Elsevier

Mechanics Research Communications

Volume 29, Issue 5, September–October 2002, Pages 327-338
Mechanics Research Communications

Effects of SH waves in a functionally graded plate

https://doi.org/10.1016/S0093-6413(02)00316-6Get rights and content

Abstract

A computational method is presented to investigate SH waves in functionally graded material (FGM) plates. The FGM plate is first divided into quadratic layer elements (QLEs), in which the material properties are assumed as a quadratic function in the thickness direction. A general solution for the equation of motion governing the QLE has been derived. The general solution is then used together with the boundary and continuity conditions to obtain the displacement and stress in the wave number domain for an arbitrary FGM plate. The displacements and stresses in the frequency domain and time domain are obtained using inverse Fourier integration. Furthermore, a simple integral technique is also proposed for evaluating modified Bessel functions with complex valued order. Numerical examples are presented to demonstrate this numerical technique for SH waves propagating in FGM plates.

Introduction

Functionally graded materials (FGMs) are widely used in modern industries. Owing to the inhomogeneous nature of the materials, damage of the material often occurs in the inner part of the structure. Therefore, non-destructive evaluation (NDE) techniques are expected to be used to determine the integrity and serviceability of FGM. Ultrasonic techniques play a much more important role in NDE technique. To use ultrasonic technique more efficiently and to develop new ultrasonic techniques, more detailed investigation of wave phenomena in FGM is required. A brief review can be found in Liu et al. (1999) on the existing methods for analyzing elastic–dynamic responses of the FGM plate. Methods using properly formulated layered elements (Liu et al., 1991a, Liu et al., 1991b) have been developed for analyzing different types of waves in FGM plates. These methods have also been extended to surface acoustic waves in piezoelectric FGM plate for applications of surface acoustic waves devices (Liu and Tani, 1994).

An analytical–numerical method (Han et al., 2000) was recently presented to investigate elastic waves in FGM plates excited by plane pressure waves. In this method, the FGM plate is first divided into quadratic layer elements (QLEs). In an element, both the elastic constant and mass density are assumed varying quadratically in the thickness direction. As the special attribute of the FGM that the material properties vary through the thickness has been taken into account, this method is very efficient for wave problems in FGM plates. In this paper, this method is extended for analyzing SH waves in FGM plates. First, the FGM plate is divided into QLEs, then the general solution of the governing equation of motion of SH waves for a QLE is derived. This solution is then used together with the boundary and continuity conditions to obtain the dynamic response of an arbitrary FGM plate in the wave number domain. The displacements in the frequency domain are obtained using inverse Fourier integration.

Modified Bessel functions in the one case of the solution for the governing equation of motion of the QLE are with complex valued order and argument, and must be evaluated. Numerical evaluation of Bessel functions is usually made using subroutines in MATLAB, and FORTRAN libraries such as NAG and IMSL. However, all these subroutines can only handle modified Bessel functions of a real valued order. There is no single subroutine available for calculating the modified Bessel functions with a complex valued order. There are techniques presented on evaluating Bessel functions with complex valued order or argument, using complex techniques of mathematics. The details of these techniques can be found in publications (Thompson and Barnett, 1986, Thompson and Barnett, 1987; Nordin et al., 1992). It has been found that these algorithms cannot be easily and efficiently coded for the purpose of this work. Here a simple and novel method is presented to evaluate these modified Bessel functions with complex valued order and argument. An error-controlled division procedure has also been proposed to reduce the sampling points and control the accuracy of the integration in evaluating modified Bessel function.

As examples, the displacement and stress responses of different FGM plates excited by shear force are calculated.

Section snippets

Formulation

Consider a two-dimension dynamic problem for which the external excitation and the field dependencies are independent of the y-axis. For anti-plane (xz plane) motion in a FGM isotropic material, the system of governing differential equations (in the case of external body force free) is expressed as:τxyx+τyzzv̈where the dot indicates differentiation with respect to time, ρ is the mass density, and v is the displacement in the y-direction, and τxy and τyz are the shear stresses. Consider a

Boundary and interface conditions

Consider now an arbitrary FGM plate. The plate can be divided into N QLEs, whose mechanical properties are assumed changing quadratically in the thickness direction. For generality, we first assume that the plate is loaded on the two surfaces as well as the (N−1) interfaces. Hence the amplitude of the external force vector can be written as follows:TT={T1,T2,T3,…,TN,TN+1}where Tj is the amplitude of the external force acting at the jth interface, j=1 is for the lower surface, and j=N+1 is for

Response in the wave number domain

We introduce the Fourier transformation with respect to the horizontal coordinate x as follows:ṽ(z,k)=∫−∞+∞v(z,x)eikxdxwhere i=−1, and k is the wave number. The application of the Fourier transformation to Eq. (8) leads to the following equation in the wave number domain:(1+bGz)22ṽz2+2(1+bGz)bGṽz+[ρ0/G0(1+bρz)2ω2−(1+bGz)2k2]ṽ=0In deriving Eq. (15), the time-dependence exp(−iωt) has been used, where ω is the angular frequency. Using the substitutionξ=1+bGzbGthe Eq. (15) can be rewritten

Response in the frequency domain

The displacement response in the frequency domain can be obtained by the following inverse Fourier integral (Liu et al., 1995)v(z,x)=1−∞+∞ṽ(z,k)eikxdx

Evaluation of modified Bessel functions

The modified Bessel functions Iν(z), Kν(z) in Eq. (24) can be represented by the following integral forms (Abramowitz and Stegun, 1964)Iv(z)=1π0πexp(zcost)cos(νt)dt−sin(νπ)π0exp(−zcosht−νt)dtargz<π2Kν(z)=∫0exp(−zcosht)cosh(νt)dtargz<π2

Here a simple technique is presented to evaluate the modified Bessel functions with complex order directly from the integral form of modified Bessel functions.

Consider the first item on the right-hand side of Eq. (42):I1=∫0πf(t)dtwhere the integrand isf(t)=exp

Numerical examples

A program is developed in FORTRAN to compute the wave field in a FGM plate excited by an incident SH wave pressure on the surface of the plate. In the computation, the following dimensionless parameters are usedx̄=x/H,z̄=z/H,Ḡ=G/G0,v̄=vG0/q0,k̄=kH,cr=G00ω̄=ωH/cr,t̄=tcr/H,ρ̄=ρ/ρr,b̄G=bGH,b̄ρ=bρHwhere G0, cr,ρ0 are the reference shear modulus, shear wave velocity, and mass density, respectively. H and q0 are the thickness of the plate and the amplitude of the external force, respectively.

The

Conclusions

In this paper, a computational technique has been presented to analyze the SH waves propagating in FGM plates. QLEs are employed in this technique, the material properties of each QLE are assumed as quadratic functions in thickness direction. The general solution for SH waves in a QLE has been derived. The final solution for SH waves in a FGM plate is obtained by assembling all the QLEs. The displacements in the frequency domain is obtained using the inverse Fourier integration. A novel and

References (15)

There are more references available in the full text version of this article.

Cited by (0)

View full text