Experiment and evaluation of wrinkling strain in a corner tensioned square membrane

Prediction of wrinkling characteristics is strongly correlated with the strain perpendicular to wrinkling direc- tion. In this paper, the strain field of wrinkled membrane is tested by VIC-3D system based on the digital image correlation technique. Experimental results are validated by the tension wrinkling simulation. The experimental strain perpendicular to wrinkling direction is analyzed in depth. The wrinkling strain of a square wrinkled membrane under corner tension is extracted from experimental strain perpendicular to wrinkling direction. A quantitative characterization format of the experimental wrinkling strain is proposed. A modified prediction method of wrinkling amplitude is presented based on the experimental wrinkling strain. The re- sults show that the precision of modified prediction model has improved 13.2% compared with the classical prediction model. The results reveal that the modified model can give an accurate prediction of the wrinkling amplitude.     

New nonlinear multi-scale models for wrinkled membranes

A new macroscopic approach to the modelling of membrane wrinkling is presented. Most of the studies of the literature about membrane behaviour are macroscopic and phenomenological, the influence of wrinkles being accounted for by nonlinear constitutive laws without compressive stiffness. The present method is multi-scale and it permits to predict the wavelength and the spatial distribution of wrinkling amplitude. It belongs to the family of Landau–Ginzburg bifurcation equations and especially relies on the technique of Fourier series with slowly varying coefficients. The result is a new family of macroscopic membrane models that are deduced from Föppl–von Kármán plate equations. Numerical solutions are presented, giving the size of the wrinkles as a function of the applied compressive and tensile stresses.     

Constitutive analysis of thin biological membranes with application to radial stretching of a hollow circular membrane

The constitutive analysis of the mechanical response of thin elastic membranes under inplane deformation is presented by using the multiplicative decomposition of the deformation gradient into its areal and distortional parts. Specific results are obtained for the Evans-Skalak form of the strain energy function. The solution to the problem of radial stretching of a hollow circular membrane obeying this constitutive model is then derived. The stress concentration factor is determined as a function of the relative hole size and the magnitude of the applied tension. The tension boundary is identified above which no compressive stress appears in the membrane. The limit boundary is introduced below which the membrane cannot support the applied loading without unstable wrinkling. For the loading between the tension and the limit boundary, nonuniformly distributed infinitesimal wrinkles appear within the inner portion of the membrane, carrying radial tension but no circumferential stress (tension field). The specific form of the strain energy function is used to describe this behavior, and to calculate the amount of the membrane area absorbed by infinitesimal wrinkles. The wrinkled portion is surrounded by the outer portion of the membrane carrying both radial and circumferential stresses. The limit boundary is reached when wrinkles spread throughout the membrane. It is shown that for a sufficiently large tension at the outer boundary, the wrinkling does not spread throughout the membrane no matter how large the applied tension at the inner boundary of the membrane is, provided that no rupture takes place. The limiting extent of the tension field in such cases is calculated. The linearized version of the analysis is characterized by a closed form solution.     

A new model for wrinkling and collapse analysis of membrane inflated beam

A new model is proposed to accurately predict the wrinkling and collapse loads of a membrane inflated beam. In this model, the pressure effects are considered and a modified factor is introduced to obtain an accurate prediction. The former is achieved by modifying the pressure-related structural parameters based on elastic small strain considerations, and the modified factor is determined by our test data. Compared with previous models and our test data, the present model, named as shell-membrane model, can accurately predict the wrinkling and collapse loads of membrane inflated beams.     

Asymptotic limits and wrinkling patterns in a pressurised shallow spherical cap

The asymmetric bifurcation problem for a shallow spherical cap is examined. The applied pressure can act either external or internal to the cap and both cases are treated here. Assuming a non-linear axisymmetric basic state, the linearised bifurcation equations for the pressurised shell are investigated in the limit when the thickness of the cap is much less than the maximum rise of the shell mid-surface. Within this regime the wrinkling patterns in both cases are confined to a narrow zone near the edge of the shell, making it possible to solve asymptotically the corresponding equations and derive analytical predictions for both the critical pressure and the corresponding number of wrinkles. Some comparisons with direct numerical simulations are included as well.     

