Sinusoidal Wrinkles Research Articles

Plasticity-mediated deformation instabilities in thin film-compliant substrate systems: direct three-dimensional simulations

AbstractSurface wrinkles driven by mechanical instability commonly form in thin-film structures attached to a compliant substrate. In this study, a recently developed computational approach is employed to simulate the formation and transformation of wrinkles involving plastic yielding of the thin film. The three-dimensional (3D) finite element models contain an embedded imperfection at the film-substrate interface, serving to trigger the bifurcation modes. Successful application of this technique to allow for film plasticity is demonstrated, including the evolution of 3D surface patterns and their correlation with the overall load–displacement response. The simulations reveal that plastic yielding transforms the surface instability patterns into more localized forms. Under uniaxial loading, the sinusoidal elastic wrinkles undergo the wrinkle-to-fold transition. With equi-biaxial loading, the initial square-checkerboard array turns into continuous tall ridges along the 45° directions. In both loading modes, the plasticity-induced instability patterns are only partially relieved upon unloading, leaving permanent features on the surface.

Buckling of a Stiff Thin Film Embedded Between Two Compliant Substrates

Layered structures consisting of stiff thin films and compliant matrixes are widely observed in flexible electronics, geology and composite materials. In this work, through analytic modelling and numerical simulations, we investigate the buckling instability of a stiff thin film embedded between two compliant substrates under uniaxial compression. Considering the shear stresses at the film/substrate interfaces and the finite geometry change of the film, the critical compressive strain, buckling wavelength and amplitude of sinusoidal wrinkles are analytically derived by the energy method. The analytic predictions agree well with the results obtained by finite element analysis. The post-buckling morphology may be sinusoidal or multiple-period, depending on the compressive strain and the modulus ratio of two compliant substrates. Our results aid in understanding the buckling of compliant substrate/film/compliant substrate tri-layer structures.

Controlled Wrinkling Analysis of Buckled Thin Films on Gradient Elastic Substrate System: A Numerical Study

Continuous change in wrinkle patterns process of thin films to a gradient substrate is the most challenging problem regarding applying reliable and robust numerical methods for postbuckling analysis of the film/substrate system. For example, in the finite element method, the postbuckling simulation suffers from the convergence issue, while, in spectral methods, it is very difficult to capture the localized behavior in soft matters when the boundary conditions are complex. When a thin film is compressed, it can form a wrinkle of a certain amount of wavelength when the compression exceeds a critical value. The compressed compliant substrate system translates to sinusoidal wrinkles and then to period-doubling wrinkling after further compression. In this work, we investigate the mathematical model arising from the changing nature of wrinkle patterns of postbuckled thin films using a robust and efficient numerical algorithm based on the spectral method to evolve the wrinkle patterns. We consider the gradient substrates of three typical variations in the modulus, namely, the symmetry, exponential, and power-law model. It has been observed that the stable equilibrium path has two bifurcation points. At the first bifurcation point, the buckling instability of wrinkling occurs, while the period-doubling buckling instability occurs at the second bifurcation point. For the substrates of material gradients of various types, the amplitude and wavelength are obtained. This study may help in better understanding of wrinkle patterns formation which could be very useful for the designing of stretchable and flexible electronic devices of most substrate systems and to avoid resonance in the noise environment.

Direct numerical simulations of three-dimensional surface instability patterns in thin film-compliant substrate structures

A comprehensive numerical study of three-dimensional surface instability patterns is presented. The formation of wrinkles is a consequence of deformation instability when a thin film, bonded to a compliant substrate, is subject to in-plane compressive loading. We apply a recently developed computational approach to directly simulate complex surface wrinkling from pre-instability to post-instability in a straightforward manner, covering the entire biaxial loading spectrum from pure uniaxial to pure equi-biaxial compression. The simulations use embedded imperfections with perturbed material properties at the film-substrate interface. This approach not only triggers the first bifurcation mode but also activates subsequent post-buckling states, thus capable of predicting the temporal evolution of wrinkle patterns in one simulation run. The state of biaxiality is found to influence the surface pattern significantly, and each bifurcation mode can be traced back to certain abrupt changes in the overall load–displacement response. Our systematic study reveals how the loading condition dictates the formation of various instability modes including one-dimensional (1D) sinusoidal wrinkles, herringbone, labyrinth, and checkerboard.

