Two-dimensional transient dynamic response of orthotropic layered media

In this study, two-dimensional transient dynamic response of orthotropic plane layered media is investigated. The plane multilayered media consist of N different generally orthotropic, homogeneous and linearly elastic layers with different ply angles. In the generally orthotropic layer, representing a ply reinforced by unidirectional fibers with an arbitrary orientation angle, the principal material directions do not coincide with body coordinate axes. The solution is obtained by employing a numerical technique which combines the use of Fourier transform with the method of characteristics. The numerical results are displayed in curves denoting the variations of stress and displacement components with time at different locations. These curves clearly reveal, in wave profiles, the scattering effects caused by the reflections and refractions of waves at the boundaries and at the interfaces of the layers, and also the effects of anisotropy caused by fiber orientation angle. The curves properly predict the sharp variations in the response at the neighborhood of the wave fronts, which shows the power of the numerical technique employed in the study. By suitably adjusting the elastic constants, the results for multilayered media with transversely isotropic layers, or layers with cubic symmetry, or isotropic layers can easily be obtained from the general formulation. Furthermore, solutions for some special cases, including Lamb’s problem for an elastic half-space, are obtained and compared with the available solutions in the literature and very good agreement is found. Preliminary version presented at the Second International Congress on Mechatronics (MECH2K3), Graz, Austria, July 14-17, 2003.     

Holographic study of propagating transverse waves in plates

The technique of double-exposure holographic interferometry was used to record the propagation of transverse waves in a plate. The waves were initiated by striking the plate in the center with a ballistic pendulum; this technique produced axisymmetric response of the plate. Several pulsed-laser interferograms were obtained, each of which showed the bending wave at a different time after the initiation of impact. From the interferograms, plots of the transverse displacement vs. distance from the center of impact were obtained. The experimental data compared favorably with analytical results for the problem.     

Degenerate weakly non-linear elastic plane waves

Weakly non-linear plane waves are considered in hyperelastic crystals. Evolution equations are derived at a quadratically non-linear level for the amplitudes of quasi-longitudinal and quasi-transverse waves propagating in arbitrary anisotropic media. The form of the equations obtained depends upon the direction of propagation relative to the crystal axes. A single equation is found for all propagation directions for quasi-longitudinal waves, but a pair of coupled equations occurs for quasi-transverse waves propagating along directions of degeneracy, or acoustic axes. The coupled equations involve four material parameters but they simplify if the wave propagates along an axis of material symmetry. Thus, only two parameters arise for propagation along an axis of twofold symmetry, and one for a threefold axis. The transverse wave equations decouple if the axis is fourfold or higher. In the absence of a symmetry axis it is possible that the evolution equations of the quasi-transverse waves decouple if the third-order elastic moduli satisfy a certain identity. The theoretical results are illustrated with explicit examples.     

Implicit and explicit secular equations for Rayleigh waves in two-dimensional anisotropic media

This paper is concerned with the derivation of implicit and explicit secular equations for Rayleigh waves polarized in a plane of symmetry of an anisotropic linear elastic medium. It has been confirmed, in accord with Ting’s paper [2], that the Rayleigh waves propagate with no geometric dispersion. Numerical evaluations of both the implicit and explicit equations give the same values of Rayleigh wave velocities. In the case of orthotropic material (thin composites) it has been found that Rayleigh wave velocity depends significantly (as with bulk waves) on the directions of principal material axes. For the same material model the analytical solutions, based on implicit and explicit secular equations, were compared against the finite element and experimental data that had been published by Cerv et al. [4] in 2010. It emerged that the theory was in accordance with the experiment.     

