PROGRESS IN THE DEVELOPMENT OF A THERMOELASTIC GAUGE
FOR STRESS SEPARATION
P. Stanley1, T.S. Phan2 and J.M. Dulieu-Barton2
University of Manchester, School of Mechanical, Aeronautical and Civil Engineering, Simon
Building, Oxford Road, Manchester, M13 9PL, UK
2
University of Southampton, School of Engineering Sciences, Highfield, Southampton, SO15 5BY,
UK
1
Abstract
The paper deals with work carried out to pursue the exploration of the practical feasibility of a bonded coupon gauge for
use in the stress separation of the stress-sum data obtained in thermoelastic stress analysis (TSA). The questions of
reinforcement and possible non-adiabatic behaviour are examined and an experimental validation procedure is described.
Introduction
The thermoelastic stress analysis technique provides a quantitative assessment of the sum of the principal stresses (i.e.
the first stress invariant) over the surface of a cyclically loaded solid. A wide range of applications of the technique
covering the stress analysis and related work for components in both homogeneous and composite materials has been
described [1]. The stress analyst/designer may, however, require information on the individual surface stresses and
techniques for their derivation from the stress sum data are required; the process is referred to as “stress separation”. A
review of such techniques, mainly numerical, up to the mid-1990s is given in Ref [2]. Other studies include the use of a
hybrid photoelastic-thermoelastic technique [3] and further numerical work [4, 5].
The present paper describes development work on an earlier approach [2] which largely dispenses with supplementary
experimental or numerical work and relies on thermoelastic data. (In the general case it may be necessary to establish
principal stress directions independently.) The method requires a small device {the “thermoelastic strain gauge”) to be
bonded to the surface of the stressed body so as to provide an additional thermoelastic signal which permits the
determination of the individual surface stresses in the body. A particularly versatile form of the device is the “coupon”
gauge; this consists of a small thin coupon of material which is bonded over its entire surface to the specimen or
component to be analysed. Assuming a complete strain transfer from the specimen through the bond and into the
coupon, then an independent relationship can be developed between the specimen stresses and the thermoelastic signal
from the coupon surface, from which, together with the thermoelastic signal from the specimen itself, individual stress
values may be derived. The paper examines the possibilities of local reinforcement and non-adiabatic behaviour, and
proposes an experimental validation procedure.
General Theory
The coupon gauge is essentially a strain transfer device, i.e. the action of the bond between the gauge and the specimen
is such that the strains in the specimen are “copied” into the gauge. The gauge and specimen notation is defined in Figure
1. The gauge is shown as rectangular; strains in the longitudinal direction of the gauge are denoted ε11g and strains
transverse to the gauge are denoted ε22g. This terminology applies for both isotropic and orthotropic gauge materials; in
the orthotropic cases the gauge axes (i.e. 11 and 22) are assumed to be aligned with the principal material axes of the
gauge. The principal strains in the specimen are denoted ε1s and ε2s. The maximum principal strain in the specimen (i.e.
ε1s) is inclined at an angle φ to the direction of the longitudinal principal axis of the gauge material. A further set of axes is
also shown in Figure 1 inclined at an angle β to the longitudinal axis of the gauge. These axes, denoted 11s and 22s, are
specifically for an orthotropic specimen and coincide with the principal material axes of the specimen. (In the general case
both φ and β are unknown. Depending on the knowledge of the specimen and the prevailing symmetry, however, one or
both of these angles may be known.)
Assuming adiabatic conditions, a general expression for the thermoelastic response of an orthotropic gauge bonded to the
surface of a stressed orthotropic solid has been developed in the form:1
Sg =
α g Q g [T ]−1 [C s ]{σ s }
(1)
A g/
{ }[ ]
/
where Sg is the gauge signal, A g is the gauge calibration constant, {σs} is the column vector of the specimen stresses
referred to the principal material axes of the specimen, [Cs] is the compliance matrix of the specimen referred to the
principal material axes of the specimen, [T] is the transformation matrix from the principal material axes of the orthotropic
specimen to those of the gauge, i.e.
cos2 β
sin2 β
2 cos β sin β
sin2 β
cos2 β
− 2 cos β sin β
− cos β sin β cos β sin β (cos2 β − sin2 β)
[Qg] is the stiffness matrix of the gauge referred to the principal material axes of the gauge and {αg} is the row vector of the
principal coefficients of thermal expansion of the gauge, i.e. {α11g, α22g, α12g}.
ε2s
ε22g
ε11s
ε1s
ε11g
ε22s
φ
β
Figure 1 Coupon gauge and specimen notation
The thermoelastic signal from the specimen itself Ss, is given by
Ss =
1
A s/
{α s }{σs }
(2)
where A s is the specimen calibration constant, and αs is the row vector of the principal coefficients of thermal expansion of
the specimen material.
