Dissertations / Theses on the topic 'Peridynamics'

Since the last century a lot of effort has been spent trying to analyze damage and crack evolution in solids. This field is of interest because of the many applications that require the study of the behavior of materials at the micro- or nanoscale, i.e. modeling of composites and advanced aerospace applications. Peridynamics is a recently developed theory that substitutes the differential equations that constitute classical continuum mechanics with integral equations. Since integral equations are valid at discontinuities and cracks, peridynamics is able to model fracture and damage in a more natural way, without having to work around mathematical singularities present in the classical continuum mechanics theory. The objective of the present work is to show how peridynamics can be implemented in finite element analysis (FEA) using a mesh of one-dimensional truss elements instead of 2-D surface elements. The truss elements can be taken as a representation of the bonds between molecules or particles in the body and their strength is found according to the physical properties of the material. The possibility implementing peridynamics in a finite element framework, the most used method for structural analysis, is critical for expanding the range of problems that can be analyzed, simplifying the verification of the code and for making fracture analysis computationally cheaper. The creation of an in-house code allows for easier modifications, customization and enrichment if more complex cases (such as multiscale modeling of composites or piezoresistive materials) are to be analyzed. The problems discussed in the present thesis involve plates with holes and inclusions subjected to tension. Displacement boundary conditions are applied in all cases. The results show good agreement with theory as well as with empirical observation. Stress concentrations reflect the behavior of materials in real life, cracks spontaneously initiate and debonding naturally happens at the right locations. Several examples clearly show this behavior and prove that peridynamics is a promising tool for stress and fracture analysis.
Master of Science

Prediction of fracture and failure is a challenging research area. There are various methods available in the literature for this purpose including well-known finite element (FE) method. FE method is a powerful technique for deformation and stress analysis of structures. However, it has various disadvantageous in predicting failure due to its mathematical structure since it is based on classical continuum mechanics (CCM). CCM has governing equations in the form of partial differential equations. These equations are not valid if the displacement field is discontinuous as a result of crack occurance. In order to overcome this problem, a new continuum mechanics formulation was introduced and named as Peridynamics. Peridynamics uses integrals equations as opposed to partial differential equations of CCM. Moreover, it does not contain any spatial derivatives. Hence, its equations are always valid regardless of discontinuities. In this thesis, the applications of Peridynamics for marine structues are demonstrated. Particularly, the Peridynamic equations are rederived for simplified structures commonly used in marine structures including beams and plates. Furthermore, underwater shock response of marine composites is investigated. Finally, the peridynamic formulation for contact analysis which can be used for collision and grounding of ship structures is demonstrated. In order to reduce the computational time, several solution strategies are explained.

This study presents peridynamic field equations for mechanical deformation, thermal diffusion, moisture diffusion, electric potential distribution, porous flow and atomic diffusion in either an uncoupled or a coupled manner. It is a nonlocal theory with an internal length parameter. Therefore, it can capture physical phenomenon for the problems which include non-local effects and are not suitable for classical theories. Moreover, governing equations of peridynamics are based on integro-differential equations which permits the determination of the field variable in spite of discontinuities. Inherent with the nonlocal formulations, the imposition of the boundary conditions requires volume constraints. This study also describes the implementation of the essential and natural boundary conditions, and demonstrates the accuracy of their implementation. Solutions coupled field problems concerning plastic deformations, thermomechanics, hygrothermomechanics, hydraulic fracturing, thermal cracking of fuel pellet and electromigration are constructed. Their comparisons with the finite element predictions establish the validity of the PD field equations for coupled field analysis.

The classical theory of solid mechanics is rooted in the assumption of a continuous distribution of mass within a body. It employs partial differential equations (PDEs) with significant smoothness to obtain displacements and internal forces of the body. Although classical theory has been applied to wide range of engineering problems, PDEs of the classical theory cannot be applied directly on a discontinuity such as cracks. Peridynamics is considered to be an alternative and promising nonlocal theory of solid mechanics that, by replacing PDEs of classical theory with integral or integro-differential equations, attempts to unite the mathematical modelling of continuous media, cracks and particles within a single framework. Indeed, the equations of peridynamic are based on the direct interaction of material points over finite distances. Another concept, derived from the peridynamic approach to cope with engineering problems with discontinuities, is that of the peridynamic differential operator (PDDO). The PDDO uses the non-local interaction of the material points in a way similar to that of peridynamics. PDDO is capable to recast partial derivatives of a function through a nonlocal integral operator whose kernel is free of using any correction function. In this dissertation, application of peridaynamics and PDDO, to three different important engineering problems including fatigue fracture, thermo-mechanics and sloshing phenomena, is examined comprehensively. To cope with fatigue fracture problems, an algorithm has been developed in such a way that the increment of damage due to fatigue is added to that due to the static increment of the opening displacement. A one degree of freedom cylinder model has been used to carry out an efficient comparison of the computational performance of three fatigue degradation strategies. The three laws have been implemented in a code using bond based peridynamics (BBPD) to simulate fatigue crack propagation. Both the cylinder model and the bond base peridynamics code provide the same assessment of the three fatigue degradation strategies. To deal with thermo-mechanical problems, an effective way is proposed to use a variable grid size in a weakly coupled thermal shock peridynamic model. The proposed numerical method is equipped with stretch control criterion to transform the grid discretization adaptively in time. Hence, finer grid spacing is only applied in limited zones where it is required. This method is capable of predicting complex crack patterns in the model. By introducing fine grid discretization over the boundaries of the model the surface (softening) effect can be reduced. The accuracy and performance of the model are examined through problems such as thermo-elastic and thermal-shock induced fracture in ceramics. Finally to investigate sloshing phenomena, the PDDO has been applied to the solution of problems of liquid sloshing in 2D and 3D tanks with potential flow theory and Lagrangian description. Moreover, liquid sloshing in rectangular tanks containing horizontal and vertical baffles are investigated to examine the robustness and accuracy of PDDO. With respect to other approaches such as meshless local Petrov-Galerkin (MLPG), volume of fluid (VOF) and and local polynomial collocation methods the examples are solved with a coarser grid of nodes. Using this new approach, one is able to obtain results with a high accuracy and low computational cost.
La teoria classica della meccanica dei solidi, formulata tramite equazioni differenziali alle derivate parziali (PDEs), è basata sull'assunzione di una distribuzione continua di massa all'interno di un corpo. Sebbene la teoria classica sia stata applicata ad un'ampia gamma di problemi ingegneristici, le equazioni differenziali su cui è basata non possono essere risolte agevolmente in presenza di una discontinuità come, ad es., una cricca. La peridinamica è considerata un'alternativa ed una promettente teoria non-locale della meccanica dei solidi che, rimpiazzando le equazioni differenziali con equazioni integrali o integro-differenziali, unisce in un’unica formulazione la modellazione dei solidi continui e quella di discontinuità (ad es. cricche). Le equazioni della peridinamica sono basate sull'interazione diretta di punti materiali all’interno di una regione di influenza di dimensioni finite. Un altro concetto, derivato dall'approccio peridinamico è l'operatore differenziale peridinamico (PDDO). Questo operatore è in grado di valutare le derivate parziali di una generica funzione per mezzo di una opportuna funzione integrale non-locale. In questa tesi viene esaminata l'applicazione della peridinamica e del PDDO a tre problemi ingegneristici: la frattura per fatica, i fenomeni termo-meccanici ed i fenomeni di sloshing. Per simulare i problemi di frattura per fatica, è stato sviluppato un algoritmo che valuta sia l'incremento del danno per fatica, legato al numero dei cicli di carico, che l’incremento del danno statico, legato all’aumento dell’apertura della cricca. Sono state proposte tre leggi di danneggiamento per fatica le cui prestazioni computazionali sono state valutate per mezzo di un modello ad un grado di libertà. Inoltre le stesse tre leggi sono state implementate in un codice basato sulla formulazione peridinamica di tipo bond-based, per simulare la propagazione delle cricche per fatica. Sia il modello ad un grado di libertà che il codice scritto utilizzando la formulazione peridinamica individuano la stessa legge di danneggiamento per fatica (fra le 3 studiate) quale più efficiente ed accurata da un punto di vista numerico. Per affrontare problemi di natura termo-meccanica, viene proposto un approccio alternativo che utilizza una griglia di nodi di dimensione variabile all’interno di un modello peridinamico. Il modello numerico proposto modifica in maniera adattiva la dimensione di griglia per garantire una elevata accuratezza dei risultati ed un minore sforzo computazionale: la griglia più raffinata è usata soltanto nelle aree in cui le cricche si propagano. L’approccio proposto è stato utilizzato in un primo momento per lo studio di fenomeni termo-elastici quindi per l’analisi di fenomeni di propagazione di cricche a seguito di sollecitazioni termo-meccaniche. Infine, il PDDO è stato impiegato per investigare i fenomeni di sloshing di liquidi in serbatoi bi-dimensionali e tri-dimensionali studiati con la teoria del flusso a potenziale e la descrizione Lagrangiana. Rispetto ad altri approcci, come ad esempio il metodo locale meshless Petrov-Galarkin, il metodo dei volumi di fluido ed il metodo locale di collocazione polinomiale, l’approccio PDDO si rivela particolarmente efficace dato che fornisce risultati di accuratezza analoga (rispetto ai risultati ottenuti con gli altri approcci) impiegando un numero minore di nodi per descrivere il sistema.

The classical continuum theory is not capable of predicting failure without an external crack growth criteria and treats the interface having zero thickness. Alternatively, a nonlocal continuum theory referred to as peridynamic theory eliminates these shortcomings by utilizing formulation that uses displacements, rather than derivatives of displacements, and including material failure in its constitutive relations through the response functions. This study presents a new response function as part of the peridynamic theory to include thermal loading. Furthermore, an efficient numerical algorithm is presented for solution of peridynamic equations. Solution method relies on the discretization of peridynamic equations at collocation points resulting in a set of ordinary differential equations with respect to time. These differential equations are then integrated using explicit methods. In order to improve numerical efficiency of the computations, spatial partitioning is introduced through uniform grids as arrays of linked lists. Furthermore, the domain of interest is divided into subunits each of which is assigned to a specific processor to utilize parallel processing using OpenMP. In order to obtain the static solutions, the adaptive dynamic relaxation method is developed for the solution of peridynamic equations. Furthermore, an approach to couple peridynamic theory and finite element analysis is introduced to take advantage of their salient features. The regions in which failure is expected are modeled using peridynamics while the remaining regions are modeled utilizing finite element method. Finally, the present solution method is utilized for damage prediction of many problems subjected to mechanical, thermal and buckling loads.

Peridynamics, a recently proposed non-local continuum theory, is particularly suitable to describe fracture phenomena in a wide range of materials. One of most common techniques for its numerical implementation is based on a mesh-free approach, in which the whole body is discretized with a uniform grid and a constant horizon, the latter related to the length-scale of the material and/or of the phenomenon analysed. As a consequence of that, computational resources may not be used efficiently. The present work proposes adaptive refinement/scaling algorithms for 2D and 3D peridynamic grids, to reduce the computational cost of peridynamic based software. Adaptive refinement/scaling is here applied to the study of dynamic crack propagation in brittle materials. Refinement is activated by using a new trigger concept based on the damage state of the material, coupled with the more traditional energy based trigger, already proposed in the literature. The use of a varying horizon and grid spacing over the grid may introduce some anomalies on the numerical peridynamic solution, such anomalies are investigated in detail through static and dynamic analyses. Moreover, while the scientific community is working to assess the full potential of peridynamics, few researchers have observed indirectly that the evolution of crack paths can follow, in an unphysical way, the axes of symmetry of the grid. The main parameter affecting such a numerical phenomenon seems to be the value of the m ratio, namely the ratio between the horizon and the grid spacing. The dependence of the crack path on the grid orientation would be a serious drawback for peridynamic based software since it would undermine what is believed to be one of its most important advantages over other computational methods, i.e. its capability to simulate (multiple) crack nucleation, propagation, branching and interaction in solids in a simple way. Finally, in order to show the effectiveness of the proposed approach, several examples of crack propagation in both 2D and 3D problems are presented. Then, the results obtained are compared with those obtained with other numerical methods and with experimental data.
La Peridynamica, una teoria non locale del continuo proposta recentemente, è particolarmente adatta a descrivere fenomeni di frattura in una vasta gamma di materiali. Una delle tecniche più comuni per la sua implementazione numerica è basata su un approccio senza mesh, in cui l'intero corpo viene discretizzato con una griglia uniforme e un orizzonte costante, essendo quest'ultimo in relazione con la lunghezza di scala del materiale e/o del fenomeno analizzato. Di conseguenza le risorse computazionali possono non essere utilizzate in modo efficiente. Il presente lavoro si propone di sviluppare gli algoritmi per l’implementazione dell’adaptive grid refinement and scaling per griglie peridinamiche 2D e 3D, con lo scopo di ridurre il costo computazionale dei software basati sulla peridynamica. Questo approccio viene applicato allo studio della propagazione dinamica di cricche in materiali fragili. Il refinement viene attivato utilizzando un nuovo concetto di “innesco” che si basa sullo stato di danneggiamento del materiale, accoppiato con il più tradizionale innesco basato su un criterio energetico, già proposto in letteratura. L' utilizzo di un orizzonte e di un passo di griglia variabile può introdurre nella soluzione numerica della peridynamica alcune anomalie, che vengono analizzate dettagliatamente tramite analisi statiche e dinamiche. Inoltre, mentre la maggior parte della comunità scientifica sta lavorando per valutare a pieno le potenzialità della peridynamica, solo alcuni ricercatori hanno osservato indirettamente come il percorso della cricca possa seguire, in modo chiaramente non realistico, gli assi di simmetria della griglia. Il principale parametro che influisce su tale comportamento sembra essere il valore assunto dal rapporto m, definito come il rapporto tra l'orizzonte e il passo della griglia. La dipendenza del percorso della cricca dall'orientamento della griglia sarebbe un grave ostacolo per lo sviluppo di un software basato sulla peridynamica, poiché ciò porterebbe a pregiudicare quella che si ritiene essere uno dei suoi vantaggi più importanti rispetto ad altri metodi di calcolo, ossia la sua capacità di simulare la nucleazione (anche multipla), la propagazione, la ramificazione e l’interazione di cricche in materiali solidi in modo semplice. Successivamente, al fine di dimostrare l'efficacia del metodo proposto, vengono presentati alcuni esempi di propagazione di cricche per problemi 2D e 3D. Infine, i risultati ottenuti sono confrontati con quelli ottenuti con altri metodi numerici e con dati sperimentali.