Transient bending waves in plates studied by hologram interferometry

Propagating bending waves are studied in plates made of aluminum and wood. The waves are generated by the impact of a ballistic pendulum. Hologram interferometry, with a double pulsed ruby laser as the light source, is used to record the out of plane motion of the waves. Elliptic-like fringes visualize differences in wave speed for different directions in the anisotropic plate and circular ones are obtained for the isotropic plate. The experimental data for the isotropic plate compare favorably with analytical results derived from the Kirchhoff-plate equation with a point impact of finite duration. A similarity variable is found when starting conditions are modeled as a Dirac pulse in space and time, that brings new understanding to the importance of specific parameters for wave propagation in plates. A formal solution is obtained for a point force with an arbitrary time dependence. For times much larger than the contact time, the plate deflection is shown to be identical to that from a Dirac pulse applied at the mean contact time. A method for determining material parameters, and the mean contact time, from the interferograms is hence developed.     

Diffraction of stress waves by a free or reinforced hole in an orthotropic plate

Photoelastic plates made of an orthotropic material are used to model the dynamic stress state near free and reinforced circular holes under blast loading. The diffraction of stress waves by holes in a thin-walled plate is studied. Experimental data are used to analyze the dynamic stress concentration in a plate with a hole in which quasilongitudinal and quasitransverse waves propagate __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 7, pp. 73–78, July 2007.     

Near-Boundary Stress Wave Field in Anisotropic Plates under Impulsive Load

The stress wave field and the behavior of waves near the free boundary in an orthotropic plate and at the interface between two anisotropic media are studied. The results presented were obtained using the dynamic photoelastic method and optically sensitive fibrous models. Experimental data for impulsively loaded plates with various boundary conditions are analyzed     

Triangular grid method for stress-wave propagation in 2-D orthotropic materials

A triangular grid method is presented to calculate propagation problems of elastic stress waves in 2-D orthotropic materials. This method is based on the dynamic equilibrium equations of the computational cells formed among the auxiliary triangular grids. The solution is obtained by calculating alternately the nodal displacements and the central point stresses of the spatial grids. The numerical results are compared with the corresponding solutions of the finite element method. Comparisons show that the triangular grid method yields a higher calculational speed than the finite element method. The stress concentrations are investigated from wave-field analyses when the stress wave propagates within an orthotropic plate with a hole. Finally, the presented numerical method is used to study the features of wave propagation and diffraction in a square orthotropic plate with a hole when an impact load is applied to the top of the plate.This work was supported by National Natural Science Foundation of China (Nos. 10025212 and 10232040) and Natural Science Foundation of Liaoning province (No. 20021070).     

Photoelastic stress analysis of orthtropic materials by using an isotropic plate

Based on the two-dimensional theory of elasticity for an orthotropic plate, relations between stress components produced in two different orthotropic plates are considered and the conditions to realize the similar stress fields in different orthotropic plates are studied. On the basis of the similarity law, a convenient photoelastic method to analyze stress fields in an orthotropic plate, using an isotropic plate, is presented. Two examples are treated. One deals with the stress-concentration problem around a circular hole in a strip with edges parallel to the symmetric axis of elasticity. In the second example, the edges of the strip are assumed to be inclined by 30 deg to the elastically symmetric axes. The experimental results are compared with theoretical calculations. Paper was presented at the SEM VII International Congress on Experimental Mechanics held in Las Vegas, NV on June 8–11, 1992.     

Holographic interferometry measurements of transient bending waves in tubes and rings

Propagating bending waves are studied in a tube of steel and in a ring of aluminum. The waves are generated by the impact of a ballistic pendulum. Holographic interferometry, with a double-pulsed ruby laser as light source, is used to record the waves. A conical mirror is placed axially inside the tube. Axial illumination and axial observation directions, make it possible to view all sides of the tube simultaneously with a high sensitivity to radial deformation. The interferograms, which have an unusual perspective, are captured with a CCD-camera and then spatially transformed into an unwrapped strip of the tube wall. This makes the interpretation of the measurements simpler. The geometry of the tube causes the wave pattern to propagate with different speed and amplitude along and across the tube, even when the material itself is isotropic. A finite-element simulation of the impact is compared to the corresponding experiment. An impact on a ring with a defect is performed in order to study the effect on the wave pattern. The proposed method could be used in nondestructive testing of pipes.     

