# Pressure and folding effects on the buckling of a freestanding compressed thin film

Julien Durinck, Sami Hamade, Jean Grilhé, Jérôme Colin

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Julien Durinck, Sami Hamade, Jean Grilhé, Jérôme Colin. Pressure and folding effects on the buckling of a freestanding compressed thin film. Physical Review E, 2023, 107 (3), pp.035003. 10.1103/Phys-RevE.107.035003. hal-04233922

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Pressure and folding effects on the buckling of a free-standing compressed thin film

J. Durinck,¹ S. Hamade,² J. Grilhé,¹ and J. Colin¹

¹Institut P', UPR 3346 CNRS, Université de Poitiers-ENSMA-CNRS, France*

²Polytech'Lab, EA 7498, Université Nice Sophia Antipolis, France

(Dated: February 27, 2023)

The combined effects of a compressive stress, applied pressure and edge folding of a free-standing thin film have been theoretically investigated on the buckle morphologies of the structure. In the framework of the Föppl-von Kármán theory of thin plates, the different buckle profiles have been first analytically determined and two buckling regimes have been identified for the film, one regime where the transition from upward to downward buckling is continuous and one which is discontinuous (snap-through). The critical pressures characterizing the different regimes have been then determined and an hysteresis cycle has been identified through the study of the buckling versus pressure. The case where the thin film is deposited on a substrate has been also discussed.

# I. INTRODUCTION

A large number of micro-systems of various shapes and materials, are currently manufactured with the aim of accessing different functionalities, ranging from mechanical to electronical functionalities, including chemical and optical ones. In a large number of applications in the engineering field, the micro-fabricated devices, are composed of slender structures whose mechanical behavior must be understood and controlled. It is now well-admitted that structures under strain can undergo mechanical instabilities when geometrical non-linearities are considered. It corresponds to the film buckling, multilayered structures wrinkling and layer folding [1-7]. Significant efforts have therefore been made to characterize the threshold of these instabilities in terms of buckling pattern, shape and critical load [8-14], in order to control the mechanical behavior of the recently elaborated metamaterials [15-19]. For example, a numerical energy-based method has been developed to analyze the condition for manufacturing free-standing films when pre- and post-buckling conditions apply [20]. Likewise, the elastic snap-through phenomenon that has been identified for structures under various conditions such as mechanical loads [21] or capillarity forces [22, 23], has been used to control the viscous flow in a channel by means of an elastic arch [24].

In the field of materials science and metallurgy, it is also well-established that an internal compressive stress [25, 26] in the films can be the cause, when it exceeds a critical value, to interface delamination and thin film buckling leading to various patterns such as circular blisters, straight-sided and worm-like wrinkles (see [13, 14, 27-35] and references therein). The problem of blister folding has been also investigated and the configurational stability of the buckled patterns has been characterized [36]. Later, the folding of buckles in the case of Y₂O₃ and Au thin films deposited on Si substrates has been studied considering that the top side or the circumference of the buckles undergoes a plastic folding

[37-39]. This folding can be described in terms of low angle tilt-boundaries modeled using vertical rows of edge dislocations [40]. For Al thin films deposited on rigid substrates, molecular dynamic simulations at finite temperature have been then carried out and the role of the grain boundaries in the plasticity mechanisms leading to the film folding have been identified [41].

The effect of pressure has been also considered and the pressure-induced redeposition of the buckled film has been characterized [42]. More recently, the combined effects of plastic folding and atmospheric pressure have been considered through finite element simulations [43]. The formation of donut- and croissant-like patterns has been thus explained when the films are deposited on a substrate. Likewise, the pressure on the free-surface of Ta thin films deposited on a Si substrate has been found to be responsible for the shape transition from circular blisters to ring-shaped buckles, when blister diameters reach a critical value [44].

From the previously cited literature, it thus appears that the study of the combined effects of pressure and folding due to plasticity on the mechanical behavior of metallic layered materials is a topic of research that deserves attention. It is the purpose of the present work to investigate the combined effect of an applied pressure and edge folding in case of a clamped free-standing thin film under compressive stress. The possibility of upward and downward buckling has been then discussed as a function of the pressure and folding angle. The configuration where the film is also deposited on a substrate has been discussed.

