TECHNICAL PAPERS Fig. 4. Comparison of center line triaxiality for Pass 4.
Fig. 5. Comparison of different criteria for maximum failure in Pass 6.
the current finite element and nnd = the number of nodes in element. A determination of the next points of the flow line is based on the calculation of the components of metal velocity for current point k per the follow equation:
ty ψ is > 1, the material will experience fracture. The function εp (k, μσ) is obtained through experiments. Bogatow proposed the following form for this function:
Eq. (7) Eq. (4)
and the integration of coordinates:
where α1 = empirical coefficients for steel. If the deformation process is multistep, then Eq. (6) is describe by the following:
Eq. (5) Eq. (8) The flow lines are estimated as described above, which allows easy visual analysis of the deformation state during the drawing processes. The boundary problem was solved as non-isothermal. Fracture theory: Kolmogorov and Bogatow This theory was developed especially for metal working processes such as drawing. A detailed description of this theory can be found in literature6-7. The main concept of the fracture theory described above is as follows. The key parameter representing fracture is called “resource of plasticity” given by the following equation: Eq. (6) where ψ = the resource of plasticity, εi = the intensity of deformation in metal working processes and εp = the critical deformation before fracture of metal as a function of the triaxity factor
and the Lode coefficient where σ = mean stress and σ1, σ2, σ3 = the principle stress components of the stress tensor. If the resource of plastici-
where ξi = the strain rate and τ = the time of deformation. The FEM code Drawing2d uses integration procedure along the flow line (2) to evaluate the above integral. This results in:
Eq. (9) where: Δτ(k) = the current time increment, ξi(k) = the values of the strain rate in the current time and k = is a time step during integration along the flow line. For axisymmetric deformation (as drawing of wire) the Lode coefficient is equal to zero. Therefore, one can propose a more simple equation for calculating the resource of plasticity: Eq. (10) and and an equation for the εp (k, μσ) function: Eq. (11)
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