Nonlinear Finite Element Analysis of Nanoindentation of Viral Capsids

Recent Atomic Force Microscope (AFM) nanoindentation experiments measuring mechanical response of the protein shells of viruses have provided a quantitative description of their strength and elasticity. To better understand and interpret these measurements, and to elucidate the underlying mechanisms, this paper adopts a course-grained modeling approach within the framework of three-dimensional nonlinear continuum elasticity. Homogeneous, isotropic, elastic, thick shell models are proposed for two capsids: the spherical Cowpea Chlorotic Mottle Virus (CCMV), and the ellipsocylindrical bacteriophage $\phi 29$. As analyzed by the finite element method, these models enable parametric characterization of the effects of AFM tip geometry, capsid dimensions, and capsid constitutive descriptions. The generally nonlinear force response of capsids to indentation is shown to be insensitive to constitutive details, and greatly influenced by geometry. Nonlinear stiffening and softening of the force response is dependent on the AFM tip dimensions and shell thickness. Fits of the models capture the roughly linear behavior observed in experimental measurements and result in estimates of Young's moduli of $\approx$280--360 MPa for CCMV and $\approx$4.5 GPa for $\phi 29$.Comment: 24 pages, 10 figures, submitted to Biophysical Journa

Flapping, swirling and flipping: Non-linear dynamics of pre-stressed active filaments

Initially straight slender elastic rods with geometrically constrained ends buckle and form stable two-dimensional shapes when compressed by bringing the ends together. It is also known that beyond a critical value of the pre-stress, clamped rods transition to bent, twisted three-dimensional equilibrium shapes. Recently, we showed that pre-stressed planar shapes when immersed in a dissipative fluid and animated by nonconservative follower forces exhibit stable large-amplitude flapping oscillations. Here, we use time-stepper methods to analyze the three-dimensional instabilities and dynamics of pre-stressed planar and non-planar filament configurations when subject to active follower forces and dissipative fluid drag. First, we find that type of boundary constraint determines the nature of the non-linear patterns following instability. When the filament is clamped at one end and pinned at the other with follower forces directed towards the clamped end, we observe only stable planar (flapping) oscillations termed flapping result. When both ends are clamped however, we observe a secondary instability wherein planar oscillations are destabilized by off-planar perturbations and result in fully three-dimensional swirling patterns characterized by two distinct time-scales. The first time scale characterizes continuous and unidirectional swirling rotation around the end-to-end axis. The second time scale captures the rate at which the direction of swirling reverses or flips. The overall time over which the direction of swirling flips is very short compared to the long times over which the filament swirls in the same direction. Computations indicate that the reversal of swirling oscillations resembles relaxation oscillations with each cycle initiated by a sudden jump in torsional deformation and then followed by a period of gradual decrease in net torsion until the next cycle of variations

On the Statics, Dynamics, and Stability of Continuum Robots: Model Formulations and Efficient Computational Schemes

This dissertation presents advances in continuum-robotic mathematical-modeling techniques. Specifically, problems of statics, dynamics, and stability are studied for robots with slender elastic links. The general procedure within each topic is to develop a continuous theory describing robot behavior, develop a discretization strategy to enable simulation and control, and to validate simulation predictions against experimental results.Chapter 1 introduces the basic concept of continuum robotics and reviews progress in the field. It also introduces the mathematical modeling used to describe continuum robots and explains some notation used throughout the dissertation.The derivation of Cosserat rod statics, the coupling of rods to form a parallel continuum robot (PCR), and solution of the kinematics problem are reviewed in Chapter 2. With this foundation, soft real-time teleoperation of a PCR is demonstrated and a miniature prototype robot with a grasper is controlled.Chapter 3 reviews the derivation of Cosserat rod dynamics and presents a discretization strategy having several desirable features, such as generality, accuracy, and potential for good computational efficiency. The discretized rod model is validated experimentally using high speed camera footage of a cantilevered rod. The discretization strategy is then applied to simulate continuum robot dynamics for several classes of robot, including PCRs, tendon-driven robots, fluidic actuators, and concentric tube robots.In Chapter 4, the stability of a PCR is analyzed using optimal control theory. Conditions of stability are gradually developed starting from a single planar rod and finally arriving at a stability test for parallel continuum robots. The approach is experimentally validated using a camera tracking system.Chapter 5 provides closing discussion and proposes potential future work

