportional reduction in the allowable pressure which can be applied to the defective plate.     

Eshelby's tensor fields and effective conductivity of composites made of anisotropic phases with Kapitza's interface thermal resistance

Eshelby's results and formalism for an elastic circular or spherical inhomogeneity embedded in an elastic infinite matrix are extended to the thermal conduction phenomenon with a Kapitza interface thermal resistance between matrix and inclusions. Closed-form expressions are derived for the generalized Eshelby interior and exterior conduction tensor fields and localization tensor fields in the case where the matrix and inclusion phases have the most general anisotropy. Unlike the relevant results in elasticity, the generalized Eshelby conduction tensor fields and localization tensor fields inside circular and spherical inhomogeneities are shown to remain uniform even in the presence of Kapitza's interface thermal resistance. With the help of these results, the size-dependent overall thermal conduction properties of composites are estimated by using the dilute, Mori–Tanaka, self-consistent and generalized self-consistent models. The analytical estimates are finally compared with numerical results delivered by the finite element method. The approach elaborated and results provided by the present work are directly applicable to other physically analogous transport phenomena, such as electric conduction, dielectrics, magnetism, diffusion and flow in porous media, and to the mathematically identical phenomenon of anti-plane elasticity.     

相位失配弹性平板复合波导中的缺陷态*

为了在弹性波波导中实现缺陷态的调控,该文基于相位失配原理设计了一种周期性平板波导结构。以正弦边界弹性波导为例,通过连接具有不同相位的两段波导结构,形成了弹性板中不同程度的缺陷,并分析了其谱带特性和能量局域化特征。结果表明,禁带中存在两个不同模式的缺陷态,其以透射峰形式出现并随着相位的改变产生频移。与此同时,两个缺陷态在空间上对应的应力场和位移场分布也具有不同的模态特征。该文提出的复合弹性波导缺陷态调控方法,不仅为研究弹性波与结构之间的内在联系提供了关键的理论支持,也为实际弹性波探测器件的设计提供重要参考。     

金属薄板弹性各向异性检测

当平面声波入射到液体中的金属薄板上时，透射系数将随声波入射角变化，根据这一规律，能够准确地测定声波入射面与金属薄板交线方向上金属板材料的弹性常数，测得金属薄板面内不同方向的弹性常数，对金属薄板弹性各向异性情况将有一清蜥的认识，由于测试过程采用微机控制、数据采集和处理，实现了各向异性测量的全自动化，该方法测试时间短、所需样品小且加工方便，并可测试金属薄板弹性各向异性的均匀度。     

Periodic geometrically nonlinear free vibrations of circular plates

The geometrically nonlinear free vibrations of thin isotropic circular plates are investigated using a multi-degree-of-freedom model, which is based on thin plate theory and on Von Kármán's nonlinear strain-displacement relations. The middle plane in-plane displacements are included in the formulation and the common axisymmetry restriction is not imposed. The equations of motion are derived by the principle of the virtual work and an approximated model is achieved by assuming that the in-plane and transverse displacement fields are given by weighted series of spatial functions. These spatial functions are based on hierarchical sets of polynomials, which have been successfully used in p-version finite elements for beams and rectangular plates, and on trigonometric functions. Employing the harmonic balance method, the differential equations of motion are converted into a nonlinear algebraic form and then solved by a continuation method. Convergence with the number of shape functions and of harmonics is analysed. The numerical results obtained are presented and compared with available published results; it is shown that the hierarchical sets of functions provide good results with a small number of degrees of freedom. Internal resonances are found and the ensuing multimodal oscillations are described.     

On Virtual Phonons,Photons, and Electrons

A macroscopic realization of the peculiar virtual particles is presented. The classical Helmholtz and the Schrödinger equations are differential equations of the same mathematical structure. The solutions with an imaginary wave number are called evanescent modes in the case of elastic and electromagnetic fields. In the case of non-relativistic quantum mechanical fields they are called tunneling solutions. The imaginary wave numbers point to strange consequences: The waves are non-local, they are not observable, and they are described as virtual particles. During the last two decades QED calculations of the solutions with an imaginary wave number have been experimentally confirmed for phonons, photons, and electrons. The experimental proofs of the predictions of non-relativistic quantum mechanics and the Wigner phase time approach for the elastic, electromagnetic and Schrödinger fields will be presented in this article. The results are zero time in the barrier and an interaction time (i.e. a phase shift) at the barrier interfaces. The measured tunneling time scales approximately inversely with the particle energy. Actually, the tunneling time is given only by the barrier boundary interaction time, as zero time is spent inside a barrier.     

Emission of stress waves during fracture

Emission of both longitudinal and surface (Rayleigh) waves during fracture of plates under conditions of plane stress and plane strain were studied experimentally. The non-equilibrated tensile stress in the fractured section of the plate creates an elastic wave, which travels radially along the plate at the sound speed. Moreover, the high surface deformation around the crack tip, due to the high stress concentration there, propagates as a surface wave following fracture of this zone, at the respective Rayleigh wave speed with a circular wavefront. The influence of the thickness of the plate and the type of fracture (brittle or ductile) was examined and interesting results were derived, by utilizing a high speed photography technique.     

