Simulations of plastic processing elasto-plastic deformation课件-SimulationsMaterials1.pptVIP

Stress tensor Von Mises stress in principal stress basis (i.e. if i≠j, σij=0) The constant von Mises stress is a cylinder (圆柱体) tilted at 45o off the axis σy σx σz (111) Yuan zhu ti Stress tensor Compute σα and τα as a function of α Mohr circle in plane stress τα is maximum for α=45o σxx σyy σα τα τα α 2α σxx σyy σα σα= σxsin2α+σycos2α τα= σysinαcosα-σxcosαsinα change of basis for Mohr circle T= σxyz= σ123=T-1σxyzT = sinα -cosα 0 cosα sinα 0 0 0 1 σx 0 0 0 σy 0 0 0 0 σxsin2α+σycos2α σysinαcosα-σxcosαsinα 0 σysinαcosα-σxcosαsinα σxcos2α+σysin2α 0 0 0 0 Strain tensor We saw the body of materials and the material elements are submitted to forces. In material science, as in the real life, the materials can deform We need to measure and quantify these deformations Usually these deformations are very small… Strain tensor Origin of the strain tensor: the displacement of material Displacement field (位移场) u(x,y,z) for each point X(x,y,z) of the material u= ey ez before after u(x,y,z) 0 X(x,y,z) ex ux uy uz Wei yi chang Strain tensor Displacement gradient tensor (位移梯度张量) G = grad(u)= The symmetric (对称的) part of G is ε=1/2(G+GT) ε is the strain tensor (or deformation tensor) The anti-symmetric (非对称的) part of G is ω=1/2(G-GT) ω is the rotation tensor (or spin tensor) ?ux/?x ?ux/?y ?ux/?z ?uy/?x ?uy/?y ?uy/?z ?uz/?x ?uz/?y ?uz/?z Wei yi ti du zhangliang Fei duicheng de Strain tensor ε= ?ux 1 ?ux ?uy 1 ?ux ?uz ?x 2 ?y ?x 2 ?z ?x 1 ?uy ?ux ?uy 1 ?uy ?uz 2 ?x ?y ?y 2 ?z ?y 1 ?uz ?ux 1 ?uz ?uy ?uz 2 ?x ?z 2 ?y ?z ?z ( + ) ( + ) ( + ) ( + ) ( + ) ( + ) Strain tensor Principal strain: as the strain tensor is symmetric, th