On the effect of shape on soft structures

Abstract

Compliance has emerged as a powerful tool in structural engineering to program complex mechanical behavior. Compliant systems can be designed through two main avenues: controlling the soft constitutive components at the molecular level or leveraging geometric effects at the mesoscale by shaping the internal architecture. This dissertation focuses on the latter. It demonstrates that controlling the shape of mechanical systems determines how mechanical loads are transferred, leading to heterogeneous deformations even when a single material is used. However, increasing the complexity of the underlying geometry results in highly complex deformation modes. This leads to an overload of degrees of freedom and the introduction of nonlinearities in the underlying models. While complexity in the material's architecture enables a larger class of functionality, it renders traditional analytical techniques intractable. This calls for the development of advanced numerical methods. By developing state-of-the-art numerical methods, in this thesis we explore the effect shape has on soft structures. Specifically, the numerical methods we develop allow us to discover novel properties, better understand fundamental behavior, and guide the design in two model systems: a deformable lens and an architected material. In the first part of this thesis we develop a higher-order finite element model of a deformable lens and couple it to a ray tracing simulation to discover how deformation can be harnessed to tune spherical aberration. Spherical aberration occurs in spherical surfaces and causes incident off-center light rays to be refracted or reflected to different degrees, generating a diffused focal point. As the light rays do not converge towards a single point, the optical quality of the projected images is diminished. By deforming lenses affected by spherical aberration under axial tension, we find that an originally diffused focal point becomes sharper, and spherical aberration is significantly reduced. The identified results are then confirmed over a variety of geometries. For each of the shapes considered we predict the optimal amount of deformation needed to achieve minimal spherical aberration. In the second part of this thesis we apply a novel continuation method in order to develop a deeper understanding of how simple structures generate complex behavior. In applying the new deflated continuation method to a finite sized cellular solid, we uncover a rich bifurcation diagram that illustrates a cascading series of instabilities that lead to multiple stable branches and unstable branches. Guided by our bifurcation analysis, we are able to replicate the results in a physical experiment and find that newfound multistability can be further harnessed to generate a structure with reprogrammable behavior. We then expand our bifurcation analysis to explore the role shape perturbations have on the underlying solution structure. Finally, guided by our newfound understanding of cellular solids, we develop shape optimization methods for architected materials with target properties. Using the higher-order moving-mesh shape optimization method, we discover shapes that achieve target nonlinear properties. Starting from originally regular and periodic shapes, we find that the method consistently converges to aperiodic designs that are often unthinkable and not immediately logical. By considering structures under different boundary conditions and architectures, we find that aperiodic designs may be more flexible in achieving target behaviors than their periodic counterparts. Having higher geometric flexibility, i.e.@ being able to access different shapes, we are able to access different target behaviors.