Elsevier

Thin-Walled Structures

Volume 47, Issues 6–7, June–July 2009, Pages 692-700
Thin-Walled Structures

Mechanics of highly deformed elastic shells

https://doi.org/10.1016/j.tws.2008.11.009Get rights and content

Abstract

Emergence of new technological applications, in addition to the constantly growing interest in biological materials has accentuated the importance of studying the mechanics of highly deformed shells. The key challenge is the intricate interplay of physics and geometry, which leads to a mechanical response much different from the response of solid objects. The quest to understand the underlying phenomena has spawned theoretical and experimental studies, which have helped in understanding the underlying mechanisms of deformation and response of shells. Here, we use numerical simulations to study the response of shells when they are deformed deeply into the nonlinear regime. We use computational models to study the mechanics of highly deformed elastic shells in several classical problems: indentation of elastic spherical caps by a flat rigid plate and a rigid sharp indenter and pure bending of circular and oval cylinders. These assays are used to highlight some of the key aspects of the mechanics of highly deformed elastic shells, while an overview of the current state-of-the-art and suggestions for future research on this subject are also provided.

Introduction

The response of naturally curved elastic shells when they are highly deformed into the (geometrical) nonlinear regime is explored by focusing on two common shell configurations; cylindrical shells and spherical shells. In addition to thin shells which are normally considered for conventional materials, we have also probed the behavior and mechanics of some relatively thick shells, in view of the current interest in biological and small-scale structures. The approach here is employment of continuum-based computational models for solving shells governed by linear elasticity and fully nonlinear geometry using ABAQUS. Our material choice is restricted to that of an isotropic linear elastic material. Moreover, qualitative experiments have also been carried out on hemispherical shells to unravel some of the physical mechanisms of their response under indentation.

Shell structures have been widely used in pipelines, aerospace and marine structures, automotive industry, large dams, shell roofs, liquid-retaining structures and cooling towers [1]. Recent advancements in micro-electromechanical systems and nanotechnology have opened new avenues for applications of shells at much smaller scales. Examples are many and vary from carbon nanotubes and nanometer-sized buckyballs to microcapsules for drug delivery, colloidal armors, flexible electronics, tissue engineering and regenerative medicine [2], [3], [4], [5]. Shell structures are also ubiquitous in nature and arise at a range of length scales from the earth's crust to microtubules and biomembranes, as well as in plants [6], [7], [8]. The current interest in understanding the behavior of living cells and subcellular components has further accentuated the importance of studying the behavior of shells. For example, much attention has been directed recently towards understanding the behavior of microtubules, which are often highly curved and buckled because of the state of stress in the cytoplasm. These studies have direct implications in understanding the physiological forces applied to microtubules, their mechanical coupling with the cytoskeleton and their role in altering cell mechanics and function [9], [10], [11], [12]. Other examples are the mechanics of the cell membrane [13], [14], [15], [16], [17], the nuclear envelope [18], [19] and even nanometer-sized viruses and retrovirus particles [20], [21].

Despite their significance, many phenomenological aspects of the behavior of naturally curved shells are still ambiguous and pose fundamental challenges for applications of mechanics in new areas such as nanostructures and biology. Many of the nonlinear shell studies conducted in the past have been motivated by failure concerns related to conventional shell structures, and for that reason no considerable effort has been made to probe the response of these structures deep into the nonlinear regime. Emergence of novel applications at micron and submicron scales, where the material failure becomes less influential, motivates investigating the response of these structures deeply into the nonlinear regime.

The highly nonlinear behavior of elastic shells is mainly governed by inextensible or almost inextensible deformations, which are energetically preferred by the shell [22], [23], [24], [25], [26]. In large deformations, this leads to the appearance of structural features such as dimensionless developable cone and curvature condensates and almost inextensible one-dimensional ridges [27], [28], [29]. These features are ubiquitous both in nature and in technology, as well as in everyday human life. Examples are kinking of a straw and indentation of a plastic bottle using a sharp pen, as well as the crushed coke-can and dried resin. Fig. 1 provides several examples of such phenomenon in shells over a range of length scales. The interplay of physics and geometry, which indeed leads to the appearance of the localized features shown in Fig. 1, can even play a critical role at the early stage of shell response. An example is the persistence of a pinch in a circular pipe which manifests itself at the earliest stage of deformation due to the dominant role of nearly inextensible deformations [30].

In this study, first, we study the mechanics of elastic spherical caps (i.e. segment of a spherical shell) under both point-like and flat plate indentation. Then, the response of elastic circular and oval cylindrical shells under pure bending is investigated when they are deformed deep into the nonlinear regime. Our study complements a wide range of previous scaling approaches based on continuum elasticity [31], [32], [33]; and continuum-based and molecular dynamics simulations for understanding the response of highly deformed plates and shells [34], [35], [74]. The detailed numerical computations enable a high-fidelity exploration of highly deformed states of elastic shells.

Section snippets

Indentation of elastic spherical caps

In this section, we study the deformation and mechanics of spherical caps clamped along the edge with radius R, thickness t, and center angle α, under both flat plate and point indentations—see Fig. 2A. First, the response of these structures under rigid flat indentation is analyzed, Fig. 2. In our numerical simulations, free sliding has been assumed between the flat plate and the spherical cap. The computations were carried out with the following material parameter values: Young's modulus E=100

Pure bending of circular cylindrical shells

The response of cylindrical shells under pure bending has been the subject of much research during the last century. The pioneering work of [75] on elastic elbow ovalization under pure bending and its extension towards studying pipes and elbows (for example see: [42], [43], [44]) has motivated many further studies. Ovalization instability, which is one of the basic mechanisms of response, was introduced and studied by Brazier [45]. The role of material nonlinearity on the instability of

Pure bending of oval cylindrical shells

In this section, we have extended our study to the case of oval cylindrical shells, Fig. 6A. This work complements the earlier theoretical and experimental work on buckling of oval cylindrical shells under various loading conditions [54], [55], [56], [57], [58]. Fig. 6B displays the moment-rotation responses of oval cylindrical shells with various ellipticity e, defined as the ratio of the major-to-minor axes of the cylinder cross-section, denoted by b and a, respectively (i.e. e=b/a). The

Discussions and concluding remarks

The inextensible or almost inextensible deformation of elastic shells under mechanical loading leads to the formation of intricate structural features at a much smaller scale than the overall structure size. These features, which can be seen by simple experiments in everyday life as well as in biological and engineering systems, are associated with high-energy density and evolve in intricate ways as the shell is further loaded deep into the nonlinear regime. The qualitative response of thin

Acknowledgements

The author is indebted to John W. Hutchinson for his mentorship and invaluable help with this work. The author also thanks L. Mahadevan, Hosam Ali, Franco Tamanini, Louis Gritzo and Alexandre Kabla for many insightful discussions, Neda Movaghar for her help with the illustrations, and the support by the NSF under Grant CMMI-0736019.

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