# Assessment of principal strain uncertainties in stereo-DIC due to uncontrollable change of initial boundary conditions during vacuum-assisted thermoforming

Abderrahmane Ayadi, Marie-France Lacrampe, Patricia Krawczak

<figure><img src="image_1.png" /></figure>

## To cite this version:

Abderrahmane Ayadi, Marie-France Lacrampe, Patricia Krawczak. Assessment of principal strain uncertainties in stereo-DIC due to uncontrollable change of initial boundary conditions during vacuum-assisted thermoforming. 22nd International Conference on Material Forming (ESAFORM 2019), May 2019, Vitoria-Gasteiz, Spain. pp.130002, 10.1063/1.5112662. hal-02175866

HAL Id: hal-02175866

https://hal.science/hal-02175866v1

Submitted on 11 Nov 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Assessment of Principal Strain Uncertainties in Stereo-DIC
Due to Uncontrollable Change of Initial Boundary
Conditions During Vacuum-Assisted Thermoforming

A. Ayadi¹, ², a), M-F. Lacrampe¹, ² and P. Krawczak¹, ²

¹IMT Lille Douai, Institut Mines-Télécom, Polymers and Composites Technology & Mechanical Engineering Department, 941 rue Charles Bourseul, 59508 Douai, France.

²Université de Lille, 59000 Lille, France.

a)Corresponding author: abderrahmane.ayadi@imt-lille-douai.fr

**Abstract.** Full-field measurement of principal mechanical strains via the stereo-digital images correlation technique (stereo-DIC) during thermoforming of thin thermoplastic sheets was recently reported as a potential method to control the wall-thicknesses under the assumption of material's incompressibility. The accuracy of mechanical strains during forming thermoplastic sheets are not only dependent of reconstruction and correlation errors of stereo-DIC, but also of the considered reference state of boundary conditions following the temperature-regulation of thermoplastic sheets. This paper introduces an experimental method based on high speed imaging techniques such as IR-thermography and stereo-DIC techniques to detect damped out-of-plane undulations which are associated with the initiation of dynamic inflation operations of thin sheets. The aim is to rely on the characteristic durations of these short-time events to objectively separate between mechanical strains which are affected by the uncontrollable change of boundary conditions from the global pressure-induced strains. An application case is presented to demonstrate that damped out-of-plane undulations are beneficial to avoid ill-posed stereo-DIC problems particularly when the boundary conditions are sensitive to regulation operations of temperature.

INTRODUCTION

Vacuum-assisted thermoforming of thin thermoplastic sheets is a high-potential forming process which is suitable for manufacturing hollow parts with complex shapes and surface textures. To withstand large deformations with no excessive thinning nor breakage, thermoplastic sheets pass through a sequence of consecutive operations including heating and pre-stretching by bubble inflation [1]. During the regulation of the thermoplastic's temperature within the thermoforming window, relaxation of residual stresses and viscous effects can induce some stochastic surface out-of-plane deformations altering the initially imposed boundary conditions of the thermoforming process [2]. Moreover, the pressure load which is applied to the softened sheet during the stretching operation induces not only out-of-plane large deformations of the sheet but also geometrical non-linearity [3]. With development of imaging technologies more performant thermo-mechanical characterization procedures are introduced to monitor surface deformations in thermoforming industrial conditions. Within this context, *Van Mieghem et al.* have recently demonstrated the potential of using the stereo-digital image correlation (stereo-DIC) technique to measure principal strains in negative mold vacuum-assisted thermoforming and to assess full-fields of strains and of thicknesses under the assumption of material incompressibility [4, 5]. In fact, the stereo-DIC technique relies on recording the deformations of a targeted surface using a stereo-vision technique and then on image-correlation algorithms to compute displacements and strains with consideration of a well-defined reference state of deformations. The accuracy of stereo-DIC results are mostly related to reconstruction and correlation errors in the framework of large deformations [6, 7]. According to the considered mechanical problem of the thermoforming cycle, this reference state should be representative of the effective boundary
conditions of the corresponding operation of the process. Thus, by continuously recording the deformations of the sheet during the heating and then the stretching operations, the computed global stereo-DIC strains correspond to an accumulation of thermal strains and pressure-induced mechanical strains if the reference state is attributed to the initial conditions imposed during the clamping operation of the sheet. However, thermal strains cannot always be measured due technical issues such as the use of sliding IR-heaters which may totally obstruct the stereo-vision system from recording the deformations of the thermoplastic sheet during the heating operation. In this case, the uncontrollable change of initial conditions during the temperature regulation affects the identification of the effective reference state of image-correlation computations and the stereo-DIC problem is considered as mechanically ill-posed [8, 9].

