# Applied/ACMS/absS20

# ACMS Abstracts: Spring 2020

### Hung Tran

Title: Coagulation-Fragmentation equations with multiplicative coagulation kernel and constant fragmentation kernel

Abstract: We study a critical case of Coagulation-Fragmentation equations with multiplicative coagulation kernel and constant fragmentation kernel. Our method is based on the study of viscosity solutions to a new singular Hamilton-Jacobi equation, which results from applying the Bernstein transform to the original Coagulation-Fragmentation equation. Our results include wellposedness, regularity and long-time behaviors of viscosity solutions to the Hamilton-Jacobi equation in certain regimes, which have implications to wellposedness and long-time behaviors of mass-conserving solutions to the Coagulation-Fragmentation equation. Joint work with Truong-Son Van (CMU).

### Curt A. Bronkhorst

Title: Computational Prediction of Shear Banding and Deformation Twinning in Metals

Abstract: The high deformation rate mechanical loading of polycrystalline metallic materials, which have ready access to plastic deformation mechanisms, generally involve an intense process of several deformation mechanisms within the material: dislocation slip (thermally activated and phonon drag dominated), recovery (annihilation and recrystallization), mechanical twinning, porosity, and shear banding depending upon the material. For this class of ductile materials, depending upon the boundary conditions imposed, there are varying degrees of porosity or adiabatic shear banding taking place at the later stages of the deformation history. Each of these two processes are as yet a significant challenge to predict accurately. This is true for both material models to represent the physical response of the material or the computational framework to represent accurately the creation of new surfaces or interfaces in a topologically independent way. Within this talk, I will present an enriched element technique to represent the adiabatic shear banding and deformation twinning process within a traditional Lagrangian finite element framework. A rate-dependent onset criterion for the initiation of a band is defined based upon a rate and temperature dependent material model. Once the bifurcation condition is met, the location and orientation of an embedded field zone is computed and inserted within a computational element. Once embedded the boundary conditions between the localized and unlocalized regions of the element are enforced and the composite sub-grid element follows a weighted average representation of both regions. Continuity in shear band growth is ensured by employing a non-local level-set technique connected to the displacement field within the finite-element solver. The material inside the band is able to be represented independent from the outside material and the thickness of the band can be assigned by any appropriate method. Dynamic recrystallization (DRX) is often observed in conjunction with adiabatic shear banding (ASB) in polycrystalline materials and is believed to be a critical softening mechanism contributing to the material instability. The recrystallized nanograins in the shear band have few dislocations compared to the material outside of the shear band. We reformulate a recently developed continuum theory of polycrystalline plasticity and include the creation of grain boundaries. While the shear-banding instability emerges because thermal heating is faster than heat dissipation, recrystallization is interpreted as an entropic effect arising from the competition between dislocation creation and grain boundary formation and is a significant softening mechanism. We show that our theory closely matches recent results in sheared 316L stainless steel. The theory thus provides a thermodynamically consistent way to systematically describe the formation of shear bands and recrystallized grains therein. The numerical tool has recently been applied to the modeling of deformation twinning in high-purity Ti which will be briefly discussed.