This study concerns the development of the peridynamic strain energy density function for a Neo-Hookean type membrane under equibiaxial, planar and uniaxial loading conditions. The material parameters for each loading case are determined by equating the peridynamic strain energy to those of the classical continuum mechanics. Therefore, the peridynamic equations of motion are derived based on the Neo-Hookean model under the assumption of incompressibility. Numerical results concern the deformation of a membrane without and with a defect in the form of a hole, an inclusion and a crack under equibiaxial, planar and uniaxial loading conditions. As part of the verification process, the peridynamic predictions are compared with those of finite element analysis. For all defect types and loading conditions, the comparisons indicate excellent agreement.

Geopolymer is a novel cementitious material which can be a potential alternative to ordinary Portland cement (OPC) for all practical applications. However, until now research on this revolutionary material is limited mainly to experimental studies, which have the limitations in considering the details of the atomic- and meso-scale structure and atomic scale mechanisms that govern the properties at the macro-scale. Most experimental studies on geopolymer have been conducted focusing only on the macroscopic properties and considering it as a single-phase material. However, research has shown that geopolymer is a composite material consisting of geopolymer binder (GB), unreacted source material, and, in the presence of Ca in the source material, calcium silicate hydrate (CSH). Therefore, in this research, a multiscale/multiphysics modeling approach has been taken to understand geopolymer structure and mechanical properties under varying conditions and at different length scales. First, GB was prepared at the atomic scale using molecular dynamics (MD) simulations with varying Si/Al ratios and water contents within the nano voids. The MD simulated geopolymer structure was validated based on comparison with experiments using X-ray pair distribution function (PDF), infra-red (IR) spectra, coordination of atoms, and density. The results indicate that the highest strength occurs at a Si/Al ratio of 2-3 and the presence of molecular water negatively affects the mechanical properties of GB. The loss of strength for GB with increased water content is linked to the diffusion of Na atoms and subsequent weakening of Al tetrahedra. The GB was also subjected to nanoindentation using MD and the effect of indenter size and loading rate was investigated at an atomic scale. A clear correlation between the indenter size and observed hardness of GB was observed which proves indentation size effects (ISE). Realizing the composite nature of geopolymer, the presence of unreacted and secondary phases such as quartz and CSH in geopolymer was also investigated. To do that, the mechanical properties of GB, the secondary phases and their interfaces was first determined from MD simulations. Using the MD generated properties, a meso-scale model of geopolymer composite was prepared in Peridynamics (PD) framework which considered large particles of GB and secondary phases of nanometers in size which cannot be easily modeled in MD. The meso-scale model provides a larger platform to study geopolymer in the presence of large nano-voids and multiple phases. Results from the PD simulations were directly comparable to experimentally observed mechanical properties. Findings of this study can be directly used in future to construct more advanced and sophisticated models of geopolymer and will be instrumental in designing the synthesis condition for geopolymer with superior mechanical properties.

This thesis explores the science of solid mechanics via the theory of peridynamics. Peridynamics has several key advantages over the classical theory of elasticity. The most notable of which is the ease with which fractures in the the material are handled. The goal here is to study the two theories and how they relate for problems in which the classical method is known to work well. While it is known that state-based peridynamic models agree with classical elasticity as the horizon radius vanishes, similar results for bond-based models have yet to be developed. In this study, we use numerical simulations to investigate the behavior of bond-based peridynamic models under this limit for a number of cases where analytic solutions of the classical elasticity problem are known. To carry out this study, the integral-based peridynamic model is solved using the finite element method in two dimensions and compared against solutions using the classical approach.
Master of Science

Peridynamics is a reformulation of classical continuum mechanics that utilizes integral equations in place of partial differential equations to remove the difficulty in handling discontinuities, such as cracks or interfaces, within a body. Damage is included within the constitutive model; initiation and propagation can occur without resorting to special crack growth criteria necessary in other commonly utilized approaches. Predicting damage and residual strengths of composite materials involves capturing complex, distinct and progressive failure modes. The peridynamic laminate theory correctly predicts the load redistribution in general laminate layups in the presence of complex failure modes through the use of multiple interaction types.This study presents two approaches to obtain the critical peridynamic failure parameters necessary to capture the residual strength of a composite structure. The validity of both approaches is first demonstrated by considering the residual strength of isotropic materials. The peridynamic theory is used to predict the crack growth and final failure load in both a diagonally loaded square plate with a center crack, as well as a four-point shear specimen subjected to asymmetric loading.This study also establishes the validity of each approach by considering composite laminate specimens in which each failure mode is isolated. Finally, the failure loads and final failure modes are predicted in a laminate with various hole diameters subjected to tensile and compressive loads.

A recently introduced nonlocal peridynamic theory removes the obstacles present in classical continuum mechanics that limit the prediction of crack initiation and growth in materials. It is also applicable at different length scales. This study presents an alternative approach for the derivation of peridynamic equations of motion based on the principle of virtual work. It also presents solutions for the longitudinal vibration of a bar subjected to an initial stretch, propagation of a pre-existing crack in a plate subjected to velocity boundary conditions, and crack initiation and growth in a plate with a circular cutout. Furthermore, damage growth in composites involves complex and progressive failure modes. Current computational tools are incapable of predicting failure in composite materials mainly due to their mathematical structure. However, the peridynamic theory removes these obstacles by taking into account non-local interactions between material points. Hence, an application of the peridynamic theory to predict how damage propagates in fiber reinforced composite materials subjected to mechanical and thermal loading conditions is presented. Finally, an analysis approach based on a merger of the finite element method and the peridynamic theory is proposed. Its validity is established through qualitative and quantitative comparisons against the test results for a stiffened composite curved panel with a central slot under combined internal pressure and axial tension. The predicted initial and final failure loads, as well as the final failure modes, are in close agreement with the experimental observations. This proposed approach demonstrates the capability of the PD approach to assess the durability of complex composite structures.

Doctor of Philosophy
Department of Mechanical and Nuclear Engineering
Kevin B. Lease
Xiao J. Xin
Peridynamics is a non-local continuum theory that formulates problems in terms of integration of interactions between the material points. Because the governing equation of motion in the peridynamic theory involves only integrals of displacements, rather than derivatives of displacements, this new theory offers great advantages in dealing with problems that contain discontinuities. Integration of the interaction force plays an important role in the formulation and numerical implementation of the peridynamic theory. In this study two enhanced methods of integration for peridynamics have been developed. In the first method, the continuum is discretized into cubic cells, and different geometric configurations over the cell and the horizon of interaction are categorized in detail. Integration of the peridynamic force over different intersection volumes are calculated accurately using an adaptive trapezoidal integration scheme with a combined relative-absolute error control. Numerical test examples are provided to demonstrate the accuracy of this new adaptive integration method. The bond-based peridynamic constitutive model is used in the calculation but this new method is also applicable to state-based peridynamics. In the second method, an integration method with fixed Gaussian points is employed to accurately calculate the integration of the peridynamic force. The moving least square approximation method is incorporated for interpolating the displacement field from the Gaussian points. A compensation factor is introduced to correct the soft boundary effect on the nodes near the boundaries. This work also uses linear viscous damping to minimize the dynamic effect in the solution process. Numerical results show the accuracy and effectiveness of this Gaussian integration method. Finally current research progress and prospective directions for several topics are discussed.

Orientador: Edison Zacarias da Silva
Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Fisica Gleb Wataghin
Made available in DSpace on 2018-08-23T16:18:47Z (GMT). No. of bitstreams: 1 Pereira_ZennerSilva_D.pdf: 11614247 bytes, checksum: d882692dfdab010e889cb1717d250a1f (MD5) Previous issue date: 2013
Resumo: Nas últimas décadas uma geração de nanodispositivos foi desenvolvida. Estes dispositivos nanoeletrônicos são fabricados por novas técnicas fundamentadas em física, química e engenharia. Muitos desses nanomateriais têm suas propriedades físicas alteradas pelo efeito de tamanho, por causa desses novos efeitos é importante entender como estes dispositivos trabalham propriamente a fim de encontrarmos formas de obter novas aplicações baseadas nestes novos efeitos. Nanofios metálicos estão sendo largamente estudados tanto teoricamente como experimentalmente. Recentemente uma nova possibilidade de soldagem foi mostrada experimentalmente entre nanofios de ouro em temperatura ambiente, sem necessidade de aplicação de calor adicional e com baixa pressão, chamada de solda fria (cold welding). Usando Dinâmica Molecular (MD) com potenciais efetivos, nós simulamos o processo de soldagem fria em nanofios de ouro, prata e ouro-prata com diâmetros de 4.3nm em 300 K. Nós mostramos que a soldagem fria é um processo possível até mesmo quando os nanofios sofrem fortes deformações e defeitos antes do processo de soldagem. Durante o processo de soldagem os nanofios resultaram com poucos defeitos. Pequenas pressões foram necessárias para que a soldagem fosse atingida. Nós também realizamos cálculos de Dinâmica Molecular com embedded-atom-method para modelar o crescimento de filmes-finos de paládio depositados em um substrato de ouro para um sistema de aproximadamente 100 mil átomos. Nós mostramos que o filmes-finos de paládio cresceu sob stress sobre o substrato de ouro. Após a deposição de 9 monocamadas o stress armazenado no filmes de paládio relaxou formando defeitos na estrutura do cristal. Defeitos do tipo falhas de empilhamento surgiram nos filmes de paládio formando um padrão de deformação no mesmo. Para quantificar o stress nós também calculamos a evolução do tensor de stress durante o crescimento. Existem fenômenos físicos como fraturas em materiais que são caracterizados pela quebra das ligações atômicas que levam a efeitos macroscópicos. Para estudarmos este tipo de problema, nós desenvolvemos um código inicial que acopla Dinâmica Molecular com Peridyvii namics (PD) (uma recente teoria de contínuo). A ideia básica para acoplar Dinâmica Molecular e Peridynamics está baseada no teorema de Schwarz. Este teorema fornece uma maneira de resolver equações diferenciais em diferentes subdomínios conectados por uma interface. O acoplamento é feito trocando condições de contorno entre subdomínios conectados por esta interface. A parte mais difícil deste acoplamento encontra-se em tratar os dados com ruídos oriundos da Dinâmica Molecular e passá-los para a Peridynamics. Para isto nós usamos uma interpolação estatística chamada interpolação de Kriging. Desta forma nós pudemos alcançar um acoplamento entre MD e PD
Abstract: Over the last decades a new generation of nanoeletronic devices have been developed. These nanoeletronic devices have been made by new techniques based on physics, chemistry and engineering. Many of these nanomaterials have shown changes in their physical properties and therefore, it is very important to understand how they work properly in order to find ways to obtain new applications supported by these new effects. Metallic nanowires have been largely studied theoretical and experimentally. Recently a new possibility of welding was experimentally shown in the case of gold and silver nanowires (NWs) at ambient temperatures, without need of additional heat and with low pressures, called cold welding. Using molecular dynamics with effective potentials, we simulated cold welding of gold, silver, and silver-gold NWs with diameters of 4.3 nm at 300 K. We show the cold welding is a possible process in metal NWs and that these welded NWs, even after losing their crystalline structure after breaking, can reconstruct their face-centered-cubic structure during the welding process with the result of very few defects in the final cold welded NWs. The stress tensor shows a low average value during welding with oscillations indicating tension and relaxation stages. Small pressures are required for the process to occur, resulting in a fairly perfect crystal structure for the final NW after being broken and welded. We have also performed Molecular Dynamics calculations with embedded-atom-method to model the growth of a Pd thin film deposited on Au(100) for a system with approximately 100,000 atoms. We showed that the Pd film grew under stress on the Au substrate. After the deposition of 9 monolayers, the stress stored in the Pd film relaxed with the formation of defects, stacking faults in the structure of Pd forming a pattern of deformation in the film. To quantitatively access the defect formation we also measured the stress tensor evolution during growth. There are physical phenomena like brittle fracture that is characterized by breaking of atomic bonds leading to macroscopic effects. In order to study this kind of problems, we developed the initial programming code that couples molecular dynamics (MD) and Peridyix namics (PD) (a new model to continuum). The basic idea to coupling Molecular Dynamics and Peridynamics is based on a mathematical theorem that is known as Schwarz theorem. It gives a way to solve differential equations in different subdomains that are connected by an interface (overlap). The coupling is made by exchanging boundary conditions through of the interface between subdomains. The hardest part is to treat noise molecular dynamics data and after that to pass those data to continuum theories. In order to pass data from MD to Peridynamics we have used a statistical interpolation called Kriging interpolation. This way we can achieve an algorithm to coupling DM with PD
Doutorado
Física
Doutor em Ciências

