Hysteretic cycles via thermodynamic and variational approaches

Modelling of hysteretic phenomena is a challenge from a mechanical and engineering points of view. The reason is that structures, systems, materials, and devices are ruled by complex hysteretic behaviour. In the first part of the talk we will present the theoretical framework for modelling an hysteretic spring from two points of view: the thermodynamic and the hemivariational ones [3,4,5]. In particular, we will propose how to derive kinematic and isotropic hardening hysteretic behaviour with and without the related flow rules assumption. In the second part of the talk we will show how it is possible to adjust the hysteretic shape of the force-displacement diagram by changing the functional shape of the dissipation energy. Thus, we will show how the coupling, in the dissipation energy, between damage and plasticity induces fatigue and therefore the change of the hysteretic cycle along the time evolution. Thus, we will show the derivation of the S-N curve (or Whöler diagram). Finally, we will show how the use of granular micromechanics (in the same form Navier [1] and Cauchy [2] used almost two hundred years ago to derive linear elasticity constitutive equations) yields an immediate generalization of the spring's behaviour to that of a 3D continuum.