Anomaly Edge Effects in Thermographic Nondestructive
Testing of Polymeric Composite Sandwich Panels
by
Anton F. Thomas
B.S., Mechanical Engineering (1999)
Florida Agricultural & Mechanical University
Submitted to the Department of Mechanical Engineering
in Partial Fulfillment of the Requirements for the Degree of
Master of Science in Mechanical Engineering
at the
P A 9 K77
Massachusetts Institute of Technology
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
June 2002
OCT 25 2002
©2002 Massachusetts Institute of Technology
All rights reserved
LIBRARIES
Signature of Author
Department of Mechanical Engineering
May 8, 2002
Certified by
James H. Williams, Jr.
Professor of Engineering
Thesis Supervisor
Accepted by
Ain A. Sonin
Chairman, Department Committee on Graduate Students
1
Anomaly Edge Effects in Thermographic Nondestructive
Testing of Polymeric Composite Sandwich Panels
by
Anton F. Thomas
Submitted to the Department of Mechanical Engineering
on May 8, 2002 in Partial Fulfillment of the Requirements for the Degree of
Master of Science in Mechanical Engineering
ABSTRACT
A two-dimensional
thermographic nondestructive testing (NDT) model of a
polymeric composite sandwich panel containing a planar anomaly within its core is
presented, and its surface temperature distributions in the vicinity of its thermographic
transition zone are obtained via the finite element method. A characteristiclength of the
surface temperature transition zone is defined for anomalies that can be modeled as semiinfinitely long. The effect on the characteristic length of varying the geometrical and
thermal anomaly parameters is investigated numerically.
Thermographic NDT
procedures are devised to characterize the geometrical and thermal parameters of the
anomaly.
Thesis Supervisor:
James H. Williams, Jr.
Title:
SEPTE Professor of Engineering
2
ACKNOWLEDGMENTS
First of all, I would like to thank God for a continual renewal of strength,
direction, and purpose especially during the dark, discouraging times. Thank you for
keeping promises.
To my advisor and mentor, Professor James H. Williams, Jr., whose unwavering
support, encouragement, patience, and advice have kept me focused on a promising
future, thank-you very much.
To Professor Liang-Wu Cai, whose technical assistance was invaluable in the
shaping of this thesis, thank-you.
To Dr. Yapa D. S. Rajapakse of the Office of Naval Research, who supported this
research, thank-you.
To the 3M Company and the Graduate Student Office, whose financial support
allow me to pursue my educational and professional goals, thank-you.
To my family and friends, whose constant well wishes, words of encouragement,
and prayers helped keep me in good spirits, thank-you.
To my daughter, Ayren Joi, whose giggles and smiles of pure love helped me to
place stress in its proper place, thank-you. Daddy loves you.
And to my cornerstone, my earth, my joy and indeed my wife, Jocelyn, whose
love is my foundation, whose strength is my solid ground and whose beauty is my peace,
thank-you for being my partner and complementing my weaknesses. I love you very
much.
3
To my parents
4
TABLE OF CONTENTS
Page
Ab stract..........................................................................................
2
A cknow ledgm ents............................................................................
3
T able of C ontents..............................................................................
5
N omenclature.................................................................................
7
L ist of F igures.................................................................................
9
Introdu ction ....................................................................................
12
Heat Transfer in Sandwich Panels.........................................................
14
Problem D efinition..................................................................
14
Nondimensional Parameters.......................................................
17
M aterial Properties............................................................................
19
Finite Length Anomalies...................................................................
20
Semi-Infinitely Long Anomalies............................................................
24
Characteristic Length of Transition Zone........................................
24
Effects of Biot Numbers on Characteristic Length.............................
24
Effects of Anomaly Geometry on Characteristic Length......................
26
Effects of Fourier Number on the Transient Characteristic Length...........
28
Proposed Measurement and Data Processing.............................................
31
C onclu sion s....................................................................................
35
R eferen ces......................................................................................
36
F ig ures.....................................................................................
5
..
38
TABLE OF CONTENTS (cont.)
Page
Table 1
Sandwich panel constituents and anomaly thermal properties at 348'K (167'F)..... 49
APPENDIX A
Finite Element Equations for the Analysis of Two-Dimensional Heat Transfer...... 50
Matrix Notation of Interpolation Functions......................................
50
Steady-State Heat Transfer Equilibrium.........................................
51
Transient Heat Transfer Equilibrium...............................................
52
Two-Dimensional Four-Node Quadrilateral Elements........................
54
Two-Dimensional Nine-Node Quadrilateral Elements........................
56
APPENDIX B
Verification and Error Analysis of the Finite Element Solution........................
61
E xample D ata.........................................................................
61
Finite Element Mesh...............................................................
62
Comparison with the One-dimensional Solution...............................
63
Convergence of Characteristic Length...........................................
64
C onclusions.........................................................................
66
APPENDIX C
Development of Nondimensional Parameters...........................................
6
76
NOMENCLATURE
Latin
Bil
BiN
Biot number on front surface
Blot number on back surface
da
thickness of anomaly
Da
depth of anomaly
dc
d,
Fo
h,
hN
kc
k,
thickness of core
thickness of face sheet
Fourier number
heat transfer coefficient on front surface
heat transfer coefficient on back surface
thermal conductivity of anomaly
thermal conductivity of core
thermal conductivity of face sheet
La
length of anomaly
Lra
minimum detectable anomaly length
minimum anomaly length required to approximate the temperature
distribution due to an anomaly of infinite length
temperature
ambient temperature
axis parallel to thickness of sandwich panel
through-thickness locations
axis parallel to surface of sandwich panel
ka
T
x
xi
y
Greek
aa
thermal diffusivity of anomaly
ac
thermal diffusivity of core
a,
thermal diffusivity of face sheet
$8,
)6,
A,8
S
thermal resistance of region I
thermal resistance of region II
deviation of thermal resistance of region II compared to region I
experimental detection resolution
tolerance associated with relative temperature contrast
temperature distribution throughout sandwich panel (rise above ambient)
temperature distribution on front surface
temperature distribution on back surface
O
OF
OB
7
NOMENCLATURE (cont.)
F*
B*
61
E),
A
TI
Ti,
normalized temperature distribution on front surface
normalized temperature distribution on back surface
one-dimensional steady-state temperature in region I
one-dimensional steady-state temperature in region II
characteristic length of transition zone
characteristic time in region I
characteristic time in region 11
8
LIST OF FIGURES
Figure
Page
1
Mathematical model of a portion of a sandwich panel containing an
anomaly.
38
2
(a) Normalized front surface temperature distribution for case one
(air void), when the anomaly thickness is 10 mm (0.4 in.), located
at the mid-thickness of the panel, and the normalized anomaly
length, La / de, is varied. (b) Based upon the experimental detection
resolution, 8, a schematic of limiting anomaly lengths, Lmi and Lo,
are shown.
39
3
Given the detection resolution 5 = 0.02, the minimum detectable
anomaly length, La, normalized by the core thickness, dc, is
plotted as a function of anomaly geometry for cases one and two.
40
4
Given the detection resolution 5 = 0.02, the minimum anomaly
length required to approximate an anomaly of semi-infinite length,
L, normalized by core thickness, dc, is plotted as a function of
anomaly geometry for cases one and two.
41
5
Schematic of the characteristic length, A, of the surface
temperature transition zone normalized by the core thickness, dc,
42
given a tolerance, e = 0.05.
6
Steady-state normalized characteristic length for cases one and two
as a function of front and back Biot numbers, Bi, and BiN ,
respectively; where, for both cases the anomaly is 10 mm (0.4 in.)
thick and located at the mid-thickness of the core.
43
7
Steady-state characteristic
44
length as a function of anomaly
geometry for cases one and two.
8
(a) Given measurements A1 = A and AN = B for the front and back
steady-state characteristic lengths, respectively, the intersection of
the corresponding isosurfaces is shown. (b) The intersection of a
third isosurface, A/I= C, reveals the anomaly geometry and thermal
conductivity.
9
45
LIST OF FIGURES (cont.)
Figure
Page
9
Characteristic length histories for case one, where the anomaly is
10 mm (0.4 in.) thick and located at the mid-thickness of the panel.
The front surface is heated continuously and one of the anomaly's
parameter is varied. (a) Thermal conductivity ka is varied. (b)
Thermal diffusivity aa is varied. (c) Thickness da is varied. (d)
Depth Da is varied.
The dashed black lines represent the
characteristic length history of case one with no variation of
parameters.
46
10
Characteristic length histories for case two, where the anomaly is
10 mm (0.4 in.) thick and located at the mid-thickness of the panel.
The front surface is heated continuously and one of the anomaly's
parameter is varied. (a) Thermal conductivity ka is varied. (b)
Thermal diffusivity aa is varied. (c) Thickness da is varied. (d)
Depth Da is varied.
The dashed black lines represent the
characteristic length history of case two with no variation of
parameters.
47
11
Transient-state characteristic length, normalized by core thickness,
evaluated at the corresponding region II characteristic time.
48
A. 1
Schematic of two-dimensional four-node quadrilateral element.
54
A.2
Schematic of two-dimensional nine-node quadrilateral element.
55
B. 1
Steady-state solution of the finite element model of case one using
four-node element.
67
B.2
Steady-state solution of the finite element model of case one using
nine-node element.
68
B.3
Steady-state solution of the finite element model of case two using
four-node element.
69
10
LIST OF FIGURES (cont.)
Figure
Page
B.4
Steady-state solution of the finite element model of case two using
nine-node element.
70
B.5
Transient solution for the front surface using four-node elements for
71
case one
B.6
Transient solution for the back surface using four-node elements for
71
case one.
B.7
Transient solution for the front surface using four-node elements for
72
case two.
B.8
Transient solution for the back surface using four-node elements for
72
case two.
B.9
Transient solution for the front surface using nine-node elements
73
for case one.
B.10
Transient solution for the back surface using nine-node elements for
73
case one.
B.11
Transient solution for the front surface using nine-node elements
74
for case two.
B.12
Transient solution for the back surface using nine-node elements for
74
case two.
B.13
Steady-state characteristic length convergence for case one.
75
B.14
Steady-state characteristic length convergence for case two.
75
11
INTRODUCTION
Sandwich panels comprise a class of engineering structural materials in which the
inner core is generally thicker and more flexible than the outer face sheets. By separating
the stiffer face sheets with a relatively low density and thicker core, sandwich
construction is capable of achieving high structural efficiencies. However, sandwich
panels may experience adverse effects such as strength and stiffness deterioration when
anomalies such as delaminations, inclusions or voids exist within the panel. To detect
these anomalies, thermographic nondestructive testing (NDT) can be used. Due to the
alteration of heat flow by the embedded anomaly, surface temperature variations can be
observed. These observations can be used to identify the geometric and thermophysical
properties of the anomaly.