Stability boundaries for wrinkling in highly stretched elastic sheets

We determine stability boundaries for the wrinkling of highly unidirectionally stretched, finely thin, rectangular elastic sheets. For a given fine thickness and length, a stability boundary here is a curve in the parameter plane, aspect ratio vs. the macroscopic strain; the values on one side of the boundary are associated with stable unwrinkled (flat) states, while stable wrinkled configurations correspond to all values on the other. In our recent work we demonstrated the importance of finite elasticity in the membrane part of such a model in order to capture the correct phenomena. Here we present and compare results for four distinct models: (i) the popular Föppl–von Kármán plate model (FvK), (ii) a correction of the latter, used in our earlier work, in which the approximate 2D Föppl strain tensor is replaced by the exact Green strain tensor, (iii) and (iv): effective 2D finite-elasticity membrane models based on 3D incompressible neo-Hookean and Mooney–Rivlin materials, respectively. In particular, (iii) and (iv) are superior models for elastomers. The 2D nonlinear, hyperelastic models (ii)–(iv) all incorporate the same quadratic bending energy used in FvK. Our results illuminate serious shortcomings of the latter in this problem, while also pointing to inaccuracies of model (ii) – in spite of yielding the correct qualitative phenomena in our earlier work. In each of these, the shortcoming is a due to a deficiency of the membrane part of the model.     

Analyses of wrinkled and slack membranes through an error reproducing mesh-free method

A mesh-free approximation of large deformations of flexible membrane structures within the tension field theory is considered in this paper. A modification of the wrinkling theory, originally proposed by Roddeman et al. (1987) [Roddeman, D.G., Drukker, J., Oomens, C.W.J., Janssen, J.D., 1987, The wrinkling of thin membranes: Part I—theory; Part II—numerical analysis. ASME J. Appl. Mech. 54, 884–892.], is proposed to study the behaviour of an isotropic membrane under the mixed state of stress (taut, wrinkled and slack). Using the facts that the state of stress is not uniform across an element and that the deformation gradient is a spatially continuous (and possibly non-differentiable) tensor, the proposed model uses a continuously modified deformation gradient to capture the location and orientation of wrinkles more precisely. While the deformation gradient need not be everywhere-differentiable in a wrinkled membrane, it is argued that the fictive non-wrinkled (non-slack) surface may be looked upon as an everywhere-taut surface in the limit as the minor (and major) principal tensile stresses over the wrinkled (slack) portions go to zero. Accordingly, the modified deformation gradient is thought of as the limit of a sequence of everywhere-differentiable tensors. The weighted residual from the governing equations are presently solved via a mesh-free method, where the entire domain is discretized only by a set of grid points. A non-uniform-rational-B-spline (NURBS) based error reproducing kernel method (ERKM) has been used to approximate the field variable over the domain. The first step in the method is to approximate a function and its derivatives through NURBS basis functions. However, since NURBS functions neither reproduce any polynomial nor interpolate the grid points (also referred to as control or nodal points), the approximated functions result in uncontrolled errors over the domain including the grid points. Accordingly the error functions in the NURBS approximation and its derivatives are reproduced via a family of non-NURBS basis functions. The non-NURBS basis functions are constructed using a polynomial reproduction condition and added to the NURBS approximation of the function obtained in the first step. Several numerical examples on wrinkled and/or slack membranes are also provided.     

New spatial curved beam and cylindrical shell elements of gradient-deficient Absolute Nodal Coordinate Formulation

Based on previous studies, a new spatial curved slender-beam finite element and a new cylindrical shell finite element are proposed in the frame of gradient-deficient Absolute Nodal Coordinate Formulation (ANCF). The strain energy of the beam element is derived by using the definition of the Green?CLagrange strain tensor in continuum mechanics so that the assumption on small strain can be relaxed. By using the differential geometry and the continuum mechanics, the angle between two base vectors of a defined local coordinate frame of the cylindrical shell element is introduced into the strain energy formulations. Therefore, the new shell element can be used to model parallelogram shells. The analytical formulations of elastic forces and their Jacobian for the above two finite elements of gradient-deficient ANCF are also derived via the skills of tensor analysis. The generalized-alpha method is used to solve the huge set of system equations. Finally, four case studies including both static and dynamic problems are given to validate the proposed beam and cylindrical shell elements of gradient-deficient ANCF.     

