U.S. patent number 5,115,673 [Application Number 07/556,584] was granted by the patent office on 1992-05-26 for non-destructive method for determining elastic moduli of material.
This patent grant is currently assigned to The Board of Regents of the University of Oklahoma, The United States of America as represented by the United States. Invention is credited to Ronald A. Kline, Eric I. Madaras.
United States Patent |
5,115,673 |
Kline , et al. |
May 26, 1992 |
**Please see images for:
( Certificate of Correction ) ** |
Non-destructive method for determining elastic moduli of
material
Abstract
A non-destructive method for determining elastic moduli of
isotropic or anisotropic or homogeneous or nonhomogeneous material.
The material is subjected to x-radiation for determining the
density of the material at a sufficient number of discrete
measurement points over the material to create an image of local
material density variation. Ultrasonic waves are propagated through
the material to determine transit times for each wave at points
corresponding to the measurement points. Using the determined
density and transit times for each of the measurement points, the
elastic moduli at each measurement point is determined. The elastic
moduli provides a means for analyzing the mechanical performance of
the material. In one aspect, the determined elastic moduli are
inputted into a finite element method code for determining
mechanical response of the material.
Inventors: |
Kline; Ronald A. (Norman,
OK), Madaras; Eric I. (York Town, VA) |
Assignee: |
The United States of America as
represented by the United States (Washington, DC)
The Board of Regents of the University of Oklahoma
(N/A)
|
Family
ID: |
24221961 |
Appl.
No.: |
07/556,584 |
Filed: |
July 20, 1990 |
Current U.S.
Class: |
73/601; 73/597;
73/602 |
Current CPC
Class: |
G01N
9/24 (20130101); G01N 29/07 (20130101); G01N
29/2431 (20130101); G01N 29/4472 (20130101); G01N
2291/0428 (20130101); G01N 2291/02818 (20130101); G01N
2291/02827 (20130101); G01N 2291/0421 (20130101); G01N
2291/0422 (20130101); G01N 2203/0005 (20130101) |
Current International
Class: |
G01N
29/24 (20060101); G01N 29/04 (20060101); G01N
29/07 (20060101); G01N 29/44 (20060101); G01N
9/24 (20060101); G01N 3/00 (20060101); G01N
029/04 () |
Field of
Search: |
;73/597,601,602,624,627,787,798 ;250/252.1R |
References Cited
[Referenced By]
U.S. Patent Documents
Other References
The Analysis of Fibre-Reinforced Porous Composite Materials by the
Measurement of Ultrasonic Wave Velocities--Ultrasonics, Jul. 1978,
5 pages, .COPYRGT.1978 IPC Business Press Ltd..
|
Primary Examiner: Williams; Hezron E.
Assistant Examiner: Arana; Louis M.
Attorney, Agent or Firm: Dunlap, Codding & Lee
Claims
What is claimed is:
1. A non-destructive method for determining elastic moduli of an
isotropic or anisotropic or homogenous or nonhomogeneous material,
comprising:
subjecting the material to x-radiation to produce an image,
digitizing the image and determining density at a sufficient number
of discrete measurement points over the material to substantially
create an image of local material density variations;
propagate ultrasonic waves through the material at multiple angles
of incidence and determine transit times for each wave at points
corresponding to the measurement points; and
determining for each measurement point all of the independent
elastic moduli at each measurement point using the densities
determined by subjecting the material to x-radiation and the
transit times determined by propagating ultrasonic waves through
the material.
2. The method of claim 1 further comprising:
inputting the elastic moduli for each measurement point into a
finite element method code for determining mechanical response of
the material.
Description
FIELD OF THE INVENTION
The present invention relates generally to a method for determining
elastic moduli of material using x-radiation to determine the
density of the material at discrete measurement points and
ultrasonic waves propagated through the material to determine
transit times for each of the measurement points and determining
elastic moduli at each of the measurement points from the
determined transit times and densities of the respective
measurement points.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 diagrammatically illustrates the method of the present
invention.
FIG. 2 is a diagrammatic illustration of a stator coordinate
system.
FIG. 3 is a schematic, diagrammatic view of an experimental
apparatus used to determine transit times in accordance with the
method of the present invention.
FIG. 4A-B is a diagrammatic, schematic view of transducer
orientations used for scans in an experimental test of the methods
of the present invention.
FIG. 5 is an image showing local density variations in a material
and a standard or calibration reference.
