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

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.

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.