Thermographic NDT of sandwich panels have been studied experimentally (Qin and
Bao, 1996; d'Ambrosio et. al, 1995; Vickstrum, 1991; Vickstr6m, et. al, 1989) and a few
analytical models of multi-layer heat transfer problems have been analyzed (Williams et.
al, 1980; Williams, 1981; Williams and Lee, 1982; Williams et. al, 1983; Williams and
Nagem, 1983; Mikhalov and Ozisik, 1984; Vavilov and Finkel'shtein, 1986). The onedimensional transient and steady-state heat transfer through a polymeric sandwich panel
containing a planar anomaly was investigated by Cai et. al (2001). A thermographic NDT
procedure was proposed to characterize such anomalies; however, access to both front
and back surfaces of the sandwich panel was required.
To circumvent the limitations of the NDT procedure proposed in the previous paper,
two-dimensional heat transfer is considered. The applicability of the one-dimensional
solution to the two-dimensional model is studied. Of primary interest is the surface
12
temperature transition in the vicinity of the edge of an embedded anomaly, which can be
described as a thermographic halo. A measurable length parameter characterizing this
surface temperature transition is defined in order to develop alternative NDT procedures.
13
HEAT TRANSFER IN SANDWICH PANELS
Problem Definition
A sandwich panel containing a planar anomaly is modeled as a multi-layer structure
as illustrated in Figure 1. The region of the sandwich panel that is free from damage,
denoted as region I, consists of a thicker core bonded between two identical face sheets.
The damaged region, denoted as region II, contains an additional planar anomaly, either a
void or inclusion, embedded within the core. In this model, the x and y axes define the
thickness and length directions, respectively, where x = x
defines the front surface of
the panel, y = 0 defines the mid-length of the embedded anomaly, and
x2
through
x6
define other through-thickness locations. Significant thicknesses are denoted by d,, de,
and d,; thermal conductivities are denoted by k,, ke, and k,; and thermal diffusivities
are denoted by a, a, and a,; where the subscripts s, c, or a, designate the parameters
of the face sheet, core, and anomaly, respectively. Additionally, the depth of the anomaly
Da is measured from the back of the front face sheet to its mid-thickness; and the length
of the anomaly is defined as La
Due to symmetry about the anomaly mid-length, y = 0,
there is no heat transfer across this symmetry line; hence, only an upper portion of the
panel is considered and an adiabatic boundary condition is defined along the symmetry
line.
Initially, the temperature throughout the panel, T(x, y, t), is uniform at the ambient
temperature T.
At time t = 0, a uniform and constant heat flux q" is applied to the front
14
surface of the panel. The heat transfer coefficients on the front and back surfaces are
hiand hN respectively.
The governing equation throughout each constituent material is given by (Arpaci,
1966)
D2 0 (x, y,t)
2
0, (x, y,t) _ 1
x2
ay
2
0, (x, y,t)
a
(1)
where 01 (x, y, t) is the temperature increase above the ambient temperature, defined by
0, (x, y,t0
T,(x, y,t0- T.
(2)
The initial condition is
0 1 (X, y,O) = 0
(3)
The adiabatic condition at y = 0 is
D0 1(x,y,t)
ay
(4)
-0
Y=O
where the subscript I in Equations (1) through (4) can represent any of s, c, or a.
The boundary conditions on the front and back surfaces are:
at x =x,
=0
q" - h s (xx, y,t)+k a
(5)
and at x =x6 ,
hNOs
6
,yt+s
(
0s (x, y,t)
=0
(6)
< 0O
(7)
The interlayer continuity equations are:
at x =
x2
O(x 2 , y,t)= 0 (x 2 , y,t)
0<y
and
D0a(x,y,t )
ax
XX2
D0 (x, y, t)
ax
15
0
X=X 2
< y < 00
(8)
at x= x3,
0, (x3,y,0 = Oa (X3 ,Y,t)
La
(9)
0< y < L
2
and
a0(X, y,t)
k a
3x
at x =x
La
(XYt)
2
xax
(10)
,
La
Oa(X4 ,y,t) = 0,(x 4,y,t)
0
< y<
(11)
"
2
and
a0(x,Y ,t )
ax
X=X
k
ka
La
(XY,()
2
ax
4
(12)
at x = x.,
OC(x,
y,t)= 0, (x,y,t)
0
(13)
< y<oo
and
a
a0,(X, Y,t)
x y(X,t )
ax
ax
0
< y < oo
(14)
x=x 5
and, at y = La
2
0,C
La
2
,t) = Oa (X,
(15)
< x < x4
and
y,t)
a0, (x,
ay
kaa0, (x, y, t)
ay
2
<
x
< x4
(16)
YLa
2
The solution to the two-dimensional heat transfer problem, O(x, y, t), is obtained
using the finite element method. Details about the finite element discretization, solution,
and verification of the heat transfer problem are given in Appendices A and B. Although
the mathematical model is semi-infinite in the y-direction, the two-dimensional solution
needs only to be obtained in the vicinity of the transition zone; that is, the region of the
16
panel in which there is significant heat flow in the y-direction. The two-dimensional
solution of the surface temperature was validated by comparing appropriate solutions
with the corresponding one-dimensional solutions in Cai et. al (2001).
Nondimensional Parameters
The front and back surface temperature distributions are denoted, respectively, as
F
(yt) and
B(y,t).
A normalized temperature distribution is defined as the difference
between the surface distribution of the two-dimensional model and the temperature
obtained for region I in the one-dimensional solution divided by the difference in the
region II one-dimensional temperature and the region I one-dimensional temperature. For
example, for the front surface the normalized temperature is denoted as
o* (y,t),
and
defined by
6(,t=OFyF-f~)(7
F
FF
of'(t)-Of(t)
where O (t) and
0 (t) are the one-dimensional temperature on the front surface of
region I and region II, respectively.
Accordingly, there are two significant limiting
values of a normalized temperature. A normalized temperature equals zero when the
surface temperature of the two-dimensional model equals the surface temperature of the
one-dimensional solution for region I, and a normalized temperature equals unity when
the surface temperature of the two-dimensional model equals the surface temperature of
the one-dimensional solution for region II.
The resulting normalized temperature distribution can be written as
6*(y,t,h,hN,
s )a
a a)
ks k k, ds
17
daDa
,D,La)
(18)
These parameters can be divided into the following six groups of nondimensional
parameters (see Appendix C):
at
number, Fo -4d2,
numbers, Bi,
the normalized surface coordinate,
Y ; the Fourier
dc
which is a nondimensional time; the front and back surface Biot
h d and BiN
, which are the ratios of the surface heat transfer
h
resistance to the internal conduction resistance of the core; the presumably known
d k
a
and
dimensions associated
dc kc
ac
geometric and thermal properties of the panel, -i--
d
D
dc
dc
with the anomaly, -- , -
and
L
" ; and the thermal properties of the anomaly,
dc
k
"' and
kc
It is noted that the system is linear with respect to the heat flux; hence, the
ac
normalized surface temperature distribution is independent of q.
18
MATERIAL PROPERTIES
The thermal properties of the panel constituents are listed in Table 1. For the
sandwich panel under consideration, the thicknesses of the face sheets and core are 3.8
mm (0.15 in.) and 25.4 mm (1.00 in.), respectively. In the numerical examples that
follow, two cases of anomalies are studied. Case one is a void filled with air, and case
two is an inclusion of an aluminum alloy. Each case is exposed to a heat flux
q= 0.5 kW/m2 (1.58x 10 3 BTU/h-ft 2 ).
coefficients are h, =
hN = 10W/(m 2 -K) (57
Unless otherwise noted, the heat transfer
BTU/[h-ft 2 -F]).
19
FINITE LENGTH ANOMALIES
For various values of anomaly length normalized by the core thickness, La / dc ,
Figure 2a shows the front surface steady-state normalized temperature distribution for
case one when the anomaly is 10 mm (0.4 in.) thick and located at the mid-thickness of
the core.
For all values of La , the temperature distribution approaches the one-
dimensional region I temperature as y -+ oo. As the anomaly length increases, the
surface temperature above the mid-length of the anomaly, OF (o), increasingly closely
approximates the one-dimensional region 11 temperature.
Figure 2b shows the corresponding normalized temperature above the anomaly
mid-length (y = 0) as the anomaly length increases. Two major limiting cases of the
anomaly length can be identified.
As La -> 0, the surface temperature distribution above the mid-length of the
anomaly also approaches the region I temperature.
The experimental
relative
temperature resolution, denoted by 8, is normally defined as the minimum detectable
temperature difference divided by a certain experimental measurement range. It is
assumed that the experimental measurement range is tuned to maximize the surface
temperature difference resulting from a semi-infinitely long anomaly.
At a particular
value of La, the normalized temperature above the mid-length of the anomaly is equal to
85.
All values of anomaly length that are less than this particular value of anomaly length
can not be detected.
Subsequently, this value of anomaly length is defined to be the
minimum detectable anomaly length, Lmin.
20
For example, when the experimental relative temperature resolution is 2% of the
overall temperature difference resulting from a semi-infinitely long anomaly, S equals
0.02. Using this value of 6, Figure 3 shows Ls, as a function of anomaly geometry for
the front and back surfaces of cases one and two. A few observations are noted.
For case one, when the anomaly depth is held constant, Ls, monotonically increases
as the anomaly depth approaches the observation surface.
For case two, when the
anomaly depth is held constant, L,,, monotonically increases as the anomaly thickness
increases. However, because the normalized anomaly conductivity,
k
a
, is large for an
ke
aluminum inclusion compared to that of an air void, the temperature gradients near the
ends of an aluminum inclusion are much larger than the temperature gradients resulting
from air inclusion. As La -> L,
the surface temperature rise caused by aluminum
anomalies are significantly higher than those resulting from an air inclusion. Hence, in
general, an aluminum anomaly can be detected at a length down to four orders of
magnitude smaller than the core thickness, while an air void can be detected at a length
three orders of magnitude smaller than the core thickness.
As La -> O0, the normalized temperature above the mid-length of the anomaly
approaches one, which corresponds to the region II steady-state temperature. At a certain
value of La, the temperature difference in Figure 2b is equal to a 1-6'. This value of
anomaly length is defined to be the minimum anomaly length that results in a temperature
distribution that approximates the temperature transition zone of a semi-infinitely long
anomaly, L..
