Pattern transformation induced waisted post-buckling of perforated cylindrical shells

Highlights

• Waisted post-buckling pattern transformation is concluded for clamped cellular cylindrical shells.

• Shell theory can be used to analyze negative Poisson's ratio formed by waisted post-buckling.

• Load-carrying capacities of cellular cylindrical shells undergo sudden drop but recover subsequently.

• Waisted post-buckling behavior could appear for thin-walled shell without external support.

• Waisted post-buckling patterns may be found on other curved structures such as cylindrical panels.

Abstract

A comprehensive investigation on the extremely large post-buckling deformation of perforated cylindrical shells is conducted using experiments and verified with an analytical shell model and nonlinear finite element simulations. A “waisted” post-buckling configuration, which is characterized by uniform shrinking in the middle section of the perforated cylindrical shell, is identified. The waisted behavior is attributed to the triggering of a pattern transformation under compressive load that shows special hyperelastic metamaterial characteristics. The load-carrying capacity of waisted post-buckling suffers a sudden drop and then recovers when the holes are completely collapsed and closed. Plenty of design parameters can be utilized to enrich variations of the waisted post-buckling responses. The negative Poisson's ratio induced by pattern transformation plays a key role in forming the waisted post-buckling modes. This special hyperelastic metamaterial behavior can be easily achieved by fixing the boundaries and adjusting the geometric parameters of the shell. By comparing the characteristics of a porous cylindrical shell with those appeared for an equivalent porous panel, it is highlighted that pattern transformation can occur in a thinner porous cylindrical shell without lateral support. The waisted post-buckling modes of a perforated cylindrical shell are stable, and the shell is invulnerable to the progressively increasing applied loads. In comparison, an ordinary cylindrical shell may snap from one mode to another in the post-buckling process. Moreover, we find that some non-closed cylindrical panels can also buckle into the waisted-like modes. These findings can be applied to the construction of functional devices for soft robotics, actuators, and structural protection for facilities, etc.

Introduction

Metamaterials belong to a novel class of artificial periodic materials that can exhibit numerous counterintuitive physical properties that surpass (or complement) those ubiquitous characteristics in nature. Over the past years, many creative metamaterials have emerged in many research fields, including electromagnetism (Caloz and Itoh, 2005; Shi and Akbarzadeh, 2020), optics (Soukoulis and Wegener, 2011), acoustics (Cummer et al., 2016), and mechanics (Coulais et al., 2016). Many other unconventional materials that can achieve inconceivable properties have also been proposed (Vescovo and Giorgio, 2014). Typically, these intriguing properties include qualities such as a negative refractive index (Valentine et al., 2008), acoustic cloaking (Zhu et al., 2015), sub-wavelength focusing (Scalora et al., 2007), and unusual mechanical behaviors (Yu et al., 2017). These properties are derived using specific designs of geometric configurations and deformation mechanisms of materials. Due to their amazing performance in a wide range of physical responses and their great innovation in material design, metamaterials present a formidable perspective for potential applications in telecommunications (Enkrich et al., 2005), optoelectronics (Esfandyarpour et al., 2014), biomedicine (Kolken et al., 2018), and aeronautic engineering (Alu and Engheta, 2005).

Classified as a primary branch of metamaterials, “mechanical metamaterials” are engineered materials that are rationally constructed in a very specific manner to achieve unheard-of, counterintuitive, and previously inconceivable mechanical properties, including a negative Poisson's ratio (Lakes, 1987; Bertoldi et al., 2010; Lim, 2015; Li et al., 2016), vanishing shear modulus (Kadic et al., 2012; Bückmann et al., 2014), and negative swelling (Liu et al., 2016). During the last few years, these innovative materials have caught the eye of thousands of scientists and engineers. These metamaterials have been implemented extensively in soft robots (Yang et al., 2015; Mark et al., 2016), energy absorption (Frenzel et al., 2016), and seismic isolation (Li et al., 2017). In the evolution of mechanical metamaterials, auxetic materials, which have a macroscopic negative Poisson's ratio, have occupied an important place in research due to the diversified styles of periodic architectures that can be constructed using these materials, including various lattices (Tancogne-Dejean et al., 2019), tessellations (Alderson et al., 2010), and cellular solids (Mullin et al., 2007) by soft polymers (Bertoldi et al., 2008; Niknam and Akbarzadeh, 2018), metals (Ghaedizadeh et al., 2016; Ho et al., 2019,) or composites (He et al., 2017; Hu et al., 2017). Under the action of a uniaxial tensile (or compressive) force, metamaterials with a negative Poisson's ratio can demonstrate expansion or constriction in the transverse direction (Bertoldi et al., 2010). Sometimes, this distinctive reaction needs to be triggered by a pattern transformation that exhibits a regular local buckling of ligaments between two holes (Bertoldi et al., 2008). The auxetic behaviors of this new type of metamaterial can provide users with strategies to acquire switchable or tunable effective material properties by adjusting pore shapes (Overvelde & Bertoldi, 2014), porosities (Bertoldi et al., 2010), and boundary conditions (Florijn et al., 2014). In addition, the methods of stimulating pattern transformation can be easily established by applying hydraulic pressure (Yang et al., 2015; Lazarus & Reis, 2015) or electromagnetic drive systems (Tipton et al., 2012). In recent years, with the help of 3D printing technology, metamaterials have been fabricated in a wide range of sizes from macroscale to nanoscale (Walia et al., 2015; Bertoldi et al., 2017), and they can be architected in 3D space (Babaee et al., 2013; Frenzel et al., 2017). By exploiting the distinctive properties of 3D auxetic structures, a vast terrain for designing more practicable metamaterial-based devices can be made possible. In this aspect, perforated hollow cylinders that can induce a negative Poisson's ratio through pattern transformation were conceived by Bertoldi's research group (Javid et al., 2016). Another kind of crucial structure is a 3D shell that is characterized by a large span-to-thickness ratio. A spherical shell with various patterned circular holes was constructed by Shim et al. (2012), and a striking volume compressibility induced by pattern transformation was demonstrated.

