DissertationIncompressibility and Global Injectivity in Second-Gradient Non-Linear Elasticity
We prove the energy minimizers for a broad class of physically reasonable secondgradient non-linear elastic materials satisfy weak equilibrium equations with incompressibility and/or self-contact constraints. In the case of incompressibility, this is equivalent to proving the existence of a distributional pressure, and for selfcontact, we prove the existence of a measure-valued surface traction supported on the coincidence set. We find that coercivity of the second-gradient energy yields enough regularity on the minimizer to rigorously take variations within the class of incompressible and/or globally injective deformations. A major difficulty lies in constructing sufficiently regular incompressible/globally injective variations with prescribed boundary conditions. We discuss some of the necessary regularity theory with an approach that has applications to broader mixed-order non-linear elliptic systems. The admissible deformations are globally injective if Dirichlet boundary conditions are imposed on the entire boundary. However with mixed boundary conditions, minimizers may lose injectivity without an additional constraint. The self-contact constraint gives an example of an infinite-dimensional variational inequality. For the self-contact problem, a significant portion of the work goes into characterizing the tangent cone at an admissible deformation.