fe5b98b8a5-p-1844.pdf - 3.1.1 Finite element formulation r ij(3.4 ij The rigid-viscoplastic finite element formulation is based on the variational

fe5b98b8a5-p-1844.pdf - 3.1.1 Finite element formulation r...

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34 3.1.1 Finite element formulation The rigid-viscoplastic finite element formulation is based on the variational approach. According to this approach, the actual velocities (i.e. the actual solution) among all admissible velocities u i that satisfy the conditions of compatibility and incompressibility as well as the velocity boundary conditions, give the following functional a stationary value: V S i i ij v dS u F dV ) ( E (3.1) where E is the power function which gives: ij ' ij E . The incompressibility constraint on admissible velocity fields can be removed by introducing a penalized form of the incompressibility as, V V S i i v v F 0 dS u F dV K dV (3.2) where ) , ( , and ii v is the volumetric strain-rate. K can be very large positive constant. 3.1.2 Thermal effects The temperatures generated in high speed machining process can be very high and have a considerable influence on the mechanical response. The heat transfer equation is expressed in the form of energy balance as follows, 0 , 1 T c r T k ii (3.3) where K 1 T, ii
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