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.

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.

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

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.

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 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.

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.

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.

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.