A modified Weibull cumulative distribution function, which accounts for the effect of fiber length on the probability of failure, was used to characterize the variation in fiber tensile strength in a SCS-6/ TIMETAL 21S material system and was implemented within the framework of the NASA code MAC/GMC. A parametric study investigating the effect of repeating unit cell architecture and fiber strength distribution on the RUC-averaged ultimate composite strength and failure was performed. Multiscale progressive failure analyses of a tensile dogbone specimen were performed using FEAMAC/ ABAQUS to assess the effect of local variations in fiber strength on the global response. The effect of the RUC architecture, fiber strength distribution, and microscale/ macroscale discretization on the global response was determined. The methodology developed in this work for accounting for statistical variations in microscale properties that feed into macroscale progressive failure analyses can readily be applied to other composite material systems.

We observe a stochastic process 𝑌 on [0,1]^𝑑 (𝑑 ≥ 1) satisfying 𝑑𝑌(𝑡)=𝑛¹/²𝑓(𝑡)𝑑𝑡 + 𝑑𝑊(𝑡), 𝑡 ∈ [0,1]^𝑑, where 𝑛 ≥ 1 is a given scale parameter (`sample size'), 𝑊 is the standard Brownian sheet on [0,1]^𝑑 and 𝑓 ∈ L₁([0,1]^𝑑) is the unknown function of interest. We propose a multivariate multiscale statistic in this setting and prove that the statistic attains a subexponential tail bound; this extends the work of 'Dumbgen and Spokoiny (2001)' who proposed the analogous statistic for 𝑑=1. In the process, we generalize Theorem 6.1 of 'Dumbgen and Spokoiny (2001)' about stochastic processes with sub-Gaussian increments on a pseudometric space, which is of independent interest. We use the proposed multiscale statistic to construct optimal tests (in an asymptotic minimax sense) for testing 𝑓 = 0 versus (i) appropriate Hölder classes of functions, and (ii) alternatives of the form 𝑓 = 𝜇_𝑛𝕀_{𝐵_𝑛}$, where 𝐵_𝑛 is an axis-aligned hyperrectangle in [0,1]^𝑑 and 𝜇_𝑛 ∈ ℝ; 𝜇_𝑛 and 𝐵_𝑛 unknown. In Chapter 3 we use this proposed multiscale statistics to construct honest confidence bands for multivariate shape-restricted regression including monotone and convex functions.

In 1998, the United States National Committee on Biomechanics (USNCB) established an evolving discipline called Functional Tissue Engineering (FTE). In establishing this discipline, the goals of the USNCB were to advance FTE by increasing awareness among tissue engineers about the importance of restoring function when engineering tissue constructs. Another goal was to encourage tissue engineers to incorporate these functional criteria in the design, manufacturing and optimization of tissue engineered constructs. Based on this motivation, an investigation of the structure and mechanical properties of the rabbit patellar tendon will be executed, with the ultimate goal of creating a multiscale soft tissue model based on internal state variable (ISV) theory. Many continuum scale models, mostly phenomenological and microstrucutral, have been created to contribute to the understanding of the complex functional properties of the tendon, such as its anisotropy, inhomogeneity, nonlinearity, and viscoelasticity. However, none of these models have represented the mechanical behavior of the tendon in the presence of internal structural change on a multiscale level. The development of a multiscale ISV model will allow the capture of the irreversible, path history dependent aspects of the material behavior. The objective of this study is to contribute to the multiscale ISV model development by quantifying the structure- property relations. In particular, the fibril distribution at the microstructural level and the resultant multiaxial stress states (longitudinal and transverse compression and longitudinal tension) will be examined).

