Plastic deformation,Plastic deformation by slip,Dislocation motion,Plastic deformation is volume conservative,Slip systems,Slip planes: crystallographic planes of maximum atomic density easier atom motion when distance a is bigger Slip directions: close-packed directions less shear stress required if smaller distance between two atoms,Slip Plane & Slip Direction,FCC crystals Slip planes (111) - close packed planes Slip direction 101 - close packed direction,BCC Crystals, slip direction (close-packed direction) Any plane containing is a potential slip plane Experimentally observed in (110), (112) & (123) planes,Slip Plane & Slip Direction,HCP Crystals,Slip Plane & Slip Direction,c/a 1.633 - basal slip dominant c/a 1.633 - prism slip dominant,Some exceptions,Schmid factor for uniaxial stress,Shear stress acting on the slip system along the slip direction: Slip system activated when: RSS= CRSS Schmid factor = cos cos,Non-uniaxial stress,General Stress State: t(n): traction per unit area t(n)= n RSS:shear stress,Pure shear: Kinematics,Displacement u=(x.n)m,Strain rate tensor,Rotation rate tensor,Gives the rotation rate of the crystal,Schmid tensor,Total strain and total rotation,Several slip systems can be activated at the same time: How to calculate ?,Slip law: Rate-Independence,If CRSS then : there is no plastic strain If = CRSS then,Taylors model,Imposed strain rate tensor Write the strain rate tensor as a combination of all the slip systems,Taylors model,5 equations Number unknowns= Number slip systems,If more than 5 slip systems: problem cannot be solved Therefore, need 5 independent slip systems,For FCC: 12 slip systems How to choose the 5 independent slip systems?,Taylors model,Find all the sets of 5 independent slip systems Find the set of 5 independent slip systems with the minimum energy dissipation For non hardening material: = CRSS Addresses ambiguity in some cases but not in all,Dependence on the rate sensitivity parameter,Viscoplastic slip law,Example 1,FCC material Applied stress:,Slip systems,12 possible slip systems,P matrices:,1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12:,Activated slip systems,Solution example 1,Total strain rate: Total rotation rate:,Example 2,FCC material Applied stress:,Change of basis,Q: transformation matrix to go from the global frame to the crystal frame,To modify a vector xg: xc= Q xg To modify a second order tensor Ag: Ac=Q Ag QT,e1c,e2c,e3c,e3g,e2g,e1g,Stress tensor,Transformation matrix: Stress tensor in crystal frame:,Activated slip systems,Solution example 2,Total strain rate: Total rotation rate:,In crystal frame,In global frame,Before deformation,After deformation,Example 3,FCC material Applied stress:,Stress tensor,Transformation matrix: Stress tensor in crystal frame:,Activated slip system,Solution example 3,Total strain rate: Total rotation rate:,In crystal frame,In global frame,In crystal frame,In global frame,Crystal rotation,Rotation tensor: R(t)=(I + t) for small time period In our example: We can calculate:,Crystal rotation,New transformation matrix: Qnew=R(t) Q,