Adnan H. Nayfeh

The general problem of elastic wave propagation in multilayered anisotropic media

Exact analytical treatment of the interaction of harmonic elastic waves with n-layered anisotropic plates is presented. Each layer of the plate can possess up to as low as monoclinic symmetry and thus allowing results for higher symmetry materials such as orthotropic, transversely isotropic, cubic, and isotropic to be obtained as special cases. The wave is allowed to propagate along an arbitrary angle from the normal to the plate as well as along any azimuthal angle. Solutions are obtained by using the transfer matrix method. According to this method formal solutions for each layer are derived and expressed in terms of wave amplitudes. By eliminating these amplitudes the stresses and displacements on one side of the layer are related to those of the other side. By satisfying appropriate continuity conditions at interlayer interfaces a global transfer matrix can be constructed which relates the displacements and stresses on one side of the plate to those on the other. Invoking appropriate boundary conditions on the plates outer boundaries a large variety of important problems can be solved. Of these mention is made of the propagation of free waves on the plate and the propagation of waves in a periodic media consisting of a periodic repetition of the plate. Confidence is the approach and results are confirmed by comparisons with whatever is available from specialized solutions. A variety of numerical illustrations are included.

Wave Propagation in Layered Anisotropic Media with Applications to Composites

Introduction - historical background. Part 1 Field equations and tensor analysis: the stiffness tensor material symmetry matrix forms of stiffness engineering constants transformed equations expanded field equations planes of symmetry. Part 2 Bulk waves: an overview the Christoffel equation material symmetry computer aided analysis group velocity energy flux. Part 3 Generalized Snells law and interfaces: boundary conditions characterization of incident waves critical angles two fluid media two isotropic media. Part 4 Formal solutions: common form of solutions triclinic layer the monoclinic case higher symmetry materials formal solutions in fluid media the alpha-c relation and the Christoffel equation. Part 5 Scattered wave amplitudes: notation reflection from a free surface scattering from fluid-solid interfaces scattering from solid-solid interface. Part 6 Interface waves: surface waves pseudo-surface waves Scholte waves. Part 7 Free wave in plates: free waves in triclinic plates free waves in monoclinic plates higher symmetry material plates numerical computation strategy. Part 8 General layered media: geometric description of unit cell analysis properties of the transfer matrix free waves on the layered cell waves in a periodic medium bottom bounding solid substrate. Part 9 Propagation along axes of symmetry: geometry SH waves motion in the sagittal plane free waves on the layered cell waves in a periodic medium bottom bounding solid substrate. Part 10 Fluid-loaded solids: reflection from a substrate plates completely immersed in fluids higher symmetry cases leaky waves experimental technique. Part 11 Piezoelectric effects: basic relations of piezoelectric materials simplified field equations analysis formal solutions higher symmetric materials remarks on the monoclinic-m case reflection and transmission coefficients sample illustration remarks on layered piezoelectric media. Part 12 Transient waves: theoretical development source characterization integral transforms of formal solutions isotropic media anisotropic media Cagniard-de Hoop transformation semi-space media. Part 13 Scattering from layered cylinders: field equations formal solutions in isotropic cylinders characterization of incident waves formal solutions for a layer scattering amplitudes. Part 14 Elastic properties of composites: general description of fibrous composites the model the layered model the square fibrous case anisotropic fibre and matrix strain energy approach undulated fibre appendix.

Leaky Lamb waves in fibrous composite laminates

Results of experimental measurements and theoretical calculations on ultrasonic leaky Lamb‐wave propagation in fiber‐reinforced, unidirectional composite laminates are presented. With the Lamb wave vector oriented parallel to the fiber direction, dispersion curves of phase velocity versus frequency and plate thickness have been constructed from measurements of ultrasonic reflection from fluid‐loaded composite plates. The experimental results are supported by a theoretical model of Lamb‐wave propagation in the composite plate. The model begins with an approximate calculation of the effective, homogeneous, transversely isotropic elastic behavior of a unidirectional composite laminate in the long‐wavelength limit, using a two‐step procedure based on alternating layered media. This intermediate continuum result is then incorporated into a calculation of the ultrasonic reflection coefficient of a fluid‐loaded anisotropic plate, which is assumed to approximate the fibrous composite laminate. Good quantitative a...

Thermomechanically induced interfacial stresses in fibrous composites

Abstract Static analyses similar to recently developed continuum mixture theories of wave propagation in structural composites are carried out in order to estimate the interfacial stresses that develop in composites subjected to various mechanical and thermal loadings. A variety of composite models are treated. These include the multi-cylindrical periodic fibres, the single cylindrical fibre and the single planar fibre reinforcement models. Results are compared with existing theoretical and experimental findings.

A continuum mixture theory of wave propagation in laminated and fiber reinforced composites

Abstract A binary mixture theory is developed for wave guide-type propagation in laminated and undirectional fibrous composites. In particular, a rational construction of both mixture interaction and constitutive relations is given. The resulting theory contains microstructure. The domain of validity of the mixture theory is determined by comparison of the phase velocity spectrum with exact and/or experimental results. The utility of the model is demonstrated for both laminated and fibrous composites by correlating theoretical and experimental transient pulse data on boron-carbon phenolic and Thornelcarbon phenolic laminates, and uni-directional fibrous quartz phenolic.