# II. MODELING OF THE BUCKLING PROBLEM AND GOVERNING EQUATIONS

An initially planar thin film of thickness $h$, Young's modulus $E$ and Poisson's ratio $\nu$ is submitted to in-plane compressive stress $\sigma_{xx}^0 = \sigma_{yy}^0 = -\sigma_0$, with $\sigma_0 > 0$ (see Fig. 1(a) for axes). In a first time, it is assumed that each of the two rigid parts of the film is lying on a semi-infinite substrate assumed to be infinitely stiff, the free-standing part being of length $2b$, with $h/b \ll 1$. The

<footnote>* julien.durinck@univ-poitiers.fr</footnote>
<figure><img src="image_2.png" /><figcaption>FIG. 1. (Sketch not to scale) Thin film of thickness h deposited on a semi-infinite substrate submitted to an in-plane stress $\sigma_{xx} = \sigma_{yy} = -\sigma_0$. The free-standing part of the film of length 2b is submitted to an external applied pressure p. (a) Planar configuration. (b) Upward buckling of the free-standing film when the pressure is smaller than a critical value $\tilde{p}_c$. (c) Downward buckling when $\tilde{p} > \tilde{p}_c$.</figcaption></figure>

case where the film is deposited on a continuous substrate and delaminated on a 2b length will be discussed later. Depending on $\sigma_0$ values, the infinitely long free-standing film in the (Oy) direction may buckle in the (Oxz) plane [13]. It is thus considered that, once buckled, the free-standing part of the film undergoes an applied overpressure p on its upper free-surface corresponding to the mismatch between the pressures applying on its upper and lower free-surfaces. In the following, the cases of buckling in the upward direction in Fig. 1(b) or in the downward direction in Fig. 1(c) are considered, assuming that, due to plastic deformation, both extremities of the film are folded, in the upper part of the (Oxz) plane, of an angle $\{\mp\theta, \theta > 0\}$ with respect to the horizontal axes [13]. It is emphasized at this point that the study of the plasticity mechanisms at the microscopic scale responsible for this folding effect including defects such as dislocations and/or grain boundaries is beyond the scope of the present analysis [36–39, 43, 44]. Consequently, the values of the folding angle $\theta$ will be arbitrarily chosen in the following. Considering the plane strain hypothesis of the isotropic elasticity theory [45], the two components of the elastic displacements, only depending on x variable and labelled u and w along the (Ox) and (Oy) axes, respectively, satisfy in the framework of the Föppl-von Kármán (FvK) theory of thin plates [13], the following

set of Eqs.:

$$ \frac{d^4 w}{dx^4} - \frac{h}{D} \sigma_{xx} \frac{d^2 w}{dx^2} = \frac{p}{D}, \quad (1) $$

$$ u(\pm b) = 0, \quad (2) $$

$$ w(\pm b) = 0, \quad (3) $$

$$ \left. \frac{dw}{dx} \right|_{x=\pm 1} = \mp \tan \theta, \quad (4) $$

to which the strain-stress relation:

$$ \frac{1 - \nu^2}{E} (\sigma_0 + \sigma_{xx}) = \frac{du}{dx} + \frac{1}{2} \left( \frac{dw}{dx} \right)^2, \quad (5) $$

must be added, with $\sigma_{xx}$ the constant stress in the buckled film and $D = Eh^3/[12(1 - \nu^2)]$. Introducing the dimensionless parameters and variables $\tilde{h} = h/b$, $\tilde{\sigma}_{xx} = \frac{1-\nu^2}{E}\sigma_{xx}$, $\tilde{\sigma}_0 = \frac{1-\nu^2}{E}\sigma_0$, $\tilde{p} = \frac{1-\nu^2}{E}p$, $\alpha^2 = -\frac{12}{\tilde{h}^2}\tilde{\sigma}_{xx}$, $\alpha_0^2 = \frac{12}{\tilde{h}^2}\tilde{\sigma}_0$, $\tilde{x} = x/b$, $\tilde{u} = u/b$ and $\tilde{w} = w/b$, the system of Eqs. (1), (2), (3), (4), and (5) has been re-casted as:

$$ \frac{d^4 \tilde{w}}{d\tilde{x}^4} + \alpha^2 \frac{d^2 \tilde{w}}{d\tilde{x}^2} = \frac{12}{\tilde{h}^3} \tilde{p}, \quad (6) $$

$$ \tilde{u}(\pm 1) = 0, \quad (7) $$

$$ \tilde{w}(\pm 1) = 0, \quad (8) $$

$$ \left. \frac{d\tilde{w}}{d\tilde{x}} \right|_{\tilde{x}=\pm 1} = \mp \tan \theta, \quad (9) $$

$$ \frac{\tilde{h}^2}{12} (\alpha_0^2 - \alpha^2) = \frac{d\tilde{u}}{d\tilde{x}} + \frac{1}{2} \left( \frac{d\tilde{w}}{d\tilde{x}} \right)^2, \quad (10) $$