According to Verron *et al.*, dynamic inflation operations of thin thermoplastic membranes can involve the occurrence of damped out-of-plane undulations which are associated to the sudden initiation of pressure increase [10]. Within this line of thought, this paper suggests the use of high-speed imaging techniques to detect these damped out-of-plane undulations at the initiation of dynamic stretching operation. Based on a previous work by the authors [11], the reference-state of image-correlation computations can be objectively shifted at the moment of attenuation of 63% of their initial amplitudes to separate computed global mechanical strains into strains which are affected by the uncontrolled change of initial boundary conditions and pressure-induced mechanical strains which correspond to the effective stretching of the thermoplastic sheet. The computed principal strains before and after shifting the reference of image-correlation computations will be used to assess uncertainty levels of major and minor principal strains.

# EXPERIMENTAL PROTOCOL

## Material and Equipment

The thermoforming rig that was introduced in [11] is used to form semi-spherical domes of 250 mm in diameter from initially flat sheets of high impact polystyrene (Athpol P91®) of 1.5 mm thickness (Fig.1a). The considered vacuum-assisted thermoforming cycle is constituted of a sequence of three operations which are: clamping, radiative heating and dynamic inflation. All these operations are time controlled and the forming cycle is interrupted before a positive mold comes into contact with the formed domes (Fig.1b). The thermoforming cycle starts automatically by clamping the sheet ($\Omega_{sheet}$) on top of the forming cabinet using hydraulic jacks and a clamp-frame. By the end of the clamping operation, the initial boundary conditions $BC_0$ correspond to constrained displacements in principal directions at edges of the HIPS sheet and the reference of out-of-plane displacements ($V_z = 0$ mm) is attributed to the top surface of the clamped sheet. The heating operation initiates by sliding an IR-heater which is located at 150 mm above the clamped thermoplastic sheet. The regulation temperature of the IR-heater was fixed to 330 °C and was maintained for 70 s to bring the temperature of the core of the sheet above the temperature of glass transition ($T_g$ = 100 °C). During this operation, the viscous effects gradually become significant and gravity force causes descent of the sheet under the effect of its own weight ($V_z(t) \le 0$ mm). After a complete retraction of the IR-heater, the pre-stretching operation starts by activating an electric valve to inflate pressurized air inside the previously sealed forming cabinet for 1.5 s. The relatively prompt build-up of pressure forces the sheet to deform out-of-plane and to form a bubble-like shape.

<figure><img src="image_3.png" /><figcaption>**FIGURE 1.** a) Illustration of the considered boundary conditions on the edges of the sheet $\Gamma_{sheet}$ and locations of the considered region of interest $\Omega_{ROI}$ and the polar region $\Gamma_{Polar}$. b) Representative schema illustrating the three constitutive operations of the considered thermoforming cycle. c) Overview of the used experimental equipment.</figcaption></figure>
During the stretching operation surface deformations are followed by using a couple of optical cameras which constitute the stereo-vision system (LaVision). Both cameras allow synchronized recording of 180 pairs of frames per second at a resolution of 2048 × 2048 pixels. The temperature of the thermoplastic sheet was also recorded using a thermal camera (Telops) at a resolution of 320 × 256 pixels and an acquisition rate of 200 fps. For reasons of limited space and degrees of freedom, the thermal camera was positioned by the side of the forming plate at an angle of inclination of about 60° relative to the revolution axis of the formed bubble. The used equipment as well as the considered positions of the imaging devices are reported in Fig.1c.