/
Equations (1) and (2) involve the specimen stresses and the angle φ (Figure 1). If the angle φ is known or has been
found independently, then in principle, the specimen stresses can be found by simultaneous solution. This approach has
been further studied for the following four coupon/specimen material combinations [6], with the results indicated
(i)
Isotropic coupon/Isotropic specimen
The signal from the coupon gauge is simply a linearly scaled version of the specimen signal and the combination is
of no use for stress separation purposes.
(ii)
Orthotropic coupon/Isotropic specimen
/
/
The calibration constants A g and A s must be known. In cases where the angle φ is not known signal readings
from a second coupon gauge, with a different orientation, will allow a full solution. Since accurate values of the
relevant material properties may not be available, a means of gauge calibration is desirable. However, this
combination offers the greatest potential for development.
(iii)
Isotropic coupon/Orthotropic specimen
With this combination, the specimen stresses in the principal material directions can be determined, but the angle φ
cannot be found. Again a suitable calibration technique is required [6].
(iv)
Orthotropic coupon/Orthotropic specimen
This gauge/specimen combination has little to offer, in practical terms, for stress separation purposes.
Reinforcement
The possibility that the bonded coupon gauge may reinforce the specimen, thereby significantly modifying the specimen
stresses (and the coupon signal), has been examined for the case of an isotropic coupon bonded to an isotropic specimen
subjected to uniaxial tension (see Figure 2).
Coupon (c)
tc
tb
Bond (b)
σapp
ts
Specimen (s)
Figure 2 Coupon, bond and specimen (not to scale)
Assuming identical Poisson’s ratio values in the coupon, bond and specimen, an analytical treatment has been developed,
based on the equivalent section approach [7], in which the widths of the coupon and bond are scaled down by factors of
Ec/Es and Eb/Es respectively (E is Young’s modulus and the suffices c, b and s denote coupon, bond and specimen). This
gives a modified ‘equivalent’ section with a uniform modulus Es. The specimen is subjected to a uniform uniaxial stress.
The centroid position and second moment of area of the equivalent section are determined, and the bending moment
developed and thence the bending stress on the outer surface of the coupon are readily derived. To complement the
analysis a reinforcement factor, R (defined as the ratio of the outer surface bending strain in the coupon to the specimen
surface strain in the absence of the coupon) was evaluated from the calculated results. Finally an approximate value of R,
obtained by eliminating minor terms in the analytical expressions, was derived in the form:
R=3
t
t c Ec
(1 + 2 b )
ts
ts Es
(3)
The range of variables studied, governed largely by the experimental coverage, was as presented in Table 1; Poisson’s
ratio was taken as 0.3 throughout.
Table 1 Reinforcement analysis parameters
Material
Coupon
Bond
Specimen
Aluminium or Copper
Epoxy
Steel
Young’s Modulus
(GPa)
70 or 120
5, 2 or 1
207
Thickness
(mm)
0.1, 0.2 or 0.5
0 or 0.1
8
Rather than presenting the results in full detail, it is convenient to use Equation (3) for the purpose of summarising the
outcome of the analytical work:
•
The values of the reinforcement factor R obtained from Equation (3) were close approximations to the actual
values.
•
R was practically independent of the bond Young’s modulus.
•
R was approximately proportional to the Young’s modulus and the thickness of the coupon; general trends are
shown in Figure 3.
•
For copper as the coupon material, R<5% if tc<0.24 mm, R<2% if tc<0.10 mm.
•
For aluminium as the coupon material, R<5% if tc<0.40 mm, R<2% if tc<0.16mm.
The effects of possible Poisson’s ratio (ν) differences in the coupon and specimen materials were assessed numerically
for the cases tc = 0.1 mm and Eb = 2 or 1 GPa using the ABACUS code with generalised plane strain conditions. Coupon
νc values of 0.25, 0.3 and 0.35 were studied; νs and νb were held at 0.30. For these examples the maximum R value for the
copper was 5.1% and for the aluminium 3.2%. A decrease in νc from 0.30 to 0.25 caused an increase in R from the range
3 to 4% to the range 4 to 5% for the copper and from the 1 to 2% range to the 3 to 4% range for the aluminium. An
increase in νc from 0.30 to 0.35 caused similar decreases in R.
It was concluded that, with the proper choice of materials and thickness, reinforcement effects may be reduced to an
insignificant level and possible differences between the Poisson’s ratios of the coupon and specimen will cause no
significant additional reinforcement.