Modeling of engineering objects with complex heterogeneous material structure at nanoscale level has emerged as an important research problem. In this research, we are interested in multiscale modeling and analysis of mechanical properties of the polymer structures created in the Step and Flash Imprint Lithography (SFIL) process. SFIL is a novel imprint lithography process designed to transfer circuit patterns for fabricating microchips in low-pressure and room-temperature environments. Since the smallest features in SFIL are only a few molecules across, approximating them as a continuum is not completely accurate. Previous research in this subject has dealt with coupling discrete models with continuum hyperelasticity models. The modeling of the post-polymerization step in SFIL involves computing solutions of large nonlinear energy minimization problems with fast spatial variation in material properties. An equilibrium configuration is found by minimizing the energy of this heterogeneous polymeric lattice. Numerical solution of such a molecular statics base model, which is assumed to describe the microstructure completely, is computationally very expensive. This is due to the problem size – on the order of millions of degrees of freedom (DOFs). Rapid variation in material properties, ill-conditioning, nonlinearity, and non-convexity make this problem even more challenging to solve. We devise a method for efficient approximation of the solution. Combining numerical homogenization, adaptive finite element meshes, and goaloriented error estimation, we develop a black-box method for efficient solution of problems with multiple spatial scales. The purpose of this homogenization method is to reduce the number of DOFs, find locally optimal effective material properties, and do goal-oriented mesh refinement. In addition, it smoothes the energy landscape. Traditionally, a finite element mesh is designed after obtaining material properties in different regions. The mesh has to resolve material discontinuities and rapid variations. In our approach, however, we generate a sequence of coarse meshes (possibly 1-irregular), and homogenize material properties on each coarse mesh element using a locally posed constrained convex quadratic optimization problem. This upscaling is done using Moore-Penrose pseudoinverse of the linearized fine-scale element stiffness matrices, and a material independent interpolation operator. This requires solution of a continuous-time Lyapunov equation on each element. Using the adjoint solution, we compute local error estimates in the quantity of interest. The error estimates also drive the automatic mesh adaptivity algorithm. The results show that this method uses orders of magnitude fewer degrees of freedom to give fast and approximate solutions of the original fine-scale problem. Critical to the computational speed of local homogenization is computing Moore-Penrose pseudoinverse of rank-deficient matrices without using Singular Value Decomposition. To this end, we use four algorithms, each having different desirable features. The algorithms are based on Tikhonov regularization, sparse QR factorization, a priori knowledge of the null-space of the matrix, and iterative methods based on proper splittings of matrices. These algorithms can exploit sparsity and thus are fast. Although the homogenization method is designed with a specific molecular statics problem in mind, it is a general method applicable for problems with a given fine mesh that sufficiently resolves the fine-scale material properties. We verify the method using a conductivity problem in 2-D, with chessboard like thermal conductivity pattern, which has a known homogenized conductivity. We analyze other aspects of the homogenization method, for example the choice of norm in which we measure local error, optimum coarse mesh element size for homogenizing SFIL lattices, and the effect of the method chosen for computing the pseudoinverse. / text

Advances in multiscale methods are presented from two perspectives which address the issue of computational complexity of optimizing and inverse analyzing nonlinear composite materials and structures at multiple scales. The optimization algorithm provides several solutions to meet the enormous computational challenge of optimizing nonlinear structures at multiple scales including: (i) enhanced sampling procedure that provides superior performance of the well-known ant colony optimization algorithm, (ii) a mapping-based meshing of a representative volume element that unlike unstructured meshing permits sensitivity analysis on coarse meshes, and (iii) a multilevel optimization procedure that takes advantage of possible weak coupling of certain scales. We demonstrate the proposed optimization procedure on elastic and inelastic laminated plates involving three scales. We also present an adaptive variant of the measure-theoretic approach (MTA) for stochastic characterization of micromechanical properties based on the observations of quantities of interest at the coarse (macro) scale. The salient features of the proposed nonintrusive stochastic inverse solver are: identification of a nearly optimal sampling domain using enhanced ant colony optimization algorithm for multiscale problems, incremental Latin-hypercube sampling method, adaptive discretization of the parameter and observation spaces, and adaptive selection of number of samples. A complete test data of the TORAY T700GC-12K-31E and epoxy #2510 material system from the NIAR report is employed to characterize and validate the proposed adaptive nonintrusive stochastic inverse algorithm for various unnotched and open-hole laminates. Advances in Multiscale methods also provides us a unique tool to study and analyze human bones, which can be seen as a composite material, too. We used two multiscale approaches for fracture analysis of full scale femur. The two approaches are the reduced order homogenization (ROH) and the novel accelerated reduced order homogenization (AROH). The AROH is based on utilizing ROH calibrated to limited data as a training tool to calibrate a simpler, single-scale anisotropic damage model. For bone tissue orientation, we take advantage of so-called Wolff’s law. The meso-phase properties are identified from the least square minimization of error between the overall cortical and trabecular bone properties and those predicted from the homogenization. The overall elastic and inelastic properties of the cortical and trabecular bone microstructure are derived from bone density that can be estimated from the Hounsfield units (HU). For model validation, we conduct ROH and AROH simulations of full scale finite element model of femur created from the QCT and compare the simulation results with available experimental data.