This work explores the computational modeling of electromechanical problems using peridynamics and in particular, its application in studying the potential of carbon nanotube (CNT) reinforced nanocomposites for the purpose of sensing deformation and damage in materials. Peridynamics, a non-local continuum theory which was originally formulated for modeling problems in solid mechanics, has been extended in this research to electromechanical fields and applied to study the electromechanical properties of CNT nanocomposites at multiple length scales. Piezoresistivity is the coupling between the electrical properties of a material and applied mechanical loads, more specifically the change in resistance in response to deformation. This can include both, a geometric effect due to change in dimensions as well as the change in resistivity of the material itself. Nanocomposites referred to in this work are materials which consist of CNTs dispersed in a binding polymer matrix. The origins of the extraordinary piezoresistive properties of nanocomposites lie at the nanoscale where the non-local phenomenon of electron hopping plays a significant role in establishing the properties of the nanocomposite along with CNT network formation and inherent piezoresistivity of CNTs themselves. Electron hopping or tunneling allows for a current to flow between neighboring CNTs even when they are not in contact, provided the energy barrier for electrons to hop is small enough. This phenomenon is highly nonlinear with respect to the intertube distance and is also dependent on other factors such as the potential barrier of the polymer matrix. To investigate this in more detail, peridynamic simulations are first employed to study the piezoresistivity at the CNT bundle scale by considering a nanoscale representative volume element (RVE) of CNTs within polymer matrix, and by explicitly modeling electron hopping effects. This is done by introducing electron hopping bonds and it is shown that the conductivity and the non-local length scale parameter in peridynamics (the horizon) can be derived from a purely physics based model rather than assuming an ad-hoc value. Piezoresistivity can be characterized as a function of the deformation and damage within the material and thereby used as an in-situ indicator of the structural health of the material. As such, a material system for which real time in-situ monitoring may be useful is polymer bonded explosives. While these materials are designed for detonation under conditions of a strong shock, they can be damaged or even ignited under certain low magnitude impact scenarios such as during accidental drop or transportation. Since these materials are a heterogeneous system consisting of explosive grains within a polymer matrix binder, it is proposed that CNTs can be dispersed within the binder medium leading to an inherently piezoresistive hybrid nanocomposite bonded explosive material (NCBX) material which can then be monitored for a continuous assessment of deformation and damage within the material. To explore the potential use of CNT nanocomposites for this novel application, peridynamic simulations are carried out at the microscale level, first under quasistatic conditions and subsequently under dynamic conditions to allow the propagation of elastic waves. Peridynamics equations, which can be discretized to obtain a meshless method are particularly suited to this problem as the explicit modeling of crack initiation and propagation at the microscale is essential to understanding the properties of this material. Moreover, many other parameters such as electrical conductivity of the grain and the properties of the grain-binder interface are studied to understand their effect on the piezoresistive response of the material. For example, it is found that conductivity of the grain plays a major role in the piezoresistive response since it affects the preferential pathways of current density depending on the relative ease of flow through grain vs. binder. The results of this work are promising and are two fold. Peridynamics is found to be an effective method to model such materials, both at the nanoscale and the microscale. It alleviates some of difficulties faced by traditional finite element methods in the modeling of damage in materials and can be extended to coupled fields with relative ease. Secondly, simulations presented in this work show that there is much promise in this novel application of nanocomposites in the field of structural health monitoring of polymer bonded explosives.
Ph. D.

The classical theory of solid mechanics employs partial derivatives in the equation of motion and hence requires the differentiability of the displacement field. Such an assumption breaks down when simulation of problems containing discontinuities, such as cracks, comes into the picture. peridynamics is considered to be an alternative and promising nonlocal theory of solid mechanics that is formulated suitably for discontinuous problems. Peridynamics is well designed to cope with failure analysis as the theory deals with integral equations rather than partial differential equations. Indeed, peridynamics defines the equation of motion by substituting the divergence of the stress tensor, involved in the formulation of the classical theory, with an integral operator. One of the most common techniques to discretize and implement the theory is based on a meshless approach. However, the method is computationally more expensive than some meshless methods based on the classical theory. This originates from the fact that in peridynamics, similar to other nonlocal theories, each computational node interacts with many neighbors over a finite region. To this end, performing realistic numerical simulations with peridynamics entails a vast amount of computational resources. Moreover, the application of boundary conditions in peridynamics is nonlocal and hence it is more challenging than the application of boundary conditions adopted by methods based on the classical continuum theory. This issue is well-known to scientists working on peridynamics. Therefore, it is reasonable to couple computational methods based on classical continuum mechanics with others based on peridynamics to develop an approach that applies different computational techniques where they are most suited for. The main purpose of this dissertation is to develop an effective coupled nonlocal/local meshless technique for the solution of two-dimensional elastodynamic problems involving brittle crack propagation. This method is based on a coupling between the peridynamic meshless method, and other meshless methods based on the classical continuum theory. In this study, two different meshless methods, the Meshless Local Exponential Basis Functions and the Finite Point Method are chosen as both are classified within the category of strong form meshless methods, which are simple and computationally cheap. The coupling has been achieved in a completely meshless scheme. The domain is divided in three zones: one in which only peridynamics is applied, one in which only the meshless method is applied and a transition zone where a transition between the two approaches takes place. The coupling adopts a local/nonlocal framework that benefits from the full advantages of both methods while overcoming their limitations. The parts of the domain where cracks either exist or are likely to propagate are described by peridynamics; the remaining part of the domain is described by the meshless method that requires less computational effort. We shall show that the proposed approach is suited for adaptive coupling of the strategies in the solution of crack propagation problems. Several static and dynamic examples are performed to demonstrate the capabilities of the proposed approach.
La teoria classica del continuo applicata ai solidi è basata sull’uso di equazioni differenziali del moto, pertanto viene ipotizzata la differenziabilità del campo degli spostamenti. Questa ipotesi viene meno quando nella risoluzione del problema compaiono delle discontinuità quali le cricche. La Peridynamics è una promettente teoria non locale del continuo che permette di gestire la presenza e l’evoluzione di discontinuità in un corpo solido. Questa nuova teoria è formulata per mezzo di equazioni integrali e pertanto può essere applicata anche allo studio della propagazione di cricche e dei fenomeni di rottura.La teoria Peridynamics utilizza un operatore integrale in sostituzione della divergenza del tensore delle tensioni, impiegata invece nella formulazione classica. Una delle tecniche di discretizzazione della Peridynamics maggiormente utilizzate è basata su un approccio di tipo meshless. Da un punto di vista computazionale questo approccio si rivela, però, più oneroso rispetto a quanto richiesto dall’uso di altri metodi meshless comunemente impiegati per lo studio della teoria classica del continuo. Questo è dovuto al fatto che in Peridynamics, in maniera analoga a quanto avviene in altre teorie non locali, ogni nodo interagisce con molti altri nodi vicini ed all’interno di una regione di dimensioni finite. Un altro svantaggio della Peridynamics è legato alla modalità con cui le condizioni di vincolo e di carico sono applicate rispetto ai metodi abitualmente utilizzati nella teoria classica. Questo problema è ben noto agli scienziati che studiano questa nuova teoria non locale. È quindi conveniente, accoppiare metodi basati sulla teoria classica con quelli che utilizzano la Peridynamics. Si svilupperà, in tal modo, un approccio in grado di utilizzare le diverse strategie computazionali nelle regioni del corpo solido in cui queste teorie si rivelano più adeguate. Lo scopo principale di questa tesi è di sviluppare una tecnica efficace per accoppiare metodi meshless di tipo non locale e locale per lo studio di problemi elasto-dinamici bidimensionali in materiali fragili anche in presenza di propagazione di cricche. Questa tecnica sarà basata sull’accoppiamento della teoria Peridynamics (nella sua formulazione meshless) con la teoria classica del continuo studiata per mezzo di metodi di tipo meshless. In particolare sono stati scelti due metodi meshless, basati sulla teoria classica del continuo, il “Meshless Local Exponential Basis Functions” ed il “Finite Point Method”. Entrambi appartengono alla categoria “strong form meshless methods”, sono di semplice implementazione e non sono particolarmente onerosi da un punto di vista computazionale. L’accoppiamento è stato realizzato grazie ad un unico schema meshless. Il dominio è suddiviso in tre regioni: una in cui si usa solamente la teoria Peridynamics, una in cui si utilizza solo il metodo meshless ed infine è prevista una regione di transizione che realizza il passaggio fra i due approcci di calcolo. La strategia proposta e sviluppata in questo elaborato, è in grado di accoppiare la teoria del continuo locale con la teoria del continuo non locale permettendo di trarre vantaggio dagli aspetti positivi delle due teorie e di superarne le rispettive limitazioni. La porzione del dominio in cui sono presenti delle cricche o in cui potrebbero propagarsi, è descritta impiegando la Peridynamics, mentre la restante porzione del dominio è descritta impiegando uno dei due metodi meshless citati in precedenza, che richiedono un minor onere computazionale. L’efficienza dell’approccio proposto sarà migliorata dall’uso di tecniche di accoppiamento adattativo: quest’ultima estensione permetterà di studiare diversi fenomeni di propagazione delle cricche con un limitato utilizzo di risorse di calcolo. Le prestazioni del metodo di accoppiamento ideato saranno descritte per mezzo di molteplici esempi con analisi sia statiche che dinamiche.

Peridynamics, presented by Silling in 2000 [1], is a reformulation of the elastic theory from differential equations to integral equations, which are more equipped to handle discontinuities, such as crack initiation and propagation. Because of this, peridynamics is an effective tool to address many of the problems relevant to the aerospace and defense industries. For example, airborne sand particles and raindrops cause local damage to aircraft in flight. This damage manifests itself as radial and subsurface lateral cracking, as well as increased surface roughness. All of these damage morphologies may result in undesired degradation of mechanical and optical properties. This dissertation aims to address the question of how peridynamics (PD) can be used as a tool to help understand impact problems and resultant damage. Three main types of problems will be discussed: (1) modeling of quasi-static nano- and micro-indentation in PD; (2) solid impact experiments and simulations involving glass micro-spheres impacting coated and uncoated advanced ceramics, and sand particles impacting optical glasses; and (3) the implementation of a new, fully three-dimensional hyperelastic material model in state-based PD to simulate nylon bead impact and capture the damage patterns relevant to raindrop impact. In the first portion, a new method for modeling indentation in PD is presented using the principle of viscous damping and automatic convergence checking. In these simulations, depth-controlled indentation is performed by splitting up the total indentation depth into multiple stages, and applying damping at each stage to ensure the system reaches equilibrium before allowing for failure. PD results show good agreement to experimental data, in terms of crack lengths and force-displacement curves. In a chapter about solid particle impact, two studies are presented. In the first, glass spheres with diameters ranging from 200 to 700 um impact multi-spectral zinc sulfide (MS-ZnS) with various coating systems. It was found that samples containing the REP coating had better resistance to damage than those without. This resistance was evident in all three damage metrics used: impact pit diameter, radial crack length, and lateral crack size. Simulations were carried out in bond-based PD, with good agreement to experiments regarding damage metrics and rebound velocity. The second solid particle impact study involved sand particles impacting four different types of optical glasses: BK7, alumino-boro-silicate, fused silica, and Pyrex. First, data from experiments was analyzed, and a multi-variable power law regression was performed to show that sand particle shape plays a significant role in resultant damage. This was confirmed via bond-based PD simulations, with damage quantities agreeing well with experimental values. Finally, the problem of how to model raindrop impact using nylon beads was examined. Due to the large amounts of elastic strain experienced by the nylon beads during impact experiments, it was determined that a hyperelastic material model could be a good fit. Based on elastic theory and classical continuum mechanics, a new, fully three-dimensional Neo-Hookean material model was implemented in nonordinary state-based peridynamics. This model was verified against results and finite element analysis, with very good agreement. Preliminary simulations including damage show good results, consistent with experiments.

