Upcoming Event: CSEM Student Forum

Geometric and Thermodynamic Structures in Continuum Multiphysics - Foundational Contexts for Models

Blake Christierson, Graduate Student, Oden Institute

Abstract

This talk represents the convergence of several seemingly distinct research questions into a unified framework. The main driving force behind this work is my thesis application problem, which involves modeling thermoviscoplasticity coupled to phase change. In the model, the core engine driving the constitutive response is the thermodynamic framework of implicit standard materials (ISMs). ISMs span a rich and widespread class of constitutive laws despite their relatively simple components. However, by the time the ISM framework is typically invoked, the complexity of the multiphysics obscures its underlying thermodynamic structure in a sea of symbols.

To address this, I began abstracting the model into a more general, interpretable form to better interface with collaborators. This evolved into a broader exploration of whether ISMs could be seamlessly generalized to a true multiphysics context—spanning mechanics, chemistry, and electromagnetism. Along the way, the topological structure of Tonti diagrams provided the inspiration needed to formulate a generalized kinematics framework, extending the differential geometric description of mechanics to multiphysics.

In this talk, I will first unpack this generalized kinematics framework, highlighting compatibility conditions and objectivity. Next, I will construct rate-form balance laws for the evolution of external configurational states using Green's theorem and the first law of thermodynamics. Finally, I will derive the evolution laws for internal variables using the standard ISM procedure. Throughout the presentation, finite-strain anisotropic thermoelastoplasticity and electromagnetism with remanence will serve as illustrative examples. If time permits, I will conclude by discussing regularization approaches inspired by information geometry.

Biography

Blake is from the Sacramento, California area where he grew up and received his undergraduate degree in mechanical engineering from UC Davis. His research is motivated by the inverse problems and uncertainty quantification applications that arise in complex engineering systems. He is currently investigating surrogate models that leverage low-rank representations of tensors.