A comparative analysis of numerical approaches to the mechanics of elastic sheets

Numerically simulating deformations in thin elastic sheets is a challenging problem in computational mechanics due to destabilizing compressive stresses that result in wrinkling. Determining the location, structure, and evolution of wrinkles in these problems has important implications in design and is an area of increasing interest in the fields of physics and engineering. In this work, several numerical approaches previously proposed to model equilibrium deformations in thin elastic sheets are compared. These include standard finite element-based static post-buckling approaches as well as a recently proposed method based on dynamic relaxation, which are applied to the problem of an annular sheet with opposed tractions where wrinkling is a key feature. Numerical solutions are compared to analytic predictions of the ground state, enabling a quantitative evaluation of the predictive power of the various methods. Results indicate that static finite element approaches produce local minima that are highly sensitive to initial imperfections, relying on a priori knowledge of the equilibrium wrinkling pattern to generate optimal results. In contrast, dynamic relaxation is much less sensitive to initial imperfections and can generate low-energy solutions for a wide variety of loading conditions without requiring knowledge of the equilibrium solution beforehand.     

薄膜拉伸褶皱失稳力学进展

薄膜拉伸起皱现象在自然界和现代工业中普遍存在,在过去二十年里,引发了学者们极大的研究兴趣.这种起皱失稳行为源自薄膜能与弯曲能之间的非线性竞争.我们回顾从本世纪初至今薄膜拉伸褶皱失稳力学研究进展,将其分为两个阶段:前十年的研究主要局限于薄膜小应变(～1%)起皱现象,而后十年的兴趣主要集中在有限应变起皱-再稳定(孤立中心分岔)行为,在过度拉伸(～30%)时褶皱最终消失.定量理解、预测和追踪这种强非线性力学行为的需求推进了有限应变板壳理论和数值计算方法的发展,不仅促进了对薄膜起皱-消皱机理的深入理解,也为无褶皱膜结构设计和薄膜表面形貌调控提供了新的思路.     

Ratcheting,wrinkling and collapse of tubes under axial cycling

Circular tubes compressed into the plastic range first buckle into axisymmetric wrinkling modes. Initially the wrinkle amplitude grows with increasing load. The wrinkles gradually induce a reduction in axial rigidity eventually leading to a limit load instability followed by collapse. The two instabilities can be separated by strain levels of a few percent. This work investigates whether a tube that develops small amplitude wrinkles can be subsequently collapsed by persistent cycling. The problem is first investigated experimentally using SAF 2507 super-duplex steel tubes with D/t of 28.5. The tubes are first compressed to strain levels high enough for mild wrinkles to form; they are then cycled axially under stress control about a compressive mean stress. This type of cycling usually results in material ratcheting or accumulation of compressive strain; here it is accompanied by accumulation of structural damage due to the growth of the amplitude of the initial wrinkles. The tube average strain initially grows nearly linearly with the number of cycles, but as a critical value of wrinkle amplitude is approached, wrinkling localizes, the rate of ratcheting grows exponentially and the tube collapses. The rate of ratcheting and the number of cycles to failure depend on the initial compressive pre-strain and on the amplitude of the stress cycles. However, collapse was found to occur when the accumulated average strain reaches the value at which the tube localizes under monotonic compression. A custom shell model of the tube with initial axisymmetric imperfections, coupled to a cyclic plasticity model, are presented and used to simulate the series of experiments performed successfully. A sensitivity study of the formulation to the imperfections and to key constitutive model parameters is then performed.     

