*Article* **Strain-Gradient Crystal Plasticity Finite Element Modeling of Slip Band Formation in** α**-Zirconium**

**Omid Sedaghat and Hamidreza Abdolvand \***

Department of Mechanical & Materials Engineering, Western University, Spencer Engineering Building, London, ON N6A 5B9, Canada; osedagha@uwo.ca

**\*** Correspondence: hamid.abdolvand@uwo.ca

**Abstract:** Two methods for the determination of geometrically necessary dislocation (GND) densities are implemented in a lower-order strain-gradient crystal plasticity finite element model. The equations are implemented in user material (UMAT) subroutines. Method I has a direct and unique solution for the density of GNDs, while Method II has unlimited solutions, where an optimization technique is used to determine GND densities. The performance of each method for capturing the formation of slip bands based on the calculated GND maps is critically analyzed. First, the model parameters are identified using single crystal simulations. This is followed by importing the as-measured microstructure for a deformed α-zirconium specimen into the finite element solver to compare the numerical results obtained from the models to those measured experimentally using the high angular resolution electron backscatter diffraction technique. It is shown that both methods are capable of modeling the formation of slip bands that are parallel to those observed experimentally. Formation of such bands is observed in both GND maps and plastic shear strain maps without pre-determining the slip band domain. Further, there is a negligible difference between the calculated grain-scale stresses and elastic lattice rotations from the two methods, where the modeling results are close to the measured ones. However, the magnitudes and distributions of calculated GND densities from the two methods are very different.

**Keywords:** zirconium; strain-gradient crystal plasticity; slip band; geometrically necessary dislocation; statistically stored dislocation; high angular resolution electron back scatter diffraction
