Mechanical instability in the form of surface wrinkles is of growing interest in modern technologies involving soft materials in thin-film substrate systems [1,2]. Surface wrinkles commonly exist in both natural and synthetic systems, e.g., constrained swelling hydrogels under environmental stimuli (e.g., pH, humidity, and temperature) [3–6], dried fruits and vegetables [7,8], and anatomical structures [9,10]. The wrinkling behavior depends on material properties, geometric configurations, boundary conditions, and mechanical forces such as compression, shear, pre-strain, swelling, drying, shrinking, applied voltage, and growth [11–15]. Under uniaxial or biaxial compression, a film-substrate structure may deform into various surface morphologies, e.g., sinusoidal [16–20], checkerboard [21–24], hexagonal [22–25], herringbone [21,23,25,26], and labyrinth [21,26] patterns or period-doubling wrinkles and folds [27–31]. These complex features are a direct mechanical response when the deformation of the thin film reaches a critical level [32,33]. Predicting wrinkle inception and its evolution is important for applications in adhesion, wettability, and other functionalities [34–37]. Although wrinkle formation is traditionally considered an undesired surface deformation, in recent years, wrinkling has also emerged as a viable mechanism to improve device performance in advanced technology. Its controlled application is now key for the development of stretchable electronics [38–42], sensor development [43,44], smart medical devices [45–47], and functional surfaces [48–52] with tailored properties.

For structures consisting of a stiff thin film bonded to a compliant substrate, surface wrinkling is generally dictated by the minimization of the total potential energy. Wrinkle formation releases a part of the elastic strain energy in the film but increases that of the substrate. Therefore, both the mechanical properties of the film and substrate contribute to the onset of instability. The role of linear elastic behavior in the wrinkling of film-substrate systems has been explored extensively [36,37,53–55]. In general, uniaxial compression beyond a threshold triggers the occurrence of sinusoidal wrinkles. This paradigm was clearly demonstrated in early studies on metal and oxide films on pre-tensioned elastomeric substrates [56,57]. Upon relaxation of the pre-strain, elastic recovery of the substrate exerts a compressive stress state on the confined film. When it exceeds the critical buckling strength, the flat and uniform state becomes mechanically unstable and bifurcates into a periodic out-of-plane undulated low-energy state [58,59]. This simple concept has been developed into a metrological technique for the extraction of thin film modulus, adhesion energy, and fracture toughness [60]. The transition of wrinkles, for instance, from the sinusoidal mode to period-doubled, herringbone-like, or localized ridge forms, introduces further complexity involving the constitutive behavior of the film-substrate layers [61–63].

While numerous foundational studies have generated insights into instabilities in thin-film systems, a direct comparison of loading pathways has not been well studied. Jiang et al. [64] investigated the post-buckling behavior of thin film on a pre-tensioned and subsequently compressed substrate, analyzing the resulting changes in amplitude and wavelength of the wrinkles. Wang and Zhao [63] mapped out the instability phase diagram of extended systems by the pre-stretch/release technique. Arafat et al. [65] investigated the relationship between cracking, delamination, and buckling for thin metallic films on elastomers under stretching. Lipomi et al. [66] developed stretchable structures based on compliant electrical conductors using wrinkling for strain accommodation under compression. Sekitani et al. [67] developed printable elastic conductors using single-walled carbon nanotubes in rubber for high conductivity and stretchability under tension. Generally speaking, attention has usually been placed on the functionalities of the wrinkled state instead of a mechanistic side-by-side comparison between deformation pathways (compression-tension).

