Mathematical Modelling in Mechanics

The Physical Mechanism

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Plasticity is a mechanism of deformation of solids that allows a body to deform beyond the elastic limit without failure.

Materials such as metals first deform in a reversible manner (the elastic regime). Beyond a critical threshold known as the yield stress, microscopic defects called dislocations become active. These dislocations move along specific glide planes and enable the material to undergo permanent deformation while remaining structurally intact.

Understanding the dynamics of dislocations is essential for predicting material performance, failure mechanisms, and manufacturing processes.

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Industrial Applications

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1. Manufacturing and Forming Processes

• Metal Forming: forging, rolling, extrusion, and deep drawing rely on the plastic behavior of metals to achieve desired shapes and mechanical properties.
• Polymer Molding: thermoplastics can be molded into complex geometries due to their plasticity when heated.

2. Numerical Simulation of New Materials

• Ni-Al and Ti-Al superalloys for jet engines and power-generation turbine discs and blades operating at high temperatures.
• High-strength, lightweight alloys with enhanced ductility, contributing to reduced greenhouse gas emissions.
• Stress and defect management in semiconductor thin films, aiming to maximize epitaxial layer thickness while avoiding interface and threading dislocations.
• Thermal mismatch stress management in chip interconnects and advanced electronic devices.

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Our Model

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The proposed model is fundamentally different from classical plasticity theories. In particular, it relies on high-order tensorial equations that explicitly incorporate the concept of incompatibility of deformation.

This incompatibility is directly related to the density and distribution of dislocations within the material. The model has been under continuous development since 2014 and combines:

• Advanced mathematical analysis.
• Sophisticated numerical methods.
• Thermodynamic consistency.
• Physical modeling of dislocation mechanics.

The research has already resulted in four publications in internationally recognized journals since 2016, with two additional papers currently in preparation.

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Research Team

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The project is led by Prof. Nicolas Van Goethem, member of CEMS.UL and professor in the Department of Mathematics at the Faculty of Sciences of the University of Lisbon since 2008.

He has more than twenty-five years of experience in Applied Mathematics and Mechanics and has worked with several leading international institutions, including:

• UCLouvain
• University College London
• École Polytechnique (France)
• University of Pisa
• SISSA Trieste

The model was originally developed by Nicolas Van Goethem and Samuel Amstutz, Professor at the University of Avignon and former professor at École Polytechnique.

A third senior member is Thien-Nga Lê, CNRS researcher at École Polytechnique (LMS Laboratory, Materials Science).

Additional collaborators are based at CIMNE (Barcelona) and the University of Notre Dame (USA).

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Current Needs

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The project seeks support to expand the research team through:

• 1 PhD student.
• 1 Postdoctoral researcher.
• 2 Master's students for complementary research projects.

Future work will focus on numerical simulations and the physical understanding of key phenomena such as hardening effects, size-dependent behavior, three-dimensional problems, and complex geometries.

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