Coupled phenomena in thin layers

Mechanical and transport behavior of thin constrained layers has been an important area of research in solid mechanics. Emerging engineering applications, involving advanced materials, require one to model and analyze the response of thin layers in which mechanical and transport phenomena are coupled. Further, those thin layers are often integrated into larger complex systems, for which fundamental understanding is lacking. This gives rise to numerous challenging problems in applied mathematics, scientific computing, and modeling.

We first analyze thin perfectly constrained linear elastic layers. A new asymptotic approach has been developed that, unlike all existing methods, does not invoke any assumptions other than the layer thinness and Saint-Venant's principle. Asymptotic solutions for the compressible and incompressible cases are relatively simple to construct, as they involve only one small parameter. However, these solutions are problematic for the nearly incompressible regime, which involves two small parameters. The problem is challenging both conceptually and in terms of matching accurate numerical solutions. Our approach closes the conceptual gaps, and is in excellent agreement with accurate numerical solutions across the entire physically admissible range of compressibilities.

Next, we extend our asymptotic analysis to thin imperfectly constrained linear elastic layers. This problem is interesting due to the presence of interphases, whereby two additional small parameters are introduced. In particular, the interplay between up to four small parameters can significantly affect the mechanical response of thin constrained layers. Our new accurate asymptotic solution establishes conditions under which interphases associated with thin layers may be detected. Further, the solution for compressible layers leads to a new and remarkably simple experimental setup, which has yielded promising results.

For thin linear chemo-elastic layers, we consider stress-assisted solid-state diffusion of solute through a deformable solid under isothermal conditions. We show that a thermodynamically-consistent characterization of stress-assisted diffusion should be in terms of the chemical potential, rather than the concentration, of the solute. Further, we formulate new thought experiments for measuring the chemo-elastic properties of solids. In particular, we propose a method for measuring the chemical potential of a solid, without imposing essential restrictions on its composition and external conditions. This enables the solid-state diffusion community to prescribe chemical potential boundary conditions, which as of yet is a non-intuitive task.

Finally, we analyze a full-scale turning process involving a workpiece made of a difficult-to-machine advanced material. Lubrication was represented by varying the friction coefficient between the cutting tool and workpiece. Here we wanted to assess the existence of an anomalous trend observed experimentally by our project partners, whereby lubrication was less effective at certain cutting speeds. The extremely complex thermo-mechanical phenomena associated with this problem lead us to rely on a commercial finite element code for our simulations. Here our main challenge from the perspective of scientific computing was balancing available experimental data, computing resources, and accuracy of simulations.