Ratcheting and wrinkling of tubes due to axial cycling under internal pressure: Part I experiments

Under compression, pressurized tubes thick enough to deform plastically buckle into an axisymmetric wrinkling mode. The wrinkle amplitude is initially small but with persistent compression grows inducing a gradual reduction in axial rigidity and eventually causing a limit load instability, which is followed by collapse. The onset of buckling and collapse can be separated by a strain level of a few percent. This work investigates whether a tube that develops small-amplitude wrinkles can be subsequently collapsed by persistent axial cycling. Part I presents the results from a set of experiments on super-duplex tubes with D/t of 28.5 loaded as follows. A tube is pressurized and then compressed into the plastic range to a level that initiates wrinkling. It is then cycled under stress control about a compressive mean stress while the pressure is kept constant. The combined loads cause simultaneous ratcheting in the hoop and axial directions as well as a gradual growth of the wrinkles. At some stage the amplitude of the wrinkles starts to grow exponentially with the number of cycles N leading to localization and collapse. The rate of ratcheting and the number of cycles to collapse depend on the initial compressive pre-strain, the internal pressure, and the stress cycle parameters all of which were varied sufficiently to generate an adequate data base. Interestingly, in all cases collapse was found to occur when the accumulated average strain reached the value at which the tube localizes under monotonic compression. A shell model coupled to a specially calibrated plasticity model that can reproduce the biaxial ratcheting exhibited in the problem are presented in Part II. The model is first evaluated by comparison to the experiments and then used to study parametrically cyclic loading histories seen in buried pipelines.     

On an analogy between the deformation gradient and a generalized shell shifter tensor

This short paper deals with an analogy between the deformation gradient in three-dimensional continuum mechanics and a generalized shell shifter tensor being employed in volume shell formulations. This analogy is very well suited to derive relations between the line, area and volume elements in the shell continuum and those located at the reference surface of the shell. Furthermore, strain and stress tensors can be derived in a very elegant way. Finally, the three-dimensional and two-dimensional strain-energy density is obtained in a very clear manner. The analogy found therefore gives a new and very powerful interpretation of the shell shifter tensor.     

New computational approaches for wrinkled and slack membranes

The static response of thin, wrinkled membranes is studied using both a tension field approximation based on plane stress conditions and a 3D nonlinear elasticity formulation, discretized through 8-noded Cosserat point elements. While the tension field approach only obtains the wrinkled/slack regions and at best a measure of the extent of wrinkliness, the 3D elasticity solution provides, in principle, the deformed shape of a wrinkled/slack membrane. However, since membranes barely resist compression, the discretized and linearized system equations via both the approaches are ill-conditioned and solutions could thus be sensitive to discretizations errors as well as other sources of noises/imperfections. We propose a regularized, pseudo-dynamical recursion scheme that provides a sequence of updates, which are almost insensitive to the regularizing term as well as the time step size used for integrating the pseudo-dynamical form. This is borne out through several numerical examples wherein the relative performance of the proposed recursion scheme vis-à-vis a regularized Newton strategy is compared. The pseudo-time marching strategy, when implemented using 3D Cosserat point elements, also provides a computationally cheaper, numerically accurate and simpler alternative to that using geometrically exact shell theories for computing large deformations of membranes in the presence of wrinkles.     

空间薄膜结构的褶皱研究进展

大型柔性薄膜结构是目前国际上十分关注的可满足多项空间任务需要的新型空间结构. 由于薄膜材料柔性且抵抗压缩应力作用的能力十分有限, 因此这些空间结构在轨时多处于带有褶皱的工作状态. 褶皱的存在会改变结构的力学性能且会影响结构表面形状, 因此开展薄膜褶皱研究对柔性结构的设计和力学性能评价等具有重要意义. 本文综述了空间薄膜结构褶皱研究的发展和现状, 重点介绍了薄膜褶皱的理论研究概况, 并讨论了其中存在的问题.根据褶皱研究的最新进展对其未来的发展趋势进行了展望.      