A multiscale approach, coined as the High Order Computational Continua (HC2), has been developed for efficient and accurate analysis and design of reinforced concrete structures. Unlike existing homogenization-like methods, the proposed multiscale approach is capable of handling large representative volume elements (RVE), i.e., the classical assumption of infinitesimally is no longer required, while possessing accuracy of direct numerical simulation (DNS) and the computational efficiency of classical homogenization methods. The multiscale beam and plate elements formulated using the proposed HC2 methodology can be easily incorporated into the existing reinforced concrete design practices. The salient features of the proposed formulation are: (i) the ability to consider large representative volume elements (RVE) characteristic to nonsolid beams,waffle and hollowcore slabs, (ii) versatility stemming from the ease of handling damage, prestressing, creep and shrinkage, and (iii) computational efficiency resulting from model reduction, combined with the damage law rescaling methods that yield simulation results nearly mesh-size independent. The multiscale formulation has been validated against experimental data for rectangular beams, I beams, pretensioned beams, continuous posttension beams, solid slabs, prestressed hollowcore slabs and waffle slabs.

Dans le contexte du « Tout Electrique », les fabricants de l’aéronautique cherchent à augmenter la puissance embarquée tout en limitant la masse de ces dispositifs électriques. Une des solutions envisagées est d’augmenter la densité de flux magnétique des matériaux magnétiques de ces appareils. L’inconvénient de l’emploi de ces matériaux réside dans leurs déformations sous l’effet du champ magnétique. Dans le cas des noyaux magnétiques de transformateurs, ceux-ci sont composés d’un empilement d’une centaine de tôles magnétiques d’épaisseur variant entre 0,2 et 0,5mm. La déformation successive des tôles du transformateur est à l’origine d’un bruit acoustique indésirable. La source principale de ces déformations est la magnétostriction qui provient du réarrangement sous champ magnétique de la structure en domaines du matériau. Dans le cadre de ces travaux de thèse, nous nous intéressons à l’alliage Fe-27%Co produit par la société APERAM Alloys Imphy, commercialement appelé AFK1. Le choix de cet alliage provient du fait qu’il présente une aimantation à saturation la plus élevée de tous les matériaux ferromagnétiques (2,4T). Son emploi permettrait alors un gain certain de densité de puissance. Selon une gamme métallurgique particulière, l’AFK1 présente une basse magnétostriction isotrope, qui s’illustre par une déformation nulle jusqu’à 1,5T puis par une déformation à saturation de l’ordre de 10ppm. L’objectif principal de ces travaux de thèse consiste à déterminer l’origine d’un tel comportement et les mécanismes associés. Les résultats expérimentaux montrent que les conditions de traitements thermiques semblent avoir un effet sur le comportement magnétostrictif. On montre par ailleurs que la magnétostriction est indépendante de l’orientation cristallographique de l’AFK1. Des essais de magnétostriction sous contrainte mécanique ont permis de supposer que l’AFK1 disposait d’une structure en domaines principalement composée de parois à 180°. La mise en place de cette structure a pu être confirmée par microscopie magnéto-optique (effet Kerr). Afin de mieux comprendre l’origine de l’orientation des domaines dans le matériau, l’influence de la géométrie d’échantillon sur le comportement magnétostrictif a également été étudiée au cours de ces travaux de thèse. Une modélisation du comportement faiblement magnétostrictif a finalement été proposée par le biais d’une approche multi-échelle. Le modèle met en évidence la nécessité de considérer une proportion non négligeable de domaines séparés par des parois à 180° pour restituer la basse magnétostriction de l’AFK1. / The main challenge in the aeronautical field concerns the increase of higher power density electrical devices onboard aircrafts. One of the solutions proposed is to increase the magnetic flux density of magnetic materials which compose these devices. The main drawback of this solution leads in the high deformation the materials concerned exhibit under magnetic field. For example, the core of onboard transformers is composed of a stack of about hundred of magnetic steel sheets, with a thickness range between 0.2 and 0.5mm. The deformation of the entire structure leads to an unwanted acoustic noise that originates from the high magnetostriction deformation of the material deriving from the change of magnetic domains configuration under magnetic field. In this thesis work, the magnetostrictive behaviour of the Fe-27$%$Co alloy is studied. This magnetic alloy is produced and marketed by APERAM Alloys Imphy as AFK1. This material leads to a low and isotropic magnetostrictive behaviour after an appropriate metallurgical process. The deformation is null up to 1.5T and the magnetic saturation is reached with a deformation lower than 10ppm. The main goal of this thesis is to understand the origin of the low magnetostrictive behaviour and to model it. The experimental results show that thermal annealing changes significantly the magnetostriction. In addition, we prove that low magnetostriction exhibits no crystallographic orientation dependence. Magnetostriction tests carried out under a mechanical loading show that a microstructure mainly composed magnetic domains separated by 180$^circ$ domain walls can explain the behaviour. The presence of this magnetic configuration was confirmed by magneto optical microscopy observations (Kerr effect) associated with a macroscopic geometry effect and residual magnetic field in the furnaces. A multiscale modeling of the low magnetostriction has been proposed next. This modeling helps us to confirm the requirement of about 80% of grains composed of a bi-domain magnetic structure to simulate low magnetostrictive behaviour in accordance with experiments.

