Wave propagation in transversely isotropic porous piezoelectric materials

Wave propagation in porous piezoelectric material (PPM), having crystal symmetry 6 mm, is studied analytically. Christoffel equation is derived for the propagation of plane harmonic waves in such a medium. The roots of this equation give four complex wave velocities which can propagate in such materials. The phase velocities of propagation and the attenuation quality factors of all these waves are described in terms of complex wave velocities. Phase velocities and attenuation of the waves in PPM depend on the phase direction. Numerical results are computed for the PPM BaTiO₃. The variation of phase velocity and attenuation quality factor with phase direction, porosity and the wave frequency is studied. The effects of anisotropy and piezoelectric coupling are also studied. The phase velocities of two quasi dilatational waves and one quasi shear waves get affected due to piezoelectric coupling while that of type 2 quasi shear wave remain unaffected. The phase velocities of all the four waves show non-dispersive behavior after certain critical high frequency. The phase velocity of all waves decreases with porosity while attenuation of respective waves increases with porosity of the medium. The characteristic curves, including slowness curves, velocity curves, and the attenuation curves, are also studied in this paper.     

Wave propagation in an incompressible transversely isotropic fibre-reinforced elastic media

The boundary conditions at free surface of an incompressible, transversely isotropic elastic half-space are satisfied to obtain the reflection coefficients for the case when outer slowness section is re-entrant. Two quasi-shear waves will be reflected for an angular range of direction of incident wave. The numerical illustrations of reflection coefficients are presented graphically for three arbitrary materials.     

Wave propagation in a transversely isotropic solid cylinder of arbitrary cross-sections immersed in fluid

The wave propagation in an infinite, transversely isotropic solid cylinder of arbitrary cross-section immersed in fluid is studied using the Fourier expansion collocation method, within the framework of the linearized, three-dimensional theory of elasticity. The equations of motion of solid and fluid are respectively formulated using the constitutive equations of a transversely isotropic cylinder and the constitutive equation of an inviscid fluid. Three displacement potential functions are introduced to uncouple the equations of motion along the radial, circumferential and axial directions. The frequency equations of longitudinal and flexural (symmetric and antisymmetric) modes are analyzed numerically for an elliptic and cardioidal cross-sectional transversely isotropic solid cylinder of arbitrary cross-section immersed in fluid. The computed non-dimensional wavenumbers are presented in the form of dispersion curves for the material zinc. The general theory can be used to study any kind of cylinder with proper geometric relations.     

Wave propagation in an inhomogeneous transversely isotropic material obeying the generalized power law model

The analytical solutions for body-wave velocity in a continuously inhomogeneous transversely isotropic material, in which Young’s moduli (E, E′), shear modulus (G′), and material density (ρ) change according to the generalized power law model, (a+b z) ^c , are set down. The remaining elastic constants of transversely isotropic media, ν, and ν′ are assumed to be constants throughout the depth. The planes of transversely isotropy are selected to be parallel to the horizontal surface. The generalized Hooke’s law, strain-displacement relationships, and equilibrium equations are integrated to constitute the governing equations. In these equations, utilizing the displacement components as fundamental variables, the solutions of three quasi-wave velocities (V _SV , V _P ,?V _SH ) are generated for the present inhomogeneous transversely isotropic materials. The proposed solutions are compared with those of Daley and Hron (Bull Seismol Soc Am 67:661–675, (1977)), and Levin (Geophysics 44:918–936, (1979)) when the inhomogeneity parameter c?=?0. The agreement between the present results and previously published ones is excellent. In addition, the parametric study results reveal that the magnitudes of wave velocity are remarkably affected by (1) the inhomogeneity parameters (a, b, c); (2) the type and degree of material anisotropy (E/E′, ν/ν′, G/G′); (3) the phase angle (θ); and (4) the depth of the medium (z). Consequently, it is imperative to consider the effects of inhomogeneity when investigating wave propagation in transversely isotropic media.     