1.8
1.6
R=10%, tb/ts=0.0125
1.4
Ec/Es
1.2
R=5%, tb/ts=0.0125
1
0.8
R=2%, tb/ts=0.0125
0.6
0.4
0.2
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
tc/ts
Figure 3 Reinforcement effects
Thermal conduction effects
It is of crucial importance in the design of the thermoelastic coupon gauge that the thermal conditions at the gauge are
adiabatic, i.e. the heat transfer between the specimen and the emitting surface of the coupon should be negligible; only
then will Equations (1) and (2) be valid. The exact analysis is complex and is being pursued. As an alternative, an
approximate approach has been developed based on the treatment of ‘Period change of the temperature of both surfaces
of a plane plate of finite thickness’ given in Ref. [8]. ‘One half’ of this configuration, i.e. from one surface to the mid-plane
of the plate, can be taken to represent a layer on a substrate subjected to cyclic temperature variations. (The approach
has been used in earlier work [9] on the thermal response of a coated body.) In the present context the ‘layer’ represents
the bond between the specimen and the coupon; the orders of magnitude difference in the thermal diffusivities of the bond
and coupon materials justifies the omission of the latter from this approximate treatment. From the theory [8] the following
expression can be derived relating the minimum layer thickness (hmin) to the thermal diffusivity (γ) of the layer material and
the frequency (f, in Hz) of the applied surface temperature change such that the amplitude of the temperature change of
the free surface of the layer is less than that of the applied temperature cycle by a factor m as follows:
γ
hmin = g(m)( )0.5
f
(4)
The multiplying coefficient g(m), a function of m, in Equation (4) can be found from data tabulated in Ref. [8]; values for m
values of 0.1 and 0.05 are 1.97 and 2.20 respectively.
The variation of hmin with frequency for an epoxy resin (the basis of the coupon-specimen bond) as derived from Equation
-6
2 -1
(4) is shown in Figure 4 for the two m values of 0.1 and 0.05. A thermal diffusivity value of 0.160 x 10 m s for the
epoxy, derived from the relevant material property values obtained from various sources, was used in the calculations. (By
-6
-6
-6
2 -1
way of contrast, diffusivity values for steel, copper and aluminium are 15 x 10 , 112 x 10 and 94 x 10 m s respectively
[10]. The exceptional effectiveness of epoxy resin as a thermal insulating material is evident from these figures.) It can be
seen from Figure 4 that with a layer (i.e. bond) thickness of the order of 0.2 mm then attenuating factors of 5 to 10% can
be expected at a frequency of 10 Hz.
Noting that the foregoing treatment is approximate, and that the influence of the coupon material has not been allowed for,
it would seem reasonable to infer that, at this stage, possible non-adiabatic effects associated with the coupon gauge are
not a major concern. It is accepted that this matter may require some further attention as the gauge is refined and, as a
possibility, some form of small correction factor is not ruled out.
0.4
m = 0.05
m = 0.1
h min (mm)
0.3
0.2
0.1
0
0
10
20
30
f (Hz)
Figure 4 Minimum epoxy resin layer thickness versus frequency for attenuation factors of 0.1 and 0.05
Validation
The expression for the thermoelastic signal (Sg) from an orthotropic coupon bonded to an isotropic specimen is derived
from Equation (1) [2] as:
(
)
(
S g = σ1s W cos 2 φ − Y + σ 2s W sin 2 φ − Y
)
(5)
where σ1s and σ2s are the principal stresses in the specimen,
W=
Y=
(1 + ν s ) {(Q
A g/ E s
1
A g/ E s
Q11g =
11g
)(
+ αQ12g - Q12g + αQ 22g
)}
{ν s (Q11g + αQ12g ) - (Q12g + αQ 22g )}
E11g
1 − ν 12g ν 21g
, Q 22g =
E 22g
1 − ν 12g ν 21g
and Q12g = ν 12g Q 22g
E11g and E22g are the longitudinal and transverse Young’s modulus of the gauge material,
ν12g and ν21g are the major and minor Poisson’s ratio of the gauge,
Es and νs are the Young’s modulus and Poisson’s ratio of the specimen material,
α = α22/α11, the ratio of the transverse to the longitudinal coefficient of linear thermal expansion of the gauge material,
/
A g is a calibration constant, and
φ is the angle between σ1s and the major principal axis of the gauge material (see Figure 1).
For the case of a circular disc (diameter, d and thickness, t) subjected to a diametral compressive force (P) [12],
σ1s = 2P / πdt and σ 2s = −6P / πdt . Substituting into Equation (5) it follows readily that
Sg =
P
( 4 W cos 2 φ − X)
πdt
where X = 3W -2Y.
