Optimai determination of the elastic constants of ...

Optimai determination of the elastic constants of composite materials from ultrasonic wave .. speed measurements Bernard Castagnede,&) James T. Jenkins, and Wolfgang Sachse Department oJ Theoretical and Applied Mechanics, Cornell University. Ithaca, New York 14853

Stephane Baste Laboratoire de Mi!Canique Physique, Universite de Bordeaux L 33405 Talence, France

(Received 2 June 1989; accepted for publication 6 December 1989) A method is described to optimally determine the elastic constants of anisotropic solids from wave-speeds measurements in arbitrary nonprincipal planes. For such a problem, the characteristic equation is a degree-three polynomial which generally does not factorize. Bv developing and rearranging this polynomial, a nonlinear system of equations is obtained. The elastic constants are then recovered by minimizing a functional derived from this overd~termined system of ~quations. Calculations of the functional are given for two specific cases, l.~., the orthorhom~lc and the hexagonal symmetries. Some numerical results showing the efficIency of the algonthm are presented. A numerical method is also described for the recover~ of t~e orienta:ion of the principal acoustical axes. This problem is solved through a dou~l~-lter~t1Ve numen~al scheme. Numerical as well as experimental results are presented for a umduectlonal composIte material.

I. INTRODUCTION

The propagation of bulk elastic waves in homogeneous anisotropic solids is a well-established field with apparently little space for refinement and improvement of the analytical and numerical tools already available. Several textbooks 1-3 and exhaustive research references 4 •5 have been available for some time. The initial applications were to the determination of elastic constants of single crystals from ultrasonics. An alternative for small single crystals (less than 1 mm 3 ) is the Brillouin scattering technique which has the advantage to be a non contact measurement. With the advent of structurally anisotropic materials such as synthetic composites, a new classes of problems emerged that are significantly different. The irregular structure of these materials makes the application of crystals acoustics inherently approximate, viscoelastic properties may be responsible for attenuation and dispersion of the acoustical waves, and heterogeneity of the properties of the fiber/matrix interface further complicates their characterization. Despite these general differences, the theoretical tools provided by crystal physics continue to be used to describe the structural anisotropy of these materials in terms of their dynamical properties. Some specific and basic differences exist between their applications to single crystals and anisotropic engineering materials. First, it is generally much easier to generate a fairly large amount of data from the engineering materials as there is no limitation in terms of the size of the samples. Thus least-squares techniques and optimization procedures are straightforward approaches to reduce experimental data. Second, in single crystals the angular parallax (Eulger angles) between the alignment of the crystallographic axes and the geometric axes can be monitored and corrected to a fraction of arc by xray techniques. Such precision is delusive with anisotropic d)

Current address: Laboratoire de Mecanique Physique, Universite de Bordeaux I, 33405 Talence, France.

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J. Appl. Phys. 67 (6).15 March 1990

engineering materials. Nevertheless, the problem of angular parallax is important for some natural anisotropic materials such as wood, rock, and bone. Furthermore, some materials exhibit induced anisotropy effects during loading. This is the case, for example, in granular materials 6 in which the orientation of the principal acoustical axes evolves during loading. Thus, in some cases at least, the recovery of the Euler angles is of importance in the inverse wave propagation problem. In many cases, the measurements of the elastic constants of anisotropic solids have been made along principal directions to obtain the diagonal terms and along particular "well-chosen" non principal directions to determine the remaining nondiagonal terms. To invert the experimental data obtained in the non principal directions, several methods have been proposed, including a numerical procedure,? series expansion, K and approximations valid near the principal axes ofsymmetry.9 In addition, a perturbation series expansion has been used to design an iterative numerical method for an arbitrary direction ofpropagation. 10,!! This approach was used with some success with experimental data, 12 but it is approximate in the sense that the convergence of the algorithm is a function of the direction of propagation, making its use in some cases erroneous. 13 More recently, an alternative rigorous method was proposed by considering three invariants of the Christoffel matrix. 14. 15 For the characterization of various composite materials, an indirect approach with no approximation has been described to solve the specific inverse propagation problem in a principal plane of symmetry. 16 In this case, there is a factorization of the characteristic equation which then simplifies to the product of a trivial solution and a second-degree equation. With this major simplification, a very simple and efficient algorithm was designed to take into account an arbitrary large amount of data over a broad range of angles in the principal plane under investigation. Using an advanced ul-

