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This thesis presents two studies on elastic plates. In the first study, we discuss the choice of elastic energies for thin plates and shells, an unsettled issue with consequences for much recent modeling of soft matter. Through consideration of simple deformations of a thin body in the plane, we demonstrate that four bulk isotropic quadratic elastic theories have fundamentally different predictions with regard to bending behavior. At finite thickness, these qualitative effects persist near the limit of mid-surface isometry, and not all theories predict an isometric ground state. We discuss how certain kinematic measures that arose in early studies of rod mechanics lead to coherent definitions of stretching and bending, and promote the adoption of these quantities in the development of a covariant theory based on stretches rather than metrics. In the second work, the effects of in-plane swelling gradients on thin, anisotropic plates are investigated. We study systems with a separation of scales between bending energy terms. Warped equilibrium shapes are described by two parameters controlling the spatial "rolling up'' and twisting of the surface. Shapes within this two-parameter space are explored, and it is shown that shapes will either be axisymmetric or twisted depending on swelling function parameters and material anisotropy. In some axisymmetric shapes, pitchfork bifurcations to twisted solutions are observed by varying these parameters. We also show that a familiar soft mode of the catenoid to helicoid transformation of an isotropic material no longer exists with material anisotropy. / Master of Science / This thesis presents two studies on the subject of thin, elastic bodies, otherwise known as plates. Plate theory has important applications in many areas of life, ranging from the design and construction of civil structures to the mechanics of wrinkling sheets. In the first work, we discuss how different elastic plate theories have qualitatively different predictions on how a plate will behave when bent. We discuss the different physical implications of each model, and relate our findings to previous studies. Additionally, we promote the use of certain technical measures in the study of plates corresponding to the most coherent definitions of bending and stretching. In the second work, we study the effects of in-plane swelling gradients on elastic plates whose material stiffnesses vary with direction. Inspired by wood panels that warp when exposed to moisture, we model elastic plates exposed to various swelling patterns and determine the resulting warped shapes. We find that some shapes are axisymmetric, while others prefer to twist when exposed to moisture-induced swelling. By varying certain parameters of the swelling functions, or by increasing the material fiber stiffness, we also find a qualitative change in shape from an axisymmetric to a twisted surface.

<p>Elastic energy plays important roles in biology across scales, from the molecular to organismal level, and across the tree of life. The ubiquity of elastic systems in biology is partly due to the variety of useful functions they permit such as the simplification of motor control in running cockroaches and the efficient recycling of kinetic energy in hopping kangaroos. Elastic energy is also responsible for ultrafast movements; the fastest movements in animals are not powered directly by muscle, but instead by elastic energy stored in a spring. By demonstrating that the power required to generate ultrafast movements exceeds the limits of muscle, many studies conclude that energy storage is necessary; but, what these studies do not explain is how the properties of a biological structure affect its capacity for energy storage. In this dissertation, I test the general principles of energy storage by investigating elastic systems at three hierarchical levels of organization: a single structure, multiple connected structures, and a spring system connected to muscle. By using a multi-level approach, my aim is to demonstrate, at each of the mentioned levels, how properties of the spring system affect where or how much energy is stored in the system as well as how these conclusions can be combined to inform our understanding of the biomechanics of hierarchical elastic systems.</p><p>When considering spring systems at the level of a single structure, morphology is one major structural aspect that affects mechanics. Continuous changes in morphology are capable of dividing a structure into regions that are responsible for the two contradicting functions that are essential for spring function: energy storage (via deformation) and structural support (via resistance to deformation). Using high quality micro computed tomography scans, I quantify the morphology of the mantis shrimp (Stomatopoda) merus, a single structure of the raptorial appendage hypothesized to store the elastic energy that drives ultrafast strikes. Comparing the morphology among the species, I find that the merus in smashers, species that depend heavily on elastic energy storage, have relatively thicker ventral regions and more eccentric cross-sections than spearers, species that strike relatively slower. I also conclude that differential thickening of a region can provide structural support for resisting spring compression as well as facilitate structural deformation by inducing bending. This multi-level morphological analysis offers a foundation for understanding the evolution and mechanics of monolithic systems in biology.