## Stereo-DIC Computations

The commercial software DaVis (LaVision) was used to post-process the pairs of stereo images that were recorded by conducting a total of four forming cycles at the considered regulation temperature of the IR heater (where each cycle corresponds to the sequence of three operations in Fig.1b). All stereo-DIC computations considered a circular field-of-view of 120 mm in radius that was concentric to the circular opening of clamp-frame (Ω<sub>ROI</sub> in Fig.1a). The used subset and the step-size corresponded respectively to 81×81 pixels and 21×21 pixels. Due to the large out-of-plane deformations of the softened HIPS sheets the sum of the differential method was considered to conduct the images-correlation computations in a Lagrangian coordiantes system [6]. Because the aim of the current study is related to the reliability of the measured strains related to the choice of the reference state of correlation computations, reconstruction and correlation sources of error were assumed within the tolerance range of the calibrated stereo-DIC equipment and were not evaluated separately. The main focus was attributed to displacement vectors and major and minor principal Green-Lagrange strains as out-put of the image-correlation computations. Where the major principal strain (and the minor principal strain) measures the local deformation in the direction of the highest (respectively the lowest) deformation of the sheet.

## Evaluation of Principal Strains

Because the stereo-vision measurements were only possible after softening the HIPS samples, the adopted approach consisted on evaluating pressure-induced mechanical strains with consideration of two steps. The first step assumed that the reference of displacement vectors corresponded to the deformed speckle observed on the sagged surface at the moment of complete retraction of the IR-heater. This assumption was required to qualitatively analyze the evolution of the out-of-plane components of displacement vectors (w) in order to get more insight about the damped out-of-plane oscillations associated with the transition from the sagged to the strained shapes of the softened HIPS sheet. The second step consisted on shifting the reference of displacement vectors to the moment of attenuation of the damped out-of-plane undulations to evaluate pressure-induced mechanical strain corresponding to a quasi-stable stretching regime of the sheet. The mathematical difference between strains from both sub-steps corresponds not only to the shift of the reference of image-correlation computations but also represents pressure induced mechanical strains which are more dependent of the uncontrolled change of initial conditions during the heating operation. Thus, the global mechanical strain ε<sup>DIC</sup> measured by stereo-DIC during the stretching operation with consideration of the sagged surface as a reference of image-correlation computations can be expressed as indicated in equation (1):

$$ε^{DIC} = ε^{unst} + ε^{st} + ε^{*} \qquad (1)$$

Where, ε<sup>unst</sup> is the pressure-induced mechanical strain during the attenuation of the damped oscillations; ε<sup>st</sup> is the pressure-induced mechanical strain after the attenuation of oscillations; and ε<sup>*</sup> is a mathematical function which represents the correlation and reconstruction errors. In the current work, the main focus is attributed to the evaluation of ε<sup>unst</sup> which represents the effect of uncontrolled change of initial conditions caused by the heating operation. Since the same calibration configuration of the stereo-DIC system and the same post-processing parameters are used to compute ε<sup>DIC</sup>, the computation of ε<sup>*</sup> is not considered. For more details about the evaluation of correlation and reconstruction errors the reader can refer to the following studies [6, 7, 12, 13].
## Assessment of Strain Uncertainty Levels

A home-developed routine was implemented on Matlab to extract and match the spatial coordinates of the nodes of the meshes that were used in both previous image-correlation computations. Then, for each node i of the coordinates(xᵢ; yᵢ), a scalar βᵣᵢ was calculated as indicated in equation (2):

$$ \beta_{qi} = 100 \times \left| \frac{\varepsilon_{qi}^{\text{DIC}} - \varepsilon_{qi}^{\text{st}}}{\varepsilon_{qi}^{\text{DIC}}} \right| \quad (2) $$

Where, i ∈ {1, 2, ..., n} and q ∈ {1, 2}, n the total number of computing nodes (which is equal to 83 × 83 nodes); εDIC 1i and εDIC 2i represent the major and minor principal strains on the surface of the sheet before shifting the reference state; εst 1i and εst 2i represent the major and minor principal strains on the surface of the sheet after shifting the reference state. Each scalar represented the local fraction of εunst qi strains, which is affected by the uncontrolled change of local thermal and geometric conditions taking place during the heating step. The median of overall calculated βᵣᵢ scalars were then considered as a collective measurement to statistically quantify the uncertainty level of evaluated strains if a shift of the reference state would not be considered. In fact, the median was preferred over the average because the latter is more sensitive to local imprecisions which can be related to the correlation errors at the edges of the considered virtual gauge or to an excessive distortion of the corresponding mesh due to large deformations of the speckle [14]. The median of β̃q was defined based on equation (3):

$$ \tilde{\beta}_q = \text{median of } \{\beta_{q1}, \beta_{q2}, \dots, \beta_{qn}\} \quad (3) $$

