## Colloquium

**Date: ** November 28, 2023

**Time: ** 4:00PM - 5:00PM

**Location: ** BLOC 117

**Speaker: **Christian Goodbrake

**Description: **Geometric and group theoretic approaches to theoretical and applied solid mechanics
Within the context of mathematical physics, elastic solids are characterized by their ability to "remember" an intrinsic geometry, storing and releasing elastic energy through the stretching and relaxing of material. As such, approaching the associated governing equations through a geometric lens can reveal and clarify material and structural symmetries, formulate constraints and enable generalizations. Geometric elements such as curvature can concisely express compatibility conditions that would otherwise appear opaque, and this geometric framework readily generalizes to describe anelastic materials, those with nonvanishing residual strains, a feature relevant to plasticity, accretion, biological growth and remodeling. We show how this approach provides a natural explanation for the classification of universal solutions of incompressible hyperelasticity by the subgroup structure of SE(3). This is accomplished by extending the domain of the classification problem by allowing nonzero curvature in the relaxed state of the body, then examining the induced group action of the ambient space's isometry group on the material strain tensor fields. We demonstrate applications of this group action to the numerical simulation of soft tissues and the automatic determination of constitutive model form from DTMRI image data. Moving forward, we show how neural networks enable the rapid estimation of solutions to parameterized hyperelasticity problems, how a geometric framework enables the design of natural training algorithms for physics driven machine learning, and a potential framework for exploring the limiting dynamics of neural network training.