Nonlinear Dynamics

This volume covers a diverse collection of topics dealing with some of the fundamental concepts and applications embodied in the study of nonlinear dynamics. Each of the 15 chapters contained in this compendium generally fit into one of five topical areas: physics applications, nonlinear oscillators, electrical and mechanical systems, biological and behavioral applications or random processes. The authors of these chapters have contributed a stimulating cross section of new results, which provide a fertile spectrum of ideas that will inspire both seasoned researches and students

Modeling of Masonry Structures at Multiple Scales

Zdivo je materiál použitý ve většině stavebních památek na celém světě. Spolehlivé nástroje pro analýzu zděných konstrukcí jsou zapotřebí nejen pro vyhodnocení jejich seismické zranitelnosti, ale také při návrhu opatření směřujících k obnovení či zvýšení únosnosti existujících budov, které si zaslouží ochranu. Zdivo je nelineární, heterogenní a anizotropní materiál, jehož vlastnosti silně závisejí na základních stavebních jednotkách, tedy blocích (cihlách) a maltě, a na jejich prostorovém uspořádání. Pro simulaci mechanického chování zděných konstrukcí byla vyvinuta řada modelů, které se liší mírou rozlišení. Pro velké konstrukce vede snaha o výpočetní efektivitu ke zjednodušeným modelům, charakterizovaným rozdělením zděných stěn na makroprvky. Významným zástupcem této skupiny modelů je metoda ekvivalentního rámu. Její podstatou je nahrazení zděné stěny idealizovaným rámem, přičemž panely jsou modelovány jako nosníky charakterizované odpovídajícím mechanickým chováním. Míra rozlišení může být zvýšena tím, že se každý makroprvek uvažuje jako homogenizované kontinuum s vlastnostmi, které reprodukují celkovou odezvu určitého výseku heterogenní mikrostruktury. Formulace vhodného konstitutivního zákona ale není lehkou úlohou. Tento zákon by měl fenomenologicky reprodukovat mechanické chování materiálu, včetně vzniku tahových trhlin, smykového pokluzu, drcení v tlaku a dalších jevů. Navíc tento přístup vyžaduje těžkopádnou identifikaci mechanických parametrů, které není vždy snadné určit na základě běžných laboratorních testů materiálu. K popisu role základních stavebních jednotek a jejich interakce může posloužit model formulovaný na mikroúrovni, který explicitně bere v úvahu jednotlivé bloky, maltu a rozhraní mezi nimi. Tato práce se zabývá zděnými konstrukcemi na několika úrovních rozlišení. Problémy s formulací modelů ekvivalentního rámu v případě nepravidelného rozmístění otvorů se zkoumají na základě porovnání výsledků pro ekvivalentní rámy s výsledky získanými metodou konečných prvků, o které lze předpokládat, že lépe postihuje skutečné chování nepravidelných stěn. Provedená parametrická analýza zděných pilířů modelovaných jako homogenizované kontinuum je zaměřena na posouzení vlivu tvaru a svislého tlakového zatížení na nelineární statické chování. Pozornost se pak přesouvá na jemnější úrovně rozlišení, na nichž se zkoumá lokalizace nepružného přetváření, která ovlivňuje konstitutivní zákony pro modelování zdiva na makro a mikroúrovni. Provádí se lokalizační analýza ortotropního makroskopického modelu formulovaného podle teorie plasticity s více plochami plasticity, v jejímž rámci jsou odvozeny analytické podmínky lokalizace potvrzené simulacemi metodou konečných prvků. V závěru je vyvinut mikromechanický model pro pravidelné zdivo a pomocí něj se na reprezentativním objemu materiálu analyzují lokalizační vlastnosti, ovlivněné velikostí tohoto objemu a předpokládanými směry periodicityMasonry represents the material used in the great majority of the world building heritage structures. Reliable tools for analysis of masonry structures are needed not only for seismic vulnerability assessment but also to properly design interventions to restore and strengthen existing buildings, which deserve to be preserved. Masonry is a nonlinear, heterogeneous, and anisotropic material whose properties strongly depend on its microstructure, typically composed of two phases, blocks and mortar, and on the way it is assembled. To simulate the mechanical behavior of masonry structures, numerous models have been developed, characterized by different detailing levels. For large structures, the need for computational efficiency leads to simplified models characterized by the subdivision of masonry walls in macro-elements. A notable example of this group of models is the equivalent-frame method, which consists of identifying the masonry wall with an ideal frame, where panels are modeled as beams characterized by proper mechanical behavior. The detailing level can be increased by considering each macro-element as a homogenized continuum, assuming that, at the scale of representation, masonry can be treated as a continuum having mechanical properties that reproduce the overall response of a certain portion of the heterogeneous microstructure. However, the formulation of a suitable constitutive law is not an easy task. It should phenomenologically reproduce the material mechanics, including tension cracking, shear sliding, compressive crushing, and many other aspects. Moreover, this approach requires a cumbersome identification of mechanical parameters that are not always easy to determine from basic experimental tests on the material. To consider the role of each constituent and the effects of their interactions, a microscale model can be set up, where blocks, mortar joints, and mortar-block interfaces are represented explicitly. In this work, masonry structures are studied at several detailing levels. An issue affecting equivalent-frame models, namely the presence of irregularity in the wall opening layout, is addressed by comparing equivalent-frame results with finite-element ones, which are assumed to better represent the actual behavior of irregular walls. A parametric analysis on masonry piers, modeled as a homogenized continuum, is carried out, aimed to assess the influence of the height-to-width ratio and the vertical compression load on the nonlinear static behavior. The focus is then shifted to finer scales. The localization analysis of an orthotropic macro-scale model in the framework of multi-surface plasticity is presented, deriving analytical localization conditions corroborated by finite element simulations. Finally, a microscale model for regular masonry is developed to analyze the localization properties of the representative volume element, also by investigating the role of its size and periodicity directions

