A Fourier based reduced model for wrinkling analysis of circular membranes
Introduction
Membrane structures have been widely used in aerospace and civil engineering, such as solar sail, large membrane roof and wall, because of their high performance in resisting external tensional loads with much lighter weight compared with traditional structures [1]. In recent years, various kinds of membrane structures have been developed and applied in different engineering fields to meet specific requirements. For example, through mounting electronical elements on membrane structures [2], various kinds of stretchable electronics have been fabricated, including stretchable batteries [3], supercapacitors [4] and wireless epidermal sensors [5], which provide superior mechanical properties inaccessible in traditional electronics. Recent studies found wide existence of membrane structures in biological organism [6], which play essential roles in realizing specific functions, such as morphogenesis of bacterial biofilms [7], cell spreading [8], and deformation of red blood cell [9]. In some composite manufacturing processes, such as Vacuum Assisted Resin Transfer Molding (VARTM) [[10], [11], [12]] and Flexible Injection (FI) [[13], [14]], membranes play a key role in inducing the through-thickness impregnation for faster composite manufacturing process.
As a soft material with an almost negligible bending stiffness, a membrane easily loses its mechanical stability and wrinkles when subject to in-plane compressive stress. Such phenomena are widely observed both in nature and engineering. One common example is the wrinkles emerging in the direction perpendicular to tensional direction in clamped rectangular elastic membrane [15], which can be explained by the strain incompatibility due to the Poisson effect. Additionally, as presented in Li et al. [16], membrane wrinkling can be also observed in film–substrate system.
Wrinkling can have significant effects on the mechanical and optical properties of membranes [[17], [18]]. This can lead to undesired problems in some engineering applications, such as the production defects in strip conveying process [19] and deep drawing process [20]. To explore the mechanism for the instability in membranes, a large amount of modeling methods have been proposed. In a pioneering work, Wagner [21] developed a tension field theory, which assumed that there was no in-plane compressive stress throughout the membrane. This approach involves a uniaxial tension field with only one nonzero principal stress component, which is perpendicular to the wrinkled direction. To model partly wrinkled membranes, Pipkin [22] proposed a relaxed energy density to replace the strain energy function in tension field theory. This approach was then employed by Steigmann and Pipkin [23] to study the wrinkling of pressurized membranes. By introducing a modified deformation tensor based on a fictive non-wrinkled mode, Roddeman et al. [24] further considered the effects of large deformation and anisotropy in their model, which is capable to precisely describe the stress situation of anisotropic membrane. Schoop et al. [25] transformed the wrinkling conditions of Roddeman [24] into a reference configuration to determine the stress state, and implemented the algorithm into a FE code for analyzing general wrinkling problem.
Although the essential reason for membrane wrinkling is the negligible bending stiffness and the in-plane compressive stress, the occurrence and evolution of the wrinkles differ among structures with different geometric shapes and materials distributions. For example, wrinkles in annular membrane usually show periodic oscillation in the circumferential direction due to its axisymmetry, while those in rectangular domain are highly dependent on specific loads and boundary conditions. The wrinkling problems in annular and circular membranes have been widely studied. Plaut [26] analyzed the large unwrinkled axisymmetric deformations of circular membranes under several different loadings using three theories, including Generalized Reissner theory, Reissner theory, Föppl–von Kármán (FvK) theory. The analytical solutions of wrinkling phenomena in annular membranes under different boundary conditions were thoroughly studied by Coman and Haughton [27], Coman [[28], [29]]. Géminard et al. [30] experimentally investigated the main features of wrinkles that form in a pre-stretched annular membrane under axisymmetric traction at the center. Wang et al. [31] proposed a theoretical model to predict the wrinkling and post-buckling behaviors, which is verified by experimental measurement.
Most studies mentioned above are based on analytical solutions, which correctly predict the critical wrinkling loads and modes and stress distribution before bifurcation. However, the post-buckling analysis is also very important, because it is essential to predict and control the evolution of wrinkles in its engineering application. The finite element method has been proved to be the most robust method to capture the complex evolution of wrinkles and stress distribution. For example, Argyris et al. [32] developed a TRIC (TRIangular Composite) element with aim to model post-buckling of shells based on the natural mode finite element method. Combescure [[33], [34]] developed a new computational scheme which permits to consider the effects of initial geometrical imperfection on the post-buckling behaviors. However, these methods either present intensive computation especially for the case with small wave length or suffer the difficulties of choosing a proper imperfection.