(6)
Equation (6) requires that the thermoelastic signal from an orthotropic coupon at the centre of a circular disc in diametral
2
compression should vary linearly with cos φ as the disc is rotated through an angle φ. The attraction of this simple
specimen for the work in hand is evident. Initial experimental work was carried out with prime objective of demonstrating
the overall validity of Equation (6). The specimen used was a circular steel disc 100 mm in diameter and 20 mm thick,
loaded in diametral compression with a small bonded orthotropic coupon at the centre. The coupon itself consisted of a
flattened tow of E-glass fibre, bonded to the surface with epoxy resin so that the E-glass was exposed, i.e. with no resin
rich layer on the emitting surface. By changing the orientation of the disc in the test machine for successive tests, the
variation of the thermoelastic signal from the coupon for different coupon alignments relative to the same stress field was
established. The Deltatherm TSA equipment was used throughout. A typical plot is shown in Figure 5 in which the
2
thermoelastic signal is plotted against cos φ for test frequencies of 10 Hz and 20 Hz. It can be seen whilst there are
irregularities in the trends both plots show a reasonably consistent linear variation of signal with cos2φ. The close similarity
in the data at 10 and 20 Hz is an indication of the absence of non-adiabatic behaviour. Clearly a fuller range of possible
coupon materials needs to be investigated; this work is in hand. However, the demonstration provides an encouraging first
step in the design and development of the thermoelastic coupon gauge.
500
450
400
350
Signal
300
250
200
10 Hz
20 Hz
Linear (10 Hz)
Linear (20 Hz)
150
100
50
0
0.00
0.25
0.50
0.75
1.00
cos2φ
Figure 5 Results from validation specimen
Conclusions
1.
With the proper choice of materials and thickness, reinforcement effects may be reduced to an insignificant level and
possible differences between the Poisson’s ratios of the coupon and specimen will cause no significant additional
reinforcement.
2.
Possible non-adiabatic effects in the coupon gauge are, at present, not a major concern. This may warrant some
further work, but the initial experimental work has indicated that for glass epoxy coupons adiabatic conditions prevail.
3.
The experimental work has provided an initial demonstration of the validity of the underlying theory.
Acknowledgements
The authors are grateful to T. Emery, University of Southampton, for his assistance in the TSA and to G. Hall, University of
Manchester, for his help with the numerical work. They are also indebted to the Leverhulme Trust for the award of a
Leverhulme Emeritus Fellowship to PS, which enabled them to pursue the work.
References
[1]
Dulieu-Barton, J.M. and Stanley, P., “Development and application of thermoelastic stress analysis”, Journal of
Strain Analysis, 1998, 33, 93-104.
[2]
Stanley, P. and Dulieu-Smith, J.M., “Devices for the experimental determination of individual stresses from
thermoelastic data”, Journal of Strain Analysis, 1996, 31, 53-63.
[3]
Barone, S. and Patterson, E.A., “Polymer coating as a strain witness in thermoelastic stress analysis”, Journal
of Strain Analysis, 1998, 33, 223-232.
[4]
Barone, S. and Patterson E.A., “An alternative finite difference method for post-processing thermoelastic data
using compatibility”, Journal of Strain Analysis, 1998, 33, 437-447.
[5]
Murakami, Y. and Yoshimura, M., “Determination of all the stress components from measurements of the
stress invariant by the thermoelastic stress method”, International Journal of Solids and Structures, 1997, 34,
4449-4461.
[6]
Stanley, P. and Dulieu-Barton, J.M., to be published.
[7]
Rees, D.W.A., “Mechanics of Solids and Structures”, 1990, Mc-Graw-Hill Book Company (UK) Ltd.), p 146.
[8]
Jakob, M., “Heat Transfer”, Vol. 1, John Wiley and Sons Inc., New York, p 299.
[9]
McKelvie, J. “Consideration of the surface temperature response to cyclic thermoelastic heat generation”,
SPIE Vol 731 (Stress Analysis by Thermoelastic Techniques), 1987, 44-53.
[10]
Rogers, G.F.C. and Mayhew, Y.R., “Engineering Thermodynamic Work and Heat Transfer”, 4 Edition, 1992,
Longman Scientific and Technical, p 512.
[11]
Dulieu-Barton, J.M., Stanley, P. and Phan, T.S. “An experimental approach to stress separation in
thermoelastic stress analysis”, Presented at Photomechanics 2006, Clermont-Ferrand, France, July 2006.
[12]
Den Hartog, J.P., “Advanced Strength of Materials” 1952, McGraw-Hill Book Company, Inc., New York, p 200.
th