Three-dimensional mixed mode I plus II plus III singular stress field at the front of a crack weakening a diamond cubic mono-crystalline plate with crack turning and step/ridge formation

A heretofore-unavailable mixed Frobenius type series, in terms of affine-transformed x-y coordinate variables of the Eshelby-Stroh type, is introduced to develop a new eigenfunction expansion technique. This is used, in conjunction with separation of the z-variable, to derive three-dimensional mixed-mode I+II+III asymptotic displacement and stress fields in the vicinity of the front of a semi-infinite through-thickness crack weakening an infinite diamond cubic mono-crystalline plate. Crack-face boundary conditions and those that are prescribed on the top and bottom (free, fixed or lubricated) surfaces of the diamond cubic mono-crystalline plate are exactly satisfied. Explicit expressions for the mixed mode I+II+III singular stresses in the vicinity of the front of the through-thickness crack, are presented. Most important mixed modes I+II+III response is elicited even though the far-field loading is only mode I or II or III or any combination thereof. Finally, atomistic modeling of cracks requires consideration of both the long range elastic interactions and the short range physico-chemical reactions, such as bond breaking. The Griffith-Irwin approach does not take the latter into account, and nano-structural details such as bond orientation must be accounted for. A new mixed-mode I+II+III crack deflection criterion elucidates the formation of steps and/or triangular ridges on the crack path. The planes of a multiply deflected crack are normal to the directions of broken bonds. Additionally, the mixed-mode (I+II+III) crack deflection and ridge formation are found to be strongly correlated with the elastic stiffness constants, and , of the diamond cubic single crystal concerned.