Understanding and Controlling Hexagonal Patterns of Wrinkles in Neo-Hookean Elastic Bilayer Structures

A controlled surface wrinkling pattern has been widely used in diverse applications such as stretchable electronics, smart windows, and haptics. Here, we focus on hexagonal wrinkling patterns because of their great potentials in realizing anisotropic and tunable friction and serving as a dynamical template for making non-flat thin films through self-assembling processes. We employ large-scale finite element simulations of a bilayer neo-Hookean solid (e.g., a film bonded on a substrate) to explore mechanical principles that govern the formation of hexagonal wrinkling patterns and strategies for making nearly perfect hexagonal patterns. In our model, the wrinkling instabilities are driven by the confined film expansion. Our results indicate robust hexagonal patterns exist at a relatively small modulus mismatch (on the order of 10) between the film and substrate. Besides, the film expansion should not exceed the onset of wrinkling value too much to avoid post-buckling patterns. By harnessing the imperfection insensitivity of one dimension sinusoidal wrinkles, we apply a sequential loading to the bilayer structure to produce the nearly perfect hexagonal patterns. Lastly, we discuss the connection between the simple bilayer model and the gradient structures commonly existed in experiments.

A secure version of asymptotic numerical method via convergence acceleration

The paper deals with the numerical computation of difficult problems requiring many steps and many degrees of freedom such as the finite element analysis of wrinkling of film–substrate systems. The asymptotic numerical method (ANM) is well adapted to such computations but with a progressive loss of accuracy during step chaining. Thus, correction phases are necessary, which are rarely carried out within ANM. A convergence acceleration algorithm and a step-length adaptation have been included to limit the growth of computation time and to strengthen the reliability of the procedure. This modified version of the ANM is assessed by simulating the appearance and evolution of sinusoidal wrinkles under uniaxial compression.

Articulated surface wrinkling of a patterned film with periodic stiffness distribution on a compliant substrate

Articulated surface wrinkling patterns can be induced through the mechanical instability of patterned stiff films with periodic stiffness distributions (including periodic material and/or film thickness distributions) on a compliant substrate. To explore this articulated surface wrinkling, the governing equations are established by energy minimization, and the theoretical results are consistent with those from experiment and finite-element simulations. Morphology transitions from articulated wrinkling to trapezoidal or sine–trapezoid hybrid wrinkling are observed when the geometrical or material property parameters beyond a specific interval. Phase diagrams are established for the formation of articulated wrinkling morphology with respect to the geometrical and material parameters. The results show that articulated wrinkling is observed for a patterned film on a compliant substrate, depending on the periodic segment dimensions of the patterned film compared to the sinusoidal wrinkle wavelength of the corresponding homogeneous film. Compared to the sinusoidal wrinkling of a homogeneous film, articulated wrinkling has notably larger wrinkling wavelength and normal traction at the interface (approximately double when L1=L2), however, has slightly smaller amplitude and film strains (membrane and bending peak strains). This study is a good reference for designing articulated surfaces with potential applications in many areas, especially energy harvesting devices, because of the controlled deformation location and strain distribution of such surfaces.

Wrinkling and folding patterns in a confined ferrofluid droplet with an elastic interface.

A thin elastic membrane lying on a fluid substrate deviates from its flat geometry on lateral compression. The compressed membrane folds and wrinkles into many distinct morphologies. We study a magnetoelastic variant of such a problem where a viscous ferrofluid, surrounded by a nonmagnetic fluid, is subjected to a radial magnetic field in a Hele-Shaw cell. Elasticity comes into play when the fluids are brought into contact, and due to a chemical reaction, the interface separating them becomes a gel-like elastic layer. A perturbative linear stability theory is used to investigate how the combined action of magnetic and elastic forces can lead to the development of smooth, low-amplitude, sinusoidal wrinkles at the elastic interface. In addition, a nonperturbative vortex sheet approach is employed to examine the emergence of highly nonlinear, magnetically driven, wrinkling and folding equilibrium shape structures. A connection between the magnetoelastic shape solutions induced by a radial magnetic field and those produced by nonmagnetic means through centrifugal forces is also discussed.

