This thesis focused on the mechanisms involved in negative thermal expansion of 2D/3D lattice structures. The effects of varying the constituent materials and geometry were explored. The lattices had geometries similar to those found in light-weight structures in many transport applications, including aerospace and spacecraft. One specific case was to determine how to reduce the coefficient of thermal expansivity (CTE) of such structures to near zero, by using two constituent materials with contrasting CTEs, without incurring penalties in terms of other elastic and failure properties, mass and manufacturability. The lattice geometries able to exhibit altered CTE were explored, and penalties in terms of other elastic properties were quantified. The results were scale-independent and so were generic to all such lattices. Analytical prediction and generic relationships between the geometries of the lattices and their performance were proposed. Experimental validation of the model predictions was undertaken using physical samples. The thermomechanical properties were simulated by commercial finite element method (FEM) codes (Ansys 11, Ansys, Inc.). Ansys parametric design language was adopted to generate large sets of solutions to be evaluated against chosen criteria. Results show small or, in some cases, no penalties to be paid in terms of stiffness and mass for implementing dual-material lattices with near-zero CTE. Such lattices may compete favourably with high-cost and high-density materials (e.g. Invar) and the manufacture of dual-material lattices can be by standard processes or alternative new process such as Additive Layer Manufacturing (ALM). An example of truss core sandwich application for aerospace application was modelled by FEM. Applications as cores in sandwich panels might be the first route by which the ALM manufacturing process is required to develop dual-material capability.

Solid foams have been studied for years for their ability to mitigate damage from sudden impact. Small explosive attacks threaten to damage or destroy key structures in some parts of the world. A newly developed material, carbon foam, may offer the ability to mitigate the effects of such blasts. This project investigates the energy absorbing properties of carbon and polyurethane based foams in dynamic compression to illustrate their viability to protect concrete structures from the damaging effects of pressure waves from a small blast. Cellular solid mechanics fundamentals and a survey of the microscopic cellular structure of each type of foam are discussed. Experiments were performed in three strain rate regimes: low strain rate compression testing, middle strain rate impact testing, and high strain rate blast testing to reveal mechanical behavior. Experiments show a 7.62 cm (3”) thick hybrid composite layered foam sample can protect a concrete wall from a small blast.