All values of anomaly length that are greater than L. have transition
regions that are indistinguishable from that of a semi-infinitely long anomaly. The heat
21
transfer through the mid-length of such anomalies can be modeled as one-dimensional
through the thickness.
Figure 4 illustrates L. for case one (air void) and case two (aluminum alloy
inclusion) when 9 = 0.02.
Figures 3 and 4 reveal an interesting symmetry for these panels. An anomaly has a
complementary relationship about the mid-plane of the panel, such that if the anomaly
were relocated to a mirror-image location about the mid-thickness of the panel, the
relative temperature distributions on the front and back surface would be reversed.
Unfortunately, knowledge of the region II one-dimensional temperature, which is
needed to compute the normalized temperature, is not normally known a priori,and the
luxury of running experiments on anomalies of varying lengths are neither common nor
practical. However, to determine whether an anomaly can be considered of semi-inifinite
length without knowledge of the region II temperature, a more subtle observation of
Figure 2a is necessary. As the anomaly length increases, the temperature distribution
becomes increasingly uniform over the mid-length of the anomaly and the curvature
approaches zero.
Recall that the curvature, K, of a function of one independent variable , f = f(x), is
determined by the first and second derivatives of that function with respect to the
independent variable,
f'
and
f",
respectively, according to the following equation
(Kreyszig, 1993)
K(x)
(1+
f2)
3
/
2
(19)
According to Equation (19), the curvature of the surface temperature at the position
above the mid-length of the anomaly depends only on the absolute value of the second
22
derivative of temperature relative to the surface coordinate, because according to the
adiabatic boundary condition, Equation (4), the first derivative vanishes. For the front
surface temperature distribution, for example, the necessary condition that curvature
approaches zeros above the anomaly mid-length is expressed as
«1
kcd C [a2O(Y]
q"f
ay 2
(20)
Y=
Assuming the expression in Equation (20) remains small for finite region near the
surface, the governing equation of heat transfer, Equation (1), via an order of magnitude
argument, locally reduces to
a20
ax
2
iao
1
a at
(21)
Recall, that Equation (21) is the governing equation for one-dimensional heat transfer;
therefore, for such cases the anomaly can be considered of semi-infinite length.
23
SEMI-INFINITELY LONG ANOMALIES
Characteristic Length of Transition Zone
In the vicinity of the transition zone, consider a surface temperature distribution due
to a semi-infinitely long anomaly. To quantify this transition zone, a characteristiclength
is defined as the length required for the surface temperature to approach, within a certain
tolerance e, each temperature asymptote. These asymptotes are predicted by the onedimensional heat transfer solution in Cai et. al (2001). According to this definition, there
is exactly one characteristic length on each surface at each instant.
In Figure 5, a
schematic of the characteristic length, denoted by A, is shown.
The limits of e are zero and 0.5. Due to the asymptotic nature of the temperature
distributions, as F -> 0 the characteristic length would approach infinity. On the other
hand, as e -> 0.5, the characteristic length would approach zero. This indicates that there
is a tradeoff between the sensitivity-higher as c -> 0 and lower as e -> 0.5 -and
the
required detection resolution-more demanding as e -+ 0 and less demanding as
e -> 0.5. In the numerical examples that follow, e is chosen to be 0.05. Consequently, in
order to resolve A, the minimum resolvable temperature can not be larger than 5 percent
of the overall surface temperature difference.
Effects of Biot Numbers on Steady-State Characteristic Length
Figure 6 shows the characteristic length for cases one and two as functions of front
and back surface Biot numbers, where the anomaly is 10 mm (0.4 in.) thick and located at
the mid-thickness of the core. A few observations are noted.
24
First, for both case one and case two, there is a symmetric relationship between the
front and back contours of characteristic length about the line Bil = BiN ; i.e., for an
anomaly centered at the mid-thickness of the core, a transposition of the front and back
Biot numbers results in a corresponding transposition of the front and back characteristic
lengths. Thus, when Bil = BiN the characteristic length on both surfaces will be equal
for an anomaly located at the mid-thickness of the core. This implies that the normalized
temperature distributions on the front and back surfaces for such cases are identical.
Second, there are regions where the characteristic length is a function of only one
Biot number. On the front surface of both the air and aluminum inclusions, in the region
bounded approximately by logo(BiN )>1.5 and log 0 (Bi)<1.5,(that is, conditions such
1
that the front surface heat transfer resistance per unit area, I, is sufficiently low relative
hi
d
to the thermal resistance of the core per unit area, - ), the characteristic length is a
kc
function only of Bi1 . On the back surface, when logo (Bi )<1 and logo (Bi)>1 for an
air void, or logo(BiN )<2 and log 10 (Bi)>2 for an aluminum alloy inclusion,(that is,
conditions such that the back surface heat transfer resistance,
, is sufficiently low
hN
relative to the thermal resistance of the core), the characteristic length depends only on
BiN
*
Third, in the region bounded by logo(BiN )<O and logo(Bi)<O, the characteristic
length is a monotonic function of BiN / Bi which reduces to hNI h1 .
25
For reference, for cases one and two the front and back surface Biot numbers are
Bil= BiN = 5.81, or loglo (Bi)=logo (BiN) =0.76.
Effects of Anomaly Geometry on Steady-State Characteristic Length
The front and back characteristic lengths, normalized by the core thickness, as a
function of anomaly geometry are presented in Figure 7.
For both cases, symmetry about the mid-thickness of the core is observed; that is, if
the anomaly location is moved to a mirror-image location about the mid-thickness of the
panel, the front and back characteristic lengths are transposed.
For case one, the anomaly thickness has very little effect on the characteristic length;
as the depth approaches the observation surface, either front or back surface, the
corresponding characteristic length decreases. For both cases, the larger the depth of the
anomaly is from an observation surface, the larger is the observed steady-state
characteristic length.
Figure 7 suggests the following procedure for determining the geometric properties of
an anomaly when its thermal properties are known. First, the steady-state front surface
temperature of region II, denoted by (6 1) 11, and a front surface characteristic length, A1 ,
are measured. The following equations (after Cai et. al, 2001)
= q "(1+ hN II)
hi +hN +hhN3II
,=2 ds+
c +d2
k, ke
26
ka
k,
2
can be solved for the thermal resistance of the damaged panel /,
anomaly thickness
and the unknown
d.. The intersection of the contour line for Al with da , in the
anomaly thickness-anomaly depth plane, gives the actual depth of the anomaly.
Furthermore, the following graphical procedure can be used for steady-state anomaly
characterization when the geometric and thermal properties of the panel are known, but
the anomaly depth, anomaly conductivity, and the surface heat transfer coefficients are
not known.
First, four steady-state surface temperatures, one for each region on each surface, and
two steady state characteristic lengths, one for each surface, are measured. Along with
Equation (22), the following equations (after Cai et. al, 2001)
(00
(O
q"(1 + hN/1)
hi + hN + hhN/I
hN + hN + hlhN
hN,
/3,
(24)
q=
N+1I
N+1JIl
can be used to solve for hl,
_
=EN1I
25)
(
q
h + hN + hlhN /3II
(26)
and 6I ;where /, represents the thermal resistance
of the undamaged region, given by
d
d
,8 = 2 d' + d'
k,
kc
(27)
Second, after the boundary heat transfer coefficients are found, a series of figures
analogous to Figure 7 can be generated for a range of ka.
Suppose the value of the
characteristic length on the front surface, A,, is equal to A. For each figure generated, a
contour line corresponding to A, = A is identified. These contours are combined to form
a continuous surface in the anomaly thickness-anomaly depth-anomaly conductivity
27
space. This surface, which is defined in this triangular prismatic space, can be called an
isosurface for Al=A.
Third, the same procedure can be used to form an isosurface for the characteristic
length from the back surface, AN= B. The two isosurfaces are placed in the same
triangular prismatic space as shown in Figure 8a. The resulting intersection reveals all
possible anomaly parameter coordinates.
Fourth, another parameter, Afl, is computed by subtracting Equation (27) from
Equation (23), or
A/ = 8,1 -8,
= d.
(28)
kc
which represents the deviation of the thermal resistance of region II from region I.
ka
According to Equation (28), a corresponding isosurface, Af8= C, is generated, where it is
noted that A$8 is independent of anomaly depth. As shown in Figure 8b, the coordinates
of intersection of this third isosurface with the intersection of Al= A and AN= B
determine the values of anomaly thickness, depth, and conductivity.
Effects of Fourier Number on the Transient Characteristic Length
Figure 9 shows colorized maps of front and back surface characteristic length
histories for case one, where the anomaly is 10 mm (0.4 in.) thick and located at the midthickness of the panel, and when one of the following anomaly parameters is varied:
thermal conductivity ka , thermal diffusivity aathickness da, or depth Da . The panel is
heated continuously from t = 0 to a time corresponding to a Fourier number of 20,
approaching the steady state. The dashed black lines represent the characteristic length
28
history of case one with no variations of anomaly parameters. A few observations are
noted.
By the definition of the characteristic length, when region I and region II
temperatures are equal, the characteristic length is undefined. The discontinuities in
otherwise smooth contours of characteristic length distributions represent these times.
According to the front surface characteristic length histories in Figure 9c and 9d,
there are certain regions where the characteristic length is not a monotonic function of
Fo. This indicates that the maximum characteristic length in these cases may occur
during a transient state.
There are certain regions where the characteristic
remarkably simple shapes.
length histories show
In Figure 9b on the front surface, in regions where the
anomaly diffusivity is larger than that of the core and further when the thermal diffusivity
of the anomaly is held constant, the characteristic length is a monotonically increasing
function of Fo on both surfaces.
For histories in which there are regions where there has been an equalization of
region I and region II temperatures-front and back surfaces of Figure 9a and 9b and the
back surfaces of Figures 9c and 9d-the surface characteristic length is a monotonically
increasing function of Fo as the anomaly parameters are held constant for all times after
the equalization of region I and region II surface temperatures. This indicates that the
maximum characteristic length in these cases is reached only when the system
approaches the steady-state.
The corresponding series of histories for case two is given in Figure 10.
Observations similar to Figure 9 can be made for Figure 10.