New Tools for Viscoelastic Spectral Analysis, with Application to the Mechanics of Cells and Collagen across Hierarchies

Viscoelastic relaxation spectra are essential for predicting and interpreting the mechanical responses of materials and structures. For biological tissues, these spectra must usually be estimated from viscoelastic relaxation tests. Interpreting viscoelastic relaxation tests is challenging because the inverse problem is expensive computationally. We present here (1) an efficient algorithm and (2) a quasi-linear model that enable rapid identification of the viscoelastic relaxation spectra of both linear and nonlinear materials. We then apply these methods to develop fundamental insight into the mechanics of collagenous and fibrotic tissues. The first algorithm, which we term the discrete spectral approach, is fast enough to yield a discrete spectrum of time constants that is sufficient to fit a measured relaxation spectrum with an accuracy insensitive to further refinement. The algorithm fits a discrete spectral generalized Maxwell (Maxwell-Wiechert) model, which is a linear viscoelastic model, to results from a stress-relaxation test. The discrete spectral approach was tested against trial data to characterize its robustness and identify its limitations and strengths. The algorithm was then applied to identify the viscoelastic response of reconstituted collagen and engineered fibrosis tissues, revealing that cells actively adapted the ECM, and that cells relax at multiple timescales, including one that is fast compared to those of the ECM. The second algorithm, which we term the discrete quasi-linear viscoelastic (DQLV) approach, is a spectral extension of the Fung quasi-linear viscoelastic (QLV) model, a standard tool for characterizing biological materials. The Fung QLV model provides excellent fits to most stress-relaxation data by imposing a simple form upon a material\u27s temporal relaxation spectrum. However, model identification is challenging because the Fung QLV model\u27s “box” shaped relaxation spectrum, predominant in biomechanics applications, because it can provide an excellent fit even when it is not a reasonable representation of a material\u27s relaxation spectrum. The DQLV model is robust, simple, and unbiased. It is able to identify ranges of time constants over which the Fung QLV model\u27s typical box spectrum provides an accurate representation of a particular material\u27s temporal relaxation spectrum, and is effective at providing a fit to this model. The DQLV spectrum also reveals when other forms or discrete time constants are more suitable than a box spectrum. After validating the approach against idealized and noisy data, we applied the methods to analyze medial collateral ligament stress-relaxation and sinusoidal excitation data and identify the strengths and weaknesses of an optimal Fung QLV fit. Taken together, the tools in this dissertation form a comprehensive approach to characterizing the mechanics of viscoelastic biological tissues, and to dissecting the micromechanical mechanisms that underlie a tissue\u27s viscoelastic responses