While the structure and mechanical properties of natural cellular solids such as wood and trabecular bone have been extensively studied in the past, the structural design and underlying deformation mechanisms of natural cellular solids with very high mineral contents (> 90 wt%), which we term as natural ceramic cellular solids, are largely unexplored. Many of these natural ceramic cellular solids, despite their inherent brittle constituent biominerals (e.g., calcite or aragonite), exhibit remarkable mechanical properties, such as high stiffness and damage tolerance. In this thesis, by carefully selecting three biomineralized skeletal models with distinctly different cellular morphologies, including the honeycomb-like structure in cuttlefish bone (or cuttlebone), the stochastic open-cell structure in sea urchin spines, and the periodic open-cell structure in starfish ossicles, I systematically investigate the mechanical design strategies of these natural ceramic cellular solids. The three model systems are cuttlefish Sepia officinalis, sea urchin Heterocentrotus mammillatus, and starfish Protoreaster nodosus, respectively. By investigating the relationship between their mechanical properties and structural characteristics, this thesis reveals some novel structural design strategies for developing lightweight, tough, strong, and stiff ceramic cellular solids. The internal skeleton of S. officinalis, also known as cuttlebone, has a porosity of 93 vol% (constituent material: 90 wt% aragonite), which is a multichambered structure consisting of horizontal septa and thin vertical walls with corrugated cross-sectional profiles. Through systematic ex-situ and synchrotron-based in-situ mechanical measurements and collaborative computational modeling, we reveal that the vertical walls in the cuttlebone exhibit an optimal waviness gradient, which leads to compression-dominant deformation and asymmetric wall fracture, accomplishing both high stiffness (8.4 MN∙m/kg) and high energy absorption (4.4 kJ/kg). Moreover, the distribution of walls reduces stress concentrations within the horizontal septa, facilitating a larger chamber crushing stress and more significant densification. For the stochastic open-cell foam-like structure, also known as stereom (porosity: 60-80 vol%, constituent material: 99 wt% calcite) in H. mammillatus, we first developed a computer vision-based algorithm that allows for quantitative analysis of the cellular network of these structures at both local individual branch and node level as well as the global network level. This open-source algorithm could be used for analyzing both biological and engineering open-cell foams. I further show that the smooth, highly curved branch morphology with near-constant surface curvature in stereom results in low-stress concentration, which further leads to dispersed crack formation upon loading. Combined synchrotron in-situ analysis, electron microscopic analysis, and computational modeling further reveal that the fractured branches are efficiently jammed by the small throat openings within the cellular structure. This further leads to the formation of damage bands with densely packed fracture pieces. The continuous widening of the damage bands through progressive microfracture of branches at the boundaries contributes to the observed high plateau stress during compression, thereby contributing to its high energy absorption (17.7 kJ/kg), which is comparable and even greater than many synthetic metal- and polymer-based foams. Lastly, this thesis leads to the discovery of a unique dual-scale single-crystalline porous lattice structure (porosity: 50 vol%, constituent material: 99 wt% calcite) in the ossicles of P. nodosus. At the atomic level, the ossicle is composed of single-crystal biogenic calcite. At the lattice level, the ossicle's microstructure organizes as a diamond-triply periodic minimal surface (TPMS) structure. Moreover, the crystallographic axes at atomic and lattice levels are aligned, i.e., the c-axis of calcite is aligned with the [111] direction of the diamond-TPMS lattice. This single crystallinity co-alignment at two levels mitigates the compliance of calcite in the c-axis direction by utilizing the stiff <111> direction of the diamond-TPMS lattice. Furthermore, 3D in-situ mechanical characterizations reveal that the presence of crystal defects such as 60° and screw dislocations at the lattice level suppresses slip-like fracture along the {111} planes of the calcitic diamond-TPMS lattice upon loading, achieving an enhanced energy absorption capability. Even though the skeleton of echinoderm is made up of single-crystal calcite, the structure fractures in a conchoidal manner rather than along the clipping plane, which can continuously fracture the fragments into small pieces and enhance energy dissipation. / Doctor of Philosophy / The application of engineering ceramic cellular solids as structural components is limited by their brittleness and flaw sensitivity. In contrast, nature has evolved ceramic cellular materials such as sea sponge, sea urchin spine, and diatom shells that are simultaneously lightweight, strong, and damage-tolerant. These properties are thought to be achieved by the structure design of the component of those materials. Learning design strategies from these natural ceramic cellular solids is significant for developing lightweight bio-inspired ceramic materials with improved mechanical performance. In this thesis, I investigated mechanical design strategies from natural ceramic cellular solids in three model systems, i.e., cuttlebone from cuttlefish Sepia officinalis, spines from sea urchin Heterocentrotus mammillatus, ossicles from starfish Protoreaster nodosus. These three natural ceramic porous solids have high mineral content in the constituent materials (> 90 wt%) and have a highly porous structure (porosity: 50 vol%-93 vol%). These three model systems are selected to represent the analogs of three typical structure forms of synthetic cellular solids, i.e., honeycomb-like structures, stochastic and periodic open-cell structures, respectively. This thesis aims to establish a quantitative relationship between the 3D multiscale structure and deformation/toughening behavior for these selected natural ceramic cellular solids via a combination of different experimental and computational approaches.

