Quasistatic evolution problems for material failure due to fatigue

Project Information

FatiguEvoPro

Grant agreement ID: 792583

DOI

10.3030/792583

Project terminated on 27 February 2021

EC signature date 3 December 2018

Start date 1 December 2019

End date 30 November 2021

Funded under

• EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions

Total cost

€ 159 460,80

EU contribution

€ 159 460,80

Coordinated by

TECHNISCHE UNIVERSITAET MUENCHEN
Germany

Periodic Reporting for period 1 - FatiguEvoPro (Quasistatic evolution problems for material failure due to fatigue)

Reporting period: 2019-12-01 to 2021-11-30

Summary of the context and overall objectives of the project

In Materials Science, fatigue is the process by which a solid exhibits a loss of its elastic properties as a consequence of repeated cycles of loading and unloading. Such failure phenomena occur because microscopic defects appear where the strain oscillates during the evolution, resulting in a considerable macroscopic weakening of the material and, in drastic cases, in rupture.

Fatigue is responsible for most of unforeseen mechanical failures. These affect people and society at all levels, encompassing all possible structures: everyday-life items, human body prostheses, buildings, bridges, ground vehicles, ships, aeroplanes, oil platforms, etc. In most dramatic situations, the consequences of fatigue failure can be catastrophic, causing tremendous financial losses, serious injuries, and deaths.

Investigating fatigue phenomena is a crucial aspect in the Engineering community, however so far scarcely investigated by the Mathematics community. This results in the lack of proofs of consistency and well-posedness for variational models of material fatigue.

The overall aim of the project has been to prove the existence of evolutions for mechanical models of material fatigue.

This has been pursued by examining fatigue in the different contexts of damage, plasticity, and cohesive fracture.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

The research work carried out in the period covered by the action has been focused on the main aim of the project: proving the existence of evolutions for mechanical models of material fatigue. This has been done in the different contexts of damage, plasticity, and cohesive fracture.

More precisely, the following objectives have been tackled:
– Objective 1: Proving an existence theorem for quasistatic evolutions in regimes of high-cylcle fatigue;
– Objective 2: Proving a well-posedness theorem for models involving the coupling between damage and plasticity;
– Objective 3: Proving the existence of quasistatic evolutions for cohesive fracture due to fatigue.
For Objective 1, we considered a gradient damage model featuring a term depending on the cumulated elastic strain that appears in the dissipation of damage. As a consequence of this term, the energy dissipated to produce a damage increment becomes lower in the regions where the material has experienced oscillations of the elastic strain. We proved existence of evolutions via a vanishing viscosity approach.
For Objective 2, we considered a model for linearised elasto-plasticity coupled with damage and we proved the lower semicontinuity of the plastic potential term in the energy only requiring only H1 regularity of the damage.
For Objective 3, we studied a cohesive fracture model with fatigue in the one-dimensional setting, where fracture concentrates on points.

Finally, we additionally studied discrete-to-continuum limits of atomistic systems, with applications to dislocation theory.

The results have been disseminated with seminars and talks in conferences.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

We expect the results of the project to give a deeper insight into mechanical models of fatigue, beside validating their consistency. Well-posedness theorems for mechanical models are crucial for Engineers to run simulations with a solid foundation. The analysis carried out with the project will give a deep insight into fatigue phenomena and will further encourage Engineers to explore the problem of fatigue. The achievement of the project will allow for a substantial improvement on the mechanical models, thus making these completely reliable for exploitation in industry.

The results obtained open a new line of research in the variational modelling of Fracture Mechanics and create many future perspectives, like the following problems: i) brittle fracture due to plastic fatigue; ii) brittle fracture as sharp-interface limit of damage in the context of fatigue phenomena; iii) discrete-to-continuum motivation of fatigue, etc.

Some of the techniques developed have great potential for applications in different contexts, as shown by the applications of tools from Geometric Measure Theory to discrete-to-continuum limits of atomistic systems.

The outcome of the project will also improve the perception among the general public of the work done by mathematicians. Indeed, the common thought is that research in Mathematics only deals with abstract theories and that the produced results are only exploitable by mathematicians themselves. Instead, it concerns also real problem like fatigue failure.

Fatigue is usually classified into two large categories: low-cycle and high-cycle fatigue.