The axisymmetric end problem for transversely isotropic circular cylinders

Energy-decay inequalities are applied in investigating the decay of end effects in a transversely isotropic circular cylinder subject to torsionless axisymmetric end loads. A lower bound (in terms of the elastic constants) is obtained for the rate of exponential decay of stresses and this is compared with results of other authors. For a highly anisotropic medium, a slow decay rate is predicted thus anticipating disagreement with Saint-Venant's principle in this case.     

Axisymmetric wave propagation in a layered transversely isotropic medium

Summary In this paper, the method of numerical integration along bicharacteristics is generalized to the case of layered transversely isotropic medium for analysing the axisymmetric stress wave propagation. The stability of the present scheme is studied. The advantages and limitations of the method are discussed. Received 12 June 1996; accepted for publication 6 May 1997     

Transient torsional wave propagation in a transversely isotropic tube

Summary By use of the separation of variables method and the Laplace transformation, the two-dimensional transient torsional wave propagation problem in a transversely isotropic tube is studied when a torque is suddenly applied to its end surface. The results show that, for the discontinuous distribution of the impact shear stress, the region of the 2D stress distribution in a transversely isotropic tube becomes large with the increase of the anisotropy of the material. Received 13 June 1997; accepted for publication 17 June 1998     

Wave propagation in transversely impacted composite laminates

An experimental investigation was conducted to determine wave-propagation characteristics, transient-strain distributions and residual properties for unidirectional and angle-ply boron/epoxy and graphite/epoxy laminates impacted with silicon-rubber projectiles at velocities up to 250 ms^?1 (820 ft/s). Tests were conducted at normal and 45-deg oblique impact. Strain signals obtained from surface and embedded strain gages were recorded and analyzed to determine the types of waves, propagation velocities, peak strains, strain rates and attenuation characteristics. The predominant wave is a flexural on propagating at different velocities in different directions. The flexural wave velocity is higher in the higher-modulus direction. In general, measured wave velocities were higher than theoretically predicted. The amplitude of the in-plane wave is less than ten percent of that of the flexural wave. Peak strains and strain rates in the transverse to the (outer) fiber direction are much higher than those in the direction of the fibers. Strain rates up to 640 s^?1 were measured. Under oblique 45-deg impact, the flexural wave is still the predominant one. Peak strains under this oblique impact range between 36 and 56 percent of those under normal impact of the same velocity. Residual elastic properties and strength were measured around the point of impact. The most significant result was a reduction in the transverse strength of the unidirectional laminates. The dynamics of impact were also studied with high-speed photography. The projectile is completely flattened within 50–70 μs and the total contact time is of the order of 300 μs.     

Study of stress concentration in short hollow cylinders of transversely isotropic materials

The axisymmetric stress state of hollow cylinders made of tranversely isotropic materials with a longitudinal axis of isotropy is examined in a three-dimesnional formulation. An analytical solution is obtained by using homogeneous solutions written in Kosmodamianskii's form. Results of calculations of stress concentration are given for different materials as a function of the relative thickness and relative external radius of the cylinder. Donetsk State University, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 7, pp. 43–48, July, 1999.     

多铁性扇环截面柱体中的波传播

论文基于线性磁电弹性理论,研究了具有扇环形截面的多铁性柱形波导中的弹性波传播问题.利用波动势函数法,解析推导获得波动特征方程,进而得到弥散关系.通过算例研究了波传播的关键特性,深入分析了弥散曲线、相速度曲线和截止频率变化情况.结果显示,波的相速度和截止频率非常依赖于波导结构的扇环截面半顶角、内外径比和层合界面的弱界面系数,对于给定材料的波导结构,这些参数也是控制其弥散特性的重要影响因素.值得指出的是,在柱面应力自由的边界条件下相速度曲线中存在独特的频率带隙,而这通常是在周期结构中才会出现.     

