Numerical plasticity

Course summary

The course focuses on the numerical modelling of plastic and viscoplastic material behaviour. It assumes that the basic concepts of plasticity and viscoplasticity have already been introduced during the first year of the programme.

The first part of the course recalls the main mechanical concepts needed to describe inelastic behaviour, including stress and strain measures, yield criteria, hardening variables, flow rules and rate-dependent effects. This review is then extended through an energy-based viewpoint. The thermodynamic structure of constitutive modelling is introduced, and the framework of generalized standard materials is briefly presented as a useful background for formulating elastoplastic and viscoplastic models in a consistent way.

The second part of the course is dedicated to the numerical integration of constitutive laws. Classical incremental methods are introduced, with particular emphasis on stress update procedures, return mapping algorithms, internal variable evolution and the solution of nonlinear constitutive equations. The course aims to clarify both the structure of these algorithms and their practical role in computational mechanics and finite element simulations.

Practical sessions are carried out in Python. Students implement simple elastoplastic and viscoplastic models, test the influence of material parameters, compare rate-independent and rate-dependent responses, and analyse the numerical behaviour of integration algorithms. These practical sessions provide a direct link between theoretical constitutive modelling and numerical implementation.

By the end of the course, students are expected to understand the structure of classical inelastic constitutive models, the principles of their numerical integration, and the main difficulties associated with their implementation in computational mechanics.