# RESULTS AND COMMENTS

## Detection of Out-of-plane Surface Undulations Using Stereo-DIC

During each of the conducted four thermoforming cycles, the used stereo-vision system allows continuous recording of the speckled images applied to the sheet surface using synchronized acquisitions with air pressure measurements under the bubble. The calculated out-of-plane components of displacement vectors show that the formed domes reach, under the defined experimental conditions (average strain rate of 11 s⁻¹), an average height of 65 mm at the end of the stretching operation. In the polar zone (ΓPolar in Fig.1a) of the bubble in particular, the out-of-plane components of displacement vectors show that during its expansion, the sheet passes through two different regimes: a regime lasting around 250 ms characterized by the presence of damped undulations followed by a regime where the out-of-plane deformation becomes more uniform until the end of the dynamic-inflation process (Fig.2a).

<figure><img src="image_4.png" /><figcaption>FIGURE 2. a) Out-of-plane displacement at the pole of the formed bubbles. b) Iso-values of the out-of-plane components of displacement vectors illustrating the damped undulations of the surface of the sheet.</figcaption></figure>

Figure 2b presents representative projections of iso-values of the out-of-plane components of the displacement vectors illustrating the attenuation of damped undulations. The time-evolution of the deformation vectors shows that the transition from the collapsed form to the stretched form (at Z=0) of the softened sheet corresponds to symmetrical buckling of the surface with respect to the revolution axis of the formed bubble. The initial amplitude of these undulations corresponds to the sagging level of the sheet by the end of the heating operation (Fig. 2b-i). The observation of these damped undulations is limited to less than 250 ms from the moment of the initiation of the pre-stretching operation. Such phenomenon can be explained based on the work of Verron et al. which demonstrated that
relatively fast increase of pressure during dynamic inflation operations of thin circular membranes is associated with damped undulations [10]. Thus, these damped out-of-plane undulations can constitute an objective indicator of the moment of transition to the quasi-static stretching regime of the thermoplastic sheet. Consequently, identifying the characteristic time of their attenuation constitutes an essential step to evaluate the deformations after uncontrolled change of the initial boundary conditions during the heating operation.

## Detection of Out-of-plane Surface Undulations Using IR-Thermography

The accuracy of the thermal measurements is strongly related to the depth of field, the resolution of the camera sensor and its orientation relative to the observed surface. With the chosen configuration, the temperature distributions can be reconstituted for the two detected regimes. During the first regime, the variation of the thermal fields of the surface of the bubble confirms the existence of damped undulations marked by distortions of these fields. Due to the position of the thermal camera, these undulations result in an alternation between increase and decrease in the local distributions of the thermal fields, which correspond to a change in the angle described between the axis of acquisition of the thermal camera and the surface during its rapid expansion. Once the quasi-static regime is established, the temperature fields no longer fluctuate and stabilize (Fig.3). During the bubble inflation operation using unconditioned pressurized-air the sheet's temperature drop was limited to almost 10 °C.

<figure><img src="image_5.png" /><figcaption>**FIGURE 3.** a) Average out-of-plane displacement at the polar region during a vacuum assisted thermoforming cycle. b) Temperature fields measured at the surface of the deforming HIPS sheet.</figcaption></figure>

## Principal Strain Uncertainties: Case of Application

In this section, one single stretching test corresponding to the thermoforming cycle of the reference Test 1, was considered. Thus, two reference states were defined: the initial reference state corresponded to the sagged shape of the sheet at the initiation of the stretching step and the second reference state was defined based on a time-shift based on velocity characteristic-time according to the method described in [11]. Displacement vectors and both major and minor principal mechanical strains on the surface of the formed bubbles were evaluated according to the previously indicated two-step image-correlation method. $\epsilon_1^{PIC}$ and $\epsilon_1^{st}$ (respectively $\epsilon_2^{PIC}$ and $\epsilon_2^{st}$) represent the evaluated major (respectively minor) principal strains before and after shifting the reference state of the correlation computations. The considered case of application corresponds to a fixed pressure level of 385 Pa.

