# Towards strain gauge 2.0: Substituting the electric resistance routinely deposited on polyimide film by the optimal pattern for full-field strain measurement

Adrien Vinel, Michel Grédiac, Xavier Balandraud, Benoît Blaysat, Thomas Jailin, Frédéric Sur

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Adrien Vinel, Michel Grédiac, Xavier Balandraud, Benoît Blaysat, Thomas Jailin, et al.. Towards strain gauge 2.0: Substituting the electric resistance routinely deposited on polyimide film by the optimal pattern for full-field strain measurement. Strain, 2025, 61 (1), pp.e12488. 10.1111/str.12488. hal-04456869v2

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Submitted on 16 Sep 2024

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<figure><img src="image_1.png" /></figure>
Towards strain gauge 2.0:
Substituting the electric resistance routinely
deposited on polyimide film by the optimal pattern
for full-field strain measurement

A. Vinel¹, M. Grédiac¹†, X. Balandraud¹, B. Blaysat¹,², T. Jailin¹ and F. Sur³

¹Clermont Auvergne Université, CNRS, Clermont Auvergne INP, Institut Pascal, F-63000 Clermont-Ferrand,
France

²IUF - Institut Universitaire de France, 1 rue Descartes, 75231, Paris Cedex 5 France

³LORIA, UMR 7503, Université de Lorraine, CNRS, INRIA, Nancy, France

† corresponding author, michel.grediac@uca.fr

**Abstract:** The checkerboard constitutes the best pattern for full-field strain measurement because it maximizes image gradient. In the experimental mechanics community, employing this pattern is currently strongly limited because depositing it on the surface of specimens raises practical difficulties. A recent study shows, however, that it is technically possible by using a laser engraver. The present paper aims to push this solution forward by printing the same pattern on a thin polymeric film and then gluing the resulting laser-engraved film on the specimen surface. The underlying idea is to separate the manufacturing process of this optical strain gauge on the one hand and its use on the other hand to help spread this strain measuring tool in the experimental mechanics community. The polymeric film employed here is the same as that used in the manufacturing process of classic electric gauges, so one can rely on the classic strain gauge bonding know-how to glue this optical strain gauge on the specimen surface. The main difference between the proposed tool and classic electric gauges is that the strain field beneath the polymeric support is measured instead of localized strain values. The paper is a proof of concept for this strain field measuring tool. The manufacturing and bonding processes are described in the paper. The Localized Spectrum Analysis, a spectral technique developed for processing images of periodic patterns, is used to retrieve the strain fields from checkerboard images. Through two complementary examples, we show the ability of this new type of strain gauge to detect and quantify local details in the strain field beneath. A simplified 1D model is also proposed to assess the minimum width of the strain peak that can reliably be measured with this technique.

**Keywords:** checkerboard, crack, digital image correlation, full-field measurement, laser marking, localized spectrum analysis, optimal pattern, shape memory alloy, strain gauge 2.0, wood

**Introduction**

Full-field measurement techniques are now widespread in the experimental mechanics community. Undoubtedly, the most popular one is Digital Image Correlation (DIC) [1]. This technique relies on surfaces marked with random patterns. With the classic version of DIC, the displacement is calculated pointwise at the centre of the subsets used to mesh the zone of interest. DIC retrieves this displacement by iteratively minimizing the so-called optical residual. In its simplest form, this
quantity is the squared difference between the grey-level distributions of the images of the current and the deformed configurations. This minimization is performed subset by subset.

Defining the best pattern for such measurements is a key issue because the pattern carries the sought information. Optimizing this pattern in terms of metrological performance is the aim of several papers, see [2, 3, 4, 5, 6, 7, 8, 9, 10, 11] for instance. In the case of in-plane displacements and as discussed in [3, 4], the best option is to use a periodic pattern, such as a checkerboard, which is not random. However, classic versions of DIC generally fail to find the displacement field with periodic patterns because of the presence of numerous local minima in the cost function to be minimized. A solution can be to use a particular version of DIC called virtual DIC [11], to seed the solution with a manually defined initial guess, or to switch the minimization of the optical residual from the spatial to the frequency domain [12, 13]. This latter option eventually leads to a quasi-direct link between the sought displacement and the phase change of the periodic pattern between current and reference configurations. The metrological performance is the same for these three approaches, but the second one is much faster [13], so it will be adopted in this study. The so-called Localized Spectrum Analysis (LSA) is used to extract phase change from the checkerboard pattern images discussed in this study, and then to deduce the corresponding displacement and strain fields. As discussed in [14], this is, with the Geometric Phase Analysis [15] and its windowed version [16], one of the spectral techniques which can be used to perform this task. It leads to a good tradeoff between calculation time and various parameters characterizing the metrological performance.

With such optimized patterns, a major hurdle is to deposit the tiny dots forming the checkerboard on the surface of the specimen to be tested, the challenge being to keep the size and the spatial distribution of those dots as regular as possible. It is, for instance, possible to print the pattern beforehand onto a polymeric sheet and then to transfer the black marking on the surface of interest by using a suitable white glue, as described in [17] for 2D grids, for instance. However, a printing device featuring a resolution greater than 10,000 dots per inch must be employed, which is much higher than that of classic office printers. In addition, complete adhesion is not always observed, and tiny bubbles in the glue may corrupt the regularity of the pattern. It is also challenging to keep a uniform and negligible thickness of the glue over the surface, which may cause the measurement technique to be intrusive in the case of high-strain gradients. Another route recently presented in [18] is to directly engrave the pattern on flat surfaces with a laser engraver, the surface being spray-painted in white beforehand. Indeed, the laser beam hits the coat of paint, and the colour of the paint locally turns from white to black, eventually ensuring a good contrast between the black “printed” dots and the white layer of paint. Checkerboards with squares as small as 50 [µm] were engraved in the examples discussed in Ref. [18]. Further developments then enabled us to reduce this size to 30 [µm] [19]. However, a disadvantage of this solution is that quite an expensive laser engraver was employed (55 keuros minimum when it was purchased). Cheaper engravers are now commercially available, but they should be tested to check that their resolution and the regularity of the pattern period they print remain acceptable.

The main contribution of this paper is to show that checkerboard patterns can be engraved on thin polymeric films, which can then be cut and bonded on the surface of any flat specimen. This piece of film is called the “optical strain gauge” in the following. The underlying idea is to separate the manufacturing process of this optical strain gauge on the one hand, and its use on the other hand, to help distribute this strain measuring tool in the experimental mechanics community, in particular to perform and develop tests within the framework of Material Testing 2.0 [20, 21]. Indeed, considering preprinted films produced at the industrial level would help improve the repeatability of the measurements by reducing the fluctuations caused by a more user-dependent solution like painting and printing a pattern. In this study, a laser engraver was used to print the tiny dots forming the checkerboard on the film. The polymer of the film used here is the same as that classically used as backing material for electric strain gauges, namely polyimide (or Kapton® [22]). The advantage of using the same type of film is that one can rely on a bonding process similar to that used for bonding classic electrical gauges, the latter being widely proven for strain measurement. This opens up the prospect of a “strain gauge 2.0” that provides the whole in-plane strain maps under it, as opposed
to “strain gauge 1.0”, namely the classic electrical strain gauge, which returns localized values of the strain.

The paper is organized as follows. The materials and methods used to print and bond the optical strain gauge are presented in Section 1. This also includes a brief description of how the images of such patterns are processed to extract displacement and strain fields. The results obtained with the optical strain gauge bonded on two different specimens are presented in Sections 2 and 3. They are made of a monocystal of shape memory alloy and wood. In both cases, the main challenge is detecting and reliably measuring sharp details in the strain distributions under the optical strain gauge. The experimental results are discussed, in particular, through the prism of an analytical shear-lag model, which predicts the “damping” of localized details caused by the presence of a film and a layer of adhesive. This discussion is in Section 4.

# 1. Materials and methods

## 1.1. *Selecting the constitutive material for the optical strain gauge*

The first step of this study was to choose a suitable flexible support that meets different requirements. Indeed, the film must be:

1. Sensitive to the hit of the laser beam so that tiny dots forming the highest possible contrasted checkerboard pattern are visible after engraving.

2. Non-intrusive or as least intrusive as possible so that it reliably reproduces on its free surface the state of strain which takes place under it.

3. Easily bondable onto the specimen surface, the latter being assumed to be flat, at least for this first attempt.

Various stretchable and adhesive bandages, polymer stickers, and tattoo papers were first tested unsuccessfully in a preliminary study (results are not reported here). Checkerboard patterns could nearly systematically be successfully engraved after spray-painting these types of film in white. However, the problem was how to correctly tackle requirement 2 above in the case of high strain gradients because strain peaks were generally significantly “damped” by the presence of the film. For instance, this phenomenon was observed with the narrow martensite needles that appear in the shape memory alloy specimen discussed in Section 2. This hurdle was overcome by choosing the film routinely used to produce classic electrical strain gauges. Indeed, electrical strain gauges are generally obtained by etching constantan film and sealing it on a support made of polyimide. Therefore, the idea was to employ the same backing material as that used for electric strain gauges to benefit from the long experience gained when bonding them on flat surfaces. A 25 [µm]-thick polyimide film was used for this purpose. This film, supplied by ADDEV Materials, Saint-Chamond, France (https://addevmaterials.com/en/), is manufactured by DuPont [22]. It is worth mentioning that the raw colour of polyimide is yellow/brown. To avoid needing a first layer of white paint sprayed on the film, this latter was chosen directly white in its bulk. In addition, choosing a white film limits possible spatial fluctuations of the white colour across the surface of the specimen. This solution was, therefore, adopted in this study.