Several aspects of fracture nucleation and growth in brittle porous ceramics and in thin films are investigated, through analytical, numerical modelling, and experimental validation. A mechanical experimental characterization has been developed for a porous ceramic, namely, a 3D apatite, characterised by an oriented porosity and used for biomedical applications. The ceramic is produced from wood, so that the resulting porosity evidences a multi-scale nature, a feature determining peculiar failure mechanisms and an unprecedented porosity/strength ratio. In particular, the material exhibits an exfoliation-type failure, resulting in a progressive loss in mechanical properties, occurring for compression tests parallel to the grains and for highly slender specimens. Similar cohesive-brittle behaviour is also found when the compression is applied in the direction orthogonal to the porous channels, regardless of the shape ratio of the specimen. An in-depth analysis of this response is performed by means of a phase-field model. After calibrating the model, stress-strain curves and fracturing patterns are accurately reproduced. Furthermore, the effects of multi-scale porosity on mechanical behaviour are determined. Various strategies available in the literature for evaluating the properties of porous materials are compared to the proposed phase-field approach. The results open new possibilities for the prediction and characterization of complex fracturing phenomena occurring in highly porous ceramics, so to facilitate medical applications as structural bone repair. An application of the peridynamic theory of continuum mechanics is developed to obtain a dimensional reduced formulation for the characterisation of through-thickness delamination of plates. The kinematic of the plate is carefully chosen to be composed of an absolutely continuous part and a zone where jumps in the displacements are allowed; in this way, the reduced form of the elastic bond-based peridynamic energy and the reduced Lagrangian are explicitly retrieved in a closed-form. The reduction generates a hierarchy of terms, characterizing the energy stored inside the plane element. A semi-analytical solution, obtained by means of a minimization procedure, is obtained for a test case and compared with finite element simulations. Despite the fact that the numerical model is fully three-dimensional (in other words, it is not reduced), this model leads to the same moment-curvature diagrams and nucleation/growth of the delamination surface found with the reduced formulation. Finally, the convergence of the proposed reduced model to local elastic theory at vanishing internal length is determined, so that a reduced-localized cohesive model for fracture is retrieved.

In order to maintain the quadratic convergence properties of the first-order Newton's method in quasi-static nonlinear analysis of solid structures it is crucial to obtain accurate, algorithmically consistent tangent-stiffness matrices. For an extremely small class of nonlinear material models, these consistent tangent-stiffness operators can be derived analytically; however, most often in practice, they are found through numerical approximation of derivatives. A goal of the study de- scribed in this thesis was to establish the suitability of an under-explored method for computing tangent-stiffness operators, referred to here as 'complex-step'. Compared are four methods of nu- merical derivative calculation: automatic differentiation, complex-step, forward finite difference, and central finite difference in the context of tangent-stiffness matrix calculation in a massively parallel computational peridynamics code. The complex-step method was newly implemented in the peridynamics code for the purpose of this comparison. The methods were compared through in situ profiling of the code for Jacobian accuracy, solution accuracy, speed, efficiency, Newton's method convergence rate and parallel scalability. The performance data was intended to serve as practical guide for code developers and analysts faced with choosing which method best suit the needs of their application code. The results indicated that complex-step produces Jacobians very similar, as measured by a low l 2 norm of element wise difference, to automatic differentiation. The values for this accuracy metric computed for forward finite difference and central finite differ- ence indicated orders of magnitude worse Jacobian accuracy than complex-step, but convergence vstudy results showed that convergence rate and solution was not strongly affected. Ultimately it was speculated that further studies on the effect of Jacobian accuracy may better accompany experiments conducted on plastic material models or towards the evaluation of approximate and Quasi-Newton's methods.

Thesis (MEng)--Stellenbosch University, 2014.
ENGLISH ABSTRACT: Current numerical methods in the eld of hydraulic fracturing are based mainly on continuum methods, such as the Finite Element Method (FEM) and the Boundary Element Method (BEM). These methods are governed by Linear Elastic Fracture Mechanics (LEFM) criteria, which su er from the inherent aw of a non-physical stress representation at the fracture tip. In response to this, a non-local method is proposed, namely the peridynamic theory, to model sleeved hydraulic fracture. A 2D implicit quasi-static ordinary state based peridynamic formulation is implemented on various benchmark problems, to verify the ability to capture constitutive behaviour in a linear elastic solid, as well as, the quanti cation of adverse e ects on the accuracy of the displacement solution, due to the nature of the non-local theory. Benchmark tests consist of a plate in tension, where convergence to the classical displacement solution, non-uniform re nement and varying cell sizes are tested, as well as, a thick walled cylinder with internal pressure, where three di erent loading techniques are tested. The most accurate loading technique is applied to the sleeved fracture model, in order to simulate fracture initiation and propagation. This model is then veri ed and validated by using the Rummel & Winter hydraulic fracturing model and experimental results, respectively. Displacement error minimisation methods are implemented and as a result, the displacement solutions for a plate in tension converges to the analytical solution, while the thick walled cylinder solutions su er from inaccuracies due to an applied load on an irregularly discretized region. The fracture initiation test captures the fracture tip behaviour of the Rummel & Winter model and the fracture propagation test show good correlation with experimental results. This research shows that the peridynamic approach to sleeved hydraulic fracture can yield a realistic representation of fracture initiation and propagation, however, further research is needed in the area of a pressure load application on a solid using the peridynamic approach.
AFRIKAANSE OPSOMMING: Huidige numeriese metodes in die veld van hidrouliese breking is hoofsaaklik gebaseer op kontinuum metodes, soos die Eindige Element Metode (EEM) en die Rand Element Metode (REM). Hierdie metodes word beheer deur Linie^ere Elastiese Breukmeganika (LEB) kriteria, wat ly aan die inherente gebrek van 'n nie- siese voorstelling van die spanning by die fraktuur punt. Om hierdie probleme aan te spreek, word 'n nie-lokale metode voorgestel, naamlik die peridinamiese teorie, om gehulsde hidrouliese breking te modelleer. 'n 2D implisiete kwasi-statiese ordin^ere toestand gebaseerde peridinamika formulering word ge mplimenteer op verskeie norm probleme, om te veri eer of dit oor die vermo e beskik om die konstitutiewe gedrag van 'n linie^ere elastiese soliede materiaal te modeleer, asook die kwanti sering van nadelige e ekte op die verplasings oplossing as gevolg van die natuur van die nie-lokale teorie. Normtoetse bestaan uit 'n plaat in trek spanning, waar konvergensie na die klassieke verplasings oplossing, nie-uniforme verfyning en vari^eerende sel groottes getoets word, asook 'n dikwandige silinder onder interne druk, waar drie verskillende belasting aanwendingstegnieke getoets word. Die mees akkurate belasting aanwendingstegniek word dan gebruik in die gehulsde hidrouliese breking model, om fraktuur aanvangs en uitbreiding na te boots. Die model word dan geveri- eer deur die Rummel & Winter hidrouliese breking model en eksperimentele resultate, onderskeidelik. Fout minimering metodes word toegepas en as 'n resultaat, konvergeer die verplasing oplossing vir die plaat na die analitiese oplossing, terwyl die oplossing van die dikwandige silinder onakuraathede toon as gevolg van 'n toegepaste belasting op 'n onre elmatig gediskretiseerde gebied. Die modellering van die fraktuur inisi ering by die fraktuur punt, stem goed ooreen met die Rummel en Winter voorspelling en die fraktuur uitbreiding stem goed ooreen met eksperimentele resultate. Hierdie navorsing toon dat die peridinamiese benadering tot gehulsde hidrouliese breking wel die fraktuur inisi ering en uitbreiding realisties kan modelleer, maar nog navorsing word wel benodig in die area waar 'n druk belasting op 'n peridinamiese soliede model toegepas word.

A new experimental method to characterize the mechanical properties of metallic nanowires is introduced. An accurate and fast mechanical characterization of nanowires requires simultaneous imaging and testing of nanowires. However, there exists no practical experimental procedure in the literature that provides a quantitative mechanical analysis and imaging of the nanowire specimens during mechanical testing. In this study, a customized atomic force microscope (AFM) is placed inside a scanning electron microscope (SEM) in order to locate the position of the nanowires. The tip of the atomic force microscope cantilever is utilized to bend and break the nanowires. The nanowires are prepared by electroplating of nickel ions into the nanoscale pores of the alumina membranes. Force versus bending displacement responses of these nanowires are measured experimentally and then compared against those of the finite element analysis and peridynamic simulations to extract their mechanical properties through an inverse approach.The average elastic modulus of nickel nanowires, which are extracted using finite element analysis and peridynamic simulations, varies between 220 GPa and 225 GPa. The elastic modulus of bulk nickel published in the literature is comparable to that of nickel nanowires. This observation agrees well with the previous findings on nanowires stating that the elastic modulus of nanowires with diameters over 100nm is similar to that of bulk counterparts. The average yield stress of nickel nanowires, which are extracted using finite element analysis and peridynamic simulations, is found to be between 3.6 GPa to 4.1 GPa. The average value of yield stress of nickel nanowires with 250nm diameter is significantly higher than that of bulk nickel. Higher yield stress of nickel nanowires observed in this study can be explained by the lower defect density of nickel nanowires when compared to their bulk counterparts.Deviation in the extracted mechanical properties is investigated by analyzing the major sources of uncertainty in the experimental procedure. The effects of the nanowire orientation, the loading position and the nanowire diameter on the mechanical test results are quantified using ANSYS simulations. Among all of these three sources of uncertainty investigated, the nanowire diameter has been found to have the most significant effect on the extracted mechanical properties.

