SIMULATIONS OF LASER-GENERATED GUIDED WAVES IN
TWO-LAYER BONDED PLATE WITH A WEAK INTERFACE
Jun Du and Jianchun Cheng
Institute of Acoustics and National Laboratory of Modern Acoustics
Nanjing University, Nanjing 210093, RR. China
ABSTRACT. Numerical simulations of laser-generated guided elastic waves in two-layer
bonded plates are presented by modeling adhesive bond layer as a spring model. The transient
waveform is expressed by the two-layer normal mode expansion method (NME), and then the
sensitivity of each mode and transient waveform on the stiffness coefficients characterizing
cohesive quality are analyzed in details. This method provides a new promising way for the
characterization of the cohesive quality in bonded plates.
INTRODUCTION
It is known that ultrasonic guided wave is powerful in the characterization of the
cohesive quality in bonded plates. But its success in measuring bond strength largely
depends on the understandings of the nature of the imperfect interface and propagating
characteristics of ultrasonic guided waves. For thin bond layers, the spring model has been
widely used and extensively analyzed by many researchers [1-3]. In most discussions on
the guided waves propagating in a sandwich plate, main works are concerned with effects
of spring model on dispersion of guided waves. Xu and Datta [4] have done a parametric
study for dispersion of elastic guided modes. Wang and Ning [5] derived the characteristic
equation for Lame waves in a two-layered solid composite considering rigid and slip
boundary conditions. In recent years, there are some of studies analyzing transient guided
waves in bond plates, most of which regard interfaces perfectly joint. Seifried, et al [6]
have developed a quantitative understanding of the propagation of guided Lamb waves in
multi-layered, adhesive bonded components, where they considered the continuity of
displacement and stress at each interface. Also Cheng, et al [7] applied the continuity
conditions at layer interfaces when they simulated laser-generated ultrasonic waves in
layered plates using the numerical inversion of the Hankel-Laplace transform solution.
In an effort to simulate laser-generated guided waves by employing the normal mode
expansion (NME) method, which has applied successfully to single plates [8], we have
analyzed the orthogonality of elastic guided wave modes for two-layer bonded plates [9].
The results showed that the elastic operator is a self-adjoint operator under the spring
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/S20.00
165
model, and this model is more reasonable than the density model in physical intrinsic sense
to model thin bond layers, because the spring model does not alter the adjoint feature of the
elastic operator.
In this paper, we present a quantitative theory to simulate laser-generated transient
waveforms by using the two-layer NME method for two-layer bonded plates with a weak
interface modeled by the spring model. The sensitivity of each mode and transient
waveform on the stiffness coefficients characterizing cohesive quality is analyzed in details
in the numerical simulations. The results show that the theory provides a quantitative tool
to characterize the elastic stiffness properties of the bond layer by laser-generated Lamb
wave detection.
THEORY
Basic Equations
We consider an infinite sandwich plate as shown in Figure 1 together with a
coordinate system (x,y,z). The materials in top and bottom layers are assumed to be linearly
elastic, homogenous, and isotropic. The middle layer, i.e., cohesive layer, is modeled by
weak interface with the spring model. The layer parameters are denoted by /?/ (density), /I/
and jUi (elastic moduli), where the subscripts /= 1,0,2 are associated with the top layer,
bonding layer, and bottom layer, respectively.
For an isotropic elastic material, the time-dependent elastic wave equation is
) + //V2u - /7u-f (r,0
L(u) =
(1)
where u(r,0 is the time-dependent displacement at position r=(x,y,z), and f(r,0 is the
applied body force. For propagation of elastic waves with cylindrical symmetry, the
displacement vector can be expressed by
u(r, z, 0 = ur (r, z, f )er + M Z (r, z, f )ez
(2)
where er and ez are two unit vectors. The ur and uz satisfy
-hi
0
Material 1
Interface 0
Material 2
+fa
FIGURE 1. A sandwich plate
166
The spring model has been used to study propagating characteristics of elastic waves in
such sandwich plate. In this model, the interface is able to transmit continuous stresses, but
allows a jump in displacement components, so that the condition at the interface z=0 is
ala=ff2a,
<=<T*
<=(«! -«*)*„. <=(«!-«?)*,
Here arz and azz are components of the stresses, and the superscripts "1" and "2" denote the
top layer and bottom layer, respectively. The kn and kt are the stiffness coefficients
characterizing cohesive quality. The boundary conditions at upper surface z--h\ and rear
surface z=fe are traction-free. Two kinds of limiting interface boundary can obtained by
changing the stiffness coefficients & n and kh i.e., the slip interface (^approaches infinite
and kt approaches zero) and the rigid interface (kn and kt approach infinite).
