Kazumi Watanabe

Transient Waves in Anisotropic Laminated Plates, Part 1: Theory

A hybrid numerical method in which the finite element method and the method of Fourier transforms are combined is proposed. The plate is divided into N plate elements, and the principle of virtual work is used to develop approximate dynamic equilibrium equations for three- and two-dimensional problems

Lamb Wave Propagation in Anisotropic Laminates

The wave propagation in arbitrary anisotropic laminates is investigated on the basis of an exact theory. The dispersion relations of Lamb waves are determined for graphite/epoxy symmetric angle-ply laminates and hybrid composite ones which consist of carbon/epoxy and glass/epoxy layers. The dispersion and anisotropy of phase velocities for fundamental modes are discussed in detail. The energy distributions in the thickness direction of laminates are calculated for each kind of Lamb wave. A hybrid composite laminate is found to have better capability in absorbing impact energy by analyzing the strain energy distribution during the wave propagation. The results of the strain energy distribution are useful in determining the arrangement and the fiber orientation of the layers of hybrid composite laminates.

Characteristic Wave Surfaces in Anisotropic Laminated Plates

A numerical method is used to determine the dispersion relation (an eigenvalue equation) of plane wave propagation in an anisotropic laminated plate. A phase velocity surface, phase slowness surface, phase wave surface, group velocity surface, group slowness surface, and group wave surface are defined and their formulae are deduced from the Rayleigh quotient and the orthogonality condition of the eigenvectors of the eigenvalue equation. The six characteristic surfaces can be used to illustrate the characteristics of plane waves and waves generated from a point source in an anisotropic laminated plate. As numerical examples, the characteristic surfaces are computed for graphite/epoxy angle ply laminated plates and for a hybrid one. The results for the graphite/epoxy laminated plates are compared with those obtained by Moon’s approximate theory.

Harmonic wave propagation in anisotropic laminated strips

Abstract A numerical method is presented for investigating harmonic wave propagation in arbitrary anisotropic laminated strips. The laminated strip is divided into N plate elements in the thickness. The displacement field within each element is approximated by a linear expansion in the thickness direction and by a series in the width direction. The principle of virtual work is applied to establish the eigenvalue equations, and since the number of the equations is usually much smaller than that obtained by the conventional finite element method, the computing time is much shorter for solution of comparable accuracy. As an example, the dispersion relations are determined for unidirectional carbon/epoxy laminated strips and hybrid composite laminated ones which consist of carbon/epoxy and glass/epoxy layers.

Transient Waves in Anisotropic Laminated Plates, Part 2: Application

Both the two-dimensional problem (line step-impact load) and the three-dimensional problem (point step-impact load) are considered. First, the displacement responses of isotropic plates are computed by using the hybrid numerical method and the results are compared with those obtained by the method of the head of the pulse approximation and the method of generalized rays. Next, the displacement responses of a hybrid composite laminated plate excited by a line step-impact load and a point load are calculated

A semi-exact method for the propagation of harmonic waves in anisotropic laminated bars of rectangular cross section

Abstract A semi-exact method is presented for investigating the propagation of harmonic waves in laminated bars of arbitrary anisotropy and rectangular cross section. In the thickness direction, the laminated bar is divided into N plate elements. The displacement field within each element is approximated by a linear expansion in the thickness direction and by exponential functions in the other two directions. The principle of virtual work is used for the determination of the characteristic values in the width direction. The dispersion relations are obtained by using the boundary conditions. As numerical examples, the natural frequencies are calculated for an isotropic bar, a unidirectional carbon/epoxy laminated bar and a hybrid composite laminated bar made of carbon/epoxy and glass/epoxy. It is found that in comparison with the conventional finite element method, the method using the plate elements is more effective for the characteristic analysis of laminated composites.

Characteristics of surface wave propagation along the edge of an anisotropic laminated semi-infinite plate

Abstract Characteristics of surface wave propagation along the edge of semi-infinite plates (edge waves) are investigated by means of the semi-exact method proposed by the present authors. First, the characteristics of edge waves for semi-infinite isotropic plates are calculated and the results are compared with those obtained by other published approximate theories and by the FEM. The semi-exact method is found to completely describe symmetric and antisymmetric modes of edge wave propagation. An approximate relation between the velocity of energy propagation and Poissons ratio is obtained for the first antisymmetric mode in the high frequency range. Next, the frequency spectra are discussed for a semi-infinite unidirectionally reinforced carbon/epoxy laminated plate and for a hybrid composite laminated plate which consists of carbon/epoxy and glass/epoxy layers.

Propagation of SH-wave in an Off-Angled Anisotropic Elastic Half Space.

Propagation of antiplane SH wave in an anisotropic elastic half space is discussed. The plane surface does not coincide with the principal axis of the anisotropy, that is, the off-angled surface. The present paper composed of three parts. The first is the discussion of the power flow direction of SH wave, second is that of the reflection coefficient at the off-angled surface. The last part gives exact closed-form solutions for the transient and steady-state responses of the half space to a line antiplane load. Results obtained here are very basic, however, they would be useful for researchers of nondestructive inspection in the field of composite materials.

Wave Propagation on the Boundary Plane of an Anisotropic Body with Inclined Anisotropic Principle Axis.

In this paper, SH (horizontally polarized shear) wave propagation in an inclined anisotropic elastic body is investigated. It is analyzed as a basic boundary value problem as the interface between an inclined anisotropic half space and an isotropic half space. Direction of wave propagation is determined by the use of power flow density. It is shown that power flow density determines the direction of the principle axis of an inclined anisotropic body. Reflection and transmission coefficients in the interface are also discussed. The reflection coefficient in the interface is large when inclined angle in large.

Energy propagating surface due to a point wave source in an anisotropic laminated plate.

A numerical method is presented to determine the dispersion relation of wave propagation in an anisotropic laminated plate. A phase velocity surface, phase slowness surface, phase wave surface, group velocity surface, group slowness surface and group wave surface are defined and obtained using Reyleighs quotient. The six characteristic surfaces can be used to illustrate the characteristics of the wave propagation in the anisotropic laminated plate due to the point wave source in it. As an example, the characteristic surfaces are obtained for graphite /epoxy angle-ply laminated plates and a hybrid composite laminated plate which consists of carbon/epoxy and glass/epoxy layers. The results for the graphite/epoxy laminated plates are compared with those obtained by the other approximated theory.