The conducting polymer poly(3,4-ethylenedioxythiophene): poly(styrenesulfonate) (PEDOT:PSS) thin film deposited on the substrate of poly(dimethylsiloxane) (PDMS) elastomer is an ideal model system for investigating the mechanical instabilities from the perspective of organic stretchable devices [66]. PEDOT:PSS is typically glassy and has a much higher elastic modulus compared to PDMS which is very soft and stretchable [68,69]. Such a large mechanical mismatch, with a modulus difference of several orders of magnitude, makes the system particularly prone to instability. Under compression, a well-adhered film can form ordered, reversible wrinkles. However, when stretched, the PEDOT:PSS film tends to form channel cracks, which can electrically isolate sections of the film and lead to device failure [65]. The interfacial adhesion energy is a dominant factor, governing whether the system relieves stress through conformal buckling, delamination, or film cracking [70,71]. Bi et al. [72] presented a numerical study on the dynamic wrinkling behavior of stiff film/compliant substrate structures, analyzing factors like pre-strain and substrate modulus on the system’s potential energy and dynamic response. Recent studies on the electromechanical coupling of such systems showed that, despite their reversible nature at moderate strains, mechanical instabilities can strongly affect electrical conductivity—improvement through wrinkling-driven alignment of conductive pathways or degradation by cracking [73,74]. Analytical as well as numerical models have been developed to predict these modes of buckling in such bilayers or tri-layers, thus establishing a strong connection between observed morphologies and underlying material properties [75,76]. The particular pattern that occurs is very sensitive to the stress state (e.g., biaxial or uniaxial) as well as geometric confinement of the system [77]. Shan et al. [78] investigated wrinkling deformation of square films subjected to in-plane torsion revealing multiple wrinkling configuration transitions during evolution expanding beyond compression-tension. Yang et al. [79] showed that wrinkle wavelength decreases with increasing pre-strain, and uniform wrinkling occurs only within a specific range of yield properties—too low a yield stress suppresses wrinkling, while too high a yield stress leads to non-uniform patterns.

An embedded imperfection technique was recently developed to simulate wrinkle formation within the finite element modeling framework [80–83]. The temporal wrinkling evolution can be captured in a straightforward manner, and the methodology has been employed to study complex surface patterns in three-dimensional settings [84,85]. In contrast to the well-established compressive route, wrinkle formation under tensile strain involves a more complex and less explored mechanical pathway. When a film-substrate system is stretched, the initial response to the stretching is a uniform expansion. Beyond a critical strain, however, the system can bifurcate, leading to a variety of instabilities including sinusoidal wrinkles, delamination blisters, and counter-intuitive wrinkle-like patterns that form along the stretching direction [65,86,87]. Gao et al. [88] investigated a new wrinkling system where tension creates localized wrinkles near the edge of films on PDMS, offering insights into how tension drives wrinkling through different mechanisms from compression. Qiu et al. [89] presented an energy-based theory for stretch-induced wrinkling in rectangular films, and introduced power-law shape functions and energy barrier concepts to describe wrinkle bifurcation and evolution. While the thin film is stretched in one direction, the mismatch of Poisson’s ratios between film and substrate may induce lateral compressive stress in the film, causing eventual wrinkle formation. The asymmetric behavior of thin films under compressive and tensile loading has not been systematically examined, and a unified modeling framework capturing both deformation modes is needed. The present work builds upon our previously developed embedded imperfection technique to numerically study the compression vs. tension-induced wrinkling instabilities, using the PEDOT:PSS thin film on PDMS substrate as a model system. It follows a continuum-based numerical approach using the finite element method, assuming a linear-elastic constitutive behavior. The specific objectives include:

To demonstrate that the numerical approach is capable of predicting the transformation of parallel wrinkles from prior compressive loading into a new set of parallel wrinkles in the perpendicular direction under reversed tensile loading, as observed in experiments.

In the remainder of this paper, we first describe experimental observations in Section 2, which is followed by the numerical model description in Section 3. Section 4 includes the simulations of wrinkle transitions following the loading sequence used in the experiment, along with relevant discussion. Section 5 presents parametric numerical analyses on the film properties, to contrast their different influences on wrinkling induced by compression and tension. Important findings are summarized in Section 6.

The film-on-substrate structure was produced by spin-coating a PEDOT: PSS thin film onto the pre-fabricated and pre-stretched PDMS substrate. Relaxing the pre-tension effectively compresses the film, and the critical state can be easily reached to trigger sinusoidal wrinkles. Subsequent pulling of the structure along the same direction essentially reduces the film compression and then moves further into the tensile regime. Fig. 1 shows the schematics based on the optical microscopic images of the wrinkles at different stages. In the relaxed state (most compressed), wrinkles are aligned in the direction perpendicular to the loading axis. As reversed deformation progresses, the compressive strain experienced by the film is reduced, and the wrinkles gradually flatten. A new set of wrinkles oriented along the tensile loading direction starts to take shape.