29
In the one-dimensional transient solution (Cai et. al, 2001), the time needed for
region I or II to reach 63.2 percent of its steady-state surface temperature is called the
system characteristictime and denoted by 'z for region I and ri for region II. Figure 11
shows the transient characteristic length evaluated at rm as a function of anomaly
thickness and anomaly depth for cases one and two. For the air void, the transient
characteristic length, is highly dependent upon the depth: as the depth gets closer to the
observation surface, the characteristic length decreases. For the aluminum inclusion,
(except for thin anomalies), when the thickness is held constant, the characteristic length
is observed to monotonically increase as the depth gets farther from the observation
surface. For both cases, it is noted that, unlike the steady state results, there is no
relationship observed between the front and back surface transient characteristic lengths.
30
PROPOSED MEASUREMENT AND DATA PROCESSING
Assuming an anomaly is of a length greater than L., based upon the above analysis
and the previously proposed procedure based upon the one-dimensional analysis, the
following procedures for the thermographic NDT of composite sandwich panels are
proposed.
When both the front and back surfaces are accessible:
1. Make seven (7) measurements: one front surface system characteristic time; two
steady-state characteristic lengths, one for each surface; four steady-state
temperatures, one for each region on each surface.
2. Solve for 83 , /3$1 , A$3, h, and hN using the steady-state surface temperatures
measured in the previous step, the known thermal properties of the undamaged
core, and Equations (23), (25) ,(26), (27), and (28).
3. A series of color maps, analogous to Figure 7, is generated. For each color map
ka is selected so that the entire series covers a range within which the actual k,
would likely fall .
4. From the front and back steady-state characteristic lengths measured in step 1,
two isosurfaces are created in the anomaly thickness-anomaly depth-anomaly
conductivity space. The intersection of the front and back characteristic length
isosurfaces is found according to the procedure illustrated in Figure 8a.
5. According to the definition of Alin Equation (28) and its value as found in step
2, a third isosurface is computed.
31
6.
Analogous to Figure 8b, the intersection of isosurface A/f
with the curve
generated from the intersection of the previous isosurfaces is found. This
intersection gives the actual values of thermal conductivity, depth, and thickness
of the anomaly.
7. Using
hl, hN,
kal da
and Da as calculated in the previous steps, the front
surface characteristic time is computed via the one-dimensional solution (Cai et.
al, 2002) for a range of anomaly thermal diffusivity, where the thermal diffusivity
covers a range within which the actual thermal diffusivity would likely fall.
8. The point where the curve of the previous step equals the front surface
characteristic time, as measured in the first step, is the actual value of anomaly
thermal diffusivity.
9. Finally, Chart 9 of Ashby (1992) can be used to identify the probable material of
the anomaly.
In comparison with the NDT procedure proposed in Cai et. al (2001), the above
procedure has distinct advantages and disadvantages. First, the above procedure captures
the geometry and thermal conductivity of the anomaly via steady-state measurements,
without processing transient information; however, a moderate computational effort is
required, particularly in the formation of isosurfaces as illustrated in Figure 8. Second, in
order to measure each characteristic length, a more stringent requirement is imposed on
the minimum detection resolution. In the numerical cases that have been presented, the
minimum resolvable temperature needs to be a fraction of that required for the procedure
in Cai et. al (2001).
32
In our previous paper (Cai et. al, 2001), back surface observations are necessary to
characterize an anomaly.
By substituting the back surface transient temperature
observations with the front surface transient characteristic lengths observations, the above
procedure can be modified so that back surface dependence can be eliminated.
When only the front surface is accessible, and the front heat transfer coefficient h,
and the thermal resistance of the undamaged panel 8,J,
defined in Equation (24), are
known, the following procedure is proposed:
1. Make four (4) measurements: one front surface system characteristic time of
region II; one front surface characteristic length, at the characteristic time of
region II; two steady-state front surface temperatures, one for each region.
2. Solve for
hN
and A/8 via Equations (23), (24), (25), and (28) using the values
measured in the previous step.
3. According to the definition of Af8 in Equation (28) and its value as found in the
previous step, an anomaly thickness versus ka curve is plotted.
4. A series of color maps for the front surface characteristic time versus anomaly
thickness and anomaly depth are generated. A series of color maps for the front
characteristic length, evaluated at the front characteristic time versus anomaly
thickness and anomaly depth, are also generated. For each color map, ka is
selected so that the entire series covers a range within which the actual k, would
likely fall, and a,is selected such that the a, / k, ratio is constant.
5. For each pair of color maps generated in the previous step (for each selected ka
value), the contours corresponding to the observed values of the front surface
33
region II characteristic time and the characteristic length evaluated at the
characteristic time are identified.
The intersection of overlaying each pair of
contours on the same axes is found and anomaly thicknesses and depth are
determined.
6. Anomaly thicknesses found in the previous step are plotted as a second anomaly
thickness versus ka curve.
7. The two anomaly thickness curves intersect. The intersection of the anomaly
thickness curves gives the true values of ka and anomaly thickness.
8. A color map of the front surface characteristic time as a function of anomaly
diffusivity and anomaly depth is generated. A color map of the front surface
characteristic length at the region II characteristic time as a function of anomaly
diffusivity and the anomaly depth is also generated. For each colormap, the now
determined values of anomaly thickness and ka are used.
9. The intersection of the contours of the characteristic time and the characteristic
length measured at that characteristic time determines a, and the anomaly depth.
10. Finally, Chart 9 of Ashby (1992) can be used to identify the probable material of
the anomaly.
34
CONCLUSIONS
The two-dimensional heat transfer of composite sandwich panels has been
modeled and numerical solutions are obtained via the finite element method. Limiting
cases of anomaly length were identified and discussed. The characteristic length of the
surface temperature transition was defined and, through numerical examples, the effects
due to changes in various parameters were observed. In combination with the previous
one-dimensional analysis (Cai et. al, 2001), two procedures to characterize anomalies of
a well-defined minimum length have been proposed, including cases when the back
surface is inaccessible.
35
REFERENCES
Arpaci, V.S., Conduction Heat Transfer, Reading, Massachusetts, Addison-Wesley Co.,
1966.
Ashby, M.F., Materials Selection in Mechanical Design: Materialsand Process
Selection Charts, Oxford, England, Pergamon Press, 1992.
Cai, L.-W., A.F. Thomas, and J.H. Williams, Jr., "Thermographic Nondestructive Testing
of Polymeric Composite Sandwich Panels," Materials Evaluation, Vol. 59, 2001, pp.
1061-1071.
d'Ambrosio, G., R. Massa, M.D Migilore, G. Cavaccini, A Ciliberto and C. Sabatino,
"Microwave Excitation for Thermographic NDE: An Experimental Study and Some
Theoretical Evaluations," MaterialsEvaluation, Vol. 53, 1995, pp. 5 0 2 -5 0 8 .
Kreyszig, E., Advanced EngineeringMathematics, 7t Edition, New York, John Wiley &
Sons, 1993.
Mikhalov, M.D. and M.N. Ozisik, Unified Analysis and Solutions of Heat Mass
Diffusion, New York, John Wiley & Sons, 1984.
Qin, Y.W. and N.K. Bao, "Infared Thermography and its Application in the NDT of
Sandwich Structures," Optics and Lasers in Engineering, Vol. 25, 1996, pp. 205-211.
Vavilov, V.P. and S.V. Finkel'shtein, "Calculation of the Sensitivity of Active Thermal
Inspection on the Basis of Solving an Unidimensional Problem of Heating a Threelayer Sheet with a Constant Heat Flow," Soviet Journalof Nondestructive Testing
(English translation of Defektoskopiya), Vol. 22, 1986, pp. 422-428.
Vickstr6m, M., J. Backlund and K.A. Olsson, "Non-destructive Testing of Sandwich
Constructions Using Thermography," Composite Structures, Vol. 13, 1989, pp. 4965.
Vickstr6m, M., "Thermographic Non-destructive Testing of Sandwich Structures with
Simulated Debonds, Sandwich Constructions 2," Proceedings of the Second
InternationalConference on Sandwich Constructions,D. Weissman-Berman and
K.A. Olsson eds., Gainsville, Florida, 1991, pp. 757-775.
Williams, J.H., Jr., S.H. Mansouri and S.S. Lee, "One Dimensional Analysis of Thermal
Nondestructive Detection of Delamination and Inclusion Flaws," British Journal of
Non-Destructive Testing, Vol. 22, 1980, pp. 113-118.
Williams, J.H., Jr., "Thermal Nondestructive Testing of Fiberglass Using Liquid
Crystals," MIT Sea GrantProgramReport MITSG 81-16, M.I.T., Cambridge,
Massachusetts, 1981.
36
Williams, J.H., Jr., and S.S. Lee, "Thermal Nondestructive Testing of Fiberglass
Laminate Containing Simulated Flaws Orthogonal to Surface Using Liquid Crystals,"
Journalof Nondestructive Testing, Vol. 24, 1982, pp. 76-81.
Williams, J.H., Jr., B.R. Felenchak and R.J. Nagem, "Quantitative Geometric
Characterization of Two-dimensional Flaws Via Liquid Crystals Thermography,"
Materials Evaluation, Vol. 41, 1983, pp. 190-201.
Williams, J.H., Jr. and R.J. Nagem, "A Liquid Crystals Kit for Structural Integrity
Assessment of Fiberglass Watercraft," MaterialsEvaluation, Vol. 41, 1983, pp. 961966.
37
y
Face Sheet,
ks, as
Orl
K._
E
hN
E
Core,
kc, ac
q
if
Region I
Da
t
Anomaly,
ka,aa
Region II
La
dc
da
X1
X2
I
X4
X3
1
X5
1
X6
x
Adiabatic
Boundary
Figure 1-Mathematical model of a portion of a sandwich panel containing an anomaly.
38
-La
1
Id =
16
1
a
1-8
a)
12
o
8.
0.8
4 .. .....
II
0
.
0.8
SC
.
-- -
2
-..
. . . .. . . ... . . ..
. . ..
0.6
U-
1
0.4
.. .
-
..................
0.6
-
. .. .
-.
-.-.
- - . -
.
. .. .. ..
>
0.4
0
UC:
-. . . ... . ..-.. .. . ...-. ..
0.2
oZ
0.
6
0
0
0.2
2
4
6
y/(Core Thickness)
8
10
12
0
OL .
d
C
2
L.
4
6
d
C
(Anomaly Length)/(Core Thickness)
(a)
(b)
Figure 2-(a) Normalized front surface temperature distribution for case one (air void), when the anomaly thickness is 10 mm
(0.4 in.), located at the mid-thickness of the panel, and the normalized anomaly length, La / de, is varied. (b) Based upon the
experimental detection resolution, 6, a schematic of limiting anomaly lengths, Lmin and L., are shown.
