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Capillaries in tumours are often severely buckled (in a plane perpendicular to the axis) and / or chaotic in their direction. We develop a model of these phenomena using nonlinear solid mechanics. Our model focusses on the immediate surrounding of a capillary. The vessel and surrounding tissue are modelled as concentric annulii. The growth is dependent on the concentration of a nutrient (oxygen) diffusing from the vessel into the tumour interstitium. The stress is modelled using a multiplicative decomposition of the deformation gradient F=F_e F_g. The stress is determined by substituting the elastic deformation gradient F_e (which gives the deformation gradient from the hypothetical configuration to the current configuration) into a hyperelastic constitutive model as per classical solid mechanics. We use a Blatz-Ko model, parameterised using uniaxial compression experiments. The entire system is in quasi-static equilibrium, with the divergence of the stress tensor equal to zero. We determine the onset of buckling using a linear stability analysis. We then investigate the postbuckling behaviour by introducing higher order perturbations in the deformation and growth before using the Fredholm Alternative to obtain the magnitude of the buckle. Our results demonstrate that the growth-induced stresses are sufficient for the capillary to buckle in the absence of external loading and / or constraints. Planar buckling usually occurs after 2-5 times the cellular proliferation timescale. Buckles with axial variation almost always go unstable after planar buckles. Buckles of fine wavelength are initially preferred by the system, but over time buckles of large wavelength become energetically more favourable. The tumoural hoop stress T_{ThetaTheta} is the most invariant (Eulerian) variable at the time of buckling: it is typically of the order of the tumoural Young's Modulus when this occurs.

Slender structures are elegant aesthetically. The insufficiency in knowledge of the real resistance to buckling of very slender reinforced concrete columns leads to an exaggeration of the sizes of the columns. _The examples of concrete compression members cited and constructed in Industry on a global basis suggest that very slender columns have inherent safety both from the point of view of the ultimate strength and stability. The strengths of columns given. by the British codes would seem to be exceeded by many of the long slender reinforced concrete columns and struts which have been used Internationally. Both the theoretical and the experimental short term investigations have been carried out to establish the behaviour of hinged, very slender reinforced concrete columns at various stages'of axial loading. Forty three very slender reinforced concrete columns of two different square cross sections with two sizes of longitudinal reinforcements with lateral ties were cast. Slenderness rates, L A, were varied from 30 to 79. Special factors were obtained to relate the actual modulus of elasticity of concrete in columns at buckling failure to a knowledge of the initial modulus of elasticity of concrete in control cylinder specimens. Both theoretical and experimental graphs of load against moment, made dimensionless for critical sections of columns have been obtained. Dimensionless load-moment interaction diagrams using material failure as the criterion have been superimposed on these graphs to show considerable inherent material strength of the tested columns near buckling collapse failures. A theory using the fundamental approach has, been developed to predict the deflected shape and moments along the, heights of the columns at various stages of loading. The proposed theory predicts with good correlations the experimental deflections and moments of any loading stages of the columns. The theory has been used to obtain the required variables, to arrive at the initial predicted design loads of the investigated columns. Good correlations of the moments derived from observed strains have also been obtained. The developed theory predicts satisfactorily the buckling collapse loads of the columns. Although the theory has been derived for axially I loaded very slender reinforced concrete-columns, it seems to accept satisfactorily eccentricities of up to about 10 mm. This was confirmed after extensive comparisons of the theoretical buckling collapse loads with the applicable tests of other authors. Creep In the columns investigated was discovered to be one of the major factors for serious consideration. This was conclusively revealed from the observations on the last two very long term creep tests on columns. The actual safe sustained loads for these very slender columns of slenderness ratios, L/H, between 40 and 79 seem to be between 33% and 19% of the short term buckling collapse loads. The reduced modulus approach to predict the safe long term sustained loads seems to give reasonable values for L/H ratios of 40 and 50. The recommendations given for the proposed design of very slender reinforced concrete columns seem to be adequate and simple to use in practice. They are further simplified by the derivation of two equations for the reduction factors, R, for the slenderness ratios between 36 and 40 and between 40 and 79 respectively. The investigation has proved that very slender reinforced concrete columns are very dangerous structural members, as they tend to have violent buckling failures. Nevertheless, It must be prudent not to design against disaster at any cost. This Investigation seemed to have enhanced considerably knowledge of the design of very slender reinforced concrete columns.

