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Novel techniques for modeling 3D cracks and their evolution in solids are presented.Cracks are modeled in terms of signed distance functions (level sets).Stress, strain and displacement field are determined using the extended finite elements method (X-FEM).

This paper proposes a simplified approximation of the velocity fields close to the contact edge using non-local intensity factors and reference fields. This non-local description is not affected by size or gradient effects and leads to unique crack initiation boundaries. Linear intensity factor I represents the contribution of the elastic field on the total one and can be determined thanks to the macroscopic load even with coarse mesh. While the complementary intensity factor I c describes the friction effect and is proportional to the size of the partial slip zone. The prediction of I c using I through an incremental approach allows to predict the velocity fields for complex loadings and can be used to set up criteria to predict crack initiation.

Predicting fatigue crack growth in metals remains a difficult task because available models are based on cycle-derivative equations, such as the Paris law, while service loads are often far from being cyclic. The main objective of this paper is therefore to propose a set of time-derivative equations for fatigue crack growth. The model is based on the thermodynamics of dissipative processes. For this purpose, three global state variables are introduced in order to characterize the state of the crackthe crack length a, the plastic blunting at crack tip ρ and the intensity of crack opening C. Thermodynamics counterparts are introduced for each variable. Special attention is paid to the elastic energy stored inside the crack tip plastic zone, because, in practice, residual stresses at crack tip are known to considerably influence fatigue crack growth. The stored energy is included in the energy balance equation, and this leads to the appearance of a kinematics hardening term in the yield criterion for the cracked structure. No dissipation is associated with crack opening, but to crack growth and to crack tip blunting. Finally, the model consists in two laws: a crack propagation law, which is a relationship between dρ dt and da/dt and which observes the inequality stemmed from the second principle, and an elastic-plastic constitutive behaviour for the cracked structure, which provides dρ dt versus applied-load. The model was implemented and tested. It reproduces successfully the main features of fatigue crack growth as reported in the literature, such as the Paris law, the stress-ratio effect and the overload retardation effect.

This chapter contains sections titled: Introduction Superposition principle Modes of crack straining Singular fields at cracking point Crack propagation criteria

This chapter contains sections titled: Introduction Going back to discretization methods X‐FEM discontinuity modeling Technical and mathematical aspects Evaluation of the stress intensity factors

This chapter contains sections titled: Introduction Fatigue and non‐linear fracture mechanics eXtended constitutive law Applications

This chapter contains sections titled: Energy conservation: an essential ingredient Examples of crack growth by fatigue simulations Dynamic fracture simulation Simulation of ductile fracture

This chapter contains sections titled: Geometric representation of a crack: a scale problem Crack representation by level sets Simulation of the geometric propagation of a crack Prospects of the geometric representation of cracks