THREE-DIMENSIONAL STEADY-STATE GREEN'S FUNCTION
FOR A LAYERED ISOTROPIC PLATE
H. Bai1, J. Zhu2, A.H. Shah1 and N. Popplewell1
faculty of Engineering, University of Manitoba, Winnipeg, Manitoba, Canada, R3T 5V6
Integrated Engineering Software, Winnipeg, Manitoba, Canada, R3H 0X4
ABSTRACT. The elastodynamic response of a layered isotropic plate to a source point load having
an arbitrary direction is studied in this paper. A decomposition technique is developed within each
homogeneous isotropic lamina to simplify the general three-dimensional plane-wave propagation
problem as a separate plane-strain problem and an anti-plane-wave propagation problem. Accuracy of
computation is assured by cross-checking the numerical results by different methods. Results are
checked numerically for a vertical point load acting on a homogeneous and a layered plate by a hybrid
method. On the other hand, results are checked for a horizontal point load by using dynamic
reciprocal identities. Results are presented for both a homogeneous as well as a layered plate.
INTRODUCTION
Wave propagation in an infinite, elastic plate has been studied since the last century.
Practical applications include the ultrasonic nondestructive evaluation of defects and the
characterization of a material's properties. Two different approaches have been generally
employed. One is a wave spectra analysis based on the frequency equation. The other
approach involves fundamental elastodynamic solutions arising from the Green's function
which corresponds to a structure's dynamic responses to a source load. It is envisioned that
a Green's function is essential for applying the boundary element method (BEM) to study
the effects of cracks and other flaws in a plate.
The elastodynamic Green's function for a homogeneous isotropic plate has been
investigated for many years. See, for example, the comprehensive literature review by
Miklowitz [1]. Most previous studies deal with two-dimensional problems (i.e. the planestrain or plane-stress cases). Weaver and Pao [2], for instance, studied the dynamic
responses of an isotropic plate to a vertical load in which axisymmetry holds. Fundamental
three-dimensional transient solutions, on the other hand, have been developed by
Ceranoglu and Pao [3]. Their analysis is based on a generalized ray theory and Cagniard's
method. The latter is used to invert the double transform arising in the solution. In the
generalized ray method, however, only a finite number of rays are presumed to be
important at some finite instant. However, many rays have to be considered in practice so
that numerical calculations are tedious for late instants. The transient solution to an
embedded dislocation can be found in Vasudevan and Mal [4]. They employed an integral
transform technique that is also very cumbersome numerically.
CP657, Review of Quantitative Nondestructive Evaluation Vol. 22, ed. by D. O. Thompson and D. E. Chimenti
© 2003 American Institute of Physics 0-7354-0117-9/03/$20.00
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The dynamic response due to a source load in an anisotropic plate has also been
reported in the literature. For example, two-dimensional transient wave propagation in an
anisotropic plate caused by a line load was studied by Scott and Miklowitz [5, 6] as well as
by Willis and Bedding [7]. Green and Green [8] and Green [9, 10] investigated the
transient stresses in a four-ply fiber-composite plate for both inextensible and extensible
fibers. Their solution was based on the contributions of solely the propagating modes. Liu
et al. [11] proposed a hybrid numerical method to compute the transient waves propagating
in a laminated anisotropic plate from a short duration line load and a vertical point load.
The displacement response was determined by employing a Fourier transform and modal
analysis. However, only 10 modes were considered in their computation. Zhu et al. [12]
and Zhu and Shah [13] presented a modal representation of the plane-strain, elastodynamic
Green's function for laminated composite plates. Contributions from all the modes gave
accurate results in both the near and far fields. Recently, the axisymmetric transient
dynamic response in a transversely isotropic plate was found by Weaver et al. [14]. In a
fundamental three rather than two dimensional solution, Mal and Lih [15] as well as Xu
and Mal [16, 17] employed a wavenumber integral approach to find the elastodynamic
response of a unidirectional composite plate to a vertical point load or a line load.
However, the wavenumber integral used involves fairly time consuming computations,
even for a single layer. To the authors' knowledge, no efficient numerical method has been
derived to compute the three-dimensional Green's function.