Defect detection and localization in orthotropic wood slabs by inversion of dynamic surface displacements

The nondestructive evaluation inversion and generalized force-mapping techniques developed and demonstrated for isotropic thin plates by Bucaro et al. [(2004). "Detection and localization of inclusions in plates using inversion of point actuated surface displacements," J. Acoust. Soc. Am. 115, 201-206] are extended to the case of orthotropic plates. The extended techniques are applied to a finite-element generated numerical database for point excited wooden slabs with and without an internal defect at 5 and 10 kHz. Operation of the original isotropic algorithms on the wood surface displacements is shown to fail in recovering the uniform elastic parameters or in detecting and locating the defect. The new algorithms based on the wave equation for a thin, orthotropic plate successfully convert the surface displacements on the uniform wooden slab to elastic parameter maps which serve to detect and localize the defect in the flawed plate. The results, particularly at the higher frequency, indicate that the onset of failure in the thin plate approximation impacts both the inversion and the generalized force-mapping accuracy. However, in this case use of the inversion algorithm to obtain modified wave equation coefficients followed by operation of the force-mapping algorithm with these new parameters inserted is shown to successfully mitigate this effect.     

Axisymmetric vibrations of a circular plate of linearly varying thickness on an elastic foundation according to Mindlin theory

Free axisymmetric vibrations of an elastic circular plate of linearly varying thickness on an elastic foundation have been studied on the basis of shear theory [1,2]. The transverse displacement and local rotation are expressed as an infinite series. The frequencies corresponding to the first two modes of vibrations are obtained for a circular plate with clamped and simply supported edge conditions for various values of the taper constant and the foundation modulus. The results have been compared with those of reference [3].     

Instability of interstitial dislocation loops in electron-irradiated dielectrics

The experiments on electron irradiation of yttrium-stabilized zirconium oxide samples show the formation of strong elastic fields near interstitial dislocation loops. The fields increase with an increase in the loop radius and, when the loop radius reaches a certain critical value, the loops became unstable due to the beginning of plastic deformation and the formation of a dislocation network. The mechanism of the occurrence of this instability is suggested. It is based on the accumulation of charges at dislocation loops due to ionization processes in an electron-irradiated dielectric. It is shown that the accumulation of the electric charge at growing dislocation loops in dielectrics may be responsible for an increase in elastic stresses near dislocation loops and for their instability because of the beginning of plastic deformation near the loops when stresses at growing loops become close to the theoretical yield stress of the material.     

弯管对末端带弹性障板充液管路辐射声能量的影响

基于声固耦合有限元方法建立了末端带弹性障板的充液管路数值模型,重点分析了不同激励下弯管对管口辐射声能量的影响.结果表明:弯管引入的高阶周向模式耦合使结构振动和流体声传播都发生明显改变,以致系统辐射声能量及主要能量贡献源也发生转移,并随激励方式和频率而不同.对本文管路模型,平面波激励下弯管系统在低频的结构辐射声能量明显增...     

Propagation in an elastic wedge using the virtual source technique

The virtual source technique, which is based on the boundary integral method, provides the means to impose boundary conditions on arbitrarily shaped boundaries by replacing them by a collection of sources whose amplitudes are determined from the boundary conditions. In this paper the virtual source technique is used to model propagation of waves in a range-dependent ocean overlying an elastic bottom with arbitrarily shaped ocean-bottom interface. The method is applied to propagation in an elastic Pekeris waveguide, an acoustic wedge, and an elastic wedge. In the case of propagation in an elastic Pekeris waveguide, the results agree very well with those obtained from the wavenumber integral technique, as they do with the solution of the parabolic equation (PE) technique in the case of propagation in an acoustic wedge. The results for propagation in an elastic wedge qualitatively agree with those obtained from an elastic PE solution.     

Elastic instability of grain boundaries and the physical origin of superplasticity

Matter between contiguous crystallites is assimilated to a thin elastic plate immersed in a different elastic medium. It is shown that a shear stress exceeding a critical value should corrugate the boundary and induce periodic normal stress fields in the two adjacent crystal surfaces, which cause motion of vacancies in closed loops between the two crystals. The consequent cyclic transport of atoms in the opposite sense determines crystal sliding at a temperature dependent relative speed. Most of the phenomenology of superplastic allows follows in a quantitative manner.     

Verzerrung anisotroper Körper durch Versetzungen mit Anwendung auf die Berechnung der Röntgenintensität

An elastic state in arbitrarily anisotropic cylinders is considered for the case in which the displacement does not vary in the direction of the axis. The first and second boundary-value problem are solved for the half-plane and the infinite plane with an elliptic hole as cross sections of the cylinders. With the aid of these solutions one can treat further domains by the method of successive approximation. For a general cross section the problem leads to a system of integral equations. The strain fields about straight dislocation lines parallel to the surface (which may be free from applied stress) are special cases of the treated state of deformation. The solution is given for 1) dislocations in a half-space, 2) dislocations in an infinite plate (through a recursion formula), 3) a dislocation in an infinite solid out of which an elliptic cylinder is cut containing the dislocation line. — From the strain one obtains a formula for the intensity distribution of X-rays reflected under certain conditions by a dislocation parallel to the surface of a half-space. Numerical results are given for two dislocations in Germanium.