As indicated in Fig.4, the evaluated median $\tilde{\beta}_2$ corresponded to 15.47 % and was almost three times higher than $\tilde{\beta}_1$ (4.49 %). According to the adapted identification method of the reference of the images-correlation calculations after heating the surface of the material, all the calculated scalars $\beta_1$ and $\beta_2$ represented not only the strain imprecisions caused by variation of the thermal level but also imprecisions related to an amplification of the damped oscillation caused by the sagging of the sheet. Thus, strain fields evaluated from step 2 of stereo-DIC computations are more representative of the effective stretching of the material as the stereo-DIC computations are performed based on an objectively defined reference state of the sheet deformations. In fact, based on a previous work of the authors [11], this defined the reference of stereo-DIC computations is identified based on an exponentially decayed expression of time and corresponds to the attenuation of 63 % of the amplitudes of the damped out-of-plane undulations with respect of the considered heating conditions and pressure level.
<figure><img src="image_6.png" /><figcaption>**FIGURE 4.** Principal strains obtained using the stereo-DIC technique before (step1) and after (step2) filtering the transient effects. Histograms correspond to the distribution of the calculated uncertainties of the principal minor and major strains.</figcaption></figure>

CONCLUSIONS

By using high speed stereo-DIC and IR-thermography techniques, the existence of damped out-of-plane undulations of the surface sagging were detectable. To quantify the effective stretching of the sagged surface that takes place during bubble-inflation by stereo-DIC, an objective identification of the reference of deformations was required. Stereoscopic measurements were performed at a frequency of 180 fps during bubble-inflation posteriori to radiative heating operations. The suggested method allowed comparison between the principal strains before and after shifting the reference of correlation computations. After separating mechanical strains which are affected by the uncontrollable change of boundary conditions and the global pressure-induced strains uncertainty levels of stereo-DIC strains were evaluated. This work constitutes a first step towards using high speed imaging techniques notably stereo-DIC to identify the thermo-mechanical parameters of thermoplastic sheets in thermoforming conditions.

ACKNOWLEDGEMENTS

The authors acknowledge the European Regional Development Fund FEDER, the French state and the Hauts-de-France Region council for co-funding the ELSAT 2020 by CISIT project (POPCOM action).

REFERENCES

1. W. Leite, J. Campos Rubio and F. Mata Cabrera, *Polymers*. **10**, 143–60 (2018).

2. Y. Li, J.A. Nemes and A. Derdouri, *Polym Eng Sci.* **41**, 1399–1412 (2001).

3. E. Verron, G. Marckmann and B. Peseux, *Int. Journal for Num. Methods in Engineering*. **50**, 1233–51 (2001).

4. B. Van Mieghem, P. Lava and D. Debruyne, *Key Eng Mater.* **554–557**, 1583–91 (2013).

5. B. Van Mieghem, F. Desplentere, A. Van Bael and J. Ivens, *Express Polym. Letters* **9**, 119–28 (2015).

6. T. Pottier, P. Vacher and F. Toussaint, *Exp Mech.* **52**, 951–63 (2012).

7. M. Cali and L. Savio, “Accurate 3D reconstruction of a rubber membrane inflated during a Bulge Test to evaluate anisotropy”, in *Advances on Mechanics, Design Engineering and Manufacturing*, edited by B. Eynard (Springer, 2016), pp. 1221–31.

8. O. Rokoš, J. Hoefnagels, R. Peerlings and M. Geers, *Int J Solids Struct.* **146**, 241–59 (2018).

9. A. Ayadi, M-F. Lacrampe and P. Krawczak, *Int J Mater Form. Accepted* (2019).

10. E. Verron, R. Khayat, A. Derdouri and B. Peseux, *J Rheol.* **43**, 1083–97 (1999).

11. A. Ayadi, M-F. Lacrampe and P. Krawczak, “A comprehensive study of bubble inflation in vacuum- assisted thermoforming based on whole-field strain measurements”. In *ESAFORM-2018: AIP Conference Proceedings 1960*. (American Institute of Physics, Melville, NY, 2018), pp. 120004–10.

12. X. Shao, F. Zhu, D. Su and X. Dai, *Opt. Eng.* **57** (3), 034102 (2018).

13. XD. Ke, HW. Schreier, MA. Sutton and YQ. Wang, *Exp Mech.* **51**, 423–41 (2011).

14. W. Liu and R. Long, *J Appl Mech.* **83**, 011006–21, (2015).