Solving this system of Eqs. (6), (7), (8), (9) and integrating Eq. (10) on the interval $\tilde{x} \in [-1, +1]$ lead to the following elastic solution:

$$ \tilde{w}(\tilde{x}) = \frac{1}{\alpha^3 \tilde{h}^3} \left\{ \begin{aligned} & 6\alpha \tilde{p}(\tilde{x}^2 - 1) \\ & + (\alpha^2 \tilde{h}^3 \tan \theta + 12\tilde{p}) \\ & \times (\cos \alpha \tilde{x} - \cos \alpha) \sin^{-1} \alpha \end{aligned} \right\}, \quad (11) $$

$$ \tilde{u}(\tilde{x}) = \frac{\tilde{h}^2}{12}(\alpha_0^2 - \alpha^2)\tilde{x} - \frac{1}{8\alpha^6\tilde{h}^6}\left\{ \begin{aligned} & 192\alpha^2\tilde{p}^2\tilde{x}^3 \\ & + (\alpha^2\tilde{h}^3 \tan \theta + 12\tilde{p}) \sin^{-1}\alpha (96\alpha\tilde{p}\tilde{x} \cos \alpha\tilde{x} \\ & - 96\tilde{p} \sin \alpha\tilde{x} + \alpha(\alpha^2\tilde{h}^3 \tan \theta + 12\tilde{p}) \sin^{-1}\alpha \\ & \times (2\alpha\tilde{x} - \sin 2\alpha\tilde{x})) \end{aligned} \right\}. \quad (12) $$

The parameter $\alpha$ related to the stress in the film, once it is buckled, is implicitly determined by Eq. (10) recasted as:

$$ \frac{\tilde{h}^2}{12}(\alpha_0^2 - \alpha^2) - \frac{1}{4\alpha^6\tilde{h}^6}\left\{ \begin{aligned} & -48\tilde{p}(\alpha^2(\tilde{h}^3 \tan \theta - 2\tilde{p}) \\ & + 12\tilde{p}) + \alpha(\alpha^2\tilde{h}^3 \tan \theta + 12\tilde{p})((36\tilde{p} - \alpha^2\tilde{h}^3 \tan \theta) \\ & \times \tan^{-1}\alpha + \alpha(\alpha^2\tilde{h}^3 \tan \theta + 12\tilde{p})\sin^{-2}\alpha) \end{aligned} \right\} = 0, \quad (13) $$

the smallest absolute value of $\alpha$ being the one selected, since it corresponds to the more relaxed configuration. It
has to be noted that, because of the imposed angle $\pm\theta$ at the extremities of the film, the relaxed buckle configuration can be found to be either in compression ($\tilde{\sigma}_{xx} < 0$ and $\alpha^2 > 0$) or in tension ($\tilde{\sigma}_{xx} > 0$ and $\alpha^2 < 0$). Thus, solutions have to be sought for $\alpha \in \mathbb{R}$ in compression and for $\alpha \in \mathbb{S}$ in tension.

From Eq. (11), the buckle profile parameter $\tilde{\delta} = \tilde{w}(0)$ defined as:

$$ \tilde{\delta} = \frac{-6\alpha\tilde{p} + (\alpha^2\tilde{h}^3 \tan\theta + 12\tilde{p}) \tan\frac{\alpha}{2}}{\alpha^3\tilde{h}^3}, \quad (14) $$

has been also considered in the following to characterize the thin film morphology.