FIG. 6 represents the experimental results for the elastic moduli
(see C.sub.11 vs position).
FIG. 7 represents the experimental results for the elastic moduli
(see C.sub.22 vs position).
FIG. 8 represents the experimental results for the elastic moduli
(see C.sub.33 vs position).
FIG. 9 represents the finite element mesh applied to the stator
geometry (see Finite Element Mesh-Stator).
DESCRIPTION OF THE PREFERRED EMBODIMENTS
The present invention provides a method for analyzing the
mechanical performance of isotropic or anisotropic; or homogeneous
or nonhomogeneous materials in a non-destructive manner. For
example, the method of the present invention provides a means for
analyzing the load bearing capability of the material in its
service environment.
The method of the present invention generally is illustrated in
FIG. 1 of drawings. The material is subjected to x-radiation to
produce an image, the image is digitized and the density at a
sufficient number of discrete measurement points over the material
is determined to substantially create an image of local material
density variations. In this x-radiation technique, a calibration
reference also is imaged which provides a standard. The image is
then corrected for machine dependent artifacts and the density is
determined at a sufficient number of discrete measurement points
over the material, the number being sufficient to create an image
of local material density variations.
Ultrasonic waves are transmitted through the material at multiple
angles of incidence. Transit times are determined for each wave at
points on the material corresponding to the measurement points.
Using the previously determined densities and the transit times,
all of the independent elastic moduli at each measurement point
then are determined. The elastic moduli then can be used to analyze
the mechanical performance of the material.
In one embodiment, the determined elastic moduli for each
measurement point are inputted into a finite element method code
for determining mechanical response of the material.
It should be noted that the ultrasonic waves propagated through the
material in accordance with the method of the present invention can
be via through transmission or pulse echo. Further, the x-ray
imaging technique can be a two dimensional or a three dimensional
type of system. However, the ultrasonic wave portion of the present
invention as described herein is limited to a two dimensional
analysis.
The use of x-radiation for determining density values at discrete
measurement points over a material is well known in the art and a
detailed description of the apparatus and methods used in
connection with such systems is not deemed necessary. Further, the
use of finite element methods and the use of finite element method
codes are well known in the art and a detailed description of such
methods and codes is not deemed necessary. One such finite element
method code suitable for use with the present invention is referred
to in the art as "NASTRAN".
The introduction of high strength/low weight composite materials
has dramatically changed aircraft design in recent years. And the
situation is not static with new materials constantly being
introduced along with increased demands on the performance
capabilities of composite materials. This is particularly true for
emerging aircraft where expected performance requirements exceed
the capabilities of currently available materials. While high
strength composite materials are commonly utilized for weight
critical applications and there are existing materials with
elevated temperature capability, there is no accepted material
system which combines these two required attributes. In particular,
as illustrated in FIG. 1. While carbon-carbon composite structures
have been routinely utilized at elevated temperatures, these
materials have principally been used as thermal protection rather
than as load-bearing structural elements since mechanical
properties of conventional carbon-carbon structure will degrade
substantially at elevated temperatures, principally due to
oxidation. One possible solution to this problem is to employ a
thin surface coating to serve as an oxidation barrier. This has led
to a new class of carbon-carbon materials which hopefully will
preserve the intrinsic performance capabilities of carbon-carbon
composites at extreme temperature and offer a possible solution to
the material selection problem presented by proposed aircraft
designs. The purpose of the present invention is to provide a
nondestructive method for characterizing the material properties of
these composites.
The governing equations for wave propagation in anisotropic media
are relatively straightforward. Linear momentum considerations
require that the following ligenvalue equation be satisfied:
where
I=identity matrix
.rho.=density
m=1 /V=slowness vector
1=wave normal
V=phase velocity
.alpha.=direction cosines of particles displacements
the elements of the tensor .lambda. are given by .lambda..sub.ik
=.sup.c ijkl.sup.m j.sup.m l/.rho.
Next, one must find the waves which must be generated to satisfy
the boundary conditions at the interface between the two media.
Continuity of particle displacement and surface traction at the
interface requires the typical case): ##EQU1## where:
t=.alpha..nu.
v=normal to interference (direction cosines)
The superscripts in, re, and t correspond to the incident,
reflected and transmitted waves and the subscript i=1, 2, 3
corresponds to the three possible reflected and transmitted waves.