Attention is devoted in this paper to a modal representation of the three-dimensional,
elastodynamic Green's function in a layered isotropic plate due to a source point load
which may be arbitrarily oriented. The plate is composed of perfectly bonded lamina, each
of which has a distinct thickness and isotropic material properties. For each isotropic
lamina, the three-dimensional plane-wave propagation problem is decomposed into a
plane-strain problem in addition to a separate anti-plane problem. The complexities of an
arbitrarily laminated profile are circumvented for the layered plate by employing a finite
element modeling in the thickness direction (Dong and Huang [18]) to determine the wave
modes. For a plane-wave, the wave modes are obtained by using eigendata extracted from
two independent algebraic eigensystems; one for the plane-strain problem and the other for
the anti-plane problem. Explicit forms of the Green's function are constructed by first
summing the wave modes of a plane-wave and then superposing the plane-wave solutions
in the circumferential direction. The procedure detailed here will be designated as planewave superposition technique. A similar procedure can also be used to construct the
Green's function for laminated composite plates.
As a cross check, the axisymmetric Green's function of a homogeneous as well as
layered plate excited by a vertical point load is derived by using a hybrid model. On the
other hand, a cross check for a horizontal point load is made by considering the dynamic
reciprocal relation. The numerical results are in excellent agreement.
FORMULATION
The time-harmonic elastic wave propagation in a laminated isotropic plate is
considered next. The laminated plate has perfectly bonded isotropic laminas that are
separate entities, each enjoying distinct mechanical properties and thicknesses. The two
outer surfaces of the plate are considered to be traction free with a separation of 2H.
Assume that the waves travel in the x' direction, which has an angle ^ to the x direction of
the global (x, v, z) coordinate system shown in Figure 1. The three-dimensional, timeharmonic wave motion equation for each isotropic lamina can be written in the frequency
domain as
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Direction of
traveling wave
FIGURE 1. Plate configuration.
//V2u + (;i + //)V(Vni) + y9<y 2 u+b = 0
(1)
where /I and ju are Lame's constants, p is the mass density and CO is the circular frequency.
u = (u, v, w) and b = (bx, by, bz) are the displacement and body force vectors. Here, u, v,
and w represent the displacements whilst bx, by, and bz correspond to the body forces in the
jc, 3; and z directions, respectively.
Decomposition of the Problem
A semi-analytical finite element method is used to formulate the governing equations.
With this method, the x and 3; dependencies of u are accommodated by means of an
analytical double integral Fourier transform, with the z dependence approximated by finite
elements. The Fourier transform employs transform pairs that are defined in terms of the
wavenumbers in the x and 3; directions, kx and ky, respectively i.e.
+
v)'dkdk
(2)
where i = ^^-l. By applying transform (2) to the wave Eq. (1) and by defining k as a
wavenumber in the x direction which is oriented at angle <f> to the x axis (See Figure 1),
the transformed wave motion equation in the (x', y', z) coordinate system can be written as
dz2
dz
(3)
(4)
dz
with ks=k cos (Z* and fe = k sin 0. Eq. (3) describes a plane-strain problem in the x'-z
plane. Eq. (4), on the other hand, represents an independent anti-plane problem in which
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waves travel in the x' direction. Consequently, the three-dimensional wave propagation
problem in isotropic plates is decomposed into separate plane-strain and anti-plane
problems. This decomposition is important because a complicated three-dimensional
problem is solved by merely superimposing the solutions of two simpler problems having
lower dimensions. Moreover, the plane-strain wave propagation problem has been
investigated already by efficiently using two-dimensional elastodynamic Green's functions
in modal representation (Zhu et al. [12, 19]).
Construction of the Green's Functions of Plain-Strain and Anti-Plane Problems
To solve Eqs. (3) and (4), the z-dependence is found by using the finite element
modeling detailed by Dong and Huang [18]. Then, by using a conventional finite element
procedure, the discretized governing equations over the complete thickness profile of the
plate can be derived for the plane-strain and anti-plane problems as
(fcX -ikpKp2 +K p 3 -02Mp)Qp =¥A
(5)
(fcX,+K a 3 -ft> 2 M a )Q 0 =F y .