### III. FREE-STANDING FILM BUCKLING

#### A. Buckled configurations with respect to pressure

Although no explicit analytical solution for $\alpha$ versus pressure could be obtained by the Authors from Eq. (13), an analytical expression of the pressure versus $\alpha$ can be on the other hand provided. More specifically, two solutions labelled $\tilde{p}_1$ and $\tilde{p}_2$ (with $\tilde{p}_1 < \tilde{p}_2 < 0$) have been explicitly derived from Eq. (13) but their expressions have not been displayed in this Paper for the sake of compacity. The pressures $\tilde{p}_1$ and $\tilde{p}_2$ have been plotted in Fig. 2 with respect to $\alpha$, for $\alpha_0 = 1, 2, 3$ and $4$. It is seen that, for $\alpha_0 = 1$, the solutions correspond to buckled configurations in tension ($\alpha^2 < 0$) for any pressure. For the other considered values of $\alpha_0$ (2, 3 and 4), the solutions can correspond to either buckled configurations in tension or in compression, depending on the value of the pressure. Another result highlighted by the Fig. 2 is that for $\alpha_0 = 1, 2$ and $3$, one pressure value corresponds only to one buckled configuration, i.e. one value of $\alpha$. On the contrary, for $\alpha_0 = 4$, when $\tilde{p}_1$ and $\tilde{p}_2$ take values in the range [$\tilde{p}_l, \tilde{p}_r$], three different buckled configurations can be found, underlining a possible snap-through between two stable configurations (the third one being unstable). Before studying the stability of the configurations which is required to fully understand the snap-through, the critical value $\alpha_{0c}$ at which the system transits from a stable behaviour ($\alpha_0 < \alpha_{0c}$) to an unstable one ($\alpha_0 > \alpha_{0c}$) is determined in the following paragraph.

The study of $\tilde{p}_1$ and $\tilde{p}_2$ variations versus $\alpha$ shows that there exists a particular value $\alpha^*$ such that $\tilde{p}_1 = \tilde{p}_2$ at $\alpha = \alpha^*$. In Fig. 2, they are labelled $\alpha_i^*$ for $\alpha_0 = i$ with $i = 1, 2, 3, 4$. When $0 < \alpha^* < \pi$ or when $\alpha^* \in \mathbb{S}$ (see $\alpha_1^*, \alpha_2^*$ and $\alpha_3^*$ in Fig. 2), it is found that the buckled configuration evolves in a continuous and stable way with the pressure. When $\alpha^* > \pi$ (see $\alpha_4^*$ in Fig. 2), a shape instability or snap-through from upward to downward buckling may occur for the film. The critical value $\alpha_{0c}$ is then given by the following expression determined by

<figure><img src="image_3.png" /><figcaption>FIG. 2. Evolution of pressures $\tilde{p}_1$ and $\tilde{p}_2$ with respect to $\alpha$ for different values $\alpha_0 = 1, 2, 3$ and $4$ and for $\theta = \pi/20$ and $\tilde{h} = 0.1$. The pressures corresponding to buckle configurations in tension (compression) are in dotted lines (solid lines).</figcaption></figure>

finding $\alpha_0$ which verifies $\alpha^* = \pi$:

$$ \alpha_{0c} = \sqrt{\pi^2 + 2\frac{\pi^2 - 6}{\tilde{h}^2\pi^2} \tan^2\theta}. \quad (15) $$

In the cases where $\alpha_0 > \alpha_{0c}$, it is found that the limits of $\tilde{p}_1$ and $\tilde{p}_2$ when $\alpha$ tends to $\pi$ are equal to $\tilde{p}_c$:

$$ \tilde{p}_c = \lim_{\alpha \to \pi} \tilde{p}_1 = \lim_{\alpha \to \pi} \tilde{p}_2 = -\frac{\pi^2}{12} \tilde{h}^3 \tan \theta. \quad (16) $$