Assuming plane wave propagation of the form:
u.sub.j =A.sub.o a.sub.j e
e.sup.i(k(ljxj-.omega.t))
for each of the incident waves as well as transmitted and reflected
waves, it can be shown that the boundary conditions are satisfied
providing the frequencies of the waves are equal and Snell's law
holds. Numerical procedures have been developed to solve this
problem for the reflected and transmitted amplitudes and
velocities.
Based on the above considerations, the nature of ultrasonic waves
generated at a particular incident angle may be explicitly
determined. One important practical aspect of this problem needs to
be discussed, the differences between phase and group velocities.
All considerations to this point have been for the phase velocity.
However, experimentally one measures group velocity. Therefore, one
must employ some means of correcting for the fact that the energy
flux vector s (.vertline.s.vertline.=group velocity) where ##EQU2##
does not coincide with the phase velocity wave normal 1. However,
it can be shown that
Hence, one can directly relate the phase and group velocities if
the shape of the slowness surface is known.
For mechanical property measurement, we are particularly concerned
about wave speed (or transit time measurement) as all pertinent
information regarding the elastic moduli (C.sub.ijkl) may be
obtained from the phase velocity. ##EQU3##
Since there are 3 possible values for each subscript, this means
that 81 distinct measurements are, in principle, needed. However,
because of symmetry considerations in the stress and strain tensors
as well as energy consideration, it may be argued that
C.sub.ijkl =C.sub.jikl
C.sub.ijkl =C.sub.ijlk and
C.sub.ijkl =C.sub.klij
This reduces the number of independent constants from 81 to 21. In
order to simplify the calculations, the following index
simplification may be employed:
11.fwdarw.1, 22.fwdarw.2, 33.fwdarw.3, 23.fwdarw.4, 13.fwdarw.5,
12.fwdarw.6 so C.sub.1111 .fwdarw.C.sub.11,C.sub.1112
.fwdarw.C.sub.16, etc.
where C.sub.in now represents a 6.times.6 symmetric array whose
elements represent the various elastic stiffness associated with
the material. Internal symmetry allows further reduction. For the
woven carbon-carbon structure illustrated here (assuming orthotropy
with the radial and tangential directions being the two in-plane
orthotropic axes) considered here, there are 9 pertinent elastic
moduli to be determined (C.sub.11, C.sub.22, C.sub.33, Chd 12,
C.sub.13, C.sub.23, C.sub.44, C.sub.55 and C.sub.66 ; see FIG. 2
for coordinate system). Note that direction 1 corresponds to this
tangential direction, 2 represents the through thickness direction,
and 3 the radial direction.
Thus, 9 independent measurements must be made. One choice is a
longitudinal wave traveling normal to fiber reinforcement plane.
This yields C.sub.22 directly. Propagation of shear waves in a
direction normal to the surface could be used to measure C.sub.55
and C.sub.13 directly. Unfortunately, coupling problems associated
with shear wave propagation preclude efficient scanning. Since
access often is available only to one of the panel surfaces, mode
conversion is the preferred approach. This brings oblique incidence
and energy flux consideration into play as the energy flux vector
will, in general, deviate from the wave normal unless the
propagation direction is a 2-, 4-, or 6-fold symmetry axis or
perpendicular to a plane of material symmetry (as appropriate for
all waves generated by normal incidence in a symmetric composite
laminate). However, as demonstrated above, these considerations can
be efficiently included in the data analysis. One must simply
choose appropriate angles for the measurement of the 9 moduli.
Ultrasonic velocity measurements were carried out using the
apparatus illustrated in FIG. 3. Test geometry is illustrated in
FIG. 4. The transducers were mounted in a fixture attached to a
computer-driven scanning bridge which allowed the transducer
assembly to be indexed both radially and tangentially (angular
displacement). Transducer angle of incidence was fixed in a
separate jig for each angle of interest. The transducer-specimen
distance was also adjusted for each angle to insure that the
sensing transducer would receive the desired ultrasonic reflections
(front surface and back surface). Separate scans were performed for
each inspection angle. (Note: To increase inspection speed, it may
be desirable to make all 9 measurements for a given point in a
single pass. This can be readily achieved by incorporating motion
control including one rotation angle for each transducer.)
Inspection angles were chosen to take optimal advantage of the
assumed orthotropic symmetry of the material. First, a normal
incidence longitudinal phase velocity inspection was performed.