## 1.2. *Procedure for engraving checkerboard patterns on a white polyimide film*

The procedure used to engrave a checkerboard pattern on a polyimide film is similar to that described in [18] to engrave this type of pattern directly on the surface of specimens spray-painted in white beforehand. It relies on the use of a laser engraver. Some settings had, however, to be adjusted to avoid the film being pierced by the laser beam. It was sometimes impossible, probably because of the non-uniform contact between the film and its support while engraving. The film was pierced in this case, but the holes were as small as possible. Indeed, the glue may flow through these holes, which impairs the contrast of the printed side of the film. This may also cause some issues when removing the components used during the bonding process since they can remain bonded on the pattern; see the following section for more details on the bonding procedure. A tradeoff had, therefore, to be found
between laser power, the number of repetitions (in other words, the number of hits at the same place to progressively draw the dots), and the quality of the resulting contrast between black dots and white background. The settings used in this study are given in Appendix 1, along with those used to engrave a white paint layer directly deposited on the specimen surface. This second type of marking will be used in the examples below for comparison purposes.

### 1.3. Procedure for bonding the laser-engraved polyimide trip

A procedure similar to that used for bonding classic electric strain gauges was followed to glue pieces of laser-engraved film forming the optical strain gauge on the specimen surface. The glue employed here was a cyanoacrylate available for classic electrical gauge bonding; see the reference in Table 1. It is classically used to bond electrical strain gauges. However, with the present application, there are two main differences with classic electrical strain gauge bonding:

* The first difference is that the size of the optical strain gauge to be bonded is much greater than the average size of an electrical strain gauge. For instance, the area covered by the optical strain gauge reaches several $cm^2$ in the two examples discussed below (about 10 and 30 $cm^2$, respectively). Hence, obtaining a regular layer of glue between the optical strain gauge and the substrate is somewhat tricky. This justified the setting up of a procedure dedicated to bonding optical strain gauges larger than classic electrical gauges.

* The second difference is that tiny details in the strain distribution under the piece of the laser-engraved film must be reliably revealed. In contrast, a classic gauge only returns a mean strain value under the electric resistance. Consequently, the glue layer must be as thin as possible to limit the attenuation of the sharp details in the strain maps, and an important question is also to know to what extent the polyimide film induces a damping of these details.

After different attempts, the best way to bond an optical strain gauge on a flat specimen was first to place it on an adhesive tape with the printed face *against* the adhesive layer. This adhesive layer, made of flexible PVC to facilitate its deposition onto the engraved polyimide film and limit tiny air bubbles to form between the two, serves two roles:

1. Engraving a pattern on a thin polyimide film (thickness: 25 [$\mu$m]) causes the film to become brittle, so it easily tears. Depositing it on a flexible tape makes it easier to handle.

2. Glue flows out the adhesive joint while pressing the film onto the substrate, so the tape protects the printed surface against this excess glue.

The following procedure is used to glue the piece of film on the specimen after depositing it onto the adhesive tape. We assume these pieces of film and adhesive tape are rectangular. One of the four sides of the tape protrudes. It is stuck onto the specimen surface; see the left-hand side of Figure 1-(a) and - (b). The three remaining borders of this laminated film (tape + polyimide film) are kept free by cutting the excess tape. In the following step, they will allow the glue to flow outside the space between the surface of the specimen and the film. A thin layer of glue is then deposited on the specimen surface, and the opposite side of the printed face of the optical strain gauge is placed on it. The stack formed by the specimen, the glue, the optical strain gauge, and the tape (see schematic view in Figure 1-(a)) is pressed with a steel block featuring one side slightly convex and the other flat. The convex surface is placed on the stack, protected by a thin plate of foam rubber (thickness: 8 [mm]) coated with a release (anti-adhesive) layer deposited with a pressure canister. This foam rubber behaves like a sponge soaked with anti-adhesive. Gently pressing and oscillating the block by hand for about one minute enables the excess glue to flow outside the bonded joint; see the three arrows in Figure 1-(b). The block is then turned over, with its flat surface now pressing onto the stack, and the adhesive is allowed to cure at room temperature for about one hour. This duration is much longer than the few minutes specified in the datasheet of the glue manufacturer, but it is justified by the much larger area of polyimide film to be bounded compared to that of classic electrical strain gauges. Pressure is exerted by a clamp or a massive object while curing to reach the lowest and most uniform thickness possible for the adhesive
layer after polymerization, see the schematic view in Figure 1-(a), where all the ingredients involved in
the bonding process are reported. The adhesive tape is finally removed after curing and unclamping,
and only the optical strain gauge remains bonded on the specimen surface.

Measuring the thickness of the specimen before and after this procedure and subtracting the thickness of the optical strain gauge enables us to deduce the thickness of the glue in the bonded joint. The average thickness of the polymerized adhesive layer is equal to about 50 and 80 [µm] for the two specimens presented below, respectively. This thickness influences the ability of the present optical strain gauge to return fine details in strain maps reliably; see Section 4.

<figure><img src="image_2.png" /><figcaption>Figure 1: Schematic view illustrating the bonding process of an optical strain gauge. The engraved surface is against the adhesive side of the tape. The other side is bonded onto the specimen surface.</figcaption></figure>

Table 1 gathers the reference numbers and names of the suppliers and/or manufacturers of the ingredients used in the bonding process.

1.4. *A primer on the Localized Spectrum Analysis*

We now briefly explain how checkerboard images are processed to extract the strain maps shown
and discussed in the examples below. The so-called Localized Spectrum Analysis (LSA) was used here
for this purpose. This is one of the spectral techniques employed to process periodic patterns [14, 23].
It is shown in [14] that it leads to the best tradeoff between different constraints. While DIC relies
on the minimization, in the spatial domain, of the optical residual calculated on subsets of images of
random patterns, LSA can be regarded as the minimization of the same quantity on images of periodic
<table><caption>Table 1: Materials used in the preparation of the gauge and the bonding process.</caption><thead><tr><th>Name</th><th>Reference</th><th>Manufacturer (man.)/Supplier(sup.)</th><th>Remark</th></tr></thead><tbody><tr><td>laser engraver</td><td>MD-U1000C</td><td>man.: Keyence, Japan</td><td>-</td></tr><tr><td>film</td><td>Kapton® 100WS</td><td>man.: Du Pont, USA<br>sup.: ADDEV Materials, France</td><td>developmental film,<br>colour: white<br>thickness: 25 [μm]</td></tr><tr><td>adhesive</td><td>M-bond 200</td><td>man.: Micro-Measurements, USA</td><td>cyanoacrylate glue</td></tr><tr><td>release spray</td><td>5090</td><td>man.: Jelt, France</td><td>called “spatter release<br>spray” by<br>the manufacturer</td></tr><tr><td>foam rubber<br>in cellular<br>Ethylene Propylene<br>Diene Monomer (EPDM)</td><td>304/B</td><td>man.: EUROFOAM,<br>Italy</td><td>thickness: 8 [mm]</td></tr><tr><td>adhesive tape</td><td>NBAD0031CT</td><td>man.: Corderies Tournonaises,<br>France</td><td>support made<br>in polyvinyl chloride<br>(PVC)</td></tr></tbody></table>

patterns, but in the frequency domain [12]. The benefit is twofold: *i*- images of optimized patterns (in terms of metrological performance of the displacement/strain maps) such as checkerboards can be processed, and *ii*- the minimization is quasi-direct, thus avoiding the heavy calculations performed with the iterative minimization of the optical residual performed in the spatial domain with DIC. A consequence is that LSA returns displacement and strain maps estimated pixelwise, and the procedure is less computationally demanding than DIC [14]. In its present version, LSA only returns in-plane measurements. Out-of-plane movement can potentially induce some errors in the estimation of the in-plane displacements and strains. However, a recent study shows that this out-of-plane movement can also be estimated and accounted for in the determination of the in-plane components, [24, 25]. This option was not employed here to focus only on comparing two types of patterns. The impact of the out-of-plane movement was eliminated by using a telecentric lens in the first example shown in Section 2, and the effect of this out-of-plane movement was assumed to be negligible in the second example discussed in Section 3. As a general remark, it is also worth mentioning that compared to DIC, much fewer studies are dedicated to the development of LSA. Thus, some questions remain open, such as the reliability of the estimated strains in case of large deformation or the ability of this technique to obtain measurements at different scales. All details concerning this measuring technique can be found in previous studies [26, 27], so we only briefly describe below the main steps of LSA.

The first step is to apply to the images of the pattern a Windowed Fourier Transform (WFT) with a unique frequency equal to the mean value of the period *P* of the checkerboard along the diagonals of the unit squares forming the pattern, see Figure 2-(a). The phase distributions for both the reference and the deformed images are then calculated by taking the argument of the result of the WFT (the WFT of an image gives a distribution of complex numbers defined at each pixel). The displacement field is finally deduced by using the following expression:

$$ \underline{u}(\underline{x}) = -\frac{\mathcal{P}}{2\pi} \left( \underline{\Phi}^{\text{cur}}(\underline{x} + \underline{u}(\underline{x})) - \underline{\Phi}^{\text{ref}}(\underline{x}) \right) \quad (1) $$

where *u* is the sought displacement field, $\underline{\Phi}^{\text{cur}}$ and $\underline{\Phi}^{\text{ref}}$ both contain two phase distributions, namely one along each direction of the reference and current images, respectively. *u* is involved in both parts of Equation 1. It is retrieved using the fixed-point algorithm, which generally rapidly converges [26]. Directly subtracting the phase maps without performing this compensation of the displacement between current and reference images causes the parasitic fluctuation of the frequency of the checkerboard to
impact the quality of the strain maps negatively. The gradient function available in the NumPy library of Python was used to differentiate the displacements and to deduce the strain components. This function relies on the central difference approximation.

