Influence of nonlinearities on the accuracy of the analytical solution for the shaft loaded blister test

Finite element analysis (FEA) is employed to study the effects of nonlinearities on the accuracy of the analytical solution for the shaft loaded blister test. The FEA model was validated using constrained blister test measurements showing a good correlation between the experimental and the FEA data. The analytical solution is then compared with the energy release rate obtained from J-integral evaluation in the FEA. For small and large shaft displacements deviations larger than 20% are encountered which is explained with the violation of the membrane limit condition and the onset of plasticity for larger displacements, respectively. Simplifications of the analytical solution are discussed using a random sampling method and it is shown that the thickness ratio between film and substrate can be neglected for thin films on rigid substrates. Further, values for the angular quantity, ω, which is required to calculate the mode mix phase angle are tabulated for the case of thin, elastic films on stiff substrates using a crack surface opening displacement extrapolation method.     

BUCKLING ANALYSIS OF STRETCHABLE FERROELECTRIC THIN FILM ON ELASTOMERIC SUBSTRATES

The thin stiff films on pre-stretched compliant substrates can form wrinkles, which can be controlled in micro and nanoscale systems to generate smart structures. Recently, buck- led piezoelectric/ferroelectrie nanoribbons have been reported to show an enhancement in the piezoelectric effect and stretchability, which can be applied in energy harvesting devices, sensors and memory devices instead of polymeric polyvinylidine fluoride （PVDF）. This paper studies the buckling and post-buckling process of ferroelectric thin films bonded to the pre-stretched soft layer, which in turn lies on a rigid support. Nonlinear electromechanical equations for the buckling of thin piezoelectric plates are deduced and employed to model the ferroelectric film poled in the thickness direction. Two buckling modes are analyzed and discussed： partially de-adhered buck- ling and fully adhered buckling. Transition from one buckling mode to the other is predicted and the effect of piezoelectricity on the critical buckling condition of piezoelectric film is examined.     

Buckling of a stiff thin film on a pre-strained bi-layer substrate

Controlled buckling can impart stretchable mechanics to brittle materials when integrated as thin films on soft, elastomeric substrates. Typical elastomers are permeable to fluids, however, and therefor unable to provide robust barriers to entry of water, for instance, into devices built with the supported thin films. In addition, the mechanical strength of a system dominated by a soft substrate is often unsatisfactory for realistic applications. We show that introduction of a bi-layer substrate yields a robust, high strength system that maintains stretchable characteristics, with a soft layer on top of a relatively stiff layer in the substrate. As a mechanical protection, a soft encapsulation layer can be used on top of the device and the stretchability of the encapsulated system is smaller than that of the system without encapsulation. A simple, analytic model, validated by numerical analysis and FEA, is established for stiff thin films on a bi-layer substrate, and is useful to the design of stretchable systems.     

Buckling of thin gel strip under swelling

The buckling of thin gel film has attracted much attention due to its applications in the design of threedimensional structure from two-dimensional template. We have established an analytical model to study the swelling-induced buckling of a thin gel strip with one edge clampecd and the others free. The closed-form solutions for the amplitude and wavelength of the buckled shape are obtained by energy minimization of the total potential energy. The analytical results agree well with finite element analysis based on the inhomogeneous gel theory without any parameter fitting. The model provides a route to study complex postbuckling behaviors of thin gel films and guidelines to design the buckled configuration quantitatively by controlling the swelling ratio.     

Buckling and postbuckling of a compressed thin film bonded on a soft elastic layer: a three-dimensional analysis

The wrinkling of a stiff thin film bonded on a soft elastic layer and subjected to an applied or residual compressive stress is investigated in the present paper. A three-dimensional theoretical model is presented to predict the buckling and postbuckling behavior of the film. We obtained the analytical solutions for the critical buckling condition and the postbuckling morphology of the film. The effects of the thicknesses and elastic properties of the film and the soft layer on the characteristic wrinkling wavelength are examined. It is found that the critical wrinkling condition of the thin film is sensitive to the compressibility and thickness of the soft layer, and its wrinkling amplitude depends on the magnitude of the applied or residual in-plane stress. The bonding condition between the soft layer and the rigid substrate has a considerable influence on the buckling of the thin film, and the relative sliding at the interface tends to destabilize the system.     