</p><p>When two or more structures are connected, their relative physical properties determine whether the structures store energy, provide structural support, or some combination of both. Although the majority of elastic energy is stored via large deformations of the merus in smashers, some spearer species show relatively little meral deformation, and it is unclear whether elastic energy is stored in these systems. To determine whether the apodeme (arthropod tendon) provides energy storage in species that exhibit low meral deformation, I measure the physical properties of the lateral extensor apodeme and the merus to which it is connected. Comparisons of these properties show that in the spearer species I tested, the merus has a relatively higher spring constant than the apodeme, which results in the merus providing structural support and the apodeme storing the majority of elastic energy. Comparing the material properties of the apodemes with those of other structures reveals that apodemes and other biological spring systems share similar material characteristics. This study demonstrates that in order to understand the biomechanics of spring systems comprised of connected structures, it is necessary to compare their relative mechanical properties.</p><p>Finally, because muscles are responsible for loading spring systems with potential energy, muscle dynamics can affect elastic energy storage in a spring system. Although spring systems can circumvent the limits imposed by muscle via power amplification, they are not entirely independent from muscle dynamics. For example, if an organism has relatively low time to prepare and stretch the spring prior to the onset of movement, the limits of muscle power can dominate energy storage. To test the effects of muscle dynamics on spring loading, I implement a mathematical model that connects a Hookean spring model to a Hill-type muscle model, representing the muscle-tendon complex of the hindlimbs of American bullfrogs, in which the muscle dynamics are well understood and the duration of spring loading is low. I find that the measured spring constants of the tendons nearly maximize energy storage within the duration of in vivo spring loading. Additionally, the measured spring constants are lower than those predicted to produce maximal energy storage when infinite time is available for spring loading. Together, these results suggest that the spring constants of the tendons of American bullfrogs are tuned to maximize elastic energy for small durations of spring loading. This study highlights the importance of assessing muscle dynamics and their effect on energy storage when assessing the functional significance of spring constants.</p> / Dissertation

Throwing with power and accuracy is a uniquely human behavior and a potentially important mode of early hunting. Chimpanzees, our closest living relatives, do occasionally throw, although with much less velocity. At some point in our evolutionary history, hominins developed the ability to produce high performance throws. The anatomical changes that enable increased throwing ability are poorly understood and the antiquity of this behavior is unknown. In this thesis, I examine how anatomical shifts in the upper body known to occur during human evolution affect throwing performance. I propose a new biomechanical model for how humans amplify power during high-speed throwing using elastic energy stored and released in the throwing shoulder. I also propose and experimentally test a series of functional hypotheses regarding how four key shifts in upper body anatomy affect throwing performance: increased torso rotational mobility, laterally oriented shoulders, lower humeral torsion, and increased wrist hyperextensability. These hypotheses are tested by collecting 3D body motion data during throws performed by human subjects in whom I varied anatomical parameters using restrictive braces to examine their effects on throwing kinematics. These data are broken down using inverse dynamics analysis into the individual motions, velocities, and forces acting around each joint axis. I compare performance at each joint across experimental conditions to test hypotheses regarding the relationship between skeletal features and throwing performance. I also developed and tested a method for predicting humeral torsion using range of motion data, allowing me to calculate torsion in my subjects and determine its effect on throwing performance. My results strongly support an important role for elastic energy storage in powering humans’ uniquely rapid throwing motion. I also found strong performance effects related to anatomical shifts in the torso, shoulder, and arm. When used to interpret the hominin fossil record, my data suggest high-speed throwing ability arose in a mosaic-like fashion, with all relevant features first present in Homo erectus. What drove the evolution of these anatomical shifts is unknown, but as a result the ability to produce high-speed throws was available for early hunting and likely provided an adaptive advantage in this context. / Anthropology