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trasonic spectrointerferometer,17 various synthetic 18 and natural 19 composite materials have been tested. The quality of the fit between theory and experimental data was used to ascertain the clastic symmetry for a given material. Ig Two basic assumptions for this approach were that the principal axes of symmetry were known and coincident with the geometric axes and that the propagation was in a principal plane, i.e., a plane containing two principal axes of symmetry. This second hypothesis is too restrictive when dealing with other geometries. For instance, when considering the problem oflocation of a pointlike source of acoustic emission in an anisotropic solid,20.21 the propagation is generally in nonprincipal planes. 22 This constitutes the much easier forward problem which is a simple eigenvalue problem, where the eigenvalues (here the wave speeds in arbitrary nonprindpa] planes) arc numerically computed with the help of a Jacobi numerical procedureY·21 Extensions of the original "in-principal-plane" clastic constants optimum recovery procedure are sought in this paper. Preliminary numerical work as well as experimental data have recently been presented for materials of hexagonal symmetry.25 Here this work is extended to the orthorhombic symmetry. A comprehensive description of the numerical method, as well as some results obtained by using a numerical simulation, are discussed in the next two sections. Then, in Sec. IV, it is shown that the orientation of the principal axes of symmetry can be optimally recovered with some very simple modifications of the basic algorithm. Some experimental results arc also discussed in that section.

II. METHOD OF CALCULATION

We consider an anisotropic solid which belongs to an arbitrary Hermann-Mauguin class of symmetry. Although the method that is described is general, here we only provide the complete results for systems which have the orthorhombic symmetry or higher. The calculations are exactly the same for less symmetric systems and the derivations are just a matter of additional algebra. Let us consider the three Euler angles 1/1, ¢, and () between the crystallographic (or material) axes and the geometric (or experimental) axes. These Euler angles relate the bases vectors in an arbitrary change of Cartesian coordinates: (1)

Here n; and fl" with i = 1,2,3, are the components ofa unit vector in the ~ew and old coordinate systems, respectively, and au' with iJ, = 1,2,3, is the transformation matrix given in terms of the Euler angles by

If; sin ¢ + cos If; cos ¢ cos e, cos 1/! sin ¢.- sin 1}; cos ¢ cos (),

a! 1 = sin

a l2 =

a 13 = cos rP sin 0, 021

= sin !/J cos c/J =

-

a 32 = sin

cos 1}! sin ¢ cos (),

(2)

1/1 cos ¢ + sin !/J sin ¢ cos 0, sin dJ sin (), a31 = ~- cos 1/! sin 0,

1/; sin e, a3 .> =

cos ()

The propagation of the three bulk acoustical modes is 2754

(3)

Here lil' the propagation, or Green-Christoffel tensor, is given by (4)

1i/=Cijkllljn"

where Cijkl is the stiffness tensor of the material. p is the density of the material, 0" is the Kronecker delta, and Vare the wave speeds of the three bulk modes. Equation (3) is a cubic equation for the eigenvalues of the propagation tensor. These eigenvalues are measured experimentally, and in order to obtain the elastic constants of the material, Eq. (3) must be inverted. To do this, one can proceed as follows: (1) Expand the determinant, (2) rearrange the various terms by factorizing around the elastic constants and (3) consider as many such equations as there are different directions of wave propagation, say N, in the experiment. This method provides a new set of cubic equations for the determination of the column vector of the unknown elastic constants which has the following general form: 'P3(X) = (.n)AUKXIXJXK

+ (~j ) BUXfXJ + C/X + D~O 1

(5)

where the dimension of X varies from 3 to 21 for the cubic to the triclinic systems. The various coefficients depend upon the wave speeds, the density of the material, the direction cosines of the direction of wave propagation, and the Euler angles. These coefficients are listed in Appendix A for materials possessing hexagonal and orthorhombic elastic symmetries. Generally, N> dim(X) and Eq. (3) represents an overdetermined set of cubic equations. When dealing with the propagation of the elastic waves in a principal plane of symmetry, Eq. (3) factorizes, and there is a trivial eigenvalue, with the two nontrivial eigenvalues solutions of a quadratic equation. In this case, analytical expressions for the eigenvalues can be found in textbooks. 3.26 When considering the inverse problem and dealing with the two nontrivial modes, the elastic constants are solutions of a nonlinear equation of second degree: (6) Expressions for these coefficients are also listed in Appendix A for one arbitrary principal plane of orthorhombic symmetry. Other principal planes and higher-order systems of symmetry can be obtained by performing the appropriate cyclic permutations of the elastic constants as noted in Table 1. These coefficients have been provided in a slightly different form elsewhere. 16.!H Hi. NUMERICAL SIMULATION

an = cos

an

the solution of a classical eigenvalue problem that is based on the characteristic equaticn

J. Appi. Phys., Vol. 67, No.6, 15 March 1990

The solutions of the overdetermined systems ofEqs. (5) and (6) are numerically obtained by using a standard

Newton-Raphson method that minimize the Euclidean functionals N

tV

L 'P 3(X)

and

I

r:p ~ eX) . Castagnede et al.