Snap-through is a striking instability in which an elastic object rapidly jumps from one state to another. It is seen in the leaves of the Venus flytrap plant and umbrellas flipping on a windy day among many other examples. Similar structures that snap-through are used to generate fast motions in soft robotics, switches in micro-scale electronics and artificial heart valves. Despite the ubiquity of snap-through in nature and engineering, its dynamics is usually only understood qualitatively. In this thesis we develop analytical understanding of this dynamics, focussing on how the mathematical structure underlying the snap-through transition controls the timescale of instability. We begin by considering the dynamics of 'pull-in' instabilities in microelectromechanical systems (MEMS) - a type of snap-through caused by electrostatic forces in which the motions are dominated by fluid damping. Using a lumped-parameter model, we show that the observed time delay near the pull-in transition is a type of critical slowing down - a so-called 'bottleneck' due to the 'ghost' of a saddle-node bifurcation. We obtain a scaling law describing this slowing down, and, in the process, unify a large range of experiments and simulations that exhibit delay phenomena during pull-in. We also investigate the pull-in dynamics of MEMS microbeams, extending the lumped-parameter approach to incorporate the details of the beam geometry. This provides a model system in which to understand snap-through of a continuous elastic structure due to external loading. We develop a perturbation method that systematically exploits the proximity to pull-in to reduce the governing equations to a simpler evolution equation, with a structure that highlights the saddle-node bifurcation. This allows us to analyse the bottleneck dynamics in detail, which we compare with previous experimental and numerical data. The remainder of the thesis is concerned with the dynamics of snap-through in macroscopic systems. In particular, we explore the extent to which dissipation is required to explain anomalously slow snap-through. Considering an elastic arch as an archetype of a snapping system, we use the perturbation method developed earlier to show that two bottleneck regimes are possible, depending delicately on the relative importance of external damping. In particular, we show that critical slowing down occurs even in the absence of damping, leading to a new scaling law for the snap-through time that is confirmed by elastica simulations and experiments. In many real systems material viscoelasticity is present to some degree. Finally, we examine how this influences the snap-through dynamics of a simple truss-like structure. We present a regime diagram that characterises when the timescale of snap-through is controlled by viscous, elastic or viscoelastic effects.

Graphene is the strongest material ever discovered and has an extremely high Young's modulus (~ 1 GPa). Although stretching the sp2 covalent bonds between carbon atoms is very difficult, significant deflections do develop in graphene membranes. In fact, thermal rippling naturally emerges in graphene at any finite temperature. Moreover, the topological defects mediating plastic deformation often cause out-of-plane crumpling, an effective way to reduce the total elastic energy of the topological defect. It is even possible to design specific stress states to produce periodic wrinkles in graphene with adjustable wave lengths.This work aims to understand how buckling influences the elastic and plastic behavior of graphene-based nanostructures. While the elastic moduli may be straightforwardly computed using structure optimization techniques with applied test stresses, it is a nontrivial task to obtain elastic properties at any specific temperature. Further, linear elastic theories are not able to describe large buckling because the second order nonlinear terms in the definition of the Lagrangian finite strain tensor cannot be neglected. In addition, the existence of topological defects and constraints need to be properly treated. Finally, buckling and defects of a curved surface such as a nanotube is even more complicated and poses other intriguing challenges.To proceed, we employ Monte Carlo techniques to obtain fundamental elastic properties of graphene at desired temperatures, which supplies useful inputs for a nonlinear continuum model for graphene. This model not only takes into account, in a suitable manner, both large out-of-plane buckling and interactions among edge dislocations with periodic boundary conditions, but also serves as a handy tool for simulating nanoindentation experiments and the controlled wrinkling of graphene. At last, we focus on the Stone-Wales defect mediated plasticity of CNTs. Specifically, a kinetic Monte Carlo framework is designed to model the defect dynamics over a long time scale. We find that in large nanotubes, a chain of closely packed dislocations (called a "dislocation worm") may have less buckling and lower formation energy than the conventional dislocation glide under high tensile stresses.

Elastic filaments subjected to hydrodynamic forcing are a common class of fluid-structure interaction problems. In biology, they are crucial to cytoskeletal motions of the cell, the locomotion of micro-organisms, and mucal transport. In engineering, they are often the source of non-Newtonian rheological properties and a variety of complex fluid behavior such as hydrodynamic instabilities and chaotic mixing. At the heart of many of these dynamics is the competition between an elastic backbone and viscous forces acting to deform it, tangled with the anisotropic shape of such filaments as well as slowly decaying hydrodynamic interactions in a Stokesian fluid. In this work, we use slender-body theory for low-Reynolds-number hydrodynamics to address theoretically and computationally a few such problems of physical and biological significance. We first describe the tumbling of polymers in shear flow and the suppression of thermal fluctuations in extensional flow. A theory for the stretch-coil transition of semiflexible polymers is also developed. We then turn to the transport properties of semiflexible filaments in a flow setup that mimics some of the dynamics of biological polymers in motility assays. Thermal fluctuations frequently trap polymers within vortical cells, and are shown to lead to subdiffusive transport at long times. The mechanism behind this anomalous feature and the subtle role of flexibility is emphasized. Then we focus on the sedimentation of flexible filaments. In the weakly flexible regime, a multiple-scale asymptotic expansion is used to obtain expressions for filament shapes and peculiar trajectories. In the highly flexible regime, we show that a filament sedimenting along its long axis is susceptible to a buckling instability. Our predictions are corroborated by detailed numerical simulations. Finally, we look at suspensions of sedimenting elastic fibers, emphasizing the role of filament shape, flexibility, and long-ranged hydrodynamic interactions. We develop a mean-field theory for such a suspension, and its stability to perturbations of fiber concentration is analytically explored. Detailed numerical simulations are also performed to verify these predictions and elucidate the microstructural mechanisms tied to the growth or suppression of this instability.

