By Claude Brezinski
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Additional info for Algorithmes d'acceleration de la convergence: etude numerique
40) eλ ΔδΠ Tm = ΔλTmδg Tmn + δλTm Δg Tmn + λTm Δδg Tmn , eλ ΔδΠ Ts = ΔλTs δg Tsn + δλTs Δg Tsn + λTs ΔδgTsn . 45) pλ ΔδΠ Ts = μp s s s Δλ N δg Tsn + μp s s s λ N ΔδgTsn + μs s λ N Δp sδg Tsn . 39), the sliding orientation is controlled by the values sm and ss. It is worth to note, that in the Lagrange multipliers method the values of the multipliers corresponding to friction, λTm and λTs, can be negative. Such a situation takes place, when the sliding has the orientation opposite to the local co-ordinate.
In the Lagrange multipliers method such a situation must be recognized, when the absolute value of the tangential displacement starts to decrease. In the above presented considerations it is suggested, that the sliding criterion is checked independently for two friction forces acting along edges of beams m and s. Hence, in this approach a possibility exists, that in one contact point two different friction states – stick and slip are present at the same time. To eliminate such a problematic case a different approach might be used.
The physical solution of the problems is of a smaller importance, hence the data used in the examples are devoid of any physical units. It is understood, however, that any system of physically consistent units might be attached. In these examples the loading is applied in the form of displacements imposed simultaneously in a given number of equal increments. The loading and the deformation process is measured in a fictitious time and a given moment in this time is determined by the parameter T, which ranges from 0 to 1.
Algorithmes d'acceleration de la convergence: etude numerique by Claude Brezinski