This yields C.sub.22 directly as V.sub.L =.sqroot.C.sub.22/.rho.
for this mode of propagation. Next, three-phase velocities were
measured with the transducers aligned in the tangential direction
and incidence angles of 5.degree. (QL wave), 15.degree. (QT wave)
and 20.degree. (QT wave). This produces three nonlinear equations
involving unknowns C.sub.11, C.sub.12 and C.sub.66. Repeating the
same procedure with the transducers aligned in the radial direction
produces three coupled nonlinear equations which can be solved for
C.sub.33, C.sub.44 and C.sub.23. The remaining two moduli (C.sub.13
and C.sub.55) were determined using QT phase velocity measurements
and inspection angles at 15.degree. and 20.degree. with the
transducers oriented at 45.degree. with respect to radial.
The approach to characterizing C--C microstructure was based on
ultrasonic velocity measurements. However, since as just shown,
ultrasonic velocity measurements are sensitive to both elastic
moduli and density in order to characterize the elastic anisotropy,
it was necessary to have an additional local measure of density. In
this case, radiographic test methods were employed for this
purpose. A schematic of the test methodology is shown in FIG. 4 and
the test procedure followed is presented below.
Ultrasonic testing was performed in immersion on a full-sized
sample space shuttle brake stator. In order to prevent moisture
absorption during the inspection process, the samples were sprayed
with an acrylic coating layer. The layer was sufficiently thin that
it could be safely neglected in the time-delay measurements. For
oblique incidence, a set of specially designed test fixtures were
fabricated for pulse-echo inspection at fixed angles of incidence.
The inspection was performed by mounting the specimen on a
turntable which allowed rotation of the ring in 1.2.degree.
increments. The scan equipment also permitted radially indexing the
fixture in 0.5 cm increments. In this way, it was possible to scan
the part. It should, however, be pointed out that due to the use of
oblique incidence, only the center region of the ring could be
reliably inspected due to edge effects. This problem, of course,
increases with increasing angle of incidence. In experiments,
maximum angle of incidence of 20.degree. was used which permitted
inspection to within a distance on the order of the part thickness
from the part's edges.
One of the major problems encountered in this research effort was
the high degree of attenuation and dispersion present in the
sample. The resulting signal distortion due to a variety of effects
including porosity and scattering from the reinforcing fibers,
makes ultrasonic velocity testing rather difficult. As signal
distortion is highly dependent upon frequency, to reduce distortion
errors in the velocity measurements, relatively low frequency (500
kHz) narrow band sensors were used because of the thickness and
attenuation of the sample. This coupled with the use of a sensitive
pulsed phase locked loop (P.sup.2 L.sup.2) detection circuit,
reduced dispersion errors considerably and made accurate velocity
measurements possible.
Radiographic test methods were used for local density determination
in the composite samples. For this measurement, radiographs of the
complete stator were made along with an x-ray image of an aluminum
calibration bar (stepped in increments of 0.05 inch).
To accomplish this, it was first necessary to convert the
photographic x-ray images to digital form. This was done using an
Eiconiz digitizing camera at NASA/Langley with 8-bit resolution. It
should be pointed out that the limited field of view of the
digitizing camera along with the size of the stator (since it was
full-size image) precluded coverage of the entire part with a
single shot. Accordingly, multiple images were made and later
merged together into a single image. It also should be noted that
the light source for the digitizing camera was not uniform, hence
altering the x-ray intensity. In order to correct this problem, it
was necessary to first digitize an image of the light source itself
and develop normalizing factors to adjust each point in the image
to a constant intensity level. These same correction factors were
then used to correct the digitized radiographic images for the
light intensity variations.
Another problem arose from the fact that the x-ray absorption
values for C--C composites are not well established and had to be
determined. To do this, we used a C--C calibration sample where the
density could be readily determined which was radiographed along
with the stator and A1 stepped wedge. By comparing the C--C
calibration block to the A1 sample, the absorption coefficient for
C--C could be directly established.
Density variations, as measured using the radiographic test method
previously described, are illustrated in FIG. 5. Density variations
on the order of .+-.10% of this average value were observed.
Typically, slightly higher measured densities were observed along
the inner and outer radii of the part. The origin of this finding
has not been established, although it may be indicative of more
complete densification at the part extremities during the chemical
vapor deposition process.