The window used in the WFT is a 2D Gaussian function. Such a window gives the best tradeoff between various constraints [28]. The following equation gives the function defining this Gaussian window:

$$w(\underline{x}) = \frac{1}{2\pi\ell_{\text{LSA}}^2} e^{-\frac{\|\underline{x}\|^2}{2\ell_{\text{LSA}}^2}} \quad (2)$$

where $\ell_{\text{LSA}}$ is the standard deviation and $||\cdot||$ denotes the norm of vector $\cdot$. This is a handy parameter that the user can adjust to decrease the noise level in the strain maps since the higher the value of $\ell_{\text{LSA}}$, the lower the noise level in the maps. The price to pay is that the spatial resolution is impaired in proportion. Another remark is that at first approximation and regardless of the noise, the phases retrieved at each pixel are not exactly the true phases, but the true phases *convolved* by the window defined in Equation 2 [29]. This induces a blur of the details in the maps, especially in the strain maps, in which those details are generally sharper than in the displacement maps. It is worth noting that, up to a certain cutoff frequency, these maps can be restored with a dedicated deconvolution algorithm described in [30, 31], but this option was not used in this study. The minimum value of the standard deviation of the Gaussian window is equal to $\ell_{\text{LSA}} = \mathcal{P}$ after [27]. This gives the lowest possible (thus the best) spatial resolution, but the effect of sensor noise on the final map is the highest.

### 1.5. Description of two complementary validation tests

Two different tests were performed to validate the proposed approach. Since this optical strain gauge aims to measure the heterogeneous strain field underneath, we chose challenging cases for which strong heterogeneities occurred. The first test was performed on a phase-transforming material, namely a shape memory alloy, subjected to a thermal load. The second one was a tensile test performed on a specimen of wood in which a cracked knot was embedded.

Two patterns were deposited on the specimens, that is, with the optical strain gauge on the one hand and a classic checkerboard pattern directly engraved on its spray-painted surface on the other hand. The strain maps obtained in the second case served as ground truth since this is the technique that has been used so far. Comparing the strain distributions obtained in each case enables us to assess to what extent the optical gauge can return strain maps, which are similar to those returned by the checkerboard directly engraved on the paint layer.

## 2. Test #1: specimen in shape memory alloy

### 2.1. Material and method

Test #1 consisted of measuring strain fields on the surface of a disk specimen made of a single-crystal of CuAl$_{13.9}$Ni$_{4.6}$ (wt.%) shape memory alloy (SMA) subjected to temperature-induced phase transformation. The specimen had a thickness of 3 mm and a diameter of 33 mm. The alloy's chemical composition and the heat treatment were adjusted so that the specimen could be either austenitic, martensitic or a mixture of austenite (A) and martensite (M) at room temperature, depending on the thermal history. In the present test, starting from a purely austenitic specimen, about 60% martensite was obtained by cooling it to nearly 0°C. The microstructure that was observed was preserved after returning to room temperature. Full details on this type of experiment and a discussion on the observed phase transformation phenomena can be found in a recent paper [19]. The present study aims to analyse the strain maps returned by the measurement system in a configuration leading to highly localized gradients. Indeed, considering that the austenite is the reference state (zero strain), martensite appearance in the form of needles or bands caused local strain values up to about ±4%. The values of the strain components depend on the band orientation, as the transformation strain tensors
of the different bands are rotationally related. This leads to sharp interfaces with strain "jumps" in theory. At the present experiment's observation scale, the strain values within a given martensite band are expected to be constant or to gently spatially evolve (due to elastic strains potentially involved in addition to the phase transformation strain).

For the test, half the specimen was instrumented using the conventional method, that is, first spray-painting in white and then engraving the checkerboard pattern (CKB #1). The other half was instrumented with the proposed optical gauge (CKB #2). The repeatability of the A↔M transformation in this specimen is not perfect, but sufficient to rely on previous experiments discussed in [19] to know the orientation of some martensite bands. This enabled us to place the border between the two marking types perpendicularly to a set of parallel martensite bands. This enabled us to examine if the strain values measured within them remain continuous when changing the type of marking, i.e., when crossing the border between the two marking zones. It should be recalled that there are, theoretically, strain jumps of several percent at the transition between austenite and martensite. Hence, this situation is quite challenging for the polyimide film bonded over such interfaces. It will, therefore, be interesting to observe whether the strain distributions are similar within the martensite bands when crossing the border between CKB #1 and CKB #2. Such a property is reflected by bands or needles featuring a similar strain amplitude with both types of marking. Another point will be to examine how the borders of these needles or bands, which should correspond to a strain discontinuity, are returned by the measurement system.

Two sets of images were taken in the reference configuration (undeformed austenite) at room temperature, one for CKB #1 and the other for CKB #2. Only the exposure time changed from one case to another to optimize the contrast in the images; see Table 2 providing the testing conditions. The same was done after cooling down to 0°C and returning to room temperature, that is, when the specimen was partially transformed to martensite. This corresponds to the deformed configuration. The strain fields obtained on both halves were then merged to form a unique map covering the whole specimen.

Finally, note that the small squares forming the checkerboard pattern were slightly inclined with respect to the borders of the pixel grid of the camera to avoid aliasing. This undesired phenomenon induces parasitic fringes in strain maps; see [32] in the case of 2D grids. The tilt angle $\theta$ is defined in Figure 2. Its value is reported in Table 2.

## 2.2. Comparison between the checkerboard patterns obtained with the two procedures. Impact of contrast on sensor noise propagation

Figures 2-(a) and (b) show typical close-up views of the two checkerboard patterns, obtained with CKB #1 and CKB #2, in the reference undeformed state, and with optimal lighting conditions for each pattern. The most striking point is that both patterns have the desired aspect of a checkerboard.

The latter is, however, slightly brighter than the former when optimal lighting is used for each pattern. The grey level histograms associated with each patterns are shown in Figure 2-(c) and (d). For CKB #1, the histogram has a specific profile with three "bumps", as already established and discussed in [27]. The aspect for CKB #2 is different, with a unique peak shifted towards the right because the pattern is slightly brighter than for CKB #1. The width of the histogram is lower for CKB #2 than for CKB #1 because the contrast is lower. Since sensor noise propagation is all the more pronounced as the contrast is low [29, 33], sensor noise propagation is expected to be slightly more favored with CKB #2 than with CKB #1. The brighter aspect of CKB #2 also causes the noise expected in the strain maps to be higher because the higher the brightness at a given pixel, the higher the sensor noise [34, 35, 33]. This is confirmed in Figure 3 where the standard deviations of the noise in the strain component maps are reported (excluding outliers corresponding to localized marking defects). For a given strain component, the spatial distribution of the standard deviation of the noise was first deduced pixelwise from a set of 50 maps for each checkerboard pattern in the reference configuration. Then, an equivalent and global standard deviation was deduced for each pattern. Note that these maps were obtained from independent pairs of images, thus from 100 images.
<table><caption>Table 2: Experimental settings for Test #1 (Section 2) and Test #2 (Section 3)</caption><thead><tr><th></th><th>Test #1 (SMA)</th><th>Test #2 (wood)</th></tr></thead><tbody><tr><td>Camera</td><td>Prosilica GT 6600</td><td></td></tr><tr><td>Type of sensor</td><td>CCD</td><td></td></tr><tr><td>Image resolution</td><td>6576×4384 pixels<sup>2</sup></td><td></td></tr><tr><td>grey depth</td><td>8 bit</td><td></td></tr><tr><td>Acquisition rate</td><td>1 fps</td><td></td></tr><tr><td>Patterning technique</td><td>Laser marking</td><td></td></tr><tr><td>Type of pattern</td><td>Periodic (checkerboard)</td><td></td></tr><tr><td>Type of load</td><td>Temperature change</td><td>Tensile force</td></tr><tr><td>Exposure time of the paint layer (CKB #1)</td><td>5.0 ms</td><td>3.9 ms</td></tr><tr><td>Exposure time of the optical strain gauge (CKB #2)</td><td>7.5 ms</td><td>5.5 ms</td></tr><tr><td>Lens</td><td>TC16M048 Opto Engineering</td><td>Nikon ED AF Micro Nikkor</td></tr><tr><td></td><td>Telecentric lens</td><td>200 mm 1:4 D</td></tr><tr><td>Magnification</td><td>0.75</td><td></td></tr><tr><td>Aperture</td><td></td><td>f/8</td></tr><tr><td>Field of view</td><td>48.4×32.2 mm<sup>2</sup></td><td>69.7×46.5 mm<sup>2</sup></td></tr><tr><td>Image scale</td><td>1 pixel = 7.36 µm</td><td>1 pixel = 10.6 µm</td></tr><tr><td>Tilt angle θ (paint layer, CKB #1)</td><td>7.2°</td><td>1.5°</td></tr><tr><td>Tilt angle θ (optical strain gauge, CKB #2)</td><td>6.0°</td><td>-0.3°</td></tr><tr><td>Stand-off distance</td><td>50 cm</td><td>70 cm</td></tr><tr><td>Period P used in LSA</td><td>5.8 pixels</td><td>4.0 pixels</td></tr><tr><td>Standard deviation l<sub>LSA</sub> of the Gaussian window used in LSA</td><td>l<sub>LSA</sub> = P = 5.8 pixels</td><td>l<sub>LSA</sub> = √2 × P = 5.7 pixels</td></tr></tbody></table>

It can be checked in Figure 3 that the noise affecting the $ε_{xy}$ component is, as expected, about $\sqrt{2}$ lower than those measured for the $ε_{xx}$ and $ε_{yy}$ components.

On average, the noise level for CKB #2 is about 10% higher than for CKB #1. It can also be noted that the noise levels obtained here are significantly higher than those measured in literature [27, 36], but the sampling density, the camera, and the lighting systems were different in these two papers. In addition, the support of the standard deviation of the Gaussian window employed in the present test is smaller than in these papers to diminish (thus improve) the spatial resolution, which automatically increases the noise level [12].

Finally, the presence of some saturated pixels is revealed by the peak on the right-hand side of these histograms, especially for CKB #2. These pixels are located near the border of the checkerboard patterns. As a general remark, LSA is robust to the presence of isolated saturated pixels.