In the classical continuum theory of solid mechanics, the mathematical framework involves partial derivatives to represent the state of deformation of a solid body. A significant drawback due to derivatives is related to the unphysical results given near the discontinuities, because they are undefined wherever a continuous field of displacements is not verified, such as in the presence of dislocations, voids, cracks, interfaces between different phases within the same body and grain boundaries. Various techniques were employed for overcoming this incapability of the classical theory in describing material behavior in such conditions; in fact, spontaneous formation and growth of discontinuities are of great importance in solid mechanics: they lead to fractures and failures of systems that must be avoided, especially in aerospace structures, primarily, for safety reasons and, secondly, for economic purposes. One of these new approaches concerns employing nonlocal theories, based on integral formulations (more precisely integro-differential formulations), defined even when non-derivable displacement fields are involved. Peridynamics is one of these theories: it was suggested by Stewart Silling in 2000 [1] in order to adopt a consistent formulation describing material behavior not only when a continuous displacement field is provided, but also whenever discontinuities are present, avoiding partial differential equations or pre-setting of conditions which can influence the results. There are two versions of peridynamic models: bond-based, which was introduced first (see [1, 2]) and state-based. In the bond-based version, forces between two material points depend solely on their relative displacement, their relative initial position, and material properties. Due to its simplicity compared to the state-based version, most of the peridynamic applications have employed bond-based Peridynamics. However, bond-based models result in several limitations (the same of other atomistic or molecular dynamics models [3], although this is a continuum theory, not a discrete one), the most important of these is the fixed value of Poisson’s ratio: 1/4 in 3D or 2D plane strain, and 1/3 in 2D plane stress (see e.g. [1, 4]). This peculiarity implies other restrictions, such as the impossibility of reproducing plastic incompressibility in an accurate way. Nevertheless, for many purposes, bond-based Peridynamics fits the requirements and gives satisfying results. State-based peridynamic models remove these restrictions by allowing the interaction (“bond”) between a pair of points to potentially depend on all other bonds connected to the two points. Moreover, there are two types of state-based peridynamic formulations: ordi- nary and non-ordinary [2, 5, 6]. In the former, the forces between two material points act along the vector connecting the points in the deformed configuration. In the latter, such characteristic is not present. The ordinary state-based formulation requires specific derivation of constitutive models, see examples of viscoelasticity and plasticity models in [7, 8]. For non-ordinary state-based formulation, two approaches have been proposed: the development of an explicit model for the peridynamic force state [2] and the development of a map thanks to which classical mechanics constitutive relations are incorporated to indirectly establish the relationship between the interaction force and the deformation. The latter approach is called correspondence model [2]. The purpose of this thesis has been the investigation of possible advantages and drawbacks of this new and unexplored theory, so to identify some guidelines for choosing parameters fundamental for the analyses and the development of models for particular structural analyses. In the first year of the PhD course, the state of the art of this theory was studied and the bond-based linear and nonlinear static solvers developed in Matlabr were analyzed, employed and improved. During the second year of PhD course, the author of this thesis has focused her attention on the second version of the theory, based on concepts of advanced mathematics. She has become familiar with it, thanks to the functional analysis course that she had attended in the first year. One of the main original contributions of the present work to the existing literature is the development of the 2D linearization of the state-based “linear peridynamic solid” model in the state-based formulation. These models are useful whenever simplifying assumptions of plane stress and plane strain can be adopted for the simulation of a system, which, otherwise, would be described by a 3D model requiring high computational resources (time and memory). Particular attention is paid to this aspect, because, being a nonlocal model, implementing a peridynamic code is, in general, more computationally expensive than a code based on a local approach. The study of the state-based version started before going abroad and the development of the 2D models was completed during the six month stay at the University of Nebraska-Lincoln in USA. Both static and dynamic codes have been developed and the relevant parameters of these models have been analyzed. These linearized models are described in chapter 1.2.2. The study of failure criteria in state-based Peridynamics and the improvement of the algorithms in Matlabr to accelerate the codes and to optimize memory resources have been the main issues of the third year research. Some failure criteria, presented in section 1.2.3, have been proposed for brittle homogeneous linear elastic materials. They are criteria based on the maximum admissible stretch: a given bond fails at a critical stretch obtained by the work required to break that bond and this work is related to the fracture energy of the material. The results are compared to experimental data both for static and for dynamic cases, in bondbased and in state-based formulations. The detailed description of the algorithms can be found in chapter 3, while the results are illustrated in chapters 4 and 5.
Nella teoria classica della meccanica dei solidi, la formulazione matematica include derivate parziali, grazie alle quali si possono rappresentare stati di deformazione come funzioni degli spostamenti relativi dei nodi in cui è discretizzato il sistema continuo. Una carenza rilevante dovuto all’utilizzo delle derivate è legato ai risultati privi di significato fisico ottenuti in prossimità delle discontinuità perché le derivate non sono definite laddove manca un campo di spostamenti continuo, come può capitare in presenza di dislocazioni, vuoti, cricche, interfacce tra fasi differenti nello stesso corpo e bordi dei grani. Dato che la formazione spontanea e la crescita di discontinuità sono di grande importanza in meccanica dei solidi, diverse tecniche sono state utilizzate per superare questa incapacità della teoria di descrivere il comportamento dei materiali in tali condizioni, perché situazioni in cui le strutture sono incapaci di continuare a svolgere la propria funzione devono essere evitate, specialmente per strutture aerospaziali, in primo luogo, per ragioni di sicurezza ed, in secondo luogo, per motivi economici. Uno di questi nuovi approcci riguarda l’utilizzo di teorie non locali basate su formulazioni integrali (più precisamente formulazioni integro-differenziali), definite anche quando campi di spostamento non derivabili sono presenti. La teoria “Peridynamics” è una di queste teorie: è stata proposta da Stewart Silling nel 2000 [1] così da adottare una formulazione unica e coerente capace di descrivere i comportamenti dei materiali in corpi sia continui che discontinui, evitando l’uso di equazioni alle derivate parziali o la definizione a priori di alcune condizioni che possono influenzare (e in un certo senso favorire) dei risultati. Ci sono due versioni di modelli peridinamici: la state-based, e un suo caso particolare, la bond-based, che è stata introdotta per prima (vedi [1, 2]). Nella versione bond-based, le forze tra due punti materiali dependono unicamente dal loro spostamento relativo e dalla loro posizione relativa iniziale, oltre che dalle proprietà del materiale. Vista la sua semplicità a confronto con la seconda versione, la maggior parte delle applicazioni e degli articoli sulla Peridynamica ha adottato la formulazione bond-based. Tuttavia, i modelli nella formulazione bond-based sono caratterizzati da alcune limitazioni (le stesse dei modelli di altre teorie atomistiche e dei modelli di dinamica molecolare [3], anche se la Peridinamica è una teoria del continuo, non discreta), la più notevole di queste è il modulo di Poisson fisso: 1/4 nelle simulazioni 3D oppure in caso di deformazione piana 2D, e 1/3 nelle simulazioni in stato di tensione piana 2D (si veda per esempio [1, 4]). Questa particolarità implica altre restrizioni, come l’impossibilità di riprodurre la condizione di incomprimibilità plastica in maniera accurata. Tuttavia, per la maggior parte degli scopi, la formulazione bond-based è sufficiente e fornisce risultati approssimati soddisfacenti. I modelli della versione state-based rimuovono queste restrizioni, permettendo che le interazioni tra due punti possano dipendere da tutte le interazioni (i “bond”) connessi ad almeno uno dei due punti, tramite delle mappe avanzate chiamate “states”. Inoltre, ci sono due tipi di formulazioni state-based: la ordinary e la non-ordinary [2, 5, 6]. Nella formulazione ordinary, le forze tra due punti materiali agiscono lungo la congiungente i due punti nella configurazione deformata, mentre nella formulazione non-ordinary, questa caratteristica non è più vera. La formulazione ordinary della state-based necessita di modelli costitutivi appositamente derivati, come per esempio i modelli di viscoelasticità e platicità in [7, 8]. Per la formulazione non-ordinary della state-based, due approcci sono stati proposti: lo sviluppo di un modello esplicito per l’espressione dello state della forza peridinamica [2] e lo sviluppo di una mappa grazie alla quale le relazioni costitutive della meccanica classica sono incorporate per stabilire indirettamente la relazione tra la forza d’interazione e la deformazione. I modelli derivanti dal secondo approccio sono chiamati modelli correspondence [2]. L’argomento di questa tesi è lo sviluppo di modelli per particolari tipi di analisi e la ricerca di possibili vantaggi e inconvenienti di questa teoria nuova ed inesplorata, così da identificare alcune linee guida per la scelta di parametri fondamentali per le analisi. Durante il primo anno del corso di dottorato, lo stato dell’arte relativo a questa teoria è stato studiato e i solutori statici lineari e non lineari nella formulazione bond-based sviluppati precedentemente in ambiente Matlabr sono stati analizzati, usati e migliorati. Durante il secondo anno, l’autrice di questa tesi si è concentrata sulla seconda versione, basata su concetti di matematica avanzata con cui ha preso dimestichezza grazie al corso di analisi funzionale seguito il primo anno. Uno dei principali contributi originali alla letteratura esistente presenti in questa tesi è lo sviluppo dei modelli linearizzati 2D del modello solido lineare nella formulazione state-based. Questi modelli sono particolarmente utili quando semplificazioni di stato piano di tensione o di deformazione possono essere assunte per la simulazione di un sistema tridimensionale, che altrimenti verrebbe descritto da un modello 3D che necessiterebbe di risorse computazionali più elevate (in termini di tempo e memoria). Una particolare attenzione è richiesta per quest’aspetto, perché, essendo un approccio non locale, implementare un codice basato sulla teoria peridinamica richiede in generale più risorse computazionali di un codice basato su un approccio locale. Lo studio della versione state-based è iniziato prima di andare all’estero e lo sviluppo dei modelli 2D si è poi completato durante il soggiorno di sei mesi alla University of Nebraska-Lincoln negli Stati Uniti. Sono stati sviluppati sia un codice dinamico che uno statico. I parametri principali di questi modelli sono stati analizzati e i modelli linearizzati si possono trovare descritti nel capitolo 1.2.2. Lo studio dei criteri di frattura adottabili nella formulazione state-based e il miglioramento degli algoritmi in Matlabr per accelerare i codici e ottimizzare le risorse di memoria e gestione dei dati sono stati gli argomenti principali del terzo anno. Alcuni criteri di frattura, presentati nel capitolo 1.2.3, sono stati proposti per materiali lineari elastici omogenei e caratterizzati da frattura fragile. Sono criteri basati sul massimo allungamento: un’interazione non locale (“bond”) viene meno quando un valore critico di allungamento è raggiunto; questo valore di allungamento critico è calcolato dal lavoro richiesto per rompere il bond e questo lavoro è a sua volta legato all’energia di frattura. I risultati ottenuti sono stati confrontati con dati sperimentali per casi sia statici che dinamici, sia nella formulazione bondbased che in quella state-based. La descrizione dettagliata degli algoritmi si trova nel capitolo 3, mentre i risultati sono riportati nei capitoli 4 e 5.

Crack propagation and branching are modeled using nonlocal peridynamic theory. One major advantage of this nonlocal theory based analysis tool is the unifying approach towards material behavior modeling- irrespective of whether the crack is formed in the material or not. No separate damage law is needed for crack initiation and propagation. This theory overcomes the weaknesses of existing continuum mechanics based numerical tools (e.g. FEM, XFEM etc.) for identifying fracture modes and does not require any simplifying assumptions. Cracks grow autonomously and not necessarily along a prescribed path. However, in some special situations such as in case of ductile fracture, the damage evolution and failure depend on parameters characterizing the local stress state instead of peridynamic damage modeling technique developed for brittle fracture. For brittle fracture modeling the bond is simply broken when the failure criterion is satisfied. This simulation helps us to design more reliable modeling tool for crack propagation and branching in both brittle and ductile materials. Peridynamic analysis has been found to be very demanding computationally, particularly for real-world structures (e.g. vehicles, aircrafts, etc.). It also requires a very expensive visualization process. The goal of this paper is to bring awareness to researchers the impact of this cutting-edge simulation tool for a better understanding of the cracked material response. A computer code has been developed to implement the peridynamic theory based modeling tool for two-dimensional analysis. A good agreement between our predictions and previously published results is observed. Some interesting new results that have not been reported earlier by others are also obtained and presented in this paper. The final objective of this investigation is to increase the mechanics knowledge of self-similar and self-affine cracks.

Due to their unpredictability, rapid growth and difficulty of detection, localised forms of corrosion represent a threat to human life and the environment. The current empirical and semi-empirical approaches used by engineers to hinder corrosion damage have several disadvantages and limitations. In this regard, numerical approaches can be a valuable complement. However, the majority of the numerical techniques currently available in the literature are based on partial differential equations, which become invalid in the presence of field’s discontinuities such as cracks and sharp concentration gradients. In order to overcome these limitations, a recently introduced continuum theory of mechanics based on integro-differential equations, peridynamics, is used for the first time for the modelling of polycrystalline fracture, stress-corrosion cracking, pitting corrosion and crack propagation from corrosion pits in steels exposed to different corrosive environments. The results are validated against experimental data and other numerical results. It was found that the microstructure can have a significant impact on the fracture behaviour of the material, and that aqueous solutions of sulfuric acid can lead to an embrittlement of high-strength steels so severe that the material can fail at stress intensity factors even four times smaller than the value of the fracture toughness. It was also found that peridynamics can be successfully used to reproduce realistic pit morphologies and to model microstructural effects, such as the presence of clusters of cathodic intermetallic particles, which can channel the propagation of corrosion pits. Finally, it was demonstrated that peridynamics can also be used to simulate crack nucleation and propagation from corrosion pits, without the need for any assumption on the location of crack nucleation, which, in contrast, is needed when using other numerical techniques. In conclusion, the results of this study support the idea that the peridynamic models produced as part of this research can be helpful in failure analysis and in the microstructural design of new fracture-resistant and corrosion-resistant materials.

Peridynamics is an emerging nonlocal continuum theory which allows governing field equations to be applicable at discontinuities. This applicability at discontinuities is achieved by replacing the spatial derivatives, which lose meaning at discontinuities, with integrals that are valid regardless of the existence of a discontinuity. Within the realm of solid mechanics, the peridynamic theory is one of the techniques that has been employed to model material fracture. In this work, the peridynamic theory is used to investigate different fracture problems in order to establish its fidelity for predicting crack growth. Various fracture experiments are modeled and analyzed. The peridynamic predictions are made and compared against experimental findings along with predictions from other commonly used numerical fracture techniques. Additionally, this work applies the peridynamic framework to model heat transfer. Generalized peridynamic heat transfer equation is formulated using the Lagrangian formalism. Peridynamic heat conduction quantites are related to quanties from the classical theory. A numerical procedure based on an explicit time stepping scheme is adopted to solve the peridynamic heat transfer equation and various benchmark problems are considered for verification of the model. This paves the way for the coupling of thermal and structural fields within the framework of peridynamics. The fully coupled peridynamic thermomechanical equations are derived based on thermodynamic considerations, and a nondimensional form of the coupled thermomechanical peridynamic equations is also presented. An explicit staggered algorithm is implemented in order to numerically approximate the solution to these coupled equations. The coupled thermal and structural responses of a thermoelastic semi-infinite bar and a thermoelastic vibrating bar are subsequently investigated.

