Yalcin Mengi

A mixture theory for elastic laminated composites

Abstract A theory is developed which governs the dynamic response of a homogeneous, elastic, dispersive material. The material is used as a model of a two phase layered material in which each of the layers is isotropically elastic. The theory is derived from a general theory for all two phase periodic materials which was developed earlier [1]. The general theory was derived using the theory of mixtures. Dispersion is accommodated through the use of elastodynamic operators and for the layered material a micro model is used to establish the forms of the operators appropriate to the material. These specific operators are simplified by replacing them with truncated power series before introducing them into the equations of linear momentum. The theory for layered materials contains nineteen model constants and equations are developed from which these constants can be derived from the layer constants. The equations are derived partly using micro model analysis and partly by matching specific dynamic behaviors of the model and prototype. The ability with which the model predicts the dynamic response of the layered material is assessed in two ways. Both compare spectra reflecting the behavior of infinite trains of the principal kinds of waves. The first compares spectral lines from the model with those derived from the exact theory for layered materials. The second compares lines from the model with those obtained from experiments. Predictions from the model prove to be quite accurate.

A simple soil–structure interaction model

Abstract A simple three-dimensional soil–structure interaction (SSI) model is proposed. First, a model is developed for a layered soil medium. In that model, the layered soil medium is divided into thin layers and each thin layer is represented by a parametric model. The parameters of this model are determined, in terms of the thickness and elastic properties of the sublayer, by matching, in frequency–wave number space, the actual dynamic stiffness matrices of the sublayer when the sublayer is thin and subjected to plane strain and out-of-plane deformations with those predicted by the parametric model developed in this study. Then, by adding the structure to soil model a three-dimensional finite element model is established for the soil–structure system. For the floors and footings of the structure, rigid diaphragm model is employed. Based on the proposed model, a general computer software is developed. Though the model accommodates both the static and dynamic interaction effects, the program is developed presently for static case only and will be extended to dynamic case in a future study. To assess the proposed SSI model, the model is applied to four examples, which are chosen to be static so that they can be analyzed by the developed program. The results are compared with those obtained by other methods. It is found that the proposed model can be used reliably in SSI analysis, and accommodates not only the interaction between soil and structure; but, also the interaction between footings.

Propagation of transient out-of-plane shear waves in viscoelastic layered media

Abstract Propagation of two-dimensional transient out-of-plane shear waves in multilayered viscoelastic media is investigated. The multilayered medium consists of N different isotropic, homogeneous and linearly viscoelastic layers with more than one discrete relaxation time. The top surface of the layered medium is subjected to dynamic out-of-plane shear tractions; whereas, the lower surface is free or fixed. A numerical technique is employed to obtain the solution, which combines the Fourier transform with the method of characteristics. The numerical results are displayed in curves denoting the variations of the shear stresses with time at different locations. These curves reveal clearly the scattering effects caused by the reflections and refractions of inclined waves at the boundaries and at the interfaces of the layers. The curves also display the effects of viscous damping in the wave profiles. By suitably adjusting the material constants, the curves for the case of elastic layers are also obtained as a special case. The curves further show that the numerical technique applied in this study is capable of predicting the sharp variations at the wave fronts.

A new approach for developing dynamic theories for structural elements: Part 1: Application to thermoelastic plates

Abstract With the object of developing refined dynamic theories for plates, shells, beams and composites, a new technique is proposed. This technique eliminates any inconsistency between the assumed deformation or temperature shape and lateral boundary or interface conditions. Accordingly, it improves the dispersive characteristics of waves propagating in any of these structural elements. In this study the new technique is applied to thermoelastic plates. It is found that the dispersion curves predicted by the refined approximate theory duplicate very closely those derived from the exact theory without introducing any matching coefficients into the approximate theory.

A new approach for developing dynamic theories for structural elements: Part 2: Application to thermoelastic layered composites

Abstract In this study a continuum theory is proposed which predicts the dynamic behavior of thermoelastic layered composites consisting of two alternating layers. In constructing the theory, it is noted that the governing equations for a single layer, derived in Part I, hold in each phase of the layered composite. The theory is completed by supplementing these equations with continuity conditions and using a smoothing operation. The derivation of the continuity conditions is based on the assumption that the layers are perfectly bonded at interfaces. To assess the theory, spectra from the exact and the derived theory are compared for waves propagating in various directions of the composite. The match between the two is excellent. For waves propagating normal to layering the theory predicts both the banded and periodic structure of the spectra. The region of validity of the theory on the wave number-frequency plane can be enlargened by increasing the orders of the theory and the continuity conditions.