Ratcheting and wrinkling of tubes due to axial cycling under internal pressure: Part II analysis

Part I presented a set of experiments in which pressurized tubes were cycled axially under stress control about a compressive mean stress. This loading history causes biaxial ratcheting involving compressive axial strain and expansion of the diameter of the tube. The compressive strain in turn induces the initiation and growth of axisymmetric wrinkles. Persistent cycling resulted in localization of the wrinkles and collapse. In Part II the problem is first modeled as a shell with initial axisymmetric imperfections while the biaxial ratcheting of the material is modeled using the Dafalias–Popov two-surface nonlinear kinematic hardening model. It is demonstrated that when suitably calibrated this modeling framework reproduces the prevalent ratcheting deformations and the evolution of wrinkling including the conditions at collapse accurately for all experiments. The calibrated model is then used to evaluate the ratcheting behavior of pipes under thermal-pressure cyclic loading histories experienced by axially restrained pipelines.     

Wrinkle-crease interaction behavior simulation of a rectangular membrane under shearing

Wrinkling analysis of a rectangular membrane with a single crease under shearing is performed to understand the wrinkle-crease interaction behaviors.The crease is considered by introducing the residual stresses from creasing and the effective modulus into the baseline configuration with assumed circular cross-sectional crease geometry.The wrinkling analysis of the creased membrane is then performed by using the direct perturb-force(DP) simulation technique which is based on our modified displacement components(MDC) method.Results reveal that the crease may influence the stress transfer path in the membrane and further change the wrinkling direction.The crease appears to improve the bending stiffness of the membrane which has an effective resistance on the wrinkling evolution.The effects of the crease orientation on wrinkle-crease interaction are studied toward the end of this paper.The results show that the wrinkling amplitude,wavelength,and direction increase as the crease orientation increases,and the wrinkling number decreases with the increasing crease orientation.These re-sults will be of great benefit to the analysis and the control of the wrinkles in the membrane structures.     

A study on wrinkling characteristics and dynamic mechanical behavior of membrane

An eigenvalue method considering the membrane vibration of wrinkling out-of-plane deformation is introduced, and the stress distributing rule in membrane wrinkled area is analyzed. A dynamic analytical model of rectangular shear wrinkled membrane and its numerical analysis approach are also developed. Results indicate that the stress in wrinkled area is not uniform, i.e. it is larger in wrinkling wave peaks along wrinkles and two ends of wrinkle in vertical direction. Vibration modes of wrinkled membrane are strongly correlated with the wrinkling configurations. The rigidity is larger due to the heavier stress in the part of wrinkling wave peaks. Therefore, wave peaks are always located at the node lines of vibration mode. The vibration frequency obviously increases with the vibration of wave peaks.     

Temporal evolution and instability in a viscoelastic dielectric elastomer

Dielectric elastomer transducers are being developed for applications in stretchable electronics, tunable optics, biomedical devices, and soft machines. These transducers exhibit highly nonlinear electromechanical behavior: a dielectric membrane under voltage can form wrinkles, undergo snap-through instability, and suffer electrical breakdown. We investigate temporal evolution and instability by conducting a large set of experiments under various prestretches and loading rates, and by developing a model that allows viscoelastic instability. We use the model to classify types of instability, and map the experimental observations according to prestretches and loading rates. The model describes the entire set of experimental observations. A new type of instability is discovered, which we call wrinkle-to-wrinkle transition. A flat membrane at a critical voltage forms wrinkles and then, at a second critical voltage, snaps into another state of winkles of a shorter wavelength. This study demonstrates that viscoelasticity is essential to the understanding of temporal evolution and instability of dielectric elastomers.     

适用于壳体h型自适应有限元分析的一组新单元

推导出一组适用于h型自适应分析的四边形蜕化壳元。对于大多数壳体结构,壳单元的刚度矩阵可分为薄膜、弯曲和剪切三部分。对薄膜部分本文采用杂交应力元方法进行设计,独立假设薄膜应力场以改善其精度;弯曲部分的刚度矩阵则依然由基于位移的应变来获得;而剪切部分则采用假设自然应变的方法来获得能克服薄壳下剪切自锁的新剪应变并用于计算此部...