The peridynamic theory is a generalization of classical continuum mechanics and takes into account the interaction between material points separated by a finite distance within a peridynamic horizon δ. The parameter δ corresponds to a length scale and is treated as a material property related to the microstructure of the body. Since the balance of linear momentum is written in terms of an integral equation that remains valid in the presence of discontinuities, the peridynamic theory is suitable for studying the material behavior in regions with singularities. The first part of this work concerns the evaluation of the properties of a linear elastic peridynamic material in the context of a three-dimensional state-based peridynamic theory, which uses the difference displacement quotient field in the neighborhood of a material point and considers both length and relative angle changes. This material model is based upon a free energy function that contains four material constants, being, therefore, different from other peridynamic models found in the literature, which contain only two material constants. Using convergence results of the peridynamic theory to the classical linear elasticity theory in the limit of small horizons and a correspondence argument between the free energy function and the strain energy density function from the classical theory, expressions were obtained previously relating three peridynamic constants to the classical elastic constants of an isotropic linear elastic material. To calculate the fourth peridynamic material constant, which couples both bond length and relative angle changes, the correspondence argument is used once again together with the strain field of a linearly elastic beam subjected to pure bending. The expression for the fourth constant is obtained in terms of the Poisson\'s ratio and the shear elastic modulus of the classical theory. The validity of this expression is confirmed through the consideration of other experiments in mechanics, such as bending of a beam by terminal loads and anti-plane shear of a circular cylinder. In particular, numerical results indicate that the expressions for the constants are independent of the experiment chosen. The second part of this work concerns an investigation of the behavior of a one-dimensional linearly elastic bar of length L in the context of the peridynamic theory; especially, near the ends of the bar, where it is expected that the behavior of the peridynamic bar may be very different from the behavior of a classical linear elastic bar. The bar is in equilibrium without body force, is fixed at one end, and is subjected to an imposed displacement at the other end. The bar has micromodulus C, which is related to the Young\'s modulus E in the classical theory through different expressions found in the literature. Depending on the expression for C, the displacement field may be singular near the ends, which is in contrast to the linear behavior of the displacement field observed in classical linear elasticity. In spite of the above, it is also shown that the peridynamic displacement field converges to its classical counterpart as the peridynamic horizon tends to zero.
A teoria peridinâmica é uma generalização da teoria clássica da mecânica do contínuo e considera a interação de pontos materiais devido a forças que agem a uma distância finita entre si, além da qual considera-se nula a força de interação. Por ter o balanço de momento linear formulado como uma equação integral que permanece válida na presença de descontinuidades, a teoria peridinâmica é adequada para o estudo do comportamento de materiais em regiões com singularidades. A primeira parte deste trabalho consiste no cálculo das propriedades de um material peridinâmico elástico linear no contexto de uma teoria peridinâmica de estado, linearmente elástica e tridimensional, que utiliza o campo quociente de deslocamento relativo na vizinhança de um ponto material e leva em conta mudanças relativas angulares e de comprimento. Esse modelo utiliza uma função energia livre que apresenta quatro constantes materiais, sendo, portanto, diferente de outros modelos peridinâmicos investigados na literatura, os quais contêm somente duas constantes materiais. Utilizando resultados de convergência da teoria peridinâmica para a teoria de elasticidade linear clássica no limite de pequenos horizontes e um argumento de correspondência entre as funções energia livre proposta e densidade de energia de deformação da teoria clássica, expressões para três constantes peridinâmicas foram obtidas em função das constantes de um material elástico e isotrópico da teoria clássica. O argumento de correspondêmcia, em conjunto com o campo de deformações de uma viga submetida à flexão pura, é utilizado para calcular a quarta constante peridinâmica do material, que relaciona mudanças angulares relativas e de comprimentos das ligações entre as partículas. Obtem-se uma expressão para a quarta constante em termos do coeficiente de Poisson e do módulo de elasticidade ao cisalhamento da teoria clássica. A validade dessa expressão é confirmada por meio da consideração de outros experimentos da mecânica, tais como flexão de um viga por cargas terminais e cisalhamento anti-plano de um eixo cilíndrico. Em particular, os resultados numéricos indicam que as expressões para as constantes são independentes do experimento escolhido. A segunda parte deste trabalho consiste em uma investigação do comportamento de uma barra unidimensional linearmente elástica de comprimento L no contexto da teoria peridinâmica; especialmente, próximo às extremidades da barra, onde espera-se que o comportamento da barra peridinâmica possa ser muito diferente do comportamento de uma barra elástica linear clássica. A barra está em equilíbrio e sem força de corpo, fixa em uma extremidade, e sujeita a deslocamento imposto na outra extremidade. A barra possui micromódulo C, o qual está relacionado ao módulo de Young E da teoria clássica por meio de diferentes expressões encontradas na literatura. Dependendo da expressão para C, o campo de deslocamento pode ser singular próximo às extremidades, o que contrasta com o comportamento linear do campo de deslocamento observado na elasticidade linear clássica. Apesar disso, é mostrado também que o campo de deslocamento peridinâmico converge para o campo de deslocamento da teoria clássica quando o horizonte peridinâmico tende a zero.

In this dissertation, a novel fast modeling technique called peri-ultrasound that can model both linear and nonlinear ultrasonic behavior of materials is developed and implemented. Nonlinear ultrasonic response can detect even very small material non- linearity. Quantification of the material nonlinearity at the early stages of damage is important to avoid catastrophic failure and reduce repair costs. The developed model uses the nonlocal continuum-based peridynamic theory which was found to be a good simulation tool for handling crack propagation modeling, in particular when multiple cracks grow simultaneously. The developed peri-ultrasound modeling tool has been used to model the ultrasonic response at the interface of two materials in presence of an interface crack. Also, the stress wave propagation in a half-space (or half-plane for a 2-dimensional problem) with boundary loading is investigated using peri-ultrasound modeling. In another simulation, well-established two-dimensional Lamb's problem is investigated where the results are verified against available analytical solution. Also, the interaction between the surface wave and a surface breaking crack is studied.

This study focuses on developing novel modeling techniques for fiber-reinforced composites with polymer and ceramic matrix based on Peridynamic approach. To capture the anisotropic material behaviors of composites under quasi-static and dynamic loading conditions, a new peridynamic model for composite laminate and a modified peridynamic approach for non-uniform discretization are proposed in this study. In order to achieve the numerical implementation of the proposed model and approach, a mixed implicit-explicit solver based on GPU parallel computing is developed as well. The new peridynamic model for composite laminates does not have any limitation in fiber orientation, material properties and stacking sequence. It can capture the expected orthotropic material properties and coupling behaviors in laminates with symmetric and asymmetric layups. Unlike the previous models, the new model enables the evaluation of stress and strain fields in each ply of the laminate. Therefore, it permits the use of existing stress- or strain-based failure criteria for damage prediction. The computation of strain energy stored at material points allows the energy-based failure criteria required for delamination propagation and fatigue crack growth. The capability of this approach is verified against benchmark solutions, and validated by comparison with the available experimental results for three laminate layups with an open hole under tension and compression. The modified peridynamic approach for non-uniform discretization enables computational efficiency and removes the effect of geometric truncations in the simulation. This approach is a modification to the original peridynamic theory by splitting the strain energy associated with an interaction between two material points according to the volumetric ratio arising from the presence of non-uniform discretization and variable horizon. It also removes the requirement for correction of peridynamic material parameters due to surface effects. The accuracy of this approach is verified against the benchmark solutions, and demonstrated by considering cracking in nuclear fuel pellet subjected to a thermal load with non-uniform discretizations. Unlike the previous peridynamic simulations which primarily employs explicit algorithm, this study introduces implicit algorithm to achieve peridynamic simulation under quasi-static loading condition. The Preconditioned Conjugate Gradient (PCG) and Generalized Minimal Residual (GMRES) algorithms are implemented with GPU parallel computing technology. Circulant preconditioner provides significant acceleration in the convergence of peridynamic analyses. To predict damage evolution, the simulation is continued with standard explicit algorithms. The validity and performance of this mixed implicit-explicit solver is established and demonstrated with benchmark tests.

Bi₂Sr₂CaCu₂Ox (Bi2212) superconducting round wires are a well-known high temperature superconductor due to their isotropic properties, high fill factor, and ease of winding. There have been extensive experiments to improve the wires’ performance, yet there is little understanding of how the internal microstructure of the wires influences the mechanical behavior. This is due to the multiple phases and their complex arrangements inside the wires, making it challenging for traditional approaches to investigate and simulate the wires’ behavior. The peridynamic theory, using non-local interactions and integral constitutive equations, can provide a solution to these challenges from the Bi2212 wires microstructure. To reduce computation cost, in this study the peridynamic formulas are developed for 2D simulations. Dynamic relaxation and energy minimization methods to find the steady-state solution are used and compared. The model shows m-convergence and δ-convergence behaviors when m increases and ä decreases. Model verification shows close quantitative matching to finite element analysis results. The 2D peridynamic model is then used to simulate mechanical behavior of Bi2212 wires. Various types of natural and artificial defects are simulated and compared quantitatively. Both defect geometry and physical characteristics are investigated to study their influence on the stress concentration in the material. The results show significant stress concentration around defects and protruding growths of the Bi2212 phase.

Le travail de thèse porte sur de nouveaux compléments et améliorations pour la théorie de la péridynamique concernant la modélisation numérique de structures minces telles que les poutres et les plaques, les composites isotropes et multicouches soumis à un chargement dynamique. Nos développements ont principalement porté sur l'exploration des possibilités offertes par la méthode péridynamique, largement appliquée dans divers domaines de l'ingénierie où des discontinuités fortes ou faibles peuvent se produire, telles que des fissures. La procédure de généralisation de la méthode Peridynamics pour la modélisation des structures de poutres de Timoshenko et des structures de plaques de Reissner-Mindlin avec une large plage de rapport épaisseur sur longueur allant de structures épaisses à très minces est indiquée. Et un impact avec une faible vitesse simplifié basé sur le modèle péridynamique développé pour la poutre de Timoshenko et la plaque de Reissner-Mindlin a été proposé en utilisant une procédure de contact spécifique pour l'estimation « naturelle » de la charge d'impact. L’originalité de la méthode actuelle réside dans l’introduction avec deux techniques permettant de réduire le problème de blocage par cisaillement qui se pose dans les structures à poutres et à plaques minces, à savoir la méthode d’intégration réduite (ou sélective) et la formulation mixte. Le modèle péridynamique résultant pour les structures de poutre de Timoshenko et les structures de plaque de Reissner-Mindlin est efficace et ne souffre d'aucun phénomène de verrouillage par cisaillement. En outre, la procédure de généralisation de la méthode péridynamique pour la modélisation de structures composites minces renforcées par des fibres est introduite. L’approche péridynamique pour la modélisation d’une couche est d’abord validée en quasi-statique, ce qui inclut des problèmes de prévision de la propagation de fissures soumis à des conditions de chargement mécaniques. La méthode péridynamique a ensuite été étendue à l’analyse de structures composites minces renforcées par des fibres utilisant la théorie fondamentale d’une couche. Enfin, plusieurs applications impliquant des structures composites minces renforcées par des fibres et des résultats numériques ont été validées par comparaison à la solution FEM obtenue à l'aide d'un logiciel commercial ou à des solutions de référence de la littérature. Dans toutes les applications, Péridynamics montre que les résultats correspondent parfaitement aux solutions de référence, ce qui prouve son potentiel d’efficacité, en particulier pour la simulation de chemins de fissures dans les structures isotropes et composites
This thesis introduces some new complements and improvments for the Bond-Based Peridynamics theory concerning the numerical modeling of thin structures such as beams and plates, isotropic and multilayer composites subjected to dynamic loading. Our developments have been focused mainly on exploring the possibilities offered by the Peridynamic method, which has been widely applied in various engineering domains where strong or weak discontinuities may occur such as cracks or heterogeneous media. The generalization procedure of the Peridynamics method for the modeling of Timoshenko beam structures and Reissner-Mindlin plate structures respectively with a wide range of thickness to length ratio starting from thick structures to very thin structures is given. And A simplified low velocity impact based on the developed Peridynamic model for Timoshenko beam and ReissnerMindlin plate has been proposed by using a specific contact procedure for the estimation of the impact load. The originality of the present method was the introduction for the first time of two techniques for the alleviation of the shear locking problem which arises in thin beam and plate structures, namely the reduced (or selective) integration method and mixed formulation. The resulting Peridynamic model for Timoshenko beam structures and Reissner-Mindlin plate structures is efficient and does not suffer from any shear locking phenomenon. Besides, the generalization procedure of Peridynamic method for the modeling of fiber-reinforced thin composite structures is introduced. The Peridynamic approach for the modeling of a lamina is firstly validated in the quasi-statics including a crack propagation prediction problems subjected to mechanical loading conditions and then the Peridynamic method was further extended to analyze fiber-reinforced thin composite structures using the fundamental lamina theory. Finally, several applications involving fiber-reinforced thin composite structures and numerical results were validated by comparison to the FEM solution obtained using commercial software or to reference solutions from the literature. In all applications, the Peridynamics shows that results are matching perfectly the reference solutions, which proves its efficiency potentiality especially for crack paths simulation in isotropic and composite structures