Localized ridge wrinkling of stiff films on compliant substrates

Wrinkling of thin stiff films on thick compliant elastomeric substrates subject to plane strain compression is considered for cases in which the substrate is pre-stretched prior to film attachment. Advanced wrinkling modes are investigated that evolve as the systems are compressed beyond the onset of the primary sinusoidal wrinkling mode. If the substrate pre-stretch is greater than about 40%, an advanced mode in the form of a series of well-spaced ridges separated by relatively flat film is observed in the simulations. Our experiments reveal a localization mode in the form of alternating packets of large and small amplitude wrinkles, but not ridges, while ridge formation has been observed in other recent experiments. Measurements of undulation amplitudes have been made for wrinkle fields of stiff films formed by oxidation of the surface of pre-stretched PDMS substrates. Simulations have been performed with a finite element model and an analytical film/substrate model. The formation of the ridge mode is a consequence of the altered nonlinearity of the substrate produced by the pre-stretch. The role of the tangential substrate stiffness in suppressing localization at the ridges is also highlighted. If there is no substrate pre-stretch, or if the substrate is pre-compressed, the primary sinusoidal mode gives way to an entirely different sequence of advanced modes usually entailing period doubling followed by folding. The nature of substrate nonlinearity that leads to ridges or folds is discussed.     

Reprint of “Post-buckling analysis for the precisely controlled buckling of thin film encapsulated by elastomeric substrates” [In. J. Solids Struct. 45 (2008) 2014–2023]

The precisely controlled buckling of stiff thin films (e.g., Si or GaAs nano ribbons) on the patterned surface of elastomeric substrate (e.g., poly(dimethylsiloxane) (PDMS)) with periodic inactivated and activated regions was designed by Sun et al. [Sun, Y., Choi, W.M., Jiang, H., Huang, Y.Y., Rogers, J.A., 2006. Controlled buckling of semiconductor nanoribbons for stretchable electronics. Nature Nanotechnology 1, 201–207] for important applications of stretchable electronics. We have developed a post-buckling model based on the energy method for the precisely controlled buckling to study the system stretchability. The results agree with Sun et al.’s (2006) experiments without any parameter fitting, and the system can reach 120% stretchability.     

Post-buckling analysis for the precisely controlled buckling of thin film encapsulated by elastomeric substrates

The precisely controlled buckling of stiff thin films (e.g., Si or GaAs nano ribbons) on the patterned surface of elastomeric substrate (e.g., poly(dimethylsiloxane) (PDMS)) with periodic inactivated and activated regions was designed by Sun et al. [Sun, Y., Choi, W.M., Jiang, H., Huang, Y.Y., Rogers, J.A., 2006. Controlled buckling of semiconductor nanoribbons for stretchable electronics. Nature Nanotechnology 1, 201–207] for important applications of stretchable electronics. We have developed a post-buckling model based on the energy method for the precisely controlled buckling to study the system stretchability. The results agree with Sun et al.’s (2006) experiments without any parameter fitting, and the system can reach 120% stretchability.     

Deformability of thin metal films on elastomer substrates

Recent applications in flexible electronics require that thin metal films grown on elastomer substrates be deformable. However, how such laminates deform is poorly understood. While a freestanding metal film subject to tension will rupture at a small strain by undergoing a necking instability, we anticipate that a substrate will retard this instability to an extent that depends on the relative stiffness and thickness of the film and the substrate. Using a combination of a bifurcation analysis and finite element simulations, we identify three modes of tensile deformation. On a compliant elastomer, a metal film forms a neck and ruptures at a small strain close to that of a freestanding film. On a stiff elastomer, the metal film deforms uniformly to large strains. On an elastomer of intermediate compliance, the metal film forms multiple necks, deforms much beyond the initial bifurcation, and ruptures at a large strain. Our theoretical predictions call for new experiments.     

Buckling of a stiff thin film on a compliant substrate in large deformation

A finite-deformation theory is developed to study the mechanics of thin buckled films on compliant substrates. Perturbation analysis is performed for this highly nonlinear system to obtain the analytical solution. The results agree well with experiments and finite element analysis in wavelength and amplitude. In particular, it is found that the wavelength depends on the strain. Based on the accurate wavelength and amplitude, the membrane and peak strains in thin films, and stretchability and compressibility of the system are also obtained analytically.     

Controlled buckling of thin film on elastomeric substrate in large deformation

Electronic systems with large stretchability have many applications. A precisely controlled buckling strategy to increase the stretchability has been demonstrated by combining lithographically patterned surface bonding chemistry and a buckling process. The buckled geometry was assumed to have a sinusoidal form, which may result in errors to determine the strains in the film. A theoretical model is presented in this letter to study the mechanics of this type of thin film/substrate system by discarding the assumption of sinusoidal buckling geometry. It is shown that the previous model overestimates the deflection and curvature in the thin film. The results from the model agree well with finite element simulations and therefore provide design guidelines in many applications ranging from stretchable electronics to micro/nano scale surface patterning and precision metrology.     