Determination of the transverse elastic coefficients of bone

Abstract A method for determining the elastic coefficients of bone as a transversely isotropic material is described. The constitutive equations relating the stresses to strains were solved simultaneously for tension and hydrostatic compression tests. Cylindrical specimens with symmetry axes oriented with the long axis of the fresh bovine Haversian femur were tested in tension, torsion and hydrostatic compression, and the results were used to calculate the five independent elastic coefficients.

Propagation and decay of waves in porous media

A study is made of the propagation of wave fronts in two types of fluid‐filled porous media, anisotropic and isotropic. The characteristic equation governing wave propagation velocities in anisotropic media is obtained by using the notion of surfaces of discontinuity. It is found that the wave velocities in such media vary with the direction of propagation and that for a specified direction there are four different wave velocities with which the disturbances can propagate. When the material is isotropic, its frequency equation shows the existence of three distinct waves, two longitudinal waves and one shear wave with velocities independent of the direction of propagation. The decay equations derived for these waves indicate that there are two factors which influence the decay of waves in isotropic porous media. The first is the geometry of the wave front, and the second is the friction between the solid and fluid phases.

The propagation of initially plane waves in nonhomogeneous viscoelastic media

Abstract In this study, the propagation of an initially plane wave in a linear isotropic nonhomogeneous viscoelastic medium, where the nonhomogeneity varies transversely to the direction of propagation, is investigated. For this purpose, first the propagation of waves in a linear isotropic viscoelastic medium of arbitrary inhomogeneity is studied by employing the notion of singular surfaces. The characteristic equation governing wave velocities, and the growth and decay equations describing the change of the strength of the discontinuity as the wave front moves are obtained. In the second part of this work, the propagation of initially plane waves is studied for three types of inhomogeneities by employing the findings established in the first part. The first kind of inhomogeneity considered is of axisymmetrical type where the wave propagation velocity depends on the radial coordinate only, increasing linearly up to a certain radial distance and remaining constant thereafter. The second kind is also axisymmetrical with a wave velocity distribution decreasing linearly till a given value of the radial coordinate. In the third one, the wave velocity is assumed to vary linearly over a given interval along a certain coordinate axis only, which is perpendicular to the direction of propagation, and remain constant outside. The ray and wave front analyses are carried out and the decay or growth of stress and velocity discontinuities are studied for each of the three cases.

A theory for the formation and propagation of lüders bands in a plate subjected to uniaxial tension

Abstract It is known from experiments that when a plate, made of a material belonging to a restricted class of elastic-perfectly plastic materials, is subjected to a uniform tensile stress along opposite edges, yielding begins with the formation of Luders bands. Recent research also shows that if the load is maintained, the bands grow and gradually penetrate the elastic zones. The study presented here develops a theory for both the formation of the Luders bands and for the phenomenon of the growth of the plastic zones at the expense of the elastic. No restriction is placed upon Poissons ratio, though the incompressible material is studied as a special case. We find that for each value of Poissons ratio there are two possible angles of inclination of the bands. We also establish that when the plastic zone begins to grow for a compressible material the boundary of the plastic zone need not be straight but that the normal vector of this boundary is contained within certain bounds. We also find that the normal velocity at which a point on the boundary propagates depends both on the direction of propagation and on Poissons ratio.

Transmitting boundary conditions suitable for analysis of dam-reservoir interaction and wave load problems

Abstract Transmitting boundary conditions (tbcs) are developed, suitable for both boundary and finite element analyses, for the radiation of waves propagating in horizontal direction along a compressible inviscid fluid layer. The derivation of the proposed tbcs uses a completely continuum approach and is based on the spectral properties of radiating waves. The formulation is presented in Fourier space and accommodates the effects due to surface waves, as well as, due to the radiation of waves in viscoelastic foundation. The proposed tbcs may be used in the analyses of both dam-reservoir systems and wave load problems of floating or submerged bodies. Here, for assessment, they are used in the analyses of two simple benchmark problems. The results indicate that, when used in conjunction with boundary element analysis, the proposed tbcs not only improve the prediction of BEM, but also reduce the computational load of the analysis.