In order to reduce the computation for the case of small wrinkle wave length and meanwhile ease the sensitivity of post-buckling behaviors to the applied geometric imperfection, Damil and Potier-Ferry [35] proposed a reduced-order FE model based on the Fourier expansion of the displacement field. This approach showed good performance in both accuracy and efficiency for many kinds of instability problems, such as rectangular membranes [[36], [37], [38]], rectangular sandwich plates [[39], [40], [41]], and thin films on compliant substrates [42]. Besides, because the reduced model is based on the Fourier expansion, it can accurately trace the specific instability pattern without introducing imperfection in the initial model. In order to model the wrinkling phenomena of circular membranes, it is natural to consider the geometric axisymmetric property of membranes [[43], [44]]. However, the deformations in instability phenomena are not axisymmetric, therefore general axisymmetric elements are no longer applicable. The objective of this study is to develop a one-dimensional reduced order model based on the approach proposed by Damil and Potier-Ferry [35] to model three-dimensional wrinkling problems in circular membranes. The accuracy and efficiency of the reduced model are verified by comparing the simulation results with the ABAQUS and analytical results. Based on the modeling results of the reduced model, the instability mechanism of circular membranes for different geometries and loading conditions are thoroughly investigated. The developed model can be extended to simulate and investigate other instability phenomena in circular domain, such as wrinkling in film–substrate system [[45], [46], [47]], buckling of sandwich plates [[48], [49], [50]] or functionally graded plates [[51], [52], [53]].
This paper is structured as follows. In Section 2, several basic equations are reviewed. In Section 3, the finite element procedure is described, and the one-dimensional reduced model is constructed in cylindrical coordinates. From Sections 4 to 7, the accuracy and efficiency of the reduced model are verified, and the various instability phenomena in circular domain are investigated by the reduced model. The paper ends with a discussion on main conclusions and future work.
Section snippets
Basic equations
As mentioned above, the bending stiffness should be taken into account to solve membrane wrinkling problems. Two types of stresses are considered including the bending stress caused by the flexion of the membrane and the membrane in-plane stress determined by the conditions of static equilibrium. Let , , denote respectively the radial, circumferential and transversal components of the mid-plane displacement in cylindrical coordinates () and (). The linear strains in the
Reduced model for instability problems in circular domain
In this section, the method of Fourier series with slowly variable coefficients is briefly reviewed first. Then the finite element procedure is described to construct the one-dimensional reduced model. Finally, the nonlinear system is solved by the Asymptotic Numerical Method.
Annular membrane under in-plane radial tensile load
In this section, the instability problem of an annular membrane under radial traction is studied. As shown in Fig. 2, the external edge of the membrane is simply supported, while the transverse displacement of the internal edge is fixed. A uniformly distributed radial tensile load is applied along the internal edge. The wrinkling pattern, bifurcation curve etc. obtained by the Fourier reduced model are compared with the 3D finite element model in ABAQUS (referred to as the ‘full shell model’ in
Annular membrane under transverse load
As shown in Fig. 8, the reduced model is used to simulate a pure bending problem, which is an annular membrane under transverse load, to verify that the model is able to precisely simulate the flexion of the membrane. Linear shell element (S4R) and quadratic shell element (S8R5) are used in ABAQUS to simulate the deformation of the annular membrane under transverse load as references to verify the reduced model in this example. In order to investigate the membrane locking phenomenon, reduced
Annular membrane under diagonal tensile load
After the validation of the reduced model under in-plane and out-of-plane loads, the reduced model is used to simulate a more complex example, which is an annular membrane under diagonal radial tensile load along the internal edge as shown in Fig. 10. This example is selected to verify that the model is able to simulate the instability phenomenon consisting of a local wrinkling and a global bending. The applied force is decomposed into a horizontal component and a vertical one.
This example is
Circular plate under in-plane radial compressive load
A circular plate under uniform compressive load along the external edge, as shown in Fig. 14, is investigated in this section to verify that the reduced model can be employed in the instability problems of circular thin plates. The analytical solution of the example obtained by Alfutov [61] is introduced as a benchmark for our result. The stress distribution in the entire process is simulated by using the reduced model. The material properties and geometric dimensions are given in Table 6.
Conclusions
Based on the method of Fourier series with slowly variable coefficients [35] and the asymptotic numerical method [58], a one-dimensional reduced model is constructed to model circumferential periodic instability patterns in circular domain. The reduced model is verified to be efficient and accurate by comparing its results with those of ABAQUS for an annular membrane under various loads. With reduction of DOFs, the reduced model still precisely simulates the evolution of displacement and
Acknowledgments
This work has been supported by the National Natural Science Foundation of China (Grant Nos. 11772238 and 11702198). The authors also acknowledge the financial support of the National Science & Engineering Research Council of Canada (NSERC) .
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