8
x
1
1
5.0
4.5
0.8
0.8
4.0
0.6
0.6
JU
10-
2
3.5
4)
0.4
0
44
0.4
01
0
01
1
0.8
0.6
0.4
0.2
0
Thickness)
Thickness)/(Core
(Anomaly
(a) Air Void, Front Surface
0
0.2
0.4
0.6
0.8
1
(Anomaly Thickness)/(Core Thickness)
(b) Air Void, Back Surface
1
1
0.8
0.8
0.6
0.6
0
.2
a
0
0.4
0
0.4
0.2
0.2
0
0
0.2
0.4
0.6
0.8
1
(Anomaly Thickness)/(Core Thickness)
(c) Aluminum Alloy Inclusion, Front Surface
3.0
2.5
2.0
1.5
1.0
0.5
0.2
0.2
0
*1
01
0
-0.4
-0.8
-1.2
-1.6
-2.0
-2.4
-2.8
-3.2
-3.6
-4.0
1
0.8
0.6
0.4
Thickness)
Thickness)/(Core
(Anomaly
(d) Aluminum Alloy Inclusion, Back Surface
0.2
Figure 3-Given the detection resolution S = 0.02, the minimum detectable anomaly
length, La , normalized by the core thickness, de, is plotted as a function of anomaly
geometry for cases one and two.
40
1
1
4.)
0.8
U
0.6
U
rA
rA
0.8
4.)
0
0.6
8.0
0.
0.4
0
S.
4.)
0.2
0
0.4
7.2
6.4
0.2
0
0
0
.
0.2
0.4
0.6
0.8
5.6
01
0
1
0.2
0.4
0.6
0.8
1
(Anomaly Thickness)/(Core Thickness)
(Anomaly Thickness)/(Core Thickness)
(a) Air Void, Front Surface
(b) Air Void, Back Surface
'0
23
3.2
2.4
4.)
0
U
0.81
0.8
1.6
4.)
0
0.6
U
0.8
0.6
.0
'
E
0.
0.4
4.)
0.2
0
0.4
0.2 O
0
O
4.0
1
1
0
U
4.8
0
.
0.2
n
0.4
0.6
0.8
1
(Anomaly Thickness)/(Core Thickness)
(c) Aluminum Alloy Inclusion, Front Surface
0
0.2
0.4
0.6
0.8
1
(Anomaly Thickness)/(Core Thickness)
(d) Aluminum Alloy Inclusion, Back Surface
Figure 4-Given the detection resolution S = 0.02, the minimum anomaly length
required to approximate an anomaly of semi-infinite length, L., normalized by core
thickness, dc, is plotted as a function of anomaly geometry for cases one and two.
41
1
1-E
0x
0.8
A
d
41)
....
-.
0.4
- - . . . .- .. -
0d
-
-.
0.6
-.
-
.
-
-
Z.
0.2
E
0
-4
-3
-2
Region II (Flawed)
1
0
(y - La /2)/(Core Thickness)
-1
1
-4
2
3
4
Region I (Unflawed)
Figure 5-Schematic of the characteristic length, A, of the surface temperature
transition zone normalized by the core thickness, d,, given a tolerance, e = 0.05.
42
5
5
4
4
3
3
z
C22
Ozo2
.
2
10
0
9
1 1
8
0
0
-1
-1
-1
-1
0
1
3
2
logl 0(Bil)
4
5
0
1
3
2
logl 0(Bil)
4
5
5
5
4
4
3
3
5
4
00
0
0
0
1
3
2
logl 0(Bil)
4
5
(c) Aluminum Alloy Inclusion, Front Surface
2.
3
U.
2
1
-z
'12 2
2
6
00
(b) Air Void, Back Surface
(a) Air Void, Front Surface
-1
-1
7
0
0
01
-1
-1
0
1
2
3
4
5
logl0(Bil)
(d) Aluminum Alloy Inclusion, Back Surface
Figure 6 -Steady-state normalized characteristic length for cases one and two as a
function of front and back Biot numbers, Bil and BiN , respectively; where, for both
cases, the anomaly is 10 mm (0.4 in.) thick and located at the mid-thickness of the core.
43
1
1
C.)
0.8
0.8
C.)
0.6
U
0.4
0.
C.)
0
0
0.6
4.0
6
C.)
0.2
0.41
C.)
3.2
0.2
0
0
0
0.2
0.4
0.6
0.8
2.8
1
2.4
(Anomaly Thickness)/(Core Thickness)
(b) Air Void, Back Surface
2.0
0
1
(Anomaly Thickness)/(Core Thickness)
(a) Air Void, Front Surface
1
3.6
0.2
0.4
0.6
0.8
1 K-I
1.6
1.2
C.)
0
0.8
C.)
0.6
U
0.8
0.8
0.4
U
0
0.6
0
.0
0.
0.4
C.)
0.4
0.2
0.2
0
A
0
0.2
0.4
0.6
0.8
1
(Anomaly Thickness)/(Core Thickness)
(c) Aluminum Alloy Inclusion, Front Surface
0
0.2
0.4
0.6
0.8
1
(Anomaly Thickness)/(Core Thickness)
(d) Aluminum Alloy Inclusion, Back Surface
Figure 7-Steady-state characteristic length as a function of anomaly geometry for cases
one and two.
44
A =A isosurface
M AN
=B
isosurface
Intersection of A, = A
and AN = B isosurfaces
M AO =C isosurface
Anomaly Loc ation
5
5
4
3
3
2
2-
0
0
1
0.
8
0.6
0.4
(C 4.2
> 0
0(
0 0
a
(a)
N
04 0.2)
0.0
(b)
Figure 8-(a) Given measurements A1 = A and AN = B for the front and back steady-state characteristic lengths, respectively,
the intersection of the corresponding isosurfaces is shown. (b) The intersection of a third isosurface, AP = C, reveals the
anomaly geometry and thermal conductivity.
Back Surface
Front Surface
5
5
4
4
2
2
3
U
2.7
0
00
2.4
S
U
0
2
4
6
10
8
12
14
16
18
20
Fo(xetid2)
C
0015
4
.
-1
S3
-
(a)
2.1
-1
1.8
0
1.5
1.2
0.9
0.6
0.3
L ---j 0
4
6
10
8
12
14
16
18
20
Fo(actd2)
5
4
0
d 2
2
C~C
2
0
0
(b)
-1
0
2
4
6
8
10
12
14
16
18
0
20
2
4
6
10
8
0.6
0.6
0.4
0.4-
0.2
0.2
0
12
14
16
18
20
12
14
16
18
20
12
14
16
18
20
Fo
Fo
-
(c)
0
0
2
4
6
8
10
12
14
16
18
0
20
2
4
6
8
10
Fo
Fo
0.6
0.6
0.4
0.4
S0.2
0
(d)
0
0
0
2
4
6
8
10
Fo
12
14
16
18
20
0
2
4
6
8
10
Fo
Figure 9-Characteristic length histories for case one, where the anomaly is 10 mm (0.4 in.) thick and located at the
mid-thickness of the panel. The front surface is heated continuously and one of the anomaly's parameter is varied. (a)
Thermal conductivity ka is varied. (b) Thermal diffusivity aa is varied. (c) Thickness da is varied. (d) Depth Da is
varied. The dashed black lines represent the characteristic length history of case one with no variation of parameters.
Front Surface
Back Surface
5
5
3
3
-d2
2
0
-1
0
2
,
4
.
6
,
8
10
12
14
16
18
20
0
Fo(xCtid2C)
>3
3
2.7
2.4
2.1
1.8
0
0
-1
0
2
4
6
8
1.5
1.2
5
4
.0
3
10
12
14
16
18
20
Fo(aCtid2C)
5
0.9
0.6
0.3
0
4
S3
ets
0
0
2
2
C) 1
1
0-
0
(b)
-1
-1
0
2
4
6
8
10
12
14
16
18
0
20
2
4
6
8
0.4 0.6
06
0.2
0.4
0.2
0
0
2
4
6
8
10
10
12
14
16
18
20
12
14
16
18
20
12
14
16
18
20
Fo
Fo
12
14
16
18
20
01
0
(c)
2
4
6
8
Fo
10
Fo
0.6
0.6
"t3 0.4
0.2
0.2
0
0
0
2
4
6
8
10
Fo
12
14
16
18
20
(d)
0
2
4
6
8
10
Fo
Figure 10-Characteristic length histories for case two, where the anomaly is 10 mm (0.4 in.) thick and located at the
mid-thickness of the panel. The front surface is heated continuously and one of the anomaly's parameter is varied. (a)
Thermal conductivity ka is varied. (b) Thermal diffusivity aa is varied. (c) Thickness da is varied. (d) Depth Da is
varied. The dashed black lines represent the characteristic length history of case two with no variation of parameters.
1
1
0.8
0.8
UQ
0.6
0.6
U
0.4
U
3
0.4
2.7
J
0.2
0.2
0
2.1
-
UQ
0
01
1
0.8
0.6
0.4
0.2
0
Thickness)
Thickness)/(Core
(Anomaly
(a) Air Void, Front Surface
1
0
0.2
U
0.4
0.6
0.8
1
(Anomaly Thickness)/(Core Thickness)
(b) Air Void, Back Surface
0.8
0
0.6
0.4
0.
U
0.4
0.2
S
0
0.2
U
1.5
0.6
U
0.6
1.8
0.9
U
0.8
U0
1.2
1
U
2.4
0.3
0
A
0
0.2
0.4
0.6
0.8
1
(Anomaly Thickness)/(Core Thickness)
(c) Aluminum Alloy Inclusion, Front Surface
0
0.2
0.4
0.6
0.8
1
(Anomaly Thickness)/(Core Thickness)
(d) Aluminum Alloy Inclusion, Back Surface
Figure 11-Transient-state characteristic length, normalized by core thickness, evaluated
at the corresponding region II characteristic time.
48
Table 1-Sandwich panel constituents and anomaly thermal properties at 348'K (167'F).
Material
Thermal Conductivity (k)
Thermal Diffusivity (a)
W/(m-K)
m2/s (ft2 ls)
Face Sheet
0.4250
2.27x10-7 (2.44x10 6)
Core
0.0437
1.69x10~6 (1.82x10- 5)
Air
0.0293
0.0272 (0.293)
180.0
7.00x10~4 (7.53x10-3 )
Aluminum Alloy
49
APPENDIX A: FINITE ELEMENT EQUATIONS FOR THE ANALYSIS OF
TWO-DIMENSIONAL HEAT TRANSFER
Matrix Notation of Interpolation Functions
The solution to a two-dimensional heat transfer model is approximated by using
the finite element method [A-1]. Consider an analytical model that is discretized into an
assemblage of finite elements, with each finite element composed of n number of nodes.