Slender structures like plates and shells -- for which at least one dimension is much smaller than the others -- are lightweight, flexible, and offer considerable strength with little material. As such, these structures are abundant in nature (e.g. flower petals, eggshells, and blood vessels) and design (e.g. bridge decks, fuel tanks, and soda cans). However, with slenderness comes suceptibility to large and often sudden deformations, which can be wildly nonlinear, as bending is energetically preferable to stretching. Though once considered categorically undesirable, these instabilities are often coveted nowadays in the engineering community. They provide mechanical explanations for observations in nature like the wrinkled structure of the brain or the snapping mechanism of the Venus fly trap, and when precisely controlled, enable the design of functional devices like artificial muscles or self-propelling microswimmers. As a prerequisite, these achievements require a thorough understanding of how thin structures "shape-shift" in response to stimuli and confinement. Advancing this fundamental knowledge is the goal of this thesis. In the first two chapters, we consider the shape-selection of shells and plates that are confined by their environment. The shells are made by residual swelling of silicone elastomers, a process that mimics differential growth, and causes initially flat structures to irreversibly morph into curved shapes. Flattening the central region forces further reconfiguration, and the confined shells display multi-lobed buckling patterns. These experiments, finite element (FE) simulations, and a scaling argument reveal that a single geometric confinement parameter predicts the general features of this shape-selection. Next, in experiments and molecular dynamics (MD) simulations, we constrain intrinsically flat sheets in the same manner, so that their center remains flat when we quasi-statically force them through a ring. In the absence of planar confinement, these sheets form a well-studied conical shape (the developable cone or d-cone). Our annular d-cone buckles circumferentially into patterns that are qualitatively similar to the confined shells, despite the distinct curvatures and loading methods. This is explained by the dominant role of confinement geometry in directing deformation, which we uncover via a scaling argument based on the elastic energy. There are also marked differences between the way plates and shells change shape, which we highlight when we investigate the rich dynamics of reconfiguration. In the final two chapters, we demonstrate how mechanics, geometry, and materials can inform the design of structures that use instabilities to function. We observe in experiments that dynamic loading causes a spherical elastomer shell to buckle at ostensibly subcritical pressures, following a substantial time delay. To explain this, we show that viscoelastic creep deformation lowers the critical load in the same predictable, quantifiable way that a growing defect would in an elastic shell. This work offers a pathway to introduce tunable, time-controlled actuation to existing mechanical actuators, e.g. pneumatic grippers. The final chapter aims at reducing the energy input required for bistable actuators, wherein snap-through instability is typically induced by a stimulus applied to the entire shell. To do so, we combine theory with 1D finite element simulations of spherical caps with a non-homogeneous distribution of stimuli--responsive material. We demonstrate that restricting the active area to the shell boundary allows for a large reduction in its size, while preserving snap-through behavior. These results are stimulus-agnostic, which we demonstrate with two sets of experiments, using residual swelling of bilayer silicone elastomers as well as a magneto-active elastomer. Our findings elucidate the underlying mechanics, offering an intuitive route to optimal design for efficient snap-through.