By introducing the Hankel transforms
Equation (3) can be reduced to an operator form of matrix
L[u] = /4^-f,
(6)
o t
where u = [w r ,w z ]' is the displacement column vector (the superscript 'V denotes
transpose), f =(/ r ,/ z )' is the body force column vector, and the elements of
the 2 x 2 matrix operator L=[///] are
z-/^
The relations between stress components and \ur , uz ] are
(8)
Two-Layer NME Method
Usually, Equation (6) is solved by using the temporal Laplace transformation.
However, this technique requires a numerical inverse transform integral evaluated by the
calculus of residues, which in the first approach involves the understanding of the intricate
behavior of dispersion relations for real and complex wave numbers. Here we present the
two-layer NME method to solve this equation. We define an eigen-function series {Am, ^m,
w=l, 2, 3, . . . } by the eigen-value problem of the operator L
167
=0
(9a)
under the boundary conditions
8^ = 0, an = 0 , at z = -A, and z = ^
^=d£. o£=a£, a t z = 0±
(9b)
5L = K -"£)*,. o£ = (vl -v* )*„ at z = 0±
where com is the eigen-frequency corresponding to the eigen-mode Am =[vm,wm]'. Because
the operator L is a self- adjoint operator under the boundary conditions, the eigen-function
series {Am} form an orthogonal set with the weighting function p(z) [9]
dz = SmnMm
(10)
where is Mm the norm of the eigen-function. It has been proven that {Am} is also a
complete function series [10], so that the displacement column vector ucan be expanded
by the generalized Fourier series
9a)m,t)Am(k9G)m,z).
(11)
For non-destructive detection, a thin coating of oil at the generation spot can be
added to increase the acoustic generation efficiency from the laser pulse with lower laser
power densities. In this case, evaporation of the oil becomes dominant sources. The
generation mechanism of acoustic sources by evaporation of the oil is that momentum
is transferred from the evaporating particles to the solid surface. The evaporating source
is dominated by a normal mono-pole force with a time dependence of Dirac delta 8(0- For
simplicity, we consider that the bulk force f only is located at the surface and depends on
the energy intensity of the laser pulse
f (r, z,0 = -iy*-'2/"2 S(t)S(z + /Oez .
(12)
Here a is the laser beam radius. Finally, we can obtain the generalized coefficients
£. (*. z. 0 = -
e'*V/V (k, com , z)\ ^ sin<V
Then the time domain response can be obtained through an inverse Hankel transform from
u to 11(7; z, t).
NUMERICAL ANALYSES
In the numerical simulations, the elastic constants, thickness and densities are
Xi=54.58 GPa, m=25.61GPa, X2=106.52GPa? jLi2=45.46GPa, /z1=0.045mm, /z2=0.5mm,
pi=2.7x!03kg/m3, and p2=8.9xl03kg/m3, respectively. The stiffness coefficients kn and kt
are related to the elastic constants (^O,UQ) and thickness ho of the cohesive layer by
^-(2// 0 +^)/V,
*,=//„//%
(14)
so that sensitivities of each mode and the total waveforms on the physical parameters
characterizing cohesive quality can be analyzed by changing the stiffness coefficients.
168
Dispersion Curves
Although the specimen we analyze is not an absolutely symmetric composite, we
still use aj and Sj (/=0,1,2,...) to denote the modes of the dispersion curves for simplicity.
FIGURE 2. Dispersion curves of the Lamb waves for the first few modes in the plate, (a) the elastic
constants and thickness of the cohesive epoxy layer Ao=5.5GPa, |Oo=2.7GPa and /z0=10uin; (b) a rigid
interface; (c) slip interface.
169
Z
-0.6
0.02
0.04
0.06
Time(ms)
0.04
Time(ms)
FIGURE 3. The response of a bonded plate with a rigid interface (a); the spring model with the stiffness
coefficients kn and &,(b); a bonded plate with a slip interface (c).
