Extensional and Flexural Waves in a Thin-Walled Graphite/Epoxy
Tube
*
William H. Prosser
NASA Langley Research Center
Hampton, VA 23665
Michael R. Gorman
Aeronautics and Astronautics
Naval Postgraduate School
Monterey, CA 93943
John Dorighi
**
NASA Langley Research Center
Hampton, VA 23665
Journal of Composite Materials
Vol. 26(14), 1992, pp. 418-427
Abstract
Simulated acoustic emission signals were induced in a thin-
walled graphite/epoxy tube by means of lead breaks (Hsu-Neilsen
source). The tube is of similar material and layup to be used by
NASA in fabricating the struts of Space Station Freedom. The
resulting waveforms were detected by broad band ultrasonic
transducers and digitized. Measurements of the velocities of the
extensional and flexural modes were made for propagation
directions along the tube axis (0 degrees), around the tube
circumference (90 degrees) and at an angle of 45 degrees. These
velocities were found to be in agreement with classical plate
theory.
*
Work supported by NASA Langley Research Center
**
Mechanical Engineering student, University of Colorado,
Boulder, CO 80309-0427
Introduction
Graphite/epoxy composites, because of their high strength, high
stiffness, and light weight, have been chosen for fabrication of
the strut tubes of Space Station Freedom (SSF). While exposed to
the harsh environment of space, these tubes will be subjected to
hypervelocity micrometeriod impacts and large thermal cycling,
among other things. Thus, some method of monitoring these tubes
is needed while SSF is in orbit. Other critical parts which need
to be monitored include the various fiber wrapped pressure vessels
used for life support and fuel containment purposes.
Acoustic emission (AE) testing has been proposed as a technique
for nondestructively monitoring these structures. The global
monitoring capability of AE makes it well suited for this purpose.
However, much information about the generation and propagation of
acoustic emission signals in composites and composite tubes is
needed before AE techniques can yield useful quantitative
information.
Gorman and Ziola [1] demonstrated that simulated AE signals in
flat composite plates consist of the lowest order Lamb modes.
These modes are often referred to as plate modes. The lowest
order symmetric mode reduces to what is called the extensional
mode in plate theory, while the lowest order anti-symmetric mode
reduces to the flexural mode; these terms apply when the plate is
thin, that is, the wavelength is much greater than the thickness.
The signals were generated by pencil lead breaks (also known as
Hsu-Neilsen sources). Gorman and Ziola [1] further showed that
real AE signals generated by transverse matrix cracking in a
composite propagate as plate modes.
The propagation of Lamb and plate modes in composite plates has
been studied by numerous investigators. Noiret and Roget [2]
investigated the case of long wavelength and low frequencies.
Chimenti and Nayfeh [3, 4] have extensively studied Lamb mode
propagation for ultrasonic materials characterization
measurements. Veidt and Sayir [5] also characterized the material
properties by measuring the flexural plate mode velocity.
Stiffler and Henneke [6] investigated plate modes in an attempt to
better understand the acousto-ultrasonic technique. A number of
investigators have studied plate waves generated by the impact of
a composite including Moon [7], Chow [8], Sun and Lai [9], and
Rose and Mortimer [10].
With the exception of the study by Rose and Mortimer [10] who
also studied composite shells or tubes, the previous research has
all focussed on the simple geometry of a flat plate. The majority
of practical structures, such as the strut tubes studied in this
research, are of more complicated geometries. This research
demonstrates, however, that AE signals propagate in these thin-
walled tubes as plate waves and that classical plate theory is
adequate for the prediction of their propagation velocities.
Gorman and Ziola [1] further pointed out that an understanding
of the propagation of plate modes with different velocities is
needed to improve the location capability of AE techniques. They
also investigated the effect of source orientation by breaking
pencil leads on the surface and on the edge of the plate. An
interesting feature of these plate modes was that the amplitude of
the displacement components of the different modes was dependent
on the source orientation for the simulated AE sources. For the
composite plates, if the lead was broken on the surface, the
resulting waveform had a large out-of-plane component for the
flexural mode and a much smaller out-of-plane component for the
extensional mode. For lead breaks on the edge of the plate, the
out-of-plane component of the extensional mode was much larger
than that of the flexural mode. Gorman [11] showed similar
results on aluminum plates for surface and edge breaks.
Gorman and Prosser [12] additionally demonstrated that the
amplitudes of the displacement components of the extensional and
flexural modes were dependent on source orientation for
intermediate angles on aluminum plates. This was accomplished by
machining angled slots into the plate on which the lead was
broken. This feature of plate waves may be useful in determining
the type of source and its orientation for actual AE signals. For
example, the observed out-of-plane displacement components of
plate waves due to transverse matrix cracking in the work of
Gorman and Ziola [1] had extensional amplitudes which were much
larger than the flexural. This makes sense since transverse
matrix cracking should produce a source motion in the plane of the
plate generating a larger extensional wave.
In this research, the out-of-plane components of AE signals
produced by lead breaks on a graphite/epoxy tube of dimensions and
design to be used on SSF were measured using broad band
transducers. The signals were shown to consist of plate modes.
The amplitudes of the modes were again shown to be affected by
source orientation by creating lead breaks on the edge as well as
the surface of the tube. Since the specimen being tested is
actually a tube geometry, the plate wave predictions are only an
approximation. However good agreement might be expected at the
frequencies measured since the wavelengths are smaller than the
radius of curvature. Measurements of the velocities of both modes
were made for propagation along the tube (0 degrees), around the
tube circumference (90 degrees), and at 45 degrees. These
velocities were then compared with velocities predicted from
classical plated theory using stiffness coefficients predicted by
laminated plate theory and good agreement was demonstrated.
Theory
Classical plate theory predicts three modes of propagation in a
plate [13]. These are called the extensional, the in-plane shear
and the flexural modes. This theory is based on the assumption of
plane stress in a thin plate where the wavelength is large in
comparison with the plate thickness. The in-plane shear mode is
not detectable when using a transducer which is sensitive to out-
of-plane displacement and is mounted on one of the faces of the
plate. Thus, it was not observed previously by Gorman and Ziola
[1] and has not been detected in this research. Therefore, the
theoretical treatment of this mode is not presented here.
While classical theory predicts only an in-plane displacement
component for the extensional mode, higher order theories predict
[14, 15] and experimental measurements verify the existence of an
out-of-plane displacement component for this mode. This out-of-
plane component is due to the Poisson effect from the in-plane
motion and thus propagates at the same velocity. In a homogeneous
isotropic material the extensional mode propagates with a velocity
c
e
which is dispersionless:
c
e
=
E
r(1-n
2
)
(1)
where E is the Young's modulus,
n is Poisson's ratio, and r is the
density.
In anisotropic composite materials the extensional velocity is
dependent on the in-plane stiffness coefficients A
ij
, i, j = 1, 2,
and 6 in the usual contracted notation. These are referred to the
laminate axes, x and y, which are in the plane of the plate. The
A
ij
are defined as
A
ij
= Q
ij
(k)
dz
z = -
h
2
z =
h
2
(2)
where h is the thickness of the laminate, z is the distance in the
thickness direction from the midplane of the laminate, and Q
ij
(k)
are
the lamina stiffness coefficients for the k
th
lamina of the
composite. For a +/- angle-ply symmetric laminate, only four of
the in-plane stiffness coefficients are non-zero. These are A
11
,
A
12
, A
22
, and A
66
. The equations of motion for the in-plane
displacements for a material with this symmetry are given by
Whitney [16] as