The approximate temperature distribution throughout each element can be written as
n
(A.1)
g (x, y)O1
6' (x, y) =
where 0' (x,y) is the continuous temperature distribution throughout the m element,
0m
is the temperature of node i, located at position (xi,yi), and gi is a function that satisfies
1
g (x ,y 1 ) =(S1 1 , where
and
i! n
15j:! n
(A.2)
The temperature-gradient distribution is defined by the following equations:
n
aoaxMx,,y
a
i=1(A3
g i, (x, y)0 1
=
a (x,y)
i=1
The temperature interpolation for the mth finite element in the assemblage, according to
eqn. (A.1), is rewritten in matrix form as
Om (x, y) = H (m) 0(m)
(A.4)
where H(m) is a 1 xn row vector called the temperature interpolationmatrix, written as
H C'"
=[g1(x, y)
g2(X,
y)
'---
gn (Xy)]
(A.5)
and (m) is an nxI column vector of nodal point temperatures, written as
0(m)=
[0
02
50
__
]
OT
(A.6)
Equation (A.3) is rewritten in matrix form as
(A.7)
ax
(x,y)
=
where B(') is a 2xn matrix called the temperature-gradientinterpolationmatrix, written
as
B'
g" (xy)
g 2 ,,
(x, y)
gY (x, y) 9 2 ,y (X, Y)
-gn y,(xY)]
(A.8)
Due to boundary conditions, it is necessary to evaluate the temperature
distribution on the surface of the body. In this case, the right superscript S is used to
denote the surface temperature distribution along the finite element,
0
MS (xS, ys),
and the
The equation for the surface
surface temperature interpolation matrix, H('n)s
temperature distribution is
Oms S
s)= H(m)s
0(M)
(A.9)
where the surface temperature interpolation matrix is written as
H(,)S = [g (xSyS)
s
2 (XS
YS)
...
g(xs
yS)]
(A. 10)
Steady-State Heat Transfer Equilibrium
The steady-state equilibrium equation for linear heat transfer is expressed as
(Kk + Kc)0 = Q B + QS
51
(A.11)
where K is the conductivity matrix,
Kk =
k(m)B(M) T B(m)dV (m)
(A.12)
h(m)H S(m)TH s(m)dV (m)
(A.13)
Kc is the convection matrix,
KC
QB is the volumetric heat generationmatrix,
QB
(A.14)
T
m) H (M) q B(m)dV (")
=
and Qs is the surface heatflux matrix,
QS
=j
(A.15)
1( H s(m)q s(m)dS "m
The summation signs in eqns. (A.12) through (A.15) represent the assembly of the local
finite element matrices to the global finite element matrix. The aforementioned S refers
to the surface and script V denotes the volume of the finite element. It is noted that in the
two-dimensional Cartesian case, the volume integral is an area integral over an assumed
constant unit thickness.
Transient Heat Transfer Equilibrium
In transient heat transfer, eqn. (A. 14) can be rewritten as
QB
I
()=H "' T(q
) (m)H (m)t
B(M
) dV (")
(A.16)
where
(A.17)
at
The left superscript t, denotes the time at which the temperature time-derivative vector,
0 , and temperature vector, 0 , are evaluatated.
By substituting eqn.(A.16) into eqn.(A. 11), in the absence of volumetric heat
generation, the resulting transient heat transfer equilibrium equation for a given time t is
52
(A.18)
C'6+ (K'+ Kc)'O = Qs
where C is called the heat capacity matrix
C=
(A.19)
Hm) pcm)Hm)dV'(m)
It is noted that the volumetric specific heat, pc, can also be expressed as
k
, where a is
a
--
the thermal diffusivity.
Equation (A.18) is a system of coupled differential equations that can be solved
numerically by the implicit Euler backward method or the implicit trapezoidal rule, both
of which are special cases of the alpha-integration method [A-1]. For these methods, tO
is assumed to be known. In the Euler backward method, the nodal temperature derivative
matrix at time t+At is approximated as
t+A6
=
0
0
(A.20)
At
Substituting eqn.(A.20) into eqn.(A.18) at time t+At and solving for the unknown '"A'0
result in
C
C
At
At
(A.21)
The implicit trapezoidal rule approximates the temperature time-derivative matrix and
At
temperature matrix at time t + -
2
with the following equations
t+ -' 2 0
=
t+A
0-t 0
At
t+ A
2 0-
(A.22)
t+At O+t 0
2
Substituting eqn.(A.22) into eqn.(A. 18) at time t + - and solving for '+A' O result in
2
53
C K+ K' K
At-+-+-2
2
C -K* kKc
+t
At
2
'Kc'+A=Q+r'C(A23
o(.3
2
(
Both the Euler backward method and implicit trapezoidal rule are unconditionally stable
methods in time; however, the former method is of first-order accuracy in At, while the
latter is of second-order accuracy in At.
Two-Dimensional Four-Node Quadrilateral Elements
Consider a two-dimensional four-node quadrilateral element located in the x-y
plane as shown in Fig. A. 1. The upper right-hand node, located at point (xj,yj), is labeled
node 1. The upper left-hand node, located at point (x2,y ), is labeled node 2. Continuing in
a counterclockwise fashion, nodes at points
(x2,y4)
and (xJ,y4) are labeled node 3 and node
4, respectively.
node 1
node 2
(x 2,
X(
1 ,y )
1)
node 4
node 3
( XY
(x 2, Y4 )
4)
x
Figure A.1-Schematic of two-dimensional four-node quadrilateral element.
54
For convenience, an axis transformation from the x-y global axes to the x'- y' local axes,
with the local origin at the center of the finite element, is defined by
1+2
X/
2
(A.24)
+ Y
y'= y + Y1
2
The local temperature interpolation matrix, H("), becomes
here a1n+ b a
H
4 _
a)('
y
(
t
bi
C-JC-
-''+J
a) b)
b)_
C+X1-
b')
)
(A.25)
b
a)
where a and b are
a=
-2
b=
-
(A.26)
2
2
Given the elemental conductivity, k', elemental diffusivity, d, front and back surface
heat transfer coefficients, h, and hN, respectively, and applied heat flux on the front
surface, q", the local elemental matrices are formulated according to eqns.(A.12) through
(A.15) and eqn. (A.19).
b2 +a2
3
K(m)k = k'2
a2 -2b 2
6
b 2 +a
b 2 -2a
3
6
b +a
3
2
a b
symmetric
b2 -2a 2
6
b2 +a
6
6
+b2+2a
2
2
2
2
a
2
6
b +a
3
2
55
-2b2
2
2
(A.27)
0
K(MC fron t
0
0
b h3 00 21 21 00
0
0
2 0
0
b hN
K (m)back
c
3
0 0
1
0
0
0
0
0
0
1
0
0
2
0
0
(A.28)
(A.29)
0
Q()
0
0
C
4
2
1
2
= abk'" 2
9,m 1
4
2
2
4
1
2
2
1
2
4]
-)
(A.30)
Two-Dimensional Nine-Node Quadrilateral Elements
The formulation for a two-dimensional nine-node quadrilateral element is similar
to that for a four-node element. The corner nodes, node 1 through node 4, are identical in
label and position to a four-node element. Four nodes-node 5, node 6, node 7, and node
8-are placed at the midpoints of the top, left, bottom, and right edges, respectively. The
last node, node 9, is located in the center of the finite element. Figure A.2 illustrates the
nine-node finite element.
56
node 5
node2
node]I
([xi+ x2]/2, yi )
)
(x2,y]
(X4 j
node 8
node 9
node 6
(xi, [y+
4
4
(x2, [yI+ y4]/2)
(Ix+
y4]/2)
x2]/2,
[y]+ y4]/2)
node 3
node 4
(X2,y4
,4)
node 7
[xl+
x2]/2, Y4 )
Figure A.2-Schematic of two-dimensional nine-node quadrilateral element.
For a nine-node quadrilateral element, the local interpolation functions are
-
g
g2
1+
4ab
__
g4 =
s5
-g
4ab (
2'
a)
b)
+
-1 I+ x
4ab
93
I+-
-
I+
4ab
1+
a
b
a)
1 + b)
-+
a)
b)
- 1+
(1+
2b
b
x
1
a2
-1+
Y1)(A.
-i+KJ
-1+±
=
96
i+~ 2C±
g6
g
-+
i- +
=
9757
g 1+(
-+
2
31)
The resulting non-zero upper diagonal elements of K(m> k are listed below.
Kkll =Kk
-4k
22
=Kk
Kk55 =Kk77 =
Kk66 =Kk88 =
33
=Kk 44
M
1
45 (a
abk m 32 56)
45 (a 2b2
abk'
56
32
45 (a 2
K128k'"1k1
499a4b
Kk12 =Kk3 4 =
abk m
45
Kk13 = Kk 24 =
Kkl 4 =Kk23=
Kk1 6 = Kk3 6
=
=K
K1g =Kk
b2
b
abk m
7
4
( 22+- 2
90
a b )
Kk
28
=K
Kk47 -
abk m
16
4
45
a2
= Kk3 8 =
abk m 1
+
45
a2
=
=K
= K=K
48
a
7
2
90 (a
k25 = Kk37
Kk15 =K
2
=Kk
26
abk'" 4
45
a2
=K
=Kk 36 -5= ab k '"
45
Ka7
2
(A.32)
1
b24
16 )
b2)
Kk19 =Kk 29 = Kk39 = Kk49 = Kk56 = Kk67 = K k58 =K k78
8
k57
abk'"
45
a
abk m
45
8
8
k68
a2
b2
Kk59 = Kk7
9
Kk69
Kk89
8
b2
abk' (16
64
45 Ka 2b2)
abk'
=
+
45
64
a2
16
b2
58
abk m
45
I2
8
abj
1
The front and back surface convection matrices are listed below.
0 0
4
0
-1
4
K(m)front
0
0
0
0
0
0
0
2
2
0
0
0
0
0
0
0
0
0
0
0 0
0 0
0 0
0 0
0
= bh 2
15
0
16
symmetric
4 0 0
0 0
-1
0
0
0
4
K ) "bck-bhN
15
0
0
0
0
40
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
16
symmetric
(A.33)
2
0
0
0
0
The surface heat flux vector is given by
Q('") = bq" [1
3
0
0
1
0
0
0
4
(A.34)
0]f
The conductivity matrix is
716
-4
16
1
-4
- 4 1
16 -4
16
8
8
-2
-2
-2
8
8
-2
-2
-2
8
8
8
-2
-2
4
4
4
8
4
64
4
-16
64
4
4
-16
32
32
32
32
256
C M) = abk
225a m
symmetric
59
64
4
64
(A.35)
Reference
[A-1] Bathe, K.-J., Finite Element Procedures,Upper Saddle River, New Jersey,
Prentice Hall, 1996.
