Inversion of elastic waveform data in anisotropic solids using the delta ...

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Inversion of elastic waveform data in anisotropic solids using the delta-function representation of the Green’s function V. K. Tewary Materials Reliability Division, National Institute of Standards and Technology, Boulder, Colorado 80303

~Received 20 November 1996; accepted for publication 5 June 1998! A method for inversion of measured data on elastic waveforms in anisotropic solids is proposed, using the delta-function representation of the Green’s function. The method requires integration over a closed 2-D ~two-dimensional! space. In contrast, inversion using the traditional Fourier representation requires integration over an infinite 4-D space. The method can be used to determine the Green’s function for imaging applications and elastic constants for materials characterization. The method is illustrated by applying it to determine all six elastic constants of a model graphite fiber composite assuming a tetragonal structure and using simulated data. © 1998 Acoustical Society of America. @S0001-4966~98!04509-3# PACS numbers: 43.20.Gp, 43.20.Jr, 43.35.Cg, 43.35.Pt @DEC#

INTRODUCTION

We describe a new method for the inversion of measured data on transient elastic waveforms in anisotropic solids by using the delta-function representation of the Green’s function.1,2 The inversion of the waveform data is required for determining the elastodynamic Green’s function for applications such as imaging in elastic and acoustic scattering problems, characterization of the source in acoustic emissions, measurement of elastic constants, and materials characterization. Traditionally, inversion3 of waveform data is done by using the Fourier representation of the Green’s function. The exact inversion would require measurement of waveforms over the infinite Fourier space ~wave vector and frequency!. Since that is not possible in practice, special techniques have been developed for processing of data in the Fourier space. However, that is subject to sampling errors and statistical uncertainties. The use of delta-function representation reduces these uncertainties. We have shown earlier1,2 that the delta-function representation is computationally very efficient for forward calculations. Now we show that it is particularly suitable for the inversion problem. The inversion formula using deltafunction representation requires integration over a closed 2-D ~two-dimensional! space of the surface of a unit sphere. In contrast, the Fourier inversion formula requires integration over the 4-D infinite space of frequency and wave vectors. The main characteristic of the delta-function representation is that it uses a variable that is a linear combination of the space and time variables and maps the Green’s function 1716

in the slowness space. In the Fourier representation, the space and time variables are treated separately and are mapped into the corresponding frequency and wave vector space. Full inversion leading to the determination of the whole Green’s function is needed only for imaging applications. The elastic constants of isotropic solids have been traditionally determined by measuring the transit time or the ultrasonic phase velocities and interpreting the data using Fourier representation.3 This technique does not work very well for anisotropic solids, because of uncertainties due to energy flux deviation and path length variations.4–6 Experimentally,7 it is possible to measure the waveforms at a large number of points over the surface of a hemisphere with a transient point source at its center, or over the surface of a semicylinder with a transient line source at its axis. The waveforms can then be analyzed in terms of the Green’s function that accounts for the response of the whole solid and therefore does not suffer from the uncertainties like path length. The delta-function representation provides an efficient inverse relationship for determination of elastic constants by such measurements. We show here how to use the inversion formula of the delta-function representation to obtain the elastodynamic Green’s function and elastic constants of anisotropic solids from the transient waveforms measured over a closed space. To illustrate our method, we use simulated data to determine all the six elastic constants of graphite-fiber composite assuming a tetragonal model. We calculate the waveforms using known elastic constants over a finite set of points on the

J. Acoust. Soc. Am. 104 (3), Pt. 1, September 1998 0001-4966/98/104(3)/1716/4/$15.00 © 1998 Acoustical Society of America

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surface of a hemisphere. We treat the calculated values as the ‘‘measured values.’’ We then apply our method to determine the elastic constants of the solid by using these ‘‘measured values.’’ We find an excellent agreement between the original and the derived values of the elastic constants.

I. GREEN’S FUNCTION REPRESENTATION AND INVERSION OF WAVEFORMS

We use the same notation and frame of reference as in Ref. 2. For a step function pulse, the elastodynamic Green’s function is written as1,2 G~ x,t ! 5

1 4p3

E

g~ q! d ~ t2q–x! dq,

~1!

where g~q!, the Green’s function in slowness space, is given by g~ q! 5Im@ L ~ q! 2 ~ 12i« ! I# 21 ,

~2!

L i j ~ q! 5c ik jl q k q l ,

~3!

I is the unit matrix, c is the elastic constant tensor, q is a vector in the slowness space, x and t are the 3-D space and time variables, respectively, L is the Christoffel matrix, and «510 in the limit. The integration in Eq. ~1! is over the entire vector space of q. The function g~q!, defined as the imaginary part of @ L(q)2(12l«)I # 21 in Eq. ~2!, has deltafunction type resonances.1,2 The real and imaginary parts of g~q! are related due to causality. Equation ~1! is the Radon representation of the Green’s function.8 Using the inversion formula8 for the Radon transform yields g~ Q! 5

1 4p3

E

x51

G9 ~ x,t5Q–x! dx,

~4!

where the primes denote the second derivative with respect to time and the integration in Eq. ~4! is over the surface of a unit sphere. Equation ~4! provides a prescription for experimentally determining g~Q! for any vector Q in the slowness space. Take a hemispherical sample. Choose units such that the radius of the hemisphere is 1. Apply a localized stepfunction force at the center. Measure the waveforms as function of time at various points on the surface of the sample. At each point on the surface, extract the second derivative of the displacement field at t5Q–x for a selected value of Q. Integrate over the surface by summing over all values of x in accordance with Eq. ~4!. This would give g~Q! if the force is known, or the force if g~Q! is known. Our present objective is to determine only the elastic constants of a solid. It would be sufficient to measure the location of resonances in the elements of g~Q!. If the applied force is a point force, and if the density of the solid is uniform, the position of the resonance would depend only upon the eigenvalues of the Christoffel matrix. The eigenvalues are simple functions of the elastic constants of the solid for Q in symmetry directions. Thus, by choosing suitable values of Q, we can determine the elastic constants by locating the resonances in the elements of g~Q!. 1717

J. Acoust. Soc. Am., Vol. 104, No. 3, Pt. 1, September 1998

FIG. 1. Green’s function determined from the simulated measurements, plotted against K, a component of the slowness vector Q. Solid line— G 11(Q), dotted—G 22(Q), dashes—G 33(Q), where Q5(K,0,0). The vertical lines are locations of resonance.

II. APPLICATION TO SIMULATED DATA

To test our method, we use simulated data for a model tetragonal solid assuming uniform density. We assume that we have the specimen in the form of a hemisphere as used by Hurley et al.7 We further assume that the transient elastic waves are generated by a step-function-type point source at the center of the hemisphere and we measure the first derivative of the velocity field at various points on the curved surface of the hemisphere. We also assume that all six components of the Green’s function can be obtained by measuring the three components of the displacement field for each direction of the force. The chosen values of the elastic constants for a model2 graphite-fiber composite are given below in units of c 44 : c 1153.1,

c 33554.6,

c 6650.7,

c 1251.2,

c4451, c1351.9.

We calculate the second derivative of the displacement field at t5Q–x for selected values of Q and x by using Eqs. ~1!– ~3!. We treat these calculated values as the experimental data. We assume units such that the density of the solid, the radius of the hemisphere, and c 44 are all equal to 1. We choose 561 points over the surface of the sphere—17 points for 0