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Metal Machining - Theory and Applications Episode 2 Part 8

Tham khảo tài liệu 'metal machining - theory and applications episode 2 part 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Perfectly plastic material in plane strain 333 ds ds ds ds ds ds. _ dơ dơ ds_ ds77 A xx xz yz z 0 dx dy dz dx dy dz dx dy dz du dv dw 0 dx dy dz A1.15 where u V and w are the x y and z components of the material s velocity. The general three-dimensional situation is complicated. However in plane strain conditions and if the work hardening of the material is negligible the integration of the equilibrium and compatibility equations under the constraint of the constitutive equations is simplified by describing the stresses and velocities not in a Cartesian coordinate system but in a curvilinear system that is everywhere tangential to the maximum shear stress directions. The net of curvilinear maximum shear stress lines is known as the slip-line field. Determining the shape of the net for any application and then the stresses and velocities in the field is achieved through slip-line field theory. This theory is now outlined. A1.2.1 Constitutive laws for a non-hardening material in plane strain When the strain in one direction say the z-direction is zero from the flow rules equation A1.13 the deviatoric stresses in that direction are also zero. Then szz ơm 1 2 sxx syy . The yield criterion equation A1.12 and flow rules equation A1.13 become ơxx - ơyy 2 4s2xy 4k2 dexx -deyy dexy 1 2 Sx - ơyy - 1 s - s - Sxy A1.16 When the material is non-hardening the shear yield stress k is independent of strain. If in a plastic region the x y directions are chosen locally to coincide with the maximum shear stress directions sxx becomes equal to s and equal to sm so sxx - ơ 0. Equation A1.16 becomes a statement that i the maximum shear stress is constant throughout the plastic region and ii there is no extension along maximum shear stress directions. The consequences of these statements for stress and velocity variations throughout a plastic region are developed in the next two subsections. A1.2.2 Stress relations in a slip-line field Figure A1.4 a shows a network of slip-lines in a .