To illustrate these statements, the profiles $\tilde{w}(\tilde{x} - \tilde{u}(\tilde{x}))$ versus $\tilde{x}$ have been plotted in Figs. 3(a), (c) and (e) for $\alpha_0 = 1, 2$ and $4$, respectively, and for increasing pressure $\tilde{p}$ (in absolute value) from top to bottom profiles, which can alternatively be either $\tilde{p}_1$ or $\tilde{p}_2$, with $\tilde{h} = 0.1$ and $\theta = \pi/20$. For such values of the thickness and the folding angle, the critical value $\alpha_{0c}$ beyond which it has been suggested that the system is unstable is equal to 3.44. It is underlined that the pressure variation on Figs. 3(a), (c) and (e) corresponds to $\alpha$ variations with a step equal to 0.05 ($0.05i$, with $i^2 = -1$) for buckled configurations in compression (in tension). It is observed in Figs. 3(a) and 3(c) that for $\alpha_0 = 1$ and $\alpha_0 = 2$ respectively, the transition from upward to downward buckling is continuous. This evolution has been then confirmed in Figs. 3(b), (d) and (f), where $\tilde{\delta}$ has been displayed versus $\tilde{p}$ for $\alpha_0 = 1, 2$ and $4$, with $\tilde{h} = 0.1$ and $\theta = \pi/20$. Indeed, in Figs. 3(b) and (d), the $\tilde{\delta}$ evolution from A to B is observed to be continuous for $\alpha_0 = 1$ and $\alpha_0 = 2$ as the pressure increases (in absolute value) and $\tilde{\delta}$ continuously decreases to zero and then goes under the horizontal plane as the film buckles downward. The main difference between both cases lies in the stress state of the buckles with respect to pressure: for $\alpha_0 = 1$, the
<figure><img src="image_4.png" /><figcaption>FIG. 3. Buckle evolution under pressure for $\theta = \pi/20$ and $\bar{h} = 0.1$. Buckle profile $\tilde{w}(\tilde{x} - \tilde{u}(\tilde{x}))$ versus $\tilde{x}$ for (a,c,e) increasing pressure $\tilde{p}$ (from top profile to bottom) and (b,d,f) evolution of $\tilde{\delta}$ versus $\tilde{p}$ for (a,b) $\alpha_0 = 1$, (c,d) $\alpha_0 = 2$ and (e,f) $\alpha_0 = 4$. Curves corresponding to pressures $\tilde{p}_1$ and $\tilde{p}_2$ (with $\tilde{p}_1 < \tilde{p}_2$) are respectively in blue (dark gray) and orange (light gray). Solid lines are used for buckled configurations in compression and dotted lines for those in tension. Dashed lines correspond to buckled configurations at pressures in the range $[\tilde{p}_l, \tilde{p}_r]$.</figcaption></figure>

buckle is in tension whatever the pressure, whereas, for $\alpha_0 = 2$, the buckle is in tension at low pressures, changes into compression at intermediate pressures and goes back to tension at higher pressures.

For $\alpha_0 = 4$, Fig. 3(f) shows that, for pressures in the range $[\tilde{p}_l, \tilde{p}_r]$, three different configurations labeled A, B and C are in equilibrium at the same pressure $\tilde{p}$. It is seen in Fig. 3(e) that A, B and C correspond to upward, downward and intermediate camel back buckle shapes, respectively. In order to study the stability of the three equilibria, the energy of the buckle configurations needs to be determined.

## B. Stability analysis of the equilibrium configurations

The total energy of the system is given by:

$$E_{\text{tot}} = E_s + E_b - \tilde{p}\tilde{V}, \quad (17)$$

where the two first terms $E_s$ and $E_b$ are the stretching and the bending energies, respectively, written as [46]:

$$E_s = \int_{-1}^{1} \frac{\tilde{h}}{2} \varepsilon_{xx}^2 d\tilde{x}, \quad (18)$$

$$E_b = \int_{-1}^{1} \frac{\tilde{h}^3}{24} \left( \frac{\partial^2 \tilde{w}}{\partial \tilde{x}^2} \right)^2 d\tilde{x}, \quad (19)$$

with $\tilde{u}$ and $\tilde{w}$ given by the Eqs. (2) and (3), respectively, and the strain $\varepsilon_{xx}$ in the buckle given by [13, 46]:

$$\varepsilon_{xx} = \varepsilon_0 + \frac{\partial \tilde{u}}{\partial \tilde{x}} + \frac{1}{2} \left( \frac{\partial \tilde{w}}{\partial \tilde{x}} \right)^2, \quad (20)$$

The third term in Eq. (17) represents the work of the pressure, with $\tilde{V}$ the volume below the buckle:

$$\tilde{V} = \int_{-1}^{1} \tilde{w}(\tilde{x} - \tilde{u}(\tilde{x})) d\tilde{x} \approx \int_{-1}^{1} \tilde{w}(\tilde{x}) d\tilde{x}. \quad (21)$$