## 2.3. Strain maps

Figure 4-(a) shows the specimen with the two checkerboard patterns. The bottom part corresponds to CKB #1 and the top part to CKB #2. An unintended marking defect is visible in this figure; see Defect A at the bottom of CKB #1. This is like a bubble that causes no marking to appear and, consequently, no strain to be measured over this zone, as can be checked on the strain maps depicted in the following figures. This shows one of the limitations of using paint and engraving it. Interestingly, another type of defect appears on CKB #2; see Defect B in Figure 4-(a). It is probably caused by a film fold that induced a printing defect. No impact is visible on the corresponding zone in the strain maps.

The $ε_{xx}$, $ε_{yy}$ and $ε_{xy}$ strain maps are shown in Figure 4-(b), Figure 5-(a) and Figure 5-(b), respectively. To limit the impact of sensor noise, these strain maps were obtained using the mean phase maps of 100 images of the reference configuration and the mean phase maps of 100 images of the
<figure><img src="image_3.png" /><figcaption>Figure 2: Checkerboard engraved on a layer of paint directly sprayed on the specimen surface (CKB #1) and its counterpart engraved on a polyimide film to form the optical strain gauge bonded on the specimen surface (CKB #2). Close-up views and grey level histograms.</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Figure 3: Standard deviation of the noise affecting the strain maps.</figcaption></figure>

deformed configuration. A correction is introduced to eliminate solid-rigid-like movements that poten-
tially corrupt each phase map set because of micro-vibrations. This procedure is, therefore, different
from merely directly averaging the stack of reference and the deformed images, the averaged image
obtained in the latter case being potentially corrupted by these micro-vibrations [37].

Overall, no real difference can be noted between the global appearance of the maps obtained with both markings. The strain amplitudes are similar. So-called martensite “laminates”, that is, periodic layerings of two strain levels within the martensite bands, are visible at the bottom of the specimen marked with CKB #1: see the red box in Figure 4-(b). In [19, 31], the pattern was an engraved paint layer, and such laminates were also observed in roughly the same area of the same specimen. Martensite laminates are classically revealed at much lower scales within martensite bands. In the present test, laminates are not visible in the top part of the specimen covered by the optical strain gauge (CKB #2). In previous studies carried out on the same specimen [19, 31], laminates were observed in the middle part, within a microstructure similar to the yellow-green “V”-shaped zone shown in the white box depicted in Figure 4-(b). This is not the case here using CKB #2. However, it is not possible to state at this stage whether the optical strain gauge attenuates the potential periodic layering of two strain levels within the martensite bands in the top part of the specimen. This result is discussed further in Section 4.

The colourbars show that strain values up to some percent are reached during this test. However, the images were processed with the classic small strain assumption, and one may wonder if this assumption is not too rough in the present case. Interestingly, the strain values in the different martensite variants that appear in this specimen were estimated with a suitable model in Ref. [19]. This was possible because the lattice parameters of the constitutive material, as well as the orientation of the crystal, were known. The main conclusion is that the strain values measured in the different martensite needles are in excellent agreement with theoretical expectations, which lends credence to the fact that the strain values measured here are not affected by the small-strain assumption. However, further studies should be undertaken to estimate the ultimate strain value that can be reliably measured with this technique.

A close-up view of the ε<sub>yy</sub> map obtained with both markings is depicted in Figure 6-(a). This zone corresponds to the black box in Figure 4-(a). The idea is to check, along a given martensite band, if the amplitude of the strain components measured within this band changes or not when crossing the border between the two patterns. Indeed, this could be the illustration of a possible attenuation of the signal because of the presence of the film. Various cross-sections of the ε<sub>xx</sub>, ε<sub>yy</sub> and ε<sub>xy</sub> maps are plotted for this in Figure 6-(b)(c)(d), along the portions of lines superposed to Figure 6-(a). The same colourcode is used for the small portions of lines and for the curves representing these cross-sections. The origin of each x'-axis is at the centre of each portion of line, thus along the black line in Figure 6-(a). The width of this band, estimated at ε<sub>ij</sub>=0, ij = xx, yy, xy, is equal to about 90 [px] =
<figure><img src="image_5.png" /><figcaption>(a) Picture of the specimen marked with CKB #1 and CKB #2</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>(b) $\epsilon_{xx}$. The black box corresponds to the zone over which a close-up view is plotted in Figure 6(a)</figcaption></figure>

<caption>Figure 4: Picture of the SMA specimen with the two patterns and $\epsilon_{xx}$ strain map. Bottom part of the specimen: CKB #1 (paint layer); Top part: CKB #2 (optical strain gauge)</caption>
<figure><img src="image_7.png" /><figcaption>Figure 5: ε<sub>yy</sub> and ε<sub>xy</sub> strain maps measured on the SMA specimen surface. Bottom part: CKB #1 (paint layer); Top part: CKB #2 (optical strain gauge)</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Figure 6: Analysis of the strain distributions measured in the black box in Figure 4-(a): (a) close-up view of the ε<sub>yy</sub> strain map, showing the location of the lines along which the cross sections are plotted. (b)-(d): strain profiles along the different cross-sections. Red: CKB #1 (paint layer). Blue: CKB #2 (optical strain gauge).</figcaption></figure>
90 × 7.36 × 10⁻³ ≃ 0.66 [mm] on average. No attenuation of the signal is observed when going from the red to the blue curves for the εyy and εxy curves, their summit being nearly the same. The absolute value of εxx is even higher for the blue curves (CKB #1) than for the red ones (CKB #2), while a damping effect of the film would give the contrary. This is certainly due to physical reasons. Indeed, it can be seen that the unique band at the top separates into two parts at the bottom, just after the border between the two patterns, and one of the two smaller bands is evanescent when going toward the bottom. This attenuation is caused by LSA, but also potentially by the presence of the paint (CKB #1) or the adhesive and the film (CKB #2). Another observation is that the apparent damping of these discontinuities is more pronounced with CKB #2 than with CKB #1, which is illustrated by a shallower slope of the εyy and εxy curves in the former case (this is less visible on the εxx curve). This suggests that the adhesive and film used with CKB #2 induce a higher damping than the paint used with CKB #1. However, this conclusion should be considered with caution at least for two reasons: i- the state of strain is not rigorously the same at the places where the cross-sections are plotted because of the potential presence of a heterogeneous elastic strain, as shown in [19], and ii- the neighbouring needles have a different width at these places. In addition, some details in the strain maps are sharper with CKB #2 than with CKB #1 in the second example discussed below, which is contradictory with the present observation. A possible reason is that the attenuation of the actual strain fluctuations also depends on the thickness of the paint, which is not spatially uniformly distributed.

Finally, most of the needles/bands abruptly stop at the border of CKB #1 (see, for instance Points A, B and C in Figure 4-(a)), while they progressively attenuate along the other side, thus at the border of CKB #2 (Points A', B' and C'). This point is discussed further in Section 4.

### 3. Test #2: cracked knot embedded in a wood specimen

#### 3.1. Material and method

Test #2 was performed on a wood specimen. A knot crossed by a crack is embedded in it. Annual rings are also visible on the front face, in the zone where the optical strain gauge is bonded. These features give rise to a heterogeneous strain field. The idea here is to see to what extent these heterogeneities can be detected in a configuration complementary to the preceding one. Indeed, since the specimen is tested in its elastic regime, the strain level is much lower than that reached with Test #1 (some 10⁻³ instead of some 10⁻²). The impact of sensor noise propagation on strain maps is, therefore, higher. In addition, the crack induces a strong heterogeneity, and the influence of the film in this case will be interesting to observe.

Figure 7 shows this wood specimen. Its dimensions are 275 × 45 × 7.5 [cm³]. The red rectangle corresponds to the zone of interest. It surrounds the knot where the two checkerboard patterns were deposited in turn. Indeed, a first tensile test was performed after depositing the checkerboard pattern by spray-painting the external surface white and engraving the tiny black dots (CKB #1). The ultimate force (F = 3200 [N]) that was applied was such that the response of the specimen remained in the elastic regime. This corresponds to the deformed configuration for which the strain maps are shown below. The tensile machine used for this test was a Zwick/Roell UTS 20kN. After this first test, the specimen was sanded to remove the marked layer of paint. An optical strain gauge (CKB #2) was then glued on the specimen. The latter was tested again by following the same procedure as that used in the preceding test, apart from the exposure time, which changed from one test to another to have the highest possible contrast in the images for the pattern under interest. The values are reported in Table 2. In this case, the quality of the two checkerboard patterns was similar to that of the two checkerboard patterns deposited in the preceding experiment. The physical size of the pitch also remained unchanged, but the field of view increased, which means that the number of pixels per period decreased in proportion.

#### 3.2. Results

Figures 8-9 show the distribution of the three in-plane strain components εxx, εyy and εxy obtained with these two tests over the red box in Figure 7. The maps are rotated by 90 degrees to make them
<figure><img src="image_9.png" /><figcaption>Figure 7: Wood specimen tested in this study. The red rectangle approximately corresponds to the zone over which the checkerboard patterns are deposited in turn, and thus, strain maps are obtained. See the following figures where those maps are displayed after anticlockwise rotation of this zone by 90 degrees.</figcaption></figure>

easier to compare. There is a slight vertical shift between the maps displayed on the left and the right. The reason is that the specimen was removed from one test to another to change the type of marking. These maps are displayed in false colour. Their size is 4262 × 6457 ≃ 27.5 × 10⁶ [px], or 4.5 × 6.8 [cm²]. A close-up view of the zone of interest is also displayed in Figure 9-(c) for comparison purposes between the features in the maps and the picture (annual rings, crack, knot).

The standard deviation of the Gaussian window used in LSA to obtain these maps is chosen here to be equal to $\sqrt{2} \times P$ instead of $P$ with Test #1. This choice is justified by the fact that the strain level reached in Test #2 is lower than that reached in Test #1. The Gaussian window is, therefore, chosen to be slightly larger (in pixels) to limit the noise level in the strain maps. Note that $\sqrt{2} \times P$ is twice the size of one unit square forming the checkerboard pattern.

