Fracture toughness is one of the most important properties of a material. Being able toaccurately estimate the energy that goes into forming new crack surfaces is essential for the development of new materials, quality assurance, structural monitoring and failure analysis. Fracture toughness parameters are routinely determined by mechanical testing and are often used in numerical tools. Furthermore, fracture toughness is a common property in material specification. Numerical simulation of fracture toughness can reduce the need of mechanical testing and is sometimes the only viable alternative when mechanical testing is not an option, for example in component optimisation and in the assessment of operational structural components.  However, complex fracture is a challenge in material modelling, which comes from that a material body is assumed to remain continuous in classical continuum mechanics. Classical continuum mechanics is formulated assuming a continuous body and that spatial derivatives are defined. However, this is not the case at cracks and other dis­ continuities. Complementing continuum mechanics with supplementary procedures for modelling discontinues can also add further challenges. Besides, the assumption of locality, that each material point only interacts with is immediate neighbouring points, becomes invalid for nanoscale geometries. Thus, fracture cannot easily be modelled. An alternative is therefore of interest. Peridynamics is a nonlocal extension of continuum mechanics with the constitutive model formulated as an integro-differential equation. The advantages of using an integral expression are foremost that long-range forces can be handled and that the theory is valid even in the presence of discontinuities, such as cracks, allowing unguided modelling of fracture. Since damage is introduced to the constitutive model of peridynamics, there is no requirement of supplementary procedures that can add further complications. Due to its nonlocal formulation, the method is also capable of capturing nano-effects. However, the use and reporting of fracture toughness parameters in peridynamics is a routine in its infancy as the method is under development.In this thesis, two fracture toughness methods, the classical J-integral and the essential work of fracture (EWF), are studied with peridynamics. Also, as the nonlocality of peri­ dynamics give rise to certain boundary effects, e.g. on crack faces, homogenisation is a part of the study. The thesis consists of two parts; an introductory summary with discussion and conclu­ sions, followed by a series of appended papers. The first paper concerns application of Rice's J-integral on displacement derivatives formulation in peridynamics with comparison to an exact analytical stress-strain-displacement specimen solution. The next two papers concerns homogenisation of a peridynamic bar, to remove the end effects, arisen from the nonlocality of peridynamics, to obtain an elastic behaviour exact to a classical continuum mechanics bar. The fourth paper is an implementation of the J-area integral into peridynamics, with study of various discretisation methods. Thereafter, in the last paper, Rice's J-integral and the nonlocal peridynamic J-integral are compared on various specimens, followed by an extension of the research to study EWF with peridynamics for the first time. The study includes a novel automated calibration at the interparticle bond level to simulate nonlinear elastic behaviour, which subsequently is complemented with softening and used for EWF modelling. As a part of introducing the peridynamic J-integral, the study also includes a proof of path independence.

Major findings of the study includes:

• The classical J-integral on a displacement derivative formulation gives accurateestimations of fracture toughness in peridynamics.

• The peridynamic lD bar can be homogenised to obtain a linear elastic behaviour identical to that of an corresponding continuum mechanics body.

• The bond calibration method gives a nonlinear elastic peridynamic model that can accurately recover an experimentally obtained stress-strain response. Up to the start of material softening, the nonlinear elastic model recovered the experimentally obtained stress-strain response of two very different materials; a lower-ductility martensitic-bainitic steel and a higher-ductility bainitic steel.

• The nonlinear elastic model were able to match very well the experimentally measured EWF for the higher-ductility bainitic steel.

• The J-integral value obtained from the peridynamic model, matched the experimen­tally obtained EWF value for the higher-ductility bainitic steel.