Compressional waves in fluid‐saturated elastic porous media

This article discusses the propagation of compressional waves in fluid‐saturated elastic porous media. Both harmonic and transient pulses are considered. In general, two modes of wave propagation exist. In the case of a transient pulse, these modes lead to a two‐wave structure. It is not possible to obtain closed‐form solutions for the general case of transient loading, but considerable insight may be obtained from certain limiting cases (e.g., no viscous coupling, large viscous coupling) for which analytical solutions are derived by means of Laplace transform techniques. Strong viscous coupling leads to the coalescence of the two wave fronts into a single front; in this case the material behaves like a single continuum with internal dissipation. Solutions for the general case are obtained both by numerical inversion of the Laplace transforms and by direct finite‐difference methods.

Propagation of guided waves in fluid‐coupled plates of fiber‐reinforced composite

Guided wave propagation in fluid‐coupled plates of fiber‐reinforced composites has been investigated by studying ultrasonic reflection in these structures. From measurements of ultrasonic reflection on unidirectional graphite‐epoxy plates over a range of incident angles, experimental dispersion curves using a total‐transmission criterion for leaky plate waves have been constructed. Theoretical calculations with no adjustable parameters demonstrate excellent agreement with the experimental data. Unusual behavior observed in the fundamental total‐transmission curve led to a reexamination of the mode identification criteria. Both the total‐transmission curves and the normal modes of the fluid‐coupled plate differ significantly from the result expected on the basis of the widely used Cremer coincidence condition to identify propagating plate waves. It is found that these differences are particularly pronounced in cases where the ratio of fluid to solid densities is high, as for composite materials. The existe...

Excess attenuation of leaky Lamb waves due to viscous fluid loading

In two recent papers [J. Acoust. Soc. Am. 97, 3191–3193 (1995) and 98, 1057–1064 (1995)], Zhu and Wu presented an analytical technique to assess the effect of viscous fluid loading on the propagation properties of Rayleigh and Lamb waves in fluid-loaded solids. They modeled the viscous fluid as a hypothetical isotropic solid having rigidity c55=−iωη, where η denotes the viscosity of the fluid and ω is the angular frequency. In this way, the vorticity mode associated with the viscosity of the fluid is formally described as the shear-mode in the fictitious solid. In this paper this technique is further developed by removing certain inconsistencies that unnecessarily reduce the accuracy and the range of validity of Zhu and Wu’s results. By properly accounting for viscous effects on the bulk compressional wave in the fluid and applying a rigorous treatment of the field equations and boundary conditions, the exact dispersion equations that are not limited to low frequencies and viscosities are derived. Examples of these results are presented to illustrate the effect of fluid viscosity on the lowest-order symmetric and antisymmetric Lamb modes. One interesting feature revealed by these calculations is the presence of a sharp minimum in the viscosity induced attenuation of the lowest-order symmetric mode of thin plates either immersed in or coated with a viscous fluid. This minimum occurs at a particular frequency where the otherwise elliptical polarization of the surface vibration becomes linearly polarized in the normal direction.In two recent papers [J. Acoust. Soc. Am. 97, 3191–3193 (1995) and 98, 1057–1064 (1995)], Zhu and Wu presented an analytical technique to assess the effect of viscous fluid loading on the propagation properties of Rayleigh and Lamb waves in fluid-loaded solids. They modeled the viscous fluid as a hypothetical isotropic solid having rigidity c55=−iωη, where η denotes the viscosity of the fluid and ω is the angular frequency. In this way, the vorticity mode associated with the viscosity of the fluid is formally described as the shear-mode in the fictitious solid. In this paper this technique is further developed by removing certain inconsistencies that unnecessarily reduce the accuracy and the range of validity of Zhu and Wu’s results. By properly accounting for viscous effects on the bulk compressional wave in the fluid and applying a rigorous treatment of the field equations and boundary conditions, the exact dispersion equations that are not limited to low frequencies and viscosities are derived. Example...

Leaky Rayleigh waves on a layered halfspace

Experimental and theoretical results on the nonspecular reflection of finite acoustic beams incident at and near the Rayleigh angle onto a fluid‐solid interface loaded by an elastic layer are presented. Measurements with Gaussian‐shaped acoustic beams have been performed on the reflected field amplitude distribution and surface wave speed dispersion in copper‐loaded stainless‐steel specimens where the fluid medium is water. It is shown that existing theory can be utilized to explain the results by constructing and solving boundary condition equations for the Rayleigh wave pole appropriate for the lossless layered halfspace with the fluid. Excellent agreement with the exact treatment is observed in most aspects of the measurements. In the model calculation the imaginary part of the Rayleigh wave pole is found to display an unexpected maximum at a value of layer thickness over wavelength where the real part corresponds nearly to the transverse wave speed in the layer.

Ultrasonic reflection and guided wave propagation in biaxially laminated composite plates

This paper presents the results of numerous ultrasonic reflection measurements on plates of laminated composites in which the angle of incidence of a finite‐aperture acoustic wave and its frequency have been varied. These experiments have been carried out on biaxially laminated graphite–epoxy specimens utilizing water as a fluid‐coupling medium. The stacking sequence of the individual laminae is restricted such that in each layer a principal material axis lies in the incident plane of the acoustic wave. The data are compared to the results of a recent theoretical analysis based on an analytical treatment of wave propagation in orthotropic media in conjunction with a transfer matrix approach. Reflection and transmission coefficients are derived from which the characteristic behavior of the system is identified. Results are presented both as reflection spectra and dispersion curves. In general, very good agreement is found between prediction and experiment. Moreover, significant changes in the reflection sp...