60
APPENDIX B: VERIFICATION AND ERROR ANALYSIS OF THE FINITE
ELEMENT SOLUTION
Example Data
As an example, the verification and error analysis of two cases of semi-infinitely
long anomalies within a composite polymeric sandwich panel are considered. Case one
models a void filled with air having a thermal conductivity ka, = 2.93x10-2 W/(m-K) and
thermal diffusivity ax,=2.72x10-2 m 2/s. Case two models an inclusion of an aluminum
alloy having a thermal conductivity
aam =7x10~4
kam=
180 W/(m-K) and thermal diffusivity
m2/s. For each case, the anomaly has the following geometry:
anomaly depth:
Da= 0.0127 m
anomaly thickness:
da =
0.01 m
The anomaly is embedded in a sandwich panel having the following properties:
face sheet thickness:
d, = 0.0038 m
core thickness:
dc = 0.0254 m
face sheet thermal conductivity:
k,= 0.425 W/(m-K)
core thermal conductivity:
kc = 0.0437 W/(m-K)
face sheet thermal diffusivity:
a,= 2.27x10- m2/s
core thermal diffusivity:
ac= 1.69x10- 6 m2 /s
The panel is exposed to the following boundary conditions:
front surface heat flux:
q"= 500 W/m 2
front surface heat transfer coefficient:
h = 10 W/(m 2 .K)
back surface heat transfer coefficient:
hN = 10 W/(m 2 K).
61
Finite Element Mesh
The finite element method is implemented using Mathwork's MATLAB. For
cases one and two, Figures B. 1 through B.4 show the resulting steady-state temperature
distributions and heat flux components for a section of the sandwich panel near the end of
the anomaly. For each figure the same finite element mesh is used, with the lightweight
lines representing the boundaries of each element and the heavier lines representing the
boundaries between dissimilar materials. Progressively smaller quadrilateral elements
with a graded mesh are located near the end of the anomaly to account for the large
variations in heat flux, while longer elements are used where the model approaches onedimensional heat flow.
Comparing the temperature solutions of each case generated using four-node
elements to those using nine-node elements (Figure B.1a with Figure B.2a and Figure
B.3a with Figure B.4a), very little variation is noticed upon visual inspection. However,
the resulting heat flux distribution in the corresponding figures (parts b & c) show that
the bands of heat flux between adjacent elements in meshes produced with nine-nodes
elements are more continuous than meshes with four-node elements. Within the sandwich
panel, the heat flux distributions also reveal that the component of heat flux normal to
material boundaries is continuous across each boundary, a verification that the finite
elements are, indeed, energy balanced.
Near the end of the anomaly, the predicted heat flux components are much higher
than the values throughout the sandwich panel. If progressively finer meshes near such
sharp edges are employed, it will undoubtedly be discovered that the heat flux
components are unbounded. This is due to the fact that the anomaly is modeled as a
62
rectangle with perfectly sharp edges. Such geometries are not physically realizable, and,
if the heat transfer near the edges of a physical anomaly is desired (analogous to a stress
concentration problem in solid mechanics) the edges should be modeled with a small
radius.
However, the temperature distributions along the front and back surfaces are
insensitive to small changes in the shapes of the anomaly. For this investigation the
temperatures on the front and back surface are desired, not the heat flux in areas very
close to such edges; therefore, modeling an anomaly as rectangular is adequate.
Comparison with the One-dimensional Solution
For anomalies that are semi-infinitely long, the finite element solution of the
temperature (above ambient), a(x, y,t), must agree with the one-dimensional steady state
and transient solutions as y -> ±oo [B-1].
For cases one and two, the difference in the steady state surface temperatures
between the analytical (exact) solution and the finite element solution for region I
(undamaged region) and region II (region containing anomaly), yield a steady state error,
defined as
steady state error
=
dexact - OFEM
Oexact
which is found to be of order of 10-9 to
10-42
for the four-node elements and 10-12 for the
nine-node .
The transient solution is numerically integrated using the implicit Euler backward
method for the four-node element and the implicit trapezoidal rule for the nine-node
element. For cases one and two, Figures B.5 through B.8 show the relationship between
63
the exact solution of the transient surface temperatures and the corresponding solutions at
The transient solutions for the nine-node
various times using a four-node element.
element are shown in Figures B.9 through B.12. The time steps vary progressively from
0.15 s at the beginning to 260 s near t = 3000 s.
To evaluate the accuracy of the numerical integration, the transient state error is
defined as
jO(t)ex
transient state error
-
6(t)FEMI
=
0(t)exact
The calculated front surface temperatures using the four-node element (Figures B.5
through B.8) have transient state errors of the order of 10-2 ; however, the back surface
transient temperatures (Figures B.9 through B.12) have a transient state error that is
approximately 2% near t = 3000 s, indicating that finer time steps are needed to improve
accuracy. On the other hand, the transient error for both front and back temperatures of
both cases using the nine-node element is of the order of 10-3 as t nears 3000 s.
Convergence of Characteristic Length
In order to characterize the transition length of the surface temperature
distribution from the region I temperature to the region II temperature, a measurable
parameter called the steady-state characteristiclength, denoted as A, is introduced. It is
defined to be the length along the surface in which the temperature approaches within 5%
of each temperature asymptote. By employing an appropriate mesh, the finite element
model approximates the temperature distribution throughout the model and, based on the
approximation of the surface temperatures, an approximate characteristic length is
64
calculated. This approximate characteristic length is, in general, a function of the mesh
used to generate a solution.
(Reconsider the finite element mesh used identically in
Figures B.1 though B.4.) The successive length of the horizontal divisions follow in
geometric progression and is symmetric about the line y=O. This mesh can be
characterized by the length of the smallest elements, namely those that adjoin the line
y=O, which is denoted here as h. Subsequently, the approximate characteristic length
based on such a length is denoted A .
The error associated with the approximation of the characteristic length is
expressed as
Error=IA-AhI
A
Of course, the exact value of A is not known. For this reason, an extremely fine ninenode element mesh, with h-10-6 m, is used to approximate the exact value of A for cases
one and two. The convergence of this error for cases one and two are shown in Figures
B.13 and B.14. First, it is observed that the overall trend of the data is converging.
Second, the slopes of the linear trend line, which can be considered the order of
convergence, indicates that the nine-node elements is converging at a faster rate than the
four-node elements. Third, the slopes of the trend line for cases one and two are very
similar, indicating that the order of convergence is independent of the material of the
anomaly.
The orders of the error in the characteristic length calculated using the mesh in
Figures B.1 through B.4 are 10-2 and 10-3 for the four- and nine-node element,
respectively.
65
Conclusions
The verification and error analysis of cases one and two have been presented. The
steady state solution throughout the model for sample meshes was presented. Excellent
agreement was found between the one-dimensional exact solution and the finite element
solution when using nine-node elements meshes in both steady and transient
computations. Additionally, a convergence analysis compared and contrasted the
convergence behavior of the characteristic length, with nine-node elements converging at
a faster rate.
Although nine-node elements have a higher computational cost than four-node
elements, the associated error is lower. When compared with using four-nodes elements,
larger elements and longer time steps can be used to obtain the same error. For this
reason, computations using nine-nodes elements (as shown in Figure B.2 and B.4) are
used throughout the analysis of the sandwich panel.
Reference
[B-1]
Cai, L.-W., A.F. Thomas and J.H. Williams, Jr., "Thermographic Nondestructive
Testing of Polymeric Composite Sandwich Panels," Materials Evaluation, Vol.
59, 2001, pp. 1061 - 1071.
66
0
10
20
30
40
50
50
55
60
65
70
-20
W/m 2 per unit width into page
*C
-10
0
10
20
W/m 2 per unit width into page
0.04
0.04
0.04-
0.03
0.03
0.03-
0.02
0.02
0.02-
0.01
0.01
0.01
-
1-0RC
0
0
0~
0
0
0
0
0
0
y'
-0.01-
-0.01
-0.02-
-0.02-
-0.01
-
-0.02I
~
-h
-0.03-
-0.03-
-0.041
-0.04
-0.04
0
0.02
0.01
x-coordinate (m)
(a) Temperature Rise
0.03
-0.03-
0
0.01
0.02
x-coordinate (m)
(b) Heat Flux, x-direction
0.03
0
0.02
x-coordinate (m)
0.01
(c) Heat Flux, y-direction
Figure B.1-Steady-state solution of the finite element model of case one using four-node element.
0.03
0
10
20
30
40
50
50
55
60
65
70
-20
W/m 2 per unit width into page
OC
-
0.04
-
0.04
0.03
-
0.03
-
0.03
0.02-
0.01
0.01
-
0
10
20
W/m2 per unit width into page
0.04
0.02 -
-10
0.02
0.01
-
4.)N
0
00
0
0
0
4)
0
*0
0
-0.01-
-0.01
-0.01
-0.02-
-0.02 -
-0.02-
-0.03-
-0.03-
-0.03-
-0.04 0
-0.04 0
-0.04-
x-coordinate (m)
0.02
x-coordinate (m)
(a) Temperature Rise
(b) Heat Flux, x-direction
0.01
0.02
0.03
-
0.01
0.03
0
0.01
0.02
x-coordinate (m)
(c) Heat Flux, y-direction
Figure B.2-Steady-state solution of the finite element model of case one using nine-node element.
0.03
0
10
20
30
40
50
50
*C
60
70
80
90
100
-20
W/m2 per unit width into page
0.04-
0.04
0.03
0.03-
0.03-
0.02
0.02-
0.02-
0.01
0.01-
0.01-
0
0
0)
0
0
-0.01
-0.02
-0.02
-0.02
-0.03-
-0.03
-0.03
(a) Temperature Rise
0.03
-
-0.04- -
-0.04
0.01
0.02
x-coordinate (m)
0
0.01
0.02
0.03
0
0.01
0.02
x-coordinate (m)
c-coordinate (m)
(b) Heat Flux, x-direction
(c) Heat Flux, y-direction
Figure B.3-Steady-state solution of the finite element model of case two using four-node element.
........
..
.