(6)
Subscripts p and a indicate the plane-strain and anti-plane problems, respectively. Kpi,
K a / , M^ and Ma are the corresponding assembled global stiffness and mass matrices for
all the lamina. On the other hand, Q/; =(U' r , W' T ) T and Q f l = V ' , and F^ and fy,
represent the total (global) nodal displacement and traction vectors in the transformed
domain, respectively. Note that K/?i, K/?3, K a i, Ka3, M^ and Ma are symmetric but K^ is
antisymmetric.
The corresponding general eigen-solutions of Eqs. (5) and (6) are given in terms of the
eigenvalues (denoted as kpm and kam), the associated left eigenvectors, <j>Lpm and $f w , and the
right eigenvectors, 0*m and 0*m. By following the modal summation technique of Liu et al.
[11] as well as Liu and Achenbach [20] and applying the orthogonal conditions of the left
and right eigenvectors, the displacement vectors for the plane-strain and anti-plane
problems in the transformed domain can be constructed by a summation of the eigendata.
The detail can be found in Zhu et al. [12].
3D Green's Function
A steady-state unit load is located, without loss of generality, at the origin of a plane,
distance ZQ from the plate's top exterior surface, in order to construct the Green's function.
The spatial representation of this load, f(jc, y), takes the form
t(x,y)=tQS(x)S(y)
(7)
where f 0 = (/ x , fy, jf ) represents the amplitude of the load in the x, y and z directions,
respectively. Therefore, in the finite element formulation, the corresponding external nodal
force vectors for the plane-strain and anti-plane problems, F/z and F y /, respectively,
contain zero entries except at the load's nodal surface z = zo> The (x, y', z) components in
the plane-strain and anti-plane problems are (fx',fz) and fy>, respectively, so that,
^((V-.O^O-.O; 0,...,0,/ z ,0...,0) T , F = 0 , - , 0 , / / , 0 ^ . , 0 r
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(8)
Load in the z direction. A unit load in the z direction, f' = (0,0,1) applied at (0, 0, zo)
gives
(0 L m/ )
Fy —\l/
,
((ftaml]
F''=0.
(9)
L
Here ^zp corresponds to the element, $ pml, for the plane-strain problem. It should be noted
that, in this case, the displacements have no contribution from the anti-plane problem.
The nodal displacement vector in the transformed domain can be constructed in the
cylindrical coordinate system (r, 9, z) illustrated in Figure 1 by using plain strain Green's
functions. Applying the inverse Fourier will give the displacements in the spatial domain
as (Stratton [21])
,+ _..
(10)
I
ll ^/"ll
\ — ' / \Kp,p) •" '
i"11 \
/'
i"11 s\\
yy
n
Zhu and Shah [13] demonstrated that only half the 2MP modes associated with the waves
propagating from the source towards the field points are relevant to the plane-strain
problem. Therefore, by introducing <p = <p-0, (-x/2<(p<7r/2), and by applying
Cauchy's residue theorem, Eq. (10) can be simplified to
~ur|l
(ID
It should be noted that the Hn in Eq. (11) are not Hankel functions. They take the form
in the x direction. A unit load in the x direction, i = (l, 0,0) , is applied at (0, 0, zo)The Green's function can be derived by following the similar procedure in the last section
and it is given by
"[W'J
(13)
0
The Green's function due to a load in the y direction, on the other hand, can be obtained by
simply interchanging .s;'n#and cc«#and changing the sign of M^in Eq. (13).
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NUMERICAL RESULTS AND DISCUSSION
Numerical examples are presented in this section for the displacement Green's
functions of a homogeneous isotropic plate as well as a layered plate.
Example 1. A homogeneous isotropic plate with Poisson's ratio v= 1/3.
Example 2. A 3-ply laminated plate that is symmetric about the middle plane. Layers 1 and
3 are at the top (z = 0) and bottom (z = 2H) of the plate, respectively, with layer 2 in the
middle. The outer layers are assumed to be aluminum and the middle steel. The material
constants and geometric data are given by Poisson's ratio, v, the shear modulus, //, mass
density, /?, and laminate thicknesses, //, as
V2 = 0.28, //2 = 75xl09Mm2, p2 = 7.8xl03^/m3, H2 = 1.6H
9
2
3
= jff, = 0.2H
Vl =v3 =0.32, j i = jU3 = 25xl0 Mm , pi = p3 = 2.1xW Kg/m\
where a subscript denotes the corresponding layer.