Accurately simulating material systems in a virtual environment requires the synthesis and utilization of all relevant information regarding performance mechanisms for the material occurring over a range of length and time scales. Multiscale modeling is the basis for the Integrated Computational Materials Engineering (ICME) Paradigm and is a powerful tool for accurate material simulations. However, while ICME has experienced adoption among those in the metals community, it has not gained traction in polymer research. This thesis seeks establish a hierarchical multiscale modeling methodology for simulating polymers containing secondary phases. The investigation laid out in the chapters below uses mesoscopic Finite Element Analysis (FEA) as a foundation to build a multiscale modeling methodology for polymer material systems. At the mesoscale a Design of Experiments (DOE) parametric study utilizing FEA of polymers containing defects compared the relative impacts of a selection of parameters on damage growth and coalescence in polymers. Of the parameters considered, the applied stress state proved to be the most crucial parameter affecting damage growth and coalescence. At the macroscale, the significant influence of the applied stress state on damage growth and coalescence in polymers (upscaled from the mesoscale) motivated an expansion of the Bouvard Internal State Variable (ISV) (Bouvard et al. 2013) polymer model stress state sensitivity. Deviatoric stress invariants were utilized to modify the Bouvard ISV model to account for asymmetry in polymer mechanical performance across different stress states (tension, compression, torsion). Lastly, this work implements a hierarchical multiscale modeling methodology to examine parametric effects of heterogeneities on Polymer/Clay Nanocomposite’s (PCNs) mechanical performance under uncertainty. A Virtual Composite Structure Generator (VCSG) built three-dimensional periodic Representative Volume Elements (RVEs) coupled to the Bouvard ISV model and a Cohesive Zone Model (CZM) which featured a Traction-Separation (T-S) rule calibrated to results upscaled from Molecular Dynamics (MD) simulations. A DOE parametric examination utilized the RVEs to determine the relative effects of a selection of parameters on the mechanical performance of PCNs. DOE results determined that nanoclay particle orientation was the most influential parameter affecting PCN elastic modulus while intercalated interlamellar gallery strength had the greatest influence on PCN yield stress