Cylindrical shells can be found everywhere in our daily lives. These structures are common fundamental elements and are widely applied in lightweight engineering structures. Due to their intrinsic complexity, the post-buckling behaviors of cylindrical shells have long been one of the most challenging problems in the mechanics of solids. Post-buckling issues can be identified by investigating the nonlinear large deformation occurring after bifurcation or extremum buckling, which is characterized by a rapid increase in deflection without a sustainable increase in load. Many well-known theoretical and experimental results for large-deflection post-buckling of cylindrical shells are reported by Yamaki (1984). Meticulous attention is paid to boundary conditions and mode functions to guarantee solution accuracy. Recently, the most representative work that quantifies the knockdown effect of geometric imperfections is the worst multiple perturbation load approach (WMPLA) by Wang et al. (2013). This work represents a great advancement in the current methods of research and was validated by a series of buckling experiments (Wang et al., 2018, 2019). Generally, it is a technical requirement in the design of a thin-walled structure to avoid buckling under the expected loading conditions because of the remarkable decline in the load-carrying capacity after the loss of structural stability. Through decades of past research, structural buckling control for some previously inconceivable phenomena in physics and engineering can be established. Among these methods on capitalizing structural buckling, the most significant achievements are the innovative ways to obtain unusual material properties by taking advantage of the instabilities of periodically arranged structural members. For example, a negative Poisson's ratio can be obtained in pattern transformed metamaterials by controlling the beam-like buckling of the periodic ligaments between two holes (Kochmann and Bertoldi, 2017). In the last several years, many studies have focused on deriving these innovative macroscopic properties by exploiting the instabilities of diversified microstructures (Dykstra et al., 2019; Che et al., 2017). However, there are very few studies that concern controlling the post-buckling behaviors of cylindrical shells by using these newly emerged metamaterial properties.

In this paper, our primary purpose is to demonstrate and analyze a special post-buckling configuration of cylindrical shells by exploiting pattern transformation. Several perforated cylindrical shells with periodic circular holes are constructed using elastomeric solids. Their nonlinear waisted post-buckling behaviors are investigated by experiment, theory, and numerical simulations using the finite element method (FEM). Different from Javid et al. (2016), which mainly focuses on the cause of pattern transformation for a longer cylinder, our research effort emphasizes the ability to control pattern transformation for the global deformation of a shallow cylindrical shell. The concerned structural behavior is within the scope of nonlinear mechanics of shells and involves very large radial deflections of cylindrical shells. A waisted post-buckling configuration, different from the classical buckling behavior of ordinary cylindrical shells, exists once the porosity and outline sizes of porous cylindrical shells attain a certain combination. As explained above, “waisted” post-buckling in this context occurs while post-buckling takes place. In addition, the higher-order shell theory (Reddy and Liu, 1985; Reddy, 2004) that is applicable to thicker shells is established to validate the observation in experiments.

This paper is arranged into the following sections. Section 2 describes the geometry of perforated cylindrical shells, experimental facilities, and a typical waisted post-buckling mode. Section 3 presents the FEM settings for eigenvalue buckling and nonlinear post-buckling analyses. In Section 4, a description of the shell theory for revealing the influence of a negative Poisson's ratio is provided. Section 5 discusses several numerical examples are to explore the effects of design parameters for the perforated cylindrical shells. Finally, Section 6 concludes the paper with impactful insights.