Interfaces (when two bodies come together or a single body breaks apart) are ubiquitous in mechanical, material, and biological systems. Examples (of interest to our group) include: bolted steel connections in buildings, adhesives bonding composite repair patches onto structural components, cracks in concrete and delamination between composite laminates, ligaments that connect bones and muscles, and cavity growth along grain boundaries during creep. Although the applications vary, the typical mathematical continuum models for these systems end up requiring similar looking interface terms to handle the evolution of the relative discontinuities between the surfaces of the interface. Hence, our group has pursued the Variational Multiscale Discontinuous Galerkin Framework as a vehicle to derive rather than design numerical methods that are robust and efficient for these classes of problems. This framework has provided a rational approach for solving both old and new classes of problems in solid and fluid mechanics with enhanced stability, thereby enabling the confident design of systems containing interfaces.
Although some materials, like structural steel and titanium, appear homogenous to the naked eye, observing them under a microscope reveals thousands and thousands of grains which have their atomic crystal lattice arranged in various directions. Other materials (such carbon-fiber composites, concrete, and human bones) display their heterogeneity even to our eyes. With recent advanced manufacturing techniques, it is increasingly feasible to control these directions and phases in materials across multiple length scales, such that stronger, lighter, and more optimized properties can be achieve. Computational process models are needed to guide this expansive design space rather than heuristics. Additionally, multiscale material models that link mechanical behavior at the micro and macro scales empower the design of structural systems and components using that material in complex loading conditions and long time horizons that can’t be tested in the laboratory. These applications of interest to our group span the range of process-structure-property-performance relationships that are a hallmark of materials science research. Thus, our computational modeling group collaborates actively with experimental colleagues to address challenging problems in materials with scales, directionality, and phases to certify existing materials for longer operation and to design next-generation materials with enhanced performance.
Our group takes on and explores emerging and challenging problems that intrigue us, in collaboration with other researchers at the University of Tennessee and other universities and laboratories. For example, we are employing machine learning for developing surrogate (low-fidelity) models that complement physics-based (high-fidelity) models to accelerate analyses of problems for calibration, validation, and uncertainty quantification. Historically, we have looked at other topics including: Fast Fourier Transform based modeling of material microstructures, mixture theory for modeling growth in biological tissues, estimation of discretization error in (and thus accuracy of) nonlinear finite element models of materials, and deriving stable and efficient formulations for linear tetrahedral finite element models. These topics don’t exactly fit in the other two themes, but they have intrigued us, and one day may lead to an additional research theme for our group.