EN0175-25

EN0175-25 - EN0175 12 / 05/ 06 9. Boundary value problems...

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EN0175 12 / 05/ 06 9. Boundary value problems in plasticity Slip line theory An important theory in the plane problems of plasticity is the slip line theory. This theory simplifies the governing equations for plastic deformation by making several assumptions: 1) rigid-plastic material response (see explanations below). 2) plane strain deformation; 3) quasi-static loading; 4) no temperature change and no body force; 5) isotropic material 6) no Baushinger effect 7) No work hardening Typical experimental stress-strain relation on plastic material behavior has the form: ε σ As an idealized model, rigid-plastic model simplifies the above elastic-plastic stress-strain relation to a much simpler form described in the figure below: Y Y This indicates that the von Mises stress shall be kept constant at the yield stress 1
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EN0175 12 / 05/ 06 Y e σ = , where ' ' 2 3 ij ij e σσσ = once plastic deformation occurs. Slip lines are defined as trajectories of the directions of maximum shear stress. For a 2D stress state, the orientation of the maximum shear can be found by the techniques of stress transformation or Mohr’s circle. I II τ The above Mohr circle suggests that the 2D stress state in the orientations of the maximum shear stress can be expressed as: ββ αα = = , αβ = = zz α β φ 1 x 2 x , : slip directions corresponding to the directions of maximum shear stress. 2
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EN0175 12 / 05/ 06 Since the hydrostatic part of stress causes no plastic deformation, consider the deviatoric stress ( ) αβ βα τσ e e e e v v v v + = ' Within the plastic zone, Y ij ij e στ τ σσσ = = × = = 3 2 2 3 2 3 2 ' ' or k Y = = = const.
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EN0175-25 - EN0175 12 / 05/ 06 9. Boundary value problems...

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