20
0
-0.01
0
10
0
-0.01
-0.04
0
W/m2 per unit width into page
0.04-
0
0
-10
0.03
0
10
20
30
40
50
50
70
60
80
90
W/m 2 per unit width into page
OC
0.04
0.04
0.03
0.03
0.03
0.02
0.02
0.02
0.01
0.01
0.01
0
0
0
0
0
0
-0.01
-0.02 -
-0.02
-0.02-
-0.03-
-0.03-
-0.03-
-0.04 -1
-0.04
(a) Temperature Rise
0.03
20
0
-0.01-
0.02
x-coordinate (m)
10
0
-0.01-
0.01
0
0
0
0
-10
W/m2 per unit width into page
0.04-
7E!
-.1
-20
100
-
-0.04
0
0.01
0.02
x-coordinate (m)
(b) Heat Flux, x-direction
0.03
0
0.02
0.01
x-coordinate (m)
(c) Heat Flux, y-direction
Figure B.4-Steady-state solution of the finite element model of case two using nine-node element.
0.03
50
45
40
35
0
30
... FEM
.
su.c te praue
Reio
.....
25
-
15
10
I surface temperature, exact
-Region
Region I surface temperature, FEM
-- Region II surface temperature, exact
3 Region I1 surface temperature, FEM
.. . . . .. . . .
20 -. . .. . .
-. . .... .
. .... .
-...
- .. -.. ......
500
. ...
...
1000
......
-
.....
..
-.
..
-..
.... ..-.. ...... .... ..
2000
1500
times (s)
2500
3000
Figure B.5-Transient solution for the front surface using four-node elements for case
one.
10
9
-
O
8
-
o
7
Region I surface temperature, exact
Region I surface temperature, FEM ...................................
Region II surface temperature, exact
Region II surface temperature, FEM
. -
-'
.- -
-.
1
5.
.
S0
2 -.
. . . . . . ..
3 . . . . .
01
.... ..
. . . ..
.
500
1000
. . . . . .. . . . . . . . ..
1500
times (s)
2000
. . .
2500
. .
3000
Figure B.6-Transient solution for the back surface using four-node elements for case
one.
71
50
45F-
40
-
35
-
-
-
.
0 Reio
......
*.
.M
....
sufc te praue F
.
30
-- Region I surface temperature, exact
Region I surface temperature, FEM
-- Region IH surface temperature, exact
0 Region II surface temperature, FEM
-.-. -0
225 -.............
20
- ....
-.
-. -.
......... ...
.. .. ... .... ..
......
.-.
.-.-.
. -.
-
.... .
15
..... .... ...
... .....
.. ..
.. ..
10
- .. ..
-.
-.
. -.
.
...
....-. ...-. ... .
.. .. ...
-.
5
J'
V_
500
1000
1500
times (S)
2000
2500
3000
Figure B.7-Transient solution for the front surface using four-node elements for case
two.
9
B
Region I surface temperature, exact
0 Region I surface temperature, FEM . .
- Region II surface temperature, exact
0 Region II surface temperature, FEM
7-
--
0
-...--..--
~-~0
C
-.. . . .
5 -
00
S00
E
O0
-
4.
.. .
3 -.
2E
3
.
0
. . . .
. . . .
500
.
.
..
1000
. . . . .
. . . .
1500
times (s)
. . . .
. .. .
2000
. . .
. . .
. .
2500
. . .
. .
3000
Figure B.8-Transient solution for the back surface using four-node elements for case
two.
72
50
-
45
40
-35
OU 30
-.......
-O
E 25
-
o
Region
Region
Region
Region
I surface temperature, exact
I surface temperature, FEM
II surface temperature, exact
II surface temperature, FEM
-.
20
-
..-.
.-
15
-...
-.....
-. .. ... ....
-....
10
.
....
- -.-.....-..
-...-..-.....-..-...-
5
-1
VW
500
1000
1500
times (s)
2000
2500
3000
Figure B.9-Transient solution for the front surface using nine-node elements for case
one.
1f).
9
8
7
- Region
Region
- Region
o Region
o-
I surface temperature, exact
I surface temperature, FEM
I surface temperature, exact
II surface temperature, FEM
...................
6
0
5
CE
]
-0
4
3
-
--
2
-6-
500
1000
1500
times (s)
2000
2500
3000
Figure B.10-Transient solution for the back surface using nine-node elements for case
one.
73
50
4540
35
-
-0
0 RgionI surace emperture FE
/
9-30
2 25
-Region I surface temperature, exact
Region I surface temperature, FEM
-- Region II surface temperature, exact
0 Region II surface temperature, FEM
-- - -
-.
20
-.
..... ...
.
-.
-..
-.-.-
-.. -.
15
..
...............
.. ...... ...
10
~~
....
.. ..... ..-
-....
...
.... -..
--.
.. ... .. ...-..
......
..
5
1000
500
2000
1500
times (s)
2500
3000
Figure B.11-Transient solution for the front surface using nine-node elements for case
two.
1A
9
8-
7
Region I surface temperature, exact
0 Region I surface temperature, FEM
- Region II surface temperature, exact
0 Region II surface temperature, FEM
-
-
-
-
- -.00
5 -
4 --
-
-
- -- - - -
-
- - --.-.-.--
-
-
-.-.
3 ----
2
0
500
1000
1500
times (s)
2000
2500
3000
Figure B.12-Transient solution for the back surface using nine-node elements for case
two.
74
0
--
-
-
- -
-
--
-2 - -
0: 0
00
-30
%
-4---
0
0-0
O
AA
0
0
-L -0 --
-6 -
-
0
-1 ---
- -
-
CC
-2.5
-2.25
O
~Z~~AA
A
A
FEM,4node
Linear fit, 4 node
A FEM, 9 node
fit, 9 node
-Linear
A
A
-
--
-
-
log10 hId )
-1
-1.25
-1.5
-1.75
-2
A
-
O A0
-
AA00A9
A
AA0
04
01.
0
h = Length of Smallest Element
de = Core Thickness
Figure B.13-Steady-state characteristic length convergence for case one.
-7A
04FEM,
A
a
AA-A
O
----
o
-6
-2.5
-
-
-.-.-.---.-.-.-
-.-. ---.-.-.-
-.-.-.-.
-2.25
-Linear
-1.75
-2
node
Linear fit, 4 node
FE~M, 9node
fit, 9 node
-1.5
-
-1.25
logI(h/d,)
h
=
Length of Smallest Element
d = Core Thickness
Figure B.14-Steady-state characteristic length convergence for case two.
75
-
APPENDIX C: DEVELOPMENT OF NONDIMENSIONAL PARAMETERS
The functional form the front surface temperature distribution is
OF=
g,(y, t, q",7hl 9hN) as ac, a ks 10k ka 1ds
,d
, Da, La)
(C. 1)
The base dimensions of each variable are
[OF]=T [y]=L
[a]=L
2t
[q"]=Mt~3
[t]=t
-' [k] =MLt -3 T -1 [Da]=L
3
[h]=Mt -T'-
[d]=L
[L]=L
where M, L, t, and T represent mass, length, time, and temperature, respectively.
Because there are four independent base dimensions, four independent variables, namely
ac, kr, dc, and q", are chosen to nondimensionalize the remaining variables [C-1]. The
resulting nondimensional form of Equation (C. 1) is
OFkc
q dC
-
Is
dC
ska
)a
a
a
ds da Da La
/ kc dc
kc
dr
d
(C.2)
d)
where Fo is the Fouriernumber based upon the properties of core, which represents the
nondimensional time, defined by
(C.3)
Fo act
d2
and where Bi, and BiN and are the front and back Biot numbers defined by
Bi
kd
and
Bi
hNdc
(C.4)
(The Biot number is defined as the conduction resistance of the body divided by
its boundary heat transfer resistance. The one-dimensional conduction resistance per unit
d
d
kc
k,
area of the undamaged panel is d +2---,
76
but here it is assumed that the thermal
resistance of the core is dominant, i.e.,
d~
d
d
>> ' . Hence after dividing
by the
kc
ks
kc
-
-L-
1
boundary heat transfer resistance per unit area, -,the expressions above are adequate.)
h
The one-dimensional front surface temperatures in region I and region II [C-2],
are expressed in nondimensional form as
__k
,
=
93
q dc
(y
Y,
(dc
Fl'
a
.
a
Fo,Bil , BiN
_
kskadsdaDa
sk
sd
ac ac kc kc dc d,
a
yc
O k
a
s5a
d F B.
Bd a sa
k
_k _
k , ka d
d
)D
dc
(C.5)
D
da Da
(C.6)
Forming the normalized temperature distribution from Equations (C.2), (C.5), and (C.6),
results in
OF
d
-OF
asa
B
k s kf d s d f Df L(
i,
(C .7)
A similar normalized temperature distribution is obtained for the back surface.
The limiting steady-state anomaly lengths, Ln
and L., have the following
functional forms
Lmn =g5 (3,hl,h2 ,k,kc, ka, d,, de, da, Da)
L.
= g6
(o,hl,h 2 ,ks, 1c, ka , d,, dc, da , Da)
(C.8)
(C.9)
where it is noted that 8, the relative temperature resolution, is dimensionless. Although
there are four base dimensions represented by the variables in eqns. (C.6) and (C.7), only
two base dimensions are independent. Hence two variables, kc and dc, are chosen to
nondimensionalize the remaining variables. Consequently, the following equations give
the nondimensional functional dependence of parameters L. and L, .
77
""n = f5r, ,Bi, BiN -
dc
L*
ds d
'dc dc
a
k 'k
-
f6, Bil, BiN
s
a
d
k
k
da
(C. 10)
a
d
Da_
de d
(C. 11)
de )
The characteristic length, A, has the following functional form
A = g7(E,t,as,ac,aa,hj,hNI ksl
k a,ds,,de,da,Da)
(C.12)
where it is noted that e is a dimensionless tolerance value associated with the normalized
temperature.
Three variables, ac , k and dc , are chosen to nondimensionalize the
remaining variables. Hence, the nondimensional form of eqn. (C.12) is
(
A
-f
dC
Ae,
Fo, Bi Bi
Ia
aa ks
ka
ds
daI Da
a. ac kc k C dc dc dc
(C. 13)
References
[C-1]
Buckingham, E., "On Physically Similar Systems: Illustrations of the Use of
Dimensional Equations," PhysicalReview, Vol. 4, 1914, pp. 345 - 376.
[C-2]
Cai, L.W., A.F. Thomas, and J.H. Williams, Jr., "Thermographic Nondestructive
Testing of Polymeric Composite Sandwich Panels," Materials Evaluation, Vol.
59,2001,pp.1061
-1071.
78