Stress Separation Using Thermoelastic Data
E. A. Patterson, Department of Mechanical Engineering, Michigan State University, East Lansing,
Michigan, USA, [email protected] and R. E. Rowlands, Department of Mechanical Engineering,
University of Wisconsin, Madison, Wisconsin, USA, [email protected]
ABSTRACT
Under adiabatic and reversible conditions, a cyclically loaded structure experiences in-phase temperature variations which are
a function of the changes in the stresses or strains. Thermoelastic stress analysis (TSA, thermoelasticity) uses an infrared
radiometer to measure the local temperature fluctuations and relates these changes to the relevant dynamic stresses by
thermodynamic principles. The recorded temperatures are associated with a combination of the individual stress components.
However, engineering analyses typically necessitate knowing the values of the individual stress components, and so one must
‘separate the stresses’. The authors review various approaches which have been used to evaluate the individual components
of stress from the TSA-measured information, and speculate on anticipated coming progress and directions for future
research.
Introduction
Thermoelasticity (thermoelastic stress analysis, TSA) is an emerging stress analysis technique that can be used effectively for
design, fracture mechanics, damage detection, fatigue and residual stresses. In addition to its applicability for assessing the
reliability of other solutions, TSA is useful for stress analyzing situations for which other methods are difficult or not feasible.
The technique is based on the small temperature changes that occur when a material is subject to a time-varying stress or
strain. Thermoelastic stress analysis usually employs cyclic loading and measures the resultant temperature changes with an
infrared sensor or camera. Recent developments in commercial hardware and software render TSA a non-contacting method
for acquiring (virtually instantaneously) and processing vast amounts of stress information from actual structures in their
operating environments with a sensitivity comparable to that of strain gages.
For an isotropic material under plane stress, the linear relationship between the stresses and the signal recorded by the
system, S*, can be expressed as [1]
S * = K (∆σ p + ∆σ q )
(1)
where ∆σp and ∆σp are the changes in the principal stresses. For orthotropy,
S * = K1 ∆σ 11 + K 2 ∆σ 22
(2)
with ∆σ11 and ∆σ22 of eqn. (2) being the changes in the stresses in the directions of material symmetry. Thermo-mechanical
coefficients, K, K1 and K2 are usually determined experimentally. For plane-stress isotropy [2],
∇4 φ = ∇2 S = 0
(3)
where φ is the Airy stress function and S = (σp + σp ) = (σx + σy) = (σr+ σθ) is the first stress invariant, i.e., the isopachic
stress.
Several TSA methods have been developed to obtain individual components of stress (or strain) in isotropic and orthotropic
composite materials. This paper reviews the subject and indicates some areas of potential further research [3-47]. The cited
references, while not all encompassing, are believed to be representative. Many of the articles discussed here reference other
relevant developments. Although touching on the necessary aspects of poor edge data from TSA and the possible need to
smooth the measured information, this paper emphasizes stress-separation. It does not treat directly TSA as it relates to
topics such as fracture, fatigue or residual stresses.
Many of the approaches for converting measured temperatures into individual components of stress or strain employ
equations of compatibility and/or equilibrium, perhaps in combination with FEM-, BEM- or FD-concepts. Alternate techniques
synergize TSA with photoelastic (PSA) data, treat the situation as an inverse problem, employ bonded TSA gages and/or
utilize the nonlinear thermoelastic effect.
Stress Separation Methods
It is appropriate to acknowledge that Professor Peter Stanley and his colleagues were among the first, and continue to be very
active, contributors to evaluate individual stresses from measured temperatures [3 - 5]. Although reference [3] is limited to
axially-symmetrical stresses, the research inspired others to seek ways of converting measured temperatures into individual
stresses. The idea of bonded TSA gages is developed in refs. [4, 5]. The method involves adhering longitudinal/uniaxial
isotropic strips or rods, or an orthotropic coupon, to the surface of the structure of interest, Fig. 1. Such TSA ‘stress-gages’
are potentially applicable to components made of orthotropic as well as isotropic material. Reference [6] also discussed a
somewhat similar orthotropic TSA coating.
Fig. 1 TSA strip gage [4].
Based on some concepts by Balas [7], ref [9] evaluated the vertical stresses in a partially-loaded half plane, Fig. 2. Individual
stresses were obtained by approximating S as a third degree polynomial along the top traction-free edge of Fig. 2 and
expressing the stresses in terms of finite differences. This formulation is best suited to regions adjacent to linear boundaries
and is less convenient around holes, notches of grooves. Reference [10] uses an alternative finite-difference method based on
compatibility for post-processing thermoelastic data into individual principal stresses. Since compatibility is the only field
equation involved, the approach appears applicable to orthotropy as well as isotropy.
Fig. 2 Thermoelastically-determined vertical stress in a partially-loaded aluminum half plane [9].
It can be convenient to represent a stress function in complex variables [11-15]. While avoiding the use of unreliable TSA data
near an edge, employing analytical continuation provides edge stresses at traction-free boundaries and therefore stress
concentrations. Mapping techniques enable the handling of complicated boundaries. In addition to being applicable for either
isotropic or orthotropic constitutive response, the technique simultaneously smoothes the measured information, separates the
stresses into their three components adjacent to the edge and evaluates the non-zero stresses along a traction-free boundary.