According to the “3-sigma rule” [38], the Gaussian window used in LSA has an apparent area equal to the area of a disk with a diameter equal to six times the standard deviation. In the present case, this diameter equals 34.2 [px]. At the scale of the specimen, this length is equal to 0.36 [mm].

With LSA, the phases and the displacements are calculated pixelwise, but over a zone that is, roughly speaking, equal to this disk. The strain components are obtained with the *gradient function* of the NumPy library. They are therefore estimated over a disk with a size equal to 5.7 × 6 + 2 [px]≈ 34.2 + 2 [px]≈ 36.2 × 10.6 × 10⁻³ [mm]≈ 0.38 [mm]. It is plotted to scale in Figure 8-(d). With the present example, without taking border effects into account for simplicity, each strain map can be regarded as a set of results obtained with 27.5 millions of circular strain gauges of diameter 0.38 [mm]. These local gauges are centred at all the pixels. Consequently, each local strain gauge is shifted by 1 [px] = 10.6 [µm] from the other along both directions. This shift causes a substantial overlap between these miniature local gauges, which explains that sharp strain gradients (at the scale of these tiny distances) may be blurred in the strain maps. Employing a dedicated deconvolution algorithm limits this negative effect [30, 31], but it was not necessary in the present case.

The calculation time needed to retrieve the three strain maps from a pair of images was 160 seconds with a PC (Intel 8-core, i9-11950H @ 2.60Ghz), which seems quite reasonable given the size of the maps. This is because the calculations performed with LSA are nearly straightforward and partly performed in the Fourier domain; see previous studies [26, 39, 13] for more details.

When observing these maps, the striking point is that they are similar from one type of marking to another. On close inspection, some differences can be detected, particularly concerning the sharpness of the details (effect of annual rings, for instance). In particular, these details are more pronounced either in the maps obtained with CKB #1 or CKB #2, depending on the zone of interest. For instance, in the region denoted by $R_1$ for CKB #1 in Figures 8-a-c and 9-a, the annual ring is revealed by a band that is sharper than its counterpart obtained with CKB #2 in Region $R'_1$ in Figures 8-b-d and 9-b. The opposite phenomenon is observed in Regions $R_2$ and $R'_2$ since the details are more blurred in the former region than in the latter. This is probably because the thickness of the paint layer on the left and the thickness of the adhesive on the right may spatially fluctuate; thus, the damping of the actual signal may also spatially fluctuate. Another point is that the thickness of the paint layer is higher here (150 [µm]) than in the preceding case (85 [µm]), which certainly induces additional damping with CKB #1 in Test #2 compared to Test #1. This is due to the higher roughness of the wood specimen
<figure><img src="image_10.png" /><figcaption>Figure 8: $\epsilon_{xx}$ and $\epsilon_{yy}$ maps plotted over the red box in Figure 7 for both types of patterns. Size: $4262 \times 6457 \approx 27.5 \times 10^6$ [px]. (d): The small circle represents the apparent size of the local strain gauge around any pixel on the map. Its diameter equals $36.2$ [px] $\approx 0.38$ [mm].</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>Figure 9: ε<sub>xy</sub> maps plotted over the red box in Figure 7 for both marking techniques. Comparison with a picture of the zone of interest</figcaption></figure>

surface than that of the SMA specimen, which causes additional spray paint layers to be deposited
before obtaining a surface sufficiently smooth to be laser-engraved. It should also be noted that these
maps were obtained with two separate tests, and the specimen was removed from the tensile machine from one test to another. Consequently, the loading conditions (positioning the specimen in the grips, clearance compensation) were not exactly the same from one test to another. Another remark is the close correspondence between the features observed in the picture of the zone of interest in Figure 9-c and in the measured strain maps; see, for instance, the shape of the annual rings or the shape of the crack. The singular distribution of the strain components near the crack tip is also noticeable for the strain maps obtained with both patterns, mainly for the $\varepsilon_{yy}$ component (see Figures 8-c-d). It is also worth noting that the annual rings within the knot, which are visible in Figure 9-c, do not lead to significant strain fluctuations in this zone, which is not the case outside the knot. This is undoubtedly because the wood within the knot is stiffer than its counterpart outside it. Finally, the noise effect is not pronounced on these maps. It is mainly visible near the corners of the maps. It can be checked that the pattern is less contrasted in the corresponding zones. This is probably because the effect of the marking defects on the strain maps is less efficiently corrected near the corners than at the centre because of the lens distortion, which was not accounted for here.

## 4. A simplified model for the strain distribution along the gauge in the presence of a sharp detail in the substrate

### 4.1. Description of the model

Here, we model the strain distribution in the optical strain gauge when a sudden strain change occurs in the substrate located just below. Bearing in mind that LSA alone also causes a damping of the actual strain values, the objective is to have an idea of the size of the tiniest detail that can be reliably measured in a strain map when the optical strain gauge is employed.

We consider for this the simplified 1D model shown in Figure 10-(a).

This substrate is covered by a layer of adhesive and the optical strain gauge. This substrate is split into three domains, namely domains $D_1$, $D_2$ and $D_3$. $D_3$ is symmetric to domain $D_1$ with respect to the vertical line $\Delta$ plotted at the middle of domain $D_2$. Domain $D_2$ models a singularity. This is a portion of the substrate where the Young's modulus suddenly drops. This causes strain jumps in the substrate, mimicking those existing in Test #1 with the SMA specimen when a needle appears or in the wood specimen perpendicularly to the annual rings.

Under some suitable assumptions, it can be shown that the following equations give the strain distribution along the optical strain gauge:

$$ \left\{ \begin{aligned} \varepsilon_{xx}^{(g_1)} &= \frac{1}{E_{S_1}} \left( K_1 e^{\gamma_1 x} + L_1 e^{-\gamma_1 x} + \frac{\beta_1}{\gamma_1^2} \right) & \text{over domain } D_1 \\ \varepsilon_{xx}^{(g_2)} &= \frac{1}{E_{S_2}} \left( K_2 e^{\gamma_2 x} + L_2 e^{-\gamma_2 x} + \frac{\beta_2}{\gamma_2^2} \right) & \text{over domain } D_2 \end{aligned} \right. \quad (3) $$

where $\varepsilon_{xx}^{(g_i)}$ is the strain in the optical strain gauge in domain $D_i$, $i = 1, 2, 3$. $\varepsilon_{xx}^{(g_3)}$ over domain $D_3$ is deduced from $\varepsilon_{xx}^{(g_1)}$ by mirror symmetry. For the sake of readability and compactness, the calculations leading to this result as well as the expressions giving $K_1, K_2, L_1, L_2, \gamma_1, \gamma_2, \beta_1$ and $\beta_2$ as a function of the dimensions of the specimen and the material properties are given in Appendix 2. This model enables us to examine the influence of different parameters on the strain distribution along the gauge.

### 4.2. Damping effect induced by the presence of a sharp detail in the substrate

Figure 11 shows the ratio denoted by $r_e$ between the normal strain in the gauge $\varepsilon_{xx}^{(g_i)}$, $i = 1, 2, 3$ (for domains $D_1$, $D_2$ and $D_3$), and the strain that would take place in domain $D_2$ without the gauge. The corresponding curves are plotted in blue for domains $D_1$ and $D_3$, and in red for domain $D_2$. They were obtained with values of the $\ell_2$ lying between 1 and 10 [mm] and with the following colour code: the lighter the red and blue colours, the lower the value of $\ell_2$, thus the lower the width of domain $D_2$.
<figure><img src="image_12.png" /><figcaption>Figure 10: Substrate covered by the adhesive and the optical strain gauge.</figcaption></figure>

The strain that would take place in domain $\mathcal{D}_2$ without the gauge is equal to $\frac{\sigma_{xx}^{\infty}}{E_{S_2}}$, where $E_{S_2}$ is the Young's modulus of the substrate in domain $\mathcal{D}_2$. The $r_\varepsilon$ ratio, thus equal to $r_\varepsilon = \frac{E_{S_2} \varepsilon_{xx}^{(g_1)}}{\sigma_{xx}^{\infty}}$, is equal or lower than one. $r_\varepsilon$ should ideally be equal to one if the strain in the gauge was exactly equal to the strain in the substrate, in domain $\mathcal{D}_2$. This is not the case; thus, $1 - r_\varepsilon$ reflects the extent to which the strain measured in the gauge is affected by the presence of the optical strain gauge in case of localized change of the strain in domain $\mathcal{D}_2$ of the substrate: the higher the value of $1 - r_\varepsilon$, the higher the error induced by the presence of the gauge. Concerning $\mathcal{D}_1$ and $\mathcal{D}_3$, $r_\varepsilon$ is automatically lower than 1 in these two domains because $E_{S_1} > E_{S_2}$, thus both $\varepsilon_{xx}^{(g_1)}$ and $\varepsilon_{xx}^{(g_3)}$ are lower than $\frac{\sigma_{xx}^{\infty}}{E_{S_2}}$.

The values of the parameters used to plot these curves are reported in Table 3. $E_{S_1}$, $G_a$ are representative of the different materials involved in Test #1 above, $e_s$, $e_a$, $e_g$ of the thickness measured on the corresponding specimen. $E_{S_1} = 120$ GPa is the order of magnitude of the Young's modulus of the austenite [40]. $G_a$ is deduced from the Young's modulus given in [41] (1.26 GPa), assuming the Poisson's ratio is equal to 0.30 [-]. $E_g$ is the value given in the datasheet of the polyimide supplier [22]. The value of $E_{S_2}$ is significantly lower than $E_{S_1}$ ($E_{S_1} = 24 \times E_{S_2}$) to model a sudden softening of the substrate, thus a sudden strain peak in this substrate since the ratio between the longitudinal strain in $\mathcal{D}_1$ and $\mathcal{D}_2$ is equal to $\frac{E_{S_2}}{E_{S_1}}$, the model being unidimensional. This induces a ratio between the strain levels similar to austenite and martensite in Test #1. The objective here is to observe the influence of the width of this singularity, namely $\ell_2$, on the $r_\varepsilon$ ratio, $\ell_2$ spanning the [0,1] [mm] interval to include the values for which the longitudinal strain drops. $\ell_2$ plays the same role as the width of a martensite needle in Example 1 or an annual ring in Example 2.