Although both the film and substrate are under horizontal tension at the later stages of the process, the greater tendency for PDMS to contract laterally (due to its higher Poisson’s ratio than that of PEDOT: PSS) exerts an increasing compressive strain in the film along the perpendicular direction. Wrinkling thus develops in response to this lateral compression. This type of wrinkle transition due to cyclic compression-tension loading has also been reported in other experimental studies [39]. The numerical modeling presented below aims to capture the same process through 3D finite element analysis.

The problem domain consists of a thin film above a compliant substrate with perfect bonding. As shown in Fig. 2, the film surface was square-shaped in the xz-plane. The film thickness is taken as 0.1 μm, and the substrate was sufficiently thick in comparison with the film (about 280 times thicker). The choice of dimensions in the x- and z-directions was guided by the theoretical critical wavelength under compression (see equation below), and they are taken as ten times the critical wavelengths to ensure a sufficient number of wrinkles once formed [85]. The computational domain serves as the 3D representative volume element of the larger periodic structure, minimizing boundary effects while maintaining computational efficiency. To help trigger instability, an embedded imperfection is included, which is a regular finite element at the center of the top-layer elements of the substrate (immediately below the film/substrate interface). Although this element is a part of the substrate, it is assigned the properties of the film material, so this arrangement may be viewed as having an interface that is not perfectly flat.

Fig. 2 also includes the boundary conditions. Two vertical side surfaces (an xy-plane and anyz-plane) are references planes, where out-of-plane movement is prohibited but tangential slides are allowed. Pushing and pulling displacements are imposed on the other yz-plane along the x-direction, with tangential slides allowed to simulate compressive and tensile loading, respectively. The remaining vertical side face,an xy-plane, can move in a parallel fashion and stays vertical and planar with tangential slides allowed. The top surface of the film and the bottom surface of the substrate are traction-free. These conditions ensure a representative volume element within a large periodic system along the x- and z-directions [90].

To establish a baseline and compare the numerical results with available analytical solutions, all materials are assumed to be isotropic and linearly elastic. The Young’s modulus (E_f) and Poisson’s ratio (ν_f) of the PEDOT:PSS film are 2000 MPa and 0.35, respectively [91]. For the PDMS substrate, its Young’s modulus (E_s) and Poisson’s ratio (ν_s) are 2.97 MPa and 0.495, respectively [91]. The simulations were carried out using the finite element program ABAQUS (Version 2022). The model employs 20-noded higher-order brick elements throughout, with a uniform mesh in the film layer consisting of four elements across its thickness. In the substrate, the mesh density was gradually reduced with depth to optimize computational efficiency while maintaining accuracy near the film-substrate interface. To ensure numerical convergence, an in-plane mesh density equivalent to ten elements per analytical sinusoidal wavelength is used, following established guidelines [84].

As described above, the overall model dimensions are scaled according to the theoretical critical wavelength corresponding to the primary sinusoidal buckling mode [59,92], (1)λcr=2πtf[(1−νs2)Ef3(1−νf2)Es]13

Note that this expression pertains only to compression following the plane strain assumption (or, using the current modeling scheme, no displacement allowed in the z-direction). Under the same condition, the critical stress at the onset of primary bifurcation, σ_cr, was analytically derived as [58,59,92] (2)σcr=[Ef4(1−νf2)][3(1−νf2)Es(1−νs2)Ef]23

Dividing Eq. (2) by the plane strain modulus of the thin film, Ef1−νf2, one obtains the critical wrinkling strain e_cr as (3)ecr=(14)[3(1−νf2)Es(1−νs2)Ef]23

Eq. (3) will be used below to validate our numerical models. There is no analytical solution for uniaxial compressive loading (non-plane-strain) and tensile loading. Predicting wrinkling instabilities therefore relies on numerical simulations.

Fig. 3 shows the overall load-strain curve obtained from the finite element modeling of uniaxial compression from the initial stress-free state. Three points are labeled along the curve, with the corresponding snapshots of the model configuration also shown in the figure. At point A, the film-substrate structure is flat, and it remains flat under increasing compression throughout the beginning linear portion of the load-strain curve. There is a distinct change in slope at the strain of 0.00842, which marks the onset of surface wrinkling as the system bifurcates from a state of uniform compression to a periodized, out-of-plane buckling mode. Point B in Fig. 3 is shortly after this instability, and it can be seen that the wrinkling pattern has started to form. The wavelength and the evolution of amplitude have been shown to agree with established analytical solutions in our previous work [84]. With a greater compressive strain at point C, well developed sinusoidal wrinkles are observed. Upon unloading from point C, the overall load-strain response is fully retraced following the same path. The wrinkles gradually disappear and the structure becomes flat again after entering the pre-instability regime. The three simulation images can thus represent unloading history as well.