Figure 3 illustrates so-obtained contours of the largest rectangular component of stress in a region adjacent to a circular hole
in an aluminum plate, while Fig. 4 demonstrates the use of the method to evaluate the hoop stress on the edge of a side notch
in an orthotropic tensile member. The TSA result of Fig. 4 agrees well with that by FEM and a strain gage. These complexvariable concepts of refs. [11-14] can be extremely effect, but they typically presuppose being near a traction-free boundary.
Moreover, and unlike a subsequently-discussed approach using Airy stress functions in terms of a real variable, stress
determination around an entire geometric discontinuity typically necessitates processing several (perhaps overlapping)
independent regions. Figure 5 demonstrates an alternate complex-variable concept which employs hybrid elements.
Fig. 3 TSA-determined contours of largest normalized stress throughout a region adjacent to a
circular hole in horizontally-loaded aluminum plate [12].
Fig. 4 TSA-determined (solid line) tangential stress on the edge of a side notch
in a uniaxially loaded orthotropic Gl/E composite [13].
Fig. . 5 Use of a hybrid element to process recorded TSA data into individual stresses
near a hole in an orthotropic Gl/E composite [15]
The largest normal stress adjacent to holes or notches in orthotropic materials can be determined by representing this stress
ahead of the notch or hole as a polynomial in terms of the coordinate normal to the geometric discontinuity. Figure 6
illustrates this where the TSA-determined normalized vertical stress, ∆σy, is plotted with other available results along the
horizontal axis extending from the edge of a circular hole in a vertically loaded orthotropic composite tensile specimen. This
technique is also suitable for isotropic materials.
Fig. 6 TSA-evaluated (SFER) vertical stress along the horizontal axis extending from
edge of a hole in vertically-loaded composite strip [16]
Although complex-variable analyses were utilized to evaluate the individual stress components in Figs 3 through 5, realvariable forms of the Airy stress function, φ, of eqn (3) can be convenient and effective for separating the stresses from
measured S* in isotropic materials [17-21]. A reasonably general form of φ is available in polar coordinates, r and θ,
consisting of infinite series. However, many of the terms can often be eliminated by various conditions and/or arguments (e.g.,
symmetry about an axis; single-valued stresses, strains, displacements; self-equilibrated boundaries; whether or not the
coordinate origin is within the body; boundedness at origin or infinity) [2]. The coefficients of the remaining terms can
frequently be evaluated from measured values of S* and eqns (1) and (3), thereby providing known expressions for the
individual components of stress. If the problem motivates using rectangular coordinates, quantities in polar coordinates can
be transformed into rectangular coordinates. Figure 7 demonstrates this general approach by separating the stresses in a
loaded disk, while Fig. 8 uses an enhanced version of a somewhat similar concept to evaluate the tangential stress on the
edge of a hole in a compressively-loaded half-plane [19]. While restricted to isoptropic behavior, this general technique, like
the use of complex variables as in Figs. 3 through 5, avoids utilizing unreliable TSA data near an edge. Moreover, the method
simultaneously smoothes the measured information, separates the stresses and evaluates the non-zero stresses on a
boundary. Computationally, working with real variables can be easier than with complex variables. Reference [21] also
formulated a scheme based on evaluating the coefficients of a stress function, again looking at a partially-loaded half-plane
somewhat like that of Fig. 2. The authors minimized the differences between generated and simulated/measured isopachic
stresses, but do not report any experimental results.
2
σ /σo
0
-2
-4
-6
0
0.2
σ xx
σ xy
Fig.
7
0.4
y/R
Theor. σ xx
Theor. σ xy
σ yy
S
0.6
Theor. σ yy
Theor. S
TSA-determined rectangular components of stress compared with
along a horizontal line in a diametrally-compressed aluminum disk [17].
FEM
and
theory
20
15
ANSYS
σθ θ /σ0
10
TSA(ANS_Cen)
5
0
-5 0
45
90
135
180
-10
-15
Degree
Fig. 8
Hoop stress at the boundary of a hole in a compressively-loaded aluminum half-plane [19].
Although many of the techniques presuppose being associated with a traction-free boundary, ref [22] illustrates a BEM TSA
method applicable to loaded or unloaded boundaries. Table 1 compares results by this approach with those predicted by
ANSYS. The normalized stresses of Table 1 represent stress concentration factors at the root of side notches in a tensile strip
of aluminum. BEM-formulations can offer advantages, but boundary-element theory is more involved mathematically (than
say Airy stress functions in real coordinates) and less intuitive than some of the other approaches.
References [23] through [25] employ BEM to evaluate boundary stresses from interior TSA data, from which the individual
interior stresses are then evaluated using a forward method. In reality, many approaches described in this manuscript to
determine edge stresses (e.g., [26, 27]) can provide the boundary data for forward processing as a boundary value problem.
References [23-25] discuss the careful numerical considerations needed, the importance of filtering the initial measured TSA
data [28], and a variety of means of conducting the forward process to determine the individual stresses away from the edges.