When a solid object is placed under a load, it will deform to a new shape. Typically, these shape changes are modest and predictable, but sometimes, when a critical load is reached, the object will undergo an elastic instability and adopt a dramatically different and more complicated shape. Traditionally, elastic instabilities, such as buckling and wrinkling, have been studied as failure modes in stiff material systems. However, soft highly deformable solids, such as rubbers and biological tissues, can undergo instabilities without failure, offering the opportunity to utilize instabilities to change their shape. Furthermore, the largestrain mechanics of soft solids introduces many new instabilities not seen in traditional materials. Evolution is known to exploit soft elastic instabilities to sculpt brains, guts and other developing organs, and human engineers are interested in using them to create shape switching devices. There is a pressing need to understand what types of elastic instability exist, what shapes they form, and how these shapes can be controlled. In this thesis, I address each of these challenges. I first present a new elastic instability, in which a cylindrical channel through a soft solid adopts a peristaltic shape upon loading with sufficient internal pressure. I then present a theory of pattern selection in surface elastic instabilities, including the compressive wrinkling of a stiff sheet on a soft substrate, and the gravitational fingering of a soft gel layer. Using higher order perturbation theory, I find, in both cases, that patterns of hexagonal dents are favoured, and I present a symmetry argument that hexagonal patterns are generic. Finally, I present a technique to manipulate the patterns formed in layer/substrate buckling by patterning the system with holes. This simple technique shows that instability patterns can be designed, opening new engineering opportunities.

This thesis is concerned with a combined experimental and theoretical investigation of the static and dynamic response of an axially compressed sandwich structure. For the static response problem of sandwich structures, a two-dimensional mechanical model is developed to predict the global and local buckling of a sandwich beam, using classical elasticity. The face sheet and the core are assumed as linear elastic orthotropic continua in a state of planar deformation. General buckling deformation modes (periodic and non-periodic) of the sandwich beam are considered. On the basis of the model developed here, validation and accuracy of several previous theories are discussed for different geometric and material properties of a sandwich beam. The appropriate incremental stress and conjugate incremental finite strain measure for the instability problem of the sandwich beam, and the corresponding constitutive model are addressed. The formulation used in the commercial finite element package is discussed in relation to the formulation adopted in the theoretical derivation. The Dynamic response problem of a sandwich structure subjected to axial impact by a falling mass is also investigated. The dynamic counterpart of the celebrated Euler buckling problem is formulated first and solved by considering the case of a slender column that is impacted by a falling mass. A new notion, that of the time to buckle is introduced, which is the corresponding critical quantity analogous to the critical load in static Euler buckling. The dynamic bifurcation buckling analysis is extended to thick sandwich structures using an elastic foundation model. A comprehensive set of impact test results of sandwich columns with various configurations are presented. Failure mechanisms and the temporal history of how a sandwich column responds to axial impact are discussed through the experimental results. The experimental results are compared against analytical dynamic buckling studies and finite element based simulation of the impact event.

This thesis presents four investigations into mechanical aspects of soft thin structures, focusing on the effects of stochastic and thermal fluctuations and of material inhomogeneities. First, we study the self-organization of arrays of high-aspect ratio elastic micropillars into highly regular patterns via capillary forces. We develop a model of capillary mediated clustering of the micropillars, characterize the model using computer simulations, and quantitatively compare it to experimental realizations of the self-organized patterns. The extent of spatial regularity of the patterns depends on the interplay between cooperative enhancement and history-dependent stochastic disruption of order during the clustering process. Next, we investigate the influence of thermal fluctuations on the mechanics of homogeneous, elastic spherical shells. We show that thermal fluctuations give rise to temperature- and size-dependent corrections to shell theory predictions for the mechanical response of spherical shells. These corrections diverge as the ratio of shell radius to shell thickness becomes large, pointing to a drastic breakdown of classical shell theory due to thermal fluctuations for extremely thin shells. Finally, we present two studies of the mechanical properties of thin spherical shells with structural inhomogeneities in their walls. The first study investigates the effect of a localized reduction in shell thickness—a soft spot—whereas the second studies shells with a smoothly varying thickness. In both cases, the inhomogeneity significantly alters the response of the shell to a uniform external pressure, revealing new ways to control the strength and shape of initially spherical elastic capsules., Engineering and Applied Sciences

The bending of elastic strips and rods is a field of research that continues to offer new possibilities for exploration. This dissertation focuses on two distinct problems within this context. These are the search for the equilibrium shape of thin inextensible elastic strips, such as a M�öbius strip made out of paper, and the optimal shape of tapered columns that are stable against buckling. A theoretical approach based on the principle of virtual work is used to investigate both problems. This produces novel governing non-linear differential equations that describe both equilibrium and form. In order to discover the equilibrium shapes, numerical algorithms are developed that are based on Dynamic Relaxation. There are two ways in which they are used, one as an explicit form-finding tool, and the other as a way of solving differential equations. Results are provided that extend current theoretical models. The numerical schemes produce three-dimensional shapes for strips, going beyond the canonical Möbius strip, and solution shapes for tapered columns made from non-linear elastic materials. With the aid of analytical and numerical tools, finding the form of the M�öbius strip and the tallest possible column are interesting challenges in the search for new shapes that are driven by physical and material rules. These have applicability in structural engineering, architecture, nano-technology and even artistic endeavour.