La fissuration induite par le retrait de dessiccation est un mécanisme essentiel à prendre en compte dans l'étude de la durabilité des matériaux et structures à base de ciment. La présente thèse est consacrée à l'étude expérimentale et à la modélisation numérique de ce mécanisme en mettant l'accent sur les influences de la rigidité et de la taille des inclusions. La thèse est composée de deux parties.La première partie est consacrée à l'étude expérimentale. Une série d'échantillons de béton sont d'abord préparés avec des inclusions artificielles de différentes rigidités. Ces échantillons sont soumis à différents niveaux de séchage afin d'évaluer les fissures induites par le retrait de séchage. Les échantillons séchés sont ensuite examinés à l'aide de la méthode non destructive de micro-tomographie au rayon X. Les distributions tridimensionnelles (3D) des fissures induites dans les échantillons séchés sont identifiées, y compris leur emplacement et leur forme. L'influence de la rigidité d'inclusion sur le processus de fissuration induite par le retrait est clairement démontrée. On constate que la fissuration induite par le retrait est fortement amplifiée par la différence de rigidité entre l'inclusion et la pâte de ciment. Deuxièmement, des échantillons coulés avec des billes de verre de différentes tailles sont étudiés. En utilisant une procédure expérimentale similaire, les influences de la taille des inclusions sur le processus de fissuration induite par le retrait de séchage sont étudiées.Dans la deuxième partie, une méthode numérique basée sur la théorie de la péridynamique est proposée. La formulation et la mise en œuvre de la méthode numérique sont d'abord présentées et discutées. Son efficacité dans la modélisation de l'initiation et de la propagation de fissures multiples dans des matériaux hétérogènes est ensuite démontrée. La méthode proposée est ensuite appliquée à la description du processus de fissuration induit par le retrait de séchage et le changement de température dans les composites de béton contenant différents types d'inclusions. L'accent est mis sur les effets de la rigidité et de la taille des inclusions sur les modèles de fissuration. Une série de simulations numériques est réalisée. Des comparaisons entre les résultats numériques et les observations expérimentales sont présentées
Drying shrinkage induced cracking is an essential mechanism to be considered in the durability study of cement-based materials and structures. The present thesis is devoted to experimental investigation and numerical modeling of this mechanism by putting the emphasis on the influences of inclusion stiffness and size. The thesis is composed of two parts.The first part is devoted to experimental study. A series of concrete samples are first casted with artificial inclusions of different rigidities. These samples are subjected to different levels of drying in order to evaluate cracks induced by the drying shrinkage. The dried samples are then examined by using the non-destructive X-ray micro-tomography imaging method. Three-dimensional (3D) distributions of induced cracks in the dried samples are identified, including their location and shape. The influence of inclusion rigidity on the shrinkage induced cracking process is clearly demonstrated. It is found that the shrinkage-induced cracking is strongly enhanced by the stiffness difference between the inclusion and cement paste. Secondly, samples casted with glass balls of different sizes are considered. By using a similar experimental procedure, the influences of inclusion size on the drying shrinkage induced cracking process are investigated.In the second part, a numerical method based on the peridynamics theory is proposed. The formulation and implementation of the numerical method are first presented and discussed. Its efficiency in modelling the initiation and propagation of multiple cracks in heterogeneous materials is then demonstrated. The proposed method is further applied to the description of cracking process induced by drying shrinkage and temperature change in concrete composites containing different types of inclusions. The emphasis is put on the effects of inclusion stiffness and size on cracking patterns. A series of numerical simulations are performed. Comparisons between numerical results and experimental observations are presented

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Aeronautics and Astronautics, 2014.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 171-185).
The simulation of materials exposed to extreme loads, as is relevant in many areas of engineering, including protection materials, impact damage in turbine engines and high velocity impact in space, remains one of the key challenges in the field of computational mechanics. Despite significant advances, a fully robust and generally applicable computational framework for simulating the response of materials under a wide range of dynamic loading conditions is still lacking and the search for improved approaches continues. Existing methods suffer from a litany of limitations and drawbacks, including difficulty representing fracture, robustness issues, difficulty scaling to a large number of processors, excessive computational expense, and fundamental convergence issues for problems involving material damage. In this thesis, we conduct a thorough investigation into the theory of peridynamics and its numerical implementation as a promising alternative approach for simulating extreme material response. Peridynamics is a relatively new nonlocal formulation of continuum mechanics based on integral equations. It includes a physical length scale and naturally supports the presence of discontinuities in the solution field. As part of the work for this thesis, we uncover fundamental limitations in existing constitutive formulations of the peridynamic theory, and propose solutions to these limitations which furnish an extended constitutive theory of peridynamic for large deformations of continua. It is shown that these issues are responsible for numerical instabilities commonly observed in peridynamic particle discretizations. Specifically, unphysical deformation modes which allow for matter interpenetration, without contributing to the strain energy, are shown to exist in the original formulation. In order to address this issue, we introduce an extension of the constitutive correspondence framework based on bond-level nonlinear strain measures. It is found that numerical instabilities are suppressed by the extended theory. In addition, we address the issue of incorporating damage and fracture, as is required for modeling materials subjected to intense loads. In particular, two novel approaches for modeling damage and fracture within peridynamics are proposed. One is based on classical continuum damage models, while the other is specifically suited for brittle fracture response. A robust, scalable computational framework based on these extensions to the peridynamic theory is developed, and numerical examples are provided which demonstrate the ability to capture experimentally observed ballistic limit curves for ductile materials, as well as realistic fracture patterns in brittle materials subjected to projectile impact loadings.
by Michael R. Tupek.
Ph. D.

Le verre de silice (SiO₂) est un des matériaux les plus couramment utilisés dans notre société moderne. Il est notamment employé dans des structures à haut niveau de risque, telles que les verrières d'engins spatiaux ou les protections d'équipements optiques. Cette thèse est effectuée dans le cadre du projet ANR GLASS, qui a pour objectif de faire évoluer les moyens servant à en étudier le comportement sous chargement dynamique (hautes pressions et hautes vitesses de déformations). Elle est focalisée sur l'étude expérimentale de la silice dans ce domaine, afin notamment de permettre un dialogue efficace entre expériences et simulations. Pour cela, la silice est impactée par une impulsion laser de haute puissance, générant une onde de choc qui se propage dans le matériau. Une première étude faite avec des résultats de mesures in situ de la propagation d'ondes de choc dans le verre permet d'obtenir des points de l'équation d'état du matériau. Ensuite, des mesures de spectroscopie Raman sont effectuées sur les échantillons impactés pour observer les modifications permanentes de leur structure atomique. Elles mettent en évidence une densification du matériau et la relaxation thermique du verre dans les zones ayant subi les plus hautes pressions lors du choc. Cet effet est causé par l'importante élévation de température pendant le chargement. Ces résultats montrent une bonne correspondance avec des études numériques effectuées dans le cadre du projet ANR. Enfin, des mesures de microtomographie aux rayons X montrent l'existence de nombreuses fissures à l'intérieur de l'échantillon. Des simulations numériques de peridynamic, une formulation spécialisée dans l'étude de l'endommagement, fournissent un scénario possible pour leur formation
Fused silica (SiO₂) is one of the most commonly used materials in our modern society. Among other uses, it is the main component of highly critical structures like spacecraft windows or shields for optical equipments. This PhD thesis is done within the context of the ANR GLASS project, whose objective is to model the behavior of silica glass from the atomic cluster to the whole structure under dynamic loading (high pressures and high strain rates). Its main objective is to conduct an experimental study of this material in this loading domain to enable an efficient dialog between experiments and simulations. To this end, samples of fused silica are impacted with high-power laser impulses, generating a shockwave that propagates in the material. A first study is done with in situ results of shockwave propagation in fused silica, giving some data of the equation of state. Subsequently, Raman spectroscopy is used to observe the atomic structure modifications of shocked samples. These measurements show that silica glass is densified in the shocked area, and also that the zones where the highest pressures were applied are subjected to thermal relaxation. This last effect is caused by the important temperature increase during the shock loading. All these results are in accordance with those of numerical simulations performed within the ANR project. Finally, X-Ray microtomography highlight complex fracture patterns inside some of the shocked samples. Numerical simulations using peridynamic formulation, a method specialized to study fracture patterns, provide a possible scenario for the formation and propagation of these cracks

Thesis (MEng)--Stellenbosch University, 2015.
ENGLISH ABSTRACT: The fracture of ductile materials is currently regarded as a complex and challenging phenomenon to characterise and predict. Recently, a bond-based, non-local theory was formulated called the peridynamic theory, which is able to directly solve solid mechanics problems that include fracture. The failure criterion is governed by a critical stretch relation between bonds. It was found in literature that the critical stretch relates to the popular fracture mechanics parameter called the critical energy release rate for predicting brittle linear-elastic failure. It was also proposed that the non-linear critical energy release rate or J-integral can be used to model ductile failure using peridynamics. The aim of this thesis was to investigate the validity of using the J-integral to determine the critical stretch for predicting ductile failure. Standard ASTM fracture mechanics tests on Compact Tension specimens of Polymethyl methacrylate, stainless steel 304L and aluminium 1200H4 were performed to determine the critical energy release rates and non-linear Resistance-curves. Furthermore, a novel peridynamic-based algorithm was developed that implements a critical energy release rate based failure criterion and Digital Image Correlation (DIC) measured full surface displacement fields of cracked materials. The algorithm is capable of estimating and mapping both the peridynamic damage caused by brittle cracking and damage caused by plastic deformation. This approach was used to validate the use of an energy release rate based failure criterion for predicting linear-elastic brittle failure using peridynamics. Also, it showed a good correlation among the test results for detecting plastic damage in the alloys when incorporating the respective J-integral derived critical stretch values. Additionally, Modified Arcan tests were performed to obtain Mode I, Mode II and mixed Mode fracture load results of brittle materials. Mode I peridynamic models compared closely to test results when using the Mode I critical energy release rate, derived critical stretch and served as validation for the approach. Moreover, it was argued that Mode I failure criteria cannot in principle be used to model shear failure. Therefore, it was proposed to rather use the appropriate Mode II and mixed Mode critical energy release rates to predict the respective failures in peridynamics. Also, for predicting ductile failure loads it was found that using a threshold energy release rate derived from the R-curve yielded considerably more accurate failure load results compared to the usage of the critical energy release rate, i.e. J-integral. In this thesis it was shown that there exists great potential for detecting and characterising cracking and failure by using a peridynamic-based approach through coupling DIC full displacement field measurements and the critical energy release rate of a particular structural material.
AFRIKAANSE OPSOMMING: Duktiele breeking van materiale word tans beskou as 'n kompleks- en uitdagende fenomeen om te voorspel en te karakteriseer. 'n Binding-gebaseerde, nie-lokale teorie is onlangs geformuleer, genaamd die peridinamika teorie. Die laasgenoemde stel ons in staat om soliede meganiese probleme met krake direk op te los. Die falings kriterium word bemagtig deur die kritiese strekfaktor tussen verbindings. Daar was bewys dat die kritiese strekfaktor in verband staan met die popul^ere breek meganika parameter, genaamd die kritiese vrylatings-energie-koers vir die voorspelling van bros line^ere-elastiese faling. 'n Onlangse verklaring meen dat die kritiese strekfaktor vir duktiele falingsgedrag, bereken kan word met die nie-line^ere kritiese vrylatings-energie-koers, beter bekend as die J- integraal. Die doel van hierdie tesis was om te meet hoe geldig die gebruik van die J-integraal is om die kritiese strekfaktor te bereken, om sodoende duktiele breking te ondersoek. Standaard ASTM breukmeganika toetse op Polimetilmetakrilat, vlekvrye staal 304L en aluminium 1200H4 is uitgevoer om die kritiese vrylatings-energie-koers en Weerstandskurwes te bepaal. Verder was 'n nuwe peridinamies-gebaseerde algoritme ontwikkel. Die laasgenoemde implementeer die berekening van 'n kritiese strekfaktor, gebaseer op die kritiese vrylatings-energie-koers, sowel as Digitale Beeld Korrelasie (BDK) vol oppervlaks-verplasings veld metings van gebreekte materiale. Dit is in staat om die peridinamiese skade te bereken, tesame met die beeld wat veroorsaak was van bros krake en plastiese vervorming in duktiele materiale. Hierdie benadering is aangewend om die gebruik van 'n vrylatings-energie-koers gebaseerde falings kriterium vir bros line^ere-elastiese falings in peridinamika te bekragtig. 'n Goeie korrelasie tussen toets resultate is ook gevind vir die opsporing van skade wat veroorsaak is deur plastiese deformasie in die legerings waar die onderskeilike J-integrale gebruik was as falings kriteria. Daarbenewens, was Verandere Arcan toetse uitgevoer om die Modes I, Modes II en gemenge Modes falingsresultate te verkry. Die Modes I peridinamiese model het goed vergelyk met die toetsresultate en het gedien as bekragtiging vir die falingsbenaderings. Verder was dit aangevoer dat Modes I falings kriterium in beginsel nie gebruik kan word om skuiffaling te modelleer nie. Dus was dit voorgestel om eerder die toepaslike Modes II en gemengde Modes kritieke vrylatings-energie-koerse te gebruik om onderskeie falings te voorspel in peridinamiese modelle. Dit was ook gevind dat vir die voorspelling van duktiele falingslaste die drumpel vrylatings-energie-koers, wat verkrygbaar is vanaf die Weerstands-kurwe, aansienlik meer akkurate resultate gee, in vergelyking met die gebruik van die kritiese vrylatings-energie-koers, m.a.w. die J-integraal. In hierdie tesis was dit gewys dat daar groot potensiaal bestaan vir die opsporing en karakterisering van krake en falings met 'n peridinamies-gebaseerde benadering, deur dit te skakel met BDK vol verplasings veld metings en die kritiese vrylatings-energie-koers van 'n bepaalde strukturele materiaal.