Exact analytical solutions for the local and global buckling of sandwich beam-columns under various loadings

Sandwich structures are widely used in many industrial applications, due to the attractive combination of a lightweight and strong mechanical properties. This compromise is realized thanks to the presence of different parts in the composite material, namely the skins and possibly core reinforcements or thin-walled core structure which are both thin/slender and stiff relative to the other parts, namely the homogeneous core material, if any. The buckling phenomenon thus becomes mainly responsible for the final collapse of such sandwiches. In this paper, classical sandwich beam-columns (with homogeneous core materials) are considered and elastic buckling analyses are performed in order to derive the critical values and the associated bifurcation modes under various loadings (compression and pure bending). The two faces are represented by Euler–Bernoulli beams, whereas the core material is considered as a 2D continuous solid. A set of partial differential equations is first obtained from a general bifurcation analysis, using the above assumptions. Original closed-form analytical solutions of the critical loading and mode of a sandwich beam-column are then derived for various loading conditions. Finally, the proposed analytical formulae are validated using 2D linearized buckling finite element computations, and parametric analyses are performed.     

Buckling strength analysis of adhesively bonded aluminum hat sections

This paper provides an analytical solution for the critical buckling stress of adhesively bonded aluminum hat sections under static axial compression. The governing rectangular plate member of the structure is treated based on the differential equation for out-of-plane deflections of thin plates. Finite element eigenvalue buckling analysis is performed to verify the assumed simply supported boundary conditions for common edges between adjacent plate elements. Elastic restraint is applied to the two loaded edges of the rectangular plate, and the relative critical buckling stress is computed according to the transcendental equations. It is found from experiments that there is no adhesive bonding failure in the elastic buckling stage. The analytical solution yields buckling stress predictions which are in reasonable agreement with measured values.     

Experimental and theoretical study of the buckling of narrow thin plates on an elastic foundation under compression

The paper describes the processes of elastic deformation of thin films under mechanical loading. The film is modeled longitudinally by a compressed plate arranged on an elastic foundation. A computer model of the buckling of the narrow thin plate with a delamination portion located on an elastic foundation is constructed. This paper also touches upon the supercritical behavior of the plate–substrate system. The experiments on the axial compression of a metal strip adhered to a rubber plate are performed, and 2 to 7 buckling modes are obtained therein. The critical loads and buckling modes obtained in the numerical calculations are compared with the experimental data. It is shown that there is the possibility of progressive delamination of the metal plate from the foundation if the critical load is exceeded. It is found that the use of the proposed approach, which, in contrast to other approaches, accounts for the elastic deformation of the substrate, causes the dependence between the critical bending stress and the stiffness of the foundation.     

Determining the Elastic Modulus of Compliant Thin Films Supported on Substrates from Flat Punch Indentation Measurements

Instrumented indentation is a technique that can be used to measure the elastic properties of soft thin films supported on stiffer substrates, including polymer films, cellulosic sheets, and thin layers of biological materials. When measuring thin film properties using indentation, the effect of the substrate must be considered. Most existing models for determining the properties of thin films from indentation measurements were developed for metal and dielectric films bonded to semiconductor substrates and have been applied to systems with film-substrate modulus ratios between 0.1 and 10. In the present work, flat punch indentation of a thin film either bonded to or in contact with a substrate is examined using finite element modeling. A broad range of film-substrate modulus ratios from 0.0001 to 1 are investigated. As the substrate is effectively rigid compared to the film when the film-substrate modulus ratio is less than 0.0001, the results are also useful for understanding systems with lower film-substrate modulus ratios. The effects of the contact radius, film thickness, elastic properties, and friction between the film and the substrate on the measured stiffness were quantified using finite element modeling in order to understand how the elastic properties of the film can be extracted from indentation measurements. A semi-analytical model was developed to describe the finite element modeling results and facilitate the use of the results to analyze experimental measurements. The model was validated through analysis of indentation measurements of thin polyethylene sheets that were supported on substrates of various stiffness.     