The normalized frequency £1 = coH/^jU/p is used for Example 1. It is replaced by
Q, = G)H/^jU2/p2
for Example 2. Two loads are considered for both plates. One
corresponds to a vertical load, the other to a horizontal load in the x direction. These loads
act separately at the same location (0, 0, 0.5//). Axisymmetry about the z axis holds for the
vertical load, but not for the horizontal load. In the hybrid model, the finite element mesh
contains 20 quadratic elements and 85 nodes. The boundary is located at r0 = 0.2/f. Forty
modes are employed in the wave function expansion in the exterior region.
Results are given in Figures 2 and 3 for Example 1. Figures 2(a) and 2(b) compare the
vertical displacement (w) computed through the thickness at sections x = 2H and x = IOH
for a frequency £1 = 10. Both the hybrid method as well as the plane-wave superposition
technique are used in this Figure. It also shows that the data from the two approaches agree
well.
Figure 3 presents the vertical displacement distribution through the thicknesses at x =
2H and WH when a horizontal x direction load acts at (0, 0, Q.5H) for Q, = 10. The
dynamic reciprocal identity (Betti-Rayleigh Theorem [22]) is used to check the validity of
(b)
0.00
I
o.oo
"a.
15
-0.01
-0.02
0.0
0.5
1.0
1.5
2.0
z/H
FIGURE 2. For a vertical load, vertical displacements for a homogeneous plate for Q, = 10: (a) x = 2H\ (b) jc
= IOH (——, Re(w) from hybrid method; —, 7m(w) from hybrid method; A, Re(\v) from superposition
method; o, /ra(w) from superposition method).
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(b)
-0.1 0
2.0
z/H
FIGURE 3. Reciprocity check for a homogeneous plate's displacement for Q. = 10: (a) x = 277; (b) x =
1077 (——, Re(w) due to a horizontal load; —, 7m(w) due to a horizontal load; A, Re(w) due to a vertical
load; o, 7m(w) due to a vertical load).
(b)
0.0
0.5
1.0
1.5
2.0
z/H
FIGURE 4. For a vertical load, vertical displacements for a 3-ply laminated plate for Q. = 10: (a) x - 277;
(b) x = 1077 (——, Re(w) from hybrid method; —, 7m(w) from hybrid method; A, Re(w) from superposition
method; o, 7ra(w) from superposition method).
the results. For example, in Figure 3(a) the solid (—) and dashed (—) lines indicate the
vertical displacement through the thickness at x = 2H when a horizontal load acts at (0, 0,
0.5#). The distribution was evaluated by using Eq. (13). The same results are recovered by
applying vertical load at (2H, 0, zo) where zo is at one of the (27V+1) node points. The
results, shown by circle (o) and triangle (A) in the figure, are calculated with the aid of Eq.
(11). Similar results are presented in Figures 3(b). From these figures it can be inferred that
satisfaction of reciprocity relations confirms the correctness of the proposed superposition
algorithm. As observed earlier, the results for the case of Q = 10 exhibit oscillatory
behavior.
To demonstrate the versatility of the plane-wave superposition method, the results for a
3-ply laminated plate of Example 2 are given in Figure 4. The overall behavior of the
displacement distribution is shown to be similar to that of the homogeneous plate
considered in Example 1.
CONCLUSIONS
A plane-wave superposition method is used to compute the three dimensional, steadystate Green's function for a laminated plate. It is validated for a vertical load by the
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invariably close agreement with independently obtained results from a hybrid finite
element method. Such a validation is not possible with the hybrid modeling presented here
for a horizontal load. However, it is feasible to check the displacement reciprocity relation
which is satisfied. Moreover, the general displacement behavior of the homogeneous and
laminated plates considered is shown to be similar.
ACKNOWLEDGMENTS
The work presented in this paper is supported by the Natural Science and Engineering
Research Council of Canada (grant OGP007988).
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