Formation of grain boundaries in ductile single crystals under plane-strain simple shear: a block-coordinate finite element method
Large plastic deformation can drive an initially uniform single crystal to spontaneously subdivide into misoriented grains separated by thin dislocation walls -- a pattern-forming instability rooted in the loss of convexity of the crystal's elastic energy at large strain. We study this phenomenon for a ductile crystal in plane-strain simple shear within continuum dislocation theory, using a polyconvex (Ciarlet--Geymonat) elastic energy that guarantees existence of minimizers for the coupled deformation--slip problem. Minimizing over the plastic slip yields a condensed energy of double-well form whose non-quasiconvexity favours a lamellar microstructure; the gradient of the geometrically necessary dislocation density regularizes it, giving the grain boundaries a finite thickness and energy as functions of the misorientation angle. A block-coordinate finite element scheme -- alternating a convex non-smooth solve for the slip with a Levenberg-regularized Newton solve for the deformation -- resolves this microstructure numerically and detects its spontaneous onset, reproducing the lamellar grain structure in agreement with the closed-form analysis.