Multiscale modeling is an effective tool for integrating different computational methods, creating a way of modeling diverse chemical and physical phenomena. Presented are studies on a variety of chemical problems at different computational scales and also the combination of different computational methods to study a single phenomenon. The methods used encompass density functional theory (DFT), molecular dynamics (MD) simulations and finite element analysis (FEA). The DFT studies were conducted both on the molecular level and using plane-wave methods. The particular topics studied using DFT are the rational catalyst design of complexes for C—H bond activation, oxidation of nickel surfaces and the calculation of interaction properties of carbon dioxide containing systems directed towards carbon dioxide sequestration studies. Second and third row (typically precious metals) transition metal complexes are known to possess certain electronic features that define their structure and reactivity, and which are usually not observed in their first-row (base metal) congeners. Can these electronic features be conferred onto first-row transition metals with the aid of non-innocent and/or very high-field ligands? Using DFT, the impact of these electronic features upon methane C—H bond activation was modeled using the dipyridylazaallyl (smif) supporting ligand for late, first-row transition metal (M) imide, oxo and carbene complexes (M = Fe, Co, Ni, Cu; E = O, NMe, CMe2). To promote a greater understanding of the process and nature of metal passivation, first-principles analysis of partially oxidized Ni(111) and Ni(311) surface and ultra-thin film NiO layers on Ni(111) was performed. A bimodal theoretical strategy that considers the oxidation process using either a fixed GGA functional for the description of all atoms in the system, or a perturbation approach, that perturbs the electronic structure of various Ni atoms in contact with oxygen by application of the GGA+U technique was applied. Binding energy of oxygen to the nickel surfaces, charge states of nickel and oxygen, and the preferred binding mode of oxygen to nickel were studied to gain a better understanding of the formation of oxide layers. Using density functional theory, the thermodynamic properties for developing interaction potentials for molecular dynamics simulations of carbon dioxide systems were calculated. The interactions considered are Ni + H2O, Ni + Ni, Ni + CO2, CO2 + CO2, CO2 + H2O and H2O + H2O. These systems were chosen as the possible interactions that can occur when carbon dioxide is stored in the ocean. Molecular dynamics simulations using the results from the DFT studies were also conducted. Finally, thermal conduction analysis was performed on layered functionally graded materials (FGM) subjected to thermal shock by sudden cooling of the material in order to investigate the results obtained from three different mixing laws: linear, quadratic, and half-order. The functionally graded material considered was a composite of nickel and carbon nanotubes at different compositions varying from two to five layers. The middle layers for the three to five layers are composed of graded (i.e., gradually changing) percentages of nickel and carbon nanotube. The thermal conductivity, specific heat and density for the composites were calculated depending on the percentages of materials in each layer, and assuming different rules of mixture.

Multiscale modeling has become an important tool in material mechanics because material behavior can exhibit varied properties across different length scales. The use of multiscale modeling is essential for accurately capturing these characteristics and predicting material behavior. Mesh-free methods have also been gaining attention in recent years due to their innate ability to handle complex geometries and large deformations. These methods provide greater flexibility and efficiency in modeling complex material behavior, especially for problems involving discontinuities, such as fractures and cracks. Moreover, mesh-free methods can be easily extended to multiple lengths and time scales, making them particularly suitable for multiscale modeling. The thesis focuses on two specific problems of multiscale modeling with mesh-free methods. The first problem is the atomistically informed constitutive model for the study of high-pressure induced densification of silica glass. Molecular Dynamics (MD) simulations are carried out to study the atomistic level responses of fused silica under different pressure and strain-rate levels, Based on the data obtained from the MD simulations, a novel continuum-based multiplicative hyper-elasto-plasticity model that accounts for the anomalous densification behavior is developed and then parameterized using polynomial regression and deep learning techniques. To incorporate dynamic damage evolution, a plasticity-damage variable that controls the shrinkage of the yield surface is introduced and integrated into the elasto-plasticity model. The resulting coupled elasto-plasticity-damage model is reformulated to a non-ordinary state-based peridynamics (NOSB-PD) model for the computational efficiency of impact simulations. The developed peridynamics (PD) model reproduces coarse-scale quantities of interest found in MD simulations and can simulate at a component level. Finally, the proposed atomistically-informed multiplicative hyper-elasto-plasticity-damage model has been validated against limited available experimental results for the simulation of hyper-velocity impact simulation of projectiles on silica glass targets. The second problem addressed in the thesis involves the upscaling approach for multi-porosity media, analyzed using the so-called MultiSPH method, which is a sequential SPH (Smoothed Particle Hydrodynamics) solver across multiple scales. Multi-porosity media is commonly found in natural and industrial materials, and their behavior is not easily captured with traditional numerical methods. The upscaling approach presented in the thesis is demonstrated on a porous medium consisting of three scales, it involves using SPH methods to characterize the behavior of individual pores at the microscopic scale and then using a homogenization technique to upscale to the meso and macroscopic level. The accuracy of the MultiSPH approach is confirmed by comparing the results with analytical solutions for simple microstructures, as well as detailed single-scale SPH simulations and experimental data for more complex microstructures.