Omnipresence of heterogeneity is conspicuous in all creations of nature. Heterogeneity manifests itself in many forms at different scales, both in time and space. Engineering domain being an exotic fusion of human creativity and ever-increasing demands exemplifies the ubiquity of heterogeneity. Surprisingly, the plethora of materials we see around seem to stem from myriad combination of few base materials identified as elements in chemistry. Further, a simple rearrangement of atoms in these materials leads to allotropes with startling contrasts in properties. Similarly, micro- and meso-scales in heterogeneous materials also dis-play this phenomenon. Human requirements propelled by necessities and wants have leveraged heterogeneity deliberately or naively. In the context of engineering materials, light weight heterogeneous materials like composites and cellular solids are outstanding inventions from the last century. The present thesis highlights this phenomenon on a meso-scale to explore generalized variants of circular and elliptical honeycomb structures (HCSs) with an emphasis on their effective transverse elastic responses, a crucial pillar of engineering design and analysis. Homogenized or effective properties are an extension of continuum hypothesis, conceived for ease in analyses. E ective properties are employed in multi-scale analyses resulting in less complex models for analysis, for example, for predicting the speed of wave propogation. The thesis extends and generalizes existing close-packed circular and elliptical HCSs to more broader configurations. Simpler periodic arrangement of the unit cells from numerous exotic possibilities directly incorporates Design for Manufacture and Assembly (DFMA) philosophy and o ers a potential scope for analysis by simpler tools resulting in handy expressions which are of great utility for designer engineers. In this regard, analytical expressions for moduli having compact forms in the case of circular HCS are developed by technical theories and rigorous theory of elasticity. Regression analysis expressions for the moduli of elliptical HCS are presented, and the elasticity solutions for the same are highlighted. The thesis consists of seven chapters with Chapter 1 presenting generalized circular and elliptical HCSs as a potential avenue beyond composite materials. Following a survey of pertinent HCS literature of these HCSs, research gaps and scope are delineated. Chapter 2 briefly y summarizes the ideas, concepts and tools including analytical and numerical methods. This chapter sets the ground for the analysis of generalized circular and elliptical HCS in the following four chapters. Following the classification of the circular HCSs, Chapter 3 assesses the complete transverse elastic responses of generalized circular HCS through technical theories which are a first-order approximation. Here, thin ring theory and the more elaborate curved beam theory are employed as models to assess the moduli. Normal moduli - E and - are obtained by employing Castigliano method, while shear moduli (G ) are obtained by solving the differential equations derived in terms of displacements. Compact expressions for moduli presented wherever possible furnish the designer with a range of moduli for different configurations and modular ratios (Ey=Ex). The results show the range of applicability of technical theories within 5% of FEA. For hexagonal arrays, these results are more refined than those in literature; while the same are new for other configurations. Surprisingly, the more elaborate curved beam theory offers no better results than the thin ring theory. Chapter 4 extends the aforementioned task of assessing the complete trans-verse elastic moduli of generalized circular HCS by employing rigorous theory of elasticity (TOE) which is a second-order approximation. Utilizing Airy stress function in polar coordinates, the boundary value problems resulting from modeling of the circular HCS under different loads are solved analytically in conjunction with FEA employing contact elements. Contact elements circumvent the point loads which give finite values of displacements in technical theories and singular values in TOE. A widely used idea of employing distributed load, statically equivalent to point load, is invoked to empower TOE. The distributed load is assumed a priori and the contact length is obtained from FEA employing con-tact elements. Thus, FEA compliments the present analytical methods. Results demonstrate a very good match between analytical method in conjunction with FEA and numerical results from FEA; the error is within 5% for very thick ring (thickness-radius ratio 0.5). Further, computationally and numerically efficient expressions for displacements give better results with same computational facility. To illustrate the effect of coating on effective moduli, a limited study based on thin ring theory and elasticity theories is undertaken in Chapter 4. The study explores the effects of moduli and thickness ratios of substrate to coating on the effective normal moduli. Employing thin ring theory with only flexure as the bending mode, we get compact expressions giving good match for very thin rings in all confifigurations. The elasticity approach presented for square array demonstrates a very good match with FEA for thick rings. Coatings offer a strategy to increase the effective moduli with same dimensions. Chapter 5 broadens the scope of circular HCS by considering elliptical HCSs. While generalized circular HCS can cater to anisotropic requirement to an extent, larger spectrum is offered by considering elliptical honeycomb structures. In this regard, a generalized version of concentric thin coated elliptical HCS is investigated for transverse moduli. Thin HCSs are explored by technical theories as in circular HCS. However, a lack of exact compact-form expressions necessitates the use of regression analysis. The resulting expressions are presented in terms of ellipticity ratio describing the ovality of the ellipse and geometric parameters. Normal moduli are obtained by Castigliano method implemented in MATHE-MATICA, but shear moduli are obtained from FEA employing beam elements. The need for FEA employing beam elements stems from the subtle fact that Castigliano method implicitly assumes preclusion of rigid body motions, while shear loading for shear moduli evaluation entails rigid body motions. Interestingly, curved beam theory, as in circular HCS, offers no better refinement in assessing the moduli as compared to thin ring theory. The graphs showing the moduli with respect to thickness and modular ratios are presented as design maps to aid the designer. Chapter 6 extends the works of thin concentric coated elliptical to thicker concentric and a novel confocal elliptical HCS, a variant of elliptical HCS. In this regard, thick concentric and confocal elliptical HCS by elasticity approach are attempted for a simple case. Airy stress function in polar coordinates is tried for concentric elliptical HCS. Confocal HCS analysis employs stress function in terms of elliptical coordinate system. After proving the correctness of the stress function for both the cases by comparing the reconstructed boundary conditions with actual boundary conditions, the restrictions in solving the case of rings under load over a small region is highlighted. A parametric study for moduli is under-taken by employing FEA. These are presented as design graphs which compare and contrast the two variants of elliptical HCS on the same graphs. The modular ratio (Ey=Ex) is conspicuously more for confocal elliptical HCS than concentric elliptical HCS. Chapter 7 gives the conclusions in a nutshell, and explores the feasibility of stress evaluation of heterogeneous media on the lines of effective media theory.

Quick and accurate predictions of equivalent properties for thin-walled composite structures are required in the preliminary design process. Existing literature provides analytical solutions to some structures but is limited to particular cases. No unified approach exists to tackle homogenization of thin-walled structures such as beams, plates, or three-dimensional structures using the thin-walled approximation. In this work, a unified approach is proposed to obtain equivalent properties for beams, plates, and three-dimensional structures for thin-walled composite structures using mechanics of structure genome. The adopted homogenization technique interprets the unit cell associated with the composite structures as an assembly of plates, and the overall strain energy density of the unit cell as a summation of the plate strain energies of these individual plates. The variational asymptotic method is then applied to drop all higher-order terms and the remaining energy is minimized with respect to the unknown fluctuating functions. This has been done by discretizing the two-dimensional unit cell into one-dimensional frame elements in a finite element description. This allows the handling of structures with different levels of complexities and internal geometry within a general framework. Comparisons have been made with other works to show the advantages which the proposed model offers over other methods.