Characteristic analysis for stress wave propagation in transversely isotropic fluid-saturated porous media

According to generalized characteristic theory, a characteristic analysis for stress wave propagation in transversely isotropic fluid-saturated porous media was performed. The characteristic differential equations and compatibility relations along bicharacteristics were deduced and the analytical expressions for wave surfaces were obtained. The characteristic and shapes of the velocity surfaces and wave surfaces in the transversely isotropic fluid-saturated porous media were discussed in detail. The results also show that the characteristic equations for stress waves in pure solids are particular cases of the characteristic equations for fluid-saturated porous media.     

A numerical solution of transient torsional wave propagation in a transversely isotropic cylinder

Summary A numerical technique is proposed to obtain the solution for the transient torsional wave propagation in a transversely isotropic cylinder when a torque is suddenly applied to its end surface. The dynamic behavior in the vicinity of the impact end of the transversely isotropic cylinder is studied. The stability and convergence of the present numerical technique and the validity of the numerical solution are examined.     

Finite and transversely isotropic elastic cylinders under compression with end constraint induced by friction

This paper derives an exact solution for the non-uniform stress and displacement fields within a finite, transversely isotropic, and linear elastic cylinder under compression with a kind of radial constraint induced by friction between the end surfaces of the cylinder and the loading platens. The main feature of the present work is the introduction of a general solution form for Lekhnitskii’s stress function such that the governing equation and all end and curved boundary conditions of the cylinder are satisfied exactly. Two different solutions were obtained corresponding to the real or complex characteristic roots of the governing equation, depending on the combination of the elastic material constants. The solution by Watanabe [Watanabe, S., 1996. Elastic analysis of axi-symmetric finite cylinder constrained radial displacement on the loading end. Structural Engineering/Earthquake Engineering JSCE 13, 175s–185s] for isotropic cylinders under compression test can be recovered as a special case. Our numerical results show that both the non-uniform stress distribution and the difference between the apparent and the true Young’s moduli of the cylinder are very sensitive to the anisotropy of Young’s moduli, Poisson’s ratios and shear moduli. A more distinct bulging shape of the cylinder is expected when anisotropy in shear modulus is strong, the cylinder is relatively short, and the end constraint is large. The bulging shape, however, does not depend strongly on anisotropy of either Poisson’s ratio or Young’s modulus.     

《Elaticity of Transversely Isotropic Materials》一书评介

浙江大学土木系丁皓江教授和陈伟球教授及澳大利亚悉尼大学航空、机械与机电工程学院章亮炽教授的专著“Elasticity of Transversely Isotropic Ma- terials”(ISBN：1-4020-4033-4),2006年由Springer公司出版,该书是加拿大著名力学家G．M．L．Gladwell     

Representations for transversely hemitropic and transversely isotropic stress-strain relations

The sets of polynomial stress-strain relations for elastic points which are transversely hemitropic and transversely isotropic are presented as projections of free algebraic modules having 20 and 10 generators, respectively. Complete sets of relations for the projections are presented which allow the sets of interest to be identified as free submodules having 12 and 6 generators, respectively. The results are established using the Cartan decomposition of the representation of the adjoint action of the two-dimensional rotation and orthogonal groups on the space of three-by-three symmetric matrices. The results are compared to known representations for nonlinear transversely isotropic stress-strain relations and for linear, transversely hemitropic and transversely isotropic ones.Work supported in part by National Science Foundation Grant INT-9106519.     

Representations for transversely hemitropic and transversely isotropic stress-strain relations

The sets of polynomial stress-strain relations for elastic points which are transversely hemitropic and transversely isotropic are presented as projections of free algebraic modules having 20 and 10 generators, respectively. Complete sets of relations for the projections are presented which allow the sets of interest to be identified as free submodules having 12 and 6 generators, respectively. The results are established using the Cartan decomposition of the representation of the adjoint action of the two-dimensional rotation and orthogonal groups on the space of three-by-three symmetric matrices. The results are compared to known representations for nonlinear transversely isotropic stress-strain relations and for linear, transversely hemitropic and transversely isotropic ones.