Figure 11 shows that the strain distribution in the optical strain gauge, normalized by $\frac{\sigma_{xx}^{\infty}}{E_{S_2}}$, continuously changes (for the sake of continuity of the film). It is nearly constant over some zones and
<figure><img src="image_13.png" /><figcaption>Figure 11: Ratio $r_{\epsilon} = \frac{E_{S2} \epsilon_{xx}^{(g_1)}}{\sigma_{xx}^{\infty}}$, $i = 1, 2, 3$ for domains $D_1$ ($0 \le x \le 5$ mm), $D_2$ ($5$ mm $\le x \le 5$ mm $+ \ell_2$) and $D_3$ ($5$ mm $+ \ell_2 \le x \le 10$ mm $+ \ell_2$), respectively, and for various values of $\ell_2$ reported in the legend. Red: domains $D_1$ and $D_3$. Blue: domain $D_2$. The lighter the colour, the lower the value of $\ell_2$. Since the origin of the curves is the same $\forall \ell_2$, the location of the right-hand end of the curves changes with $\ell_2$, thus with the color shade. The values of the parameters governing the 1D model are reported in Table 3. $L_{tr}$ denotes the length of the transfer zone deduced from the 1D model.</figcaption></figure>

features some abrupt changes near both the free borders as well as at the border between $D_1$ and $D_2$ on the one hand, and between $D_2$ and $D_3$ on the other hand. Near the free borders, this phenomenon can be represented by a classic shear lag model [42]. At the boundaries between $D_1$ and $D_2$, or $D_2$ and $D_3$, the optical strain gauge “accommodates” the sudden strain change in the substrate. Hence, a transfer zone also exists between these two places. This causes a decrease of the strain peak, thus of the $r_{\epsilon}$ ratio, if the length of domain $D_2$, namely $\ell_2$, becomes lower than a threshold value. With the present values of the parameters governing the model, the curves plotted in Figure 11 show that the $r_{\epsilon}$ ratio begins to decrease for a value of $\ell_2 \approx 0.7$ mm. A loss of amplitude of $r_{\epsilon}$ equal to 10 % is reached for a value of $\ell_2$ lying between 0.3 and 0.4 mm.

The model depends on several geometric ($\ell_1$, $\ell_2$, $e_a$, $e_g$, $e_s$) and material ($G_a$, $E_g$, $E_{S1}$, $E_{S2}$) parameters, so it is out of the scope of this study to investigate the influence of all of them. However, in addition to giving typical values of $r_{\epsilon}$ for various values of $\ell_2$ in Figure 11, we examine here briefly the influence of the ratio $r_E = \frac{E_{S2}}{E_{S1}}$ between the soft and stiff parts of the substrate in Figure 10. Indeed, the preceding curves were obtained for a particular value of this $r_E$ ratio. Thanks to the model, we can easily change the values of $\ell_2$ and $r_E$ to see how the damping of the strain peak in domain $D_2$ is affected by both the width of this domain and the ratio between Young's moduli in domains $D_2$ and $D_1$ (or $D_3$). $\ell_2$ still spans the [0,1] [mm] interval. $r_E$ lies now between 2.5% and 80% instead of being constant, with a step equal to 2.5%. The other parameters remain unchanged. For all values of $r_E$, we collected the maximum value of the $r_{\epsilon}$ ratio along $x$ observed in domain $D_2$. As an example, this is the value of the summit of the curves plotted in Figure 11 in the particular case $r_E = \frac{E_{S2}}{E_{S1}} = \frac{1}{24} \approx 0.0417$. This enabled us to plot this maximum value as a function of $\ell_2$ for various values of $r_E$. These curves are shown in Figure 12. It can be seen that depending on the value of $r_E$, a damping of 10 % of the strain peak in $D_2$ is obtained for a value of $\ell_2$ lying between 0.1 and 0.35 mm: the higher the value of $r_E$, the lower this value of $\ell_2$, which seems logical.

The results obtained with the tests above can be discussed through the prism of this model, especially Test #1, for which the jumps in the strain maps are more pronounced. Points A, B, and C in Figure 6-(a) show that the strain distribution abruptly stops at the border of CKB #1, while their
<figure><img src="image_14.png" /><figcaption>Figure 12: Maximum value of the $r_e = \frac{E_{S2}e_{xx}^{(g_i)}}{\sigma_{xx}^{\infty}}$ ratio in domain $D_2$ for various $r_E$ ratios (between 0.025 and 0.800) and values of $\ell_2$ (between 0.1 and 1 mm).</figcaption></figure>

prolongations at the other side progressively stop at Points A', B' and C'. This illustrates the effect of the transfer zone described by the 1D model at the free borders. The 1D model predicts that the transfer zone, here estimated by considering it is defined by the abscissa for which 90 % of the plateau value observed in Figure 11 in $D_1$ is reached, has a length $L_{tr}$ approximately equal to $L_{tr} \approx 0.17$ [mm] (see left-hand side of Figure 11). Figure 6-(a) shows that the progressive loading of the optical strain gauge along its border is somewhat longer, say approximately between 0.5 and 1 [mm] at Points A', B', and C'. This figure depicts the $\varepsilon_{xy}$ distribution and not the longitudinal strain along the bands, which would undoubtedly be more representative for a comparison with a 1D model. Still, the three in-plane strain components are all affected similarly. This theoretical value of the transfer length $L_{tr}$ can be visually estimated with the white line segment superposed to this map. Two conclusions can be drawn from this figure:

* The transfer length fluctuates from one band to another, probably because the thickness of the glue is not constant or because the excess glue along the boundary does not form the same glue border everywhere after polymerization.

* Overall, the order of magnitude of this transfer length observed visually is greater than that predicted by the model. This difference is likely caused by the different assumptions under which the calculations are performed, which are quite rough. Indeed, the different parameters are approximated, and this is a 1D, not a 2D model.

However, this greater experimental transfer length lends credence to the fact that tiny details as small as those observed in the martensite laminates with CKB #1 in Test #1, see red box in Figure 4-(b) (their period is equal to about $\approx 14$ [px] $\times 7.2$ [$\mu$m/px] $\approx 100$ [$\mu$m] = 0.1 [mm]), cannot be revealed by CKB #2 because of the attenuation of the details caused by the optical strain gauge. This point should, however, be investigated further with dedicated experiments.

<table><caption>Table 3: Values of the parameters governing the shear lag model</caption><thead><tr><th>e<sub>s</sub><br>[mm]</th><th>e<sub>a</sub><br>[mm]</th><th>e<sub>g</sub><br>[mm]</th><th>l<sub>1</sub><br>[mm]</th><th>l<sub>2</sub><br>[mm]</th><th>E<sub>S1</sub><br>[GPa]</th><th>E<sub>S2</sub><br>[GPa]</th><th>G<sub>a</sub><br>[GPa]</th><th>E<sub>g</sub><br>[GPa]</th></tr></thead><tbody><tr><td>3</td><td>0.005</td><td>0.025</td><td>50</td><td>1 to 10</td><td>120</td><td>5</td><td>0.46</td><td>6</td></tr></tbody></table>
## Conclusion

A procedure for laser-engraving checkerboard patterns on polyimide films as thin as 25 microns is proposed in this paper. It is shown that such laser-engraved films form an optical strain gauge that can be bonded on flat specimens using a procedure similar to that used to glue electrical strain gauges, mainly because polyimide films frequently constitute the backing material of this classic measuring device. The difference with the latter is twofold. First, the polyimide film employed here is white in its bulk instead of brown to ensure good contrast with the dots forming the checkerboard pattern. Second, the present optical gauge returns the full-field strain distribution beneath the optical strain gauge instead of the mean strain value under the resistance of classic electrical gauges. The strain maps obtained in two different testing configurations show that only negligible differences can be detected compared to strain maps returned by the procedure used so far to deposit optimized patterns, which consists of spray-painting the specimen and then directly engraving the paint layer with the same type of engraver as that used in this study. Tiny details are revealed in the strain maps that are shown. This illustrates that this optical strain gauge can be an ideal tool for analysing heterogeneous strain fields in experimental mechanics. In addition, images are processed using a spectral technique, not with digital image correlation. The former technique is much less computationally demanding, which enables us to provide pixelwise-defined strain maps.

Some points should be investigated further to improve the quality and reliability of the measurements provided by this optical strain gauge. For instance, the quality of the laser-engraved checkerboard patterns depends on several parameters, and the optimal settings found for them in this study could potentially be improved by testing other combinations of these multiple parameters. Thinner and/or other types of polymeric films should also be tested. Dedicated experiments should also be designed and performed to apply on the optical strain gauge the stereo-LSA technique recently introduced in [24, 25], or to finely validate the 1D model proposed in this study.

However, this study already opens up two perspectives. First, demonstrating that the optimal pattern for full-field strain measurement, namely checkerboards with a tiny pitch, can be deposited on thin polyimide layers makes it possible to pre-print such patterns on an industrial scale like classic electrical strain gauges routinely employed in industry. This would make this type of gauge affordable and commercially available. Second, the geometry of a checkerboard pattern is easy to define, for instance, by giving the value of the size of the unit dots forming the pattern, the tilt angle when taking the images, or the number of pixels per period that should ideally be employed to encode the periodic pattern in digital images. A consequence is that checkerboard patterns would be easy to standardize. These two perspectives certainly deserve to be explored in the future. This would help spread and popularize optimal patterns for full-field strain measurement in the experimental mechanics community and contribute to making such measurements more reliable and reproducible.

