1. Grain Boundary Cohesive Laws as a Function of Geometry Valerie R. Coffman, James P. Sethna Cornell University

2. Measuring Grain Boundary Energy and Fracture Strength Stress (Lennard-Jones Units) Strain -Measure energy and fracture strength for all commensurate geometries -Cohesive Laws used in FEM simulations -2D Geometries described by 2 angles -3D described by 5 parameters

3. Grain Boundary Energy θ1θ1 θ2θ2 θ 1 =θ 2 60 Energy (LJU) 30 0 1 -Grain Boundary Energy has cusps for high symmetry geometries -Devil’s staircase analogy: cusp singularity at all rationals θ1θ1 θ2θ2

4. Grain Boundary Energy d θ 1 =θ 2 Energy (LJU) -Dislocation added to high symmetry geometry with Burger’s vector = b -For low angle grain boundaries: θ = b / d E ~ - θ log (θ) -Near high symmetry geometries: E ~ - |θ-θ 0 | log |θ-θ 0 |

5. Peak Stress θ1θ1 θ2θ2 θ 1 =θ 2 6030 2 2.5 Stress (LJU) Strain Peak Stress Perfect Crystals Fracture toughness decreases abruptly when high symmetry is broken

6. -Added flaw nucleates fracture at stress = σ c -Nucleation point feels stress of added dislocations -Volterra solution gives: σ peak (θ) = σ c - A(θ - θ 0 ) Peak Stress Stress (LJU) θ 1 =θ 2 Atomistic Volterra Soln nucleation pt

7. Thanks to Yor Limkumnerd, Anthony Ingraffea, Gerd Heber, Wash Wawrzynek, Paul Stodghill, The Adaptive Software Project -Adding a flaw to a high symmetry grain boundary is analogous to adding a flaw to a perfect crystal -Energies have cusps at high symmetry boundaries -Fracture toughness has discontinuities everywhere -Self-similarity in both functions