Wave propagation in layer with two preferred directions

This article is concerned with overall or macroscopic properties of a composite material with no distinction made between the fibres and the matrix which they are embedded in. All the properties with dimensions larger than the fibre diameter and spacing are regarded as averaged over a volume of material. The systems of particular interest here are in the fibre reinforced composites with the fibres being very much stiffer and stronger than the matrix.Laminated plates of fibre-reinforced material are often fabricated from prepreg tapes, laid up according to some specific arrangement of fibre orientation and then bonded together. An angle-ply laminate is formed by alternating plies so that the families in adjacent laminas are inclined by angle ϕ and −ϕ to given direction alternately. The process of fabricating a multilayered plate of this material gives rise to a laminate in which the plies are separated by resin rich layer, and when this layer is thin enough that its thickness is negligible it may be regarded as plate reinforced by two families of fibres. Problems shall be considered in three dimensions, but attention shall be restricted to linear elasticity theory. The plate under consideration is reinforced by two mechanically equivalent families of fibres, but with no other preferred directions, so that it is locally orthotropic with respect to the plane of the fibres and to the two planes that orthogonally bisect the fibres.In this article linear elastic stress–strain relation is employed to derive dispersion curves for plane harmonic waves propagating in a plate of finite thickness but of infinite lateral extent. Attention is restricted to waves propagating in the plane parallel to stress free plate faces where waves travelling at any angle to one of the families of very strong fibres are examined. The dispersion equations, relating the phase velocity to the wavelength, are obtained. The fundamental modes are examined for symmetric as well as for anti-symmetric deformations. This leads to full understanding of displacement field as well as stress field.     

Propagation of thickness-twist waves in a multi-sectioned piezoelectric plate of 6 mm crystals

We study thickness-twist vibrations and waves in an unbounded, multi-sectioned piezoelectric plate of crystals with 6 mm symmetry or polarized ceramics. An exact solution from the three-dimensional equations of piezoelectricity is obtained. Basic vibration and wave propagation characteristics are calculated based on the solution. The results are useful in the understanding and design of plate resonators, filters and acoustic wave sensors of ZnO, AlN and polarized ceramics.     

SH-type waves dispersion in an isotropic medium sandwiched between an initially stressed orthotropic and heterogeneous semi-infinite media

The present paper studies the propagation of shear waves (SH-type waves) in an homogeneous isotropic medium sandwiched between two semi infinite media. The upper half-space is considered as orthotropic medium under initial stress and lower half-space considered as heterogeneous medium. We have obtained the dispersion equation of phase velocity for SH-type waves. The propagation of SH-type waves are influenced by inhomogeneity parameters and initial stress parameter. The velocity of SH-type wave has been computed for different cases. We have also obtained the dispersion equation of phase velocity in homogeneous media in the absence of initial stress. The velocities of SH-type waves are calculated numerically as a function of kH (non-dimensional wave number) and presented in a number of graphs. To study the effect of inhomogeneity parameters and initial stress parameter we have plotted the velocity of SH-type wave in several figure. We have observed that the velocity of wave increases with the increase inhomogeneity parameters. We found that in both homogeneous and inhomogeneous media the velocity of SH-type wave increases with the increase of initial stress parameter. The results may be useful for the study of seismic waves propagation during any earthquake and artificial explosions.     

各向异性平板开孔动应力集中问题的研究

采用各向异性平板弯曲波动理论及摄动方法，对正交各向异性平板开孔弯曲波的散射及动应力集中问题进行了分析研究，得到了此种平板稳态弯曲波动问题的渐近形式的分析解。同时采用保角映射技术，为求解正交各向异性平板开孔弹性波的散射及动应力集中问题提供了一种统一规范的方法。     

Nonlinear stability of a shallow unsymmetrical heated orthotropic sandwich shell of double curvature with orthotropic core

Equations which govern the behavior of an elastic unsymmetrical, orthotropic sandwich shell of double curvature with orthotropic core having different elastic characteristics under uniform heating are derived. The face sheet may be of unequal thickness of different materials. However, a restriction that the radii of curvature of the shell elements be large compared with the overall thickness of the sandwich shell is imposed. The variational procedure has been used to obtain the five equations which govern the behavior of the heated orthotropic sandwich shell for the stability. In case of symmetry the equations resemble with those of Grigolyuk. Finally, the numerical results of a square or a rectangular simply supported curved plate section of a cylindrical shell under thermal loading have been computed and compared with other known results. The graphs have been drawn to show the effects of different sandwich material for immovable and movable edge conditions.     