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Les alliages de zirconium sont utilisés notamment dans les assemblages de combustible nucléaire pour leur transparence aux neutrons ainsi que pour leur tenue mécanique et leur résistance à la corrosion. La connaissance de leur microstructure et de son évolution est nécessaire pour maîtriser les différents traitements thermomécaniques de la gamme de transformation qui comporte plusieurs trempes depuis le domaine bêta. Cette microstructure présente, à l’issue d’une trempe, des lamelles dites de Widmanstätten. Ces dernières soit se disposent parallèlement entre elles (platelets parallèles), soit se croisent en vannerie. Ces morphologies jouent sur l’étape suivante de filage ; en effet, les platelets parallèles défavorisent la fragmentation des lamelles. Une méthode a été mise en place pour quantifier ces morphologies.Lors de la transformation bêta vers alpha, un grain peut générer 12 orientations alpha (variants). Les paramètres qui influencent leur sélection sont encore peu connus. Le modèle proposé minimise la déformation moyenne lors de la transformation. D’abord analytique, il a été ensuite implémenté numériquement afin d’aborder des effets tels que la relaxation d’Eshelby, l’anisotropie élastique, une contrainte extérieure ou le voisinage. En parallèle, la sélection expérimentale a été quantifiée au moyen original de l’EBSD et des fractions des variants locales dans un ex-grain bêta. La confrontation des résultats expérimentaux avec le modèle a permis de le valider en partie et de déterminer la contrainte de trempe à la surface des éprouvettes ainsi que son effet sur la sélection de variants. / Zirconium alloys are frequently used in nuclear fuel assemblies. They are chosen for their low neutron absorption, their mechanical properties and their corrosion resistance. A better understanding of the microstructure evolution of these alloys should allow a better control of their process of fabrication. During processing, several quenches, from the beta to the alpha domain take place. The resulting microstructures are lamellar and are called Widmanstätten microstructures. These lamellae are either disposed in parallel or in crisscross and are named “parallel platelets” and “basketweaves”, respectively. These various morphologies have a significant impact on the extrusion; basketweaves facilitate grain fragmentation unlike parallel platelets. In this thesis project, a methodology was developed in order to quantify these morphologies.During the phase transformation, one beta grain can generate 12 different orientations of new alpha grains. The parameters which can influence variants selection are not yet well-known. The model proposed in the present study is based on the minimization of the mean elastic energy of the system during the phase transformation. First results were obtained analytically. Then, additional effects such as the Eshelby relaxation, the elastic anisotropy and the external strain were implemented numerically. In parallel, each alpha variant was quantified within a former beta grain by EBSD analysis. The comparison between the model and the experiments helped to partially validate the model as well as determine the quench strain on the surface of the sample. It was then possible to study the effect of quench strain on the variant selection.

We study thin films with residual strain by analyzing the limit of non-Euclidean elastic energy functionals as the material’s thickness tends to 0. We begin by extending prior results (Bhattacharya et al. in Arch Ration Mech Anal 228: 143–181, 2016); (Agostiniani et al. in ESAIM Control Opt Calculus Var 25: 24, 2019); (Lewicka and Lucic in Commun Pure Appl Math 73: 1880–1932, 2018); (Schmidt in J de Mathématiques Pures et Appliquées 88: 107–122, 2007) , to a wider class of films, whose prestrain depends on both the midplate and the transversal variables. The ansatz for our convergence result uses a specific type of wrinkling, which is built on exotic solutions to the Monge-Ampere equation, constructed via convex integration (Lewicka and Pakzad in Anal PDE 10: 695–727, 2017). We show that the expression for our limit has a natural interpretation in terms of the orthogonal projection of the residual strain onto a suitable subspace. We also show that some type of wrinkling phenomenon is necessary to match the lower bound of the limit in certain circumstances. These results all assume a prestrain of the same order as the thickness; we also discuss why it is natural to focus on that regime by considering what can happen when the prestrain is larger.