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Single crystal deformation vis slip and twinning

Single crystals can deform via slip or twinning. When slip occurs we need to calculate the stress necessary to activate slip. We do this by using the critical resolved shear stress and the equations for resolved stress as a function of loading direction, slip plane, and slip direction. We can use the dot product to calculate the angle between these directions. Twinning occurs when large groups of atoms cooperatively shift in the same direction to create a grain boundary that preserves mirror symmetry.