Clinching is often used to join sheet-like parts, which can be made of different materials, to produce complex lightweight structures. The holistic prediction of the joinability, the properties and the life cycle of a joint requires the modeling and simulation of the complete clinching process chain. Therefore, (i) the design phase, (ii) the joining process itself and (iii) the resulting properties of the connection in the operational phase must be taken into account. This paper is the second part (part II) of a series of reviews summarizing the current state of research on modeling and simulation of clinching. The topic of this second part is the joining process itself, i.e. the process simulation of the clinching process. Thereby, the focus is on material models for the process simulations, on plasticity-damage models for metallic materials and on multi-scale material models for fiber-reinforced plastic composites. These complex material models are required to make clinching process simulations more accurate and predictive. More detailed process simulations allow better prediction of the properties of the clinch joints, which is essential for their distribution and optimization in the design phase (part I of the review series) and for the subsequent simulation of the operating phase considering corrosion and fatigue cracks (part III).