This paper presents a modeling framework to describe the driving mechanisms of cyclic failure in brittle and ductile materials, including cyclic plasticity and fatigue crack growth. A variational model is devised using the energetic formulation for rate-independent systems, coupling a phase-field description of fatigue fracture to a cyclic plasticity model that includes multi-surface kinematic hardening, gradient-enhanced isotropic hardening/softening and ratcheting. The coupled model embeds two distinctive fatigue effects.The first captures the characteristic features of low-cycle fatigue, driven by the accumulation of plastic strains, while the second accounts for high-cycle fatigue, driven by free energy accumulation. The interplay between these mechanisms allows to describe a wide range of cyclic responses under both force loading and displacement loading, as shown in several numerical simulations. Moreover, the phase-field approach to fracture accounts for the initiation and propagation of fatigue-induced cracks.

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In this work, the energetic formulation for rate-independent dissipative materials is extended to consider non-associative plasticity models. In associative models, the fulfilment of the principle of maximum dissipation naturally leads to a variational formulation of the evolution problem. This is no longer true for non-associative models, which are generally presented in the literature in a non-variational form. However, recent studies have unveiled the possibility to recover a variational structure in non-associative plasticity by relying on a suitable state-dependent dissipation potential. Here, this idea is further elaborated in the framework of the energetic formulation, providing a systematic variational approach to non-associative plasticity. A clear link between the classical governing equations of non-associative plasticity and the energetic formulation is established, for which a state-dependent dissipation potential is derived from a generalization of the principle of maximum dissipation. The proposed methodology is then applied to recast specific non-associative plasticity models in variational form, highlighting the flexibility of the formulation. The examples include a Drucker-Prager model with combined isotropic-kinematic hardening and a ratcheting plasticity model. Several thermomechanical insights are provided for both examples. Moreover, exploiting the flexibility of the energetic formulation, extensions to gradient plasticity are devised, leading to representative finite element simulations.

The capability to analyze structures under static and dynamic loads is an essential skill for structural engineers. Structural analysis therefore is a key component in civil and architectural engineering education, where analytical methods are traditionally complemented by the use of (commercial) software packages. The latter are often closed source, which may obscure the links to the underlying matrix structural analysis or finite element formulation. To stimulate active, cooperative, and solution‐oriented learning, we developed Stabil, an electronic learning environment implemented as a Matlab toolbox. Stabil has an open structure to elucidate the links between theory and implementation and is presently used throughout the curricula of Civil and Architectural Engineering at KU Leuven. This paper explains the main principles of Stabil and discusses teaching experiences over the past 15 years. Stabil can be downloaded from http://bwk.kuleuven.be/bwm/stabil.

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