Table 1 TSA-determined stress concentrations, σmax (centerline)/σ0, at side notches in an aluminum tensile plate [22].
Pmin
(lbs)
Pmax
(lbs)
∆P
(lbs)
1
20
280
2
20
3
4
Case
2
σmax (Center line) /σ0 (Notches)
Left
Right
Average
% Error
260
3.72
3.72
3.72
3.1
320
300
3.73
3.70
3.72
3.1
20
380
360
3.69
3.68
3.68
1.9
20
400
380
3.68
3.67
3.68
1.9
2
ANSYS, σ max (Center line) = 3.61.
Based on FEM-type concepts, ref [29] formulated an effective method for separating TSA data into individual stresses, Fig. 9.
The approach is not restricted to traction-free boundaries. Moreover, since the concept is based on equilibrium, it has merit
for non-circular local geometry, orthotropy and stress-separation in 3-D [30, 31]. It has been noted that measured TSA
information tends to be noisy and several of the separation schemes smooth the recorded input data as part of the process.
Since the stress-separation method utilized to evaluate the results of Fig. 9 does not, of itself, filter the temperature data, a
separate smoothing method [28] was employed.
Fig 9 Normalized stresses along a vertical line in a horizontal aluminum beam subjected to threepoint bending [29].
The previously-discussed stress separations have involved only measured TSA data. While TSA provides information on the
sum of the principal stresses, photoelastic isochromatics (i.e., PSA) give the difference in principal stresses. As reviewed in
ref [32], combining PSA with TSA enables one to evaluate the two (under plane stress) individual principal stresses as shown
in Fig. 10 [33-37]. The additional availability of isoclinic data would enable one to determine the three in-plane individual
components of stress. Such approaches enjoy the advantages of not necessarily being associated with a traction-free edge,
and the feasibility of recording both PSA and TSA data from a common birefringent coating since the latter are opaque in the
infrared spectrum [38]. Reference [36] describes an instrument that allows the simultaneous collection of the PSA and TSA
data from a common optical path using a beam splitter to divide the visible and infrared light, and ref [39] applied this approach
to industrial components with some success. Representing stresses associated with each of the measured TSA and PSA
data as derivatives of a stress function could potentially improve the quality of the TSA/PSA-evaluated principal stresses of
Fig. 10 as the hole is approached, as well as provide the three in-plane stresses without involving the isoclinics [17-19, 40].
normalised principal stress, σ/σ0
3.0
σ1 experimental
σ1 theoretical
σ2 experimental
σ2 theoretical
‹
2.5
⎯
‘
2.0
---
1.5
1.0
0.5
0.0
1.0
1.5
2.0
2.5
3.0
3.5
4.0
-0.5
normalised distance from hole centre, x/a
Fig 10 TSA-PSA determined principal stresses near a hole in a vertically-loaded tensile plate [35].
Motivated by a desire to evaluate stress concentrations, and the relevance of some analytical features, most developed
approaches for separating measured temperature data into individual stresses are associated with boundaries. Reference
[41], again using a stress function in real variables but recognizing the nonlinear relationship (second order effects) between
S* and the stress, formulated a stress-separation concept which does not involve boundary conditions. Although
mathematical considerations are addressed and the ideas are extended to structural optimization, no measured TSA results
are shown. Reference [42] separates stresses from measured temperatures also using the nonlinear stress-temperature
effect. This latter paper pays considerable and appropriate attention to the need to first filter the measured data.
Thermoelastic stress analyses usually assume adiabatic, reversible thermodynamics, and therefore employ cyclic mechanical
loading. Reference [43] utilized the frequency-dependent depth effect to formulate a non-adiabatic TSA theory for evaluating
individual strains in laminated composites. No experimental results are included. Most TSA stress-separation schemes
developed to date emphasize linear-elastic plane-stress. There are few 3-D formulations and fewer experimental results [4447].
Summary, Discussion and Future Considerations
Numerous techniques are available for evaluating the individual components of stress from thermoelastically-recorded
information. Many of them assume isotropic behavior and a linear relationship between temperature and stress, presuppose a
neighboring traction-free boundary in a plane-stressed component, and that the loading produces an adiabatic and reversible
thermodynamic response. However, there are methods which include non-adiabatic, three-dimensional, nonlinear and/or
orthotropic responses, or do not depend on boundary conditions. Several of the techniques simultaneously filter the
measured input data, separate the stress and provide boundary stresses (e.g., stress concentrations). Many approaches
combine experimental, analytical and computational features. In part because recorded temperature information is often
noisy, some methods are somewhat computationally intensive which necessitates being attentive to numerical stability,
robustness, and reliability. While not necessarily independent of future TSA research in general, (additional) stressseparation attention could well pursue areas such as (i) materials whose constitutive response is nonlinear and/or inelastic, (ii)
laminated composites, (iii) loaded boundaries, (iv) three-dimensional problems, (v) further advantageously synergize
temperature data with analytical and numerical tools, and other measured information, and (vi) applications to engineeringtype situations.
Acknowledgements
Drs. J. M. Dulieu-Barton, R. J. Greene, and S. T. Lin kindly supplied figures.
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