Controlling nested wrinkle morphology through the boundary effect on narrow-band thin films

We describe the formation of nested wrinkles created by the thermal mismatch between a narrow-band thin film and a compliant substrate. When a film is described as “narrow-band”, it literally means that the film band width is much shorter than its length; more precisely, it means that the width is comparable with the wavelength of the wrinkles. A silicon mask was used during film sputtering to create narrow-band films on poly (dimethylsiloxane) substrate, thus creating regular boundaries to steer local stresses and control wrinkle morphology. Disordered nano-scale wrinkles were found nested within highly ordered micro-scale sinusoidal wrinkles. The formation of nested wrinkles was explained through the amplitude and wavelength saturation of nano-scale wrinkles. The disordered morphology of nano-scale wrinkles and the highly ordered morphology of micro-scale wrinkles were explained by using the boundary effect.

Wrinkling of a stiff film resting on a fiber-filled soft substrate and its potential application as tunable metamaterials

Mechanical self-assembly of ordered patterns via spontaneous buckling of thin-film/soft-substrate systems has received considerable interests in recent years. Here we study the wrinkling of a stiff film resting on a fiber-filled soft substrate. In particular, the effects of the cross-section dimension, spacing, and positions of fibers on the wrinkling patterns in the film/substrate bilayer system are investigated. We show that diverse wrinkling patterns, including sinusoidal wrinkling, period-doubling, period-tripling and mountain ridge modes, may occur at a small or moderate overall compression strain due to the inhomogeneous deformation in the substrate and they can be well controlled by tuning geometrical and physical parameters of the system. To illustrate the potential use of the wrinkling patterns revealed in this study, we investigate the elastic wave propagation in such a wrinkled bilayer using the Bloch wave theory. Our computational results show that diverse stress patterns generated in the soft composites give rise to a rich variety of band structures. Desired bandgaps of elastic waves can be achieved and tuned by simply designing the geometric parameters and controlling the external stimuli imposed on the soft metamaterials.

Global and local interactive buckling behavior of a stiff film/compliant substrate system

This paper elucidates the global and local interactive buckling behavior of a stiff film resting on a compliant substrate under uniaxial compression. The resulting governing non-linear equations (non-autonomous fourth-order ordinary differential nonlinear equations with integral conditions) are then solved by introducing a continuation algorithm, which offers considerable advantages to detect multiple bifurcations and trace a complex post-buckling path. The critical conditions for local and global buckling and respective post-buckling equilibrium paths are carefully studied. Two different evolution mechanisms of buckling modes and processes from destabilization to restabilization (snap-back) are observed beyond the onset of the primary sinusoidal wrinkling mode in the post-buckling range. In addition, the shear modulus of an orthotropic substrate acts as a dominant role in the bifurcation portrait. Our results offer better understanding of the global and local buckling behaviors of such a bilayer system, and can open up new opportunities for the design and applications of novel nanoelectronics.

Computational aspects of morphological instabilities using isogeometric analysis

Morphological instabilities play a crucial role in the behavior of living systems as well as advanced engineering applications. Such instabilities initiate when a thin stiff film on a compliant substrate is subject to compressive stresses. For bilayer systems, the first mode of instability is sinusoidal wrinkling. While the critical conditions to induce wrinkling are extensively studied, the more complex patterns formed beyond wrinkling remain elusive and poorly understood. The objective of this contribution is to establish a generic computational framework capable of capturing various instabilities, using isogeometric analysis (IGA) enhanced with a concurrent eigenvalue analysis. It is shown that the eigenvalue analysis provides quantitatively accurate predictions for the onset of instabilities. In addition, the results are compared with the standard finite element analysis (FEA) and it is clearly observed that IGA furnishes significantly more accurate results compared to FEA, for the same number of degrees of freedom. We believe that this generic framework is widely applicable to advance our understanding of emergence and evolution of morphological instabilities for a rich variety of applications in soft materials and living systems.