The evolution of the energy $E_{\text{tot}}$ has been calculated from configuration A to B, passing through C. To do so, a path has been linearly interpolated between A, B and C in the configuration space such that:

$$\tilde{w}(x, \xi) = \begin{cases} -\xi \tilde{w}_A(x) + (1+\xi) \tilde{w}_C(x) & \text{for } \xi \le 0, \\ -\xi \tilde{w}_B(x) + (1-\xi) \tilde{w}_C(x) & \text{for } \xi > 0, \end{cases} \quad (22)$$

and:

$$\tilde{u}(x, \xi) = \begin{cases} -\xi \tilde{u}_A(x) + (1+\xi) \tilde{u}_C(x) & \text{for } \xi \le 0, \\ -\xi \tilde{u}_B(x) + (1-\xi) \tilde{u}_C(x) & \text{for } \xi > 0, \end{cases} \quad (23)$$

where the configurations A, C and B are found for $\xi = -1, 0$ and $1$, respectively. Using $\tilde{u}(x, \xi)$ and $\tilde{w}(x, \xi)$ in Eqs. (18), (19), (20) and (21), $E_{\text{tot}}(\xi)$ has been calculated for any value $\xi \in [-1.2, 1.2]$ and plotted in Fig. 4 for some pressures $\tilde{p} \in [\tilde{p}_l, \tilde{p}_r]$. It is seen that, for any pressure in the range $\tilde{p} \in [\tilde{p}_l, \tilde{p}_r]$, the configuration C is unstable (the energy is maximum) whereas the configurations A and B are stable. It is recalled that A and B are the only stable configurations for $|\tilde{p}| < |\tilde{p}_l|$ and for $|\tilde{p}| > |\tilde{p}_r|$, respectively. The energy of the configuration A is lower than that of the B for $|\tilde{p}_l| < |\tilde{p}| < |\tilde{p}_c|$ and this is the opposite for $|\tilde{p}_c| < |\tilde{p}| < |\tilde{p}_r|$, the energies of both configurations being equal when $|\tilde{p}| = |\tilde{p}_c|$. This suggests a hysteresis behavior as described in Fig. 5. At low pressures $|\tilde{p}|$, configuration A is the most stable configuration and $\tilde{\delta}$ continuously decreases as $|\tilde{p}|$ increases. At $\tilde{p} = \tilde{p}_c$, the energies of configurations A and B are equal and, as soon as $|\tilde{p}| > |\tilde{p}_c|$, the configuration B has the lower energy and A becomes metastable. The system
<figure><img src="image_5.png" /><figcaption>FIG. 4. Energy path between three buckle configurations A, B and C in the range pressure $\tilde{p} \in [\tilde{p}_l, \tilde{p}_r]$ for which the buckle can transit from the upward configuration A to the downward configuration B. The considered parameters are: $\alpha_0 = 4$, $\theta = \pi/20$ and $\tilde{h} = 0.1$.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>FIG. 5. Evolution of the maximum deflection $\tilde{\delta}$ with respect to the pressure $|\tilde{p}|$ for $\alpha_0 = 4$, $\theta = \pi/20$ and $\tilde{h} = 0.1$. The filled arrows correspond to the path from upward to downward buckling and the empty arrows to the path from downward to upward buckling.</figcaption></figure>

can remain in the metastable configuration A until the snap-through to B occurs at $\tilde{p} = \tilde{p}_r$, after the configuration B becomes the only stable configuration. The buckle is in configuration B and its maximum deflection $\tilde{\delta}$ continuously decreases as the pressure $|\tilde{p}|$ increases. From high pressure to low pressure, the configuration B is first the most stable configuration, then becomes metastable when $|\tilde{p}| < |\tilde{p}_c|$ and the snap-through from B to A finally occurs at $\tilde{p} = \tilde{p}_l$. Thus, the transitions from upward to downward buckling and from downward to forward buckling take place at two different pressures, exhibiting indeed a hysteresis behavior. It is worth noting that the two previously described regimes for the thin film buckling, i.e. the continuous and discontinuous transitions, have been also observed (but not shown) while considering different folding angles $\theta$ at constant initial stress $\alpha_0 > \alpha_{0c}$ (Eq. 15).

For convenience, the transition will be considered in the following to occur at the pressure $\tilde{p}_c$, i.e. when both upward and downward buckled configurations have the same energy.