Orientador: Marco Lucio Bittencourt
Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecanica
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Resumo: Nesse trabalho, considera-se a simulação em múltiplas escalas concorrentes, usando a teoria peridinâmica e a elasticidade clássica, para a simulação de problemas de engenharia. Primeiramente a teoria peridinâmica em uma dimensão é estudada em detalhes com o foco na aplicação de condições de contorno de Dirichlet. Problemas de estado plano de tensão em chapas com e sem furo são considerados. É proposto um método de pós-processamento dos resultados de peridinâmica para o cálculo das tensões no material. Em seguida, a peridinâmica discretizada é acoplada ao método dos elementos finitos por meio de dois diferentes programas de computador, um especializado em peridinâmica e o outro em elementos finitos. A modelagem acoplada é usada para prever a formação e a propagação de uma trinca em uma chapa com furo. O fenômeno macroscópico de formação e propagação de trincas é resultado de processos físicos com origem na escala atomística. No entanto, as simulações existentes deste tipo problema são normalmente feitas com abordagens baseadas na teoria do contínuo, como a mecânica da fratura e o dano contínuo, que não consideram aspectos atomísticos do problema. A teoria peridinâmica é uma formulação da mecânica do contínuo em termos de equações integrais, permitindo a solução de problemas que apresentam descontinuidades. Na peridinâmica, trincas se propagam autonomamente como componentes naturais da deformação do material. Há um paralelo entre a formulação peridinâmica e a dinâmica molecular, um método atomístico. Em ambas as abordagens o movimento de uma partícula é encontrado através de um processo de somatório de forças devido às partículas vizinhas. No esquema de simulação em múltiplas escalas concorrentes aqui propostos, a peridinâmica é usada em pequenas porções do domínio onde a falha do material é esperada e a elasticidade clássica, usando o método dos elementos finitos, é utilizada no restante do domínio do problema. Os resultados mostram que a metodologia proposta para cálculo de tensões é satisfatória. A importância da correta imposição de condições de contorno de Dirichlet no domínio de peridinâmica também é destacado (este aspecto é de fundamental relevância para a abordagem acoplada, peridinâmica/elementos finitos). Finalmente, o padrão de propagação da trinca está de acordo com os resultados esperados
Abstract: We consider the peridynamic theory and the theory of classical elasticity for concurrent multiscale simulation of engineering problems. First the peridynamic theory in one dimension is studied in details focusing on the application of Dirichlet boundary conditions. Two-dimensional plane stress problems in plates with or without hole are considered. We propose a methodology to post-processing the peridynamics results in order to estimate stresses in the material. Then, the discretized peridynamics is coupled to finite elements by two different computer programs; one specialized in peridynamics and the other in finite elements. The coupled approach is used to estimate the crack formation and propagation in a plate with hole. The macroscopic phenomenon of crack formation and propagation is a result of physical processes with their origin in the atomistic scale. However, computer simulations of this type of problem are usually performed with continuum based approaches, such as fracture mechanics and continuum damage, which do not consider atomistic aspects of the problem. The peridynamic theory is a formulation of continuum mechanics in terms of integral equations allowing the solution of problems with discontinuities. In peridynamics cracks progress autonomously as natural consequence of the material deformation. There is a parallel between the peridynamic formulation and molecular dynamics, an atomistic method. In both approaches the motion of a particle is found by a process of summation of forces due to neighboring particles. In our concurrent multiscale scheme, peridynamics is used in small portions of the domain where material failure is expected and classical elasticity is used for modeling the rest of the problem domain. The results show that the proposed methodology for computing stresses is satisfactory. The importance of correctly imposing Dirichlet boundary conditions in the peridynamic domain is also highlighted (this aspect is of fundamental relevance for the coupled peridynamics/finite element approach). Finally, the pattern of the cracking agrees with the expected results
Mestrado
Mecanica dos Sólidos e Projeto Mecanico
Doutor em Engenharia Mecânica

Novel numerical methods for treating fractional differential and integrodifferential equations arising in non local mechanics formulations are proposed. For fractional differential equations arising in modeling oscillatory systems incorporating viscoelastic elements governed by fractional derivatives, the devised scheme is based on the Grunwald-Letnikov fractional derivative representation, dual time meshing technique and Taylor expansion. The proposed algorithm transforms the governing fractional differential equation into a second order differential equation with appropriate effective coefficients. The enhanced efficiency of the scheme hinges upon circumventing the calculation of the non local fractional derivative operator. Several examples of application are provided. Further, the concept of non locality, specifically viscoelasticity, governed by fractional derivatives is utilized to accurately model polyester materials. Specifically, the linear standard solid (Zener model) is extended to capture non linear viscoelastic behavior. Then, experimental data of polyester ropes are utilized using the Gauss Newton and Levenberg-Marquart minimization algorithm to determine the model parameters. Next, for integrodifferential equations arising in peridynamics theory of mechanics, an approach is formulated based on the inverse multi-quadric (IMQ) radial basis function (RBF) expansion and the Kansa collocation method. The devised scheme utilizes interpolation functions and basis function expansion for the spatial discretization of the peri dynamics equation. This significantly reduces the computational effort required to numerically treat the peri dynamics equations. Further, the proposed method is extended to account for mechanical systems with random material properties operating under random excitation. For this, the stochastic peridynamics governing equation of motion is solved using the benchmark Monte Carlo analysis and tools of stochastic analysis. The stochastic analysis is done by numerical evaluation of the requisite Neumann expansion using pertinent Monte Carlo simulations. Further, the usefulness of the radial basis function (RBF) collocation method in conjunction with a polynomial chaos expansion (PCE) is explored in stochastic mechanics problems. It is shown that the proposed approach renders further solution improvements in solving stochastic mechanics problems vis-a-vis the stochastic finite element method and the element free Galerkin method.

In this work we study peridynamics, a non-local model in continuum me- chanics introduced by Silling (2000). The non-locality is reflected in the fact that points at finite distance exert a force upon each other. If, however, these points are more distant than a characteristic length called horizon, it is customary to assume that they do not interact. We compare peridynamics with elasticity, especially in the limit of small horizon. We restrict ourselves, concerning this vanishing non-locality, to variational formulation of time- independent processes. We compute a Γ-limit for homogeneous and isotropic solid in linear peridynamics. In some cases this Γ-limit coincides with linear elasticity and the Poisson ratio is equal to 1 4. We conclude by clarifying why in some situation the computed Γ-limit can differ from the linear elasticity. 1

This study proposes an approach for solving density-based multi-material topology optimization of cracked structures using Peridynamics. The alternating active-phase algorithm is utilized to transform the multi-material problem into a series of binary phase topology optimization sub-problems. Instead of the conventional mesh-based methods, the Peridynamics theory (PD) is used as a tool to model the behaviour of the materials and solve for the displacement field. The most significant advantage of PD is its ability to model discontinuities in a relatively straightforward manner. Thus, in the present work, the effect of cracks as a discontinuity is investigated on the optimal multi-material topologies. The Solid Isotropic Material with Penalty (SIMP) method is utilized to define the material properties as a function of the design variables. Also, the optimization problem is solved through the Optimality Criteria (OC) approach. The proposed method is compared to the results reported in the literature by executing three numerical examples that investigate the effect of the direction of an interior crack on the optimal topologies. Moreover, the efficiency of the proposed approach is verified by solving several examples where we aim at minimizing the compliance of the structure with and without initial cracks.
Graduate

碩士
國立臺灣海洋大學
系統工程暨造船學系
105
The thick plates (over 20 mm) are the most important structural member for the construction of ultra large ships and marine structures. One of the most essential design requirement for thick plates is the crack generation and accumulation, which greatly affect the reliability and safety of the structure. Therefore, the design rule usually demand shipyard to provide documents regarding crack accumulation for inspection. In order to satisfy the new design purpose, the method we proposed is the development of coupled peridynamics and iso-geometric analysis method for dynamic analysis. The IGA method is recently developed method in the field of solid mechanics. It is currently one of the most robust method for solving problems with complex surface geometry and deformation, which is a perfect match for the ship structural analysis. On the other hand, peridynamics is a new particle based method that can solve the crack problem efficiently. In the future, we will extend the program of this research to model the crack formation and propagation on ship. To couple these two methods, we introduced two possible approaches: subdomain interface and overlapping interface. In subdomain interface method, the IGA domain and peridynamics domain only share the interface line. We use penalty method to enforce the continuity of displacement on the interface. In overlapping interface method, the coupling zone where the IGA mesh and peridynamics particle overlapping with each other is introduced. The penalty method is also applied in the coupling area to maintain the continuity. In the end, we verify the proposed method by one- dimension wave propagation problem. The comparisons of different discrete-density, time-step size, the effect of penalty number and coupling zone are also discussed in here.

The classical continuum mechanics, which studies the mechanical behavior of structures based on partial differential equations, shows its deficiencies when it encounters a discontinuity. Peridynamics based on integral equations can simulate fracture but suffers from high computational costs. A hybrid local/nonlocal model combining the advantages of peridynamics with those of classical continuum mechanics can simulate fracture and reduce the computational cost. Under the framework of the hybrid local/nonlocal model, this research developed an approach and computational codes for fracture simulations. First, we developed the computational codes based on the hybrid model with a priori partition of the domain between local and nonlocal models to tackle engineering problems with relevant level of difficulty. Second, we developed a strength-induced approach to enhance the capability of the computational codes because the strength-induced approach can limit the peridynamic model to necessary computational steps at the time level and a relatively small zone at the space level during a simulation. The strength-induced approach also improved the hybrid models by enabling an automatic partition of the domain without manual involvement. At last, a strength-induced computational code was developed based on this research. This dissertation complemented and illustrated numerically some previous work of Cohmas laboratory, in which a new route was introduced toward simulating the whole process of material behaviors including elastic deformation, crack nucleation and propagation until structural failure.

Recent advances in non-local continuum models, notably peridynamics, have spurred a paradigm shift in solid mechanics simulation by allowing accurate mathematical representation of singularities and discontinuities. This doctoral work attempts to extend the use of this theory to a community more familiar with local continuum models. In this communication, a coupling strategy - the morphing method -, which bridges local and non-local models, is presented. This thesis employs the morphing method to ease use of the non-local model to represent problems with failure-induced discontinuities. First, we give a quick review of strategies for the simulation of discrete degradation, and suggest a hybrid local/non-local alternative. Second, we present the technical concepts involved in the morphing method and evaluate the quality of the coupling. Third, we develop a numerical tool for the simulation of the hybrid model for fracture and damage and demonstrate its capabilities on numerical model examples

The thesis is devoted to study of continuum mechanics and thermodynamics and the related mathematical analysis. It consists of four self-contained chapters dealing with different aspects. The first chapter focuses on peridynamics, a non-local theory of continuum mechanics, and its relation to conventional local theory of Cauchy-Green elasticity. Similar compar- isons has been used for proving consistency and for determining some of the material coefficients in peridynamics, provided the material parameters in the local theory are known. In this chapter the formula for the non-local force-flux is computed in terms of the peridynamic interaction, relating the fundamental concepts of these two theories and establishing hence a new connection, not present in the previous works. The second and third chapters are both devoted to Rate-Independent Systems (RIS) and their applications to continuum mechanics. RIS represents a suitable approximation when the internal, viscous, and thermal effects can be neglected. RIS has been proven to be useful in modeling hysteresis, phase transitions in solids, elastoplasticity, damage, or fracture in both small and large strain regimes. In the second chapter the existence of solutions to an evolutionary rate-independ- ent model of Shape Memory Alloys (SMAs) is proven. The model...

Peridynamics (PD), a reformulation of the Classical Continuum Mechanics (CCM), is a new and promising meshless and nonlocal computational method in solid mechanics. To permit discontinuities, the PD integro-differential equation contains spatial integrals and time derivatives. PD can be considered as the continuum version of molecular dynamics. This feature of PD makes it a good candidate for multi-scale analysis of materials. Concurrently, the topology optimization has also been rapidly growing in view of the need to design lightweight and high performance structures. Therefore, this thesis presents the potential for a peridynamics-based topology optimization approach. To avoid the gradient calculations, a heuristic topology optimization method is employed. The minimization of the PD strain energy density is set as the objective function. The structure is optimized based on a modified solid isotropic material with a penalization approach and a projection scheme is utilized to obtain distinct results. Several test cases have been studied to analyze the suitability of the proposed method in topology optimization.
Graduate