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TABLE I. Circular permutations on the elastic constants.

Orthorhombic

System Plane

(1,2)

(1,3)

Hexagonal (2,3)

(1,2)

("',3)
6'

X2

= C33 ,

X, = C ll ,

X6

=

= a,ni

+ a2n~

A=pV 2 ,

= C44 , Cu, X 7 = C23 , X3

X 4 = C55 ,

- A, K2 = a 2TlT- A,

/3 = KIK2 - 117 n~aL

Cu ,

An~,

1,2,3,

X, = C22 ,

KI

,

a = lIT

= njK2 ,

A347

= n';C ll

III

B24

(A6)

(A9)

and where all the other terms are identically zero. Note that Eqs. (A7), (A8), and (A9) reduce to Eqs. (A4), (A5), and (A6), respectively, by using the supplementary relationships, C 22 = CII' CS5 = C44 ' C Z3 = e l ], and C I2 )/2. 66 = (C II -

e

Castagnede et al.

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APPENDIXB

e cos if; cos if; sin () cos!/; sin if>

sin e sin ¢ cos if; - sin e sin!/; sin if;

- cos (3 cos ¢

cos

[ - sin

=

ae

e sin if;

-a 21

a 32

- 0 31

IJ

;a

Ja aif; = [""

The first-order derivatives are

aa

a8if; = [""

a22

A. The partial derivatives of the alj coefficients of the basis change matrix with respect to the Euler angles

__ '.1

-all

G ij

To obtain the two first derivatives of the coefficients (AS) given in Appendix A with respect to Euler angles, one has to evaluate successively.

ll

an -a 12 0

~l '

(Bl)

;a u, a"

]

1

cos () cos if; - cos e sin ¢> • - sin ()

(B2)

(B3)

The second-order derivatives are then easily calculated.

i B. The partial derivatives of the plimed direction cosines with respect to the Euler angles

D. Then the second derivatives with respect to TET23 of coefficients (A5) are easily calculated

The two first-order derivatives are easily calculated from Eqs. (1) and (Bl), (B2), and (B3). Next the derivatives of the coefficients (A4) can be calculated. When only one Euler angle is taken into account, e.g., 23 , Eqs. (1) simplify to

a, /3, K 1, K2, K', ai' a 2 , and A are defined in (A6).

e

n 2 = n2cos ()23 - n3 sin eD' n~ = n 2 sin 8 2 } + n3 cos fJ23 ,

(B4)

and the partial derivative of the primed direction cosines with respect to this Euler angle are an; 10823 = - n J and anVaf)n = n~. In that case, the two first derivatives of the coefficients (A5) are given below.

C. First derivatives with respect to TET23 of coefficients (AS) , ,5 A 1]2,19:, = l .t..n 2 n.1' ~

Am,e"

= 8n;n3 3(ni 2 -

2a),

A 233 ,o,., = ~ A223,e" ,

= 4n; n~ '(K I + K 2 )

B 12 ,O"

B 22 ,6

13

B2},e"

+ a 2 ), =4nin 3 [ni(K 1 -K2 ) +2nin!//3 + K'] - 4n~n; 3(n~.B + K2 + ala + A}, = 4nini (2nt ni 213 - n7 K2 - III IK,) -

2n; n; 5(a l

+4n;n;3( -211~!3+K, +aln~ +a 2n;2), B 33 ,6

13

= H2

},!}".

Cl,o" = 2n2 It; [(K 1K 2

-

nt n; 2[]2)

+ 02K2 - (:J2n7) ]. = - 2n2n; [K 1(K2 + ala) + ozK' - n~ 2(o]K,

Cz,o"

+A(K t +K2 -Gln;2) D,e, , = 2n; n3A(a]K,

+ a2K 2 -

-,82n~(n;2+a)],

plni),

where X

aX(i ... ) (i ...

).fi"

=

a8

23

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a

Castagnede at al.

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