Both walkers and runners know it well: Every step creates shock and vibration. Athletic shoe manufacturers are particularly interested in reducing these vibrations for comfort reasons. Antoni Joubert takes up this challenge in his research work as part of the Arkema Chair “Design and modeling of innovative materials”. A pair of hiking shoes. (iStock/Getty Images Plus – MTB Studio)
When running, the impact of the foot on the ground creates a shock that causes vibrations that travel down the legs. In order to improve the comfort of athletes, the companies that develop sports shoes are innovating in the construction of soles that absorb vibrations.
Antoni Joubert, a PhD student at the Solid Mechanics Laboratory (LMS), worked on the optimization of shoe sole shapes as part of the Arkema-supported Chair in Design and Modeling of Innovative Materials at the École Polytechnique. His goal was to determine the optimal shape to dampen vibrations after impact as quickly as possible. This is modeling work in collaboration with the Center for Applied Mathematics (CMAP).
To develop his program, the researcher worked progressively and with increasing complexity: first from a one-dimensional (1D) beam model, then from a two-dimensional (2D) plate, before adding a certain thickness to switch to three dimensions (3D). Finally, he used the 3D model of a real sports shoe sole.
He digitally simulated a vibration wave to which the sole would be subjected and observed the reduction in its amplitude by the polymer material from which it is made. This was chosen because of its viscoelastic properties, ie its dual ability to dissipate energy (viscosity), such as that transmitted during an impact, and to return to its original shape (elasticity) after deformation.
Starting with a sole pierced in several places, he searched for the optimal shape of the holes that would maximize the speed at which the vibration wave is dampened. These calculations were carried out while maintaining an equal amount of material and a certain rigidity whatever the shape of the sole, important parameters for industrial concerns.
The optimal shape for the tested material offers a 17% higher cushioning gain than the original sole, a real improvement! The look may surprise you: the optimized sole has both outward opening holes that form galleries and cavities within its thickness. (More details at the end of the article)
This image shows a cross-sectional view of the original sole (before optimization) and a bottom view of the optimized sole. The white areas show the through holes on both sides of the sole. Image courtesy of Antoni Joubert.
On December 15, 2022, Antoni Joubert successfully defended his dissertation entitled “Optimization of the forms of viscoelastic structure under dynamic stresses”. This is the first doctoral thesis to be defended as part of the “Design and modeling of innovative materials” chair at the École polytechnique. This multidisciplinary work was led by Julie Diani, researcher at the LMS and also Chair, and by Grégoire Allaire and Samuel Amstutz, both professors at the CMAP.
When asked about the special feature of doing a doctorate within the framework of a chair, Antoni Joubert emphasizes the advantage of working on an applied research topic while maintaining academic freedom. Additionally, during his thesis work, he had the opportunity to visit Arkema for two weeks a year to discuss with the Research and Development (R&D) team.
“Until then, I had mainly worked in academic research laboratories. I appreciated discovering how an R&D team works in a company and benefiting from knowledge sharing with the team on site,” he explains.
“Arkema’s support of this work reflects the importance of modeling in the development of specialized, high-performance, and environmentally friendly materials.” explains Nessim Ghamri, head of the R&D department at the Rhône-Alpes research centre. “Thanks to the models developed, Arkema will be able to optimize the mechanical behavior of polymer, composite and adhesive materials by studying the relationships between microstructure and mechanical behavior for different applications (sports, new mobility, renewable energy).”
For the future, Antoni Joubert, a new doctor, wants to continue digital simulation and plans to move to industry.
For further
These curves represent the attenuation of the vibration wave by the initial sole model (in solid line) and by the one whose shape has been optimized (in dotted lines). It can be seen that in both cases the amplitude of the wave decreases over time, which happens faster with the optimized sole that dampens the vibration more efficiently. Image courtesy of Antoni Joubert.
The surprising shape of the optimized sole is explained by the rigidity condition imposed during the optimization. This is derived from an industry standard flex test useful for testing outsole fatigue. One of the perspectives of the thesis recommends replacing this bending stress with a load condition representative of walking or running.
This work, originally aimed at improving cushioning, can be directly extended to improving the dynamics (property somewhat opposite to cushioning) of shoe soles for sports applications such as running.