Wrinkling Measurement of the Mechanical Properties of Drying Salt Thin Films

We report a time-resolved approach to probe the mechanical properties of thin films during drying and solidification based on surface wrinkling. The approach is demonstrated by measuring the modulus of a ternary system comprising an inorganic salt (aluminum chlorohydrate), a humectant (glycerol), and water across the glassy film formation pathway. The topography of mechanically induced wrinkling of supported films on polydimethylsiloxane (PDMS) is experimentally monitored during mechanical extension and relaxation cycles. Nontrivial aspects of our method include the need to oxidize the (hydrophobic) PDMS surface prior to solution deposition to enable surface wetting, which simultaneously creates a glassy-layer skin, whose wrinkling can contribute to the overall topography. Film drying is studied as a function of solution concentration and time, and a range of pattern morphologies are found: sinusoidal wrinkling, transient double-wavelength wrinkling accompanying film "crust" formation, ridging associated with stress localization, and cracking. We quantify the evolution of the elastic modulus during the sinusoidal wrinkling stage, employing bi- and trilayer models, which are independently confirmed by nanoindentation. The method provides thus a simple and robust approach for the mechanical characterization of out-of-equilibrium thin films.

Wrinkling micropatterns regulated by a hard skin layer with a periodic stiffness distribution on a soft material

A wrinkling-based method is proposed to create various surface micropatterns regulated by a hardened skin layer with a periodic stiffness distribution on a soft material. It is shown that the surface patterns generated by this technique are controlled by three fundamental surface deformation modes that involve sinusoidal wrinkling, Euler buckling, and rigid rotation of the skin. Systematic experiments and a phase diagram validate the efficacy and robustness of the proposed method.

Controlling elastic wave propagation in a soft bilayer system via wrinkling-induced stress patterns.

Compression of a film/substrate bilayer system with different surface/interfacial structures can lead to diverse buckling patterns including sinusoidal wrinkles, ridges, folds, creases and tilted sawteeth wrinkles. In this paper, we show that elastic wave band gaps in the film/substrate bilayer system largely depend on the wrinkling patterns. More interestingly, we find that different wrinkling patterns investigated here can coexist and evolve in one bilayer system and the elastic wave propagation behaviors can be controlled by manipulating the hybrid wrinkling patterns. Our analysis also reveals that the periodic stress pattern plays a dominant role in tuning the bandgap structures in comparison to geometrical patterns caused by surface instability. A careful investigation of the transmission spectra of the composite systems has validated the main findings given by the analysis based on the Bloch wave theory. Potential use of the method and materials reported here to gain wide attenuation frequency ranges and the design of nesting Fibonacci superlattices have been demonstrated.

Bloch wave approach for the analysis of sequential bifurcations in bilayer structures

A wide variety of surface morphologies can be formed by compressing a bilayer comprising a thin film bonded to a compliant substrate. In particular, as the applied strain is increased, secondary instabilities are triggered and the initial sinusoidal wrinkles evolve into new complex patterns. Here, we propose a robust numerical analysis based on Floquet–Bloch boundary conditions to detect the primary and secondary instabilities triggered upon compression. Because the proposed method is based on unit cell simulations, it is computationally very efficient. Moreover, it accurately predicts not only the critical strains, but also the corresponding critical modes and their wavelengths, enabling us to follow the evolution of the surface morphology as the applied strain is progressively increased.

Primary and secondary instabilities in soft bilayered systems

AbstractWrinkling phenomena emerging from mechanical instabilities in inhomogeneously compressed soft bilayered systems can evoke a wide variety of surface morphologies. Applications range from undesired instabilities in engineering structures such as sandwich panels, via fabricating surfaces with controlled buckling patterns of unique properties, to wrinkling phenomena in living matter such as lungs, mucosas, and brain convolutions. While moderate compression evokes periodic sinusoidal wrinkles, higher compression induces secondary instabilities ‐ the surface bifurcates into increasingly complex morphologies. Periodic wrinkling has already been extensively studied, but the rich pattern formation in the highly nonlinear post‐buckling regime remains poorly understood. Here, we establish a computational model of differential growth to explore the evolving buckling pattern of a growing layer bonded to a non‐growing substrate. Our model provides a mechanistic understanding of growth‐induced primary and secondary instabilities. We show that amongst all possible secondary bifurcations, the mode of period‐doubling is energetically favorable. We experimentally validate our numerical results by examining buckling of a compressed polymer film on a soft foundation. Our computational studies have broad applications in the microfabrication of distinct surface patterns as well as in the morphogenesis of living systems, where growth is progressive and the formation of structural instabilities is critical to biological function. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Design of a Compact Biaxial Tensile Stage for Fabrication and Tuning of Complex Micro- and Nano-scale Wrinkle Patterns