Interfacial imperfection on reflection and transmission of plane waves in anisotropic micropolar media

This study is concerned with the reflection and transmission of plane waves at an imperfectly bonded interface between two orthotropic micropolar elastic half-spaces with different elastic and micropolar properties. There exist three types of coupled waves in xy-plane. The reflection and transmission coefficients of quasi-longitudinal (QLD) wave, quasi-coupled transverse microrotational (QCTM) wave and quasi-coupled transverse displacement (QCTD) wave have been derived for different incidence waves and deduced for normal force stiffness, transverse force stiffness, transverse couple stiffness and perfect bonding. The numerical values of modules of the reflection and transmission coefficients are presented graphically with the angle of incidence for orthotropic micropolar medium (MOS) and isotropic micrpolar medium (MIS). Some particular cases of interest have been deduced from the present investigation.     

Elastodynamic behavior of laminated orthotropic plates under initial stress

A finite element method of analysis of the vibrational and wave propagational characteristics is presented for a laminated orthotropic plate under initial stress. The plate may have an arbitrary number of bonded elastic orthotropic layers, each with distinct thickness, density and mechanical properties, and the analysis is capable of treating a completely arbitrary three-dimensional state of initial stress. Biot's theory for incremental elastic deformations of a stressed solid forms the basis for this study. A homogeneous, isotropic plate under two different states of initial stress was analyzed and their numerical results showed excellent correlation with those from an exact solution. Further examples of a three layer composite plate and a sandwich plate are offered to add some general insight to the physical behavior of such plates.     

Ultrasonic scattering in polycrystals with orientation clusters of orthorhombic crystallites

Scattering-induced ultrasonic attenuation and backscattering in a polycrystalline medium with orientation clusters composed of orthotropic crystallites are studied theoretically, aiming to improve understanding of ultrasonic wave interaction with such clustered microstructures for application to the modeling of titanium alloys. Both orthorhombic crystallites and their arrangements in orientation clusters (also termed microtexture regions, MTRs) are of general ellipsoidal shape. The preferred orientation of orthotropic crystallites in the clusters is represented by a generalized Gaussian orientation distribution function with three independent texture parameters. The effective elastic properties of the clusters, which have in general orthorhombic symmetry, are determined by a volume average of elastic constants weighted by orientation distribution functions and then used to obtain the cluster-scattering-induced attenuation and backscattering in the polycrystalline medium. In the model the wave propagation direction is arbitrary relative to the ellipsoidal axes of the clusters. The contribution of crystallite-scattering-induced attenuation is estimated by the untextured attenuation coefficient factored by a texture transition function. The total attenuation and backscattering are determined by combining scattering by the clusters and crystallites. Drastic effects of clustering and the transition to unclustered polycrystals are demonstrated. Reasonable agreement is observed between the model’s prediction and measurements on two Ti-alloy samples with different crystallite clustering.     

Natural harmonic oscillations of a heavy fluid in basins of complex shape

The natural harmonic oscillations of a heavy fluid in uniform-depth basins of complex shape, including those with three or more symmetry axes, are investigated within the shallow-water approximation. The waves are assumed to be gently sloping. The modes found are compared with the similar modes in an elliptic basin representing the class of basins with two symmetry axes. The mode characteristics associated with the number of basin symmetry axes and the basin shape are explored. Basins with two symmetry axes whose shape differs considerably from the elliptic, in particular, nonconvex basins, are considered. Both rotating and non-rotating basins are studied. The possibility of approximating the amplitudes of certain rotating-basin mode classes by Bessel functions is discussed.