## Acknowledgements

Mr. Jacques, from the ADDEV Materials company, Saint-Chamond, France (https://addevmaterials.com/en/), is gratefully acknowledged for providing the white polyimide film used to prepare the optical strain gauges tested in this study, and Pr. Jean-Benoît Le Cam for supplying silicone plates during the development of the bonding procedure of the gauge. This work has been partially funded by the IDEX-ISITE initiative program (16-IDEX-0001).

## Data availability

The LSA code used to process the images is available online: https://gitlab.ip.uca.fr/expmech/openlsa; see [43] for a complete description. The images used in this study are available on the following ZEN-ODO repository: DOI10.5281/zenodo.11185324
Conflict of interest statement

The authors declare no potential conflict of interest.
<table><caption>Table 4: Settings used for engraving the checkerboard patterns on the two specimens tested in this study.</caption><thead><tr><th></th><th>Paint (SMA)</th><th>Paint (wood)</th><th>Polyimide</th></tr></thead><tbody><tr><td>Scan speed [mm/s]</td><td>3600</td><td>3600</td><td>3600</td></tr><tr><td>Pulse frequency [kHz]</td><td>60</td><td>60</td><td>60</td></tr><tr><td>Number of repetitions [-]</td><td>35</td><td>40</td><td>35</td></tr><tr><td>Percentage of laser power [-]</td><td>50%</td><td>80%</td><td>80%</td></tr></tbody></table>

## Appendix 1: Procedure and settings used for engraving the checkerboard pattern on the paint or the polyimide film

The procedure consists first of cutting a piece of white polyimide strip and placing it in the chamber of the laser marker. The checkerboard pattern is then engraved on this film using the same procedure described in [18]. Four parameters directly influence the quality of the resulting pattern:

1. **Scan speed:** This is the movement speed of the laser spot. Its maximum value is 12,000 mm/s. This is relatively high, but it is worth remembering that no mechanical system is involved thus no inertia effect may slow down this movement.

2. **Pulse frequency:** This is the frequency at which the system emits a pulse. The higher this frequency, the lower the spot energy. This quantity lies between 40 and 400 kHz.

3. **Repetition:** This is the number of laser passes, that is, the number of times the pattern is printed at the same place. The higher this number, the more marked the contrast of the resulting pattern, but this number must also be limited in the present case to avoid piercing the film or to pierce it with the smallest possible holes.

4. **Laser power:** The maximum power equals 3 W at 40 kHz, but the value at the focal point is 2.5 W only. The laser power can be adjusted between 0 and 100 % of this maximum value.

A fifth parameter, the spot variable, can be mentioned. This parameter corresponds to a slight defocus, but the present study did not use this option. The settings used for engraving the checkerboard pattern on the SMA and wood specimens are gathered in Table 4.
# Appendix 2: Shear lag model

The objective is to model the 1D stress and strain distributions along an optical strain gauge bonded on a substrate. The latter is split into three zones, namely $D_1$, $D_2$ and $D_3$ (see Figure 10). The latter domain is mirror-symmetric to $D_1$ with respect to the midline $\Delta$ of $D_2$.

## Assumptions

The assumptions under which the calculations are performed are the same as those employed in the shear lag model presented first in the seminal paper by Volkersen [42]. The only difference is that the substrate is divided into three zones, while classic shear lag models focus on the transfer zone at the free edge of a unique zone. These assumptions are as follows:

* The model for the stress distribution in the substrate, the adhesive, and the optical strain gauge is unidimensional.

* Traction is applied on the left- and right-hand sides of the substrate. Its intensity is denoted by $\sigma_{xx}^{\infty}$.

* The substrate and the optical strain gauge are subject to normal stress along $x$ only, and the adhesive to shear stress only. Their distributions over $D_1$ and $D_2$ are respectively denoted by $\sigma_{xx}^{(s_1)}$, $\sigma_{xx}^{(g_1)}$, and $\sigma_{xz}^{(a_1)}$, $i \in 1, 2$. The distributions over $D_3$ are deduced by mirror-symmetry from their counterparts calculated over $D_1$.

* The constitutive material of the three components is linear elastic. The Young's modulus of the film used for the optical strain gauge is denoted by $E_g$, the Young's modulus of the substrate by $E_{S_1}$ and $E_{S_2}$ over domains $D_1$ and $D_2$, respectively. The shear modulus of the adhesive is denoted by $G_a$, $e_s$, $e_a$, and $e_g$ represent the thickness of the substrate, the adhesive and the optical strain gauge, respectively.

The model developed here should also satisfy the four conditions listed below. There are three boundary conditions (BCs) and a symmetry condition:

* BC1: The boundary of the optical strain gauge is free at $x = 0$

* BC2: The normal stress distribution in the optical strain gauge is continuous at $x = \ell_1$

* BC3: The shear stress distribution in the adhesive is continuous at $x = \ell_1$.

* The solution is strictly symmetric with respect to the middle of $D_2$.

## Development of the model

The equilibrium of the small portion of substrate over $D_1$ shown in Figure 10-b reads as follows:

$$ \left(-\sigma_{xx}^{(s_1)} + \sigma_{xx}^{(s_1)} + d\sigma_{xx}^{(s_1)}\right) e_s + \sigma_{xz}^{(a_1)} dx = 0 \quad (4) $$

Thus,

$$ \frac{d\sigma_{xx}^{(s_1)}}{dx} e_s + \sigma_{xz}^{(a_1)} = 0. \quad (5) $$

In the same way, we have for the portion of the optical strain gauge:

$$ \left(-\sigma_{xx}^{(g_1)} + \sigma_{xx}^{(g_1)} + d\sigma_{xx}^{(g_1)}\right) e_g - \sigma_{xz}^{(a_1)} dx = 0, \quad (6) $$

which gives
$$ \frac{d\sigma_{xx}^{(g_1)}}{dx} - e_g - \sigma_{xz}^{(a_1)} = 0. \qquad (7) $$

Substituting $\sigma_{xz}^{(a_1)}$ from Equation 5 in Equation 7 leads to:

$$ \frac{d}{dx} \left( \sigma_{xx}^{(s_1)} e_s + \sigma_{xx}^{(g_1)} e_g \right) = 0. \qquad (8) $$

Integrating with respect to $x$ and considering the boundary conditions of domain $D_1$ leads to:

$$ \sigma_{xx}^{(s_1)} e_s + \sigma_{xx}^{(g_1)} e_g = \sigma_{xx}^{(s)\infty} e_s. \qquad (9) $$

Thus

$$ \sigma_{xx}^{(s_1)} = \sigma_{xx}^{(s)\infty} - \sigma_{xx}^{(g_1)} \frac{e_g}{e_s}. \qquad (10) $$

The thickness of the adhesive layer is small, so the through-thickness engineering shear strain $\gamma_{xz}^{(a_1)}$ is reasonably assumed to be homogeneous, which enables us to express it as a function of the displacement along $x$ in the film ($u_x^{(g_1)}$) and in the substrate ($u_x^{(s_1)}$):

$$ \gamma_{xz}^{(a_1)} = \frac{u_x^{(g_1)} - u_x^{(s_1)}}{e_a}. \qquad (11) $$

The adhesive is linear elastic, thus

$$ \sigma_{xz}^{(a_1)} = G_a \gamma_{xz}^{(a_1)} = G_a \frac{u_x^{(g_1)} - u_x^{(s_1)}}{e_a}. \qquad (12) $$

Substituting this expression of $\sigma_{xz}^{(a_1)}$ in Equation 7 leads to:

$$ \frac{d\sigma_{xx}^{(g_1)}}{dx} e_g - \frac{G_a}{e_a} (u_x^{(g_1)} - u_x^{(s_1)}) = 0. \qquad (13) $$

This equation is then derivated with respect to $x$, which gives:

$$ \frac{d^2\sigma_{xx}^{(g_1)}}{dx^2} e_g - \frac{G_a}{e_a} (\varepsilon_{xx}^{(g_1)} - \varepsilon_{xx}^{(s_1)}) = 0. \qquad (14) $$

Since $\varepsilon_{xx}^{(g_1)} = \frac{\sigma_{xx}^{(g_1)}}{E_g}$ and $\varepsilon_{xx}^{(s_1)} = \frac{\sigma_{xx}^{(s_1)}}{E_{s_1}}$, Equation 14 becomes:

$$ \frac{d^2\sigma_{xx}^{(g_1)}}{dx^2} e_g - \frac{G_a}{e_a} \left( \frac{\sigma_{xx}^{(g_1)}}{E_g} - \frac{\sigma_{xx}^{(s_1)}}{E_{s_1}} \right) = 0. \qquad (15) $$

After introducing in the preceding equation the expression of $\sigma_{xx}^{(s_1)}$ given in Equation 10 and dividing by $e_g$, we have

$$ \frac{d^2\sigma_{xx}^{(g_1)}}{dx^2} - \frac{G_a}{e_a} \left( \frac{1}{e_g E_g} + \frac{1}{e_s E_{s_1}} \right) \sigma_{xx}^{(g_1)} + \frac{G_a}{e_a e_g E_{s_1}} \sigma_{xx}^{\infty} = 0. \qquad (16) $$

The same reasoning can be applied to domain $D_2$, thus

$$ \frac{d^2\sigma_{xx}^{(g_2)}}{dx^2} - \frac{G_a}{e_a} \left( \frac{1}{e_g E_g} + \frac{1}{e_s E_{s_2}} \right) \sigma_{xx}^{(g_2)} + \frac{G_a}{e_a e_g E_{s_2}} \sigma_{xx}^{\infty} = 0. \qquad (17) $$
In conclusion, the stress distribution in the optical strain gauge is modeled by the following two differential equations:

$$ \left\{ \begin{aligned} & \frac{d^2 \sigma_{xx}^{(g_1)}}{dx^2} - (\gamma_1)^2 \sigma_{xx}^{(g_1)} = -\beta_1 \quad \text{over } \mathcal{D}_1 \\ & \frac{d^2 \sigma_{xx}^{(g_2)}}{dx^2} - (\gamma_2)^2 \sigma_{xx}^{(g_2)} = -\beta_2 \quad \text{over } \mathcal{D}_2 \end{aligned} \right. \qquad (18) $$

with:

$$ \left\{ \begin{aligned} \beta_1 &= \frac{G_a}{e_a e_g E_{s_1}} \sigma_{xx}^{\infty} & \beta_2 &= \frac{G_a}{e_a e_g E_{s_2}} \sigma_{xx}^{\infty} \\ \gamma_1 &= \sqrt{\frac{G_a}{e_a} \left( \frac{1}{e_g E_g} + \frac{1}{e_s E_{s_1}} \right)} & \gamma_2 &= \sqrt{\frac{G_a}{e_a} \left( \frac{1}{e_g E_g} + \frac{1}{e_s E_{s_2}} \right)}. \end{aligned} \right. \qquad (19) $$

We now give the solutions of these two differential equations using the classic resolution method of linear differential equations of second order and first degree.