Figure 2(a) depicts the dispersion curves of the Lamb waves for the first few modes
in the plate with the elastic constants and thickness of the cohesive epoxy layer Xo=
5.5GPa, uo=2.7GPa and /z0=10|im5 respectively. The normalized phase velocity and
frequency are given by c = c/c rl and ffi = 0)h21 ctl, respectively. In this case, we can see
that the curves are quite similar to the general ones for a classical single plate, except that
there is an additional curve denoted as Sm below the a0 mode. By calculating the spatial
distribution of the displacement in this mode, we found that it is mainly due to the particle
170
vibration in the weak interface.
Figure 2(b) and Figure 2(c) present the normalized phase velocity distributions for
the guided wave modes in the bonded plate with a rigid and slip interface, respectively. It
can be seen that the interface condition has great influence on the dispersion characteristics
of the Lamb wave modes in a two-layer bonded plate. The dispersion curves for the
composite plate with a rigid interface are quite similar to those of a single plate. However,
when the interface become slip, the curves change greatly and there exist two s0 modes.
Transient Response
Equations (11) and (13) is used to calculate the transient response of the two-layer
bonded plate with a weak interface modeled by the spring model. The material properties
used in our calculation are the same as those listed above. In addition, the
source-to-receiver distance is 80mm and the laser beam radius is 0.2mm.
Firstly, we calculate the response of a bonded plate with a rigid interface. The time
domain waveform is shown in Figure 3(a). The waveform looks very similar to the ones
propagating in a single plate. The initial arrival shows the s0 mode and then the aa mode.
Next, the normal displacement at the top surfaces is calculated with the following
configurations: aluminum(0.045mm)/epoxy(0.01mm)/copper(0.5mm). The weak interface
is modeled by the spring model with the stiffness coefficients kn and kt decided by Equation
(14). From Figure 3(b), we can see that the Lamb wave of the sm mode have arrived before
the arrival of the s(, mode. Also we calculate the normal displacement of a bonded plate
with a slip interface. In the waveform in Figure 3(c), we find that the Lamb wave of the sm
mode is the first arrival, and then sol arrives.
The differences among three laser-generated transient waveforms show that the
presence of the epoxy layer has marked effects on the transient waveform. This sensitivity
analysis of the waveforms to variations of the physical parameters is crucial to evaluate
unknown material properties of bond layers in composite plates. Based on this study, we
can extract properties of bond layer from measured data by using an inverse technique.
CONCLUSION
The transient laser-generated guided waves in two-layer bonded plates with a weak
interface modeling by the spring model have been studied by developing the two-layer
NME method in this work. The waveforms in bonded plates with different weak interfaces,
namely, rigid interface, slip interface and spring interface, are obtained and compared. The
sensitivity of each mode and the total waveform on the stiffness coefficients characterizing
the cohesive quality are discussed in details. This method provides a new promising way
for the characterization of the cohesive quality in bonded plates. In the future work, we will
use the model to extract the unknown properties of the bond layer with an inverse method.
ACKNOWLEDGMENTS
This work is supported by the National Natural Science Foundation of China.
171
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
Liu Sheng-Xing and Wang Yao-Jun: Chin. Phys. Lett. 17(2000) 277.
Y Tsukahara and K. Ohira: Ultrasonics. 27(1989) 3.
S. Rokhlin and D. Marom: /. Acoust. Soc. Am. 80(1986) 585.
RC. Xu and S.K. Datta: /. Appl. Phys. 67 (1990) 6779.
Wang Yao-jun, Ning Wei, and Ou Xian-hua: ACTA. Physica. Sinica. 3(1994) 561.
R. Seifried, L.J. Jacobs and J. Qu: NDT&E Int. 35 (2002) 317.
A. Cheng, T.W. Murray and J.D. Achenbach: /. Acoust. Soc. Am. 770(2001) 848.
J.C. Cheng, S.Y. Zhang, and L. Wu: Appl Phys. A61(1995) 311.
J. Du and J.C. Cheng: in Rev. Prog. Quant. Nondestru. Eval. Vol. 21, ed. By D. O.
Thompson and D. E. Chimenti (2001)R189.
10. A.C. Eringen and E.S. Suhubi: Elastodynamics, Linear Theory, (Academic, New York
1975) Vol.2.
172