## IV. STUDY OF THE BUCKLING BEHAVIOR WITH SUBSTRATE

The configuration where the film is delaminated on a length $2b$ after deposition on a continuous substrate has been also briefly addressed. Indeed, some information on the buckling state can be derived in this case from the free-standing analysis, considering the limit configuration where the snap-through or the continuous deposition leads to the situation where the buckle amplitude is zero, i.e. $\delta = 0$, such that the buckled film touches the

substrate under pressure. It is underlined that the full redeposition process of the film and the contact modeling with the substrate are beyond the scope of the present analysis. Consequently, two other critical pressures $\tilde{p}_0$ and $\tilde{p}_c^0$ can be defined for the film touching the substrate. The pressure $\tilde{p}_0$ corresponds to the case where $\tilde{\delta}$ goes to 0 without any abrupt transition as $\alpha \to 0$, i.e. when the film is fully released (without compression or tension). The other pressure $\tilde{p}_c^0$ is the pressure at which the buckle touches the substrate just after the abrupt transition, i.e. when $\alpha \to \pi$. It yields:

$$ \tilde{p}_0 = -\sqrt{\frac{35}{22}} \tilde{h}^4 \alpha_0, \qquad (24) $$

$$ \tilde{p}_c^0 = -\frac{1}{3} \pi^2 \tilde{h}^4 \sqrt{\frac{\alpha_0^2 - \pi^2}{3\pi^2 - 16}}. \qquad (25) $$

The critical pressure $\tilde{p}_c$ at which the snap-through phenomenon occurs can also be considered as a function of $\alpha_0$, without any consideration on the film position with respect to the substrate (contrary to $\tilde{p}_c^0$). Its expression is given by combining the equations (15) and (16):

$$ \tilde{p}_c = -\frac{1}{12} \pi^3 \tilde{h}^4 \sqrt{\frac{\alpha_0^2 - \pi^2}{2\pi^2 - 12}}. \qquad (26) $$

In Fig. 6(a), the thin film behavior diagram, only valid in the case where $\delta = 0$, has been displayed in the $(\alpha_0, |\tilde{p}|)$ plane for $\tilde{h} = 0.1$. The limits defined by the Eqns. (24), (25) and (26) split the diagram into four regions A, B, C and D. In region A, the deflection $\delta$ is equal to 0 as soon as the snap-through phenomenon from upward to downward buckling has taken place. In regions B and C, the
transition is continuous, the buckle being in compression in region B and in tension in region C for any value of $\theta$ when the condition $\tilde{\delta} = 0$ is reached. From this configuration where $\delta$ is zero, the free-standing film can then go from the upper part of the plane to the lower part with increasing pressure while in presence of a substrate, the redeposition should take place. The region D defines a region where the buckle can undergo an abrupt transition at $\tilde{p} = \tilde{p}_c$ and can reach $\tilde{\delta} = 0$ at a pressure $|\tilde{p}| > |\tilde{p}_c|$. In order to highlight the differences between A, B and D regions, the evolution of the deflection $\tilde{\delta}$ with respect to $|\tilde{p}|$ is plotted in Fig. 6(b) for $\tilde{h} = 0.1$, $\tan \theta = \pi/20$ and $\alpha_0 = 3.40, 3.45$ and $3.50$ corresponding to the regions B, D and A displayed in Fig. 6(a), respectively. It can be seen that for $\alpha_0 = 3.40$, the buckle touches the substrate ($\tilde{\delta} = 0$) after $\tilde{\delta}$ has continuously decreased with increasing $|\tilde{p}|$, whereas it undergoes an abrupt upward to downward transition at $|\tilde{p}_c| \approx 1.30$ for $\alpha_0 = 3.45$ and $3.50$. The contact between the buckle and the substrate occurs at $\tilde{p} = \tilde{p}_c$ for $\alpha_0 = 3.50$ and at $\tilde{p} > \tilde{p}_c$ for $\alpha_0 = 3.45$. It should be emphasized here that this analysis is valid only if the snap-through is assumed to occur at pressure $\tilde{p}_c$. In the case where the hysteresis behavior is taken into account, the region D can disappear under certain conditions of applied stresses $\alpha_0$ and folding angles $\theta$, making the diagram much more difficult to read.