Size effects in out-of-plane bending stiffness of honeycomb cellular materials were studied using analytical mechanics of solids modeling, fabrication of samples and mechanical testing. Analysis predicts a positive size-effect relative to continuum model predictions in the flexure stiffness of a honeycombed beam loaded in out-of-plane bending. A method of determining the magnitude of that effect for several different methods of constructing or assembling square-celled and hexagonal-celled materials, using both single-walled and doubled-walled construction methods is presented. Hexagonal and square-celled honeycombs, with varying volume fractions were fabricated in Nylon 12 using Selective Laser Sintering. The samples were mechanically tested in three-point and four point-bending to measure flexure stiffness. The results from standard three-point flexure tests, did not agree with predictions based on a mechanics of solids model for either square or hexagonal-celled samples. Results for four-point bending agreed with the mechanics of solids model for the square-celled geometries but not for the hexagonal-celled geometries. A closed form solution of an elasticity model for the response of the four-point bending configuration was developed, which allows interpretation of recorded displacement data at two points and allows separation the elastic bending from the localized, elastic/plastic deformation that occurs between the loading rollers and the specimen’s surface. This localized deformation was significant in the materials tested. With this analysis, the four-point bending data agreed well with the mechanics of solids predictions. / text

Des méthodes micromécaniques spécifiques ont été développées pour la détermination du comportement effectif de matériaux cellulaires dotés d’une architecture discrète à l’échelle microscopique. La méthode d’homogénéisation discrète a été appliquée à des structures tissées monocouches ainsi qu’à l’os trabéculaire. La topologie discrète initiale de ces milieux est remplacée à l’échelle mésoscopique par un milieu effectif anisotrope micropolaire, qui rend compte des effets d’échelles observés. Ces méthodes d’homogénéisation permettent d’accéder à des propriétés classiques et non classiques dont la mesure expérimentale est souvent difficile. Des modèles 3D ont été développé afin de décrire la rupture fragile et ductile de l’os trabéculaire, incorporant des effets de taille des surfaces d’écoulement plastique. Nous avons construit par des analyses éléments finis de la microstructure de l’os trabéculaire un milieu de substitution 3D homogène, orthotrope de type couple de contraintes, sur la base d’une équivalence en énergie. Les tissus osseux ont la capacité d’adapter leur densité locale et leur taille et forme aux stimuli mécaniques. Nous avons développé des modèles de remodelage interne et externe dans le cadre de la thermodynamique des processus irréversibles, aux échelles cellulaire et macroscopique. Finalement, le remodelage interne anisotrope a été couplé à l’endommagement de fatigue, dans le cadre de la théorie continue de l’endommagement / A methodology based on micromechanics has been developed to determine the effective behavior of network materials endowed with a discrete architecture at the microscopic level. It relies on the discrete homogenization method, which has been applied to textile monolayers and trabecular bones. The initially discrete topology of the considered network materials results after homogenization at the mesoscopic level in anisotropic micropolar effective continuum, which proves able to capture the observed internal scale effects. Such micromechanical methods are useful to remedy the difficulty to measure the effective mechanical properties at the intermediate mesoscopic level scale. The bending and torsion responses of vertebral trabecular bone beam specimens are formulated in both static and dynamic situations, based on the Cosserat theory. 3D models have been developed for describing the multiaxial yield and brittle fracture behavior of trabecular bone, including the analysis of size-dependent non-classical plastic yield. We have constructed by FE analyses a homogeneous, orthotropic couple-stress continuum model as a substitute of the 3D periodic heterogeneous cellular solid model of vertebral trabecular bone, based on the equivalent strain energy approach. Bone tissues are able to adapt their local density and load bearing capacities as well as their size and shape to mechanical stimuli. We have developed models for combined internal and external bone remodeling in the framework of the thermodynamics of irreversible processes, at both the cellular and macroscopic levels. We lastly combined anisotropic internal remodeling with fatigue continuum damage

Functional polymer foams are considered as a promising field which could potentially produce foams with added value. Specifically, functionally graded foams are materials which are expected to provide better mechanical properties while preserving low density in comparison with regular foams. In this thesis, a preparation process of such foams is designed, examination of prepared structure and comparison of mechanical properties with regular foams is performed. The obtained results are discussed and further research in this field is proposed.