**Solution of the associated homogeneous equation.** The roots of the characteristic polynomial of these equations are $\gamma_1$ and $-\gamma_1$ for the former, and $\gamma_2$ and $-\gamma_2$ for the latter, so the solutions of the homogeneous equations associated with the two complete equations read as follows:

$$ \left\{ \begin{aligned} \sigma_{xx}^{(g_1)} &= K_1 e^{\gamma_1 x} + L_1 e^{-\gamma_1 x} & K_1, L_1 \in \mathbb{R} \\ \sigma_{xx}^{(g_2)} &= K_2 e^{\gamma_2 x} + L_2 e^{-\gamma_2 x} & K_2, L_2 \in \mathbb{R}. \end{aligned} \right. \qquad (20) $$

**Particular solution of the complete equation.** $\sigma_{xx}^{(g_1)} = k_1$, $\sigma_{xx}^{(g_2)} = k_2$ are particular solutions of the complete equations. Feeding the complete differential equations with these particular expressions of $\sigma_{xx}^{(g_1)}$ and $\sigma_{xx}^{(g_2)}$ leads to $k_1 = \frac{\beta_1}{\gamma_1^2}$ and $k_2 = \frac{\beta_2}{\gamma_2^2}$.

**Solution of the complete equation.** The solution of each equation is obtained by adding the last two expressions, thus:

$$ \left\{ \begin{aligned} \sigma_{xx}^{(g_1)} &= K_1 e^{\gamma_1 x} + L_1 e^{-\gamma_1 x} + \frac{\beta_1}{\gamma_1^2} \\ \sigma_{xx}^{(g_2)} &= K_2 e^{\gamma_2 x} + L_2 e^{-\gamma_2 x} + \frac{\beta_2}{\gamma_2^2} \end{aligned} \right. . \qquad (21) $$

**Fixing the value of $K_1$, $K_2$, $L_1$ and $L_2$.** The four conditions listed above give equations that are now used to fix the value of $K_1$, $K_2$, $L_1$, and $L_2$.

* BC1 (the boundary of the optical strain gauge is free at $x = 0$): $\sigma_{xx}^{(g_1)}(0) = 0$

* BC2 (the normal stress distribution in the optical strain gauge is continuous at $x = \ell_1$): $\sigma_{xx}^{(g_1)}(\ell_1) = \sigma_{xx}^{(g_2)}(\ell_1)$

* BC3 (the shear stress distribution in the adhesive $\sigma_{xz}^{(a)}$ is continuous at $x = \ell_1$): $\frac{d\sigma_{xx}^{(g_1)}}{dx}(\ell_1) = \frac{d\sigma_{xx}^{(g_2)}}{dx}(\ell_1)$ according to Equation 7

* mirror-symmetry condition (the normal stress distribution in the optical strain gauge is symmetric at $x = \ell_1 + \frac{\ell_2}{2}$): $\frac{d\sigma_{xx}^{(g_2)}}{dx}(\ell_1 + \frac{\ell_2}{2}) = 0$
This leads to the following linear system:

$$ egin{cases} K_1 + L_1 = -\frac{\beta_1}{\gamma_1^2} \\ K_1 e^{\gamma_1 \ell_1} + L_1 e^{-\gamma_1 \ell_1} - K_2 e^{\gamma_2 \ell_1} - L_2 e^{-\gamma_2 \ell_1} = -\frac{\beta_1}{\gamma_1^2} + \frac{\beta_2}{\gamma_2^2}. \\ \gamma_1 K_1 e^{\gamma_1 \ell_1} - \gamma_1 L_1 e^{-\gamma_1 \ell_1} - \gamma_2 K_2 e^{\gamma_2 \ell_1} + \gamma_2 L_2 e^{-\gamma_2 \ell_1} = 0 \\ K_2 e^{\gamma_2 (\ell_1 + \frac{\ell_2}{2})} - L_2 e^{-\gamma_2 (\ell_1 + \frac{\ell_2}{2})} = 0 \end{cases} \quad (22) $$

This system of four linear equations was inverted with XCAS, an open-source symbolic computation software [44]. After reordering the raw results, the following expressions were obtained for $K_1$, $L_1$, $K_2$, and $L_2$:

$$ egin{cases} K_1 = \frac{e^{\gamma_2 \ell_2} [-\beta_1 \gamma_1 \gamma_2 + \beta_1 \gamma_2^2 (1 - e^{\gamma_1 \ell_1}) + \beta_2 \gamma_1^2 e^{\gamma_1 \ell_1}] - \beta_1 (\gamma_1 \gamma_2 - \gamma_2^2 e^{\gamma_1 \ell_1} + \gamma_2^2) - \beta_2 \gamma_1^2 e^{\gamma_1 \ell_1}}{\gamma_1^3 \gamma_2 [(e^{\gamma_2 \ell_2} + 1) (e^{2\gamma_1 \ell_1} + 1)] + \gamma_1^2 \gamma_2^2 [(e^{\gamma_2 \ell_2} - 1) (e^{2\gamma_1 \ell_1} - 1)]} \\ K_2 = \frac{(\beta_1 \gamma_2^2 e^{2\gamma_1 \ell_1} - 2\beta_1 \gamma_2^2 e^{\gamma_1 \ell_1} + \beta_1 \gamma_2^2 - \beta_2 \gamma_1^2 e^{2\gamma_1 \ell_1} - \beta_2 \gamma_1^2) e^{\gamma_2 \ell_1}}{\gamma_2^2 \gamma_1 [(e^{2\gamma_1 \ell_1} e^{2\gamma_2 \ell_1} + e^{2\gamma_2 \ell_1 + \gamma_2 \ell_2}) (\gamma_1 - \gamma_2) + (e^{2\gamma_1 \ell_1} e^{2\gamma_2 \ell_1 + \gamma_2 \ell_2} + e^{2\gamma_2 \ell_1}) (\gamma_2 + \gamma_1)]} \\ L_1 = -\frac{e^{\gamma_1 \ell_1} [\beta_1 \gamma_1 \gamma_2 (e^{\gamma_2 \ell_2} e^{\gamma_1 \ell_1} + e^{\gamma_1 \ell_1}) + \beta_1 \gamma_2^2 (e^{\gamma_2 \ell_2} e^{\gamma_1 \ell_1} - e^{\gamma_2 \ell_2} - e^{\gamma_1 \ell_1} + 1) + \gamma_1^2 \beta_2 (e^{\gamma_2 \ell_2} - 1)]}{\gamma_1^3 \gamma_2 [(e^{\gamma_2 \ell_2} + 1) (e^{2\gamma_1 \ell_1} + 1)] + \gamma_1^2 \gamma_2^2 [(e^{\gamma_2 \ell_2} - 1) (e^{2\gamma_1 \ell_1} - 1)]} \\ L_2 = \frac{e^{2\gamma_2 \ell_1 + \gamma_2 \ell_2} e^{\gamma_2 \ell_1} [\beta_1 \gamma_2^2 (e^{\gamma_1 \ell_1} - 1)^2 - \beta_2 \gamma_1^2 (e^{2\gamma_1 \ell_1} + 1)]}{\gamma_2^2 \gamma_1 [(e^{2\gamma_1 \ell_1} e^{2\gamma_2 \ell_1} + e^{2\gamma_2 \ell_1 + \gamma_2 \ell_2}) (\gamma_1 - \gamma_2) + (e^{2\gamma_1 \ell_1} e^{2\gamma_2 \ell_1 + \gamma_2 \ell_2} + e^{2\gamma_2 \ell_1}) (\gamma_2 + \gamma_1)]} \end{cases} \quad (23) $$

The model is unidimensional, so the strain distribution along the optical strain gauge is merely obtained by dividing the stress distribution by the value of the Young's modulus over each of the two domains $D_1$ and $D_2$. Thus:

$$ egin{cases} \varepsilon_{xx}^{(g_1)} = \frac{1}{E_{S_1}} \left( K_1 e^{\gamma_1 x} + L_1 e^{-\gamma_1 x} + \frac{\beta_1}{\gamma_1^2} \right) \\ \varepsilon_{xx}^{(g_2)} = \frac{1}{E_{S_2}} \left( K_2 e^{\gamma_2 x} + L_2 e^{-\gamma_2 x} + \frac{\beta_2}{\gamma_2^2} \right) \end{cases} \quad (24) $$

In conclusion, the strain distribution along the optical strain gauge over $D_1$ and $D_2$ is given by Equations 24 above. Coefficients $K_1$, $K_2$, $L_1$ and $L_2$ are given by Equations 23, and coefficients $\beta_1$, $\beta_2$, $\gamma_1$ and $\gamma_2$ by Equations 19. The strain distribution along the optical strain gauge over $D_3$ is mirror-symmetric to its counterpart over $D_1$. This gives Equations ?? in Section 4.
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