Fig. 4 shows the overall load-strain cyclic curve, starting from an initial flat and stress-free state (point A) into compression, with subsequent unloading and reversed loading into tension, following the history of A-X-A-B-C as labeled along the curve. Note that the A-X-A portion is the same as in Fig. 3 but it is repeated here for completeness. The surface patterns corresponding to points X, A, B and C are also shown in the figure. From X back to A, the compression-induced wrinkles vanish. Further loading leads to elastic stretching of the structure. The greater Poisson contraction of the substrate imposes lateral compression on the film. Upon reaching a critical tensile strain of 0.1712, the second instability commences in response to this lateral compression. A new set of wrinkles in the perpendicular direction, aligning with the tensile axis, takes shape. From B to C the wrinkles grow more prominent with a greater amplitude.

The above results illustrate that the current modeling approach is able to capture the orientational transition of wrinkles due to loading in opposite directions, consistent with experimental observation [39]. Furthermore, the entire process is predicted using one smooth simulation run without any ad hoc treatment. The comparison of modeling with experiment is limited to qualitative features due to insufficient quantitative information of the actual measurement. It is also noted that the tensile strain to induce wrinkles seen in Fig. 4 is relatively large, which also implies nonlinear behavior in actual materials. The viscoelastic response, nonlinear elasticity (especially for the elastomeric substrate), plastic yielding of the film, and structural damage such as local interfacial delamination are all possible reasons contributing to the actual deformation behavior. The rate-dependent viscous effect will contribute significantly to relaxation and alter the stress state during both compression and tension. These will be left for future studies. The present work focuses on the straightforward elastic instability to demonstrate the model capability, establish fundamental characteristics of compression/tension wrinkle formation, and examine the effects of film/substrate elastic properties through parametric analyses.

A separate simulation is conducted, using only pure tensile loading directly from the initial stress-free state. The result is shown in Fig. 5 (as “pure tension”), along with the tensile portion of the load-strain curve in Fig. 4 (“tension after comparison”) for comparison. The two curves coincide over the entire strain range. In addition, in both cases, the tension-induced wrinkles start to develop at the same strain (0.1712), and the wrinkle patterns are also identical (as the deformation images of points B and C shown in Fig. 4). Therefore, the tensile instability is unaffected by the prior deformation history.

Numerical finite element analyses were conducted to simulate the wrinkling instabilities on the thin film-compliant substrate structure, induced by both compressive and tensile loading. The study is motivated by experimental observations that previously formed parallel wrinkles under compression gradually transform into a new set of parallel wrinkles in the perpendicular direction, as the structure is loaded in the reversed direction into tension. We demonstrate that this transition of wrinkle patterns during the compression-tension cycle can be obtained through the embedded imperfection technique using a single continuous simulation run. Wrinkling induced by tension is caused by the mismatch in Poisson contraction between the film and substrate, forcing the film into lateral compression, which can exceed the critical state as deformation continues. Due to the reversibility of elastic wrinkles, the histories of “tension after compression” and “pure tension” were found to generate the same overall deformation response. Parametric numerical analyses showed that an increasing film-to-substrate modulus ratio Ef/Es leads to easier wrinkle formation under compression, but an opposite trend is true for tension. Tension-induced wrinkles are also heavily influenced by the difference in Poisson’s ratio, with close values of νs and νf drastically increasing the difficulty of triggering instability. Compression-induced wrinkle formation, on the other hand, is insensitive to Poisson’s ratio. The present study assumes linear elastic material behavior and perfect bonding between the film and substrate. Effects such as nonlinear elasticity, viscoelasticity, plastic deformation, potential interfacial damage, as well as film cracking under various loading conditions are not considered. Incorporating these effects, as well as further investigations of the multiaxial loading effect leading to complex surface patterns, will be explored in future analyses.