The results from the local phase velocity measurements at multiple
angles of incidence and the density determination were analyzed
using the technique described earlier to determine the anisotropic
elastic properties of the material and how these properties varied
from position to position While all nine orthotropic moduli were
determined, three are presented in FIGS. 6-8. Of these
measurements, the most reliable is the through thickness modulus
(C.sub.22) determination as only one measurement is required
(through thickness time delay) and the desired modulus can be
obtained directly (not through numerical analysis techniques as
required for the remaining moduli). Accordingly, the variation in
this measurement is somewhat less than that found for all the
remaining moduli, and significantly less than that observed for the
two in plane moduli (C.sub.13 and C.sub.55) which involve all seven
of the other moduli (previously determined) in their calculation.
Some degree of error propagation is inherent in this approach and
further study is needed to ascertain the accuracy of the velocity
and density measurement techniques utilized and the ultimate effect
of this measurement uncertainty on the moduli calculations.
Upon examination of the results, one finds that the ultrasonic
results have much in common with the x-ray density measurements.
One of the most prominent features of the x-ray scan is a low
density region near the top of the scan. This feature can be
observed in virtually all of the C.sub.ij plots as a region of
significantly lower modulus than the surrounding materials. This is
supportive of the conjecture that incomplete densification has
taken place in this region. A second possibility is an increased
microcrack density in this region as this type of microstructural
flaw would result in lower density as well as decreased moduli
(particularly in-plane). There are, however, other regions which
exhibit somewhat different behavior. Consider, for example, the
anomalous region in the center of the ring at the 75.degree.
position with respect to vertical in the Figure This area does not
appear to be significantly different from the surrounding media
from the x-ray results yet modulus measurements are, in fact,
dissimilar. Of particular note is the observation that while some
of the properties are degraded in this region, others are improved.
The in-plane normal stiffness (C.sub.11, C.sub.33) are lowered in
this region while the out-of-plane stiffness properties (C.sub.22)
are enhanced. Possible explanations for this is a warping of the
prepreg fabric during layup or fiber breakage in the region. While
it is not possible to test the accuracy of these assertions without
sectioning the sample, it is clear that multiple ultrasonic modulus
measurements do provide a direct method of characterizing local
variation in material microstructure. With additional experience in
interpreting the results, we may be able to precisely
nondestructively classify material anomalies and their origin in
carbon-carbon samples.
Structural analysis was performed on the brake ring using the
finite element method (FEM) using the measured local stiffness as
inputs into the model. The finite element mesh for the
carbon-carbon composite brake disk is shown in FIG. 9. The disk was
modeled with 10800 nodal points and 5250 HEXA elements and was
analyzed using MSC/NASTRAN Version 65B2 run on an IBM 3081K
mainframe.
The brake disk was subjected to diametric compression. This was
modeled by fixing the disk in the r and theta directions at two
grid points (one on each of the upper and lower surfaces) along the
outer edge of the disk. Then a force was applied radially inward to
the grid points on the upper and lower surfaces along the outer
edge at the point 180 degrees from the fixed nodes, the magnitude
of this force was 5000 pounds, distributed equally between the two
grid points. This distribution was chosen to keep the outer surface
of the disk as near as parallel as possible with the x axis. For
stability purposes, it was also necessary to fix the grid points in
the theta direction at the points where the force was applied. This
analysis was repeated for similar loading and boundary conditions
rotated by 36, 72, 90, 108 and 144 degrees counterclockwise from
the original case.
For each of the loading cases, the output generated by NASTRAN
included the displacement and the normal stresses and strains in
the r, theta, and x directions, as well as the shear stresses and
strains in the r-theta plane. Also output were the strain energy
and strain energy density for each element. In order to compare our
results with those for an homogeneous carbon-carbon disk, another
analysis was run for a disk with the same geometry, loading and
boundary conditions. However, for this disk, the stiffness matrix
components were constant throughout the disk. This analysis allowed
us to better visualize the effects of material inhomogeneity on the
stress/strain distribution.
Using the procedure outlined above, the local densities and elastic
stiffness were determined for the space shuttle stator. Typical
results are illustrated in FIG. 7 where the through thickness
normal stiffness is shown. Of particular interest is the low
stiffness regions near the top of the Figure. Typical variations in
moduli were 10%. These results were used as inputs to the NASTRAN
finite element code. This allows one to analyze the response of a
model with actual, not idealized, mechanical properties.
Changes may be made in the construction and the operation of the
various components, elements and assemblies described herein
without departing from the spirit and scope of the invention as
defined in the following claims.
* * * * *