Isogeometric and Meshfree Analysis of Nonlocal Damage, Fracture, and Contract Part II: Mathematical Issues

• Are nonlocal phase-field fracture formulations compatible with time-explicit analysis? My proposal is to formulate a hyperbolic partial differential equation governing the damage field, with constraints on its coefficients following by analogy to mathematical literature on the telegraph equation. It may also be derived from microforce balance theory, by associating a kinetic-like energy to the damage field.  
• How should nonlocal damage models be formulated on thin shells? The problem of gradient-enhanced damage is re-cast in the language of differential geometry, which provides the appropriate framework for deriving a shell theory. Because isogeometric analysis operates directly on manifold geometries, discretization follows naturally in that setting.
• Can nonlocal contact formulations include friction in a way that preserves objectivity and angular momentum balance? Yes; the concept of nonlocal Taylor extrapolation is introduced for this purpose, and may prove useful elsewhere.

Several open mathematical questions for future work on nonlocal damage and contact modeling will also be posed. 

David Kamensky is an Assistant Professor in the Mechanical and Aerospace Engineering department at UC San Diego, specializing in computational mechanics. He received a BS from the University of Virginia in 2011, where he studied Computer Science and Physics, and a PhD from the University of Texas at Austin in 2016, in Computational Science, Engineering, and Mathematics. Before joining the faculty at UC San Diego in 2019, he held postdoctoral positions in the Structural Engineering department at UC San Diego and in the School of Engineering at Brown University.