Stimuli-responsive shell theory
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Abstract
Shell structures are used widely in nature and human-made systems due to their ability to efficiently carry load using a relatively light structure, where the shell is a curved surface with small, but finite thickness compared to the other dimensions. The geometric characteristic of thin shells, i.e. slenderness, makes the shells able to rapidly change shape via elastic instabilities. Despite the traditional view of instabilities as deleterious and undesirable events for structural stability, nature has successfully harnessed these instabilities to morph the shape of soft thin shell systems for several purposes such as movement, development, and morphogenesis, which can result from non-mechanical stimuli, e.g. heating, swelling, and biological growth, in the absence of external, mechanical loadings. Through the shape-morphing process, lilies bloom during spring, and Venus flytraps catch prey. Additionally, this process has been applied to design soft robotics actuators and adaptive metamaterials. Therefore, this thesis aims to understand the deformation and the instabilities induced by non-mechanical stimuli for soft thin shells and to present a novel stimuli-responsive shell theory.
First of all, two-field partial differential equations are derived for deformation problems under non-mechanical stimuli, based on the linear momentum balance for the mechanical field and the non-mechanical quantity balance for the non-mechanical field. For the mechanical field, the Kirchhoff-Love shell kinematics are utilized to describe the behavior of soft thin surfaces. In addition, the concept of the multiplicative decomposition of deformation gradient is applied to calculate the internal stress state induced by non-mechanical stimuli, and the projection method is employed in the stress constitution to consider the general multi-layer surface as an exact 2D shell. Moreover, the effect of mass addition from biological growth is dealt with on the stress constitution. For the non-mechanical field, the internal diffusion flux is described through the well-established diffusion law of gradient forms depending on the stimulus type, such as Fick's law and Darcy's law. The derived two-field equations are discretized to develop a computational tool based on the Isogeometric analysis, which satisfies the continuity requirement on the shape function for the shell analysis.
The developed two-field shell formulation is applied to two example problems. First, a pressure buckling problem of closed spherical shells in which a homogeneous natural curvature exists is dealt with as numerical simulations are performed with a thickness-direction differential swelling of a bi-layer spherical shell under pressure. To analyze this problem, it is studied that as the homogeneous natural curvature acts like pressure on the shell, the natural curvature impedes or accelerates the pressure buckling depending on its sign. Second, the optic cup morphogenesis during embryonic development is studied, indicating that the optic cup formation is dictated by elastic instabilities depending on the initial geometry of the optic cup. From our numerical simulation results, it is also shown that accounting for mass change from biological growth is crucial to capturing the morphogenesis process through growth.
For the last portion of this thesis, a novel stimuli-responsive shell theory is proposed as non-mechanical stimuli are considered as effective external loadings which are functions of mid-surface fundamental forms that describe natural stretch and curvature associated with the change in rest length and curvature of the mid-surface under non-mechanical stimuli. In this novel theory, analytical equations are derived to calculate the mid-surface fundamental forms that describe natural stretch and curvature, with consideration for the effect of mass addition for biological growth cases. The variational method is applied to this novel shell theory to derive the equilibrium equation as well as a stability criterion of the linear stability analysis. In the derived equilibrium equation, the standard equilibrium balance is solved between the standard internal stresses due to deformations from the initial body and external forces, which enables a simpler physical interpretation of how non-mechanical stimuli work on the initial body to deform it. As the weak form of the novel stimuli-responsive shell theory has the same form as the conventional shell analysis, it is expected that mechanicians and non-mechanicians alike can easily modify existing computational tools with the effective external loadings calculated in this novel theory, in order to study how various types of non-mechanical stimuli impact the mechanics and physics of thin shell structures.
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Attribution 4.0 International