Wrinkling of thin films is a strain-driven process that enables scalable and low-cost fabrication of periodic micro- and nano-scale patterns. In the past, single-period sinusoidal wrinkles have been applied for thin-film metrology and microfluidics applications. However, real-world adoption of this process beyond these specific applications is limited by the inability to predictively fabricate a variety of complex functional patterns. This is primarily due to the inability of current tools and techniques to provide the means for applying large, accurate, and nonequal biaxial strains. For example, the existing biaxial tensile stages are inappropriate because they are too large to fit within the vacuum chambers that are required for thin-film deposition/growth during wrinkling. Herein, we have designed a compact biaxial tensile stage that enables (i) applying large and accurate strains to elastomeric films and (ii) in situ visualization of wrinkle formation. This stage enables one to stretch a 37.5 mm long film by 33.5% with a strain resolution of 0.027% and maintains a registration accuracy of 7 μm over repeated registrations of the stage to a custom-assembled vision system. Herein, we also demonstrate the utility of the stage in (i) studying the wrinkling process and (ii) fabricating complex wrinkled patterns that are inaccessible via other techniques. Specifically, we demonstrate that (i) spatial nonuniformity in the patterns is limited to 6.5%, (ii) one-dimensional (1D) single-period wrinkles of nominal period 2.3 μm transition into the period-doubled mode when the compressive strain due to prestretch release of plasma-oxidized polydimethylsiloxane (PDMS) film exceeds ∼18%, and (iii) asymmetric two-dimensional (2D) wrinkles can be fabricated by tuning the strain state and/or the actuation path, i.e., the strain history. Thus, this tensile stage opens up the design space for fabricating and tuning complex wrinkled patterns and enables extracting empirical process knowledge via in situ visualization of wrinkle formation.

Large‐Area, Periodic, Hexagonal Wrinkles on Nanocrystalline Graphitic Film

Sinusoidal wrinkles develop in compressively stressed film as a means to release stored elastic energy. Here, a simple way to fabricate large‐area, periodic, hexagonal wrinkled pattern on nanocrystalline graphitic films grown on c‐plane sapphire (<50 nm thick) by the spontaneous delamination–buckling of the as‐grown film during cooling is reported. According to the continuum mechanics calculation, strain‐relief pattern adopting the hexagonal wrinkled pattern has a lower elastic energy than that of the telephone cord wrinkle at thickness regime below 50 nm. A high‐fidelity transfer method is developed to transfer the hexagonal wrinkled films onto arbitrary substrates. Nanoindentation studies show that hexagonal wrinkle film engineered this way may act as shock absorber. The hexagonal wrinkled carbon film is able to selectively promote the differentiation of human mesenchymal stem cell toward the osteogenic lineage in the absence of osteogenic inducing medium.

From period-doubling to folding in stiff film/soft substrate system: The role of substrate nonlinearity

Uniaxial compressed stiff films on soft substrates can evolve into the period-doubling and folding instabilities, beyond the onset of sinusoidal wrinkling. The substrate is modeled as a neo-Hookean solid with a pre-stretch prior to film attachment, and its nonlinearity is obtained. Both the pre-stretch and the external nominal strain imposed on the film/substrate system can induce different substrate nonlinearity, and thus have different effects on the post-buckling mode evolution of the system. This study shows that the critical strain of period-doubling instability is linear to the pre-stretch. As the compressive nominal strain increases, the folding mode occurs beyond the onset of period-doubling in both the pre-tension and the pre-compression case, due to the softening/hardening effects for the inward/outward displacements generated by the positive substrate nonlinearity.