Buckling of delaminated bi-layer beam-columns

An improved analytical model is presented to analyze the delamination buckling of a bi-layer beam-column with a through-the-width delamination. Both the transverse shear deformation and local delamination tip deformations are taken into consideration, and two delaminated sub-layers as well as two substrates in the intact (un-delaminated) regions are modeled as individual Timoshenko beams. A deformable interface is introduced to establish the continuity condition between the two substrates in the intact regions. Consequently, a flexible joint is formed at the delamination tip, and it is different from the conventional rigid joint given in most of studies in the literature, in which the local delamination tip deformations are completely ignored. In contrast to the local delamination buckling in our previous study (Qiao et al., 2010), the present model accounts for the global deformations of the intact region in the delaminated composite beam-column, thus capable of capturing the buckling mode shape transitions from the global, to global–local coexistent, and to local buckling for asymmetric delamination as the interface delamination increases. Good agreement of the present analytical solutions with the full 2-D elastic finite element analysis demonstrates the local deformation effects around the delamination tip and verifies the accuracy of the present model. Parametric studies are conducted to investigate the effects of loading eccentricity, delaminated sub-layer thickness ratio, and interface compliance on the critical buckling load for the delaminated composite beam-column. Transitions of buckling modes from the global to local delamination buckling are also disclosed as the thickness of one sub-layer reduces from the thick sub-layer to a thin film. The developed delamination buckling solution facilitates the design analysis and optimization of laminated composite structures, and it can be used with confidence in buckling analysis of delaminated composite structures.     

Nonlinear analyses of wrinkles in a film bonded to a compliant substrate

Subject to a compressive membrane force, a film bonded to a compliant substrate often forms a pattern of wrinkles. This paper studies such wrinkles in a layered structure used in several recent experiments. The structure comprises a stiff film bonded to a compliant substrate, which in turn is bonded to a rigid support. Two types of analyses are performed. First, for sinusoidal wrinkles, by minimizing energy, we obtain the wavelength and the amplitude of the wrinkles for substrates of various moduli and thicknesses. Second, we develop a method to simultaneously evolve the two-dimensional pattern in the film and the three-dimensional elastic field in the substrate. The simulations show that the wrinkles can evolve into stripes, labyrinths, or herringbones, depending on the anisotropy of the membrane forces. Statistical averages of the amplitude and wavelength of wrinkles of various patterns correlate well with the analytical solution of the sinusoidal wrinkles.     

Inhomogeneous large deformation study of temperature-sensitive hydrogel

In this paper, inhomogeneous deformation of a temperature-sensitive hydrogel has been studied and analyzed under arbitrary geometric and boundary conditions. We present the governing equations and equilibrium conditions of an isothermal process based on the monophase gel field theory of hydrogel via a variational approach. We have adopted and implemented an explicit form of energy for temperature-sensitive hydrogel in a three-dimensional finite element method (FEM) using a user-supply subroutine in ABAQUS. For verification purpose, a few numerical results obtained by the proposed approach are compared with existing experimental data and analytical solutions. They are all in good agreement. We also provide several examples to show the possible applications of the proposed method to explain various complex phenomena, including the bifurcation, buckling of membrane, buckling of thin film on compliant substrate and the opening and closure of flowers.     

超弹性薄膜与可压缩基底双层结构表面失稳分析

当上层超弹性硬质薄膜和下层可膨胀基底构成的双层结构受压时,薄膜的自由表面可通过形成褶皱降低系统能量.研究表明,上下两层的模量比不同时,上层弹性硬质薄膜将表现出不同的表面失稳模式.本文提出了一种新颖的方法可有效抑制双层软材料的表面失稳,即改变基底材料的泊松比,这种方法同时适用于不具有应变硬化的软材料.首先基于Neo-Hookean模型发展了小变形条件下双层结构表面失稳的理论模型,通过半解析的方法得到了表面失稳的临界应变;然后通过有限元计算与模拟,进一步验证了负泊松比基底可延缓表面失稳.结果表明:(1)当双层结构基底泊松比为正且趋于0.5(不可压缩)时,双层结构在较小的压缩应变下出现表面失稳;(2)当基底的泊松比为负且趋于-1时,可被压缩至46%而不出现表面失稳,即可膨胀基底能有效抑制薄膜的表面失稳.本文发展的方法及主要结果可为延展性电子器件的设计提供指导.     

Surfactant spreading on thin viscous films: film thickness evolution and periodic wall stretch

Surfactant spreading on thin viscous films is of interest in the context of surfactant and liquid transport in the lungs, for both normal lung function and treatment of disease, as well as for many industrial processes. This paper presents experimental techniques for the measurement of film deformations due to spreading surfactant and for the investigation of the effects of periodic stretching of the wall supporting the thin film to mimic airway wall motion in the lung due to breathing. Additionally, we present results from both types of experiments, which agree favorably with our theoretical work.