Solving together Eqns. (13) and (14) for $\tilde{\delta} = 0$, it is possible to plot, in regions B and C of the $(\alpha_0, |\tilde{p}|)$ diagram, iso-$\theta$ curves for which the position of the buckle center reaches $z = 0$. Other iso-$\theta$ curves corresponding to the abrupt transition at $\tilde{p} = \tilde{p}_c$ with respect to $\alpha_0$ (see Eq. 16) have also been displayed in the region A of Fig. 6. To sum up, the transition from upward to downward buckling can be described by following iso-$\theta$ curves in the $(\alpha_0, |\tilde{p}|)$ diagram. Taking $\theta = \frac{\pi}{20}$ as an example, it can be seen that the buckle can redeposit at $z = 0$ in a tensile stress state (region C) for $\alpha_0 < 1.25$ and in a compressive stress state (region B) for $\alpha_0 > 1.25$, the state at $\alpha_0 = 1.25$ corresponding to a completely released buckle. The pressure $|\tilde{p}|$ required to maintain $\tilde{\delta} = 0$ decreases from 1.61 to 1.30 as the applied stress $\alpha_0$ increases from 0 to 3.44. For $3.44 < \alpha_0 < 3.46$, the buckle configuration is found in the region D (see the zoom in Fig. 6) where an abrupt transition occurs at $\tilde{p}_c \approx 1.30$ but $\tilde{\delta} = 0$ for a pressure $\tilde{p} > \tilde{p}_c$. Finally, for $\alpha_0 > 3.46$, the buckle configuration is found exclusively in region A where the transition from $\tilde{\delta} > 0$ to $\tilde{\delta} < 0$ occurs suddenly at $\tilde{p} = \tilde{p}_c$.

V. CONCLUSION

The study of the buckling of a free-standing thin film clamped at its edges has been investigated in the framework of the Föppl-von Kármán theory of thin plates when the film is submitted to a compressive stress and to an applied pressure on its upper free-surface. Assuming that due to the thin film plasticity including dislocations and/or grain boundaries, the film can undergo an up-

<figure><img src="image_7.png" /></figure>

FIG. 6. (a) $(\alpha_0, |\tilde{p}|)$ diagram describing the thin film behavior for $\tilde{h} = 0.1$. The straight black solid line is given by $|\tilde{p}_0|$ variation with respect to $\alpha_0$ (Eq. 24) and delimits the region C (buckles in a tensile stress state) from the region B (buckles in a compressive stress state), the buckle touching the plane $z = 0$ by continuously increasing the pressure. The dashed black curve, given by $|\tilde{p}_c|$ variation with respect to $\alpha_0$ (Eq. 26), delimits the region B (continuous transition) from the regions D and A (abrupt upward to downward transition). The dotted black curve, given by $|\tilde{p}_c^0|$ variation with respect to $\alpha_0$ (Eq. 25), delimits the region D, where the buckle touches the substrate after a snap-through for a pressure $|\tilde{p}| > |\tilde{p}_c|$, from the region A, where the buckle touches the substrate immediately after a snap-through at $|\tilde{p}| = |\tilde{p}_c|$. Dashed blue (dark gray) lines in the regions B, C and D are the iso-$\theta$ curves indicating the pressure $\tilde{p}$ required for the buckles to touch the plane $z = 0$ ($\tilde{\delta} = 0$) versus $\alpha_0$. Dashed orange (light gray) lines give the critical pressure $|\tilde{p}_c|$ (Eq. 16) at the abrupt transition in the regions A and D. (b) Evolution of $\tilde{\delta}$ with respect to $|\tilde{p}|$ for different values of the applied stress $\alpha_0 = 3.40, 3.45$ and $3.50$. The hysteresis is displayed with dotted lines.

ward folding at its edges, the buckling profile of the film has been determined and two different buckling regimes have been identified. Depending on the applied pressure and initial stress, the transition from upward to downward buckling can be either continuous or discontinuous. This transition (continuous or abrupt) has been theoretically characterized from a linear stability analysis as a
function of the different parameters of the problem such as the applied pressure, the compressive stress and the folding angle, focusing on the conditions required for the buckle to snap from upper to lower plane. An hysteresis cycle for the thin film buckling versus pressure has been also idenbtified and the strain state of the film (cmpres- sion, tension) has been discussed. As such, these films could be considered for the design of nano/micro-devices such as sensors sensitive to pressure and stress. When the film is deposited on a continuous substrate and delami- nated, the critical pressures for the redeposition onto the substrate (continuous or abrupt) have been also derived.

The next step of this work could be the study of
the post-buckling regime and the dynamics of the snap-

through. To do so, numerical approach could be used to
characterize this transition (continuous or abrupt) as a
function of the different parameters of the problem iden-
tified in the present work such as the applied pressure,
the compressive stress, and the folding angle. When the
film is lying on a continuous substrate, its full redeposi-
tion should be also investigated numerically.

ACKNOWLEDGMENTS

This work pertains to the French Government pro-
gram "Investissements d'Avenir" (EUR INTREE, refer-
ence ANR-18-EURE-0010).

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